Internship Program for Minority Students

NCAR/TN-1 19+PROC NCAR TECHNICAL NOTE

IlP Ab March 1977

Fellowship Program in Scientific Computing; Internship Program for Minority Students

Computing Facility Summer 1976

Editors: Jeanne C. Adams Russell K. Rew

ATMOSPHERIC TECHNOLOGY DIVISIONhi

8 as b 1 -e - -- I _ - :,, NATIONAL CENTER FOR ATMOSPHERIC RESEARCH i BOULDER, COLORADC I iii

PREFACE

The papers contained in the 1976 Technical Note NCAR/TN-119+PROC represent the research and programming carried out by 12 students in the Computing Facility summer programs during the summer of 1976. The papers were written by the students, and reviewed by the scientific and programming staff who worked with them. A wide variety of topics are included, since the students were assigned to many different projects according to their interests and their academic background. Many staff members participated--scientists, programmers and technical staff--supervising students, . consulting, giving lectures and courses for the students.

The Computing Facility Staff

Summer Fellowship Program Russell Rew, the Computing Facility Librarian, was co-director of the Jeanne Adams program. Linda Besen was administrative Program Dixector assistant for the program as well as editorial assistant for the collection of papers presented in this report. Internship Program Richard Valent, Jo Walsh and Fred Glare supervised the interns in their work assignments. Richard Valent monitored y ii the course credit hours for which the students had registered under a cooperative >7 education arrangement with their univ- ersities.

Russell Rew Program Co-Director iv

The Students

Two groups of students participated, one group of nine in the Summer Fellowship Program in Scientific Computing; the other group of three in the Internship Program Left to right: Eric Barron, Jim for minority students. The Thrasher, Kerry EmanueZ, Camp Tones Summer Fellowship students were:

Student University

Jon Ahlquist University of Wisconsin at Madison Eric Barron University of Miami Kerry Emanuel Massachusetts Institute of Technology Lynn Hubbard (UCAR Fellow) University of California at Riverside Iluei-Iin Lu Florida State University Curtis Mobley University of Maryland Gary Newman Pennsylvania State University Joelee Normand University of Oklahoma James Thrasher University of California at Davis

The three students listed below were selected to participate in the Internship Program.

Karen Kendrick Atlanta University Arleen Kimble Prairie View A & M University Campanella Tones Prairie View A & M University v

Summer Fellowship Program The students arrived at NCAR around June 14. These students, most of whom are graduate students, are chosen on the basis of their interest in the atmospheric sciences, as well as their academic background. A committee of three staff members at NCAR examined the applications and made the selections. The students spent the first two and a half weeks in an intensive programming review, so that they would have an introduction to the most unfamiliar features of the NCAR operating system before work was begun on the scientific projects. The topics covered during this first part of the summer were varied: Russell Rew conducted seminars that reviewed FORTRAN and the NCAR operating system for half a day; the other half day covered topics in input/output, which I presented. The students all wrote and designed a program which required the buffering techniques usually needed for a large simulation model that is not core-contained. An introduction to direct access was presented as an alternate approach to complicated problems with out-of-core data arrays. The topics in the first two and a half week review are listed.

Marie Working lectures on "Terminal Command Language; John Gary 's talk is on "Numerical Solution of Hyperbolic Equations." vi

FORTRAN Review Input/Output

1st Week NCAR System Orientation I/O and Control Cards I/O--Word Formats, Computer Arithmetic, Core Dumps Physical Characteristics of I/O Devices and Tapes

2nd Week FORTRAN Review Files and Sequential Access Blocking and Use of LCM Datasets and Direct Access I/O Overlays

3rd Week Programming Style SAVE and RESTART Use of Graphics Special Routines for Use in I/O Access Time and Transfer Rates

Computing Facility Summer Seminar Series

For the remainder of the summer there were two lecture series given weekly. One series introduced simulation models that have been designed and implemented at NCAR. The other series included special topics of general interest to users, as well as a discussion of Computing Facility

:;:::...... supporting university visitors.

*..r :...... _...... These lectures, the Computing Facility 0000iiis~ ~ Summer Seminar Series, were primarily for the students in the Fellowship program; ...... ,~l- however, the lectures were ...... announced weekly in NCAR Staff Notes and other visitors and NCAR staff

.....li....I...... Summer were invitedSe to attend.

Russell Rew gives the students an introduction to the NCAR System vii

Cicely Ridley explains how to apply for computer time in a lecture given by her and Jeanne Adams, "Applications for CRU and Site Initiation." Left to right are Gary Newman, Joelee Normand, Curt Mob ley, Cicely Ridley, Camp Tones and Eric Barron.

Tuesday Series - Special Topics

Topic Speaker Overview of Communications Hardware and Software Gary Aitken Mass Storage System Hardware and Software Jeanne Adams and Bernie O'Lear Special Routines for Use in I/O Will Spangler FORTPAN Standards and Program Portability Jeanne Adams The Status of Mathematical Software Alan Cline TLIB and NEWVOL Will Spangler Initial Boundary-Value Problems in Fluid Dynamics Joe Oliger The FRED Preprocessor Dave Kennison Sorting on Vector Computers Harold S. Stone Random File I/O in Higher Level Language-A Comparison Gary Aitken Terminal Command Language Marie Working Applications for CRU and Site Initiation Jeanne Adams and Cicely Ridley Data Archiving Paul Mulder Making Mini Computers Look Like RJE Terminals Dave Robertson viii

Thursday Series- Computing in the Atmospheric Sciences Topic Speaker Introduction to Computing in the Physical Sciences Cecil Leith Data Structures for Large Models Dick Sato Processing Results from Large Models Dave Fulker The NCAR GCM Warren Washington Numerical Solution of One-Dimensional Non-Linear Jordan Hastings Parabolic Equations Numerical Solution of Hyperbolic Equations John Gary Numerical Solution of Elliptical Equations Roland Sweet Fast Fourier Transformations Paul Swarztrauber A Coronal Magnetic Field Model John Adams Numerical Solution of Integral Equations Ben Domenico

Rob Gerritsen, consultant to the Advanced Methods Group of the Computing Facility, gave a course that met three times a week on Data Management. The students also attended other lectures offered at NCAR in scientific topics by a variety of visitors and staff.

En g i efeie ng (e ft) and Warren Washington of NCAR were among ;i:-l-j ithe, i speake2rs in- ,he seminar series eat tZed "Com autingin 'i the Atmospheric Scinces." ix

Participating Scientists

The participating scientists supervised the theoretical aspects of the research projects of the students. Without the support and co- operation of these scientists, the program could not use real projects for study. I want to acknowledge and thank the scientists for their participation in the program and their interest in the students. They provided many hours of consultation and scientific support.

Scientist Student

Grant Branstator Joelee Normand Jim Deardorff Jim Thrasher Tzvi Gal-Chen Eric Barron Jack Herring Gary Newman Akira Kasahara Huei-Iin Lu C. S. Kiang Lynn Hubbard Stephen Schneider Jon Ahlquist Roland Sweet Curt Mobley Ed Zipser Kerry Emanuel

Foreground, left to right: Joelee Normand and Jon AhZquist, Summer Fellowship students; background, Mary Trembour, NCAR. x

Summer Internship Program

The interns arrived at NCAR on June 1. There were two under- graduates from the Computer Science Education department of Prairie View A & M University and one undergraduate from the Computation Center of Atlanta University. Clyde Christopher, Director of Computer Science Education at Prairie View A & M University, was a guest lecturer during the first week of the program. Grover Simmons, Director of the Computation Center Clyde Christopher at Atlanta University, lectured and provided consulting for the students during the latter part of the program.

Internship Courses

During the first few weeks of the program, Jo Walsh taught Beginning FORTRAN for the Computing Facility Interns and the student interns active in the Advanced Study Program. This class was essentially a review for the interns since they already had programming experience through their course work. During this same period, I taught a course in "Computer Organization" for which the three Computing Facility Interns received 3 credit hours. The second credit course was offered during the later part of the summer. The following Computing Facility Staff gave lectures in Numerical Calculus; Richard Valent organized the course as well as lectured.

Grover Simmons xi

Topic Speaker

Error Discussion Jo Walsh Derivatives Ben Domenico Matrix and Linear Simultaneous Equations Ben Domenico Polynomial Approximation Fred Clare Roots of Equations Dick Valent Integrals Russ Rew Euler's Method Jo Walsh

The interns attended a number of the sessions in the lecture series as w.ell as their courses and the testing project meetings throughout the summer.

...... r

(ASP Student) and Karen Kendrick (Intern) attend a lecture.

The Computing Facility staff have enjoyed the student visitors over the years. JeanneAdamsThey have contributed many fresh ideas and ansnew waysa of solvingin ProgramDirector problems. They bring with them an enthusiasm for their research. Many very hard and learning new things.

Jeanne Adams Program Director

xiii

TABLE OF CONTENTS

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model ...... * * 1 by Jon Ahlquist (University of Wisconsin at Madison) Stephen Schneider, Scientist

Experimentation with Meridional Heat Transport Formulations in the Schneider and Gal-Chen Energy Balance Climate Model ...... 25 by Eric J. Barron (University of Miami) Tzvi Gal-Chen, Scientist

Preliminary Investigation of a Tropical Squall Mesosystem as Observed by Aircraft During Gate ...... 39 by Kerry Emanuel (Massachusetts Institute of Technology) Ed Zipser, Scientist

Numerical Simulation of Photochemical Processes in the Troposphere ...... o 73 by Lynn M. Hubbard (University of California at Riverside) C. S. Kiang, Scientist

Testing NSSL Routines KURV and RTNI at the Demonstration 97 Driver Level ...... by Karen Kendrick (Atlanta University) Dick Valent, Scientist

The NCAR Scientific Subroutine Library and Computer Solutions to Linear Systems ...... 107 by Arleen Kimble (Prairie View A&M University) Fred Clare, Scientist

On the Balance Assumption of Zonally Averaged Dynamical Model for the Annulus ...... e ...... 115 by Huei-Iin Lu (Florida State University) Akira Kasahara, Scientist

Investigation of Algorithms for the Solution of the Nonseparable Helmholtz Equation ...... 133 by Curtis D. Mobley (University of Maryland) Roland Sweet, Scientist

A Test Field Model Study of a Passive Scalar in Isotropic Turbulence . 157 by Gary R. Newman (Pennsylvania State University) Jack Herring, Scientist XlV

TABLE OF CONTENTS (Cont'd.)

Processing, Display, and the Use of the Results of a Numerical Model .. *...... 203 by Joelee Normand (University of Oklahoma) Grant Branstator, Scientist

An Adapted One-Layer Model of the Convectively Mixed Planetary Boundary Layer ...... 2217 by James Thrasher (University of California at Davis) Jim Deardorff, Scientist

Testing NSSL Routines ADQUAD and SIMPSN ...... 243 by Campanella Tones (Prairie View A&M University) Jo Walsh, Scientist 1

CLIMAT: A SIMPLE ZONALLY AVERAGED ENERGY BALANCE CLIMATE MODEL by Jon Ahlquist University of Wisconsin at Madison Stephen Schneider, Scientist

ABSTRACT This report is a description of and a users' guide for CLIMAT, a rewritten version of the simple zonally averaged energy balance climate model described in Schneider and Gal-Chen (1973) and Gal-Chen and Schneider (1976). This model is highly modular and should run on almost any FORTRAN compiler without modification. Procedures for acquiring a copy of CLIMAT are described in the conclusion of this report.

INTRODUCTION Modeling is important in climate research because it enables count- less impossible and/or undesirable experiments to be simulated. We can further state that small simple climate models have a place in climate research. Simple climate models have at least two advantages over complicated general circulation models (GCM's). One, they are much cheaper to run, being perhaps a million or more times faster than a big GCM. Two, they are more useful in gaining insights into physical processes, since cause and effect can often be easily isolated. The results of GCM's are frequently almost as difficult to interpret as the processes active in the Earth's real climate. Small models have disadvantages, though. Bluntly, their results may be wrong, because parameterizations are often based on semi-empirical formulations rather than on rigorous physics. (What exacerbates this problem is that, since there is seldom any way to check the predictions of any climate model, one cannot know for certain when predictions are wrong.) Also, one will never discover anything very subtle from small models because of their simplicity. At least at present, though, many very basic questions about our climate have not yet been answered, and the subtle questions can wait. As for the first disadvantage, one can try a number of different 2

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model ..

parameterizations and sensitivity tests for any particular quantity in a climate model. If the model's predictions are similar and not too sensitive to the exact values of parameterization coefficients, one can have some confidence in the predictions even if one cannot know positively that they are correct. This report describes a rewritten version of the simple zonally averaged climate model originally written by Schneider, Gal-Chen and others. See Schneider and Gal-Chen (1973) and Gal-Chen and Schneider (1976) for information on the original model and its results. The rewritten version, named CLIMAT, was designed to be easy to understand and modify. This report, along with a program listing, should contain sufficient information for the reader to use and modify this model. The Appendices to this report contain some of the details needed to understand CLIMAT.

THE BASIC EQUATION FOR CLIMAT In CLIMAT, the Earth is divided into eighteen zonal bands, each ten degrees of latitude wide. The bands are 900 North to 80° North, 800 North to 70° North, etc. The only prognostically evaluated quantity is the average surface temperature for each zonal band. The prognostic equation is the zonally averaged, vertically integrated, thermodynamic equation for the Earth - atmosphere system. Specifically, we have for each zonal band:

DT R = (1-a)Q - F. - div(F + F + F ) 3tir A q o where R = thermal inertia coefficient (J/K/m2) T = surface temperature (Kelvin) t = time (seconds) a = albedo Q = incoming solar flux (W/m2 ) F. = outgoing infared flux (W/m2 ) ir 3

. .. * * * ...... ee.....e 0 0 0 * * .J. Ahlquist and div(FA + F + F ) is the vertically integrated divergence of atmos- heat flux (FA), latent heat flux (F ), and oceanic sensible pheric sensible q heat flux (Fo). (Units are W/m 2 for div(FA + F + Fo).) In the present version of CLIMAT, R is constant, Q is a specified function of time, and the remaining variables are all parameterized as functions of temperature. The reader is urged to study Budyko (1969), Sellers (1969), Schneider and Gal-Chen (1973), and Gal-Chen and Schneider (1976). These articles describe various parameterizations which are applicable to CLIMAT. In its basic form, CLIMAT is a time dependent version of the Sellers model, but the Budyko parameterizations and other parameterizations can be used with equal ease. The reader should be told that Schneider and Gal-Chen in their two articles have some canceling sign errors in their definitions of fluxes and divergence.

GENERAL ASPECTS OF CLIMAT CLIMAT is written in nearly standard FORTRAN and should run on almost any FORTRAN compiler without modification. The program is structured, and subroutines are extensively used in order to isolate the various stages of computation and to improve the readability of the FORTRAN code; modularity and clarity were deemed more important than operational efficiency and speed. CLIMAT offers few frills. This was done since modifications are more simply performed by changing an occasional card or subroutine rather than by redirecting program flow through a mass of IF statements. Thus, a knowledge of standard FORTRAN is necessary to work with CLIMAT. If the

user is familiar with the articles cited in the previous section, especially those of Sellers, and Schneider and Gal-Chen (1973), CLIMAT should be fairly easy to understand, use, and modify. Every variable in CLIMAT is either a mnemonic or corresponds to notation used in the literature. Throughout the program, the MKS system of units is used without exception, and all the temperatures are in Kelvin. 4

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model ..

SEQUENCE OF EVENTS IN THE EXECUTION OF CLIMAT There are four main sections in CLIMAT. (A more detailed breakdown of the steps in CLIMAT appears in Appendix A.) In the order in which these four sections occur: 1. read in values for parameters which control the integration of the model 2. tune the model to the Earth's present climate 3. read in an array of initial temperatures (optional) 4. integrate the eighteen zonally averaged thermodynamic equations and compute and print average statistics which are generated as the model is integrated. We shall now look at each one of these points in more detail. 1. Read values for the integration parameters. The user must supply values for six parameters which CLIMAT requires when integrating any model. In this section, we shall list these parameters using CLIMAT's FORTRAN names for them, describe each one, and look at the READ and FORMAT statements through which they enter CLIMAT. The six parameters are: PRNTYR an integer greater than or equal to one, specifying how often, in years, the user wants CLIMAT to print a summary of its current calculations. For example, if PRNTYR = 50, then CLIMAT will print information regarding every fiftieth year of model integration time. Setting PRNTYR equal to 50 or to MAXYRS (one of the upcoming parameters) is generally a good choice. NSEASN an integer between one and twelve, inclusive, which specifies the number of periods per year (loosly, the number of "seasons") into which the user wants the year divided. CLIMAT knows nothing about months, so NSEASN need not be a divisor of 12. Given PRNTYR and NSEASN, CLIMAT will compute and print average statistics for each period during each of the years designated by PRNTYR. For a model using annual average solar fluxes, one would set NSEASN = 1, since "seasonal" averages would be identical with one another. For a seasonal model (i.e., one in which solar fluxes vary during the year), CLIMAT's year begins 5

...... e . . .e o e . e...... J. Ahlquist

on 1 December. (This date is set by a data card in the seasonal version of subroutine SOLAR.) This way, if the user specifies NSEASN = 4, the four periods will roughly correspond with winter, spring, summer, and fall. As a specific example, if PRNTYR = 50 and NSEASN = 6, CLIMAT will compute and print bimonthly average temperatures, albedos, heat transports, etc. for model years 50, 100, 150, etc. Whatever the value of NSEASN, CLIMAT will also compute and print annual average temperatures, etc., for the years designated by PRINTYR. Setting NSEASN = 4 is generally a good choice for a seasonal model. NSTEPS an integer greater than or equal to one, specifying the desired number of integration time steps per "season." For stability, there should be at least eighty time steps per year. So, if NSEASN = 4, one might set NSTEPS = 20. MAXYRS an integer greater than or equal to one, specifying the maximum number of years over which CLIMAT will integrate before stopping. Setting MAXYRS = 300 is often a good choice. EPSILN a real number which CLIMAT will use in testing the integrated model for convergence to steady state. Although CLIMAT computes average statistics for most variables only every PRNTYR years, it computes seasonally averaged temperatures every season of every year for its own use. CLIMAT will signal convergence to steady state as soon as all the zonally averaged temperatures for any season in any year differ by less than EPSILN degrees Kelvin from those for the same season in the previous year. let represent the zonally and seasonally Symbolically, Tijijkk averaged temperature for the i-th zonal band during the j-th season of the k-th year. CLIMAT will signal convergence as soon as max T< -T < EILN ilax Tijk ij(k-1) I

for any j and k. Once convergence has been reached, CLIMAT will integrate the model for one more complete year in order 6

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model ...

to gather and print steady state zonal temperatures, etc.; then CLIMAT will shut itself off. If the climate model does not converge after CLIMAT has integrated through MAXYRS years, CLIMAT will shut itself off. Setting EPSILN = 0.001 is often a good choice. CSOLAR a real number specifying the ratio of the solar constant to be used when integrating the thermodynamic equation to the Earth's present solar constant. For example, if CSOLAR = 1.01, CLIMAT will multiply the Earth's present solar constant by 1.01 to obtain a new solar constant which will be used when integrating the model. (CLIMAT always uses CSOLAR = 1.00 when tuning the model.) Because of the infrared flux "consistency factor" which is computed when tuning the model,* the value which CLIMAT uses for the Earth's present solar constant is not too critical. (CLIMAT uses a solar constant of 1358 W/m 2 . This value is fixed by a DATA statement in subroutine SOLAR.) However, once the model is tuned, CLIMAT is very sensitive to the value of the solar constant. If the user specifies CSOLAR = 0.97 when running the time dependent Sellers (1969) model, the entire Earth will glaciate in less than two centuries. These, then are all six parameters required as user input to CLIMAT. They are read by the main program in CLIMAT by the following READ statements: READ 110,PRNTYR,NSEASN, NSTEPS 110 FORMAT (I3,7X,I2,8X,I3) READ 120,MAXYRS,EPSILN 120 FORMAT (I4,6X,F10.5) READ 130, CSOLAR 130 FORMAT (F10.3) If the user is interested only in steady state results, he can set PRNTYR equal to the same integer which he chooses for MAXYRS. This way, CLIMAT

* See Schneider and Gal-Chen (1973) for more information about the infrared flux "consistency factor." 7

...... o* ...... o c * e J. Ahlquis t will print only when the model has reached steady state or MAXYRS years. This saves computer paper. If EPSILN = 0.001, most models will converge in a century or two. 2. Tune the model. Data statements in the main program of CLIMAT contain experimentally measured temperatures, time rates of change of temperature, meridional winds, and energy fluxes. Using these values, CLIMAT computes parameterization coefficients which are tuned to the Earth's present climate. Once these parameterization coefficients have been initially computed, they are held constant for the remainder of the program. Details of this tuning operation appear in Appendix B. 3. Initial conditions. Because our thermodynamic equation is an ordinary differential equation, we need only initial conditions with our thermodynamic equation to complete closure of the problem. CLIMAT does not use the polar boundary condition discussed in Schneider and Gal-Chen (1973). See Schneider and Gal-Chen (1973) regarding sensitivity of the model to initial conditions. If the Earth's present annual average zonal temperatures are close enough to the desired initial temperatures, the user need do nothing since CLIMAT knows the Earth's current temperature from the tuning operation. If different initial conditions are desired, the user can easily enter them by deleting the "C"s from column one of the READ and FORMAT statements in the "Read initial temperatures" section of CLIMAT's main program. Then type the eighteen desired temperatures onto data cards with the most northern temperature first and the most southern temperature last. See the "Read initial temperatures" section in CLIMAT for specifics. 4. Integration and computation of average statistics. The method of integrating the thermodynamic equation is crucial because of the sensitivity of this equation. The author has tried the following explicit methods of integration: leapfrog, fourth order Adams-Bashforth multistep, fourth order Adams-Moulton predictor-corrector, fourth order Runge-Kutta, and two fancy "canned" integration schemes from the NCAR computer library (which automatically adjust the integration time step). Of these, the 8

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model. . author found the Runge-Kutta method to be the fastest for reasonable accuracy. No implicit integration schemes were tried. (The implicit Crank-Nicholson method is used in the original Gal-Chen and Schneider model.) If the reader wishes to try his own integration method, he should bear three points in mind. First, different integration schemes are not too hard to plug into CLIMAT. Second, the integration time step must be an unvarying constant because of calls to two subroutines within CLIMAT's integration loop. These subroutines compute average statistics and check for convergence of the climate to steady state. Third, the equations in CLIMAT are very touchy as to stability with respect to time step size.* The author became an expert on the error message "FLOATING POINT OVERFLOW" in his experiments with integration methods and time step sizes. When instability occurs, the explosion in temperature takes place near the equator. This sensitivity is apparently due to the large sensible and latent heat fluxes in the subtropics, the fluxes being sensitive functions of the very small temperature gradient. Certainly, improved parameterization for fluxes near the equator would make a noble modification to CLIMAT.

GETTING STARTED WITH CLIMAT The basic time dependent Sellers model requires the following fourteen subroutines: ALBED FSENAT AVGVAL FSENOC CONVER OUT DERIVV SOLAR DIVERG TEMADJ FLATEN VPRESS FLUXIR WINDD

* For example, on a Control Data 7600, the model suggested in the next section, "Getting Started with CLIMAT," goes unstable in less than ten years of model time if NSEASN = 1 and NSTEPS = 75, but seems quite stable if NSEASN = 1 and NSTEPS = 80. The temperatures predicted when NSTEPS = 80 coincide with those predicted when NSTEPS = 90 or 120 to at least five significant digits, which is all the accuracy that is printed. 9

...... e ...... J. Ahlquist

A brief summary of each subroutine as well as flow charts for CLIMAT appear in Appendices C and D. Several versions are available for some of the subroutines. For instance, Sellers and Budyko versions of DIVERG and FLUXIR exist. DIVERG computes the vertically integrated, zonally averaged divergence of sensible and latent heat fluxes, and FLUXIR the outgoing infrared flux. The Sellers version of DIVERG requires subroutines FLATEN, FSENAT, FSENOC, VPRESS, and WINDD to compute the various fluxes. Because of the simplicity of Budyko's parameterization for divergence, the Budyko version of DIVERG uses none of the five subroutines required by the Sellers version of DIVERG. In general, two CLIMAT subroutines with the same name represent different parameterizations for the same quantity. Such subroutines always have exactly identical parameter lists, so that one or the other can be used without having to modify CLIMAT in any way. A glance at the first few comment cards of any CLIMAT subroutine will tell the user the purpose of that subroutine and the parameterization form. For the user's first run of CLIMAT, the author suggests assembling the fourteen subroutines listed above, using the annual average version of SOLAR, the Sellers version of DIVERG, and the linearized Sellers version of FLUXIR. Set: PRNTYR = 25 NSEASN = 1 NSTEPS = 90 MAXYRS = 200 EPSILN 0.001 CSOLAR = 1.01 The resulting climate generated by this run should converge to steady state in about three-quarters of a century.* On a Control Data 6600, this model compiles and executes in about eighty seconds. After the user has run this model and received his output, he should sit down in a comfortable chair and take a long look at the output and program listing for CLIMAT. Hopefully, CLIMAT does not contain too many

* A copy of the results produced by a run using this set of values appears in Appendix G. 10

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model ... constructions where the user reacts by thinking, "What on Earth does this do?" or "Why did that crazy programmer do it that way?"

CONCLUSION The original Schneider and Gal-Chen model was developed, modified, and remodified over a period of several years by a number of people. The result was a model which operated fairly well and formed the basis for the Schneider and Gal-Chen, and Gal-Chen and Schneider papers previously cited. However, this model takes literally weeks and weeks for the user to understand because of its monolithic structure and length (over 2000 FORTRAN cards). Little program bugs also kept popping up. Faced with this situation, the author decided that this climate model should be rewritten from scratch. The primary goal for the new model (CLIMAT) was modularity, both for ease of understanding and for ease of modification. This reduces operational efficiency, but efficiency is not too important for a small model which will be frequently modified. The run suggested in "Getting Started with CLIMAT" is about 1000 FORTRAN cards long, including hundreds of comment cards. On a Control Data 7600, it compiles in less than 0.8 seconds, fits in about 20K of core, and executes in 16.2 seconds. If the seasonal version of SOLAR is used instead of the annual average version, compilation and execution take only a second longer; this includes three extra years of integration required by the seasonal model to reach steady state. To obtain a copy of CLIMAT, contact the NCAR Computing Facility and ask for a Software Request Form. Complete this form, requesting a taped copy of the PLIB file named CLIMAT which is on project number 03010017. Then return this form to NCAR along with a blank tape on which a source image version of the CLIMAT file will be written. This file contains several subroutines which have the same name. The user is certainly encouraged to modify CLIMAT but to do so by modifying subroutines which are as low on the structure tree as possible. That is, keep the main program and subroutines ignorant of as much as possible. This makes the various stages of the program easier to understand. The author is open to any questions or comments regarding CLIMAT. 11

...... J Ahlquist

APPENDIX A: STRUCTURE OF THE CLIMAT MAIN PROGRAM The main program in CLIMAT performs the following tasks (in order): 1. Start. 2. Dimension arrays and establish common blocks. 3. Using DATA statements, define all the constants which are used in the main program. 4. Using DATA statements, load observed annually averaged climatic data into arrays. This data will be used in tuning the model. 5. Compute the areas of the eighteen zonal bands and the lengths of the latitude circles which bound them. Load these values into arrays. 6. Read in values for the six integration parameters. 7. Tune the model so that it can reproduce the Earth's current annual average temperatures. (See Appendix B for details.) 8. (Optional) Read in an array of temperatures to be used as initial conditions for the thermodynamic equation. 9. Print parameterization coefficients and a summary of the specified climate at time zero. 10. Integrate the thermodynamic equation. Compute and print seasonal averages as the integration progresses, and check to see if the simulated climate has reached a steady state. 11. Stop.

APPENDIX B: TUNING THE MODEL Tuning the model consists of adjusting parameterization coefficients so that the model can reproduce the Earth's present annually and zonally averaged temperature field in perpetuity if the solar "constant" were to remain fixed at its present value. The comments below assume familiarity with Budyko (1969), Sellers (1969), and Schneider and Gal-Chen (1973). Subroutines which are mentioned are briefly explained in Appendix C. If the Budyko version of subroutine DIVERG is used in CLIMAT, the tuning operation is very simple. Only the infrared flux consistency factors are computed, since they are the only free parameters. 12

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model . ..

If the Sellers version of DIVERG is used, the first step is to adjust Sellers' wind coefficients, "a," and the three diffusity coefficients, K H , Kw , and K , so that the observed annual average sensible and latent heat transports are obtained. In order to perform this operation, subroutines SOLAR, WINDD, FLATEN, FSENAT, and FSENOC have special sections which are executed only the first time they are called. Then energy balance is achieved by computing the infrared flux consistency factors. Refer to CLIMAT's main program and to the subroutines in question for specifics. If the seasonal version of subroutine SOLAR is used, an additional step occurs within SOLAR the first time it is called (which is during the tuning stage). Since seasonally varying solar fluxes are time consuming to compute, they are computed only once, during the first call to SOLAR, and stored in an array for future reference.

APPENDIX C: SUBROUTINE SUMMARY Subroutine Purpose

ALBED Computes albedo using the Sellers formulation. Called once inL the main program during model tuning and subsequently by DERIVV. AVGVAL Computes seasonal and annual average values of all time dependent quantities except temperature. Called by the main program. CONVER Computes seasonal and annual average temperatures and checks for convergence of climate to steady state. Called by the main program. DERIVV Computes the right hand side of DT 1 ( R1 (l-a)Q - - div(FA F+ + F) at_t R (1irFr A q 0) Called by the main program during integration. Calls TEMADJ, ALBED, SOLAR, FLUXIR, and DIVERG.

DIVERG Computes div(FA + F + F ). Two versions of DIVERG are available, the Budyko version and the Sellers version. The Sellers version calls WINDD, FSENAT, FLATEN, and FSENOC. DIVERG is called once in the main program during model tuning and subsequently by DERIVV. 13

...... eo v ...... J. Ahlquist

FLATEN Computes vertically integrated flux of latent heat using the Sellers formulation. Called by the Sellers version of DIVERG; calls VPRESS. FLUXIR Computes emitted infrared flux. Three versions of FLUXIR are available: Budyko, Sellers, and linearized Sellers versions. The linearized Sellers formulation agrees with the complicated original Sellers formulation to within 1% between 200 and 300 Kelvin. Called by DERIVV. FSENAT Computes vertically integrated flux of sensible heat carried by the atmosphere using the Sellers formulation. Called by the Sellers version of DIVERG. FSENOC Computes vertically integrated flux of sensible heat carried by the oceans using the Sellers formulation. Called by the Sellers version of DIVERG. OUT Prints values of all time varying quantities for all zones and latitude circles. Called by the main program. SOLAR Computes incoming solar flux. Two versions of SOLAR are available. One version returns only the annual average solar flux for each zonal band, while the other version returns a seasonally varying solar flux. Called once by the main program during model tuning and subsequently by DERIVV. TEMADJ The Sellers climate model uses "sea level" temperatures (T ) for some calculations and "ground level" temperatures (T ) for other calculations. Sellers related these two temperatures by the formula T = T - 0.0065(Z), where Z is the average elevation g s in meters of the zonal band in question. TEMADJ performs this transformation between T and T , i.e., it "adjusts" the temperatures. TEMADJ is called twice in DERIVV. VPRESS Computes the vapor pressure in each zonal band as a function of temperature using the Clausius-Clapeyron equation and an assumed relative humidity of 75%. Called by FLATEN. WINDD Computes the northward wind using the Sellers formulation. Called by the Sellers version of DIVERG. 14

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model .....

APPENDIX D: SUBROUTINE NESTING The largest structure tree in CLIMAT stems from the integration loop in climate, which computes t dT f a- dt o for each zonal band. The integration loop calls DERIVV to compute the right hand side of FT F at =R (l-a)Q- Fi - div(F A + F + F). DERIVV calls: 1. TEMADJ to adjust temperatures to sea and ground levels as needed; 2. ALBED to compute albedos; 3. SOLAR to compute solar fluxes; 4. FLUXIR to compute infrared fluxes; and

5. DIVERG to compute div(FA + F + F ). If the Sellers version of DIVERG is used, DIVERG calls: 1. WINDD to compute northward winds; 2. FLATEN to compute vertically integrated fluxes of latent heat; (FLATEN calls VPRESS to compute vapor pressures.) 3. FSENAT to compute vertically integrated fluxes of sensible heat carried by the atmosphere; and 4. FSENOC to compute vertically integrated fluxes of sensible heat carried by the oceans.

APPENDIX E: COMMON BLOCKS CLIMAT uses four labeled common blocks. Blank common is not used. The names of these common blocks and the subroutines which access them appear below. All common blocks are also accessed by the main program. Common Block Accessed by subroutine AVRAGE AVGVAL DIFFUS WINDD, FLATEN, FSENAT, FSENOC INFO AVGVAL, DERIVV, DIVERG LEVEL TEMADJ, OUT 15

...... * . .* . .* .* * * ...... J. Ahlquist

AVRAGE holds average values of quantities; DIFFUS holds diffusion and wind constants; INFO holds general information on what is going on within CLIMAT; and LEVEL holds a variable which remembers whether the temperatures at any particular moment are at ground or sea level.

APPENDIX F: SELLERS' "FLUXES" The Sellers (1969) definitions of atmospheric and oceanic "fluxes" technically are not fluxes. Strictly speaking, a flux is a vector, and a flux, F, of any extensive variable, W, points in the direction of the flow of W and has a magnitude equal to the quantity of W per unit time which passes through a unit area which is perpendicular to the flow of W. In contradistinction, Sellers' definition of "flux," F', is 27T H f | F(4,X,z) dz a cos ( dX F'() .... 2Tr S a cos 4 dX

where F(A,X,z) = northward component of the true flux a = radius of the Earth = latitude X = longitude H = depth of the atmospheric or oceanic layer through which the flux is passing. For example, Sellers' definition of oceanic sensible heat "flux" across latitude 4 is the total amount of oceanic sensible heat crossing latitude 4 per second divided by the length of the latitude circle at latitude (, i.e., 27T a cos 4.* The MKS units for Sellers' "fluxes" are Watts per second.

* There is an error in Sellers (1969) formula for oceanic sensible heat "flux." Equation (13) should read F = -Ko Az (M'/ki) (AT/Ay) cw p

where cw is the specific heat of sea water, and p is the density of sea water. The other two "flux" formulae are correct. 16

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model . ..

For simplicity, this author chose to have his subroutines compute Sellers-type "fluxes," which are actually quasi vertically integrated fluxes. However, CLIMAT's output labels these "fluxes" as "transports" to avoid confusion.

APPENDIX G: SAMPLE OUTPUT FROM CLIMAT The following pages form the complete printed output from the run of CLIMAT suggested in the section "Getting Started with CLIMAT." This run was made on the Control Data 7600 at NCAR. *._------.------. -- -- -I~------.------

PARAMETERIZATION COEFFICIENTS _0

-3C 40 -50 -60 -70 -80 LAT 80- 70 60 -- 50 40 -33 2- li -1 -20 _ .0084 .0092 .0117 .0133 .271 .0325 .G301 .G299 .03?9 .0603 .0331 .0424 .0113 .0113 .0051 .0054 A .010 4E+05 KH 1E+ E+06 2E+06 1E,06 9E+05 iE+06 E E+JE 7E+07 3E+? 7E+06 2E0+6 2E+06 2E+06 1E406 9E+05 5E+05 2E+05 O KI 4E+05 2E+05 1E+05 7+05 7E+05 4E+.5 3E+05 3E+36 6E+05 -4F+^5 4E+04 6E+05 2E+06 _E+06 tE+06 9E+ui 3E+01 1E+01 0 KC 6E+01 7E+02 9E+02 9E+02 7E+02 t1E+3 2E+'3 E+073 !E+02 2E+C3 8E+C2 5E+C2 4E-32 C .98 1.02 .98 .94 i.04 .95 .97 .94 .9 .9_ .93 .97 1.00_ 1.0 ..91 .12 .9_ ._97.

