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Heating and Kinetic Energy Dissipation in the NCAR Community Atmosphere Model

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Heating and Kinetic Energy Dissipation in the NCAR Community Atmosphere Model 1DECEMBER 2003 BOVILLE AND BRETHERTON 3877 Heating and Kinetic Energy Dissipation in the NCAR Community Atmosphere Model BYRON A. BOVILLE National Center for Atmospheric Research,* Boulder, Colorado CHRISTOPHER S. BRETHERTON Department of Atmospheric Sciences, University of Washington, Seattle, Washington (Manuscript received 2 August 2002, in ®nal form 24 March 2003) ABSTRACT Conservation of energy and the incorporation of parameterized heating in an atmospheric model are discussed. Energy conservation is used to unify the treatment of heating and kinetic energy dissipation within the Community Atmosphere Model, version 2 (CAM2). Dry static energy is predicted within the individual physical parame- terizations and updated following each parameterization. Hydrostatic balance leads to an ef®cient method for determining the temperature and geopotential from the updated dry static energy. A consistent formulation for the heating due to kinetic energy dissipation associated with the vertical diffusion of momentum is also derived. Both continuous and discrete forms are presented. Tests of the new formulation verify that the impact on the simulated climate is very small. 1. Introduction izations in CAM2 are clearly separated from the solution of the resolved adiabatic dynamics. In fact, CAM2 sup- The solution of the equations of motion in an atmo- ports three ``dynamical cores'': an Eulerian spectral spheric model requires the parameterization of processes core, as in CCM0 through CCM3 (CCM0±3); a semi- that occur on scales smaller than those explicitly repre- Lagrangian spectral core based on Williamson and Ol- sented by the model. Parameterizations are typically in- son (1994); and a ®nite volume core based on Lin and cluded for several processes (e.g., radiation, convection, Rood (1997). Horizontal diffusion is the only parame- vertical diffusion), which are treated as physically dis- terized process that acts across columns in CAM2 and tinct, even though they are intimately coupled. Because it is treated within the dynamical cores. Conservation of the difference between horizontal and vertical length of energy within the resolved dynamics is the respon- scales, the parameterized processes typically operate en- sibility of the dynamical core and the latter two (which tirely in the vertical direction, treating the atmosphere as contain implicit diffusion) include ad hoc ®xers to en- a set of distinct columns, which can be treated indepen- force conservation. dently. This paper discusses conservation of energy in Energy conservation was not a signi®cant consider- the Community Atmosphere Model, version 2 (CAM2; ation in developing the original CCM0, which contained more information available online at http://www.ccsm. an energy imbalance of ;10 W m22 (Williamson 1988), ucar.edu/models/atm-cam/), particularly within the pa- primarily due to inconsistencies in the vertical numerical rameterization suite, and a uni®cation of the application approximations. Energy conservation was taken more of parameterized thermodynamic processes in terms of seriously in CCM1±3, which conserved energy to ;0.4 dry static energy. Wm22 (e.g., Boville and Gent 1998). This level of CAM2 is the successor to the National Center for conservation was partly due to a cancellation of errors Atmospheric Research (NCAR) Community Climate since individual processes conserved energy only to ;1 Model, Version 3 (CCM3; Kiehl et al. 1998). In common Wm22. Energy conservation is now a serious issue with many atmospheric models, the column parameter- because of the application of CAM2 in multicentury coupled ocean±atmosphere simulations in which even * The National Center for Atmospheric Research is sponsored by imbalances of 0.4 W m22 can cause spurious long-term the National Science Foundation. trends. For historical reasons, the parameterizations of ther- Corresponding author address: Dr. Byron Boville, NCAR, P. O. modynamic processes in CCM0±3 have involved eval- Box 3000, Boulder, CO 80307-3000. uating tendencies for a mixture of temperature, potential E-mail: [email protected] temperature and dry static energy. Both computational q 2003 American Meteorological Society Unauthenticated | Downloaded 09/30/21 02:12 PM UTC 3878 JOURNAL OF CLIMATE VOLUME 16 constraints and the evolution of the conceptual frame- is dI/dt 1 dW/dt 5 Q, where I is the internal energy work for the individual parameterizations have contrib- per unit mass, W is the work done by the ¯uid through uted to this diversity. In addition, the forcing terms have expansion, Q is the heating rate, and d/dt represents a been applied using a mixture of time and process split- substantive derivative. For a variable mixture of ideal ting (see below). Recently, Williamson (2002) has re- gases, each with a ®xed speci®c heat at constant volume formulated the CCM3 parameterization suite entirely in cy , pa 5 RT, and I [ cy T, where R and