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A Description of the Fifth-Generation Penn State/NCAR Mesoscale Model (MM5)

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A Description of the Fifth-Generation Penn State/NCAR Mesoscale Model (MM5) NCAR/TN-398 + STR NCAR TECHNICAL NOTE 1. December 1994 A Description of the Fifth-Generation Penn State/NCAR Mesoscale Model (MM5) Georg A. Grell1 Jimy Dudhia2 David R. Stauffer3 MESOSCALE AND MICROSCALE METEOROLOGY DIVISION I I 2NATIONAL CENTER FOR ATMOSPHERIC RESEARCH BOULDER, COLORADO 1FORECAST SYSTEMS LABORATORY, NATIONAL OCEANIC AND ATMOSPHERIC ADMINISTRATION BOULDER, COLORADO 3DEPARTMENT OF METEOROLOGY, THE PENNSYLVANIA STATE UNIVERSITY UNIVERSITY PARK, PENNSYLVANIA Table of Contents Preface . ........... e 0 Acknowledgments .............. · * 1 ix Chapter 1 Introduction ......... .... .... ·. 1 Chapter 2 Governing equations and numerical algorithms .1 2.1 The hydrostatic equations .. .... ... 1 2.2 The nonhydrostatic equations . 2.2.1 Complete Coriolis force option .. * 2.3 Nonhydrostatic finite difference algorithms 2.4 hydrostatic finite difference algorithms . .. .. 10 2.5 Time splitting ........ * . .. .11 2.5.1 The nonhydrostatic semi-implicit scheme * . .... 11 2.5.2 The hydrostatic split-explicit scheme . .. .. 13 2.6 Lateral boundary conditions ... ......-16 2.6.1 Sponge boundary conditions .. ... .... .. 16 2.6.2 Nudging boundary conditions .. .. .. 16 2.6.3 Moisture variables . ... · ·.......17 2.7 Upper radiative boundary condition . · ·.......17 Chapter 3 The mesh-refinement scheme .... .1 .. 20 3.1 The monotone interpolation routines . 1.. 20 3.2 Overlapping · and * · * * moving grids . · · · 24 iii11 3.3 The feedback ........... .. 24 3.3.1 A nine-point averager ....... .. .. .*. 25 3.3.2 A smoother-desmoother ....... 25 Chapter 4 Four-dimensional data assimilation * 0* * * * * * * 26 4.1 Analysis nudging ........... 27 4.2 Observational nudging ........ 33 Chapter 5 Physical parameterizations . .. 38 5.1 Horizontal diffusion .......... 38 5.2 Dry convective adjustment 38 5.3 Precipitation physics ........... 39 .... 40 5.3.1 Resolvable scale precipitation processes and ic 5.3.1.1 Explicit treatment of cloudwater, rainwater, snow, and ice 40 5.3.1.2 Mixed-phase ice scheme ..... 46 5.3.2 Implicit cumulus parameterization schemes 47 5.3.2.1 The Kuo scheme ....... 47 5.3.2.2 A modified Arakawa-Schubert scheme .. 51 5.3.2.3 The Grell scheme ....... 65 5.3.3 Parameterization of shallow convection .. · · · a * 70 5.4 Planetary boundary layer parameterizations . 73 5.4.1 Surface-energy equation ......... 73 5.4.1.1 Net Radiative flux Rn ............ * v 74 5.4.1.2 Heat Flow into the Substrate H, ......... * v 78 5.4.1.3 Sensible-Heat Flux H. and Surface Moisture Flux E, 78 5.4.2 Bulk-aerodynamic parameterization ....... * v 78 5.4.3 Blackadar high-resolution model ........ 80 5.4.4 Vertical diffusion ........ .. .... 84 5.4.5 Moist vertical diffusion ............ 86 iv 5.5 Atmospheric radiation parameterization 9 0 & a 0 87 5.5.1 Longwave radiation ......... 0 a 0 a 0 87 5.5.2 Shortwave radiative scheme ..... a 0 0 0 0 0 0 a 90 Appendix 1 Glossary of symbols ....... 0 a 92 Appendix 2 Look-up table for transmissivity 111 Appendix 3 Map Projections ......... 112 Appendix 4 Land-Use categories ....... 116 References ............ · a * * a* 117 v I PREFACE This technical report describes the fifth generation Penn State/NCAR Mesoscale Model, or MM5. It is intended to provide scientific and technical documentation of the model for users. Source code documentation is available as a separate Technical Note (NCAR/TN-392) by Haagenson et al. (1994). Comments and suggestions for improvements or corrections, are welcome and should be sent to the authors. vi vii ACKNOWLEDGMENTS A large number of scientists, students, and sponsors, too numerous to mention, have made significant contributions towards developing and improving this model. We would like to thank everyone who directly implemented changes and corrections into the code, as well as everyone who helped to identify errors and problems. We would also like to extend our gratitude to the two reviewers (Prof. Tom Warner, and Dr. Jian-Wen Bao), whose comments improved the manuscript significantly. Computing for this model development was undertaken on the CRAY YMP at NCAR, which is supported by the National Science Foundation. The writing of this technical document was in part supported by the Department of Energy through grant DEA105-90ER61070, and the California Air Resouces Board through subcontract AG/990-TS05-S02 from Alpine Geophysics. ix 1. Introduction This technical report