LAT"LATITUOE -- 'A -SELLERS WINO COEFFICIENT (M/S/K) KHM =IFFUSIVITY COEFFICIENT FOR ATMOSPHERIC SENSIBLE HEAT FLUX (MKS UNITS) . KW =DIFFUSIVITY COEFFICIENT FOR LATENT HEAT FLUX (MKS tNITS) KO =OIFFUSIVITY COEFFICIENT FOR OCEANIC SENSIBLE HEAT FLUX (MKS UNITS) 4* '"C sCONSISTENCY FACTOR FOR INFRARED FLUX C =CONSISTE______CYFACTOR FOR NFRARD FLUX ------

N UMlER OF AVERAGING PERI03S PER YEAR =__ _ NUMIER -OF TIME STEPS PER AVERAGING PERIOD = 90 NAXt#UM NUMBER OF YEARS FOR WHICH MODEL WILL RUN =200 "COIERGENCE CRITERION FOR TEMPERATURE CHANGE = .0010 KELVIN

- _ ----- __------, 8__,______

SCALI N FACTOR FOR SOLAR CONSTANT = 1.CIG

K, ______------,------

---- e------:;'' '. , ------

* ' - _------_ _ __------_ _------;;,ii_------__------

_-__-_ _- _- __ ------,ft : -^------

C~~~~~~~~~ ::~~~~~~~~~~-1··~~~~~~~~~~~~~~~4~~~~~~~~~~

*. · ::~;....: :~.:~.... .za~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~e En C3 *- = 2 YEAR 0 AVERAGING PERIOD NO. r * - - M ** 3 -- .-~~-- Ir u A tf%,& A,.·, ne. d, A LIMI A 0 Irm9 u·El C-\ 0T 1-1 *IlUA ~ ~~C·AYiri C UeUNA·C LMFt i · It.R Vr'UK SOULAR I1RKIU K TiMU StN LA i N UIN atN * L AT LEVEL) OTEMP/OT PRESS AL9EDO FLUX FLUX 9 IV (F) WIND HT TRANS HT TRANS HT TRANS

90 c.000 0.0 _0.0_ 0.0 rn * 252.60 1.9E-09 74 .6593 174 173 -115 80 . SQ1 6. COE+C7 2.C0E+06 2.0OE+06 11 . - - - - II - 25B.60 2.3E-0g9 t29 .612 1 5b 191 -120 ? H 0 70 . '33 1.20E+C8 3. OCE+06 _1.00E+07 - - '- -50-- 7T0----3-m -5-9-4-----ZiE -5- *- --5X_ .03 0 1.50E+08 4. OOE+06 2.OOE+07 o (DN * "------275.0 --- E-0-9--- 3- "4Z9 260 2t2 .330 C' O '"-i- o 0 50 .030 1.30E+08 3.00E+07 3.00E+07 M 09 9 n n~~~~~~C--nVIn 7l -rM- .. . .; Z 1 I OI Jl1U OeUt-U"J I I .271 * 40 . C30 1.20E+C 8 5.00E+07 4.00E+07 H HPH ~------7I-. 7- ' .-5'E-O . 3 -7 '34q C2' 'r 30 1.GOE+08 3.00E+07 5.5.OOE+07 00E07 * I2 ------381 -244 20 -. 06j 9.00+C 7 -4.00E+06 5. 00E+ 07 12- rD 2 - -. -.. .- - ".. --. 1 .~1 - 1. 29. EU 4'o1-UU e-15(i' -4LU 4- 441 * 10 - 143 7. OOE+07 -3.OOE+07 3.00 E+ C7 -9-1- TF 246 -5-7- (D 0 --2-vcr 16i 1 30 -3. (OE+C7 4. 0CE+07 -6.00E+06 "~~L~~- .4iT 57- 1 ---0 2- 252 -10 .110 -S.00E+O7 5.00E+07 -3.00E+C7 1¢ -19 -- - 97o20 9.E6-09 2781- CD 'CO -420 - w .C045 -. QOE+08 -1.00E+G71.OOE+G7- -4.G0E+07 41 i 2 li klC-----~-~~T--~TT * 26751 -..185- 23P- 09 I - 30 -. C15 -1.30E+C8 -3.00E+07 CD * '7------' f 2y---2-5-- -*317 --,--s5-3 '- 2316 3- -.121 1.7 · n r r ru · 40 ~ -. -15 -1.30E+C8 -7.0OE+07 -3.00E+07 a .-3.. - - . 1- I 9 =# -Z a 1.4 - _ - . . ZO<*UU b!lt"U" O-)t .464 31 -44 O

9 '50 __vgE_ -. C3 -1.60E+08 -5 00E+C7 -1.00F+07 '9 T-27 3-----vr.E--.----.- 25T- 1-95 -i.60E+C 8 -. C5 -1. 80C+08 -4. OE+06 20 -60 - - - ~-698 -3.00DE+07 * --- ''7-- ---2-3- 22-2- --rI- -70 -. 35 -1.40E+ 8 -8.00E+06 21 -- -1.00Eo06 Z--5 2.8U . L-U9 - --1. . -1. - 01 (5--- .dU 4 -1 1 /5 -15 4 F- * -80 -. .25 -7.00E+C7 -1.0OE+06 -0.0 2------8- N^---gE_-1------2~8--- 831~'10'--' -55 ---- 7-2..7--- . ~8~45 _~~~~~~a _ _ cc7 -90 C C 0.0 0.0 023------r'0d

LA I LAIMIUt l UtGKLt(~ASJ CD 0 T =TEMPERATURE (KELVIN) H "'TE-E7oT--- --T T---EINE--EaUVM - - - F uF-P--TF--PTURE--11KELVT -PE'--SFr ------VAPQR PRESS =WATER VAPOR PRESSURE (N/M+*2) 26 BEfOLBEO ------* SO.AR FLUX =SOLAR FLUX (W/M+ 2) 1"%RKrlunj-- =UIbUtlbrU NG NIRAFt- FUX ---(H"M-U )rI * OV(F) =VERTICALLY INTEGRATED FLUX OIVERGENCE (W/M*+2) WERIEIUDWINUH ---- ~ERT WRTHRWARD T rr------ATMQ SEN HT TRANS=ATNOSPHERIC SENSIBLE HEAT TRANSPORT (W/M) * RtATENTH-TTr TMRTSPTJT TEW M ------OCM SEN HT TRANS =OCEANIC SENSIBLE HEAT TRANSPORT (W/M) C/3

,YEAR= 25 ANNUAL _AVERAGE VALUES M

I - ---- .ArriT npu cr 1 SOLAR IR MERID NTMT S LAtNITN TRAn S- UTM I T (SEA VAPOR L u V (F) WINO HT TRANS HT TRANS HT TRANS P-3 I* LAT LEVEL) OTEMP/OT PRESS____ AiLBEDO FLUX FLUX --4 c::2 0.000 0.0 0.0 90 .6?6 174-i7 1...... 8 --- 115------*, 2 56. 20 i.9E-09 103 0 c-- . 01 5.96E+C?7 2.638E+06 1.98E+06 .581 -- , I -- . -1 0--.dI 262.13 1.8E-09 178 .581 185 197 .030 1.20E+08 3.90E+06 9,79E+06 I0 -»- 70 -87 270.55 - .6E-09 369 .455 203 .3$0 1.49E+C8 4.81E+06 1.95E+07 60 : ~d ' -52 ._.. 678 262 207 u '- 278.66 1.4E-09 - ,,m;m- I -P P~c .; .t3F +*.r7 7 V.F8f6E7 eU-.e 1. _.TLa a, I, · ,, C & rt--i-- ...... 50 U b 242 1.4 4 C ' 1109- :--rSbU ---- 7ir 5 0 leltC-U 1109 *5bO .1U .v1 .£27 loiiE+08 5.46E.07 3. 77E+07 * . 40 29 2-92.78 9.2E-10 1732 ,322 343 232 4 .£14 9.50E+07 3.26E+C7 4.80E+07 30 - ; -C--- - -?.8E-10 *.278 381 0 " 2-98.15 -.- -c7 6. 89EF4C7 -4.26E+06 4.85E+07 20 -- 30i.25 -7.1E-10 29i5.. 13 E------3E - 301.64 7.0E-10 2984 *263 416 256 .1325 -37.E+07 .3426E+07 -25.9E+07 0 300.96 7T2E-10 2864 417 255 56 ;256 .135 -8.56E+C7 5.42E+07 -2.95E,07 1 -10 7 299.297 5-- 7.5E-10 2590 ...... --...... 341 -1.12E+08 1.04E+07 -3.88E+07 0 -20 295.11 8.6E-i0 20.5 .267 385 242 43 -. C15 -1.26E+08 -4.40E407 -2.89E407 -30 - ...... -.-. ._.-...... -...-- - -- 7 - -T . 0E-09 1433 353 240-. * ' - ~289.81 -2.86E+C7 -40 44 -. 0131 -1.20E+C8~ -7,74E+07 '-- 284.66 1.2E-09 102 -44 -. r27 -1. 50E+C8 -5.64E+07 -9.46E+06 -51 277.37 1.4E-09 620 .441 267 _.C~-. C49 -1.74E+C8 -3.50E+07 -3. 85E+ 06 -60 1-ft64 * - T268.13 1.7E-09 301 222 -, -J. -1.38E+C8 -1.05E+07 -9.74E+05 - 70 I. 21 .--- 7--- -A I - 256.55 2.1E-09 10o7 -T~1-- -. 025 -6.99E+07 -1.39E+06 0.0 · 22 -6o ------0 2-47 .06 -.-3-E--39-- - -. 43 ..... '9 183i'$Z- 0.200 0.0 0.0 -90 0.0 03 LA =LIIUO UJ4KE "24T . .T- =LATITUIE {DEGREE) » T* =TEMPERATURE (KELVIN) CHA8 OF-oT81PE^AT rrE<€LViTN PErT 'StOa "'OTEJg7W r----- ^TT 'RERArTE 4 VAPOR PRESS =WATER VAPOR PRESSURE (N/M *2) ' SOLAR FLUX =SOLAR FLUX (W/M 2) INFRARED FLUX (W/M *2) ='IR FLUX --- =OUTGOING 44 aOIV(F) =VERTICALLY INTEGRATED FLUX OIVERGENCE (W/M 2) r_ " RWDIM ------IDrlaNKL (OTTHWAR)'iN ( ) ATNQ SEN HT TRANS=ATMOSPHERIC SENSIBLE HEAT TRANSPORT (W/M) H' M * AENf-HTRNS =--ATENT-ET-TRN-RT---- w/-( ) (A OCN SEN HT TRANS =OCEANIC SENSIBLE HEAT TRANSPORT (W/I) -- -- I- 3^ ..- AN L A E E Hn * *1-3

P .. .4 .. I~~~~~~~~~L~~0I ,, I (tA- ..I . VrAUR SOULA I ntKl.u AMl U SEtN uG m NtNr *i. LAT LEVEL) OTEMP/OT PRESS ALREDO FLUX FLUX OIV (F) WINO HT TRANS HT TRANS HT TRANS

