1. Parameterization of convective momentum transport and energy conservation in GCMs N. McFarlane Canadian Centre for Climate Modelling and Analysis (CCCma )

2. Parameterization in larger-scale atmospheric modeling General parameterization problem: Evaluation of terms involving averaged products of (unresolved) deviations from (resolved) larger-scale variables and effects of unresolved processes in larger-scale models (e.g. NWP models and GCMs). Examples: (a) Turbulent transfer in the boundary layer (b) Effects of unresolved wave motions (e.g. gravity-wave drag) on larger scales (c) Cumulus parameterization Other kinds of parameterization problems: radiative transfer, cloud microphysical processes, chemical processes ** The parameterization scheme(s) should preserve integral constraints of the basic equations – e.g. conservation of energy and momentum

3. ; Basic Equations Motion Mass continuity Thermodynamic Or: Vapour Condensed water Equation of State (ideal gas)

4. (kinetic energy) (moist static energy) Energy Conservation (e.g., Gill, 1982, ch. 4) For airat 15C, 100hPa Kolmogorov scales These are small for the atmosphere (~ 1mm,.1 m/s) => Permissible to neglect viscous terms for modeling/parameterization purposes but not to ignore effects/processes that lead to dissipation and associated heating (k.e. dissipation rate) Energy conservation:

5. Hydrostatic primitive equations and parameterization (GCM resolved) Background state: -hydrostatically balanced - slowly varying (on the smaller, unresolved horizontal and temporal scales). - deviations from it are small enough to allow linearization of the equation of state (ideal gas law) to determine relationships between key thermodynamic variables: =>

6. GCM equations [hydrostatic p.e + parameterized terms] (+ other terms)

7. Boville and Bretherton (2003) in the context of turbulent transfer (PBL) in CAM2: Neglect the terms Include a tke equation (Stull,1988; Lendrick & Holtslag,2004)

8. ALAL

9. Cumulus effects on the larger-scales Start with basic equations, e.g. for a quantity (i) Average over the larger-scale area (assuming fixed boundaries): Mass flux (positive for updrafts): “Top hat” assumption: Quasi-steady assumption: effects of averaging over a cumulus life-cycle can be represented in terms of steady-state convective elements. Pressure perturbations and effects on momentum ignored [Some of these effects have been reintroduced in more recent work, but not necessarily in an energetically consistent manner]

10. (usually parameterized) (2) Average over convective areas (updrafts/downdrafts). Note: (K.E.1)

11. top-hat: neglect K.E. equation from the cumulus momentum equation: (K.E.2) (K.E.1 – K.E.2): and=> (top hat)

12. From the mean equations: Kinetic energy: + cumulus k.e. eq. (for top-hat profiles) (1)

13. Total energy:: The R.H.S. should be in flux form. Q R is the radiative flux divergence.

14. In summary, assuming top-hat cumulus profiles (a) (b) (c) (d) (e)

15. Parameterization of convective (horizontal) momentum transport in GCMs Schneider & Lindzen, 1976 (SL76) - ignored the PGF Helfand, 1979 – Implemented SL76 in the GLAS GCM Zhang & Cho, 1991 (ZC91), Wu&Yanai, 1993– parameterized PGF using an idealized cloud model Zhang & McFarlane, 1995 (JGR) – ZC91 in CCC GCMII Gregory et al., 1997 (G97) – parameterized the PGF on the basis of CRM results (implemented in the UKMO GCM) Grubisic&Moncrieff, 2000; Zhang & Wu, 2003 – evaluated G97 and other possibilities from CRM results Richter & Rasch, 2008 – effects of SL76 and G97 using NCAR CAM3

16. The cumulus pgf term must be parameterized, e.g. Gregory et al, 1997 propose the following for the horizontal component associated with updrafts: For the vertical component, the pgf is often assumed to partially offset the buoyancy and enhance the drag effect of entrainment. Since Let Typical choice: (Siebesma et al, 2003) (H) (V) Need to ensure that (H) and (V) are consistent with eachother.

17. Gregory et al. parameterization for updraft PGF  in  (Zhang&Wu, 2003:  ) Grubisic & Moncrieff alternative - add dependence on detrainment for PGF:

18. Zhang & McFarlane, 1995 (JJA)

19. ZM95

20. (Richter&Rasch, 2008; CAM3, Z-M) SL:  = 0; GR:  =.7

21. ZM95

22. Control (no CMF)  =.2 

23. Summary 1.Momentum transport by convective clouds has important effects in GCM simulations of the large-scale atmospheric circulation 2. Introduction of parameterized momentum transport by cumulus clouds may result in an unbalanced total energy budget if associated dissipational heating effects are ignored