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General Disclaimer One Or More of The General Disclaimer One or more of the Following Statements may affect this Document This document has been reproduced from the best copy furnished by the organizational source. It is being released in the interest of making available as much information as possible. This document may contain data, which exceeds the sheet parameters. It was furnished in this condition by the organizational source and is the best copy available. This document may contain tone-on-tone or color graphs, charts and/or pictures, which have been reproduced in black and white. This document is paginated as submitted by the original source. Portions of this document are not fully legible due to the historical nature of some of the material. However, it is the best reproduction available from the original submission. Produced by the NASA Center for Aerospace Information (CASI) ° i .R , .. a ,... y..., , a ..,tea.. ,w, ..^._...a ....p-'^,.., , r ,.^y +..+a ,^-R-".'.'^ 7 ► NASA Technical Memorandum 80608 Documentation of the Fourth Order Band Model Eugenia Kalnay-Rivas David H o itsma , (N ASA-TH-606 06) DOCUMENTATION O.F THE FOURTH N80-17638 ORDER SAND MODEL (NASA) 117 p HC A06/MP A01 CSCL 048 a. Unclas G3/47 12902 I'. DECEMBER 1979 -3 r National Aeronautics and a ` Space Administration ^^^ x> ° °' '^ Goddard Space Flight Center q © Greenbeit, Maryland 20771 a. 6 J ^ - tti 4^ .a J rrl ^^. 9 a p^ `3 f^xua....^.^.•+-+.^-^n - rn...w.....h+ +Frv'mr M^!^4nNaNY - ^a r^.+gtK#d^'.v+.ry+'^.H.k^as>dMY.HiN. y..:. i.^YS •.. Yr#+aYLi'kCA"^^NYbWUStRWY=^.fidYSiM.S#bk'u i^Yr.Nl.eu+3n:.n-.. x..L.1 +,wY^.:^rn^.I Y Y 6i1.1. Documentation of the Fourth Order Band Model Eugenia Kalnay-Rivas David Hoitsma Sigma Data Services Corp. November 1979 Modeling and Simulation Facility Laboratory for Atmospheric Sciences NASA/Goddard Space Flight Center Greenbelt, MD 20771 r e ^. f TABLE OF CONTENTS Page Preface i I. Introduction I_1 II. Primitive Equations of Motion II-1 Derivation of the Equations at the Poles II-3 Computation of ZI.e Horizontal Pressure Gradient II-7 as suggested by N. A. Phillips ' III. Finite Difference Variables and Grid IIt-1 ^i Periodic filtering of Short Waves II1-3 M v IV. Boundary Conditions IV-1 t V. Finite Difference Equations V-1 The Zonal (U) Momentum Equation V-1 The Meridional ( v) Momentum Equation V-2 .: The Thermodynamic Energy Equation V-3 The Moisture Equation V-4 The Pressure - Tendency Equation V-5 Vi. The Forecast Equations at the Poles VI-1 Zonal (U) Momentum Equation (Poles) VI-3 Meridional ( V) Momentum Equation ( Poles) VI-3 The Thermodynamics Energy Equation ( Poles) VI-4 The Moisture Balance Equation ( Poles) VI-4 ~ The Pressure Tendency Equation ( Poles) VI-5 The Vertical Velocity ,Equation ( Poles) VI-5 xr . VII. Diagnostic Equations (^,a,p) VII-1 VIII. The Time Differencing Scheme VIII-1 IX. Documentation of the Code ( Preliminary) IX-1 X. Flow Charts X-1 Appendix A Reprints of Three Papers A-;I. A Fourth-Order Forecasting Model IN-1 The 4th Order GISS Model of the Global A-3 Atmosphere The Effect of Accuracy, Conservation and A-16 Filtering on Numerical Weather Forecasting Appendix B Program Listing B-], x; it V__ Preface We have decided to compile a preliminary documentatio: of the new GLAS Fourth Order General Circulation Model. The present documenta- tion has not been subjected to a careful editing process; we hope that its possible usefulness will compensate for some of its defects. The model dynamics (COMPO, COMP1 and COMP2) is still undergoing minor improvements, especially in the time differencing scheme which . we hope will improve its efficiency. The "physics" routine (COMPS) has not been documented because it is being thoroughly revised. The present version of COMPS, similar to the one used in the GLAS/GISS models (see the documentation Iby Tsang and Karn), with modifications introduced by Y. Sud (;1979) is included in the code. Criticisms and suggestions for improvement will be reatly appreciated, since a final documentation will be prepared in 1980. We are very grateful to all the people that have helped us generously. In particular, Dr. N. Rushfield had a major impact in the process of making the model operational. W. Connelly, D. Edelmann, D. Han, S. Breining, P. Anolick and M. Almeida were very helpful, in the devel- opment of the model. The documentation was expertly and cheerfully typed by S. Mathis; D. Edelmann and D. Rosen have also