90 r_ 0 . 00 _0.C_ 0-0.- 10.09 c: 0 ------256.7 - i.8E-lO ~~~.521 1~~7V4 l l -11i-5 N 80 .C a I 5.93E+C7 2.79E+06 1.97E+06 I I I --___ - _ I Z6Z66 1.7E-10 151 .576 1B5 -119 .C30 1.1 9EC+C8 4.04E0+06 9. 72E+06 0* 7 710. -- :- 271.03 -rwiB------*3V8L4--- -2-0%- --- 5T- 60, 1 .48E+08 4.91E+06 1.94E+*07 N '130 279.f08 1-3E-10 .397 260 208 -51- H r< 50 .028 1.22F+08 3.47E+07 2.83E+07 1E-r n 4,0 2,7 W . - r , ? - .G I Z~b*Z4-.. lolt-lU 1 -55 I .32J 3'irB ,C27 1. 10E+C8 5.51E407 3. 74E07 I 29---X3.R-06 C-~PC7--8--T7iy r .27-~~ --34-a-381-- 30 . 154 9.46E+C7 3.28E+07 4.76E+07 * "t--Tq1T T.Ear -,2T3 "''-38Y~~ F-' 20 -.056 S. 89EiC7 -4.21E006 4.81E+07 oQ( . - r: . I. ,a I . II I .I_- -S4- I IVc a 2 b1 4-U--4'- 4 - 133 7, 14F+C7 -3.27E+07 2.91E+07 .2 -4 1.6- CDH- .a'. 124 -3.51E+07 4.35E+07 -5.85E+06 (DH' * 1 30 J 7 6 7 E T - --07 -- 4G- i T- - 10 .104 -8 .58E+07 5.45E+07 -2.93E+07 V -WT 7 7-, i ul CD - ifC7 0 -K-1-.' 7 *.3t-1l -4 - CtCO7 F - 7 F-il -ZO. . G41 -1.12E+C8 1.04E+07 -3.85E+07 .,267' 3'&"J5- -242- .40 (D 30- .757T--- -3953- -. 15 -1.25E+C8 -4.44E0C7 -2.87E+07 *I" 2 q w T q T7sf l -2z4- ----I -. 012 -1.19E+03 -7.81E+07 -2.84E+07 7 ~ 41-C 7. L " -7.1 *50 -.1227 -1.48E+08 -5.71E+C7 -9.35E+406 Z9-c,77 .513 1 ITWE;- 10 -- F39-WT~-~------60 - . C48 -1.73E+C8 -3.56E+07 -3. 1E+06 8"--270 Z GO7 ------f5- *5 9W ----- 22T -I W 5---B --ITT W2-2-;-- - 0170. -.034 -1.37E+C8 -1.09E+07 -9.65E+05 . I. I. -- I-~~~~- - Z- 1.-18a Z auE- Ou 11. . 7bi 194 1. -1.54 ~ ~~~~~~~~~ -. C25 -6.95E+07 -1.46E+06 * --80. - - 0, 0 22- z 4 r e7 - Z-s-T~z-IT------IW6 -.-8[30- -- 3 3 .- -T7------90 . C.C03 C.O 0.0 0.0 rt 0 -- .. ----- (D

~~~~---- I - =--L- . A 1 UuCUR tulcl...--. z ... =TEMPERATURE (KELVIN) 2 aP7DpP/DI -- =T IM AI EI NF-TUAN-O'-1 PERAT.TFT iUREtKEVTrI- PEt-SE TUOT- 00 =WATER VAPOR PRESSURE (N/M*+2) VAPIR PRESS ------* Ai.BE------e -ALB------.ftUX.^ =SOLAR FLUX (W/H+M2) ;? *'Jf:~.o =uO..X-.------UIUl INFRARLD FLUX (M,*"- -- ''=VERTICALLY INTEGRATED FLUX -IVERGENCE (W/M*-2) 7,71MIIR !Bnimr--- ~.,-.Eu Ro.A-HAOR Tlrn--w0:?-r~7^r------.: t~ TSENHT TRANS=ATMOSPHERIC SENSIBLE HEAT TRANSPORT (W/M) ri* .::i Tw TRNF"=LAtENE TRfENNU---rS-H T T---1. ..------TRANSPORT (W/M) : 1.'.i0^i* o &-S-0 TRANS =OCEANIC SENSIBLE HEAT , . ------YAeRt 73A__ANNURALAVERAGE VALUES * CLZMATE HAS REACHED STEADY STATE I-D t4ERIO~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r IC -4 SEN H SOLAR IR MFRI3 ATHO SEN LATENT OCN (SEA VAPOQ HT TRANS ALEDO FLUX 3IV(F) WIND HT TRANS HT TRANS LAT LEVEL) OTEMP/OT PRESS i-3 0.0 0.0 O.0 * 90 I I4 4 71. 4 I _ 4 rl 21.E-11 110 . 6 2 IT4 1t5I - !) 256.82 ci Q lSFl2 7 2.RfF406 1.96E+06 Ilt I -so .b-1 '1fl4 -119 262.72 2.0E-ll __188 .575 18 5 198 -. 3 1. 19E+CC 4. 05E+06 9.72E+06 4 271.07 i.8E-11 386 .450 214 204 -86 t C30 1.48E+08 4.92E+06 1 T94Ea40 -frtcI ,3._ ?rfR -... M 279.12 1.6E-i1 700 ·*0 ' t . u -6 27 i.2iF Gs 3 .4E 1M0 7 2.32E+07 242 * 286.27 1 *3E-11 1137 .363 336 3"40 1 09E+08 5.51E+07 3 .73E+7 293.08 1.OE-lt 1766 __ 322 348 232 3.28E+07 9. 8E+7V46E+076. 4.76E+07 3 nil.e 381 248 2 7 8.8E-12 2456 .278 _ - - DOAr z 7 298.41 -4.2CE+Ob - 20 3 4 245 54 2.9i1Es 07 13- 8.OE-12 2957 .261 4 7. 89E+C7 301.49 ---' ------3.27E+*7 10------10 -. 33 250 58 301.88 7.9E-12 3026 .260 416 -3.SIE+07S 4-- -- _------.124 4. 35E+07 -5. 84E'06 -5. 50Lg~i.c r -2.93E+076~ ea 255 56 417 - - .253 - . 301.19 8.1E-12 2905 V·- .4v.tsm --. 7P .104 -2.93F+-07 238 67 __8.5E-12 2630 .2.5 4U6 * 16-O------___ _299.54 . .41 ~.404E+c0 -3.85E+07 - 20 40 9.7E-12 2041 .267 385 242 295.40 I-llr5 "-i.Si-+C-1. 2f4- -8B -4-44E + 7 -2.~87E+07 0-30 4.n 1a 353 &.'"U ,.t . 290.14 1.IE-11 1464 .317 I --- - -17 -1307r-A .------~~- f_ U-n 7, ~~ -. 5B-tE+UI -. b3E40UI -40 * J.If 241 -44 285.04 _ 1.3E-11 1348 .374 33... -16.-48+0 * --. 27 - 50 641 .437 267 200 -50 --1- -72 E+ - 277.85 1.6E-11 -. £4 --3 57Et+7 -3.81E a06 * 60 208 -113 222 A S A ' 268.72 2.0E-11 316 .593 - 377r~ 02- - - ~~~~~~~--- -* LE 4 --- -1' 0 r_ u o "- ?B- -134 2.4E-11 114 .763 194 186 .._. _ -- i .47VrEr 6 * 257.24 ...... -n-2'. - -: TT-J-.O;--6.E-0: ffuO - 7- 163 -127 --D-6-2-- --- ' 247.84 2.7E-11 46 .902 183 ------...... - -' u- ...E.. 23------~-~------

L ,LAT =LATITUOE (OEGREES) 1 25-T- -t;------R TURE 0KELVT ) C- SECOND) 4 OTEWP/OT =TIME RATE OF CHANGE OF TEMPERATURE(KELVIN PER * APOTERFESS V----POR-PRESUR- (N/M ------ALBEDO =ALBEDO "S2lR FLUX =SOLAR FLUX (W/M*2) H' ::r l IR FLUX =OUTGOING INFRARED FLUX (W/M**2) 4 BTLUX V RN--F C 1/ ------28 l--YTUEVtTLLY TNRTZrrE HERPO NIND =MERIDIONAL (NORTHWARD) WINO (M/S) Ft- * '11 SENi HT TRg- TOHER -ERSILE- ET NSP-T-T-TT( i ------LATNT HT TRANS =LATENT HEAT TRANSPORT (W/M) - :-. UCn SE HT TRANS =UCEANIC SENSIBLE HEAT IRANSP'UT (HW) 0 22

CLIMAT: A Simple Zonally Averaged Energy Balance Climate Model .....

REFERENCES Budyko, M. I., 1969: The Effect of Solar Radiation Variations on the Climate of the Earth. TeZZus (21), 611-619. Gal-Chen, T., and S. H. Schneider, 1976: Energy Balance Climate Modeling: Comparison of Radiative and Dynamic Feedback Mechanisms. TeZZus (28), 108-121. Schneider, S. H., and T. Gal-Chen, 1973: Numerical Experiments in Climate Stability. J. Geophys. Res. (78), 6182-6194. Sellers, W. D., 1969: A Global Climate Model Based on the Energy Balance of the Earth-Atmosphere System. J. AppZ. Meterorl. (8), 392-400. 23

· O· · · a · 0 a It a J. Ahlquist

EXPERIMENTATION WITH MERIDIONAL HEAT TRANSPORT FORMULATIONS IN THE SCHNEIDER AND GAL-CHEN ENERGY BALANCE CLIMATE MODEL by Eric J. Barron University of Miami Tzvi Gal-Chen, Scientist

ABSTRACT Simple energy-balance climate models of the Budyko and Sellers type are extremely sensitive to variations in solar radiation. A decrease in solar input of %2.0 percent results in a catastrophic ice-covered Earth solution. During geological time glaciations have repeatedly advanced extremely close to the critical latitude of the catastrophic solution, yet the ice-covered state has never been realized. This suggests three possibilities; the solar input has varied less than 2.0 percent during geologic time or the radiation balance was different as a result of variations in atmospheric composition or oceanic circulation patterns, or the model is overly sensitive to variations in solar input. The model's sensitivity is strongly influenced by the formation of the ice- albedo positive feedback. The purpose of the modifications presented is to examine the model sensitivity to changes in solar scaling with different physical interpretations of heat transport. The modifications are a first attempt at formulating the meridional heat transport such that the transport is a function of 1) tropical sea surface temperature and 2) the average temperature structure of the atmosphere.

INTRODUCTION A major purpose of climate models is to examine the sensitivity of the climate system to various external and internal perturbations. Simple, global energy balance climate models (Budyko (1969); Sellers (1969); Schneider and Gal-Chen (1973); and Gal-Chen and Schneider (1976) are in particular useful to examine cause and effect processes and to experiment with different parametric representations. The relationship between climatic variations and scaling of the solar constant is an especially interesting aspect of these models. A decrease 26

Experimentation with Meridional Heat Transport Formulations .. ... between 1.6 and 2.0 percent in solar input results in a catastrophic ice- covered Earth solution. During the Pleistocene and Permian glacial periods, despite the repetitive nature of glacial cycles, an ice-covered earth state was never realized. This fact suggests either the solar input is rather invariant with respect to geologic time, or more than a 2.0 percent decrease is required to reach the catastrophic state. Alternatively, one of the many other theories proposed to explain climatic changes could be more valid. The present study is based on experimentation with the eddy heat flux parameterization and examination of the functional relationship of heat transport and glaciation in the Schneider and Gal-Chen model. In particular, the purpose is to examine the relationship of different parameterizations to the solar scaling required for an ice-covered Earth solution, and to examine the continuity of this relationship. Essentially, experiments with two types of variations were performed: 1) the poleward heat transport was determined as a function of changes in equatorial temperature and 2) the eddy heat transport was related to the equator to pole gradient of average potential temperature in the troposphere rather than using the surface temperature gradient.

SUMMARY OF MODEL ASSUMPTIONS A complete summary of the model characteristics is presented by Schneider and Gal-Chen (1973) and Gal-Chen and Schneider (1976). The following description is presented only for continuity of discussion. The basic modeling assumption is the long time scale equality of incoming solar radiation with outgoing infrared radiation. A time dependent version of the zonally averaged vertically integrated energy equation is

R = Qsc -)F - (1-a) F + F) (1) t sc ir sinO 3y o a q 27

·· · ·* E. Barron · · ·* · · · ·· · · · · · · · · · · · * · e where R is the thermal inertia of the ocean t = time T = sea level temperature = yearly average, zonally averaged value of solar input at Qsc( latitude ¢ = albedo FF. . is the outgoing infrared radiation flux to space iL and F , F and F are respectively the zonal heat fluxes due to ocean o q a currents, atmospheric motion and the transport of latent heat.

6 F. = c (q) T4 {l-mtanh(19T x 10 16)} (2) ir

where c(Q) is a consistency factor designed to make the present climate to an exact steady state solution of the finite difference analogue eq.(l) when no perturbations are present.

b () - CT x T T < TF g = (3)

b() - CT x TF Tg > TF where T = ground temperature g T = albedo feedback temperature (282.39) b(c) = empirical coefficients designed to fit (3) to present albedos CT = feedback rate parameter (0.009) It is assumed that a change in temperature instantaneously results in a change in albedo with the restriction: 0.25 < a < 0.85 regardless of -T I The zonal heat fluxes are 3T F =K o o 3y DT F =K - (4) a a by

F =K q(T) q q 3y 28

Experimentation with Meridional Heat Transport Formulations.. . .

where K's: are non-linear eddy diffusion coefficients as suggested by Stone (1973) and q(T) is the water vapor mixing ratio . In experiment (1) an effective eddy diffusion coefficient D* is used where D* is present day zonally averaged transport. A boundary condition is: applied such that there is no heat trans- ference across the poles (i.e., T gradient vanishes). The consistency constraint

Qs(lSC ) - Frir. = -div(F) (5) is applied at the initial time. Consistency is achieved by varying c(j) or by varying the largest divergence term. The convergence criteria for equilibrium is reached when the- energy storage terms on the left hand side of equation (1) are smaller in- absolute value than the largest transport term by a factor of 10- 4 (i.e., the storage term is. less than 10 9). The model is: not a real time dependent approach to equilibrium and is- based on how long the upper- 100 meters of the oceans adjust to an imbalance in the zonal energy balance.

THE CATASTROPHIC SOLUTION Analytical. analysis of simple. climate models of the Budyko and Sellers type (Chylek and Coakley (1975) and North (1975a,b)) indicate the present climate and: the ice-covered earth climate are stable under, small perturbations. However, the intermediate solution is unstable, and- only a few percent drop in solar input- leads to the ice-covered earth solution.. These .models are characterized:by a critical latitude- of ice- cap penetration at which point an abrupt transition to an ice-covere-d Earth occurs. The Budyko model predicts the catastrophic solution at a latitude of 500 and a decrease of the solar constant by 1.6. percent. The abrupt transition occurs because of the functional. relationship of: albedo and temperature. This feedback mechanism is important because of the large albedo contrast between ice andX ice-free areas.. The two branch analytical solution- graphed as a control axis (solar constant scaling) versus a behavior axis (the latitude of ice-- penetration) 29

· * * .· · · · e · e ·e*·*· *·e ** * *e · ...... E. Barron closely resembles a transverse section of a cusp catastrophe as described by Thompson (1975) and Zeeman (1976) and as illustrated in Figure 1. The cusp catastrophe is a three-dimensional figure with two control parameters (axes) and one behavioral axis. The mathematical theory of cusp cata- strophes suggests the possibility of a second control parameter, for instance eddy heat transport, which may yield a solution for which the latitude of ice penetration is a continuous function of solar scaling. The graphic model does not suggest that heat transport exerts an equal control in determination of the edge of the ice sheet since the transport merely distributes the solar input. Since there is no geologic evidence for the catastrophic solution despite the fact that ice has penetrated very close to the predicted critical latitude, it is desirable to experiment with different control parameterizations which potentially can yield solutions with a larger stability range. The purpose of experiment (1) is to map a cusp catastrophe where the second control parameter is a formulation of heat transport dependent on temperature changes in the tropics.

Figure 1

CUSP CATASTROPHE

w '\

, I, = LU 30

Experimentation with Meridional Heat Transport Formulations . . . .

EXPERIMENT 1 During a glacial period sea ice covers a larger portion of the oceans. Consequently poleward heat transport by surface ocean currents and in the form of latent heat will be inhibited in northern latitudes. This argument suggests meridional transport decreases during a glacial period. The arguments of Kraus (1975) suggest tropical temperatures are the controlling factor of climatic change because small reductions of tropical sea surface temperatures are associated with large reductions in latent heat release and in the temperature of the upper tropical troposphere, and consequently, in the meridional heat transport. Other arguments suggest that increased baroclinic activity during an ice age would increase the meridional transport. In this experiment, a constant, present day eddy coefficient, D*, which is a function of latitude is used during the simulation. The transport is related to equatorial temperature by the expression

Transport = D* x e ( oT * - 1) (6) where 3 is an arbitrary constant used for experimentation with the degree of tropical dependence, T is the model derived equatorial temperature and T * is the initial, present day, equatorial temperature. For present tropical temperature the transport is equal to D*, the present day transport. A decrease in solar scaling decreases transport as a function of the difference between perturbed and initial equatorial temperature. The 1 determines the degree of dependence on the temperature in the tropics. Consequently, the larger the solar scaling decrease, the smaller the poleward heat transport. Consider an extreme example of no heat transport across latitude zones for present day solar scaling. The result would be an extremely warm equator and an extremely cold polar region. A heat transport model with strong equatorial temperature dependence potentially could maintain a non-frozen climate in the tropics if heat transport decreases with decreasing solar input. The major purpose of 31

e * * l* * ** * * * * * * * E. Barron this simulation is to fit a B such that the ice-covered Earth solution occurs at a much lower solar constant and, ideally, such that the latitude of ice penetration is a continuous function of solar scaling rather than a catastrophic one. Results. A solar scaling of .99 and B equal to 20 resulted in the catastrophic ice covered Earth solution. The decrease in solar constant lowered equatorial temperature and therefore, decreased the meridional heat transport. The decreased heat transport resulted in ice formation in northern latitudes and positive ice-albedo feedback. The formulation accentuated the feedback and consequently the critical latitude of ice penetration was reached for only a one percent decrease in solar input. The solution is the opposite predicted by the theoretical model. Clearly, the model prediction is dependent on the critical latitude of the penetration rather than the temperature in the tropics. This result suggests increased meridional heat transport would reduce the significance of the ice-albedo feedback in the Schneider and Gal-Chen model. In order to experiment with increased heat transport the ration To/T * was inverted. Consequently as the ice covers more of 0 0 the Earth's surface the transport from the tropics is increased (despite the fact that the equatorial temperature is also decreasing). For a B of 20, the solar constant could be reduced to .94 before an ice covered Earth solution occurred. The critical latitude was approximately 35 . For larger B's a cool equator and slightly warmer polar region results and for B's larger than 40 the model became numerically unstable. There did not exist a B such that the latitude of the edge of the ice sheet was a continuous function of solar scaling. A similar result was derived using Budyko's formulation of D* where

D* = y (T - Tp) (7)

where y = 2.61 T = annual mean temperature at a given latitude T = planetary mean temperature 32

Experimentation with Meridional Heat Transport Formulations ....

The formulation of increased heat transport during a glacial period damped the positive ice-albedo feedback, however the physical reasoning for this type of parameterization is not readily apparent. The result is useful for examining the relationship of solar input, heat transport and glaciation. If glacial cycles are indeed caused by fluctuations in solar input, three possible conclusions are apparent from this model: 1) the climate is extremely sensitive to variations in the solar input and the model is therefore an accurate description or 2) if the theoretical model of decreased meridional heat transport is accurate, then the model's physics must be incomplete or overly simplified or 3) increased heat transport is in fact reasonable, although the physical reasoning is debatable (i.e., there is a negative feedback represented in the model).

EXPERIMENT 2 The present Schneider and Gal-Chen model formulates F , F and F o a q an an eddy coefficient multiplied by the zonally averaged surface temperature gradient. The zonally averaged potential temperature integrated over the troposphere is substituted in the calculation of F , F and F. o a q Logically, meridional heat transport is, more accurately, a function of the temperature structure of the atmosphere rather than simply the surface temperature gradient. O/by was used in the calculation of F in order 0 to prevent numerical instability in the present finite-difference scheme. Given the potential temperature at the surface

p R (To) g (8)

2 3 where R(T) = r + r1 T + r2T + r3T

r = 2.83471 x 103 o rl = -2.92257 x 101 r2 = 1.00547 x 10- 1 33

...... E. Barron

r3. = -1.153817 x 10 4 and gs = aP (saturated at To). s = r

ipQ sat-[__f _+ ~ T (9) ap sat p- pg ( surface

1000 where = RT o T = surface temperature

R = universal gas constant divided by the molecular weight of water g = acceleration of gravity and C = specific heat capacity

From Hess (1959)

l+ _Ws Rd T

P 1 P 2 CpRd T

where L = latent heat phase change Rd = gas constant for dry air

W = saturation mixing ratio s and e = ratio of molecular weights of water and dry air. The potential temperature, 0 at the tropopause is

1000

(p) = 0 + Pdp (11) o p where P = 1856.26 - 5.71 T , PT is the height of the tropopause in mb, T 0o T To oO 34

Experimentation with Meridional Heat Transport Formulation ...

The average potential temperature 0 is 1000 1 1000 - PT 0O(p)dp (12)

The calculated average potential temperature very closely matches the observed values. Based on this derivation of 0, the average potential temperature in the troposphere, the heat transport formulations Fa, F and F are a o q modified to the form

Transport = K - (13) °y The average potential temperature model resulted in an ice-covered Earth solution at a solar scaling of .972, approximately .7% lower than previous models using surface temperature.

Initially, the gradient is less than the -- gradient and consequently the heat transport would be expected to be smaller. Based on experiment (1) the decreased heat transport should enhance the ice- albedo feedback mechanism and result in a more rapid glaciation. The model result is again the opposite solution. For a temperature decrease, the moist pseudoadiabatic lapse rate decreased (larger negative number) but is much more sensitive to temperature changes in the tropics than at the poles. This tendency also reduces the equator to pole temperature gradient. The level of the tropopause, PT' is also a function of the ~T temperature and since aZ is more sensitive to a temperature change in the tropics, both these factors reduce the equator to pole temperature gradient. However, the formulation for R(T ), a factor used to match the calculated potential temperature to present day observed temperatures, changes in the opposite sense. In other words, in the formulation of a0 - where

= R(T )gs,

R(T ) changes in the opposite sense as gs and actually cancels at the equator. 0 35

· · e' e ·· e ··.e · ··. e * * * * ·* .E. Barron

R(T ) is calculated based on the observed seasonal changes which in a way is a mini-climatic change. However, the equator to pole gradient actually increased. If R(To) is maintained as a constant, a similar result as part (1) of experiment (1) occurs (i.e., enhancement of the feedback). The R(T ) formulation may be less reasonable for ice-albedo 0 larger climatic variations such as Pleistocene climates.

CONCLUSIONS The theoretical considerations presented by Kraus (1975) and discussed in this paper suggest the meridional heat transport during a glacial period should decrease. Oceanic surface currents and latent heat transport decrease because of increased extent of ice over the ocean. Secondly, the temperature of the upper tropical troposphere decreases substantially with small decreases in sea surface temperature in contrast with the polar regions. In both experiments, a decreased meridional heat transport results in a more rapid glaciation. From a geologic standpoint, this result is even less reasonable. Three possible conclusions may be reached: 1) the latitudinal extent of the ice sheet is extremely sensitive to changes in the solar constant and/or reductions of tropical sea surface temperature and consequently, these factors have been relatively stable throughout geologic time or 2) the model physics are oversimplified and do not give the correct response for a decreased temperature gradient or 3) a negative feedback mechanism is not taken into account which may oppose the ice-albedo positive feedback. Obviously, the modifications used in these experiments are only simple attempts to examine the functional relationship of meridional heat transport and glaciation. The results are encouraging even though contradictory. The problematic nature of the solutions are an incentive for formulation of a more sophisticated parameterization of heat transport during glacial periods, particularly as concerns latent and oceanic heat transport as the ice sheet advances. 36

Experimentation with Meridional Heat Transport Formulation ......

REFERENCES Budyko, M. I., 1969: The Effect of Solar Radiation Variations on the Climate of the Earth. TeZZus (21), 611-619. Chylek, P. and J. Coakley, 1975: Analytical Analysis of a Budyko-type Climate Model. J. Atmos. Sci. (32), 675-679. Gal-Chen, T., and S. H. Schneider, 1976: Energy Balance Climate Modeling: Comparison of Radiative and Dynamic Feedback Mechanisms. Tellus (28), 108-121. Hess, S., 1959: Introduction to Theoretical Meteorology. Holt, Rinehart and Winston, New York. Kraus, E., 1973: Comparison Between Ice Age and Present General Circulations. Nature (245), 129-133. North, G., 1975a: Analytical Solution to a Simple Climate Model with Diffusive Heat Transport. J. Atmos. Sci. (32), 1301-1307. ____ , 1975b: Theory of Energy-Balance Climate Models. J. Atmos. Sci. (32), 2033-2043. Schneider, S. and T. Gal-Chen, 1973: Numerical Experiments in Climate Stability. J. Geophys. Res. (78), 6182-6194. Sellers, W. D., 1969: A Global Climatic Model Based on the Energy Balance of the Earth-Atmosphere System. J. AppZ. Meteorl. (8), 392-400. Stone, P., 1973: The Effect of Large-Scale Eddies on Climatic Change. J. Atmos. Sci. (30), 521-529. Thompson, J., 1975: Experiments in Catastrophe. Nature (254), 392-400. Zeeman, E., 1976: Catastrophe Theory. Sci. American (234), 65-83. 37

†...... E. Barron

PRELIMINARY INVESTIGATION OF A TROPICAL SQUALL MESOSYSTEM AS OBSERVED BY AIRCRAFT DURING GATE by Kerry Emanuel Massachusetts Institute of Technology Ed Zipser, Scientist

ABSTRACT A squall line mesosystem is investigated using measurements obtained by aircraft during phase III of the GATE project. A preliminary analysis of the fields of motion, temperature, and moisture reveals that a) two important updraft maxima occur, one corresponding to forced ascent ahead of a surface gust front, and a second, more elevated updraft related to buoyant ascent or forcing by mesoscale heating; b) vorticity is generated in the updraft region and reaches maximum intensity in the middle troposphere, and c) the movement of the system is characterized by eastward propagation upshear and against the mean momentum field on a time scale of several hours, while it is strongly evident that the individual cells, with a lifetime of 1/2 to 1 hour, are advected northwestward with the mean flow.

INTRODUCTION A well organized, north-south oriented cumulonimbus line and associated mesosystem formed during the late morning of 14 September 1974, and propagated slowly eastward through the dense B- and C-scale ship arrays operated during phase III of the GATE project. The mesosystem was extensively surveyed by rawinsondes launched from the ship arrays, and by five aircraft operating between 990 and 190 mb. Each aircraft conducted between 4 and 12 passes through the system, flying in line patterns roughly transverse to the observed squall line orientation. The aircraft measurements permit a reconstruction of the meteorological fields for successive passes, from which certain aspects of the temporal evolution of the system may be deduced. The analysis is, however, limited to two dimensions. In the reduction of the aircraft data, it is found that the most problematic data interpretations involve the location of the aircraft with sufficient accuracy to resolve the convective scale features. 40

Preliminary Investigation of a Tropical Squall Mesosystem ......

CHARACTERISTICS OF AIRCRAFT AND MEASUREMENT SYSTEMS

A summary of the aircraft missions flown on 14 September is provided in Table 1. Unfortunately, data from the UK C-130 was unavailable at the time this research was conducted, and only limited data from the Sabreliner could be obtained. Thus, the analysis relies heavily on data collected by the DC-6, Electra, and US C-130. Each aircraft carried equipment for measuring inertial latitude and longitude, radar altitude, various aircraft control parameters, pressure, temperature, dew point temperature, apparent surface temperature, shortwave and longwave radiation, and liquid water content. In addition, the US C-130 measured the C02 temperature, and the total water content via a Lyman-Alpha instrument; and the US L-188 (Electra) measured vertical wind and boom ambient temperature. The specific quantities measured, as well as the characteristics of the measurement systems, are listed for each of the three aircraft used in the Appendix. In general, the various quantities were sampled several times per second, and in most cases were averaged over 1 second in the final output. A second of flying time corresponds to about 100 meters flying distance for the DC-6 and Electra flying at low levels, 120 meters for the C-130 at 700 mb., and 150 meters for the C-130 at 500 mb.

AIRCRAFT MISSION OF 14 SEPTEMBER The large scale flight plan for 14 September 1974, including the general area in which the flight pattern was conducted, is shown in Figure 1. Commonly, the aircraft left Dakar at about 09:00 local time, and in this instance, all aircraft had reached the pattern area by 1232Z (except for the Sabreliner, which started the flight pattern at 1311Z). Figure 1 also shows the 1200Z SMS satellite cloud outlines. The mesosystem was in range of the radars aboard the Quadra and the Oceanographer; however, data from the former has not yet been received, and the latter instrument was not operated prior to 15:50 on this day. The radar scan at this time (Figure 2) shows several banded echoes oriented SSW- NNE, moving slowly eastward. --a) 0-Cr 257-1 Zipser (US-C130) 257-2 Lazanoff (P-3A) MISSION DATE: 14 September 1974 MISSION SCIENTIST(S): Hoeber AIRBORNE 257-lb Mazin (IL-18C) 257-3 Reiff (WC-135) TAV ?i7 SCIENTIST(S): JTTTTIULINTAN X_ Lw t..- ,I r- and intercom- weather Encountreau Summary F-Mission Mission Aircraft Time of Lat. and Pattern Pattern Down and Altitude parison with Evaluation iNumber Type and Take Off Long. ( ) Flown Systems in Route and in Pattern Aircraft and of IP and Number of (feet) or Calibration, P=Prim. O 3 -' Scientist Landing Time of Circuits and number of A=Alter. (CMT Arrival each u-C n _ * _ ! t- -. - '-t~d _ (12 5,0-~Lin Sel (1)-- Planned to do a 6B add-on mis- I I4- I , I - -- I - I 1' In route: Broken trace mu, as F*LU I ! trom I I I (1) R.efractomer inop. - 0 257-1 1C2 UK-C130 1111 lOOON1000N Line (12) 5,000 Self sion, but this was cancelled 1.310 to 1317Z, 1333 to Ac. (P) Butler 2000 2200W 5,000 due to extensive heavy rain in CD D 1.340Z, 1604 to 1620Z, and In pattern: Extensive rain mainly 1232Z 5,000 de- the area. QC 5,000 ffrom 1855 to 2000Z. from As and Ac. Cloud system icn 5,000 cayed rapidly with time. 5,000 3 2,000 2,000 _ 500 CD - 500 500 500 -- CD

6½ In route: TCu, broken As and Cs. Very successful mission, O- --h 0842 100UN Line (9) 15,000 Self (4) IR radiometer questionabl 257-1 1C2 US-C130 In pattern: Line of Cbs scattered hours of continuous crossings <0 Daivs 1655 2200W 15,000 throughout. (P) Cu, overcast Ac and As. of band of Cbs in the C-array 1017Z 17,000 I! with good coordination with 17,000 ships. Life cycle of a strong C-tCD: 10,000 convective band should have 10,000 been well-documented. 10,000 £ re U) 17,000 (D 17,000

convective activit D)CD Beta vane inop. through- In route: Scattered Cu and Ci. Considerable 1C2 DC-6 0840 1000N Line (4) 1,000 Self (3) the mission. Active II 257-1 Computer inop. aftes Very hazy, poor visibility. throughout Emmanuel 1810 2200W 1,000 out. regions found. (P) 1755Z. In pattern: Line of showers marked evaporation 1030Z 1,000 turbulence. Very successful flight. 500 with wind shifts. Light of line did not change L's (2) 300 Intensity 50 at the lower levels. Some heavy l< > rain.

inop. In route: Broken Cu, As, Ac and Repeated penetrations of a L-188 0956 1000N Line (8) 2,000 Self (7) Vertical field mill Suc- 257-1 1C2 Data system overcast Ci. roughly N-S squall line. 1640 2200W 2,000 throughout. (P) LeMone inop. from 1601 to 1626Z. In pattern: Line of convection was cessful mission. 1121Z 2,000 growing, and persisted throughout 0) 2,000 mission. Line was narrow and well- CD 2,000 organized with observed convergence 2,000 First line dissipated while new 2,000 lines formed to the E and W. D 0 2,000 h -L O: This pattern was well-coordina- - 39,000 IR radiometer inop. In route: Fair weather. 257-1 1C2 Sabrelinse 1155 0835N Line (4) to ted with five other aircraft in v 3 39,000 throughout. In pattern: Line of Cb towers (P) Simpson, J. 1519 2230W the stack. Flight was success- 39,000 40,000 ft, oriented N-S line. 1311Z ful. Right altitude for cloud 39,000 tops; right pattern, correctly located. Analysis should be CD Eg very valuable.

_ 2I______- I 4 _I _ _ __ I _ __ I I__I _ _ _ 2 I C _0r (P ,^.. 42

Preliminary Investigation of a Tropical Squall Mesosystem ......

Visual observations of the squall line, by personnel aboard the aircraft, indicate a continuous band of cumulonimbus (Table 1). Apparently, the new growing cells were located near the eastern edge of the band, which was the more sharply defined in several respects, while older, decaying cells comprised the ill-defined west edge of the system. Photo- graphs reveal these characteristics, and also show anvils trailing behind the system, toward the west (Figure 3). The flight patterns were flown transverse to this system, near the southern end of the line. An attempt was made to keep the aircraft flight tracks vertically stacked, but at a given time, the individual aircraft tracks may have deviated in the horizontal up to about 10 nautical miles in the direction tangent to the squall line. The Sabreliner, flying at 39,000 feet (X198 mb.), was close to the cloud tops.* Figure 4 details the flight tracks of the DC-6 and Electra between 14:00 and 14:30; the DC-6 flying at about 980 mb., and the Electra at 940 mb. The flight tracks of the C-130 between 14:00 and 15:00 are also shown; during the first half-hour the aircraft flew at %694 mb., while during the second it operated at %522 mb. The DC-6 and Electra intercepted the gust front during these passages at 14:21, while the C-130 penetrated deep cloud at 14:13, flying at 694 mb., and again at 14:36, this time at 522 mb.

CORRECTIONS TO NAVIGATION SYSTEMS, AND LOCATION OF AIRCRAFT WITH RESPECT TO SQUALL LINE As the width of the cloud band was apparently on the order of 25 nautical miles, it is important to locate the aircraft with 1 or 2 nautical mile accuracy. The inertial systems suffered from various errors, including a systematic error which increased with time, and a "Schuler Oscillation," peculiar to inertial systems, with a period of 84.4 minutes (Kayton, 1969). The amplitude and phase of such oscillations were unknown in most instances, and thus no attempt could be made to correct for them. The total inertial position error, on landing, was as great as ten nautical miles, so that extensive attempts to correct the navigation were necessary.

* From direct observation by Joanne Simpson, aircraft scientist. 43

· Is· · · ·· · · ·e. · ·· · · ·- e. K. Emanuel

Figure 1: Flight Plan of 14 September (from "Report on the Field Phase of GATE - Aircraft Mission Summary," Gate Report No. 18, World Meteorological Organization) I__LI ______I 30o 25° 20° 15° I .....-- '1.------.-..- ' ,:..:..... >...- - -. __. - - .sAL----.---

I g50 _ _ -_ - ^_ *I. -_ -. -- " -- '-.- ...... :T" ,-4--'-AKAR

' eeeeeo' ' i' e e· eeeel~' s~~le " t~t'~:" · ··· r ·· 1 l · ~~~~~~~~~~~~~~~~~~· ......

'.. . .,. .. ,___t, · ~~~~~~~~~'"~,"-~'__,!': .. ':::'''::. " )......

110O · - 1

ale~ ~~ . ~ ~ ~. ~ ~ ~. ~~~~~~~...... , . . . ,.,,..-,. .,.....~,.. _~~~·0 ,

· ,e·~~~~~~~~~~~~~~ o ~ ~ ~ ~ _ .... a~~.....- -,,...~_- ...... _ . ~·r\ , \

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_, ______~~~~~~,______,,!.;_ .. _,_ ...... ______------_ 0

C 30° 25° 2U" I! 14 SEPT 1974 (EXCLUDING TRANS-ATLANTIC FLIGHT 257-3) 44

Preliminary Investigation of a Tropical Squall Mesosystem ......

Figure 2: Oceanographer Radar Scan at 15:02Z. Range markers are at 25 km. intervals, except the first is at 10 km.

. t m -..

\P ,

The US Electra carried a VLF system, with an accuracy of about 1 nautical mile, enabling a more precise determination of position in this case. In addition, the C-130 was equipped with Omega navigation, which was used to continuously update the inertial positions, to an accuracy of about a nautical mile. On occasion, aircraft that carry only inertial navigation systems can be located fairly accurately by comparing handwritten navigator's notes with the inertial data. (E.g., the aircraft will often fly over a ship with an accurately known position.) On this day, however, such notes were very incomplete and corrections to inertial systems of the DC-6 and Sabreliner were not possible. Fortunately, the inertial errors on landing of both these aircraft were less than .5 nautical mile, so that inertial positions were assumed correct throughout the flight. 45

...... e...... 9 ...* . .e**...... K. Emanuel

Figure 3: Top, Photograph from US C-130, panning SW - W at the east edge of the squall line. Aircraft was at 530 mb. in the top photograph and 694 mb in the bottom photograph. Bottom, looking east toward west edge of system. Note anvils overhead, sheared toward west. Photos courtesy of Dr. Edward Zipser. 46

Preliminary Investigation of a Tropical Squall Mesosystem ......

The movement of the squall line and the positioning of aircraft relative to the latter were determined in several ways, depending on the aircraft. It was found that the gust front was very well defined below 900 mb., so that this feature could be used to define the movement of the system as measured by the DC-6 and Electra. Figure 5 shows the normal velocity profile of the squall line as measured by the DC-6 and Electra during one pass each of the system. It was found that the Lyman-Alpha total water measurement device provided the most consistent method for locating the C-130 with respect to the eastern cloud edge, and that the radiation instruments aboard the Sabreliner were most useful in the same connection. As the aircraft flew at slightly different latitudes, and in directions not exactly normal to the squall line, it is necessary to project the aircraft positions onto a reference plane normal to the squall line. This reference plane is oriented in a 110 - 290 degree direction, intersecting the east edge of the squall line at all times at 8.16 degrees latitude. Thus the reference plane crosses the line at different points in the transverse direction, depending on the time. Hereafter, all aircraft positions will refer to this coordinate system. The projected longitude of the squall line east edge is plotted as a function of time, for each aircraft, in Figure 6. The correlation between the line movement, as measured by each aircraft, is quite good, especially after 13:30. Apparently, there were two line propagation regimes, one before and one after 13:00. The eastward propagation velocity during the latter was about 2.7 m sec 1 . It is apparent, in Figure 6, that the east edge of the line slopes westward with altitude. If the slopes of the regression lines are averaged for the four aircraft, during the period after 13:00, and the resulting slope is fitted to each aircraft line position set, the vertical slope of the line may be determined from the four intercepts. This slope is plotted in Figure 7. The line appears to slope about 52 degrees westward from the vertical, probably as a result of moderate easterly shear, at least below 500 mb. 47

...... e o ...... o . . . . K, Emanuel

Figure 4: Flight Tracks of Aircraft, 14 September. Upper left, DC-6 at 980 mb. between 14:00 and 14:30. Upper right, Electra at 940 mb. between 14:00 and 14:30. Lower left, C-130 at 694 mb. between 14:00 and 14:30. Lower right, C-130 at 522 mb. between 14:30 and 15:00. DAY 257 N6539C DAY 257 N595KR

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B 8.5

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-71 A -23i.5 -25.0 -Z.5 -'z. u -,. * -4.,J -. ... LONGITUDE (DEG) P= 3 LONGITUDE (DEG) P- 17

-55L 1 C JA', 25- N6541 C DA'' 257 IO192 I "

i:e c : - 5: I 1: 10.0 - i I I I I I I

(\, N

9.5 h

.*** I " * I i ! ' ! i

9.0 *. .- . , ! ' ' '

- 8.5 -- --- t- t --- i I t :

-2 .0 -23.5 -23.0 -22.5 -22. P= 3 LONGITUDE (DEG) P= 3 LONGITUDE (DEG) 48

Preliminary Investigation of a Tropical Squall Mesosystem ...

The horizontal wind components are computed, also using the aircraft inertial systems, and projected onto the aforementioned reference plane. It is observed that, in general, such measurements of wind velocity are subject to two errors: Heading-dependent errors, and errors inherent in the inertial system (including Schuler oscillation). An estimate was made of the former by examining the measured wind components just before, and just after a 180 degree turn. The results of this analysis are presented in Figure 8. No attempt could be made to correct for the Schuler oscillation in the inertial system.*

Figure 5: Left, Example of Normal Velocity Profile for DC-6. Wind shift is at zero nm, east is on the right. Velocity is in msec - . Right, Example of Normal Velocity Profile for Electra.

:. r -TT r T-T-r- -TrT-T-r [ iI i -rr-r- TTI' f T-17-T 7 T-- T-'T- I r I| I I rn 1.-5 -I- TT T I,I , I I II I i I I I I" l I I I I 1-

-i -4

iC -1.5 19i_

.: -i C- Z -3. - -3.s -

-i -3.5 -Xa i,, -4v 3

uI I I I I I I I I I I I I I I I I , iI I I I . .I I I I I II l. .II It I,, I I , LLII J -5.0 -14 -35 -30 -25 -20 -15 -10 -5 u 5 1 15I -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 2 X (N.M.) X (N.M.)

* The period of the Schuler oscillation was, unfortunately, nearly equal to the time needed for one complete aircraft cycle through the system. There is no reason to suspect, a priori, that the phase of this oscillation is correlated between aircraft. Smooth vertical trends in the measured horizontal velocity components in the regions removed from the squall line, as viewed in the composites involving all aircraft, would tend to support the conclusion that the amplitude of the oscillation is small. 49

* ...... * e* e*.....* e-...*@** K. Emanuel

Figure 6: Longitude of Squall Line Eastern Edge vs. Time. Line defined by wind shift for DC-6 and Electra, otherwise by evidence of cloud penetration. Inertial systems corrected for landing error by linear interpolation; and all longitudes are corrected to 8.6 degrees latitude, since line is skewed from north-south direction.

w o 23.1

3- z 0 23.0 0 wlU

o 22.9 0

40 13:0020 40 14:00 2 40 15:00 TIME Figure 7: Day 257: Westward Slope of Squall Mesosystem as Determined by Wind-shift Line Measured by DC-6 and Electra and Cloud Penetration by C-130 and Sabreliner. _I 1TT71__~Y~ I .; II v I I I --- e. SabrelinerSabre liner 12000 H

9000 - E

-r 0 6000 - lLI

3000 F C-1304 (694)

Electra \ C-6 0

I II - I Ii I I I I. , I I ! I I I ---J 11 10 9 8 7 6 5 4 3 2 1 0 - -2-3-4 (n.m.) 50

Preliminary Investigation of a Tropical Squall Mesosystem ......

Figure 8: Heading-Dependent Velocity Error vs. Time. Correction to East- bound Aircraft Track. Top to bottom, DC-6, Electra, C-130.

I Il I X . - - I. I I I I I I REGRESSION DATA NORMAL X-NORMAL SLOPE = -0.006 COMPONENT 2 INTERCEPT=0.3705 O-TANGENTIAL TANGENTIAL COMPONENT SLOPE =0.2094 INTERCEPT=-2.4137 m/sec I - (0.50) (0.50) (0.60) (0.33) X 0 0 x 0 - (0.20) 0 X (0.29) (0.1 1) (0.17)

-I ! I I I I I 40 40 12:00 20 40 13:00 20 14:0014:00 20 15:00 TIME

m/sec I

-I

TIME

m/sec I

-I

TIME 51

e * e e * * * * * * * e e o * o o . . *. K. Emanuel

SCALAR MEASUREMENTS, AND AVERAGING OF DATA Direct measurements of temperature, pressure, and dew point permit analysis of the thermodynamic and moisture fields associated with the mesosystem. The measurement of pressure was, in general, quite good, although in some cases there was apparently a constant or near constant bias in the observation.* Where determinable, this was removed--in no case was it greater than two millibars. The measurement of temperature and dew point both suffered a great deal from the effects of liquid water in cloud, even though much effort had been made to keep the instruments dry. Since the air stream is brought to stagnation before the temperature is registered, it is always subsaturated at this point--permitting liquid water to evaporate from the temperature element. This results in a low bias in the temperature, after a factor related to the stagnation effect is subtracted from the raw measurement. Liquid water in contact with the reflecting surface of the dew point instrument causes the latter to heat up in an attempt to evaporate the water. Because of this, the dew point is probably warm biased in cloud regions. As a result of these two effects, the measured temperature is sometimes lower than the dew point; in such cases, the temperature is set equal to the dew point. This is essentially the only correction made to the scalar measurements; it is felt that further modifications would become too arbitrary. Rather than compositing the data by using several aircraft passes through the system, the fields measured during single passes are used to construct a two-dimensional cross-section normal to the line. For this purpose, the four passes shown in Figure 4 were examined. Each of these passes were very close in time; in no case was one observation removed from another by more than 35 minutes. The data are averaged over 20 second periods (corresponding to about 1 nautical mile flying distance) for each pass, and the 4 resulting .data levels are plotted and subjectively interpolated onto a grid with 2 n.m. spacing in the horizontal, and 100 mb. in the vertical.

* Personal communication with Alan Miller, NCAR. 52

Preliminary Investigation of a Tropical Squall Mesosystem . . ...

Figure 9: Top, Component of Normal Velocity Relative to Moving Squall Line. Velocity is in msec- 1 Bottom, same as above, but for tangential component. Positive values indicate westerly and southerly flow respectively.

500 _. 1 r. ·. 0· --C - a· L n-I -. - ·I rl /a. a. a ~ ~~ ~ ~ ~ ~ I -. a~~~~~~~~~~~~

° -7.2D. Sil s * a e f a~ ~ ~~ ~ -. a~~~~~~~ 600 . a a a -a - a a~o f-&% a a~a ~ a~ ~ ~ * a a *, * -~~~~~~l a E0 ag a 0 a -, a. a . , , , ~~ ~~ ~~ ~ a 1 , a~~I

