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

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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

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

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(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:

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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: =>

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

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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)

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ALAL

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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]

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

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

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

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

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

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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

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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.

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Gregory et al. parameterization for updraft PGF in (Zhang&Wu, 2003: ) Grubisic & Moncrieff alternative - add dependence on detrainment for PGF:

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Zhang & McFarlane, 1995 (JJA)

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ZM95

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

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ZM95

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Control (no CMF) =.2

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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

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