cooperated in its compilation. We want to express our special gratitude toward Dr. Y. Sud, who offered us generously both his advice and his time in the development of the "physics 1° routine, and most especially to Dr, M., Halem: without whose many useful suggestions, constant encour- agement, and long patience, this work would not have been finished. Eugenia Kalnay-Rivas November 1979 i mR I. Introduction The band fourth order model is a G04 which uses quadratically oonservative, fourth order horizontal space differences on an unstaggered grid and second order vertical space differences with a Matsuno (forward-backward) or a smooth leapfrog time scheme to solve the primitive equations e of motion. This program numerically solves these equations one latitude hand I at a tim which greatly reduces the amount of computer core storage needed to run the program. It also uses the same variable names, order of computations, I/O, post-processing as the standard second order GCM. Appropriate modifications have been made for she fourth order differences and leapfrog scheme. (See the 1978 Goddard Model- ing and Simulation Research Review for an overview of the fourth order band model.) The main feature of this model is that fourth order approximations are used for all the horizontal derivatives. The derivative aX is approximated by 4 ( q (x+ Ax) -q (x-Ax) , - 1 ( q (x+2Ax) -q (x-2Ax) ) 3 Ax 3 4Ax and the derivative as (qT) by x 4 (T(x)+T (x+Ax) ) (q ( x) + (x+Ax)) - (T (x) +T (x-Ax))( (x) +, (x-Ax)) 3 [ 4 x 1. (T (x) +T (x+2Ax))(q (x) +q (x+2Ax)) - (T (x) +T (x-2Ax))(q (x) +q (x-'6Ax)) , 3 ` 8Ax I-1 The second approximation is derived by averaging the flux qT to yield a conservative form of the dynamic equations. Note that if T is equal to 1 the second equation reduces to the first. The primary variables are the horizontal components of the wind velocity, %V=(u,v), the temperature, T, the specific humidity, q, and the shifted surface pressure, 7r, ('r=ps-ptop' ptop^10 mb) . The secondary variables are the geopotential,o, the vertical wind velocity,a, and the pressure,p. The following ,pages give the differential equations of motion for the GCM model with the initial and boundary conditions. This is foAllowed by the equations with the corresponding fourth order approx- imations which use the same notation as the current second order model. A complete description of the primitive equations with the a coordinate system is found in the Arakawa UCLA notes (1976). P I-2 II. Primitive Eguat, ons of Motion 1 1.&2. Horizontal momentum equations 9 a -Trvo - TT RT VW- (fiu t )kx P a f + 3. Continuity equation + V• (7AV) + aQ (TTc) = 0 , or (3.1) a 1 1 (3.2) V - (7i\V)da _ - V • 7^Vda at - - 0 4. Equation of state RT a = z P S. First law of thermodynamics at + V - (7AVT) + aQ (7r6T) _ fCa + CQ, (WAR) a P P k a IT From 6 =T/p , p=pT+Q7r, w=a7r+;G, W'ItW V7r, k=R/CP we get D 7raT = pk a 7T;O Q(xT Replacing in 5, aQ Do + C P a TrT + a 7T66 _ Tr ak T a Tr + ^V • V Tr) 7TQ V (7^VT) + pk + at as p at Cp t 6. Humidity equation a q + V • (AVq) = 0 7. Iiydrostatic equation = -Cpa (from a _ -P = -^11) a A II-1 q. s^' a`'S Y"E t'».t`GAOY'v"J^NaYiWIk:".^'_'=._- Ia:oan.. 1- 3 Of the varia bl e.^. Wr ,u,v0T , q 7 0'A WW upd ate th e 5Primary var4a es gr,u,v,T, and q using equations 1 ,2,3.2,5 and 6. From equations 3.1, 4, and 7 we can obtain ^,a, and; which are our secondary variables. Note that p=af'+Atop' Sea level pressure (used only in the smoothing routine SMSHAP) Hydrostatic eq. _ -p = A = -^ log( SLP ) _ RT cp s SLP - p(a=1) exp( a) RT 11-2 DERIVATION OF THE EQUATIONS AT THE POLES Consider the continuity equation Tr as = 0 t + V . (^►) + a^O coupled with a conservation equation ^S which can be expanded into P a t a ^r ct at + v . ( ^AVT) + a^ 71S If we .integrate this equation over a polar cap of radius AO 2 r; -ao 27r 7T p0 ' I 'r a 2 cosOdOdA = - V. (7AYT) a2cosod^& a 0 7 2fr 7 /2-A o 2n 7r/2-Ao a a^T a 2 cosododA a2cosododA 7r/2 T/2 and we assume the value of atT to be approximately constant over the polar cap 27 7T/2 —Qo 2n 7r /2 -Ao T a2 cosododA = (a ffT) a2cosodo& f12 0 7r/ 2 27ra2 U-cosAy } (a tT) NP The first term in the rhs is, using Gauss' theorem 2fr ff/2-A^ 27r C V. (PVT) a 2 cos^d^dA - f 7vT a sinAoda ; ^r/2 0 II-3 This can be approximated as IM 2 7r -IM 71ivi Ti a sinA1 i=1 The third term on the rhs is 27r-A^ a7^T f a 2 cos¢ d^d A = 27a 2 (1-cos6fl (Dn;T ff and similarly with the source term.
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