- a a a~~~~~~'t~~ a,,a a a~~~ a a a EG a, aaa~~~~~~~~A'a aa 700 a - a a a'arlaaa'a' c~~~~~~~aa,a a a a~t # . - aa a a. aM~a.Ia E ,.', a5 a a a aa' ~~~~~~~~~~~~~~Arg 'V a.',* ~~faab D a a I a~~ (MB) a a a' aa~,aa a a aG a

a, S

_ # lb 800 aa s 0 * %t ~ ~ , % %lb~~~~~~~~~~~a SC s %% ~ ~~ ~ a t ~ a , a lviba

900 I

I '

I I I .1 I I *' 1000 - L aC I a a&. a a LL --. - .- - - 30 10 U -1U -LU (N.M.)

500

600

700 (MB)

800

900

1000 (N.M.) 53

***..**** **** ... *.. * ... K. Emanuel

FIELD OF MOTION The tangential wind component, and the normal component relative to the moving surface wind shift line, are shown in Figure 9. It is immediately evident that the squall line, in its entirety, is not being advected westward with the mean flow, nor, in fact, are there many regions of west relative wind. Strong convergence is apparent just ahead of the surface wind shift, and also at the 700 mb. level several nautical miles to the rear. Weak convergence predominates in the area ahead of the wind shift line, and weak divergence covers the area to the west of the mesosystem. A stronger area of divergence is evident near 500 mb. about 5 n.m. behind the wind shift. The profile of tangential velocity reveals a remarkable area of negative relative vorticity sloping back from the wind shift line, and increasing in magnitude with altitude. Minimum relative vorticities of -1 x 10 3 sec 1 occur near 500 mb. at +9 n.m. These values indicate generation of negative absolute vorticity, which can only be accomplished by twisting of horizontal vorticity tubes. This view is supported by

Figure 10: Stream Function Corresponding to Figure 9 (top), in mb msec- for same cross-section as in Figure 9. 500

600

700

:MB)

800

900 - -

1000, I'. I I I ] 00 i n , n ininn ,-n in - ?N (N.M.) 54

Preliminary Investigation of a Tropical Squall Mesosystem ... -1 Figure 11: Vertical Motion in mb. sec

'; 600 C3~is· ,: ,

- -r(; (e ijTrt ,

(POMI~~~e~Ii 1000ofI 7 f andile andI shw i I. so a T II t aim Iq

9000 a

i 1000 40 30 20 10 0 -10 -20 (N.M.) a comparison of the unperterbed environmental tangential velocity (near the right-hand edge of Figure 9 (bottom)) with the streamlines and the w field. The latter are defined via the continuity equation integrated over the reference plane:

J100 v dp with i0ooo 0 the m-so-ircuationactsto dcreas thesheare inth"'"~'"Jion

and = _'-' with - =0 at x = -24 n.m. and x =+42 n.m. Profiles of 4 and w and shown in Figures 10 and 11. Two updraft maxima occur, one just above the gust front location, and a second at higher levels somewhat to the west. Weak upward motion prevails in the general region east of the wind shift, and downward motion are evident, one on either side of the area of strong updraft at 500 mb. A comparison of the vertical motion field with the observed distribution of tangential velocity indicates that the latter is being redistributed, to a certain degree, by the former. Note that the vertical shear of tangential velocity is far less to the rear of the system than ahead of it; apparently the meso-circulation acts to decrease the shear in this direction. 55

v..... eX...... ee.....e...... * K. Emanuel

- 1 Figure 12: Relative Vorticity in sec x 10" 5

5 0 0 ' r ·,

6000 // "' "

( OOM.l c:;~~ ~ ~ !~~ ~~·~,0~ q~9-~~·r ~ IC

8 wa00

0 0 20 C ·:30 -20

'waves.

mc u pa wat h il fveria oo s.get ha h p 56

Preliminary Investigation of a Tropical Squall Mesosystem . . ..

Figure 13: 1"- 1 v in secin -222 x 10 8. 9x ap 500

600

700 (MB)

800

900

I000 (N.M.

Figure 14: Equivalent Potential Temperature in degrees Kelvin.

500

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700

1MB)

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1000 (N.M.) 57

·· · ** Emanuel ·e * e * · † e * K.

THERMODYNAMIC VARIABLES An attempt is made to define the thermodynamic properties of the mesosystem, using the direct measurements of temperature and moisture. It is felt that a calculation of equivalent potential temperature has the greatest potential usefulness, as the cold bias in temperature and warm bias of dew point in cloud tend to be compensatory. The field of o is presented in Figure 14. It should be kept strongly in mind that e a) errors of measurement are likely in cloud (roughly between 0 and 15 nm.), and that b) measurements by different aircraft are likely to give different values. Therefore, the gradients of 0E between 1000 and 700 mb are not entirely dependable, but those between 700 and 500 mb. are, as both these levels were surveyed by a single aircraft. The values of equivalent potential temperature shown here are also uniformly too low by about 6 degrees Centigrade, since specific humidity rather than mixing ratio was unintentially used in the calculation. Although relatively high 0 air is being transported upward by the vertical motion, no values characteristic of boundary layer air (as measured by the DC-6) are found at middle levels in the updraft region. The low 0 air near the 700 mb. level, exterior to the immediate squall e circulation, is highly typical of mean tropical surroundings (Aspliden, 1976). Figure 15 (top) shows the 0 field superposed on the streamlines relative to the moving wind shift line. If the effects of mixing are, for the moment, neglected, it is apparent that large local temporal changes of e occur, especially in the region of upward motion, in association with strong horizontal advections. Evidently, the updraft region does not appear to retain a quasi-steady position with respect to the surface wind shift. If, on the other hand, the 0 field is compared with streamlines in a coordinate system moving westward with the mean momentum field* (Figure 15 (bottom)), a different pattern emerges. Strong horizontal transports of 0 do not occur, and apparently, the mean circulation to advect the 0 field upward ahead of the squall line, and downward acts e behind it. Comparison of the two streamline fields would appear to indicate that the main updraft center is drifting westward with the mean flow, but that the region of upward motion immediately above the gust front maintains a relatively stationary position with respect to the latter. *as defined by vertical average at x = 24 n.m 58

Preliminary Investigation of a Tropical Squall Mesosystem . ... .

Figure 15: Top, Oe in Degrees Kelvin, and Streamfunction Relative to 1 Surface Wind Shift Line in mb. msec- . Bottom, Oe in Degrees Kelvin, and Streamfunction Relative to Coordinate System Moving with the Mean Momentum Field.

500

600

700

'MB)

800

900

1000 (N.M.) 500

600

700

(MB)

800

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1000 (N.M.) 59

*el * e * e 0 * . * . * o K. Emanuel

0 No strong upward advections of e occur in the updraft at high levels behind the windshift; rather, upward transports appear to occur in conjunction with the smaller updraft above the wind shift line. Strong downward transports take place behind the west updraft cell. The existence of two individual updraft - downdraft doublets is implied during the time of these observations. A newly initiated updraft cell is present over the wind shift line, while an older, more elevated cell is observed about 12 n.m. behind the line. The latter is accom- panied by a downdraft of greater extent, but smaller intensity, and has ceased to transport equivalent potential temperature upward, at least through the mean flow. The younger cell, however, is actively advecting higher 0 e values toward greater altitudes. Evidence of a still younger updraft cell may be seen at low levels near -12 n.m., while what may be the remains of a very old doublet is observed at 500 mb. and near +35 n.m., 0 in both the vertical motion and e fields. Quite possibly, we are looking at an evolution of individual cumulonimbus, each with a lifetime of 1% hour, spaced roughly 10 n.m. apart. The individual cells intensify and dissipate, while being advected along by the mean flow, but the sequence itself drifts slowly eastward against both the mean flow and the shear direction. The new cells grow up along and move with the surface gust front, and evidence of both newer and older gust fronts may be seen in the profile of normal velocity near the surface. These correspond with negative perturbations in the tangential velocity field, and are also spaced about 10 n.m. apart, lagging 2 or 3 n.m. behind the updraft cells. In fact, the two gust front velocity regimes, apparent in Figure 6, may represent a new, vigorous gust front overtaking an older, decaying wind shift line. The sequence of individual cells is also evident in the profile of relative humidity, shown in Figure 16. (Bear in mind that the humidities are questionable in cloudy areas, and are probably low biased in these regions due to temperature errors.) High relative humidities occur in both the important updraft regions, and also in areas corresponding to very young and very old updraft cells. 60

Preliminary Investigation of a Tropical Squall Mesosystem.. ..

Figure 16: Relative Humidity, in percent. 500

600

700

800

900

1000 4000L 1 30 20 II0lO 0 -10 -20 (N.M.) Further evidence of discrete propagation of the mesosystem may be seen, to some degree, in the sequence of individual aircraft passes through the system. If, indeed, individual cumulus towers are advected by the mean flow, then they should also have a large tangential component of motion as they are transported across the line. It is therefore improbable that an individual cell can be tracked by the aircraft (unless these cells are also elongated in the same direction as the main line). Nevertheless, some evidence of westward advection of individual features may be seen in certain aircraft sequences. An example of such a sequence, in the mixing ratio field, is shown in Figure 17. The phenomenon of discrete propagation of convective mesosystems has been discussed in connection with previous tropical experiments (Zipser, 1969), and has recently been observed on radar (Houze, 1976; Sanders and Emanuel, 1976). In the latter case, new echoes were observed to form in the middle troposphere upshear of mature echoes, and propagate downward while drifting with the mean flow. In both instances, the squall line--defined as an entity with a lifetime of %6 hours--propagated · · ··· *· · · · · · · · e . · · · · ·e · · . · · · K Emanuel generally upshear. The mechanism by which such propagation occurs remains a mystery, and should provide an interesting area for further investigation.

CONCLUSIONS The measurement of meteorological variables in a tropical mesosystem on small time and space scales has been accomplished, essentially for the first time, by the GATE project. It is now possible to resolve individual convective elements on a time scale of %1/2 hour and space scale of 1% nautical mile, and, to a certain degree, follow their temporal evolution over the span of several hours. A preliminary analysis of the aircraft data, for this case, has revealed the existence of several previously unsuspected features, including regions of large vorticity, generation of negative absolute vorticity over substantial areas, and descretization of the squall line propagation. Further investigation of such mesosystems, using GATE data, should reveal common properties of these convective systems. Additional analysis of this and other cases, including detailed heat, momentum, and moisture budgets and inspection of turbulent fluxes and mixing, will greatly increase the understanding of the dynamics of convection. 62

Preliminary Investigation of a Tropical Squall Mesosystem ......

Figure 17: Profile of Mixing Ratio, in gm Kg-, for two passes of the DC-6 %1 hour apart.

MIXING RATI0 DC-6

1 -r'r-. -r- - I-F1-TT I ! I--F-rT I AFI 1 1 i mT T r1 T r r-rT- .I

I T-TT-rT I4 I

t4 i I

LDL 1-. 4

I 2 V5 1-

16.0-

15 .8 L I I I a I I | I II I I I I I I i I I I I I I I I a I A I I I I ' ' -40 -35 -30 -25 -20 -15 -10 -5 0 5 10 15 20 X (N.M.) 63

*. . . . .*. ..e. .... o .e. * ! e. . ... e 0. -. K. Emanuel

Figure 17, continued.

MIXING RRTI0 DC-6 13.5 -r--T--' -- r-T-r- Tr-TT-I '"' ' 1I' r ' I v -T-r-T--rT r-- T-Tr-T I ' T-1 1-/4 i A A.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~· -1

r I-

i-..5 iH

II I I- Ck i 0 I -Y. -

I -. wv

15.5

15.0 L"- -25 -20 -15 -10 -5 0 5 10 15 20 X (N.M.) 64

Preliminary Investigation of a Tropical Squall Mesosystem . ...

APPENDIX: AIRCRAFT MEASUREMENTS AND INSTRUMENTATION

NOAA DC-6 N6539C

Parameter Ins trwnent Used Operating Procedures of Instrument

Time Crystal Oscillator. Time Code Generator

Latitude Delco Carousel Inertial Navigation System

Longitude

Heading

Pitch Angle II

Roll Angle

N-S Ground Speed

E-W Ground Speed

Radar Stewart Warner APN 159A Conventional pulsed radar utilizing Altitude "leading edge-time of flight" timing (measuring) techniques.

Static Garret Pressure Transducer 470 MHz Quartz crystal oscillator whose Pressure frequency is pressure dependent.

Dew Point Cambridge Systems 137-C3-S3-P Formation of dew or frost on an electri- Temperature Hygrometer cally-cooled mirror is determined by diffusion of light. Temperature measured by platinum resistance element.

Apparent Barnes PRT-5 IR Radiometer Incoming 8-13p radiation is chopped with Surface reflective blade so that alternately Temperature incoming radiation and radiation from the internal cavity is measured by an immersed thermistor which is compared against another thermistor at cavity temperature.

Sideslip Lockheed Gust Probe System Angle

True Airspeed Calculated

Liquid Water Johnson Williams Hot Wire Flowmeter Changes in resistance of heated wire Content perpendicular to the airstream due to impingment of cloud droplets <50p dia- meter is compared to that of a wire parallel to the air stream which is not in contact with the cloud droplets.

Temperature Rosemont 102E2 Platinum Resistance A platinum resistance wire shielded from Thermometer the impingment of water drops and other particles in a boundary layer controlled housing is one leg of a linearized bridge circuit. 65

...... K. Emanuel

NOAA DC-6 N6539C (continued)

Parameter Instrument Used Operating Procedures of Instrument

U Wind Calculated Component

V Wind Calculated Component

Longwave Epply PIR Pyrgrometer (3-50p) A multijunction thermopile with an Outgoing absolute reference at the focal point of Radiation a filtered hemispheric lens whose output is a 4th power of hemispheric flux of I.R.

Shortwave Epply 2 Spectral Pyranometer (.3-3p) A hemispheric filtered thermopile Outgoing referenced to the instrument base Radiation temperature.

Longwave Epply PIR Pyrgeometer (3-50p) Same as Longwave Radiation above. Incoming Radiation

Shortwave Epply 2 Spectral Pyranometer (.3-3p) Same as Shortwave Radiation above. Incoming Radiation 66

Preliminary Investigation of a Tropical Squall Mesosystem .

NCAR Electra L-188 N595KR

Parameter In trument' Used Operating Procedures of Instrument

Time Gulton Time Code Generator NCAR Electra Data Management System (EDMS) Model DST-930 records at 5 samples per second, with aperature times 500 microseconds for syncro to digital channels, 75 microseconds for analog to digital channels.

Altitude Litton LTN-51 Inertial Navigation System

Longitude " "

INS True Heading

Sideslip Angle NCAR Gust Probe System

Attack Angle " "

N-S Ground Litton LTN-51 Inertial Navigation Speed System

E-W Ground ,t Speed

True Airspeed Rosemont 1301-B Pressure Transducer

INS Pitch Litton LTN-51 Inertial Navigation Angle System

INS Roll Angle

U Wind Calculated Component

V Wind Calculated Component

W Wind Calculated Component

Static Rosemont 1301-A Pressure Transducer The transducer is a variable capacitance Pressure device. Pressure is constant on one side of diaphragm; open to ambient on the other. Pressure changes capacitance which is electrically part of an oscillatic circuit.

Radar Sperry Rand Model AA-220 FM-CW radar altimeter receivable to 20 cm. Altitude The device sweeps a fixed frequency band linearly and compares that sent to that returned.

Ambient Rosemont 102E2AL Platinum Resistance Platinum resistance wire shielded from Temperature Thermometer impingment of aerosols in a boundary layer controlled housing one leg of a linearized bridge circuit. 67

a...... a . . a. .. lK...... o K. Emanuel

NCAR Electra L-188 N595KR (continued)

Parameter Instrument Used Operating Procedures of Instrument

Boom Ambient Rosemont 102E2AL Platinum Resistance Same as sensor for Ambient Temperature. Temperature Thermometer

Dew Point Cambridge Systems 137-C3-S3 Hygrometer Formation of dew or frost on an electri- Temperature cally-cooled mirror is determined by diffusion of light. Temperature measured by platinum resistance element.

Apparent Barnes PRT-5 Bolometric Radiometer. Incoming 8-13p radiation is chopped with Surface reflective blade so that alternately Temperature incoming radiation and radiation from the internal cavity is measured by an immersed thermistor which is compared against another thermistor in the cavity.

Liquid Water Johnson Williams LV/H Hot Wire Changes in resistance of heated wire Content Flowmeter perpendicular to the air stream due to impingment of cloud droplets <50p dia- meter is compared to that of a wire parallel to the air stream which is not in contact with the cloud droplets.

Incoming Barnes IT-3 Bolometric Radiometer An early version of a type I.R. radio- Radiation meter, e.g., PRT downward-looking Temperature radiometer.

Shortwave Epply 2 Spectral Pyranometers A hemispheric filtered thermopile Outgoing (3 sensors) referenced to the instrument base Radiation temperature.

Longwave Epply PIR Pyrgeometer (4-45p) A multijunction thermopile with an Outgoing absolute reference at the focal point of Radiation a filtered hemispheric lens whose output is o 4th power of hemispheric flux of I.R.

Shortwave Eppley Spectra Pyranometers Same as Shortwave Radiation above. Incoming (3 sensors) Radiation

Longwave Epply PIR Pyrgeometer (4-45p) Same as Longwave Radiation above. Incoming Radiation 68

Preliminary Investigation of a Tropical Squall Mesosystem ...

NOAA C-130 N6541C

Parameter Instrument Used Operating Procedures of Instrument

Time Crystal Oscillator Time Code Generator

Latitude Northrop AN/ARN99V (Omega Bound) Inertial Navigation System

Longitude

INS True Heading

Sideslip Angle Rosemont Model 858 Angle of Attack and Sideslip Sensor

Attack Angle I! It

N-S Ground Northrop AN/ARN99V (Omega Bound) Speed Inertial Navigation System

E-W Ground II Speed

INS Pitch Angle

INS Roll Angle ..

Radar Altitude Stewart Warner APN 159A Conventional pulsed radar utilizing "leading edge-time of flight" timing techniques.

Static Kollsman A4533-000001 Pressure The change of the natural frequency of an Pressure Transducer aneroid capsule is a function of the pressure differential between the interior and exterior of the capsule.

Temperature Rosemont 102CH2AF Platinum Resistance A platinum resistance wire shielded from Thermometer impingment of water drops and other particles in a boundary layer controlled housing is one leg of a linearized bridge circuit.

True Airspeed Rosemont 1301DB 13B Pressure Transducer

U Wind Calculated Component

V Wind Calculated Component

Dew Point Cambridge Systems 137 Hygrometer Formation of dew or frost on an electri- Temperature cally-cooled mirror is determined by diffusion of light. Temperature measured by platinum resistance element.

Apparent Barnes PRT-5 IR Radiometer Incoming 8-13p radiation is chopped with Surface reflective blade so that alternately, Temperature incoming radiation and radiation from the internal cavity is measured by an immersed thermistor which is compared against the thermistor in the cavity. 69

...... o . * .. . K. Emanuel

NOAA C-130 N6541C (continued)

Parameter Instrument Used Operating Procedures of Instrument

CO2 Barnes PRT-5 Filtered IR Radiometer Same as radiometer used to sense surface Temperature temperature with the exception of a 14.7-15.7u bandwidth and the addition of an optical pass band filter.

Aircraft Kalman Filtered Output from Inertial Vertical Accelerometer Velocity

Longwave Epply PIR Prygeometer (3-50p) A multijunction thermopile with an Incoming absolute reference at the focal point of Radiation a filtered hemispheric lens whose output is c 4th power of hemispheric flux of I.R.

Shortwave Epply 2 Spectral Pyranometer (.3-3p) A hemispheric filtered thermopile Incoming reference to the instrument base Radiation temperature.

Longwave Epply PIR Pyrgeometer (3-50p) Same as Longwave Radiation above. Outgoing Radiation

Shortwave Epply 2 Spectral Pyranometer (.3-3p) Same as Shortwave Radiation above. Outgoing Radiation 70

Preliminary Investigation of a Tropical Squall Mesosystem ......

REFERENCES Aspliden, C. I., 1976: A Classification of the Structure of the Tropical Atmosphere and Related Energy Fluxes. J. AppZ. MeteorZ. 15(7), 692-697. Houze, R. A., 1976: GATE Radar Observation of a Tropical Squall Line. Paper presented at the 17th Radar Meteorology Conference, American Meteorological Society. Kayton, M. and F. Fried, 1969: Inertial Navigation, Avionics Navagations Systems, J. Wiley and Sons, New York. Lemone, M. and W. Pennel, 1976: The Relationship of Trade Wind Cumulus Distribution to Subcloud Layer Fluxes and Structure. Monthly Weather Review 104(5), 524-539. Riehl, H. and J. Malkus, 1958: On the Heat Balance in the Equatorial Trough Zone. Geophysica 6(3-4), 503-507. Sanders, F. and K. Emanuel, 1976 (expected publication date): The Momentum Budget and Temporal Evolution of a Mesoscale Convective System. J. Atmos. Sci. Zipser, E., 1969: The Role of Organized Unsaturated Convective Downdrafts in the Structure and Rapid Decay of an Equatorial Disturbance. J. AppZ. MeteorZ. 8(5), 799-814. 71

.. , ...... †a...... *. * * * * K. Emanuel

NUMERICAL SIMULATION OF PHOTOCHEMICAL PROCESSES IN THE TROPOSPHERE by Lynn M. Hubbard University of California at Riverside C. S. Kiang, Scientist INTRODUCTION The chemical evolution of a polluted troposphere is dependent on the flux of primary pollutants, intensity of solar radiation, meteorological conditions and chemical kinetics of the dynamic system. The photochemical process initiated by radiation absorbing molecules is often a driving force for chains of reactions resulting in the production of secondary 1 pollutants, some of which adversely affect life. An understanding of the kinetics of the chemical system is a necessary component in evaluating the total impact of source pollutants on the environment. Chemical modeling is a straightforward and useful approach for understanding the interactions of the molecular species present in the atmosphere. The biggest limitation is the availability of measured rate constants, the verification of proposed reactions and the wide variability of meteorological conditions making it difficult to test the model. The following is a discussion of a gas phase photochemical model which includes the nitrogen, oxygen, hydrogen, methane and sulfur chemistry as shown in Table 1 . The zero dimensional time dependent model includes 3 diurnally changing photochemical coefficients for summer solstice and 40° North latitude.

1 A primary pollutant is injected directly into the atmosphere and a secondary pollutant is produced in the atmosphere through chemical 2processes. The choice of reactions was governed by the existing literature, model tests of the impact of certain reactions, comments from Doug Davis and discussions with Jack Fishman of NCAR. 3 Photochemical coefficients determine the rate of photolysis and are dependent on the molecules' absorption cross section, the direct irradiance, scattered irradiance, solar zenith angle and density of air column above the height at which the coefficient is being calculated. 74

Numerical Simulation of Tropospheric Photochemical Processes .

Table 1: Chemical Reactions and Reaction Rates

NO2 + hv J l > NO + O(3 P) jl = 5.5 x 10- 3 * Leighton (1961)

4 03 + hv -2- O(3P) + 02 J2 = 3.3 x 10- * Hampson & Garvin (1974)

- 2 NO3 + hv _ NO2 + 0(3P) j3 = 5.5 x 10 . *

0(3P) + 02 + M k 03 + M k = 6.6 x 10-35exp( 5 0 )

2 0 0 03kexpl k2 -1200) 3 ) 0 3 + NO - N02 + 02 k= 9.0 x 10l 3 2 4 5 0 k 3 k-=.X -2 4 5 . 0 3 + NO2 NO3 + 02 k3 1.1 x 10T3exp(- )

NO + N03 - 2N02 k4 = 8.7 x 10 12 ks 10 , -0 0 0 , NO2 + NO3 5 > NO + NO2 + 02 ks = 23 x 1013exp(100

CH4 + OH k6 CH3 + H20 k6= 2.36 x 10 1exp(1) " T M -12 CH3 + 02 -- CH302 k7 = 1.2 x 10 1 2 k7 CH302 + NOC30 + N2 ks = 3.0 x 10N2exp(500) "

= CH30 + 02 k9- CH20 + H02 kgk 3.0 x 10-18 10-exp(-003.0x

HO2 + NO ---k NO2 + OH ko 2.0 x 10 1 3

CH20 + hv 34 > CO + H2 j4 = 1.9 x 10- 5 CAES (1975)

CH20 + hv -5 H + HCO js = 7.2 x 10 - 5

H~ + ~J-t_02M + +HO 2 NM Jsa = 6.7 x 10exp(-) Hampson. & T -. I r."/.'\ Gtarv /

HCO + 02 5 b > H02 + CO jsb =5.7 x 102

CO + OH(+02) kl- CO + H02 kl = 1.4 x 1013 it

CH4 + O('D) -k- CH3 + OH k 12 = 3.6 x 1010

6 03 + hv J >- 02 + O('D) j6 = 7.4 x 10 8* Griggs (1968)

k O('D) + H 20 - 3 20H kl3 = 3.5 x 10-10 Hampsor 1 & Garvin (1974)

Rate Constant Units: photolysis (j) - sec bimolecular - cm 3molecule1 sec 1 termolecular - cm6 molecule -sec * Noon Time Values 75

L. Hubbard

Table 1: Chemical Reactions and Reaction Rates (continued)

CH20 + OH HCO + H20 k ==4 1.4 x 1011 Hampson & Garvin (1974)

2H02 k l 5 H202 + 02 kl5 = 3.0 x 10- exp( T

H202 + OH - k H20 + H02 kl6 = 1.7 x 10-l1exp(0) T

- H202 + hv J 7 t 20H j7 = 1.23 x 10 6* Schumb et al (1955)

= -550 20H ks-; H20 + O(3P) ki8 1.0 x 10 1exp(- T5) Hampson & Garvin (1974)

k19I OH + H0 2 -- H 20 + 02 kl9 = 1.6 x 10 11.

k = *- 1 .- 000) 03 + OH k2020 H02 + 02 k20 1.6 x 10- 1exp(-000)

k2a1 1 3exp(12 5 0 ) 0 3 + H02 ---- OH + 202 k 2 1 = 1.0 x 10

M - l l O('D) - + O(P) k 2 2 = 5.0 x 10 Griggs (1968) k22 - HN03 + hv s > N02 + OH j8 = 4.83 x 10 6* Johnston & Graham (1973)

N02 + OH - k 2 3-- HN03 k23 = 4.89 x 10-12 Hampson & Garvin (1974)

2 4 if HN03 + OH - NO3 + H20 k24 = 1.3 x 1013 t' OH + NO - k 2 5 HONO k25 = 2.0 x 1012

it 3 HONO + hv --- ' OH + NO j9 = 2 e.752 x 10 *

3 8 NO + N02 + H20 - 6 2HONO k26 = 6.04 x 10 Chan et al (1976)

NO2 + HO2 k 2 7 HONO + 02 k27 = 3.0 x 10-14 Hampson & Garvin (1974)

k 28 11 S02 + OH - - HS03 k28 = 9.0 x 10-13

3 S02 + 0(3P) + M _' S03 + M k29 = 3.4 x 10- 2exp( T)3

3 2.0 x 1022 S02 + 03 S03 + 02 k03 = 9.0 x 1016 k313 ti S02 + HO2 - S03 + OH k 3 1 = 9.0 x 10-12

k 3 x 10-12 Bell et al (1975) H20 + S03 2 H2S04 k 3 2 = 1.0

S02 + CH302 3 3 S03 + CH30 k33 = 1.0 x 10-15 Davis & Klauber (1975)

* Noon Time Values 76

Numerical Simulation of Tropospheric Photochemical Processes .....

THE MODEL The model utilizes a coupled fourth order Runge-Kutta, Predictor- Corrector numerical scheme to solve the system of first order time dependent differential equations (See Table 2, Rate Equations) of the form

d[C] = k i][B] - [C]k.[D ]4 dt i j j

where the sum over i are the production terms and the sum over j are the destruction terms for the chemical species C. Runge-Kutta is a self starting method which generates the initial seven points after which the Predictor-Corrector (not self starting) scheme is activated. The variable time step is initialized and remains constant until error checks (tests for negative distribution, single step error, and time step check) require an increase or decrease in the time step. At each change in the time step Runge-Kutta restarts the calculations with seven new points. Some of the molecules in the chemical scheme have photochemical lifetimes, T, small in comparison to the long-lived species. 5 These short-lived species (Table 3) can be considered in stationary states (Leighton, 1961) and their time derivative set equal to zero.

d[C] [C] = i i dt k[DI jj J

The time derivative of the stationary state concentrations has been set equal to zero; however, the concentrations still vary as a result of dependence on time varying concentrations. A lifetime of less than 1 second was chosen as criterion for the stationary state assumption so

[C] = The concentration of C. 5 1 T -k [D 77

...... L. Hubbard

Table 2: Rate Equations

d[N] = [NO2]{jl + ks[N03]} + jg[HN02] - [NO]{k [03] - k4[N03] 1. dt 2

- k 8[CH302] - klo[H02] - k2s[OH] - k26[NO2][H20]}

2. d [N01 = j3[NO ] + [N0]{k2[03] + k4[N03] + k[CH302] + klo[H02]} dt 3

+ js[HN03] - [NO2]{jl + k3 [N03 ] + k23[OH] + k2 7[H0 2]

+ k26[NO][H20]}

3 3. d[03 = kl[0( P)][02][M] [03]{k2[NO] + j2 + k3 [NO2] + j6 + k2o[OH] dt

+ k2l[H02] + k 30 [S02 ]}

2 . d[H20 2 = ki 5[H02] - [H202]{k 1 6[OH] + j2 + Aerosol} dt

5. d[ = k 9[0 2][CH3 0] - [CH20]{j4 + j5 + kl [CH ]} dt 4 2

6. d[HN3] = k23[N02][OH] - [HNO ]{j8 + k24[OH]} dt 3

d [ 3 3 so] = - [S2]{k 28[OH] + k29[0 P)][][3H0 + k3 0[03 + k[ 2 ] dt + k33[CH302]}

8. d[HNo = [NO]{k2s[OH] + 2k2 [N02][H20]} + k27[NO2][H02] - j 9[HNO2] dt 2 6

d[H2S0 = k28[OH][S02] + k3 2 [H20][S03] dt

that the model's time step could be kept on the order of seconds. Table 4 shows the chemical species and their lifetimes at 12:00, maximum solar intensity, and at 24:00 when no photochemical reaction can be initiated. The lifetimes are approximate since they depend on time varying concentrations. The following observation (Table 4), T(N03) < At during the day T(NO3) > At during the night where At is the time step, implies there could be error associated with 78

Numerical Simulation of Tropospheric Photochemical Processes

Table 3: Stationary State Equations

k3[NO2] [03] + k24[HN03] [OH] 1. N03] =: 1.[NO 3 ] j3 + k4[NO] + k 5[N02]

[CH4]{k 6 [OH] + kl2[0('D)]} 2. [CH30210 2. 3 [CH2 ]] = ks[NO] + k33[S02]

[CH302]{k 8 [NO] + k33[S02]} 3. [CH30] = k[0 ] k9[02] 2

2 jl[N02 ] + j 2 [03] + J3[NO3 ] + k 3 2 [M][O('D)] + k1 8 [OH] = 4. O(3p)1 3 4*P)] [(= kl 1[02][M] + k29[S02][M]

j6 [ 3] 5. [o('D)] - 5 [O('D] kl 3 [H20] + k2 [CtI4] + k22[M]

3 [S0 2 ]{k2 9 ( P)] [M] + k30[03] + k3 1 [H02] + k3 3 [CH 3 02]} = 6. IS031[S03] - k 3 2[I20]... k32 [H20]

7. [IH] = [OH + H02 + HCO + H]

-B + /BZ-4ACG [H] =-B + 2A A = Destruction of two H B = Destruction of one H C = Production of H

2 2 -(2){kl 5 [HO21 /[OH] + kl9 [1H 2 ]/[OH] + k1 8 } A = (1 + [H02]/[OH]) 2

B = -[NO]{k1 o+k2 5 } - [N0 2 ]{k23+k27} - k 6[CH4 ] - k24[HN03] - k2 8 [S02] (1 + [HO2]/[OH])

C = [O('D)] k1 2 [CH4] + 2k 1 3[H20] + 2j 7 [H202] + 2j 5 [CH20] + k [CH 30][02]

+ j 8 [HN03] + js[HN02] 79

a...... L. Hubbard

Table 4: Approximate Photochemical Lifetimes T, calculated with day 1, case 5 values.

Species T(1200) (sec) T(2400) (sec)

NO 60 4. x 10 2 N02 180 5. x 10 5 03 90 1. x 104 H202 6. x 105 no removal mechanism at night CH20 1. x 104 no removal mechanism at night at night HNO 3 2. x 105no removal mechanism

S02 2. x 106 7. x 1010 HONO 3.5 x 102 no removal mechanism at night HSO3 * H2S04 * 0( 3P) 2. x 10 - 5 0.0 O('D) 7. x 10- 0 0.0 NO3 2. x 10- 1 32 CH302 3. x 10- 1 0.0 CH30 6.5 x 10- 2 0.0 S03 4. x 10 - 6 4. x 10- 6 H=OH+H02 9 x 10 - 2 0.0 ·+HCO+H__

* No gas phase removal mechanism is included in the model; removal is by aerosol formation.

- -- - -II--- -- '- -- -- C

keeping N03 in a stationary state at night. The model was tested with

-t[N03] # 0 at night for just the nitrogen chemistry (reactions jl-j 3, dt kl-ks) and the noontime values for [N03] differed from those generated

0 by .01% on day 1 and .05% on day 2. These by the model with d[N03dt errors are smaller than most of the uncertainties in the rate constants 6 and thus NO3 was left in a stationary state during the night.

6 One reason the error is so low is that the time step at night, when the rates are slow, approaches 30 seconds. 80

Numerical Simulation of Tropospheric Photochemical Processes . . . .

Table 5

Reaction Production Term for the Reaction (Values for 1200 hours, day 2, case 1-A)

1. NO2 + hv j3_ NO + 0(3P) jl[N02] = 1.043 x 10ol molecules cc-sec. 2. 03 hv - j 2 02 + 0(3P) + j2 [03 ] = 2.649 x 108

3 3 5 3. NO 3 + hv J N02 + O( P) j3 [N03 ] = 3.484 x 10

3 4. O(3) + 02 + M k > 03 + M kl [O( P)][0 2][M] = 1.070 x 1010 "

5. 03 + NO -2 N02 + 02 k2[03][NO] = 1.039 x 10'"

k3 6. 03 + N02 N03 + 02 k3 [03 ][N02 ] = 4.492 x 107

7. NO + NO 3 k4 2N02 k4[N03][NO] = 4.448 x 107

8. N02 + N03 - 5- N02 + NO + 02 ks[N03][N02] = 9.629 x 104

It is assumed the change in the following concentrations due to chemical and photochemical reactions is negligible and therefore that their concentrations are held constant. M(density of air @ 298°K) = 2.458 x 10' 9 molecules/cc 7 H20(30% relative humidity and 298°K) = 2.32 x 1017 molecules/cc7 02 = 5.15 x 1018 molecules/cc 7 CH4 = 3.44 x 1013 molecules/cc (1.4 ppm. Ehhalt, 1974) 1 3 8 CO = 9.832 x 101310 molecules/cc (4 ppm)

7 Calculated values. Typical value for a polluted troposphere. 81

...... L. Hubbard

Table 6

Figure Case Chemistry Reactions : Specifications

1 1At Nitrogen J1-iJ3 kl-ks day 3

2 lBt Nitrogen jl-j3, kl-k 5 day 3

3 2A Nitrogen-Methane il-i 7, kl-k 1 6 day 3 4* 2A Nitrogen-Methane J 1-J7, kl-kl 6 day 3

5 3A Nitrogen-Methane il-i 7, kl-k22 day 3 6* 3A Nitrogen-Methane kl-k22 day 3 il-i 7, 7 3B Nitrogen-Methane J1-J7, kl-k22 day 3 8* 3B Nitrogen-Methane Ji1-J7, kl-k22 day 3

9 4A 1 Nitrogen-Methane J1-J8, kl-k24 day 1 Nitric Acid

10 4A 2 Nitrogen-Methane J1-j9, kl-k 2 7 day 1 Nitric and Nitrous Acids

11 4A 3 Nitrogen-Methane ji-j8, kl-k24 day 1, [CH20] = Nitric Acid 6 ppb constant

12 5A1 Nitrogen-Methane jl-j9, kl-k 3 3 day 1, [CH20] = Sulphur-Acids 2 ppb constant

13 5A 2 Nitrogen-Methane 1i-j9, kl-k33 day 1, [CH20] = Sulphur-Acids 6 ppb initially

14 5As Nitrogen-Methane J1-j9, kl-k 3 3 day 3, [CH20] = Sulphur-Acids 6 ppb constant

15* 5As Nitrogen-Methane J1-j9, kl-k 3 3 day 3, [CH20] = Sulphur-Acids 6 ppb constant 1, [CH20] 16 5A 3 Nitrogen-Methane ji-js, kl-k 3 3 day = Sulphur-Acids 6 ppb constant 17 5As Nitrogen-Methane jl-j9, kl-k33 day 1, [CH20] = Sulphur-Acids 6 ppb constant, k 2 2 = C

I._ ...... * .I .

A implies [NOx] = 110 ppb; B implies [NO ] = 55 ppb. * Concentration curves of [OH] and/or [H02j: All others are concentration curves of [NO],[NO2] and [03]. 82

Numerical Simulation of Tropospheric Photochemical Processes .....

RESULTS AND DISCUSSION One of the primary pollutants emitted from exhaust systems in significant quantities in an urban atmosphere is NO and NO2(NOx = NO + NO2).

The basic chemical cycle associated with NOx is shown by reactions 1-8, Table 1, and Figures 1 and 2. The photolysis of N02 (reaction j ) is the dominant reaction for generation of ozone. Ozone (03) is a secondary pollutant which is harmful to life when the concentration is consistently of a significant level. The EPA standard for clean air is a maximum total oxidant of 80 ppb for 1 hour. The intermediate species, 0(3P), in the sequence

N02 + hv j NO + O(3P)

0(3P) + 02 + M kl + 03 + M

has a photochemical lifetime (Table 4) on the order of 10- 5 seconds indicating the rate determining step on 03 formation is the photolysis of N02. That is, each 0( 3P) formed "immediately" reacts to give 03. The equilibrium between the dominant reactions in Table 5 is evident.

N02 + hv + 02 + M j1 > NO + 03 + M

03 + hv + 02 + M 32_ 02 + 03 + M k2 NO + 03 - NO 2 + 02 with the diurnal pattern (Figures 1 and 2) due to the diurnal change in the photolysis rates with any perturbation in the equilibrium due to reactions j3, k3, k4 and ks. Throughout a 24 hour period reactions 1 through 8 give:

d[NO] _ d[N02] dt dt

Equating expressions 1 and 2, Table 2 (making the appropriate sign change), yields:

k 5[NO2][N0 3] + k4[NO][NO 3 ] + j 3[NO 3 ] = k3 [N02 ][0 3 ] 83

...... 0 ...... O .0 ...... L. Hubbard

Figure 1 Figure 2

NO(R),N02(B),AND 03(C) CONCS N0(A),N02(B),RND 03(C) C0NCS 12:

133 ; /0-

0 5 10 15 20 25 30 55 40 :5 5: 5 10 15 20 25 30 35 4' 45 5: t(HRS»=(X-l)/2 T(HRS)=(X-l)/2

Figure 3 Figure 4

N0(R),N02(B),RND 03(C) C0NCS C0NC 0H(R)XIOO RND H02(B) VS.TIME.298K

(_) z

^./ \

0 5 10 15 20 25 30 35 40 45 50 T(HRS)=(X-1)/2 84

Numerical Simulation of Tropospheric Photochemical Processes . .

The rate of reactions j3, k4 and ks equals the rate of reaction k3 with k4 dominant (Table 5) and therefore very nearly equal to the rate of k3.

k3[O3][N0 2] ~ k4[NO][N03]

Assuming the rates are equal gives

[03] = k [No][NO3] k 3 LN0 2]

Ozone is proportional to [NO] and inversely proportional to [NO 2]. When this expression is checked by a calculation using values of [NO], [NO2] and [NO 3] generated from the model, the value for [03] matches the model generated value within 2% throughout the daylit hours and within 10% during the night. This result is interesting but cannot necessarily be generalized since this involves only eight (important) reactions representing the nitrogen cycle and in the troposphere there are many more interactions which affect the concentrations of NO, NO2 and 03. In the field measurements of [NO], [NO2] and [03] an anti-correlation between [03] and [NO] has been observed. 9 This is reasonable if one considers the magnitude of reaction k2 (Table 5). Where there exists a large source of NO, such as an urban plume or a power plant plume, one would expect to find ozone depletion near the source due to the reaction (Davis and Klauber (1975)): k2 NO + 03 --k> NO2 + 02

One possible cause of the ozone buildup downwind of a plume source due to chemistry is the initial oxidation of NO to N02 (reactions 5 and 7 in the nitrogen cycle), the transport of N02 downwind and the subsequent photolysis of NO2 to yield 03. Any chemical species present within the plume that will oxidize NO to NO2 without destruction of O3 (such as H02 and CH302, reactions of ks and kio) may be important in the buildup of 03.

9 Data collected in both the Brown Cloud I (7/6/76-7/8/76) and Brown Cloud II (8/23/76-8/24/76) field experiment conducted by the Aerosol Project, NCAR, showed a consistant anti-correlation between [03] and [NO]. 85

...... 0. . .* ...... * L. Hubbard

There is an unsettled question concerning the products of reaction j3:

NO3 + hv -- _+ NO2 + O(3P)

or NO 3 + hv - NO + 02

The first reaction would initiate 03 formation through both the NO2 photolysis and the addition of O( 3P) and 02 (reaction kl). The second reaction would cause ozone destruction by reaction k 2. The first reaction is believed to dominate and has been chosen for use in this study (Hampson and Garvin (1975), page 108). Table 6 shows the five different cases for which the model was run in order to compare the effects of the different sets of reactions on the diurnal changes in [NO], [NO2], [03], [OH] and [H02]. The concentra-' tion of OH and H02 (OH and H02 are assumed in equlibirium) are monitored since the effect of the methane oxidation chain on the concentration of 03 depends on these concentrations, as will be discussed below. Figures 1 through 17 represent this comparison graphically. Case I (reactions ji - j3, ki - ks) and Case III (reactions ji - j7, kl - k22) were run for three days 0 with [NOx] = 110 ppb and again with [NOx] = 55 ppb. The noontime peaks for [03] of 33 ppb (Figure 1) and 22 ppb (Figure 2) show a 33% decrease in [03] with the 50% reduction in [NOx]. The addition of the methane chemistry (Case III) generated peak values of [03] (at 1700 hours) of 75 ppb (Figure 5) and 51 ppb (Figure 7),

a 32% decrease in [03] with the 50% reduction in NO x corresponding to the above observation. Comparison of Figures 6 and 8 show a decrease in [OH]

and [HO 2 ] with the 50% reduction in [NOx] due to the following dependence of [OH] and [H02] on [NO]:

CH202 + NO -k8 CH30 + N02 k9 CH30 + 02 9 CH20 + H02 CH20 + hv(+202) j i 5 > 2H02 + 02 H02 + NO kl OH + NO2

1A diurnal equilibrium is reached and the day 3 initial values equal the final values. 86

Numerical Simulation of Tropospheric Photochemical Processes . ..

Figure 5 Figure 6

N0 (R),N02 B), RND 03(C): C0NCS C0NC 0H(,R )X1OO RND H02 (B)' VS'. T ME 298K ..25

,8: -

Z;2"\

20/

5 10 15 2Q 25 30 TnHRS).- (X- I.1/2

Figure 7 Figure 8

N0 (R.), N02 (B') ,.RND 03(C) C0NCS C0NC 0HI(AIXI00. AONDH022(B): VS..TI ME,,298K 55

5: .:'e ,

:5 .: t L-

4 0`·· r 35 .:F2- '

_-) Z; . It. a: a W.-rr .9 U. 11·6I. 2S

10 0.0001

0 5 10 15 20 25 30 35' 40 T(HRS)=(X-11/2 T(HRSI=(X-D/2 87

* . .. e * ...... - ...... L. Hubbard

Case II differed from Case III by exclusion of reactions kl -k22. The relaxation of O(1D) to O(3P) (reaction k22) is the dominant removal mechanism for O(1D) and exclusion of this reaction cannot be considered realistic when modelling tropospheric chemistry. Case II, however, was included for the purpose of comparing the effect of the five neglected reactions on the concentrations of 03 and OH. The peak [03] decreased by 50% (Figures 4 and 6) in Case III indicating the significance of these five reactions. Inclusion of the relaxation of O(1D) to its ground state O( 3P) decreased O(1D) by two orders of magnitude therefore decreasing ozone production through the methane oxidation chain (reactions k 6-k2 2 , j4-j7). [O(1D)] affects [03] through the following:

1 k CH4 + 0( D) - CH3 + OH

H2O + O(1D) k13 20H

CH4 + OH 6 CH3 + H20

A direct comparison of the effect of [O(1D)] on [03] is shown in Figures 16 (k22 0 0) and 17 (k22 = 0) with a 12% decrease in [03] with inclusion of the O(1D) relaxation to 0(3P). Addition of the methane chemistry shifted the peak [03] from 1200 hours to 1700 hours. With addition of nitric acid (HN03), Figure 9, and nitrous acid changes in the (HNO2), Figure 10, chemistry came reversion of the diurnal concentrations to curves similar to the nitrogen chemistry (Figure 1).

This is due to the dominance of reaction k 2 3 which causes a decrease in [OH] from 106 molecules/cc in the above cases to 104 molecules/cc. Such a low OH number density cancels the effect of the methane oxidation chain by lowering the yield of one of the chain initiating reactions k6 CH4 + OH k6- CH3 + H2 0

to a quantity which produces insignificant consequences. That is, addition of HN03 and HNO2 chemistry nullifies the methane chemistry by reducing [OH]. This implies that if the methane oxidation chain is significant on 03 generation some other mechanism(s) must be present. In particular, it is likely that the hydrocarbon chemistry associated with a polluted 88

Numerical Simulation of Tropospheric Photochemical Processes ..... urban troposphere increases the concentration of formaldehyde (CH20) and aldehydes of higher order. The importance of reaction J5 for generation of odd H radicals (H = OH + HO2 + HCO + H) becomes evident:

CH20 + hv --- ÷ HCO + H

H + 02 + M -J 5 H02 + M

HCO + 02 - h HJH02 + CO

In the cases mentioned above the concentration of CH20 was determined through a rate equation in which the only production term was due to the reaction kg CH 30 + 02 -- CH20 + H02 where the only source of CH30 was through the oxidation of methane

(reactions k6,kl 2,k7,ks). This reaction alone generated a value for [CH20] of % lppb throughout a 24 hour period. Figure 11 is the same chemistry as Figure 9 except CH20 is held constant at 6ppb, a reasonable value for a polluted troposphere (Graedel, 1975). With CH20 held constant the concentration curves again show the effect of the methane oxidation chain. Figures 12 through 15 represent the total chemistry included to date in the model with variations on the concentration of CH20. In all runs a critical value for [OH] of %l. x 106 molecules/cc is necessary for the methane oxidation to have an effect on [03] (Crutzen, 1974).

The SO2-H2S04 chemistry (reactions k2 8-k3 3, Figures 12-15) generates considerable quantities of HSO3 and H 2S04 but does not substantially affect the other concentrations. The chemistry represented by Figure 14

(Day 3, CH 20 = 6ppb constant) produced concentrations of HSO3 = 3.2 x 1010 molecules/cc and H2SO4 = 1.3 x 1010 molecules/cc. The production of H202 (hydrogen peroxide) by

2H02 k15_ H202 + 02 is dominant over the destruction reactions: k H202 + OH k-1l 66 H20 + H02

H202 + hv -Jl 20H 89

0 ..* .... * o. e* e a L. Hubbard

Figure 9 Figure 10

N0(R),N02(B),RND 03(C) C0NCS N0(R),N02(B),RND 03(C) C0NCS 12. I^ :-----

1: L- \ r 2

is _ -60 z

0 20

20 / 20

0 5 10 15 20 25 30 35 4 45 50 T(HRS):(X-11/2

Figure 11 Figure 12

N0(R),N02(B),RND 03(C) C0NCS N0(R),N02(B),RND 03(C) C0NCS 12' -

------B- 9: -

8 ;a 8: :,-

C- L ,. Z - 2- I

20 2:

15 20 25 30 35 40 T(HRSI=(X-1)/2 T(HRS)=(X-11/2 90

Numerical Simulation of Tropospheric Photochemical Processes

Figure 13 Figure 14

N0(R),N02(B).RND 03(C) C0NCS N0(R),N02(B),RND 03(C) C0NCS 1;:: - 90 -

0 ip 80

:- \ X 60 L

1.20E-04, - \ 2 \

10 R

0 5 10 15 20 25 30 35 40 , 5 50 0 5 10 15 20 25 30 35 40 45 50 T(HRSI=(X-'I/Z T (HRSr=(X- 1/2

F~~~.. Figure2.00E-0- 1515Figure FigureF 16

OH NPVN02(B)PNDND 03(C) C0NCS 1.40E-O -,

1.20E-34

1.00E-05

~8.00E-05 Ca.. LiQ~~ .

o6. 00E-05 L

4.00E-05 I

20 2.00E-05

0 5 10 15 20 25 30 35 40 45 5; T(HRS)=(X-I)/2 91

* 0 .s .0 ...... e .* . .e*...... ** . . . . L. Hubbard

After running the model three days the concentration of H202 builds up to %L101 molecules/cc. An approximation of the removal of H 202 to aerosols was added to the rate equation:

d[H202] = Z[Ai][B] - [H20o ]Ek[Dj] - [H2O ]A kT } 2 2 dt =k[ j jj TmH20 2 where = probability of the gas sticking to aerosol A = surface area of aerosol/volume of air kT J27Tr- = thermal speed of particles H 202

Figures 18 through 21 show the comparison between a = 0 (no removal to aerosol) and a = .033. The difference in the concentrations of the other chemical species changed only within 2 - 10% while [H202] changed by 3 orders of magnitude. That is, [H202] has little effect on the other gas phase chemistry in the model. A comparison of the Runge-Kutta, Predictor Corrector numerical scheme with the Euler-Backward finite difference (Matsuno, 1966) scheme (approximately 4 times faster) was done in order to evaluate the difference between the respective fourth order and first order approximations. The relative computer concentrations differ most strongly at sunrise (20% difference) when the rates change the fastest. At other times the deviation in values ranges from 0 - 4% with the maximum values (peak concentrations) being equal, indicating that for the closed box chemical model presently developed the rates are changing slow enough to justify using the Euler-Backward numerical technique.