cy are the time-split form. In this formalism, the parameterizations apparent (mass weighted) constants for the mixture, a operate sequentially, on pro®les modi®ed by the pre- is speci®c volume, p is pressure, and T is temperature. vious parameterization. Noting that the work term is dW/dt [ p(da/dt) 5 (d/ In CCM0±3, heating was considered equivalent to dt)pa 2 a(dp/dt), v [ dp/dt and cp [ cy 1 R, the ®rst temperature tendency within the parameterization suite, law is so that energy was not conserved for the intermediate d(cT) da states which follow time split parameterizations. The y 1 p 5 Q, (1) total heating due to all parameterizations was correctly dt dt reported to the dynamics and applied in a nearly con- or, equivalently, servative fashion. This manuscript describes a method for applying heating to the dry static energy so that the d(cT) RTv p 25Q. (2) energy is locally conserved at all stages within the pa- dt p rameterization suite, regardless of the splitting method. The methods are equivalent when height is the vertical Here Q may have contributions from the divergence of coordinate but the distinction becomes important for radiative ¯uxes, molecular conduction, phase change pressure-based vertical coordinates, as used in most and precipitation processes, chemical reactions, and vis- global models. The meaning of energy conservation is cous dissipation. In the lower atmosphere, molecular stated precisely, including the effects of variable gas conduction is usually replaced by turbulent transport. mixtures, although CAM2 is still somewhat inconsistent For a variable mixture of gases, cy , R, and cp are in its treatment of variations in water vapor. determined by their mass-weighted average over the The vertical diffusion in CAM2 has been reformu- gases in the mixture. In CAM2, we make the common lated using dry static energy and both the kinetic energy approximation that dry air is well mixed and that water ¯ux and the heating due to kinetic energy dissipation vapor is the only additional substance present in suf®- are derived. These terms depend correctly on the stress cient abundance to affect the total pressure. Let super- tensor as discussed by Fiedler (2000) and Becker (2001). scripts v and d denote water vapor and dry air, respec- In CCM1±3, the kinetic energy ¯uxes were ignored. tively. Then the apparent constants are We begin by reviewing the total energy equation in dwd section 2 and its application to the physical parameter- cpp5 c 1 (c p2 c p)q, (3) izations, together with the hydrostatic approximation in R 5 Rdwd1 (R 2 R )q, (4) section 3. The splitting procedures used for parameter- izations are summarized in section 4, the procedure for where q is speci®c humidity. Water vapor is suf®ciently deriving temperature and geopotential from dry static abundant in the Tropics so that vertical variations in R energy is discussed in section 5, and vertical diffusion may change the sign of the static stability, while hori- and dissipation of kinetic energy are discussed in section zontal variations may produce pressure gradients even 6. Energy conservation considerations for the dynamical with uniform temperatures. Variations in cp are often cores are commented on brie¯y in section 7, the impact considered to be less important, since they affect the of the changes on the CAM2 simulation is discussed in response to heating, but do not directly affect the density d section 8, and conclusions appear in section 9. and pressure gradients. CAM2 uses cp 5 cp, so that (2) dd becomes dT/dt 2 RTv/pcpp5 Q/ c . This is the ther- modynamic equation solved by both the spectral (Eu- 2. Total energy conservation lerian and semi-Lagrangian) cores, although cp is used Energy conservation is a important property of the in the second term due to an oversight. The ®nite volume equations of motion, which should be maintained when core converts (2) to a virtual potential temperature equa- approximations are made in order to solve them. Energy tion. We retain a variable cp in the derivations below, conservation can be guaranteed by forming an evolution but properly accounting for variations in cp in a model equation for total energy and verifying that it is satis®ed will probably require predicting an enthalpy (cpT) re- by the approximate equations. The total energy equation lated variable. This will become increasingly important is obtained by adding the kinetic energy equation to the as the thermosphere is incorporated into CAM2, since ®rst law of thermodynamics. The derivations of these diffusive separation of constituents results in large var- equations may be found in Gill (1982), among other iations of cp and R above ;120 km. sources, and will only be summarized here. The kinetic energy equation is derived from the mo- A fundamental statement of the ®rst law for a ¯uid mentum equation, Unauthenticated | Downloaded 09/30/21 02:12 PM UTC 1DECEMBER 2003 BOVILLE AND BRETHERTON 3879 dV d ]p 1 2V 3 V 52a=p 2 g 2 a= ´ t, (5) (s 1 K) 5 a 2 a= ´ F 1 Q , (9) dt dt ]t K 0 where t is the stress tensor due to molecular or turbulent where s [ cpT 1Fis the dry static energy.
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