is a description of the fifth-generation Penn State/NCAR Mesoscale Model (MM5). It is based on the original version described by Anthes and Warner (1978). Although- a few of the following details of this model are well represented in Anthes et al. (1987), extensive changes and increases in options have occurred. For completeness, those parts that have changed little or none will also be represented here. The document structure is as follows. In section 2 we will describe the governing equations, algorithms, and boundary conditions. This will include the finite difference algorithms and time splitting techniques of both the hydrostatic and the nonhydrostatic equations of motion (hydrostatic and nonhydrostatic solver). All subsequent sections will describe features common to both solvers. Section 3 will discuss the mesh-refinement scheme, section 4 the four-dimensional data-assimilation technique, and section 5 will focus on the various physics options. 2. Governing equations and numerical algorithms 2.1 Hydrostatic model equations The vertical c-coordinate is defined in terms of pressure. P - Pt PP - Pt where p. and pt are the surface and top pressures respectively of the model, where pt is a constant. The model equations are given by the following, where p* = p' - pt Horizontal momentum; ap*u 2 [ap*uu/m + 9pvu/m ap*qua &t [ : z + Oy J - o' Fc9ap* MP- Pz + + p*fv + D(2.1.1) p*v _ 2 8p*uv/m Op*vv/m 1 8p*v m _t = O + Oy J O m + + D - p I~~~~I] *-Pp ffuu +D , (2.1.2) 1 Temperature; Op*_T 2 r p*uT/m Op*vT/m _ p*T& apt a + ay - + p -+ pP + DT, (2.1.3) pcp cp where the D terms represent the vertical and horizontal diffusion terms and vertical mixing heat due to the planetary boundary layer turbulence or dry convective adjustment. The qv is capacity for moist air at constant pressure is given by cp = Cpd(l + 0.8qg), where the mixing ratio for water vapor and Cpd is the heat capacity for dry air. Surface pressure is computed from Op* _m2 ap*u/m 9p*v/m 9p* (2 1.4) a~t = - m + ay ] (2.1.4) which is used in its vertically integrated form oa = _2 p*u/m pv/m da. (2.1.5) at "axy Ja by vertical Then the vertical velocity in a-coordinates, b, is computed from (2.1.4) integration. Thus 2 (2.1.6) = -4 [ + m (a + a /m)] d', P JoL[ at 9x ay where a' is a dummy variable of integration and &(u =0) = 0. In the thermodynamic equation, (2.1.3), w = d and is calculated from w = p*a + d(2.1.7) where dp* = + r-ap[u + lavU. (2.1.8) dt 9t ax9- + y Jay virtual The hydrostatic equation is used to compute the geopotential heights from the temperature, T,: (2.1.9) 9ln~aaln(o +p^p1Pt //P* ) _-RTv + g + qv.. 2 where Tv is given by TV = T(1 + 0.608qV), and qc and q, are the mixing ratios of cloud water and rain water. 2.2 Nonhydrostatic model equations For the nonhydrostatic model we define a constant reference state and perturbations from it, as follows: p(x,y,z,t) = po(z) + p'(A,y, ,t), T(x,y,z,t) = To(z) + T'(z,y,z,t), p(x,y,z,t) = po(z) + p'(x,y,z,t). Typically the temperature profile for the reference state may be an analytic function that fits the mean tropospheric temperature profile. The vertical a-coordinate is then defined entirely from the reference pressure. Po - Pt Pu - Pt where p. and Pt are the surface and top pressures respectively of the reference state and are independent of time. The total pressure at a grid point is therefore given by P =p*r + pt + p, where p*(x,y) = pa(x, ) - Pt. The three-dimensional pressure perturbation, p', is a predicted quantity. The model equations (Dudhia 1993) are then given by the following: Horizontal momentum; ap*U _ 2 rap*u/ +m p*vu/m apu uDIV Qt - la. 9x + ay j - a9 a9pa p*O mp-p 8v p-axau--+]-- p[ + Du (2.2.1)Z p [9x p* 9x 9aa ap*v 2 rapuv/m ap*vv/m] + D + + vDIV 9t t m a9x 9 jyJau mp*\9p'r)p* 9p'] ~- ---p-- -I, --- ~ - p fu + DLf (2.2.2) P 1y p* ay aI J 3 Vertical momentum; 2 ap*w _ 2[9p*uw/m ap*vwu/m ap*wr O = -m M -+ ay - q- wDIV at 8x49 9 j + P 9 a1pp + T - - pPOg[(qC + q,)] + Dw (2.2.3) Pressure; app _ m 2 [ap*up'/m apvpm _ app + M2 p au/m a ap* 9u av/m a p* av [-mpax m p- ax'a 8a- avYaa--y ay m*m+ a yaa] aw + Pog9P- + P*Pogw (2.2.4) Temperature; ap*T 2 ap*uT/m + p*VT/m _ p*T m + + T DIV at r ax ay ao + P* - pgp* - DP# + p* + DT, (2.2.5) PCp Dt cp where 49p*p'u/, 8p~v/mO p*1 DIV = + + (2.2.6) ax ay J au and nno Tnr 49p*11 MO- op* = pPa pU - rn ap. (2.2.7) p p* ax p* ay The DIV terms are not in the hydrostatic equations and arise because p* is now constant in time. Thus the hydrostatic continuity equation no longer applies, leaving the right hand side terms in (2.2.6) uncancelled by the surface pressure tendency.
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