1 The comparison was conducted with Jack Fishman of NCAR. 92

Numerical Simulation of Tropospheric Photochemical Processes ...

Figure 17 Figure 18*

N0(),N02 (B).RND 03(C) C0NCS H202 3.OOE+s0 77~-ir~I i i i k , ] 1 !'',rT'-'T-¥r-r-r-rX r ' -

2.50E+08 a = .033

8: 2.00oE+o0 / 0- 60 I \

1 IL .L J.I LI LIL 1.A LI J -.L . 50 0 5 10 15 20 25 30 L5 40 45 50 X

Figure 19 Figure 20

03 H202 - -- - I o I , .I I I I? 1 v I F I .I I.80E+12 - F-r-T-TF-FT. ' i , , , ' 1 1-- 2.95E+11 I I , , I , I T- --T-

1.6CE+12 2.94E+ I a = .033 a = 0.0 1.40E+12 / 2.930+1l - \ .20E+12 / 2.92E+1I1 .O-12-/ \

8.00E+12 / - 2.91E+tI

2.90E+11 6.0o0E+,I 4.oo£+r/ \ 1 6t~l / \ 2.89E*11 -4

2.88E+11 2.00.E+ 1 /I

s, :-.iL A A L ...... - .1 5I , . j 35I . - I^: III u . . I ...... I . . I . I . . .I . . . I I - - 2.8T+1 0 5 10 15 ZO 25 30 35 40 45 50 5 11 15 2C 25 30 35 , I 5 57: X X

*In figures 18-21, the units of the x-coordinate are T(HRS)=(X-1)/2, and the units of the Y-coordinate are CONC(molecules/cm3 ). 93

.... *.. .. e. . .e*e ...... 6* . . . . .*L...... Hubbard

Figure 21 CONCLUSION The model is to be used as the

03 gas phase component in C. S. Kiang 1 2.00FOE2 '*-and ' ' ' Paulette Middleton's gas to part-

2 0~a=0, icle (or aerosol) model. The diurnally = 0.0 12 ' / \- changing concentrations of the gas

12- / \ -phase species will be a refined compo-

12/ \ nent in the existing aerosol model which uses fixed values. Another application of the model will be its combination with Dennis Deavon's urban scale dispersion model for both research '1 purposes and the possibility of use on a predictive basis for both an urban

5: : 15 2 25 3: 35 - .: and power plant plume. x The existing model is a foundation upon which new chemistry will be added. As noted earlier, the importance of hydrocarbon chemistry in the polluted troposphere becomes evident in Cases IV and V. However, a chemical model of an isolated power plant plume (coal burning) need not include hydrocarbon chemistry since the efficiency of most existing plants is high and emissions of hydrocarbons are insignificant. It has been suggested that S02 plays a significant role in ozone formation in the power plant plume (Davis & Klauber, 1975; Graedel, 1976) and the proposed mechanisms can be tested in the model. The addition of the loss to aerosols of these chemical species (in 12 particular, the radicals ) relevant to aerosol growth is another subject of future work.

12 A radical is a molecule with one unpaired electron and is generally highly reactive. 94

Numerical Simulation of Tropospheric Photochemical Processes .

REFERENCES Bell, J., A. W. CastlemAn, Jr., R. Davis and I. N. Tang, 1975: Association Reactions Involved in H2SO4 Aerosol Formation. Paper presented at the 68th Annual Meeting of the Air Pollution Control Assoc., Massachusetts. Center for Air Environment Studies, 1975: Publication Series. The Pennsylvania State University, Pennsylvania. Chan, Walter H., Robert J. Nordstrom, Jack G. Calvert, John H. Shaw, 1976: Kinetic Study of HONO Formation and Decay Reactions in Gaseous Mixtures of HONO, NO, NO2, H20, AND N2. Env. Sci. & Tech. (10), 675. Crutzen, Paul J., 1974: Photochemical Reactions Initiated by and Influencing Ozone in Unpolluted Tropospheric Air. Tellus (26), 47-57. Davis, D. D. and G. Klauber, 1975: Atmospheric Gas Phase Oxidation Mechanisms for the Molecule SO2. Int. J. Chem. Kinetics Symp. (1), 543-556. Ehhalt, D. H., 1974: The Atmospheric Cycle of Methane. Tellus (26), 58-70. Graedel, T. E., 1976: Sulfur Dioxide, Sulfate Aerosol, and Urban Ozone. Geo. Res. Letters 3(3), 181-184.

__ , L. A. Farrow and T. A. Weber, (1976, expected publication date): Kinetic Studies of the Photochemistry of the Urban Troposphere. Atmospheric Environment. Griggs, M., 1968: Absorption Coefficients of Ozone in the Ultraviolet and Visible Region. J. Chem. Phys. (49), 857-859. Hampson, R. F. and D. Garvin, 1974: Chemical Kinetic and Photochemical Data for Modelling Atmospheric Chemistry. NBS Technical Note 866, 112 pp. Johnston, H. S., and R. W. Graham, 1973: Gas-phase Ultraviolet Absorption Spectrum of Nitric Acid Vapor. J. Phys. Chem. (77), 62-63. Leighton, P. A., 1961: Photochemistry of Air Pollution, Academic Press, New York, 217 pp. Matsuno, T., 1966: Numerical Integrations of the Primitive Equations by a Simulated Backward Difference Method. J. Met. Soc. Japan (44). Schumb, W. C., C. N. Satterfield and R. L. Wentworth, 1955: Hydrogen Peroxide, Reinhold, New York, 266-291. 95

. . . . * ...... o...... e 0 L. Hubbard

TESTING NSSL ROUTINES KURV AND RTNI AT THE DEMONSTRATION DRIVER LEVEL by

Karen Kendrick Atlanta University Dick Valent, Scientist

There are many mathematical algorithms which are used by National Center for Atmospheric Research (NCAR) scientists on a regular basis for solving complex problems on the computer. The function and subroutine subprograms performing these algorithms are invaluable tools necessary for the jobs which must be completed. In order to avoid du- plication of effort and increase availability of these routines, the NCAR Software Support Library (NSSL) was created in March, 1974. The NSSL is a collection of some one hundred ten mathematical and input/ output functions and subroutines. They are stored on the User Library (ULIB) and are available for use by all. These routines include many of the algorithms frequently used in scientific computation. They fall under the following categories: 1) Solutions of Non-Linear Systems/Determination of Roots of a Polynomial 2) Interpolation 3) Solution of Linear Systems and Eigenvalue/Eigenvector Analysis 4) Numerical Integration 5) Solutions of Ordinary/Partial Differential Equations 6) Evaluation of Special Mathematical Functions 7) Fast Fourier Analysis 8) Statistical Analysis and Random Number Generators 9) Special Purpose Input/Output Routines 10) Data Processing Utility Routines 11) Computer Graphics 12) File Manipulation, Text Editing, Program Preprocessing and Debugging

The purpose of the NSSL Testing Project is to insure that the library routines are dependable and do what is expected of them. The 98

Testing NSSL Routines KURV and RTNI ......

actual testing effort is divided into two categories: 1) demonstration drivers 2) extensive test decks. More specifically, given a particular routine in the NSSL library, two test programs are written for it: a demonstration driver and an extensive test deck.

The demonstration drivers are simple routines which are designed to give the user an example of how an NSSL routine is to be used. It also includes a small test for accuracy and dependability. The examples used in the demonstration drivers are usually very simple, and the routine being tested is expected to work well on it. (All examples used were furnished by Alan K. Cline, Numerical Analyst, NCAR.) At present, there are some 18 demonstration drivers. The extensive test deck, on the other hand, is a more rigorous testing effort designed to point out the strengths and weaknesses of the particular routine being tested. The examples used in this deck may be complicated and are expected to push the routines to their limits in order to see how well they perform under extreme conditions. The author's part in the overall testing project was primarily to write demonstration drivers for the subroutine RTNI and the subroutine package KURV, both found in the NSSL library. A general description of the work follows: The routine RTNI is essentially a mathematical algorithm for the Newton-Raphson method of root approximation. This is an iterative method which generally has quadratic convergence. The only requirement for use of this method is that the function (F(x)) to be used must be differentiable (i.e., Fl(x) must exist). At the outset, an initial approximation (x ) of a root is made. The function is evaluated at x o o and the tangent line to the function at the point F(x ) is constructed.

The x-intercept (x1) of that tangent line is used as the next approximation to a root. This process is repeated until the desired degree of accuracy is met or the allotted number of iterations has been exceeded. 99

K. Kendrick · · e e e s e o e - · e ·*

A tolerance factor (EPS) is specified in the subroutine RTNI which allows a comparison to be made between the new and the old x-values formula (Xn-, xn) and the closeness of f(xn) to zero. The iteration for the Newton-Raphson method is:

F(x n) Xn+1 = Xn F- ()

The demonstration driver for RTNI is called TRTNI. The test func- 2 tion used for the demonstration was F(x) = x -1. This function was chosen primarily for two reasons: 1) The polynomial is relatively simple and of low order; therefore, RTNI was expected to work well. 2) The actual roots of the function are not exactly zero (a case which may cause some convergence problems). The subroutine TRTNI specifies an initial root guess (XST), the number of iterations to be performed (IEND) and a machine epsilon (EPS). The function is supplied externally. RTNI is called and returns an appro- 2 ximation (x) to one of the roots {+1., -1.} of the function F(x) = x -1. Since F is convex upward to the right of positive one, and since the initial guess was taken to be greater than positive one, it was expected that the root RTNI would converge to would be positive one. The absolute value of (x-l.) is tested against EPS in order to make sure that x is a good approximation to 1., and IER (the error parameter) is checked to see if it is zero. If those two conditions are met, then the message 'RTNI TEST SUCCESSFUL' is printed, and IER is set to zero. If those two conditions are not met, then the message 'RTNI TEST UN- SUCCESSFUL' is printed, and IER is set to one.

Upon completion of the demonstration driver TRTNI, several other test cases were fed to RTNI to see how it performed under different circumstances. The following is a brief account of the results:

Normally Newton's method has quadratic convergence i.e., the from the error term, eK+ = eK . This can be seen quite clearly 100

Testing NSSL Routines KURV and RTNI ...... output results below of RTNI using the function F(x) = x 2 . Te:fAT ION 1t25:CE+0~ -6250 FU'N.A--i TO - VLUE0 1 1.250.0000E+00 5.625000S00E-O1

3 1i*CCt0UC489E+0l 6. C984+9C,48E-34 14QO7 - S fI- S~~~~~~~~~~~~~-. mi00.! c!0E+. an 0.0

In the case where a double zero exists, however, the convergence is no 2 longer quadratic but in fact linear. For the function F(x) = x , the error term eK+1 = 1/4 eK as can be seen by the output below.

ITERATIONS ROOT APPROXIMATION FUNCTION VALUE 2.0'-0 o o000a E +o0 2.OOOOOOOOOE+O 1..000GOOO'E+QC 2 2.0000000C3E+-0 3 5 5. 03 0 0 03 F- O1-1 6.2500GOCQ0E-2 4 6 25G0'C00E-02 5 1 250000 03E-31 ' .56'25-C 00 !'E'-- 2' 6.25 00 O 33E-02 3.696&25.0'E-03 6 9. .... 4 7 3 1 25 0 0GE-02 7656 25.0.- 8 1.56250:300E-3 2 2 44140 625E-0 4 ...... 7.8125 iE0E-3 3 9 3.9-062503OE-03 1.5258739iE-C5 10 " 11 1.95312503E-03 3, 81i469q727-' 6 12 9,76562533E-04 9.53674316-0 7 13 4.88281250E-34 2. 3 84i8579E -;C7 14 2.44140625E- 4 5. 9 6 644E- 8 15 1.2207 313E-3 4 16 6.10351563E- 05 3.725 29030E-CC9 17 3,.051 75731E-0 5 9.3 32t 57'5E i3 18 1.52587891E- 5 2.328 35644E-13 19 7. 62939453.E-06 53. 907660'9£-' '11 20 3,81469727E-0 6 1 455 19152-11 i ."" 21 i.9C7348633E-0 6 .3,.6'3'7 39r8'ie- t"2 22 9.53674316E-0 7 9.09494702E-13

Another case tested was when the function (F) was a linear function. RTNI was expected to calculate the root in exactly one iteration. The function used was F(x) = x and it did, in fact, converge in one step.

The function F(x) = 4(x-3) 3 + b presented a different sort of problem when x = 3. is used as the initial guess. This is so because 101

. * 5*** e , K.Kendrick

l( when x = 3. the derivative of F (F x)) = 0. . Since the iteration formula includes division by the derivative (which in this case is zero) it is impossible to carry on any further. RTNI contains an error parameter (IER) that will detect such an occurrence and will return to the main program the message IER = 34, which indicates that at some point the derivative of the function became zero, and the calculations were thus terminated.

When a function that is concave in one direction everywhere (i.e. either upward or downward) is used, then RTNI is expected to obtain convergence from one side only, depending on the values of the initial guess. This was true in the case where the function used was F(x) = eX-l.. Specifying the initial guess at 5., the convergence was from the right, and it obtained convergence in ten iterations.

In the case of two closely spaced roots, it was found that RTNI again gives linear convergence rather than quadratic convergence. The -14 function used as an example was F(x) = x(x - 10 ), and as shown by the output below, RTNI did in fact converge linearly for the function F(x) = x(x - 10-0 ), where the roots are spaced a little farther apart than previously. 102

Testing NSSL Routines KURV and RTNI ....

ITERATIONS ROOT APPROXIMATION FUNCTION VALUE I 5 0000000 E-01 2.50000000E-01 2 6.25 60C 0 -E02 3 2...250000E-01 i.. i5625 0 i - 0 2 4 1.25000000E-02 3.9062500OE-0 3 5 9.76562500E-04 6 1 56250000E-02 2.4414L 625E-04 7 7.812500 E-03 6. i351562E-05 8 3*9062500OE- 3 1. 52587891E-05 9 .195312500E- 3 3 .81469727E-06 10 9. 7656250EE- 0 4 9 536 74316E-0 7 11 4.88281250E-04 2.384i8579E -07 12 2 4414L 625E-04 5 .96046448E-08: 13 1.22070313E-04 1.490 116i2E-08 14 6. 10351563E-05 3.72529030E-0 9 L5 3.5 -5175781E- 5 9.*3 3 22575E-10 16 1.52587891E-05 2. 3283 644E-10 17 7.62939454E-0 6 5.820 76609E-11 18 3.81469727E-0 6 1i455£9152E -1i 19 1.90734864E-06 3.63797881E-12 23 9. 536 74321E-07 9.09494702E-13 21 4.76837i63E-0 7 2.27373675E-13 22 2 3 8418584E-. 7 5. 6434189E-14 23 119209295E-0 7 1. 42108547E-14 24 5.96046498E-08 3.55271368E-15 25 2.98023274E-0 8 8. 8817842OE-16 26 1.49011662E-0 8 2. 22J44605E-16 27 7.45058560E-09 5.55ii15i2E-17

The interpolation package KURV contains two subroutines KURV1 and KURV2. The entire package performs the mapping of points on the interval (0., 1.) using splines under tension. This package differs from the package CURV in that KURV generates two splines under tension while CURV only generates one. KURV is also more complicated than CURV in that it handles parametric curves whereas CURV treats function curves only.

Subroutine KURV1 takes on the task of generating the splines under tension to be used for interpolation. These splines are developed from a set of differential equations which involve second derivative values of the splines. Although these values are unknown at the outset, 103

*e********e*...... K. Kendrick

the solution of the differential equations introduces two tridiagonal linear systems of equations which when solved yields the solution vectors (XP and YP) containing those second derivative values. These second derivative values give valuable information about the curvature of the curve; furthermore, subroutine KURV2 uses the solution vectors XP and YP which were returned from KURV1. The X and Y-splines are parametrized over Sn (the polygonal are length of the curve). (See Figures 1, 2 and 3.) A value (T) is supplied to KURV2 such that ITI <1. . T is multiplied by S after which a linear search is conducted to find which two values of S the value T * S lies between. When these values are found, the function value of (T * S) is mapped on the X and Y-splines. After mapping this value onto the two parametrized functions, this information is used to map the point T onto the interpolated curve on the interval (0., 1.).

The demonstration driver for KURV is called TKURV. The arrays X, Y, XP, YP, and TEMP are dimensioned at ten and are used both in KURV1 and KURV2. The endpoint slopes (SLP1, SLPN) are set at zero, and the tension factor (SIGMA) is set at one. KURV1 is then called and returns the arrays XP and YP (which contain the second derivative values of the X and Y splines), and S (the polygonal arc length of the curve). The XP values are compared with zero (the actual second derivative value of the X-spline), and the YP values are compared with the actual second derivative (plus or minus formula) of the Y-spline. If the comparison is successful, the message 'KURV1 TEST SUCCESSFUL' is printed, and IER is set to zero. If the comparison is unsuccessful, the message 'KURV1 TEST UNSUCCESSFUL' is printed, and IER is set to one. Next, T (the value at which interpolation is desired) is specified; afterward, KURV2 is called. KURV2 returns the interpolated point (XS, YS). XS is then compared with 5.5 and YS is compared with 0.5 to see if they are reasonably close. If the values are close enough and the test criteria are met, the message 'KURV2 TEST SUCCESS- FUL' is printed, and IER is set to zero. If the criteria are not met, the message 'KURV2 TEST UNSUCCESSFUL' is printed, and IER is set to one. 104

Testing NSSL Routines KURV and RTNI ......

1. s C ^ A A^

0.0 2. 30.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

Figure 1: Plot of Interpolating Curve (solid) and Polygonal Arc Length. (dotted).

1 w I

L-0-0-71- I '00,0007-- L ^"C. .2 .3 3.5 .i 5.8 6.9 8.1 93 1 4; 111 12.

Figure 2: Plot of X-Coordinate of the Curve XS vs. Polygonal Arc Length.

, , CR A A A

. V I1 . *. e _.'1 5. Ic i5 9 9! 1 1. 1

Figure 3: Plot of Y-Coordinate of the Curve YS vs. Polygonal Arc Length. 105

...... †* ee eeeee...... e K. Kendrick

The demonstration drivers TRTNI and TKURV have been very helpful in testing the reliability of the two NSSL routines RTNI and KURV. TRTNI introduced the possibility that there was a need for two test factors (EPS1, EPS2), so that the closeness of F(xn) to zero might be tested independently of the closeness of x 2 to xn. Furthermore, TKURV disclosed two actual omissions in the code for KURV which inhibited its ability to return the correct values. This shows that the NSSL Testing Project is of immeasurable value in ensuring that the routines therein are reliable enough for use everywhere.

107

THE NCAR SCIENTIFIC SUBROUTINE LIBRARY AND COMPUTER SOLUTIONS TO LINEAR SYSTEMS

by Arleen Kimble Prairie View A&M University Fred Clare, Scientist

The NCAR Computing Facility maintains a library of software subroutines available to all computer users. This collection of subroutines is kept on disk storage and is called the NCAR Scientific Subroutine Library (NSSL). These routines are in the public domain, and are available to anyone upon request. For the purpose of maintenance and verification of the NSSL, there is a library testing project at NCAR which requires two phases: demonstration drivers, which give examples of how the routines should be used, and extensive test decks, which perform exhaustive tests on the routines. Part of the NCAR library is devoted to the solution of linear systems. Below we discuss some !of the techniques employed in this area. The systematic elimination methods of C.F. Gauss have proven to be better in time or accuracy than any other class of solution algorithms. Let us suppose that we have n linear equations relating n variables. They can be written:

x l allx + * * * * + aln n b ' (1)

a lx + . + a x = b nl 1 nn n n

or (2) Ax = b (2)

when Eq. (1) is written in matrix form. When A is a non-singular matrix, the equations (1) and (2) have a unique solution, vector x. The algebraic basis of Gaussian elimination is the following theorem. 108

Computer Solutions to Linear Systems ......

LU Theorem. Given a square matrix of order N let Ak denote the princi- pal minor matrix made from the first k rows and columns. Assume that det(Ak) # 0 for k = 1, 2, ... n-l. Then there exist a unique lower triangular matrix L = (m), with ml = m.... =1, and a unique mij 1,11 2,2 n,n upper triangular matrix U = (u. .) so that LU = A. We will motivate the proof of this theorem by considering the 4 x 4 case.

all a12 13 a14

a21 2 a23 a24 Let A = 31 a32 33 34

a41 42 43 44

Assume this matrix can be factored in the form

1 ¥11 Y12 Y13 Y14 1 a21 Y22 '23 Y24 A = LU - a31 a32 1 Y33 Y34 1 4 1 "42 : 4 3 Y44

Y12 Y 13

+ Y22 c2 1 Y11 a2 1 y12 '2 1 Y 13 + Y23 LU = + a3 2 31 y11 c31 Y12 Y22 31 Y 13 + C32 Y23 + ¥33

+ a4 2 a a41 Y 11 a4 1 y 12 Y22 C 4 1 Y13 + 42 Y23 + 43 y33

Y 14 + a 2 1 Y14 + Y24 + + Y34 a3 1 Y 14 32 ¥24 + "43 Y34 + ¥44 a4 1 Y 14 + 42 Y24 109

...... A. * A ... .Kimble

Hence, each a and y can be written in terms of the elements of A. The general equation for ij being (3)

Hence, each a and y can be written in terms of the elements of A. The

general equation for a.. being (3) 13

j aij= Z CZik aij k= l ¥kj when i > j

j-1 a.. = Z aik Ykj + 7i j when i > j k=13 k=l

j-1 ij k= 1 ik Ykj a. - when i > j (3) 13 Yjj

The equation for yij was similarly derived by setting

i for i < j aij = -kjEik (4) k= 1

From equation (4) i-1 where a 1 aij k= ik Ykj + a°ii

i-1 for i < j aij k=1 aik ¥kj + Yij

*^jj. -i-i for i < j 121ij = iji kkj k 'ij aij~ 1 ak= 110

Computer Solutions to Linear Systems ......

Thus, the factorization of A as the product LU is the basic idea of all Gaussian elimination schemes, for then the system Ax = b can be written

LUx = b.

This represents two triangular systems

Ly = b and Jx = y which are very easily solved. The components of the intermediate solution y can be obtained directly from the first system since the first

equation involves only yl, the second only yl and Y 2 ,,and so on. Then the components of x can be similarly obtained from the second system

in the order xn, xn 1 ... x1" The calculation of L and U together with the solution of Ly = b are usually called the forward elimination, and the solution of Ux = y is the back substitution. The various methods differ in the order in which the operations are carried out in the forward elimination. Also, sometimes the matrix L is stored, and sometimes it is not, but the importance of saving L can be demonstrated easily if A is a general stored matrix. The diagonal of L need not be stored since it is known to be all l's. The below diagonal part of L together with U can occupy the space originally taken by A. No intermediate storage is needed since elements of L are created at the same time that elements A are zeroed. About 60% of the computer time required to solve Ax = b is spent finding L and U. Hence, if one might later need to solve another system with the same matrix A but with a new right-hand side b, there is every reason to retain L and U and thereby avoid repeating the triangular decomposition. The above matrix discussion can be related to ordinary elimination. Given a matrix A and a vector b, one uses elementary row operations to

put zeroes below the main diagonal of A. Assume all 0, a22 °0,

a3 3 O. If any of the numbers vanish, we cannot continue the elimination. For example, suppose the pivot (element in the first column which is largest in absolute value) all were zero. Since det(A) # 0, we know that ail # 0 for some i > 1. If we interchange any such i-th 1ll

._ . e. · ... .··. ·. .· ·. . . . · A. Kimble row of A and b with the first row of A and b, we will obtain an equivalent equation system with all # 0. Unless a pivot is exactly zero, this interchange is unnecessary in theory. Working with a zero pivot all is impossible, but working with a pivot all that is close to zero is inaccurate. To see this, consider the following. Example. Assuming three-decimal floating arithmetic, we shall solve the system

.000100 xl + 1.00 x2 = 1.00

1.00 x1 + 1.00 x 2 = 2.00

The true solution, rounded to the decimals shown, is:

10,000 -. 9998 99 1 = 1.0001.0 x 0.99999 0 l 9999 2 9999

Solution by Gaussian elimination without interchange is:

.000100 xl + 1.00 x 2 = 1.00

-10,000 x2 = -10,000

X2 = 1.00

x 1 = 0.00 (awful)

Solution by Gaussian elimination with interchange:

1.00 x1 + 1.00 x = 2.00

1.00 x = 1.00

x 2 = 1.00

x 1 = 1.00 (perfect) 112

Computer Solutions to Linear Systems ......

Pivots a(r) which are small in absolute value must be avoided; rr therefore, one should choose as a pivot the largest in absolute value (r) of the numbers a. (i > r). This is accomplished at each stage by i,r - interchanging the corresponding equations in the original system. The solution of linear equations can be a long and tedious process because there are no methods which reduce the work substantially, but large systems of equations can be handled quite easily by electronic computers. The NCAR library subroutines TRIPIV and BDSLV compute the solutions of linear systems of equations. TRIPIV computes A(K) * X(I-l) + B(I) * X(I) + C(I) * X(I+1) = Y(I) for I = 1, 2, 3, ... N, where A(1) = C(N) = 0, using Gaussian elimination with partial pivoting. The solution of the tridiagonal system is computed by factoring the coefficient matrix as the product of a unit lower triangular matrix and an upper triangular matrix. For example, given the matrix

2 1 0 0 0 0 0 0 0 0 3 2 1 00 0 0 0 0 0 0 3 2 1 0 0 0 0 0 0 0 0 3 2 1 0 0 0 0 0 0 0 0 3 2 1 0 0 0 0 A= 0 0 0 0 3 2 1 0 0 0 0 0 O 0 0 3 2 1 0 0 O0 O0 O0 O0 O0 0 23 3 2 2 1 0 0 0 0 0 0 0 0 3 2 1 0 0 0 0 0 0 0 0 3 2

and the vector 113

...... * A. Kimble

3 6 6 6 6 B 6 6 6 6 5

TRIPIV can solve the matrix equation Ax = b for x.

The routine BDSLV solves the matrix equation Ax = b, where A is a matrix of band width 2M+1, i.e., A(I,J) = 0 whenever II-JI.GT. M. BDSLV also uses Gaussian elimination with partial pivoting to solve the system of equations.

115

ON THE BALANCE ASSUMPTION OF ZONALLY AVERAGED DYNAMICAL MODEL FOR THE ANNULUS by Huei-Iin Lu Florida State University Akira Kasahara, Scientist ABSTRACT A diagnostic equation of mean meridional circulation of rotating, differentially heated fluid is derived under a quasi-geostrophic assumption. The numerical solution is obtained using the forcing functions prescribed by a three-dimensional distribution of fluid variables obtained from a time integration of Quon's numerical model (1976). The meridional circulation obtained by solving the diagnostic equation is almost identical with that originally given by Quon's time-dependent model. It proves that the fluid system in the rotating annulus has a spontaneous balance of hydrostatic and geostrophic as the state of fluid changes.

INTRODUCTION In the current of research on climatic problems, it has been popular to theorize that most climatic fluctuations are a result of changes in the external conditions. However, it is also possible for climatic change to be brought about by natural time variations of the entire climate system without any external influence. Some scale of climate change could be just the natural fluctuations arising solely from the complex nonlinear interaction among land, oceans, atmosphere and polar ice. It is our post- ulation that the interaction between zonal flow and eddies must be one of the most important internal mechanisms which cause climatic variability on the time scale of 10 days. This type of fluctuation is sometimes called index cycle. The idea of an index cycle has been criticized as an over- idealization since the variation of the circulation of the atmosphere is more random than cyclic. Nevertheless, in the laboratory experiments of rotating annulus, it has been well documented that under certain external parameters, waves form and undergo periodic vacillation which transfers energies to and from zonal basic state (Pfeffer et al., 1974). Within a hierarchy of climate models (Schneider and Dickinson, 1974) we propose a 116

Kuo-Eliassen Equation for the Annulus ...... zonally symmetric dynamic model to predict the characteristic structure of the zonally averaged state with emphasis on the feedback mechanism between zonal state and eddy processes. Dynamically, this type of model involves two fundamental problems. One is related to the closure problem of eddies. Another is the balance requirement between zonal velocity and temperature fields. Prediction of eddy transport processes has been one of the most difficult problems in hydrodynamics. One has to make certain assumptions to close the system of equations. The balance means that any redistribution of heat and momentum by eddy fluxes of heat and momentum or heating and dissipation in the fluid shall be in such a way that zonal flow, pressure, and temperature fields evolve while maintaining a hydrostatic and geostrophic balance. In order to achieve such spontaneous balance, the adiabatic heating (cooling) and the coriolis force must act to create mean meridional circulation. The balance assumption enables us to predict change of mean vertical and meridional flow without using their prognostic equations. This summer as a project of the Scientific Computing Fellowship I have investigated the above stated balance in a rotating annulus fluid. I plan to extend the present result to formulate a zonally symmetrical climate model, eventually.

DERIVATION OF THE KUO-ELIASSEN EQUATION We shall derive a diagnostic equation of mean meridional circulation which arises from the balance assumption as originally formulated by Kuo (1956) and Eliassen (1952). I have simplified the derivation considerably by assuming that the fluid system is quasi-geostrophic. The quasi- geostrophic assumption enables us to derive the Kuo-Eliassen equation into an elliptic partial differential equation solvable by numerical subprograms available at NCAR. 117

** * * * ** Huei-Iin Lu

First, partition the horizontal velocity into the rotational and divergent components. Denote all variables related to the rotational part of part of flow by the subscript 2 and those related to the divergent flow by the subscript 3. Then

V = V2 + V3

4. / V2= k x VW2

V3 = VX3

where i2 and X3 are horizontal stream function and velocity potential, respectively. Hence,

vorticity E2 = kVxV = V2 2

aw3 divergence 63 = V-V = V2 3 = -

by virtue of the continuity equation.

Also specify those variables which are determined by the equation the of state and hydrostatic equation by the subscript 1. We can write quasi-geostrophic system in cylindrical coordinates as follows.

2u v2 au 2 u2 DU2U 2U2V22 rtr+ + + + fv3 + = 0 (1) Dt rDX Dr r Po r

2 2 av2 au2v2 av2v2 v 2 u2 1 pli 2v 0 (2) + + + -- fu3 + - V 2 at rrX Dr r r Po Dr

DT Du2T1 Dv 2T1 v2T1 DT1 1 2Tl (3) t -rD ar +w 3 ( ) - kV = O at -raX r r r z

( 4 ) 1 P fu2 Po r(4)

api o (5) fv2 = - 1 r (5) 118

Kuo-Eliassen Equation for the Annulus ......

au3 av3 V3 aW3 + + -T r3'+SF^+y+T- r + =0 -0 (6)

api -=-pig (7)

P1 = Po [1 - P(T 1-To)] (8) where r is radial distance from the origin, X is azimuthal angle, f is twice the rotation rate along the vertical axis, v and k are kinematic viscosity and thermal diffusivity, respectively. po is the fluid density at a reference (mean) temperature; To, 3 being the volume expansion rate of fluid and g, the gravity. Notice that in the quasi-geostrophic system the pressure gradient terms in Eqs. (1) and (2) are one order of magni- aT tude less than those in Eqs. (4) and (5). The static stability (_ 1 is assumed independent of r and X. We also assume that viscous dissipation is applicable to the quasi-geostrophic flow. We now partition every dependent variable as a sum of the zonal average and its departure; i.e.,

q = q+ q' where

q = 2 rq dX Then take the zonal average on the above system of equations. We obtain:

au2 _ u2V2 2u52 _ + f V 3 + VrV2 U2 = (1) Dt + fv3 + r + - r

7Ti _ U-T? Dv5Tj v2T] 2 Dt W3 (--)oat + Drr+ -r r -kV T =0 (3')

fu2 = -- (4)P fu2 Ooor (4') 119

. .. 0- -0o - 0* aeceHuei-Iin Lu

v2 = 5

aV3 V3 3W3 ) + - - = (6 ar r az

apl (7') = -p g

) Pi = Po[l- (T-T)] 8

f,,~(l') _(3') g-" We now eliminate time derivatives of u2 and T1 by taking f ~ - Dr- which is actually the balance assumption stated in the previous section. We obtain

av3 aT 1 aw 3 v 2uv 2- f2 - )B( - + f--+ - v f z ar + z0 ar r

av2Ti V2T1 _2- 2 g- r Dr + --- kV T1 0 (9)

Equation (6') allows introduction of a stream function ~ such that

- = W3 = 1= - By substituting them into (9) we v3- rzA andad W3 r r

obtain the Kuo-Eliassen equation.

2 +g - fr f f2 2r + g gx Psar a=r u 2v 2 2u 2 v2 + V2

aT1 az2 ar2 %-f-zO° 2T_ r -r kr g~r_ a { _ } -- +r +---- - k'V2T1 aT, ( - ) (z o 120

Kuo-Eliassen Equation for the Annulus ......

The boundary condition is = 0 on the closed boundary of the r-z plane because of non-slip conditions on the two side walls and bottom lid and free stress on the top surface. Equation (10) is elliptic if aT, (--z)> 0. It states that under the quasi-geostrophic and quasi-static balance assumption, the mechanism of creating a mean meridional circulation is due to viscous force, heat conduction, and the convergence of geostrophic eddy fluxes of heat and momentum.

DATA SOURCE

For given heat and momentum distributions (eddy statistics and basic state) in the fluid, the stream function 1 can be solved from the Kuo-Eliassen equation with prescribed boundary conditions. In this study, we simply take the three-dimensional distribution of fluid variables obtained from a time integration of Quon's numerical model (1976) as a set of source data. Quon solved complete equations of momentum and heat by employing a spectral method in the azimuthal direction and a finite difference method in radial and vertical directions. The results of each 50 time-integrations were stored in a series of standard 7-track 800 bpi tapes. The information is unformatted and binary coded. Our task is to read an appropriate portion of data from the tapes, convert them into dimensional quantities, and interpolate their values to the grid points for difference approximation. We then obtain a numerical solution of the Kuo-Eliassen equation and compare it with the stream function inferred by v or w fields of Quon's model.

THE NET AND FINITE DIFFERENCE FORMULATION

As in Quon's model, the grid network is set up with a variable grid increment. The variable grids give higher resolution near the boundary. Let I and K represent transformed coordinates of grid points in r- and z-directions, respectively. We introduce the coordinate transformations r -+ I(r), z + K(z), together with the transformation of differential 121

* e Huei-Iin Lu

K ), where Ir = and Kz = are cal- operators r ) (I 'rr z Mz

culated at each grid point in the I-K plane.

In addition to variable grids we employ staggered coordinates. The net is depicted in Fig. 1. The cross section of the annulus is divided into uniform square cells on the I-K plane. At the center of the cells the variables v2T1 and u2v2 are defined, at the middle points of the cell walls the u2 and Ti are prescribed, at the four corners of the cell stream function i is defined. We shall represent the space locations at which 4 are evaluated by I = iAI and K = kAK, where I and K are integer, AI = AK = 1 are spatial grid intervals. By applying a two-point stencil for the first derivative, a three-point stencil for the second derivative, and five points for the third derivative, we can put the Kuo-Eliassen equation into the following finite difference form.

AN(K)*@I(I,K-1) + AM(I)*i(I-l,K) + (BN(K)+BM(I))*p(I,K) + CN(K)*4(I,K+l)

+ CM(I)*p(I+1,K) = F(I,K) (11) where 2K f ,K AN(K) = (TK+- TK½)

AM(I) = gIr I(I I-+ 1)

f2 (K +K z,K+l+Kz,K ) BN(K) = - fir T )

BM(I) = -gBIr I(Ir,I+I -r,-) 122

Kuo-Eliassen Equation for the Annulus ...... f2K2 z K+l CN(K) = -z (TK+-TK_) K+-- K-i

CM(I)= gIr I(Ir, - 2r) where T is radially averaged temperature on each level. F(I,K) is a finite difference approximation of forcing function which appears on the right-hand side of equation (10).

Eq. (11) is solvable by the numerical subroutine BLKTRI developed by Swarztrauber and Sweet (1975).

It is important to note that in order to use the BLKTRI subroutine, the number of unknowns in the K-direction must be of the form 2 -l where n is an integer greater than one. In our grid net the number of unknowns

(excluding known boundary conditions) in the K-direction is 35, which clearly doesn't satisfy the required condition. Fortunately, Dr. Swarztrauber kindly provided for me his new generalized BLKTRI subroutine which solves the same type of elliptic equation with an arbitrary number of unknowns in the K-direction. The new version of BLKTRI is expected to appear in the new edition of the NCAR Technical Note (Swarztrauber and Sweet, 1975).

DISCUSSION OF RESULTS

Solution of the Kuo-Eliassen equation was obtained at each 50 time steps of Quon's numerical integration. In the mean time the stream function was computed directly from either v or w field originally solved by Quon's model. Results are plotted on the r-z plane by using a contour subroutine developed by Mr. Zarichny who was a computer programmer of Quon's model.

The two results show good agreement even in the detailed structures (Fig. 2 and Fig. 3). This may lead to the conclusion that the balance between changing zonal flow and temperature fields exists and the motions can be approximated by quasi-geostrophic assumption. 123

.... Huei-Iin Lu ..Figure * ... : .. Relative. . .. Posiions. . .. of... the V . . ...les on. S gered Huei-Iin Grids. Figure 1: Relative Positions of the Variables on Staggered I k-35 ; 2Vv 2 , 2 T / x ,, __, 0 U2 , T. ( x Q/

Ak x x K x

Tk =x2x "^"l k S l 7//77/7rm / - 4AI K i=l i=2 i=23

further partition Since the Kuo-Eliassen equation is linear we can term of forcing the solution as a sum of solutions from each individual contributes to function. Fig. 4 to Fig. 11 illustrate how each forcing shows that friction produce the mean meridional circulation. It clearly for the direct and heating near the side wall boundary are responsible of eddy heat and momen- circulation (Hadley cell) while the convergences indirect cell (Ferrel tum fluxes are responsible for the formation of the The dynamics cell). The curvature effects of eddy fluxes are negligible. are analogous to that of the mean meridional circulation in the annulus found in the atmosphere (Kuo, 1956). observed Quon's model has simulated some features of vacillation compare the structure in the laboratory experiment. It is interesting to at two extreme time and intensity of the mean meridional circulation baroclinic eddies stages of wave activity. Fig. 12 is at the state when eddy activities. are fully intensified while Fig. 13 is a result at weak in the zonally This is one of the features we hope to be able to predict symmetrical prognostic model. 124

Kuo-Eliassen Equation for the Annulus ......

Figure 2: Contour of Stream Figure 3: Contour of Stream Function Function i(r,z) solved from w p(r,z) solved from Kuo-Eliassen equation. Field of guon's Model. The unit The unit is cm3sec 1 . is cm3sec 1 .

FUTURE IMPROVEMENT

Using NCAR software to solve the elliptic equation for this type of problem can save significant amounts of computation time over the conventional relaxation method. Yet, it will still be very expensive if one applies it in the zonally symmetrical model without taking the most important advantage of that subroutine. One important feature of BLKTRI is that it initializes computation by computing the quantities AN, BN, and CN in equation (11) first and storing them in a work array. Then in the computation stage the quantities AN, BN, and CN that were computed in the initialization are used to obtain the solution i. Since initialization takes approximately twice as much computation as the computation stage it need not be repeated unless the quantities AN, BN, and CN change. 125

...... e ...... Huei-Iin-Lu

Now as in Eq. (11), coefficients AN, BN, and CN appear to be a function of time in the zonally symmetrical model. It looks like initialization has to be recalculated at each time step. However, since AM, BM, and CM are independent of time we can rotate the coordinates by 90° in the finite differencing procedure so that the coefficient arrays AM, BM, and CM replace AN, BN, and CN. They only need to be calculated once in the entire time integration of the zonally symmetrical model. Consequently two-thirds of the computation involved in solving the elliptic equation can be saved.

Figure 4: Stream Function Figure 5: Stream Function 2 contributed by - 2 contributedcontributed by -- 2 contributed b 126

Kuo-Eliassen Equation for the Annulus ......

Figure 6: Stream Function Figure 7: Stream Function contributed by vV 2 -u. contributed by -- V 2 u V + vV2u 3r r 127

· Lu · · . . · · ···o · 0 o o · · ·o ·-· ·· · · · Huei-Iin

Figure 8: Stream Function Figure 9: Stream Function c b7 v2db r V2"Tlc contributed by -Dr contributed by 128

Kuo-Eliassen Equation for the Annulus ......

Figure 10: Stream Function Figure .1: Stream Function 2 contributed by -kVT1 contributed by V' + vT - kV2Tl Dr r 129

...... Huei-Iin Lu

Figure 12: Mean meridional Figure 13: Mean meridional circulation circulation of the annulus when of the annulus when eddy processes are eddy processes are very strong. very weak. 130

Kuo-Eliassen Equation for the Annulus ......

REFERENCES

Eliassen, A. (1952): Slow thermally or frictionally controlled meridional circulations in a circular vortex. Astrophys. Norv., 5, 19-60.

Kuo, H-L. (1956): Forced and free meridional circulations in the atmosphere. J. Meteorol., 13, 561-568.

Pfeffer, R.L., G. Buzyna, and W.W. Fowlis (1974): Synoptic features and energetics of wave-amplitude vacillation in a rotating, differentially-heated fluid. J. Atmos. Sci., 31, 622-645.

Quon, C. (1976): A mixed spectral and finite difference model to study baroclinic annulus waves. J. Comp. Physics, 20, 442-479.

Schneider, S.H. and R.E. Dickinson (1974): Climate modeling. Reviews of geophysics and space physics, 12, 447-493.

Swarztrauber, P. and R. Sweet (1975): Efficient FORTRAN subprograms for the solution of elliptic partial differential equations, Technical Note NCAR TN/IA-109, Boulder, Colorado, 139 pages. 131

...... a . . . . .0 0 .,. 0 * * * * ... . Huei-Iin Lu

133

INVESTIGATION OF ALGORITHMS FOR THE SOLUTION OF THE NONSEPARABLE HELMHOLTZ EQUATION

Curtis D. Mobley University of Maryland Roland Sweet, Scientist

INTRODUCTION

There is frequent necessity in the atmospheric sciences for solving the two dimensional Helmholtz equation

(V2 + X)4 = f. (1)

Here V2 is the two dimensional Laplacian operator, X is a nonpositive constant, f is a known function of position, and ( is the unknown function of position being sought.

A simple example is the time integration of the barotropic vorticity equation

+ f) = 0. dt ( Here t is the time, C is the vorticity of the fluid relative to the earth, and f is the vorticity due to the earth's rotation (the Coriolis parameter). With 3- af/ay, a known constant, this equation is

3C = _ 3- - 3v 9t "tx Vyy

Introducing a stream function i via

u - and v =

yields

at;~~~aS. 8 ^ ^M -MM-r, afi ~(2) at y=a axx y xx

and

v 2 9 = C. (3) 134

Nonseparable Helmholtz Equation ......

Suppose C is known at some initial time t = t . Then if one can solve (3) for i, one can evaluate 3a/at at t = t from equation (2). The vorticity at some later time t = to + At can then be approximated by

%(t + At = C(t ) + (8) At . (4)

This process can now be repeated, so that by cycling through equations (2) - (4) one can find the vorticity at any time t>t .

In problems of this type the solution of a Helmholtz equation at each time step of a numerical integration can account for a significant portion of the total computer time required. For example, in a primitive equation model used by the author, (Caponi, 1974), in which the nonhydrostatic part of the pressure field is obtained from a Helmholtz equation, the solution requires up to 30% of the total run time.

Furthermore, if the solution of (1) is required on a domain whose boundaries are irregular, many of the available solution techniques are inapplicable, and others which are in principle applicable may fail to provide accurate numerical results.

It is not surprising then, that the development of efficient numerical techniques for solving this equation has received considerable attention from applied mathematicians. Continuing this development, this paper numerically compares a recently developed algorithm (Paige, 1974) with algorithms using successive over-relaxation and cyclic reduction. The Paige algorithm is applicable, in principle at least, to the solution of (1) on domains of arbitrary shape.

A PARTICULAR PROBLEM

In order to carry out the intended numerical study, a suitable example of (1) and associated initial and boundary conditions must be formulated. The problem stated here arises in certain oceanographic studies and was suggested by Dr. J. McWilliams (personal conversation). 135

...... C. Mobley

Let x and y be cartesian coordinates, and let 3 and K be given constants. Then a particular Helmholtz equation is

2 (5) V2 (-a) = V- 2 _KV2-KV f(xy)f(x,y) where u represents the quantity of interest in some domain Q.

Now consider the coordinate transformation

x = r(l + a cos kG) cos 9 (6)

y = r(l + a cos k6) sin e with the inverse transformation

2 r = (x2 + y2 ) 1 + a cos[k tan -l()] (7)

0 = tan 1(x) .

If the parameter a is zero, equations (6) and (7) reduce to those defining polar coordinates. For nonzero a this transformation modulates a circle of radius r with a sinusoid of amplitude a and wavenumber k. By varying a and k one can study the efficiency of a solution algorithm as the domain Q becomes more and more distorted from a simple annulus. (See figure 1.)

Using (6) and (7) it is straightforward but tedious to transform the Laplacian

2 V ++2 2

Since X = 0, (5) can be referred to as a Poisson equation. If the right hand side of (5) were also zero, one would have a La- place equation. 136

Nonseparable Helmholtz Equation ...... into (r,0) coordinates. The result is

2 2 V = all() - + a 1 (r,) - + a 1 2 (r,9) r + a (r,) (8) where

2 2 al,1 () = Y [1 + (yka sin k8) ]

2 2 2 al (r,G) = - [1 + 2 (ykasin k8) + yak cos kO]

1 a l 2(r,) = 3 k a sin ke (9)

2 a22(r,e) =

and y = (1 + a cos kG)-l

Likewise,

-- (y Os e -y kosin sin k) . - sine -X (y cos e - Y2 ks sin 6 sin kD)er r g. (10)

It is to be noted that (5) is not separable when expressed in (r,e) coordinates. One also notes that there is a singularity at the origin, r = 0. In order to obviate the need for developing a special equation valid at r = 0, the domain Q is taken to exclude the origin:

O < R < r < R < oo 0 -1 0 < < 2 Tr

And as stated previously, u(r,6, t = 0) is known throughout Q. In order to have a mathematically well posed problem, only the boundary 137

C. Mobley conditions remain to be specified. These are chosen to be

at (R° oet) = t (R 1,e,t) = o Du and -t is periodic of period 2 fr in e for all t>0.

Figure 1 shows the initial field u for R = .2, R1 = 1, and for the cases a = 0 (an annulus) and a = .5, k = 3. u is expressed in nondimensional units.

a = 0 a = 0.5 k = 3

Figure 1: The Initial Conditions u(r,e,t=0) for various a and k.

FINITE DIFFERENCING

After (5) is rewritten using (8)-(10), standard finite differencing formulas can be applied.

Impose a computational grid on Q by choosing integers M and N and defining 138

Nonseparable Helmholtz Equation ......

Ar ER-R _A - M+1 ' N+l

Then let

r. - R + iAr, i = 0, 1, 2, ... , M + 1 1 o

ej jAe , j = 0, 1, 2, ..., N + 1

Examples of the grid for various values of a, k, M, and N are shown in Figure 2 on the following page.

Now let v.. = v(ri,.j,t), where vis either u or the tendency 9u T - . The derivatives in (8) and (10) can be approximated by the centered differences

Vi+l,j i-lj -r (ri, e) = 2Ar

^ 1 J

- 2vi ~- (ri ,S) = Vl+l,j + Vi-,j 1,J (Ar). (11) v - v i i,j+l i.j-l ae (riSj) = 2Ae

2aV vi,j+l + v i j- - 2v.i ~z riaj ) = (Ae)1 and

ar2 1 J+l 1 + + ara (ri j 4AAe (V l+lj+l l-lj- 139

* e * * 0 . e 0. * - 0 C * · 0 0. * a * e * * e e e a e 0 C. Mobley

a = 0 a = .1, k = 4

20 x 30 grid 20 x 30 arid

a = .4, k = 5

20 x 30 grid 25 x 55 grid

Figure 2: Examples of the Computational Grid. 140

Nonseparable Helmholtz Equation ......

Substitution of (8)-(11) into (5) and collection of common terms gives the finite difference form of (5):

C (i,J) Ti, 1 + C2 (ij) Til + C 3 (ij) (Ti+l j+l - T

-Ti+lj + Tilj ) + C( (Tij+ + T l +

C 5 (i,j) Ti = fi (1

The left hand side; of (.12) is the finite difference representation of V2T at grid point (i,j). The forcing function f. has a corresponding structure in u.

The coefficients Ci, which are functions of position, are

a a al1l al C -= 1 2 _L1 G1l (Ar) 2 2Ar 2 (Ar) 2- 2Ar

a:,. a2 21 1 2 2 3. 4ArA9 C4 (Ae)

and

2a 2a 11. 22 5s (_Ar) > (Ae)

wherein the functions. of po&sition a1 ., etc., are evaluated at (ri, .) via equation (9).

This section is closed with a. remark about computational stability. The use of centered spacial differences forces the use of a centered time difference- as well. Thus the u field is updated by

u. .(t+At) = u.. (:t-At) + 2At T. . 1,J 1J 1 , 141

* e ** e. e* *-Ise**le* * *-* *Mobley C.

Only the first time step is made by a forward time difference as in (4), since this scheme causes computational instability in the particular problem stated above. The time step is limited by the usual Courant-Friedrichs-Levy condition

Ar At < Ar as well as by a viscous condition

At< (Ar)2 K

METHODS OF SOLUTION OF THE FINITE DIFFERENCE EQUATIONS

As mentioned in the introduction, many different algorithms exist for the solution of the Helmholtz equation. Two of these, successive over-relaxation and cyclic reduction, are briefly discussed here. Then a more detailed discussion of the less well known Lanczos-Paige-Saunders algorithm follows. In the next section of this paper, these three methods are applied to the problem stated in the previous section.

Successive Over-Relaxation (SOR)

SOR is certainly the best known technique for solving (1), and it serves as the standard to which the other methods are compared in this paper. For the development of this method, the reader is referred to any standard reference on finite difference equations, for example, Varga (1962). Suffice it to say that if T() denotes the mth guess of the value Tij, then (12) can be written

(m) () (m) T +TT(m) ) + C1Ti+ij i-lj 3 i+lj+l i-lj+l i+lj-l i-lj-l

() m (mi) ) (+ (m) + - f R (13) C 4(i,j+i(T , + T ij C5 T i,j m~j 142

Nonseparable Helmholtz Equation...... (mi) th Equation (13) is the definition of the residual, R.. If the m guess at each grid point were the correct one, the residual would be zero for all (i,j) and a solution would have been found. In general, however, (m) R ( 0 at each grid point. If the (i,j) value of T is updated in such a manner as to make the residual at (i,j) zero, while the surrounding T's are held fixed, (13) becomes

(m) (m) (m) (m) (m) + (m) 1Ti+j i-lj + 3 ( i+lj+l- i-lj+l Ti+lj-1 Ti-l,jl

(m) (m) (m+H)) + C (T (in + T( + CT fC . = 0 4 ij+l + j 1 5i,j i - ,j

Subtracting this equation from (13) yields the relaxation equation

R(m) (m+1) (m) Ri. T ( j ) = TT.-( - (14) i,j 1,Cj 5 (ij)

The relaxation parameter w has been arbitrarily inserted; its value is determined experimentally to give the fastest convergence for a given problem. (0) Starting with an initial guess, say T. . = 0 for all (i,j), one

can pass through the grid computing R. . from (13) and then immediately (o) (1) a im updating T( to T( via (14). The most recent update of T is used whenever possible, rather than using all T values at the same m value as shown in (13). One continues to pass through the grid using (13) and (14) to improve the guess field until a convergence criteria

max T(m+l) T(m) < i J i,jij 1'3j Tz,

is met; c is some small number. 143

...... -...... aC...... Mobley

SOR is clearly an iterative method which provides only an approximate domains solution. However, it is applicable, in principle at least, to of arbitrary geometry, and it is quite simple to code.

Cyclic Reduction of The finite difference equation (12) represents a large system such equation for each linear equations for the Ti j , there being one of (12) grid point (i,j). Thus the problem of finding the solution field can be put into matrix form

Ax = f (15) where f is a vector whose elements are the known values of the forcing at the grid points, x is a vector whose elements are the unknowns particular T. ., and A is a matrix whose elements are determined by the i,j' of finite differencing scheme used. The dimensions of x and f are order MNxl for a grid of M by N cells, and A is of order MNxMN. those However, the fact that equation (12) for Ti j involves only unknowns at adjacent grid points, Ti+lj+' means that A contains explicit nonzero elements only on and near the main diagonal. (For an A has but simple example of such a matrix, see Sweet, 1972). Indeed, system a block tridiagonal structure. Thus the solution of the linear regular (12) is not as formidable a task as it might seem, since the structure and sparseness of A can be exploited.

The method of cyclic reduction (Sweet, 1974) is a direct method econom- based upon the block tridiagonal structure of A, and is quite to ical in both storage and running time. Furthermore, it is amenable accurate. rigorous error analysis, and the solutions provided are quite for Q The method is limited, however, to certain simple geometries and is tedious to code.

The Lanczos-Paige-Saunders Algorithm

Since this algorithm is not as well known as SOR or cyclic reduction, it is described in some detail in this section. The LPS algorithm 144

Nonseparable Helmholtz Equation ...... has greater generality than will be evident from the following discussion,, and proofs are omitted here. The reader desiring a rigorous development is referred to the previously cited work by Paige (1974).

Let our problem be formulated as in (15):

Ax = f where x and f are nxl and A is nxn, but no assumption about the structure of A need be made.

Motivation. Suppose A can be written as

A = ULVT where U and V are both orthogonal nxn matrices and where L is a lower bidiagonal nxn matrix, i.e.

c 1 0 0 0 0

62 02 0 0

0 53 (o 3 0

0 3 '. .a, Cn- 1 0 o 0 5n an,

The a i and S.are chosen to be non-negative. i

Then

Ax = f becomes

ULV Tx=x = f or

LV Tx=x = UUTf f

But this system is easily solvable due to the bidiagonal structure of L. 145

* .C. Mobley T Development. From A = ULV one gets

AV = UL T T T and from A = VL U one gets T T A U = VL

Now let U and V be composed of nxl column vectors u. and v. ,vn] U = [u1 ,u2 ,..,un] and V = [vv 2,. ..

Thus from A U = VL by comparing columns one sees that

AT = aGv1 (16) A = alV1 and T A u av + v- l

and from AV = UL

Avi = aiui + ui+lu+l (18)

To begin, choose some ul of unit euclidean norm, |ull =1. (A common choice is u = f/||f|l.) Then from (16) T T A u1 A u vl= "1 IIATul I

(18) with i=l, Thus al and v1 are determined. Now from

a A1 ll = Avl - lUl

=2 2 JIAv -cau 1 I

And now a2 and v2 can be obtained from (17):

A -2 vAu- 2 2 1 v ATu2- B2v 1 - I v2 2 I IATU - 2V111 146

Nonseparable Helmholtz Equation ......

One can obviously continue alternating between (17) and (18) until U, V, and L have been constructed.

Storage Requirements. It might appear from the above development that one needs to store the U, V, and L matrices. This is emphatically not the case. Recall the form T T L(V x) = U f or

Lz - c T Now ul known implies that al' vl, and cl = u1 f are known. From

Oa 0 0 zI ci

Lz = B2 2 Z2 = C2

*a ' z c gn n n n one gets z = c /a . But

n x = Vz = Z v.z.. i=l

So the vector v.z. can be formed and stored in x. Now al', 1' ul, and v. need be saved only until a2, 82 u2, and v2 have been computed, at which time z2 is obtained from

^^+a =c~ = Tf. 2 Z1 + 2 z2 = c2 u2 f.

Then v2 z2 can be added to x. At the last step this accumulated x is just the desired solution vector. Thus one need store only a few nxl vectors and no nxn matrices. Not even A need be stored if its elements are easily computed. 147

e * * .* * * * * * * * * C.C * Mobley

Indeed, if subroutines are written to compute A ui and Av. for each particular problem to be solved, then a general routine requiring only these subroutines and the known f vector can be written to implement the LPS algorithm.

Convergence. From the analytical development it appears that this method provides a direct solution to Ax=f. In reality though, the numerically generated sequences of vectors u. and vi are only approximately orthogonal, and thus only an approximate solution vector x is for x obtained from obtained. If y1 denotes the approximate solution solving Ax=f, then

E R1 f - Ay

is some residual vector which would be zero if yl were the true solution. Now if the system

Ay2 = R1

could be solved exactly, then x = yl + Y2 would be the desired solu- course Ay = R cannot tion since A(y 1 + Y2) = f-Ri+R = f. But of 2 1 be solved exactly either. Thus a series of solutions of Ay i = Ri. 1 for some where R. E f-AYi is made. Iteration ceases when I RI |

small E. The solution is then taken to be I x = y.. i=1

COMPARISON OF ALGORITHMS

A driver FORTRAN program for solution of the stated problem was written by the author. This program then called the appropriate subroutine to obtain the solution by either SOR, cyclic reduction, or the LPS algorithm. A routine was available for the solution by cyclic reduction (Swarztrauber and Sweet, 1975). A routine to perform the LPS calculations was kindly provided the author by Dr. Alan Cline of the University of Texas at Austin. The SOR routine was coded by the author. 148

Nonseparable Helmholtz Equation ......

The runs described below were made on the NCAR Control Data 7600 computer. The tabulated times show only the times required by the solution routines and none of the time required to set up the coefficient matrix A or to perform other calculations such as the time stepping,. In all cases, the runs were for five time steps with average solution times being shown. The parameter values shown in equation (5) and 8=1. and K=0.25. Figure 3 shows the number of SOR iterations required for a given accuracy in T.. for different values of a and for different grids. It 13 is clear that the optimum w is a function of the geometry of Q and of the resolution of the grid for a given geometry. However, it seems that a value of w=1.7 does not greatly slow down convergence of the SOR routine for any of the cases shown. This value was therefore used in all SOR runs.

--- 20 by 30 grid U) x--x 25 by 55 grid z 60 o S - a=0.5,:50 k=4

H X-/ . / / X 40 a=0.5, k=4 0::

LL20 -- D30 Z

20 1.6 1.7 1.8 RELAXATION PARAMETER w

Figure 3: Determi nation of the Relaxation Parameter w. 149

...... ** . . .* .*.*.* e e ...... C. Mobley

A series of comparison runs was then made using the initial u field The shown in Figure 1 and grids of 20 x 30 cells as shown in Figure 2. results are collected in Table 1. Cyclic l OR _SOR_LPS Reduction

0. 128 msec. 3800 msec. 21 msec. .01 128 4000 not applicabl for aOO. .05 131 4400 .1 133 failed to converge in .5 138 30000. msec.

ccuracy agrees with agrees with 9 significant cyclic red. cyclic red. digits to better to about 0.1% than 0.1%

for accuracy comparable to cyclic red. need 3 times as long.

TABLE 1. Comparison of solution methods for k=4 and variable a. The times shown are the times required to obtain solutions of the indicated accuracy.

is appli- From the case of a=0 (Q an annulus), for which each method either cable, one sees that cyclic reduction is superior to SOR, and times are method is vastly superior to the LPS algorithm. The solution to in the ratio 1 to 6 to 181. Furthermore, the LPS algorithm fails dis- provide a solution for a >.1; a value which does not cause much tortion of Q away from an annulus (see Figure 2). that The accuracy provided by either SOR or LPS is much less than those provided by cyclic reduction. The results of Table 1 agree with long as of Sweet (1972) who found that SOR requires about 25 times as 150

Nonseparable Helmholtz Equation ...... cyclic reduction to produce results of the same accuracy.

An accuracy study of the SOR routine was made by comparing its output to that of the cyclic reduction routine which is known to provide solutions of at least nine significant digits. This study used a "symmetric" initial u field similar to the "asymmetric" field shown in Figure 1, and the optimum value of w=1.745 was used. The SOR routine used an initial guess of T ( ) = 0 for the first time step, and thereafter used the solu- ij (o) tion field from the previous time step as the initial guess T (. (this procedure was used in all SOR runs). The number of SOR iterations required to get a given maximum relative error compared to cyclic reduction,

max. rel. error = maxTi (SOR) - T..(cyc.red.)I, 13 1,]ij

is shown in Figure 4. Clearly, doubling the accuracy of an SOR solution requires doubling the run time. Figure 4: Accuracy Study of SOR Compared to APPROXIMATE NUMBER OF SIGNIFICANT DIGITS Cyclic Reduction. 1 2 A 3 4 5 6 7 8 9 ... I . IIVII-An. . I I I --- -I I

120 - Time Step I 0w I110 Initial Guess is 0

100 - CD (I) ZOr 90- 0

Oc 80- IJ 70 0 w 60s- m Time Steps 2-5 Initial 50- z Guess is Solution from- Previous Step 40

I I I I I I . I 30C I . . . . . 2 3 4 5 6 7 8 9 10 n MAXIMUM RELATIVE ERROR COMPARED TO CYCLIC REDUCTION IS 10- n 151

.* . .* .**...... * * *v *e . . . .. C. Mobley

Since only SOR is capable of providing a solution in a greatly distorted domain, its behavior as a function of a and k was investigated.

Table 2 shows average solution times as a function of a for k=4, -6 w=1.7, and a convergence criterion of c =10 (agreement with cyclic reduction to better than 0.1% for the a=0 case). The initial field of Figure 1 was used.

a 0 .01 .05 .1 .5 .6 .7 .8 .9

TIME IN 128 128 131 133 139 142 145 143 141 MSEC.

Table 2. Solution times of SOR as a function of O for k = 4.

SOR is able to provide a solution even for the extremely distorted domain given by a =.9, k=4 (see Figure 5). The time required for solution is nearly independent of a.

Figure 5: The Initial Conditions u(r,9,t=0) for a = 0.5, k = 4. 152

Nonseparable Helmholtz Equation ......

Table 3 shows the SOR times as a function of k for a=0.5. Two grids were used, one of 20 by 30 cells and one of 25 by 55 cells (see Figure 2). Other parameters are as in Table 2.

20 by 30 Grid

k 0 1 2 3 4 5 6

TIME IN 128 135 143 143 138 blowup blowup TIMECNMSEC

25 by 55 Grid

k 0 1 2 3 4 5 6

TIME IN 395 384 blowup MSEC (50 for cyclic reduction)

Table 3. Solution times of SOR as a function of k and grid resolution for a=0.5.

Once again, the solution time seems to be nearly independent of the distortion of Q, until a certain number of "lobes" in Q is reached. At this point the relaxation procedure becomes numerically unstable, which soon leads to numbers too large to represent in the computer. The nature of this instability is not understood, but comparison of the k=5 runs for the two grids indicates that it is related to the resolution of the grid. A further run with the 20 by 30 grid and a=0.2 found that a solution could be found quickly for k=5 and k=6, but for this a the k=7 case blows up.

CONCLUSIONS

It is clear from Table 1 that the Lanczos-Paige-Saunders algorithm is not competitive even with relaxation, at least for the particular 153

...... C. Mobley problem studied here. Whether or not the poor performance of this method is a consequence of the algorithm itself or of the problem to which it was applied can be determined only with further study. The obvious desirability of solution techniques which are applicable to arbitrary geometries and also competitive with direct methods (which depend upon simple geometries) makes such study imperative.

The sudden, catastrophic failure of successive over-relaxation for the more distorted geometriesesand was unexpected begs further investigation. An understanding of why SOR failed for these particular cases might yield insight into the nature of the relaxation process itself. 154

Nonseparable Helmholtz Equation ......

REFERENCES

Caponi, E.A., 1974: A Three-Dimensional Model for the Numerical Simula- lation of Estuaries, Ph.D. Dissertation, Tech. Note BN-800, Inst. Fluid Dynamics and Appl. Math., Univ. of Maryland, College Park, Md., 215 pp. To appear in Advances in Geophysics, ed. by H.E. Landsberg, Academic Press, 1976.

Paige, C.C., 1974: Bidiagonalization of Matrices and Solution of Linear Equations, SIAM J. Numer. Anal., Vol. II, No. 1, pp. 197-209.

Swarztrauber, P. and Sweet, R., 1975: Efficient FORTRAN Subprograms for the Solution of Elliptic Partial Differential Equations, Technical Note NCAR-TN/1A-109, subroutine PWSPLR.

Sweet, R., 1972: A Direct Method for Solving Poisson's Equation, Faci- lities for Atmospheric Research, No. 22, pp. 10-13.

Sweet, R., 1974: A Generalized Cyclic Reduction Algorithm, SIAM J. Numer. Anal., Vol. II, No. 3, pp. 506-520.

Varga, R.S., 1962: Matrix Iterative Analysis, Prentice Hall, pp. 322. 155

...... l.. . . .l . . .I..D ... C . Mobley

157

A TEST FIELD MODEL STUDY OF A PASSIVE SCALAR IN ISOTROPIC TURBULENCE by Gary R. Newman Pennsylvania State University Jack Herring, Scientist

ABSTRACT A Test Field Model representation of an isotropic, passive scalar field in an isotropic, turbulent velocity field is developed. The model is shown to exhibit required consistency properties. Numerical simulation of heated grid turbulence data using the model is shown to compare well with existing experimental data. Results of model simulations are also compared with second-order modeling parameterizations of isotropic turbulence.

INTRODUCTION In a laminar flow, molecular diffusion acts to homogenize the spatial distribution of an admixture. In a turbulent flow, on the other hand, a contaminant diffuses at a rate much in excess of the molecular diffusion rate because of the additional stirring character of the stochastic velocity field. Consequently, turbulence profoundly influences, for example, both chemical reactions and pollutant dispersal in the atmosphere and in water bodies. In this paper, we address a stochastic flow which comprises an isotropic, passive, scalar-contaminant field imbedded in an isotropic, turbulent velocity field, and hence we do not treat the physics of the interactions between fluctuating and mean scalar and velocity fields (inhomogeneous fields). We may consider that isotropic turbulence contains features which are fundamental to all turbulence flows, and so a study of this case of turbulence is certainly of value. Additionally, practical consequence of such a study may derive from the fact that the smaller-scales of turbulence are thought to be locally isotropic. In isotropic turbulence which contains a passive scalar, the scalar and velocity variances decay with time if no external forcing is applied to the system. The large-scale eddies of the velocity field generally 158

A Test Field Model Study ...... contribute a major portion of the velocity variance; and, under the decay process, variance energy is cascaded as a result of non-linear, eddy-eddy interactions down into the smallest scales where it is destroyed by viscous dissipation. The distribution and decay of scalar variance is generally similar to that of the velocity variance, although the scalar-variance cascade is driven solely (for the case of a passive contaminant) by the stirring action of the velocity field. In our work here, we investigate the simultaneous evolution of both isotropic scalar and velocity fields through numerical simulations with a statistical model representation, the Test Field Model, of the two fields.

We develop in Section 2 the Test Field Model equations for an isotropic scalar field. The Test Field Model equations appropriate for an isotropic velocity field are given by Kraichnan (1971). Our scalar Test Field Model is developed from a Langevin representation for the scalar equation of motion. The scalar Test Field Model equations are invariant under random Galilean transformations as are the corresponding velocity equations, and this property is exhibited in truncated-wavenumber representations of the scalar and velocity equations of motion. In addition, the scalar Test Field Model exhibits required consistency properties which we also describe.

In Section 3, we discuss the numerical techniques employed in the simulations.

In Section 4, we first evaluate our simulation results through comparisons with existing heated, grid turbulence data. There, we compare both spectral quantities and quantities evaluated in configuration space. Then we compare our simulation results with second-order modeling parameterizations of Lumley and Newman (1976) and Newman, Launder and Lumley (1976), where the second-order models are formulated in configuration space and comprise closed sets of equations for various statistical quantities. 159

· . . . .· .a · .· · * * G. Newman

2. Construction of the Model Equations

In this section we shall obtain the Test Field Model for the scalar field from a Langevin model equation for the scalar equation of motion. The Direct Interaction Approximation for isotropic turbulence will serve as a reference closure for our Langevin model. The forced equation of motion for a passive scalar contaminant may be written in fourier space as:

(at +i K t)u ( t) (P, = f(K,t) (2.1) K=P+

where y is the scalar molecular diffusivity, f(K,t) is a stochastic driving force and where ui(K,t) and 4(K,t) are the fourier representations of the Eulerian velocity and scalar fields respectively and are given by the transforms:

i u i (x,t) = ui(K,t)e- , i(x,t) = Z (K,t)e' - K K

where the wave-vectors assume all possible values in a large cyclic box with sides of length L. The convolution sum in (2.1) is defined for any function F(K,P,q,t) as:

A 00 F(K,P,q,t) 6=(K-P-q)F(K,P,q,t)dPdq as L + oo (2.2) K=P+q

We shall consider only the case of an isotropic scalar field imbedded in isotropic turbulence. Thus, we may write the velocity and scalar time-displaced covariance functions in the following form where the brackets denote ensemble averages: K.K. = 1 P (K)U(K,t,t')6(K+P) P. (K)-i. - i- j _ ij ij K- (2.3) (K(K,t)i(Pt')> = y(Ktt')6(K+P) K -

We shall develop our model from a closure of the equations governing the temporal evolution of the velocity and scalar covariances. 160

A Test Field Model Study ......

We may form the Direct Interaction Approximation equations for the scalar field in the following manner (where we shall denote the Approximation as the DIA hereafter and where the DIA equations for the velocity field are given by Kraichnan (1964)). Equation (2.1) is linear in i(K,t) so that we may form a Green's function solution for the equation in terms of the forcing function f(K,t). In keeping with the usage in the literature, we refer to the Green's function, g(K,t,S), as the response Green's function for the system. C(K,t,S) represents the reponse of~ (K,t) to a unit perturbation (given by f(K,s)) in wave mode K at time s. The equations for the ensemble averaged response Green's function and scalar covariance may be written as:

' y 2) K A (t + YK )<(Kt)(K,t ')>+i Km = K=P+q

(2.4) A (3t + YK2 ) f(Ktt')>+i E Km = K=P+q 6(t-t') , f (K,t,t') =

2 dS = 0 t > t (- + yK) 9(K,t,t')+ Tn(K,t,S) g(K,S,t' )dS = 0 t > t' (2.6) t-

with W(K,t't') = 1 w (K,t,t') = 0 t

* * * * * * G. Newman

2 t (-t+ YKyK ) (Ktt + (Kt,t) t+(K ,S,t')dS = " w(t , , 2 < f(K,t)f (K,S) > (K,t',S)dS + rK dS ffqp sin (q,K(KtS)U(qtS)(P,t,S)dqdP t >t' (2.7)

2 + ( + 2YK )(K,t,t) + 2 tl(K, t, S)Y(K, S,t)dS = ( t t) A 2 2t ¶(KtS)dS + 2TrKt dSj qP sin (q,K>(K, t,S)U(q,t,S)Y(P,t,S)dqdP

(2.8)

with r(K,t,S) -KJ Pq sin2 (q,K:;(P,t,S) U(q,t, )dqdP (2.9) the bipolar integrals (z ) using where we have transformed ~~~~K=P+q A ~~~~~A A __ F(K, P, q, t, S) = 2_P F (K, P q, t, S)dPdq K=P+q which holds for any suitably integrable function, F(K,P,q,t,S), and where the integration domain ( i dPdq) spans allowed K,P,q values subject to IK - PI

A Test Field Model Study ...... shown for example by Herring and Kraichnan (1972)). However, the DIA equations yield unfaithful predictions of the intertial range of large Reynolds number turbulence. This inadequacy of the DIA model results from a divergence of the DIA response integral at low wave numbers (see Leslie (1973)) if and only if one takes E(k)a k-5/35 /3 This improper behavior of DIA has been attributed by Kraichnan (1964) to the fact that the DIA equations are not invariant to random Galilean transformations. Kraichnan (1965) rectified the intertial range problem of DIA by developing his Galilean invariant Lagrangian History Direct Interac- tion model. This model provides good agreement with inertial range data. However, the model is extremely complex, requiring significant amounts of computer time for computations, and thus we have chosen not to use this statistical model for our predictions. Instead, as mentioned previously, we have utilized a Test Field Model (referred to hereafter as TFM) for isotropic scalar transport which we discuss now. We have employed Kraichnan's (1971) velocity field TFM to simulate the evolution of turbulence in our predictions. The Test Field Models exhibit the Galilean invariance property, and additionally, the TFM equations do not contain the integration over time, which is computationally costly, which is exhibited in our DIA equations above.

We shall develop our scalar TFM from a Langevin representation of the scalar equation of motion. The philosophy of this approach is that we replace the original system which is too complex to solve exactly with one which we can solve (from a statistical point of view). However, the modal system is developed to reproduce as closely as possible the statistical evolution of the original system. We adopt the following Langevin equation as a model of the scalar equation of motion where the random forcing is now specified to be white noise in time:

[-a + yK ++ (K,t)] (K,t) = q(K,t) + f(K,t) (2.11) n(K,t) = rK ffpq sin 2 (q,K) K(t)dPdq (2.12)

q(K,t) = w(t) p i K * (q) §(P,t) pr (t) (2.13) K=P+q PKq 163

a** *. G* N a * * e * e * * * * e * * * ...... G. Newman where ( = 6(t-t')

(q) 6 (P + q) < ni () nj(P) > = Pi (f(K,t)f (K,t')> = Z(K,t)6(t - t') q) = <((P,t)_(q,t')> = Y(P,,tt')6(P + is a random, where w(t) is white noise in time as shown and where n(q) all isotropic solenoidal vector. We assume that n(q), E, w and f are initial scalar field, statistically independent of each other and of the to the time i(K,O). (KP (t) is a "memory" quantity which is related P, q. We shall scale for dynamic interactions among the wave numbers K, define KPq (t) more explicitly below. KPq time displaced From (2.11) we find that the Green's function and scalar covariance function satisfy:

[t + yK 2 + p(K,t)] G(K,t,t') = 0 t> t' (2.14)

[t + YK 2 + n(K,t)] Y(K,t,t') = 0 t>t' (2.15)

(2.16) with G(K,t',t') = 1 q and f. We obtain (2.15) because of the white noise quality of and hence Y(K,t,t') and G(K,t,t') obey the same differential equation of Fluctua- must be proportional to one another. This is a statement model system: tion-Dissipation Theory. Using (2.16) we obtain for our (2.17) T(K,t,t') = G(K,t,t')T(K,t,t'). we find: We may also obtain the equation for l(K,t,t). From (2.14) (2.18) ,(Kt) - o(KO)H(tO) = 3 [q(K,S) + f(K,S)]H(t,S)dS 2 function. where H(t,S) = exp - jt [YK + P(K,v)]dvj is a non-stochastic

Now, using (2.18) we obtain:

(- + 2YK 2 ) Y(K,t,t) + 2n(K,t)Y(K,t,t) = 2S(K,t) +

2 (2.19) 2TTK Pq sinV

A Test Field Model Study ......

If we conpare (2.19) with the DIA equation (2.8) (where we now set the initial time in (2.8) to zero and where we employ the Langevin model white noise form for the forcing function which transforms the integral forcing term in (2.8 to the term 29(K,t)) we see that we reproduce exactly the 'form' of the DIA equation (2.8) with our model scalar variance equation if we define:

t U(qtS)G(KtS) (PtS) dS (2.20) 0PKq(t) =j o T(Pt,t)

We use the term 'form' because the Green's function for our model system will in general not be equal to the DIA Green's function. We may employ (2.17) and an analogous Langevin model relation for the isotropic velocity field (obtained by Kraichnan (1971)) which is U(K,t,t') = G (K,t,t')U(K,tt't) (2.21) v to rewrite (2.20) as:

PKq(t) = U(q,tt) G ( t,t,S)G,(K,t,S)GUq(P,t,S) U ) where we denote the velocity and scalar response Green's functions by

G and GT respectively to avoid confusion. T We define 0 (t) (symmetric in P and K) as: PKq e K(t) = Gv(q,t,S)G(K,t,S)GT(P,t,S)dS (2.22) PK q o where the superscript "T" denotes the scalar field and where we note T that eT (t) has the dimension of time. We now form a modified equation PKq for T(K,t,t) as:

(at+ 2yK2 )(K,t,t) = 29(K,t) + 2rK:'Pq sin2 (q,K) e (t) U(q,tt)

LT(P,t,t) - T(K,t,t) dPdq (2.23) Equation (2.23) reproduces exactly the form of the DIA equation (2.8) for the case of statistically steady scalar and velocity fields. Con- sequently, this Langevin scalar covariance equation reproduces the form of the steady-state scalar-variance transfer which is exhibited by the DIA equations. We shall adopt (2.23) as the basis for our TFM closure, although we shall change the current choice of Green's functions 165

*. .. · a a · a * * * * a* * ** GC.Newman

Our equations (2.22) and which appear in the definition of PKqK(t). (2.23) are analogous to the equations for U(K,t,t) and the corresponding velocity memory function which Kraichnan (1971) obtained from his Langevin system. eK (t) serves as a characteristic time velocity KPq T scale for interactions between the wave modes K, P, q. 0 pq (t) represents approximate memory effects inherent in the DIA equations as a result of the explicit integrations over past history.

Our Langevin model equations are not yet invariant under random Galilean transformations. To insure invariance we must alter somewhat the forms for G and GT in (2.27), and we must insure that the corresponding velocity field model exhibits the desired invariance property.

We shall employ Kraichnan's (1971) Galilean invariant TFM for the description of the evolution of the isotropic velocity field. The velocity TFM equations we shall utilize for our scalar transport simulations are:

a-t v+ 22VK+ 2n(K,t) U(K,t,t) = 2 (K,t) +

Pqe (t)U(q,tt)U(P,t,t)PqdPdq (2.24) 2rK aKKp qKP q where v is the kinematic viscosity

K q qK(t)U(qt,t)PqdPdq (2.25) nv (K,t) = KPq PqK KPq =(l )( 2 ) bKpKP K (XY + 2) ,' bp q

aKPqaKPq = (bKpq + bKqPKq where X, Y and Z are the angles opposite K, P and q

5 2'P G vG26) = TKg2 bp pG (t)U(q,t,t)PqdPdq (2.26) r v (K,t) KPq PKq G vG c 2 (2.27) = 2TKg= bKq eKq (t)U(q,t,t)PqdPdq v (Kt)2 bKPqvg Kqp where g is a dimensionless scaling parameter which we discuss in a later section 166

A Test Field Model Study ......

dt vG (t) = 1 - tV(K 2 + p2 + q2 ) + nC(Kt) + fn(Pt) + dt KPq v ' v s v rn (qtjt Kp(t) (2.28)

KPq KPq denote velocity field quantities. The differential equations (2.28) and (2.29) are obtained from:

ev (t) = G (K,tS)G P,t,S)G (q,t,S)dS KPq otv v v

T1V(q~t K~q W (2.30) ejPq(t) = (G(KtS)Gs(qts)Gs(PtS)dS where G and G are respectively the response Green's functions for v v the solenoidal and compressive parts of the velocity test field in IKraichnan's TFM formulation. The details of the velocity TFM derivation and of the consistency properties of the velocity TFM are given in Kraichnan (1971). However, it is instructive to discuss them briefly here.

The Galilean non-invariance of the Eulerian DIA derives from the fact that the memory times inherent in the DIA are not built up along Lagrangian trajectories of the fluid particles. The memory times of the Lagrangian History DIA (which is formulated using mixed Lagrangian and Eulerian fluid mechanics) are developed along particle trajectories so that Galilean invariance is achieved The velocity Langevin model was formulated by Kraichnan in an Eulerian framework, and Kraichnan devised a scheme whereby Galilean invariance could be achieved in the model in an Eulerian frame.

Internal fluid distortions, as observed from an Eulerian frame, are caused by both inertial interactions due to pressure forces and by advection, where the latter effect may be construed as a pseudo- 167

* ·e * * * v·* e 'e* * * * *v * * * * ** G. Newman distortion (see Leslie (1973)). Kraichnan (1971) achieved a Galilean invariant formulation by removing "self advection" effects in the model Eulerian Langevin system. In order to assess the effects of pressure forces in an Eulerian frame, Kraichnan simply turned off pressure interactions by considering the advection of a passive (pressure-less) velocity field (labeled a test field) which contained both compressive and solenoidal parts by a purely solenoidal velocity field. He set up the equations of motion for this mixed velocity system and insured Galilean invariance of both fields by eliminating the self-advection term in the motion equation for the solenoidal and compressive parts of the test field. Kraichnan then developed Langevin model equations which represented the two test field motion equations, and then he formulated Green's function equations from the Langevin (Gs and Gc in equation (2.30) above) equations. These Green's functions v v were shown to be free from Galilean non-invariance effects. Kraichnan then made a correspondence between the Green's function of his original Langevin system and the solenoidal Green's function, Gv, of the test s is field system. This Green's function correspondence implies that G growth a measure of the inherent internal distortions which govern the of the triple correlations. Finally, he made correspondence between the modal velocity variance of the original Langevin system and the final velocity variance of the solenoidal part of the test field. The and Galilean invariant set of equations have been given above. Herring Kraichnan (1972) have shown that the velocity TFM predictions compare the favorably with those of the Lagrangian History DIA. Further, with shown Galilean invariance problem eliminated, the velocity TFM has been giving to behave properly in the intertial range (Kraichnan (1971-a)) reasonable values for the Kolmogorov constant which scales the three dimensional energy spectrum in the inertial range.

Now, to effect our TFM closure for the scalar field, we shall employ an idea proposed by Kraichnan (1971). We note that the scalar- gradient field is identically a compressive field since 168

A Test Field Model Study ......

V £ijK iKj (xt) E 0, Vx(VW ) = 0 (x,t). Consequently, we shall assume that the compressive Green's function, c Gv, of the velocity TFM may be associated with the Green's function for the scalar-gradient field. Further, we shall assume that G c may also v be associated with the scalar Green's function (GT) in our scalar T Langevin model equation (equation (2.22)) for K (t). That is, we KPq assume that the response Green's function for the compressive test field is a measure of the distortions which limit the growth of the triple correlations which are responsible for the non-linear transfer of scalar variance among wave-number triads. We then complete our scalar TFM by replacing the velocity Green's function in equation (2.22), Gv, with the solenoidal Green's function, G , of the velocity TFM.

If we apply the above reasoning to equation (2.22) we obtain:

ep (t) = GS(q,t,S)G (K,t,S)G (P,t,S)dS (2.31) Kq T 'vv where pq(t) is now symmetric in K and P. We may differentiate (2.31) q s and use (2.14) and an analogous equation for Gs (equation (4.6) of vT Kraichnan (1971)) to obtain a rate equation for e (t) which is: KPq

d T (t)= 1 - [2 +y (K2 + p2) + c(K,t) + c(P,t) + dt KPq

r%(q,t)J ep(t) (2.32) where pq(0) = 0 because of (2.31).

Our final scalar TFM comprises equations (2.23) and (2.32) which we solve in conjunction with the velocity TFM which comprises equations (2.24)-(2.29). The total system consists of a set of prognostic equations for the velocity and scalar modal variances and the memory functions coupled with diagnostic equations for n and n. Our development of the scalar TFM involved a number of intuitive steps which we cannot justify on a rigorous basis. However, we may examine the self-consistency of the scalar TFM system as set out below. 169

. .. e .**e.*. ********..* G. Newman

The scalar TFM equations satisfy invariance to random Galilean transformations because of the replacement of Gv and GT in (2.22) s c s c with G and Gv. Further Gv and Gv are positive, monotonic-decreasing functions of t (see Kraichnan (1971)) so that equation (2.31) insures that 6 (t) > OVt which is essential in view of our defining Langevin KPq - model equation (2.13). We may show that our scalar TFM satisfies conservation of scalar variance by examining the model's scalar variance transfer function. For isotropic scalar and velocity fields the equations for the three dimensional velocity and scalar spectra may be written as:

(2.33) [-a + +K2!2yK 2 E(K,t) ==(2.33) T(K,t)

at + 2vKr2 E(K,t)a =27 T(K,t) (2.34) velocity and scalar where E(k,t) and E6 (K,t) are the three dimensional spectra respectively and are defined as: E (K,t) = 27K 2 Y(K,t,t) (2.35) E(K,t) = 2rK 2 U(K,t,t) (2.36) with

2 '- _ h (X,t)$(xt)) = i E(K,t)dK (2.37) 00 (2.38) i2 - = E(K,t)dK. 62 and q2 /2 represent the scalar variance and mechanical energy in configuration space. The quantities T(K,t) and T6(K,t) represent the transfer of energy and scalar variance respectively to wave number K from all other wave numbers and are defined as:

T(K,t) = 4TK 2 F(K,t) (2.39)

Te(K,t) = 4TK 2 Fe(K,t) (2.40) where F and Fg are defined from the velocity and scalar variance equations as: 2 (i a + vK )U(K,t,t) = F(K,t) (2.41) + K,t,t = F(K,t) (242) (2. 4 2 ) 29 + YKTl(mKct) = Fe(Kt) 170

A Test Field Model Study ......

Thus, F and Fe represent the non-linear interaction terms which we have modeled using the Test Field Models. Conservation of scalar variance by non-linear interaction refers to the relation:

T (Kt)dK = 0 (2.43) and we note that condition (2.43) is satisfied by the transfer term formed from a truncated-wave-number representation of the scalar field equation of motion. The relation (2.43) can be shown to be satisfied by our scalar TFM by forming T (K,t) from equation (2.23) (by multiply- ing the right hand side by 2TrK 2) and then utilizing both the sylnmetry in K and P of KP(t) and the symmetry of the integration aS /dPdq specified in (2.43). The symmetric integration region for K, P, q is shown in figure 1 and is derived from the triangle relation (2.10) which is also illustrated in figure 1 for one K-P-q triad set. The figures show the truncated-wave-number system (0

The existence of the model scalar Langevin system insures that the scalar TFM additionally yields realizability of the modal scalar variance, T(K,t,t). We may see this alternatively from equation (2.23). If we assume that Y(K,t,t)> 0 V K, Vt < tl and consider the condition

T(K,tl,t1 ) = 0 for some K = K, then (2.23) shows that a-Y(K,t,t)lt

> 0 insuring T(K,t,t) > 0 for t = t1 + 0+. We note that this realizability property is satisfied for any definition of ET (t) which gives T K J q1P e (t) > OVt. Orszag (1974) presents a similar argument verifying KPq realizability of U(K,t,t) for the velocity TFM.

The scalar TFM is also consistent with the equi partitioning behavior which can be exhibited from the truncated-wave-number scalar equation of motion. In (inviscid) absolute statistical equilibrium the modal variance, T(K,t,t), of the truncated motion equation becomes constant independent of K and t. We see from equation (2.23) that, with zerio forcing and with y=O, if our TFM system achieves T(K,t,t) = constant vK at some tl, then (2.23) insures T(K,t,t) = constant vt>tl 171

·. .. .a...... * ...... G. Newman

kmax

q 1k:max

Fo-i re 1. ToTs: Truncated!-wave-nu.rmber intesration region. Bottom: Cross section of the top figure for a fixed k val..ue illstsratin-.; the re.gion sranned by the triangle relation (2,10) arnon. k,,p and q. 172

A Test Field Model Study ......

providing the velocity field has also achieved equipartition equilibrium. Additionally, the scalar TFM also satisfies the fluctuation-dissipation theorem (Kraichnan (1959)) as can be seen from equation (2.21) (this theorem is valid for the truncated motion equation system only in absolute statistical equilibrium). We note further that the forms of the 'input' and 'drain' terms of the TFM modal variance equation, (2.23), are consistent with a tendency for equipartitioning. The terms act to drain modal variance from wave number regions with excess variance and input variance into wave numbers exhibiting modal variance deficiencies. We note finally that numerical studies (see Orszag (1974)) of isotropic turbulence with the eddy-damped Markovian equations have shown that the velocity field tends toward modal-energy equipartion. Since the scalar TFM equation for T(K,t,t) has exactly the 'input' and 'drain' forms of the Markovian equation for U(K,t,t), we infer that the scalar TFM system would exhibit similar equipartitioning behavior. We now discuss the numerical procedures employed in our scalar transport calculations.

3. Numerical Techniques and Procedures

As noted, we must solve prognostic equations for U(K,t,t),e q(t), and vG s e6 (t) and vG (t) along with diagnostic equations for rn(K,t) and KPq 2 KPq v nv(K,t). The time stepping for Y, U, e , e and e is performed with two-step predictor corrector marching. This numerical procedure has been determined to be sufficiently accurate for our simulations. We illustrate here the marching scheme for T(K,t,t) where we represent the equation for T(K,t,t), equation (2.23), simply as:

(1a + 2yK 2 )YK(t) = 2F(t) (3.1) The predictor, for a time step increment of A, is generated with

2y A e yKA2 TK(t + A) = e 2K K(t) + (1(1 - e-2yK A F(t). (3.2)

A) and then the corrector formula F*(t 6+ A) is next 2AK-2Kcomputed from.K-2yK T*(t + (t A) -2K2 T(t) + i (1 - e- 2Y K 2 A ) (F (t) + Fl(t + A)) (3.3)

is obtained. The wave number integrations in the scalar and velocity TFM equations are performed in the following manner. We discreetize 173

* G. Newman 35 the wave number domain into an interpolating set, {Kii which spans maximum the range of K=O to K=100. The set {K } are distributed with representa- point density at the low wave numbers so as to provide good tions of velocity and scalar spectra. We now evaluate the continuous time marching TFM equations on this discreet set, {Ki}, and perform the values of on the {Ki}. To effect the P-q integrations we interpolate the the functions in the integrands using cubic splines and perform integration using Gaussian integrations. As mentioned, the integration part of domain for the P-q integration is illustrated in the lower trun- figure 1. Herring and Kraichnan (1972) note that the wave number wave cation scheme depicted in figure 1 guarantees energy-conserving number integrations.

The initial state of the scalar and velocity fields are represented Ee(K,O). by chosen forms for the three dimensional spectra, E(K,O) and the initial In our simulations we have employed the following forms for spectra (where H(K,O) represents either E(K,O) or Ee(K,0)): -K/B (3.4) H(K,O) = AKe , A and B = constant AK H(K,O) = K 8/3 (3.5)

it We note that H(K,O) peaks at K=B for the function (3.4) whereas spectral peaks at approximately .826B for the form (3.5). These initial shapes evolved rapidly into spectra which are very nearly identically scheme by self preserving. We determine the accuracy of our numerical scalar examining the energy balance equations for the velocity and and fields. These equations may be obtained by integrating (2.33) (2.34) over the entire range of K values and are given as: (3.6) dq 2(t) = -2E(t) 3(t) d_ 2 (t) = -2 dt in (2.37) where q2 and 82 are the velocity and scalar variances given by: and (2.38), where £ is the mechanical dissipation rate given 174

A Test Field Model Study ......

£(t) = 2V 0 K2E(K,t)dK = Viji j (3.8)

and where £g is the scalar dissipation rate which represents the rate of molecular smearing of scalar fluctuations and is given by:

ge(t) = 2yJoK2E (K,t)dK = y ji (3.9)

By numerically differentiating the data from our simulations we find that the error quantities

dq dq- 1 2 1- |andand ded2 2 1 - 1|(3.10) dt 2C dt 28 - -3 0 are less than lx10 at each time step for all of our simulations. Our simulations cover a dimensional time range from 0.0 to a maximum of 3.0. Thus, with the results (3.10) we deduce that the cumulative error growth in the balance equations is small. We now present the results of our scalar transport simulations.

4. Results and Discussion

In this section we present and discuss the results of our scalar decay simulations with two objectives. First, we compare two of our simulations with existing experimental data which pertain to scalar decay in isotropic turbulence. Then, we evaluate the second-order modeling parameterizations which we discussed in the introduction. We note that some of our statements regarding the results of the velocity TFM simply reinforce the findings of Herring and Kraichnan (1972). They serve here to provide a touchstone with which the scalar TFM results may be evaluated. Before addressing the objectives, however, we will present the initial conditions and parameter values for the simulations discussed below. s c We noted in section (2) that the TFM equations for Tv and V n V (equations (2.26) and (2.27)) were augmented by the factor g2. The inclusion of this factor provides a means of tuning the characteristic memory times which control the build-up of the triple moments. In his development of the velocity TFM, Kraichnan (1971) determined the value g = 1.064 by requiring that the TFM reproduce the results of the DIA 175

* * * * * G. Newman for a case which the DIA model is expected to predict satisfactorily. perturba- Kraichnan fit the TFM to the DIA model for the case of small of wave tions about the equilibrium state in a thin spherical shell numbers. On the other hand, Herring and Krachnan (1972) determined TFM with an 'optimal' value of g = 1.5 by comparing predictions of the g yields a predictions of the Lagrangian History DIA. This value of agreement with the value for the Kolmogorov constant which is in excellent both value obtained from the Lagrangian History DIA. We have utilized in our simu- g = 1.5 and g = 1.0 (negligibly different from 1.064) lations.

of We present in Table 1 the initial conditions and parameters the simulations which we shall consider. We define the quantities are given in the table R (the turbulence Reynolds number) and PX which as: 1 _ u'j =/2 u X = = 2 (4.1) R=X uP , - ( 3 q \IV X Y and where X and XA are the velocity and scalar Taylor microscales are given by:

-2 ~ r E (K,t)dK 1 (4.2) X = i = 5 %0E - [90 K6E (K,t)dK

9 = e~ye"' fEe e(K,t)dK (4.3) = [6 - K d Xe 12 I/KZE (K, t)d J the various The RX and P% values given represent asymptotic values from slowly with simulations, but we note that both RX and PX change very time in any simulation after self preservation is attained.

We now compare two predictions with the heated grid turbulence the data of decay data of Yeh and Van Atta (1973) and in part with contaminant in Mills, Kistler O'Brien and Corrsin (1958). The scalar thermal fluctuations each of these experiments was temperature, and the the grids were input into both of these laboratory flows by heating The thermal which were employed to generate the turbulent flow fields. effects fluctuations were relatively small in both flows so that buoyancy behaved passively. were negligible and hence the temperature contaminant 176

A Test Field Model Study ......

TABLE 1

Initial Initial R% @ Prandtlt Run Spectrum Spectral Peak Wave tina g Final { @l Shape r Final Number Final Shape Number E b* 3.3 1 i .-.-.- 34.9.- 1.0 1.0 0.6 36.7 E e b 2.5

E a 9.1 2 -. .a .. 37.2 1.0 1.5 1.0 39.8 E a 9.1 __._.__.__ 0 .. E b 3.3 3 | .b - 61.5 1.0 1. 0.8 56.8 E_ b 6.6

E a 9.1 4 - .a - 24.1 1.0 1.0 3.0 25.4 E· a 18.2

E a 18.2 5 . I 22.0 1.0 1.0 3.0 24.3 E___ a 9.1

E a 9.1 6 - . a 52.8 0.1 1.5 0.5 17.7 E I a 9.1

E a 9.1 7 - 36.9 4.0 1.5 0.6 74.9 Eg a 9.1 6 ,, , ... _ _ ,_ _ ..

* a = form (3.4), b = form (3.5) 177

· ·e · ·* ·* ·* · e· · c *e a **** G. Newman

Decaying grid turbulence is known to exhibit approximate isotropy of the flow field, i.e. near equipartition of the turbulence energy among the component energies (u2-u2 u 2 ) and to exhibit lateral and longitudinal 1 2 3 approximate self preservation of various turbulence quantities when the quantities are non-dimensionalized (normalized) with appropriate local variables (see Monin and Yaglom (1975) for further details of the properties of grid turbulence). These qualities suggest that heated, grid turbulence data may be employed for comparison with our isotropic turbulence simulations which are very nearly identically self preserving. However, after Herring and Kraichnan (1972), we note additionally that it is difficult to assess to what extent the period of temporal evolution predicted in our simulations corresponds to that of the experiments. Our simulations march forward in time from chosen initial spectral forms and achieve self preservation, whereas the grid turbulence fields evolve from coalescing heated wakes (which initiate behind the grid bars) and hence have markedly different initial spectra. On the other hand, we compare self preserving forms of the simulations results and the data, and these results may well be fairly universal in nature.

To effect simulation of the experimental data we have simply closely matched the values RX = 35.2 and P% = 32.5 which were exhibited in the Yeh and Van Atta (1973) flow at the midpoint downstream tunnel position x/M = 35 (where M is the grid mesh size). We label our two comparison simulations as Runs 1 and 2, and as shown in Table 1 above, the RX values for Runs 1 and 2 are well matched with that of Yeh and Van Atta, while the PA values exceed their value by about 15%. The PX discrepancies derive from the fact that these two computer Runs were performed with unity Prandtl numbers (where the Prandtl number is y ,the Prandtl number of the experiments for which air was the working fluid was approximately .72 so that we had slightly less control over the level of Pi. We hope to run a comparison simulation with Prandtl number = .72 soon, however, it is unlikely that the results shall differ significantly from those of the two simulations discussed here. The values of RX and PX 178

A Test Field Model Study ...... exhibited in the Mills et al flow at the midpoint downstream position (x/M = 50) were 31.5 and 24.1 respectively. We present two comparison simulations in order to illustrate the sensitivity of the predictions to changes in the value of the scaling factor 9. As seen from Table 1, the factor 9 = 1.0 in Run 1 while g = 1.5 in Run 2. Further, the initial spectral shape for Run 1 is the algebraic form given in equation (3.5) whereas the initial spectrum in Run 2 is the exponential form (3.4). We have performed an independent simulation which shows that the differences between the results of Runs 1 and 2 are attributable to the disparity between the values of g rather than to the differences in the initial spectra. In figures 2 - 11 we compare predictions with the experimental data for the following quantities (where we now drop explicit time dependence): E(k), T(k), K2E(k) - the velocity dissipation spectrum, R(u,u) - the double velocity correlation, R(uu,u) - the triple velocity correlation, Ee(k), Te(k), K2E (k) - the scalar dissipation spectrum, e R(0,0) - the double scalar correlation and R(u0,0)- the triple velocity- scalar correlation. The defining relations for the double and triple correlations are given below. The simulation profiles presented (eg. E(k) = E(k,t)) are for dimensional times of t = .6 for Run 1 and t = 1.0 for Run 2; however, they represent nearly universal shapes for all t such that self preservation is closely maintained. The corresponding figure numbers for the various plotted quantities are given in Table 2. TABLE 2 Figure # Ordinate Abscissa 2 E(k) k/k 3 Ee(k) k/k 4 -T(k)kk 5 5T(k)Te(k)e t " k/kkk/k 6 K2 E(k)k/k ? 7 K2E0 (k) k/k a 8 ' R(9,0) r/X j

B 99| R(u,u) r/ A 10 ./3 i... R(uf.Lf)\-- ~ v, j Yr/(72^,- \ , L,i / 11 i~~~~~~~~~~riI R(uu,u) r/X !0 179

...... G. Newman

In Table 2 above, the variable r represents the spatial separation of the distance in the correlation quantities while ks is the inverse 3¼ utilize local Kolmogorov microscale and is defined as k = (E/v ) . We scale) values of k and v = (Ev)¼ (where v is the Kolmogorov velocity s s s preserving to scale the plotted statistical quantities into nearly self with the velocity forms. The normalization of scalar spectral quantities we note quantities k and v follows Yeh and Van Atta (1973). However, unity Prandtl that this scaling is strictly valid only for the case of Prandtl number, number. For cases with significant departures from unity to the contaminant scalar spectra must be normalized with scales relevant of the scalar field. We have employed such scaling in our presentation scale r. Further, correlation quantities where both X and Xe are used to figures should the small-scale normalization presented in our comparison at the higher enhance agreement between predictions and empirical data the smaller scales wave numbers since isotropy is more nearly achieved at of in real turbulence flows (see Yeh and Van Atta for a discussion grid turbulence). observed departures from isotropy at low wave numbers in E(k) and As shown in figures 2 and 3, the predicted energy spectra although Run 2 Ee(k) agree well with the corresponding empirical spectra, range of shows somewhat better agreement than Run 1 over the central empirical spectra k/k values. We see that the computed spectra and the number than E(k). exhibit the quality that E (k) peaks at a lower wave E (k) between The disparity of approximately 9% in the peak values of to the the simulations and the experimental data may be attributable difference in P% values. and T(k) Agreement between simulation and data profiles of Te(k) numbers and is seen from figures 4 and 5 to be good at the high wave as indicated not as good at the low wave numbers. We note, however, map the in figures 4 and 5, that Yeh and Van Atta could not accurately We entire negative regions of T(k) and T(k) by direct measurements. of Te(k) have included in figures 4 and 5 for comparison the profiles balance and T(k) which they determined indirectly using thnde spectral Yeh and Van equations (2.33) and (2.34). Additionally, we note that 180

A Test Field Model Study ...... Figure 2 120.4oIII140.- ll---.-l-l-l I

120. -

100. - /

80. E(k)ks

v2s 6o.

20.

0.0 0.3 k/ks

Figure 3 140.

120.

100.

Ee(k) k 80. N 60.

20.

0.0 0., k,/k

Figure 2. Normalized three dimensional velocity spectra. -- - , Run 1; -, Run 2; * , Yeh and Van Atta. Figure 3. Normalized three dimensional temperature spectra. --- -- , Run 1; -- , Run 2; · , Yeh and Van Atta. 181

· ·O · · · · a · · · 0 e* · · · · · *· · · G. Newman

CE U)

H-

Figure 4. Normalized three dimensional transfer spectra of kinetic energy. _ _ Run 1; , Run 2; - - - - -, Yeh and Van Atta directly measured; - --- , Yeh and Van Atta spectral balance (2.34). 182

A Test Field Model Study ......

f Figure 5 <,.

0.0

Cy-,U) -2.0

cYEwlsl w- -400

-6.0

-8.0

Figure 5. Normalized three dimensional transfer spectra of temperature fluctuations. --- , Run 1; , Run 2; ------Yeh and Van Atta directly measured; -- - -, Yeh and Van Atta spectral balance (2.33). 183

e * * * e * e G. Newman ** ·* *· * ** · * ·· * *e * *

Atta indicate that the energy conservation property (2.43) and the analogous property for T(k) (foT(k,t)dk = 0) are not satisfied identically by their directly measured data curves with positive area contributions exceeding negative area ones by 50% for Te(k) and 10% for T(k). The absence of definitive empirical data in the regions of negative transfer may account in part for the less faithful agreement in these regions between experiment and simulation. Finally, we see from Figures 4 and 5 that the negative peak values of T(k) and Te(k) are larger for Run 1 than Run 2. The cited differences between the Run 1 and Run 2 profiles with greater scalar and for E(k), E0(k), T(k) and Te(k) are consistent velocity energy transfer efficiencies in Run 1 than in Run 2. Indeed, as indicated by Herring and Kraichnan (1972), the efficiency of spectral energy transfer in the TFM increases as the parameter "g" decreases, because a decrease in g causes an increase in the decay time for G (k,t,s). The dissipation spectra are presented in Figures 6 and 7, and we observe that the results of Run 2 are in somewhat better agreement with the empirical data than those of Run 1. Differences between these spectral curves of Runs 1 and 2 are again consistent with the increased transfer efficiency in Run 1. The agreement between simulation and data for either Run, however, is not exceptional with the greatest disparities occurring in the region of the spectral peaks. On the other hand, we note that Yeh and Van Atta compare their velocity dissipation spectrum with spectra of Uberoi (1963) (with Rx70) and Van Atta and Chen (1969) (with Rx335). It can be seen from their comparison figure that the Uberoi (1963) spectral values agree well with those of Yeh and Van Atta (1973) while the values of Van Atta and Chen (1969) are lower than those of Yeth and Van Atta over the entire spectrum. In fact, the peak value of the Van Atta and Chen (1969) dissipation spectrum (where their turbulence Reynolds number is almost identical with that of Yeh and Van Atta (1973)) is nearly equal to ours. Finally, we compare in Figures (8 and 9) and (10 and 11) longitudinal second-order and third-order correlations respectively. We define the longitudinal second-order correlations as: 184

A Test Field Model Study ......

RR(uu)(u = <0(x) (x + r)> u) ( R(e, ,) and we further define the longitudinal third-order correlations as:

2 < ( ).(xx)u(x+ _r)>r) R(uu,u) / - , R(ue,) ' ½ <2(x)> For isotropic turbulence all four correlations are functions only of the separation distance magnitude r, r = (ri r i ) . We assume here that the direction of r points along the direction of the velocity component "u". The empirical data are presented in the original papers as functions of a temporal separation variable, whereas we present them as functions of r by making recourse to Taylor's hypothesis. The correlation curves for the simulations are determined with the following formulae which are valid for isotropic turbulence (see Monin and Yaglom (1975) for derivations of these expressions):

R(u,u)(r) = 2 - C kr) + sin(kr) E(k)dk (4.6) R( (kr) 2 (kr)3

R(,e)(r)= 23 [sin(kr)]E 0 (k)dk (4.7) ' _ 1 T(k) R(uuu)(r) + cos(kr) 3 sin(kr) (4 8) R(uu,u)(r)= -22 o sin(kr) + 3 (k)S 7kr)4 ] T(k)d2 (+) .[(kr )2 (kr) (kr) 4 dk Ru )()cos(kr) - w sin(kr)1 TO(k)d R(u6,6O)(r) = ( + k 2 I (4 9) RluGE3(kr)k (kr) =J We see from the correlation formulae that R(u,u) and R(0,0) are even functions of the separation r while R(uu,u) and r(u0,0) are odd functions of r. The integrations in (4.8) - (4.11) are evaluated in the same manner as all other integral quantities by application of Simpson's method to cubic spline representations of the integrands.

From figures 8 and 9 we see that the predicted second-order correlations agree reasonably well with the empirical data. Here, only one simulation curve is presented in either figure because the R(0,0) and R(u,u) results for Runs 1 and 2 are virtually indistinguishable on the scale of the graphs. We observe that R(0,0) is negative at the higher wave numbers in both the Yeh and Van Atta data and the simulations, although the negative peak in the data is of larger magnitude than in 185

a ...... 0 . . . .. * ...... G. Newman

Figure 6 3.0

2*0 2k2 E(k) vks 1.0

0.0 k/ks

Figure ? 3.

2.0 2k 2 Ee(k)vs N v

0.0 1.0 k/ks

Figure 6. Normalized three dimensional velocity dissipation spectra. , Run 1; -- , Run 2; o , Yeh and Van Atta.

Figure 7. Normalized three dimensional temperature dissipation spectra. -- -- , Run 1; -- , Run 2; o , Yeh and Van Atta. 186

A Test Field Model Study ......

Figure 8 1.

R(e,e) 0.

0.2

0.0 -1 .0 10.0 .1A

Figure 9 1

0 R(u,u)

0.0

Figure 8. Longitudinal second-order temperature correlations. . Runs 1&2; - - - - , Yeh and Van Atta; --- , Mills et al.

Figure 9. Longitudinal second-order velocity correlations. ------, Runs 1&2; - - , Yeh and Van Atta; -- - -, Mills et al. 187

* ...... o .... .* * * G ; Newman lF0 igure 10Newman Figure 10 0.08 I ___U I_ _ ~~ _ · __IIII_* s__liI / \ / /I o0o4

R(ue,e) I & 0.0 R(9,ue) /

-0o.04 /

-0.08 I 0 o 5.0 o0o 1 5.0 20.0 25.0 . r / (;\

Figure 11

R(uu,u) & R(uuu)

Figure 10. Longitudinal third-order mixed velocity-temperature correlations. R(uG,6): -, Run 1; -- , Run 2; --- , Yeh and Van Atta; - -, Mills et al. R(e8,): -- -- , Yeh and Van Atta. Figure 11. Longitudinal third-order velocity correlations. R(uu,u): , Run 1; - , Run 2; ----- , Yeh and Van Atta. R(u,uu): ---- , Yeh and Van Atta. 188

A Test Field Model Study ......

the simulations and it occurs at a lower wave number.

Finally, we compare in figures 10 and 11 the simulation and data profiles of R(ue,e) and R(uu,u). The R(uO,e) and R(0,u6) curves from Van Atta and Yeh exhibit near antisymmetry (modulus the region near zero separation); however, the peak values of these curves are seen to be approximately 80% larger than the peak values of the computed curves and the peak value of the Mills et al (1958) data. On the other hand, the R(uu,u) and R(u,uu) profiles of Van Atta and Yeh are not antisymmetrical, and our simulation curves more nearly reproduce their R(uu,u) curve. Indeed, the peak value of the Run 1 curve agrees well with that of their R(uu,u) curve although their peak occurs further to the right. We see for both the velocity and the mixed velocity-temperature triple correlations that the results of Run 1 concur more closely with the data than the results of Run 2. This fact, of course, again reflects the difference in energy transfer efficiencies between the two simulations.

We infer from the above observations that the comparisons between the TFM predictions and the heated grid turbulence data are not unfavor- able to the scalar TFM, and we propose that the scalar TFM may well provide useful information in future studies of scalars in turbulence. In addition, we note that there is room for further developmental work with the model particularly with regard to optimization of the scale parameter g in the scalar equations. Our results described above indicate that the larger value of g is preferable to the smaller one for prediction of most of the spectral quantities, although the smaller value appears to yield better results for the triple moment. However, we cannot deduce an 'optimal' value for g from our investigations to date. One possible method for determining an optimal scalar g value would be to compare scalar TFM predictions with those of direct spectral simulations and of Lagrangian History DIA predictions in the same manner that Herring and Kraichnan (1972) compared various velocity-field statistical models. Additionally, it would be valuable to make comparisons with any existing large Reynolds number, scalar trnasport data such as atmospheric turbu- 189

...... G. Newman

scalar lence data. In this manner the intertial range behavior of the various of TFM could be evaluated. We now discuss the implications of our TFM simulation results for the second-order modeling parameterizations which are relevant to isotropic turbulence.

As noted in the introduction, the second-order modeling approach involves flow description through utilization of ensemble averaged the transport equations, where the various statistical quantities in equations are evaluated for zero spatial separation in configuration space) space. The second-order model system (written in configuration scalar is: appropriate for isotropic turbulence which contains a passive

2 2 92 aq q -2= -2£ , = -2£- (4.10)

2 /- t£ = '-2/q o - 2

where Wand bJ represent second-order parameterizations and where 2 of time q2 , and , E and e have been defined above and are functions only in light of the assume isotropy. We observe that if we represent 2 second-order i and in terms of q , c2,£ and c£ (as is done in modeling), then the system (4.10) forms a closed, predictive set.

Lumley and Newman (1976) formulate a second-order closure model for anisotropic, homogeneous turbulence using the invariant modeling techniques developed by Lumley (1970). The forms for their parameterized closures are determined in part from existing homogeneous turbulence results decay data and in part from consideration of various analytical their for limiting states of homogeneous turbulence. If we specialize representation for t to the case of isotropic turbulence we obtain: (4.11) 14 r 2.83 (4.11) = + 0.980 exp -

length where Ro is the turbulence Reynolds number based on the integral size) and scale, Z, (which is a scale representative of the large eddy is defined by:

R (q) - ( q42) (4.12) Rl = 9£v V 190

A Test Field Model Study ......

We note (as is shown in Tennekes and Lumley (1973)) that R - R 2 for large Reynolds numbers. We shall evaluate i from our simulation results below; however, we note that our TFM results for the velocity field have been obtained previously by Herring (1976).

Newman, Launder and Lumley (1976) formulate a second-order representation for he from a homogeneous scalar field in a manner analogous

to the development of the form for i by Lumley and Newman (1976). The form which Newman et al. (1976) propose which is appropriate for isotropic turbulence is: 10 6.2 ] d = 3 + 0.447 exp - 6 (4.13)

The expressions for l and he are equivalent for large values of R£, but they diverge for smaller values asymptoting to their respective 'final period' values as R tends to zero. On the other hand, Newman et al (1976) note that the existing isotropic (heated grid turbulence) scalar decay data are somewhat inconsistent. They define a convenient parameter representing the evolution of the scalar and velocity fields (which we shall utilize to illustrate the inconsistencies) as:

-- 2 R = (q /E)/(e/E) (4.14) where R is the ratio of the mechanical to scalar time scales. We may view the scales q2 / and 2 /£c as the time scales for significant evolutionary changes in the large-scale (i.e. energy containing) velocity and scalar eddies respectively (see Tennekes and Lumley (1972) for a detailed discussion of the physical interpretation of the scale q /£). The levels of R in the existing data are scattered about a value of unity. However, the data values range from .6 to 2.0, and this degree of scatter among the data is difficult to rationalize physically. Indeed, from a physical viewpoint it would seem that a value of R nearly unity should be appropriate for 'equilibrium' decay situations. That is, it is physically reasonable to propose that the energetic, large- scale velocity eddies should most profoundly affect the large-scale scalar eddies, physically distorting them on the scale of the velocity eddies. Thus, the ratio R, viewed as the ratio of time scales relevant 191

· * * · e e * * * * * ·a a e e *·· * ·* a ·e * * * * G. Newman to the large eddies, should be nearly unity in decaying 'equilibrium' flow regimes (i.e. in flow regions which are not significantly influenced by initial or boundary conditions). The expression given above for 9 has been in part formulated to concur with this premise. Isotropic decay calculations using the above closure models exhibit the trend of R-.6 (the final period value) as the Reynolds number tends to zero. We note that data for q2 and 92 from the initial period of decay in isotropic, heated grid turbulence are generally well represented by power law expressions with constant exponents. This decay behavior is closely reproduced by the above closure model over temporal periods comparable to those in the experiments. We note finally that i and Ae reduce to the following simple forms for the case of power law decay for q2 (t) and

2 (t):

= -2 -2 (1 +( ) (4.15) where n and ne are the q2 and 82 power law exponents respectively q and where these forms are valid independent of possible non-zero virtual origins for q2 and @2. We now discuss our simulations.

We have performed a number of TFM simulations of scalar decay in isotropic turbulence, and we have varied the Reynolds number, Prandtl number and initial spectral shapes in the varius simulations. The Reynolds number range spanned in the Runs is RX = 3.2 to RX = 62.5, whereas the Prandtl number varies from 0.01 to 10.0. We shall discuss the results regarding i, ie and R from five of the simulations. However, these results are representative of those for the entire set of simulations, because as we discuss below, the asymptotic behaviors of i, ie and R are very similar for the entire prediction set. The initial conditions and parameter values for the five Runs are given above in Table 1, and the Runs are denoted as Runs 3-7. Before considering the results of these Runs, however, we discuss briefly some general evolutionary behaviors exhibited in the predictions.

Our simulations depict the evolution toward self preservation of scalar and velocity fields which are given initially by specifying profiles 192

A Test Field Model Study ......

Figure 12 10.0 *~~ I ~ ~~~I I ~ ~- I - 8.0 6.0

4.0 \ / -

2.0 \_ . .

R 1.0 i Ii I II /~~~~~~ 0.8 _// 0.6 /~~~

0.2

, I I 0.1 !I i I I --I- . __ O.C)1 0.03 0.06 0.1 0.3 0.6 1.0 3.0 t

Figure 12. Time scale ratio. , Run 3; -- , Run 4; -- - , RRun -- 5; ,Ri, Run 6;----- , Run 7. 193

e eG. e e Newman

profiles are distinguishable for E(k) and E6 (k). The initial spectral in terms of the positions of their peaks. The wave numbers correspond- and we ing to the peaks of the initial spectra are included in Table 1, at dif- see from the Table that the initial E(k) and E 6(k) curves peak spectra ferent wave numbers in some of the runs. For the 'well behaved' corres- considered in our simulations, we may infer that the wave number of the ponding to a spectral peak is inversely proportional to the size space energetic, large-scale eddies in the corresponding configuration (see Tennekes and Lumley (1973) for a discussion of the distinctions the between waves and eddies). Thus, we may view the relation between the inverse wave numbers of the peaks in E(k) and E0(k) as depicting eddies. relation between the sizes of the large-scale velocity and scalar exhibit Additionally, the evolution of the velocity and scalar spectra after self (qualitatively) universal characters in all of the simulations preservation is approximately achieved. In the self preserving mode, at successively all of our simulations predict that both E(k) and E0(k) peak of the lower wave numbers as time increases, and additionally, in all than simulations the Ee(k) spectra peak at somewhat lower wave numbers are the E(k) spectra. We note that both of these characteristics (1973). exhibited in the heated grid turbulence data of Van Atta and Yeh fact that In real turbulence, the former characteristic reflects the the although eddy energy is cascaded toward the higher wave numbers, consider smaller eddies decay more rapidly than the larger ones. We now the second-order parameterizations. in In figure 12 we present the results for R from Runs 3-7, while In figures 13 and 14 we give the results for k and fifrom these Runs. these figures the quantities are given as functions of the dimensional the curves simulation time. The striking feature of these plots is that in the for the time scale ratio appear to be asymptoting to values to be neighborhood of Rl, whereas the curves for both i and Aip appear these asymptoting to values in the neighborhood of 4, ie~ 4. Further, of the value asymptotic behaviors seem independent of the level of Rk and spanned by of the Prandtl number over the ranges of these two quantities 194

A Test Field Model Study ...... Figure 13 5.0 4.0 3.0

2.0 ut 1.0

0,3 L 0.c )1 0.03 0.06 0.1 0.3 0.6 1.0 3.0 Figure 14 ------~------I I I I 5.0 ---- I i |!.

4.0 - -- -- I ·- - -· 4 - - 3.0 / - / 2.0 -. <~~~~~~~~~~ 1.0 7

I I 0,.3L _ _ _ I JI I --- I ,I I 0.0)1 0.03 o. 06 0.1 t. 0.3 0.6 1.o 3.0

Figure 13 Second-order parameterization for the velocity dissipation equat ion. , Run 3;-- - -- Run 4;- -- - , Run 5; ..... Run 6;------, Run 7. Figure 14. Second-order paranmeterization for the temperature dissipation equation.- , Run 3;- - Run 5; ------, Run 6;------, Run 7. 195

e...... · . * * e - e e ea * G. Newxman the simulations. In fact, we find similar asymptotic results fort ,'P and R from all of our simulations. We observe that changes in the levels of RX and PA only influence the rate at which predictions evolve to self preservation. Additionally, in all of our simulation results, the decays of q2 and 62 asymptote nearly to power law decays where the decay exponents are nearly unity for both quantities. These decay trends are consistent with the asymptotic approach of 4 and i toward values in the region of 4.0 as can be seen by setting n and n equal to 1.0 in equations (4.15).

The apparent insensitivity of the asymptotic values of R, 4 and

B to changes in R. and P, implies that these quantities are not influenced by changes in the levels of the scalar and mechanical molecular diffusivities in the TFM model. This phenomenon should probably be exhibited in real turbulence only for the case of large Reynolds and Peclet numbers, although lack of large Reynolds number data precludes direct evaluation of this premise. The velocity field large-eddy structure in high Reynolds number turbulence (which contains several decades of eddy sizes) is thought to lose only a negligible amount of eddy energy as a result of direct viscous dissipaton. Instead, the energy of the large scales is said to be depleted predominantly through non-linear exchange of energy with slightly smaller eddies as a result of vortex stretching, and the energy in the large-scales is 'cascaded' down to the smallest scales by numerous non-linear exchanges of this form among eddies of slightly differing sizes. In addition, the evolution of the energy containing portion of the scalar spectrum is undoubtedly influenced mainly by the large-scale velocity eddies for the case of large Reynolds and Peclet numbers; and consequently, for this case of turbulence, the dissipation rates of both q2 and 62 may be considered to be approximately independent of the values of the molecular diffusivities. Thus, we may propose that the levels of R, k and Be should be fairly insensitive to changes in RX and P. for large Reynolds and Peclet numbers. Indeed, the form for i, (4.11), indicates that ' changes by about only 6% over the range R-10 to R2-o, although we note that the asymptotic value, p = 3.78 as R+oo, 196

A Test Field Model Study ......

is extrapolated from existing isotropic turbulence data which span R values only up to R-40.

The asymptotic value, P=4.0, exhibited in the simulations is in

fair agreement with the range of i values observed from the data over a fairly wide range of Reynolds numbers. This characteristic of the velocity TFM predictions suggests that our scalar TFM predictions for 1) and R may be fairly realistic, and hence they may serve to augment the existing (somewhat inconsistent) data. Thus, the single value of l=4.0 evidenced in the simulations may be indicative of the value appropriate for large Reynolds and Peclet numbers, and it may correspond fairly well with real turbulence at moderate RX and PX. Indeed, the value, l==4.0, agrees with most of the scalar-decay data to the extent that the predicted value for i agrees with the corresponding velocity- decay data. Additionally, the prediction that R=l from the simulations makes sense physically and also concurs with most of the empirical data. On the other hand, we acknowledge the need for further investigative study with the TFM regarding the second-order parameterizations. In particular, the TFM characteristic that R, Ad and i appear insensitive to changes in RA and P% deserves further attention. It is possible, for example, that TFM simulations would be dependent on the levels of RX and PX (at moderate RX, PX) at much larger integration times than those employed in our study, although the temporal period in our simulations provides for relative decreases in q and2 which equal those in the empirical data. On the other hand, the insensitivity quality of the TFM simulations may derive from a characteristic inherent within the Test Field Model equations. This possibility is being investigated analytically, but the work has not been completed to date.

We shall close with one final comment. Second-order closure models have proven to be good predictors of various complex, turbulence flow situations. For example, Zeman and Lumley (1976) present successful predictions of the rise of an inversion in the mixed layer of the atmosphere. In the second-order modeling approach, closure is effected by parameterizing higher-order moment quantities (in a hierarchy of moment 197

· · · ·*** * · G. Newman equations) in terms of lower-order quantities. Rational closure parameterizations are then developed through consideration of a hierarchy of increasingly-complex documented flows. In this manner, the numerous physical phenomena which may be evidenced in turbulence may be considered individually, and hence good parameterizations for the higher-order terms associated with these phenomena may be developed. It is noted in both Lumley and Newman (1976) and Newman et al. (1976) that the second-order models developed in those papers could undoubtedly be improved in the light of further information regarding homogeneous turbulence. Since models for homogeneous turbulence serve as the basis closure from which more complicated closures are developed, it is desirable that homogeneous turbulence models describe accurately the physics of homogeneous turbulence. In the latter part of this section we have evaluated the second- order parameterizations appropriate for isotropic turbulence containing a passive scalar; and, although our results are not entirely conclusive, they do perhaps in part augment the existing information regarding the second-order representations for this case of turbulence. Statistical theory models (such as the Test Field Model) may be employed for simu- lating other homogeneous turbulence flows, and we suggest that future investigation regarding the implications of statistical theory predictions for second-order modeling might well prove to be fruitful endea- vors. 198

A Test Field Model Study ......

FOOTNOTES

1. Our Langevin equation representation of the scalar equation of motion is modeled after a Gradient-Based, Markovian, Lagrangian History Direct Interaction scalar field representation given by Kraichnan (1970).

2. The velocity field TFM equations were solved with a code developed by Dr. J. R. Herring, and the equations of the scalar TFM were imbedded into this code and solved concomitantly. Further, the convolution sums in the scalar and velocity TFM equations were evaluated by implementing a second code developed by Doctor Herring. The availability of these codes is gratefully acknowledged.

3. Recent experimental work by Warhaft (1976) provides further indi- cation that R=l in decaying, heated grid turbulence; and additionally, the work explains in part some of the disparities in the existing literature. 199

·* · · · · . G. Newman

REFERENCES Batchelor, G. K. (1956) The Theoty of Homogeneous TuAbulen ce (The University Press, Cambridge).

Herring, J. R. (1976) Private Communication of Unpublished Work.

Herring, J. R. and Kraichnan, R. H. (1972) Comparison of Some Approximations for Isotropic Turbulence, Statistical Models and TutbueQnce, 148-194. Springer-Verlag.

Kraichnan, R. H. (1958) Irreversible Statistical Mechanics of Incompressible Hydromagnetic Turbulence, Phyz. Rev. Vol. 109, 1407-1422.

Kraichnan, R. H. (1961) Dynamics of Nonlinear Stochastic Systems, J. Math. Phys. Vol. 2, 124-148.

Kraichnan, R. H. (1964) Decay of Isotropic Turbulence in the Direct Interaction Approximation, Phy4. Fluids, Vol. 7, 1030-1048. for Kraichnan, R. H. (1965) Lagrangian History Closure Approximation Turbulence, Phy4. Fuids, Vol. 8, 575-598.

Kraichnan, R. H. (1970) Notes on Lagrangian History Amplitude Models, Communicated by J. R. Herring.

Kraichnan, R. H. (1971) An Almost-Markovian Galilean-invariant Burbulence Model, J. F£uid Mech., Vol. 47, 513-524.

Kraichnan, R. H. (1971-a) Intertial-range Transfer in two-and-three Dimensional Turbulence, J. Fuid Mech., Vol. 47, 525-535.

Leslie, D. C. (1973) Development/s in the The.oy of TuAbutence (Clarendon Press, Oxford).

Lumley, J. L. (1970) Toward a Turbulent Constitutive Relation, J. FruLd Mech., Vol. 41, 413-434.

Lumley, J. L. and Newman, G. R. (1976) The Return to Isotropy of Homogeneous Turbulence, Submitted to J. Fluid Mech.

Mills, R. R., Kistler, A. L., O'Brien, V. and Corrsin, S. (1958) Turbulence and Temperature Fluctuations Behind a Heated Grid, NACA Tech. Note, No. 4288

Monin, A. S. and Yaglom, A. M. (1975) Stati6ttical Fuid Mechanics Vol. II (J. Lumley, ed., M.I.T. Press, Cambridge). 200

A Test Field Model Study ......

Newman, G. R., Launder, B. L. and Lumley, J. L. (1976) Modeling The Decay of Temperature Fluctuations in a Homogeneous Turbulence, To be Submitted for Publication.

Orszag, S. A. (1974) Lectures on the Statistical Theory of Turbulence, Flow RaeQcach Repo.t, No. 31.

Tennekes, H. and Lumley, J. L. (1973) A FiUAt Course in Tuabutence (M.I.T. Press, Cambridge).

Uberoi, M. S. (1963) Energy Transfer in Isotropic Turbulence, Phys. FluLids , Vol. 6, 1048-1056.

Van Atta, C. W. and Chen, W. Y. (1969) Measurements of Spectral Energy Transfer in Grid Turbulence, J. Ftui.d Mech., Vol. 38, 743-763.

Van Atta, C. W. and Yeh, T. T. (1973) Spectral Transfer of Scalar and Velocity Fields in Heated-Grid Turbulence, J. Flutid Mech., Vol. 58, 233-261.

Warhaft, Z. (1976) Private Communication of Unpublished Work at the Pennsylvania State University.

Zeman, 0. and Lumley, J. L. (1976) Modeling Buoyancy Driven Mixed Layers, To Appear in J. Atmo4. Sci., Vol. 33, No. 10. 201

. .a. a . .a . . .00 .0 0.. 6 0 0 lbG. Newman

203

PROCESSING, DISPLAY, AND THE USE OF THE RESULTS OF A NUMERICAL MODEL by Joelee Normand University of Oklahoma Grant Branstator, Scientist

INTRODUCTION In order to interpret data output from large numerical models, effective such as those models used to simulate the earth's climate, an method of presentation is needed. If carefully designed, a graphical display of model output can help the scientist, as well as the unsophis- The ticated observer, easily assimilate large quantities of data. a processor project reported on in this paper involved the preparation of for data output by a new numerical model at NCAR.

DESCRIPTION OF THE MODEL The advantages of representing fields in a model of the atmosphere of in terms of coefficients of a set of orthogonal functions instead 1960). as grid point values has been recognized for years (Platzman, (1970) However, not until the work of Eliasen, et al. (1970) and Orszag models have methods been developed which allow these so-called spectral to be efficient. A global spectral multi-level primitive equation these new model patterned after Bourke (1974) which takes advantages of methods has recently been developed at NCAR. Set in sigma coordinates in the vertical, the momentum, thermo- as dynamic, continuity and hydrostatic equations in this model are follows: 204

Processing the Results of a Numerical Model ......

dV dt -fk x V- V - RTVlnp* + F

dT RT a v( Vv))- dt Cp v ao

d lnp, = -V - dt and RT RT 9a ar

Here V is the horizontal wind, f is the Coriolis parameter, k is the vertical unit vector, 0 is geopotential height, R is the gas constant for dry air, T is temperature, p* is surface pressure, F is the horizontal frictional force, C is the specific heat at constant pressure for dry air, a = where p is pressure, a = and V is the horizontal gradient operator. In the model these equations are reformulated in spherical coordinates and then expanded in the horizontal in terms of surface spherical harmonics

iXm yn (,X) = pm (sin()e m n where ( is latitude, X is longitude and P (sink) is an associated n Legendre function of the first kind normalized to unity. The vertical coordinate is handled discretely. The model is advanced forward in time by using an extension of the semi-implicit time integration scheme of Robert (1969) with all linear terms being handled implicitly. 205

*· * · · · . · e * a *· J. Normand * .,·... · ···. 0 c * e *

Figure 1.

ZONRL MERN OF ZONRL WIND

.OQ

.2s

0 L N*1 0o "et LRTITUEa -. LRTITUOE

ITERRTIIN= 30 DRY 5 nWUR 0 MINUTE 0 CASE NL. S009 10 LEVELS FECAST Fft12ZIOEC67. FIRECAST RUN IN 06/23/76 206

Processing the Results of a Numerical Model ......

To date the model has been used as a short range prediction model with only a few physical processes parameterized. The processor described in this report is for use in possible future climate simulation runs.

THE PROCESSOR'S USES A means of looking at fields produced by numerical models of the atmosphere which scientists have found useful is to examine north-south cross-sections of these fields. A processor was written that could read the output from the NCAR spectral model and produce cross-sections of various fields. In its initial state, the processor was coded specifically for retrieving and displaying the u-component of the wind at a given time in the model integration. It was found to be a simple matter, however, to generalize the code so that it could read temperature data, the v- components of the wind, or other variables output by the model, and to produce cross-sections of these fields either at a particular instant in the model run or time-averaged over any desired interval in the simulation. Though first written to produce a cross-section at any given latitude, in most cases a zonal average of the fields is desired in order to compare simulation efforts with observed conditions, and this extension of the code was easily implemented. For use in long term simulation of the atmosphere (e.g., 30, 60 or 90 day forecasts), the time mean produced by the model and displayed by the processor can be compared to long-term observed means. Examination of differences between observed and simulated means facilitates improve- ments in the model. For example, the value of the surface drag coefficient or the parameterization of diffusion can be altered to see if such changes make the simulated atmosphere more closely resemble the real atmosphere. Simulated conditions which can be compared to observational data in the displayional of the al mean of the u-component of the wind (see Figure 1) include the location and intensity of the subtropical jet and 207

...... ·* · ·...... * a la J. Normand

Figure 2.

Z0NRL MERN OF TEMPERRTURE

.66

1 .6 / \\i v\ 29 5.**S. Wee ge .5e~

#.1»\ .t 1 .I t.11- .I -60. I -91*0 LATITUOE

ITERRTIBN = 360 AY 5 mBUR0 MINUTE 0 ,CISE N S0094 10 LEVELS FBIECAST FRIZ1ZI40EC67. FIRECAST RUN BN 06/23/76 . 208

Processing the Results of a Numerical Model ...... polar night jet and the simulated strength and shape of the tropical easterlies. In the cross-section of the time averaged v-component of the wind, it is apparent whether the location of the Hadley Cell approxi- mates reality. In comparing temperature data from the model to observed data, the position of the tropopause can be checked as well as the overall static stability of the simulated atmosphere (see Figure 2). At times it is useful to look at a cross-section through a particular longitude instead of at zonally averaged quantities (see Figures 3 and 4). From such a display one can examine the affects of local conditions such as orography or land-sea contrasts.

DESIGN OF THE PROCESSOR The goals in writing the processor were primarily ease of use and reliability. Input is therefore free-format and by name, and the processor makes sure input is valid before attempting to plot the data. Another feature of the processor is that if an error is discovered, it will go on to the next plot requested by the input rather than simply stopping. Efficiency was given a lower priority since the processing time required for each plot is not large. For example, the central processing time required to produce one frame is approximately .8 seconds on NCAR's Control Data 7600. The processor reads the history tape containing the data at time intervals specified by the programmer. By reading information off the first record the program proceeds to locate the file which contains the desired time period in the series. Since the input data is arranged as east-west cross-sections, the processor must reorder the data into north-south cross-sections. The plotting routines used are contained in the NCAR Software Support Library as described in the Library Routines Manual and Vol. 3 of the NSSL manuals. These routines are used to produce labeled contour maps of cross-sections in the vertical through a single longitude or of a zonal average of all longitudes for a desired field. Information for the labels on the plots produced is 209

· ·* o o * · · · e · · ·* J. Normand * * · · · · · · · · e ·

Figure 3.

N0RTH-S0UTH CR0SS-SECT IN OF TEMPERRTURE THROUGH -100.0 DEGREES

S.,0

.11 .a

-9,., #.0 0M., MS.I SoS *HeO LATITUDE

ITERATIIN 360 OY 5 NMBU0 MINUTE 0 CSE NO. S009 10 LEVELS FIECAST FRI121Z40EC67. FORECST R BUSN06/23/76 210

Processing the Results of a Numerical Model ......

Figure 4.

N0RTH-S0UTH CR0SS-SECTION 0F Z0NRL WIND THROUGH -100.0 DEGREES

.05

.4S

Ln m

".0 .I so.IO I. -50.0 *90.0 LATITUDE

TERATION 360 DRY 5 HOUR 0 MINUTE 0 CASE N24 I 10 LEVELS F RECAST FRB12Z14EC67. IFBOICRST RUN ON 06/23/76 211

* J. Normand e ·*·*-*ee ··*· e s · · ··· · ·· · extracted from the header record of the model output. In addition to producing plots, a one-minute movie was produced of the time evolution of the zonally averaged u-components of the wind. Data on the model output tape were recorded at twenty-four hour intervals, so that linear interpolation was required in order to produce a more smoothly moving time sequence. Complete with titles, the movie presents a unique display of how the model atmosphere evolves with time.

INTERDISCIPLINARY JUSTIFICATION FOR PARTICIPATION IN THIS PROGRAM One of the reasons for my desire to learn this display technique at NCAR was because I have seen the successful use of a 3-dimensional computer movie in which the output from an air pollution diffusion model was displayed (Shannon, 1976) using the EPA required emission data from Tulsa, Oklahoma. This movie was well received by a municipal planning group from that city. The success of this demonstration is convinced me of the use of this type of presentation technique. It possible that an interdisciplinary student could perform a great service for the urban planner by learning to apply this presentation technique to output from other scientist-engineer produced models which the non- under- scientific planner is currently using and is having difficulty standing. Some models of current interest are hydrology/flood plain of models, transportation models, urban demographic models, and a host urban geographic analyses (e.g., socioeconomic and demographic configu- rations). As an interdisciplinary student I was encouraged to attempt a pro- was ject in the Summer Fellowship for Scientific Computing at NCAR. I computer told the object of the program was to facilitate my use of the that for future research efforts. However, when I arrived I discovered be pre- I was expected to be experienced in programming techniques and pared to engage in scientific research using the computing facilities at NCAR. Notwithstanding the many people who were patient with my limitations, I found a distinct problem of communication in my efforts 212

Processing the Results of a Numerical Model ...... to learn how to use the computer and in questions concerning research done at NCAR on problems which concern the outside world, e.g., how to solve problems of depleted grain stocks, polluted environment and urbanization techniques which have created artificial climates in our cities. One question which began to haunt me was: what has atmospheric science to do with real problems in the world, and why at this marvelous facility with all the latest in research capabilities do I get the impression that people still cannot communicate their needs to one another, no matter what the level of education and technology? Speaking of the "interface" needed between the English language and machine language, and technical jargon in general, a professor of English has stated: "To prevent extreme loss of information at the boundary between two social environments requires not only the full use of the powers of [English], but something more difficult to attain..." "The something more is for the sender of messages to take into account the receiver's circumstances, basic assumptions, ignorance and knowledge. The main reason for poor transmission across environmental boundaries is failure to translate beforehand into the terms of existence holding sway at the other end." (B.R. Schneider, 1974). This author goes on to say that what we need to bridge the gap is imagination. I would add, a real interest in the good of our own individual existence is needed. Schneider's efforts at cataloguing a large literary work on the computer produced a book of some 240 pages of frustrations during his experiences as a layman in "Computerland". The conclusion he reached was that man can now talk to his machines better than to his own species. My experience in the Computing Facility at NCAR has also brought about some questions as to the need for perpetuation and multiplication of specialized occupations which, in turn, multiply the difficulties already existing in man's communication with his fellow man. The research and effort put forth in the atmospheric science work at NCAR are admirable, and the efforts are awesome even to the unsophisticated 213

. . . .·.· . ·. · · * ao * · ae J. Normand

will observer and student. However, the question keeps returning: How all the marvelous research be applied to problems which have rapidly become critical in the past decade? How does all the technology get transferred to aid in solving the imminent food shortage we hear about? Where are the people who are willing to transmit the science needed to solve problems of the environment to the agencies who have to deal with the problems from day to day? Why are there no programs designed to the teach interdisciplinary students how to bridge the gap between scientist and the policy maker and make obvious the application of the research being done to problems which affect all of us? How can con- crisis cientious scientists continue to ignore the facts of the energy and depletion of our resources? Journals abound with articles arguing on how effectively we are destroying our atmosphere, and yet arguments are continuing on whether or not there really is a crisis anywhere of such proportion critical enough to warrant concern.

At a research facility such as NCAR, I expected to find discussion of which problems should receive top priority, one of which should be the transfer of information. There is a need for the efforts devoted to pure research done at NCAR. My concern is that there is no apparent used facility for the transfer of the results of such research to be in practical applications. I expected, at least, productive efforts solu- to cooperate with other disciplines to determine priorities and tions, and programs designed to educate the student who will become a user of the facilities of NCAR on the most effective means of com- deal municating his efforts to those who make decisions and have to at with their social impact. There are certainly those individuals NCAR and elsewhere who have struggled for such a cause. But so far, con- I have not seen any concentrated effort to show how the research which ducted at a national institute of the sciences produces results directly aid the man who has to foot the bill and help in solving his most urgent problems. 214

Processing the Results of a Numerical Model ......

REFERENCES

Bourke, W., "A multi-level spectral model. I. Formulation and hemispheric integrations," Monthly Weather Review, 102, pp. 687-701.

Eliasen, E., Machenhauer, B. and Rasmussen, E., "On a numerical method for integration of the hydro-dynamical equations with a spectral representation of the horizontal fields," Institut for Teoretisk Meteorologi, Kobenhavns Universitet. Report No. 2.

Library Routines Manual, NCAR TN/IA-67, Atmospheric Technology Division, National Center for Atmospheric Research, March, 1975.

NCAR Software Support Library, Vol. 3, NCAR TN/IA-105, Atmospheric Technology Division, National Center for Atmospheric Research, March, 1975.

Orszag, S., "Transform method for calculation of vector-coupled sums; application to the spectral form of the vorticity equation," J. Atmos. Sci., 27, pp. 890-895.

Platzman, G., "The spectral form of the vorticity equation," Journal of Meteorology, 17, pp. 635-644.

Robert, A., "Integration of a spectral model of the atmosphere by the implicit method," Proc. WMO/IUGG Symposium on Numerical Weather Prediction, Tokyo, 26 November - 4 December 1968. Japan Meteorolo- gical Agency, Tokyo.

Schneider, B. R., Travels in Computer Land, Addison-Wesley Publ., Phillipines, 1974.

Shannon, J., "Tulsa Air Pollution," Computer Movie, Ph.D. Dissertation, Dept. of Meteorology, The University of Oklahoma, Norman, Oklahoma, 1976. 215

00 0 0 0 e * ·e e .e0 a e e 0 e 0 0 a *e e e e a 0 J. Normand

217

AN ADAPTED ONE-LAYER MODEL OF THE CONVECTIVELY MIXED PLANETARY BOUNDARY LAYER

James Thrasher University of California at Davis Jim Deardorff, Scientist

ABSTRACT by A steady state mesoscale numerical model which was developed the Great Lakes Ronald Lavoie (1972) to simulate lake effect storms near in winter is extended to allow the representation of time dependent scheme with phenomena. An explicit, "leapfrog" numerical integration filter elimi- second-order spatial differencing is employed. A simple Entrainment nates time splitting of solutions at successive time steps. in the present processes at the top of the mixed layer are parameterized effects are model, as well as horizontal eddy mixing. Although these may be relatively minor at any instant, their time integrated effects version of important. Some preliminary results of a one-dimensional important terms the present model illustrate how it handles the more in the governing equations.

INTRODUCTION The model to be presented here is an adaptation of a one-layer The simulation of the PBL associated with "lake-effect" storms. and was shown original model was developed by Ronald Lavoie (1972), PBL accurately. to model the gross dynamics of the neutral to unstable weather pattern His primary objective was to model a pseudo-steady state integrating by starting with relatively simple initial conditions and internal the prognostic equations with inflow characteristics and showed little forcing terms held constant until subsequent time steps minor variations change. His model reflects the influence of relatively mixed layer in surface terrain on the characteristics of an overlying from the of air. Intermediate results are invalid as actual prognoses adaptation to initial conditions. Here we wish to develop a suitable of the PBL with this model whereby one can simulate temporal evolution some validity. 218

An Adapted One-Layer Model ......

THE PHENOMENON The type of system we wish to model is in most respects very much like the Lake-effect storms, except that condensation of water vapor. and precipitation will not be considered. The- structure of the lower atmosphere is divided into three distinct strata. The lowest layer, commonly referred to as the surface layer, or friction layer;, is prescribed to be 50 meters thick. It is the stratum of the atmosphere in which surface- characteristics almost completely dominate the dyna-. mics. The mixed-layer is driven from below by the surface layer. All' fluxes into or out of the mixed layer from below depend upon the specified characteristics of the surface layer. The second stratum is the mixed layer itself. It is characterized by strong mixing and :neutral stratification, uniform horizontal wind speed and direction with depth, and a sudden discontinuity of potential temperature at its top. This. is the part of the atmosphere which accounts: for by far the majority of surface-based pollutant transport by horizontal winds. The upper level is characterized by constant geostrophic wind shear and constant- potential temperature gradient. A schematic of this situation is showni in Fig. 1. We hypothesize that this atmospheric structure is quite common, especially during the daytime hours in- spring and summer in many places. Figure 2 shows. atmospheric radiosonde data; taken at:Davis, California, during April and May, 1967. Note that there appears to be a definite unstable layer near the surface- overlain by a more or less neutral'. layer with slowly varying wind speed and direction. Above. the "hneutralT. layer, there is a thin, very stable or inverted temperature. gradient; followed by air of relatively constant static stability. Lavoie's. model assumptions are chosen for this research because only one level is involved with the, prognostic variables, instead of many levels in the vertical as is the case- with most primitive equation PBL models. Thus, we may increase horizontal resolution many times. while sacrificing relatively little by the vertical homogeneity assump- r _~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

a con-vrii C wo it f l 6~~~, c

/ =

VstalAu tyg i

o-tk df Ltet i *tuyn top t osstOr6A uvd~ Ot 7> f Ic8Qhfe

I \run~iSt~s~-ec%- e~e"- ur I., - - Je- .W -the iovey siov e Oc- oH(.Sh9 C I~ C3a -II

0 ...... j . -~FL-FLrN EL~VtflOI~'?2, -

Tbrl 05C V c~~

I CBovis-irw LV7 #o Eurvoo ] A I P3 ion o the tmodel PBL. cn Figure 1. Schematic representat PD Y 220

An Adapted One-Layer Model ......

Figure 2: Radiosonde temperature data taken at Davis, California.

6 /\A , s' - -Z-117I -A - 7- ,'/f 2'ftI'/ /5(

X/ A XV~~~'S~,j

Q'~~~~~~~~~~

W~~~~~38

50, XXI~~~~~~

-2 ~~;.,,, f~A X",, -c~-~j-~.[~:c-L~(~\·-V/ L t: A; /x~~~~~

Z5 30 35 40 45 50 55 60 65 70 75 FAHRENHEIT TEMPERATURE SCALE

I 221

e J. Thrasher tion for a given amount of computer time. Such a model will give an economical means to assess horizontal pollutant transport in mesoscale systems.

GOVERNING EQUATIONS

We now follow Lavoie's development of the governing equations as applied to an air parcel within the mixed layer with mention given to adaptations resulting from the present work. The Lagrangian time derivatives of the prognostic variables are as follows (with the exception of water substance):

dv- = k x fv - aVP + V Kv(1) dt

a dO cO- ^ + V - KV6 (2) dt C T aZ p

1 d = V . v + (3) a dt $Z where v is the horizontal velocity vector. The hydrostatic assumption is made:

a UP g (4)

and we include the definition of potential temperature used in subsequent analysis:

= a p/DK-1 (5)

Note the inclusion of horizontal eddy mixing terms. Lavoie neglected these because his numerical integration scheme (upstream, forward-time) introduced artificial viscosity amounting to about two orders of magnitude larger than physically realistic values sought in the present effort. The symbols K, T, and Q are respectively, horizontal eddy diffusivity coefficient, vertical flux of horizontal momentum, and vertical flux of thermal energy. The other symbols are standard and are defined in the symbol table in the appendix.

Assuming vertical homogeneity of wind velocity and potential temperature in the mixed layer, the parcel equations may be easily 222

An Adapted One-Layer Model ......

integrated over its entire depth. Having done this, we will have a set of equations which describes the behavior of the mixed layer as a whole.

A. Horizontal Momentum Equation This is the most difficult equation to average vertically and the final result is perhaps the weakest theoretically. Letting Z represent s the height of the top of the surface layer above sea level and denoting the height of the inversion by h, we obtain the mixed layer horizontal momentum equation upon integration between these two levels.

dv _ k x fvh-h' dt k fvZ aVPdZ - a(Th - )] (6)

We will assume a linear decrease in vertical momentum flux from its surface value to Z to 0 at z = h. That is to say that there is no s transfer of momentum through the inversion from the mixed layer as a result of surface friction. In certain cases this may be a weak assumption because it is not uncommon to see a very sharp change in momentum immediately above a strong inversion, however, that may possibly be considered in a later study. The surface value of momentum flux is derived using the bulk aerodynamic method due to Priestley (1959). T = CDrhos vlv. The drag coefficient, CD, is a function of surface roughness only in Lavoie's work, and it will be here also. In order to evaluate the horizontal pressure gradient integral, consider the pressure to be known at a level H above the top of the mixed layer. Integrate from this point down to a level z within. This gives pressure as a function of height within the mixed layer. Using the hydrostatic equation and the definition of potential temperature to eliminate a, the vertical gradient of pressure may be expressed in terms of the potential temperature. K tP -t = - gCp0P - (7)

H- h OH z- /K , a P(z) = H - - o iH-n H-+ - , where --Oh. (8) o H cO H-en 223

· · · ·. e*** · o - e J.Thrasher

Approximating ln(eH/O) h by the first two terms in its Taylor Series expansion and substituting for alpha from (5) gives the horizontal pressure gradient as a function of height without explicit reference to P except PH which is prescribed.

= - (aVP) + 29g(H-h)v(H+h) + - Oh+H Vh + -aVP(z) (oP)H + 0+e °H H H h 2 h 2

g(h-z) . (9) g (9)

This expression is easily integrated through the mixed layer and together with the definitions

F - (aVP) (10) H H H

and 0 +0 - 3- E h +H2 HJ (11)

yields

dv k x fv - F H g +) V + (0-)Vh + g (h-s) dt 0h eH

CD (h-Z )Ivv + V KVv. (12) (h-Zs)

In this equation, it is evident that the choice of H strongly affects the magnitude of the third term. It is argued that mesoscale disturbances, while being felt to great heights, tilt upward with height in such a manner that pressure perturbations which would be large from vertically-aligned disturbances turn out to cancel each other. Consequently, pressure disturbances near the ground are probably small. Therefore, H need not be a great deal higher than h. Note the thermal wind equation in the following form: 3v av (13) V-0e= fk x -a - f f(1

Here v is the geostrophic wind in the layer above the inversion and

i represents the rotated shear vector. If one assumes i to be prac- tically constant in the stable layer, then applying the definition (10) at the initial height of the inversion, h i , we have 224

An Adapted One-Layer Model ......

FH F = -(H -hi)fi (14) or

FH + (H-h)fl = F i + (hi-h) (15)

Substituting this expression into (12) eliminates all reference to H except in the fourth term which represents the restoring force due to the deformation of the inversion surface. Here for the purpose of defining 0, Lavoie defines:

H -½(h + h ). (16) i1 m This expression used for the mean effective potential temperature in the overlying stable layer is:

6 = (0h + 0H )/2 = h + (h - h).

Here r is the vertical gradient of potential temperature above the inversion. This has no rigorous physical basis, but it is not unreasonable. Equation (12) is now rewritten in its final form:

v = v ' Vv k x fv - Fi - (hi-h)f + -f- (hh -h h + 3t I i h 4 m i

-(h- Z - ivv + V KVv. (17) s)v (h-Zs)

The prognostic variables are v, h, hm, 6 and h (not included in the earlier work as such).

B. Potential Temperature Equation for the Mixed Layer

Vertical averaging of (2) through the mixed layer yields

_ __-- 1 W + at -CV ve -Cp T (h-Z) (Qh-Qs)S KV (18) (K' = 1.3K). (18) Lavoie takes Qh to be negligible, but here we will parameterize it through some empirical results obtained from mesoscale observations in convective boundary layers. We shall take Qh = -20% Qs Qs in turn is parameterized by the same method as that for momentum. We will express the vertical heat flux out of the surface layer as: 225

* -***** J. Thrasher

Qs = c psCD iv(e- ), =p spC V o 0) (19) D D The heat flux at the inversion represents that due to entrainment of warmer air from above the inversion. We obtain as the final form of the prognostic equation for potential temperature the following:

e = v Ve + 2 CD ivli (e-e0) + V KVe (20) Dt oh-Z where the horizontal diffusion term is additional to Lavoie's original model.

C. The Mass Continuity Equation

Averaging (3) through the mixed layer yields: -1 d Wh-W 1 da = V * v + hs 21) dt h-Zs a s

On the mesoscale, the second term is dominant, so we are justified in using the approximation (Haltiner, 1971):

1 da 9 w (22) adtW C, where C is the local speed of sound. This yields, upon vertical averaging through the mixed layer,

(23) dc1da ' - (W + W)

Where W is defined as v -VZ,. Solving for Wh, we get

Wh = W 1 + g(h-Zs)/2c 2 - (h-Z)V v (24) h 1 g(h-Zs) 2c At this point we depart from Lavoie's analysis by considering the effect of frictional entrainment of stable air aloft into the mixed layer. (James Deardorff, private discussion, July, 1976) suggests the following parameterizations for entrainment into the top of the mixed layer, depending upon whether a zero- or first-order discontinuity in potential temperature exists at that level: W .25W s (25) e - A z=h 226

An Adapted One-Layer Model ......

for cases in which a zero-order discontinuity exists, and

W = 8's We h (26) hp for cases in which only a discontinuity in the slope of the potential temperature graph exists at the inversion. The latter condition implies encroachment. Lavoie did not use (25), and used (26) only when A9= -=hh -<0. The prognostic equation for the height of the mixed layer may now be written: Dh = - v V h + Wh + W + V KVh. (27) Dt h e D. The Potential Temperature Immediately Above the Inversion, 9

In Lavoie's development, Oh was either held constant or else was allowed to go up as the potential temperature in the mixed layer increased once the two became equal due to heating from below reducing the magnitude of the inversion. Since we have included entrainment as a mechanism for heat transport and mass transport through the inversion, h may change even though there is a zero-order discontinuity in potential temperature at that level. We obtain for the 8h forecast equation the following:

h = _ v . V9 + W r + V KVh (28) t h e h( This completes the set of equations used in this simulation study. Now the finite-difference discretization of these equations will be presented.

FINITE DIFFERENCE ANALYSIS OF THE GOVERNING EQUATIONS

A. Discussion In order to arrive at solutions of the equations which suitably represent the physical processes of interest, it is necessary to choose a numerical integration scheme which preserves quantities that are invariant in the analytical problem. For example, a numerical integration scheme which causes spurious perturbations on the inversion surface but nevertheless preserves the total momentum of the system would be very useful if all one wishes to model is the net horizontal transport without regard to the topography of the inversion surface. It would fail, 227

v ··· · c e a *e e eI · * · ** . J. Thrasher however, to describe accurately the characteristics of gravity waves which form at the fluid density interface. It is important to choose a numerical scheme which represents the physical phenomenon of greatest interest with the highest accuracy. Lavoie's objective is to model a steady-state phenomenon. Although his numerical integration scheme includes a very large numerical damping effect, the mesoscale disturbance he models is one which is continually reinforced by surface forcing functions. He does not mind that the intermediate transient features are inaccurately represented. If one wishes to model a diurnally varying sea breeze or mountain-valley wind systems, the realistic representation of transient features is highly desirable if not essential. In the present study, this is our aim. Therefore, we use a different numerical integration scheme than Lavoie. The grid used here has spatially staggered variables at the grid points. It is illustrated in Figure 3. None of the variables are staggered in the time frame. The integration scheme is second order accurate both in time and in space. The time integration is carried out using a leap-frog scheme. Because of the time splitting of solutions, which occurs with this scheme, a three-level filter due to A. Robert (1966) is applied every time step in order to eliminate waves of period 26t. The expression is as follows:

u*n = un + (un+l _ 2un + u*n-l (29)

where the starred value is the smoothed or filtered value. c is held as small as possible, usually about .05 in the present study. Since the method used in the integration is explicit, the Courant-Friedrichs- Lewy stability criterion must be strictly satisfied at all points on the grid. That is 6t< , where C is the maximum phase propagation gC g speed. In the model equations used here, sound waves in the horizontal are not allowed, so the maximum phase speed occurs with gravity waves. In a test run with an inversion height of 100 meters and a density difference of 3% between the mixed layer and the air above the inversion, the model produces stable results using three-minute time steps. The results become rapidly unstable if the mixed layer depth is 600 meters 228

An Adapted One-Layer Model ......

Figure 3: The staggered finite difference grid. o + h,e,ehWhW ,k A u

+÷ V

+- v

I 0------I-> Ji A-- --+--I p I - .... 4. -0 A-0- A--o--"A--'--O.... t ; J t '1 - s at el,tI 229

J. Thrasher ·a* · ** * ·e ·lb· * ··* e * ·* e ·e* e e e with other parameters unchanged. This is because gravity waves propagate faster in a deep layer than in a shallow one. The other important stability criterion which must be satisfied in leap frog time integrations is the diffusion time step limit,

6t < K

In the mesoscale systems which are of interest to the present effort, this criterion is more easily met than the C-F-L restriction. The space-differencing scheme used in the model is second order centered differencing. This formulation applied to advective terms as opposed to the upwind differencing method employed by Lavoie, and also by Pielke (1973), has the disadvantage of producing computational solution modes which travel upstream from their point of origin. For- am- tunately, these disturbances are of considerably shorter wavelength and computa- plitude than the physical modes which produce them and most of the tional noise is removed by the low pass filter mentioned earlier. The reliability of the model's representation of a physically induced disturbance in the computational results is relatively good for wavelengths greater than 4Ax. It should be mentioned here that forcing fields such as terrain and initial potential temperature are smoothed using a spatial filter corresponding to the temporal one. This filter was also used by Lavoie, namely (½, ¼, ½) * )

The leapfrog differencing scheme, if unfiltered, is shown by the Von Neumann stability analysis to be free of spurious computational damping such as occurs in upstream differencing schemes. This allows us to include horizontal diffusion effects in a physically realistic manner. Lavoie does not include this term in his set of governing equations because the numerical diffusion resulting from the upwind differencing scheme is one to two orders of magnitude larger than what is physically realistic for the mixed layer. In the present model we want to model the diffusion realistically, using a form suggested by Deardorff (personal discussion, 1976) where the horizontal exchange 230

An Adapted One-Layer Model ......

coefficient for momentum, K, is of the following form:

K=(g(W)s (h-z ) + V,13/) * 12(h-Zs) (30)

The second term in the brackets is just the friction velocity cubed 3 (u*).

B. The Finite Difference Equations Which Apply to Points Interior to the Numerical Grid.

The governing equations for the prognostic variables in the interior of the finite difference grid are easily put into discretized form. It should be noted that the frictional terms are lagged by one time increment relative to the other terms in the equations because the leapfrog integration scheme is unconditionally unstable if those terms are used at the same time level. The finite difference forms used in this study are as follows: n n n n u+1 ~n+ln-=n-ll +26t. ( +lj )2-~(U u i- )2 n u. j+l Uij-l =uiJ 45x - \vMi, j 2y - I (31 a,b) hn f 3vg v - (F i+j + -Fxl, ) / 2 + (hi,+ hi+lj h.- Mi6lj -jJ ' i+l,j)2 aZ

(31 c,d,e) n n -n -n + g 0ij+0i+lj' e.ij i+lj i+l, j- hi. (31 f) 6x eh + h ij i+lj n n _Z _z n n i+lj- sij si+l,j ei+l.j - ei, i+ j+ (31 g) n n x ei,j i~~~~j~+ i+lj ,1 fn n-l n n-1 + i- (U Ui - K (Ui U 1 1 (31 h)

1 0. n n-l 46y2 1(K.[ i +K i+l +K i j+l + KKi+l, )(UUl( ,j+ljn-l -

(Ki, j+ K j+ Ki j K j-)(U (uj3-u,j j+l)1 ijn- 1 } ( 3 1 i ) 231

* ** e * ·*J. Thrasher

The subscript M in terms b and c refers to the mean value of v taken over the four nearest points at which it is defined. VM=-¼(v4 +v +v +v .. ) VM = ( j+ i+l j+ i,j-l+ i+l,j-1

This is the value of v which applies at the location of uij. In the prognostic equation for v, an analogous uM is defined.

M = (i-lj+l+ Uij+l+ Ui-l j+ i-l ,j-l) The terms F and Fy in (31) and (32) are the x and y components of

the large scale pressure gradient force, Fi, defined in (14) and (15). The y-momentum finite difference equation is: n vn vn (vn 2_ (v 1 2 a,b) v. V. + 26t -U 1 (32

n - 2 0 - n nnug f 3 d f- - (Fyi,j+l+ Fyij)/ - (hi j+ hi h hi (32 cde) e. .+ - -h 1 1 h. i j + i ij ijl gn + (32 f) hi,j hi, j+l 6

hi hi,j+l-j+ Zi x-iZji, ... __ (32 g) g I+i i.! (32 g) + n + en y +ij ij+l

rh( n-l x2 j i+l,+ Ki, j+l+ Ki j+l i+l j (32 h) + (k,46x 2 5i i+lj i j

(Ki,j+ K ij++Vi K -l+ Ki-l j+l n(vi+- i l n- J

+ L-[K( - viv )n1 Ki- v n-v1 (32 i) Ly ,j+l ij+l ij i j i(3 3

In the computer program, mneumonics for the terms a through i are: a. ADVX b. ADVY c. COTERM d. pGRAD e. SHEAR f. HPRTRB g. TMPFRC h. DIFFX i. DIFFY. 232

An Adapted One-Layer Model ......

The forecast equation for h is:

n+l n-1 _ I 2 u n n n )+n n = + 2 6 t h ij ~ i i+l i- ij) + jUil, h -hil C33 a)

n (hn n n (33 b) -CD. . .- U Vj V

1i J 1 n n + ( j (Z+l j Z i) + U (Zl,jZ j- Z V26. (Z Z,- i v.)+ (

26y Lj i,j+l ) ,j-1 Z l) (33 d)

X 1 + CD. 1 -CD. 1,j ( n-) + We. + V * KVh (33 e,f) 1i,j Here Di . represents the depth of the mixed layer, h. .- Z. where Z represents the level of the top of the surface layer. C is the constant g/2c 2 . The eddy mixing term, (33f) is formulated as follows for the x-component.

D 2h 1 - K 7(- + K.)(h - h. ax ax 26x 2 i+l j+ i i+l J i j)

-(K j ij+ K.-li-l,j j )(h LJi 1i-l i, j)

The term - K is formulated in a completely analogous fashion. The horizontal eddy diffusion terms for the 6 and eh equations are computed the same way as V * K h. In the program, these terms are calculated by a call to DIFFUS(A,K). The eddy coefficient terms are updated each time step in a call to EDDYK(CDIJ,K). Potential temperature within the mixed layer is forecast using the finite difference analog of (20), which is:

en+ = enl + 26t same form as 33 a, b (34 ab) 233

..... *** J. Thrasher

n- 1 n-1 n- 1'2 l.2GD1.2C' ¼(u +U ) + ( + vi, + ,j i,j n(hn-z 1

(34 c) ( - en-1) 0i, j 1,

+ V · KVe-} (34 d) Finally, the potential temperature at the bottom of the stable air above the inversion is calculated in the following manner:

8n+l = en-1 + same form as 33 a,b (35 a) 1,j hi,j (35 b,c) +We n r + V KVh for This completes the set of discretized prognostic equations the grid interior.

C. Boundary Conditions The boundary conditions in the problem are handled through calls to the subroutines LWRUPR and LFTRGT. These cause the boundary conditions on the prognostic variables to be met at both the lower and (I=1, upper boundaries (J=l, J=N) and the left and right boundaries I=M), respectively. At the suggestion of J. Klemp (Private discussion, NCAR, 1976), boundary conditions on u, v, and h are handled as follows: the Let c be the wave propagation velocity of a disturbance approaching g then boundary. If the feature is approaching the boundary (c > u, v), it is allowed to exist. If the disturbance is moving away from the pres- boundary, then u, v and h are set to either constant values or a cribed function which defines the variable at an inflow boundary. allowed Figure 4 illustrates the conditions under which the waves are the to exit through a boundary. This procedure will be illustrated for boundary and case in which a disturbance in u approaches the right hand u + c > 0. g 234

An Adapted One-Layer Model ......

Figure 4: The Finite Difference Grid. Conditions under which disturbances are allowed to exit through the boundaries.

Z~"WA/ Oh. 0

Nh) II H h r 0

Y 0

Figure 5: Variables v and h surrounding a value of u on the right hand boundary.

hm-L. -i,j -- h,^.j -

I I r. 235

J. Thrasher

n+l UM 1 j is calculated according to

n n ,n+l .n- _ (n_ n + n -U Cgj .4L6 M (36) M.1j - lj L:J-=2 -2j J( UM-1,J)m ) + Cgx-l, d- (36)

CgM1i corresponds to the wave speed calculated from variables located differencing at the position of hMl j . This represents an "upstream" method at the boundary when disturbances approach it. If the disturbance is moving toward the interior of the region, away from the boundary, the following form is used:

U- 1 = u (t). (specified inflow velocity) M-lj M-l,j A similar equation to (36) is used for boundary values of h. h+1 n-l [-nL(n + C + CgM, ,j = hM~j M M-1M-, 9i,j MJ CgM-i,jM-19i MJ (37) 6x

0 The other prognostic variables 0 and h on the outflow boundaries are set to their values one grid point interior to the boundary. Other- wise they are set to prescribed inflow values.

PRELIMINARY MODEL RESULTS

Although the two-dimensional computer program has been written, extensive testing is yet to be done. The boundary conditions have been the most influential in determining the overall success of a test run. The two-dimensional problem has been rewritten as a one-dimensional channel problem in order to economize on computer time and allow testing various formulations of boundary conditions. 236

An Adapted One-Layer Model ......

This approach has given encouraging results. The boundary formulations presented in the previous section are working well for the channel flow problem. In this problem, the momentum equations alone are considered. Only terms involving the horizontal velocity and inversion height are evaluated, because a dimensional analysis of the equations of motion shows that amplification of small perturbations at the boundaries is greatest for these variables, if improper boundary- condition formulations are used.

The equations solved are a version of the shallow water wave equations (with no friction).

u _ u2 Ae h (38) at ax 2 8°ax

and ah a (=(uh)9)

Initial conditions are u.=5, h.=100, I-1,2,...,11 (i6).

6 = 2.5, h6 = 200

Relevant parameters are: .04, -S= .25. 6x e The integration remains stable beyond 50 6t with no signs of boundary-originating errors. Apparently because of the low-pass time filter (29), a slight decrease in the overall momentum of the system is observed. The results of this test case are illustrated in figure 6. Notice that an apparent "wake" in both u and h forms and widens with time. The upstream propagating crest in both u and h moves more slowly than the downstream propagating crest as would be expected in a physical channel. When the initial depth is 600 meters, the waves are unstable. This is probably because the greater depth allows faster moving waves which cause the courant number to exceed unity. The time step for a typical mesoscale problem with an inversion height of one kilometer, an inversion magnitude of 3K, and a horizontal grid spacing of 1 km, is about 30 seconds to one minute, not unreasonable for a model using explicit equations. 237

· · · 0 @ @· @ @ @· ··· · · ···· ··@ @ @ · ·* e J. Thrasher

Figure 6: Numerical solutions of the shallow water wave equations with initial perturbations of +100 meters in the h-field and -2.5 m/sec in the u-field both at point 1=6.

~--~ 50 tr t--~---- /2<0 ./_------_

kI Io N-1

10R000\

3100BoO

31000 - 238

An Adapted One-Layer Model ......

CONCLUSIONS

The mesoscale model developed by Ronald Lavoie has been written in modified form and discretized through an explicit numerical scheme which allows transient features as well as standing features of mesoscale flows to be simulated with reasonable accuracy, at least in one dimension. Extension of the model to two dimensions will be achieved by merely inserting slightly modified boundary condition formulations in the existing two-dimensional code. It appears that this model may be nondimensionalized by appropriate scaling parameters and applied to a wide variety of problems. 239

*...... e.e.. **** ***** 0 J. Thrasher

SYMBOL TABLE

CDICD Drag coefficients for momentum and heat

C Ratio of gravity to 1 the speed of sound in air squared c The speed of sound in dry air at 293K. C Specific heat for dry air at constant pressure. f Coriolis parameter. g Acceleration due to gravity.

H "Undisturbed height," also subscript referring to this level. h Height of the inversion, also subscript referring to the top of the inversion. h. Initial height of the inversion. h Maximum disturbed height of the inversion m i,j,n Spatial and temporal indices. k Unit vector along z axis.

K Horizontal eddy coefficient for momentum and heat

M,N Maximum i and j subscripts in the finite difference grid. o Subscript referring to the ground surface

P Atmospheric pressure.

Po Standard-level pressure, 1000 mb.

Q Vertical heat flux.

R Gas constant for dry air

s Subscript referring to the top of the surface layer.

t Time

T Kelvin temperature

u,v,w Three-dimensional wind components along x, y and z axes. 240

An Adapted One-Layer Model ......

We Entrainment velocity at inversion height.

V Horizontal component of velocity vector. x,y Mutually perpendicular coordinate axes in the horizontal plane. z Vertical coordinate distances.

Zo Height of ground surface.

~a Specific volume. r Vertical gradient of potential temperature above the inversion.

~6 Finite increment operator.

V Horizontal vector gradient operator.

K R/Cp

6o,9,'h Potential temperature at the ground surface, within the mixed layer, and immediately above the inversion, respectively. p Atmospheric density.

T Eddy stress vector.

1( ~ Rotated shear vector of the geostrophic wind in the upper, stable layer. 241

..e... ·...... ·...... J. Thrasher

REFERENCES

Haltiner, G. J., NumeZcato WeuthQe Ptedicton, John Wiley & Sons, Inc., 1971, 317 pp.

Haltiner, C. J., and R. J. Williams, Some Recent Advances in Numerical Weather Prediction, Month¾y Weathe Reviw, V. 103, (1975), 571-590.

Lavoie, Ronald, A Mesoscale Numerical Model of Lake-Effect Storms, Journae od the Atmozpheric Science, V. 29, (1972), 1025-1039.

Pielke, Roger A., A Three-Dimensional Numerical Model of the Sea Breezes over South Florida, NOAA Technical Memo, ERL/WMPO-2, (1973), 136.

Priestley, C. H. B., TuwbuRent Tmavlnse in the LoweA Atmosphete, The Univ. of Chicago Press, Chicago, Ill, (1959), 130 pp.

Roache, Patrick, ComputacionoaL FuiLd Dynamic, Hermosa Publishers, Albuquerque, N.Mex. (1972), 434 pp.

Robert, A. J., The Integration of a Low Order Spectral Form of the Primitive Meteorological Equations, J.Meteat.Soc.Japan, Vol. 44 (1966), 237-245.

Tapp, M. C. and P. W. White, A Non-hydrostatic Mesoscale Model, Quat.tJout.Roy.Met.Soc., Vol. 102 (1976), 277-296.

243

TESTING NSSL ROUTINES ADQUAD AND SIMPSN by Campanella Tones Prairie View A&M University Jo Walsh, Scientist

The NCAR Software Support Library (NSSL) is a collection of routines available to users from the system file library called ULIB. The mathematical routines include many of the algorithms frequently used in scientific computations. There are also utility routines and special purpose routines to facilitate program input/output. The graphics routines provide easy access to a variety of on-line graphical techniques including contouring. The NSSL testing program at NCAR consists mainly of writing two programs for each of the NSSL files being tested. These programs are called the demonstration driver and extensive test deck, respectively. A short description of the purpose of these programs follows in the next two paragraphs. A demonstration driver is a sample execution program. The tester should design a sample mathematical problem which the routine is designed to solve. This problem should be simple and well-conditioned, but yet it should exercise a good portion of the code. For this problem, a test is designed which can determine whether or not the routine is executing properly. The demonstration driver must be portable and commented. The demonstration drivers have one argument parameter, IERROR. If everything was computed correctly, IERROR is set to 0; otherwise, it is set to 1. My main program printed the message IERROR=O or IERROR=1. The demonstration driver must print whether or not a certain test was successful or not. The tester should also develop a program and data which extensively exercise the routine being tested and which provide evidence for timing and accuracy statements in the documentation. This extensive test deck will generally not need to be portable, since it will be used only as a basis for certification of NCAR's implementation of routines and as a library maintenance aid. When changes are required to library routines, the compiler, the operating system, or hardware, the extensive 244

Testing NSSL Routines ADQUAD and SIMPSN ......

test decks may be rerun to discover whether the changes significantly affect the timing or accuracy of the routine. This summer at NCAR I wrote two demonstration drivers for two routines, ADQUAD and SIMPSN. Both routines do numerical quadrature. For ADQUAD and SIMPSN certain values for input arguments in both routines were tested, and the true value of the integral was compared with the machine value or computed value of the integral with a tolerance of error called epsilon. Epsilon was set to a very small number. To compute the integrand of a function, I just simply called ADQUAD or SIMPSN in my subroutine. ADQUAD is a routine written to do a method of integration called Gaussian Quadrature. (See Figure 1.) Gauss developed his method of integration from the trapezoidal method. Using Gaussian quadrature, two different points, instead of the points A and B at the ends of the interval, are chosen to determine the trapezoid. These are two points C and D which are inside the interval (a,b). A straight line is drawn through these points and extended out to the ends of the interval to complete the shaded trapezoid (Figure 1). Part of the trapezoid lies outside the curve (the upper corners), while part of the curve lies outside the trapezoid. By properly choosing the points C and D, the two areas can be balanced so that the area in the trapezoid' equals the area under the curve. The resulting approximation then gives the exact integral. Gauss' method essentially consists of a simple way of choosing C and D to get as good an answer as possible. Another note is the fact that Gauss' method can be extended to three and more points. Automatic quadrature is an iterative method of integration. A function is needed. The subintervals are always evenly spaced across the intervals (a,b). The value of the interval of the first spacing is compared to the value of the interval of the second and third spacing. When the value of the interval of any two spacings are almost identical, then the computer stops integration. The algorithm for ADQUAD is an adaptive quadrature scheme. This 245

· · · a ·ae · ·e o e * ·· e * · · · · · · · ·e * · C. Tones

Figure 1

f(X)

x a b TRAPEZOIDAL METHOD

f(X)

X a b GAUSS METHOD WITH TWO POINTS 246

Testing NSSL Routines ADQUAD and SIMPSN ...... means that the routine divides up the interval of integration according to the complexity of the function. That is, ADQUAD does not have equally-spaced abscissae and ordinate values where the function is more complex. SIMPSN is a routine that does integration by using Simpson's rule and Lagrange interpolation. In Simpson's method the integral is approximated by a series of parabolic segments, with the idea that the parabola will more closely match a given curve, f(x), than would the straight line determined by the trapezoidal method.

Figure 2

YA PARABOLA \ y=aX2+x+r I I I ~~~~~~\ I ~~~~~~\ I f(X) \- I ) BI

a cb 247

v v * v * * e * * * * * * * * * C. Tones

To integrate the function, f(x), between the limits of a and b as shown in Figure 2, a point c = (a+b)/2 midway between a and b is chosen and the function values, A, B, and C, which have the coordinates:

A: (a,f(a)) B: (b,f(b)) C: (c,f(c))

2 are computed. These three points define a unique parabola, y = ax + Bx + y which passes through all three points. It is now hoped that the area under the parabola is easier to find than the area under the curve f(x) and that the two areas are approximately equal. Simpson's method should give exact answers for any function which is either constant, or a straight line, or a parabola, since a parabola can match any of these exactly. The general formula for Simpson's rule is:

b jf(x) dx ~ x[f(xo) + 4f(xl) + 2f(x2) + 4f(x 3) + 2f(x4) +... a

+ 2f(22 + 4f(X 2 nl + f(2)]

where Ax = (b-a)/2n, and x. = a + i-Ax, i=0,...,2n.

SIMPSN provides the capability of accepting unequally-spaced data through entry point, SIMPSE. In this case Lagrangian interpolation is used to create a set of equally-spaced data. The interpolation is three-point Lagrangian interpolation. The following table summarizes some of the results obtained for SIMPSN and ADQUAD. IH (D rt

OI En

SIMPSN ADQUAD Ctl (D FUNCTION INTEGRAL TRUE VALUE CALCULATED VALUE ERROR CALCULATED VALUE ERROR

2 CO 1. x 9 0 to 2.5 2.50738455 2.473050E+00 .034334 2.5073846E+00 3.213184E-07 C.-I z 2. x2 -3x+2 0 to 3 1.5 1.499999C+00 -.0000006666 1.499999E+00 9.4739031E-15

No 3. -1.5 to 2.5 FAILED FAILED 00 2

4. eX 0 to 1 1.718281 1.718445E+00 .0000936188 1.7182818E+00 4.795991E-07

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