Parameterization of the effects of Moist Convection in GCMs Mass flux schemes Basic concepts and quantities Quasi-steady Entraining/detraining plumes (Arakawa&Schubert and similar approaches) Buoyancy sorting Raymond-Blythe, Emanuel Kain-Fritsch Closure Conditions, Triggering Adjustment Schemes Manabe Betts-Miller References Atmospheric Convection , Emanuel, 1994 Arakawa and Schubert, 1974, JAS Plus papers cited later Ae (1 ( Ac ) i i ( z , t ) AL AL

)A i i L Spatial Averages For a generic scalar variable, Large-scale average: : 1 dA AL AL Convective-scale average (for a cumulus up/downdraft) : 1 c dA Ac Ac

Environment average (single convective element): 1 e dA AL Ac Ae Where Ac AL 1 c (1 ) e Vertical velocity: c ,e / O(1) * (1 ) w wc (1 ) we wc w , we

Time average over cloud life cycle: 1 (t ) L t L / 2 t (t )dt L /2 Vertical flux w [(wc w* w )( c * )]c (1 )[(we w w )( e )]e But since 1 ; wc we ) e O( 2 ) w wc ( c ) ( w* * ) c (1 )(w

For simplicity in the following we ignore the last term (to focus on predominantly convective processes) . Also ignore sub-scale horizontal fluxes on the boundaries of the large-scale area (but account for exchanges between convective elements and the environment via entrainment/detrainment processes). Cumulus effects on the larger-scales Start with a general conservation equation for ( ) ( w ) H ( V ) Q t z Plus the assumption: (similar to using anelastic assumption for convective-scale motions) (i) Average over the large-scale area (assuming fixed boundaries): ( )

( w ) ( wc ( c )) ( ( w* * ) c ) ( V ) Q others t z z z Mass flux (positive for updrafts): M c wc Also: Q (Q ) c (1 )(Q ) e ; Top hat assumption: ( w* * ) c 0 In practice (e.g. in a GCM) the prognostic variables are also implicitly time averages over convective cloud life-cycles (ii) Apply cumulus scale sub-average to the general conservation equation, accounting for temporally and spatially varying boundaries: ( c ) ( [ wc c ( w* * ) c ]) vn b dl (Q ) c

t Ac z Mass continuity gives: ( awc ) v dl 0 ; n t Ac z vn the outward directed normal flow velocity (relative to the cloud boundary) Entrainment (inflow)/detrainment (outflow):

E vn [1 H (vn )]dl Ac Define: E EAc Top hat: D vn H (vn )dl Ac vn b [1 H (vn )]dl c E e ; D c ; D 1; f 0 H( f )

0; f 0 { DAc v H (v )dl n c b n Summary for a generic scalar, (top hat in cloud drafts): [ M c ( c ) w* * c ] w V Q other t

z z M c D E 0 t z c M c c ( w* * ) c D D E Q c t z therefore : w V M c D D (1 )Q e other t

z z When both updrafts and downdrafts are present, both entraining environmental air: M c M u M d ; E Eu Ed ; D Du Dd ; M u 0; M d 0 M c c M u u M d d ; D c Du u Dd d Large-scale equations for dry static energy and water vapour heating s Ds D su Llu s D.H . M c z Dt c moistening / drying q Dq M

D(qu lu q ) c z Dt c [ Effects on horizontal momentum and associated dynamical heating: talk by Tiffany Shaw ] Note that Ds Dq h L M u D hu h D.H . Dt Dt z M u h M u (h hu ) Dhu Eh D.H . D.H . z z At the conv. layer top: M u 0; At c.l. base: hu h

Basic cumulus updraft equations (top-hat) {Dry static energy: s=CpT+gz; Moist static energy : h=s+Lq; M w } u u M u mass conservation Du Eu 0 t z (su ) M u su dry SE Du su Eu s Lcu t z

(qu ) M u qu Du qu Eu q cu vapour t z (lu ) M u lu Du lu cu Pu condensate t z (hu ) M u hu moist SE Du hu Eh 0 t z ( M u ) ( M u wu ) ( ( Pc P ) / ) [( ) ]

Dwu Ewe 0 g v c v t z z v v Tv ( po p ) ; R / c p T T (1 .608q l ) ; v vertical velocity (virtual temperature) Quasi-steady assumption: effects of averaging over a cumulus life-cycle can be represented in terms of steady-state convective elements . Transient (cloud life-cycle) formulations: Kuo (1964, 1974); Fraedrich(1974), Betts(1975), Cho(1977), von Salzen&McFarlane (2002). Entrainment/Detrainment Traditional organized (e.g.plume) entrainment assumption: v [1 H (v )]dl w P n

c n c c [ where Pc dl (local draft perimeter)] c P 2 E c wc M c wc Rc Ac Arakawa & Schubert (1974) (and descendants, e.g. RAS, Z-M): - is a constant for each updraft [saturated homogeneous (top-hat) entraining plumes] - detrainment is confined to a narrow region near the top of the updraft, which is located at the level of zero buoyancy (determines ) Kain & Fritsch (1990) (and descendants, e.g. Bretherton et al, ): - Rc is specified (constant) for a given cumulus (not consistent with varying ) - entrainment/detrainment controlled by bouyancy sorting (i.e. the effective value of is constrained by buoyancy sorting) Episodic Entrainment and non-homogeneous mixing

(Raymond&Blythe, Emanuel, Emanuel&Zivkovic-Rothman): -Not based on organized entrainment/detrainment - entrainment at a given level gives rise to an ensemble of mixtures of undiluted and environmental air which ascend/descend to levels of neutral buoyancy and detrain zt zb Determining fractional entrainment rates (e.g. when hi i h hi z Tc Te at the top of an updraft) hi (( zt ) i , i ) h* (( zt ) i ) hi ( zb ) h ( zb ) ( zt ) i h* (( zt ) i ) h ( zb ) exp[ i (( zt )i zb )] i

h ( z) exp[ ( z ( z ) )]dz i t i zb Note that since updrafts are saturated with respect to water vapour above the LCL: * L hi h* L q Ti T (qi q * (T , p )) (Ti T )(1 ) O(Ti T ) 2 c cp c p T p This determines the updraft temperature and w.v. mixing ratio given its mse. Fractional entrainment rates for updraft ensembles (a) Single ensemble member detraining at z=zt

zb z zt Eu ( zt ) M u ; Du 0 M u M b exp ( zt )( z zb ) Detrainment over a finite depth zt : D ( zt ) M u ( zt ) / zt (b) Discrete ensemble based on a range of tops M u M ( z , i ); i (( zt ) i ) i M u hu M i hi i Eu i M i i M i Du i i zt Buoyancy Sorting Entrainment produces mixtures of a fraction, f, of environmental air and (1-f) of

cloudy (saturated cumulus updraft) air. Some of the mixtures may be positively buoyant with respect to the environment, some negegatively buoyant, some saturated with respect to water, some unsaturated v saturated (cloudy) v c positively buoyant fc v e 0 f* 1 f Kain-Fritsch (1990) (see also Bretherton et al, 2003): Suppose that entrainment into a cumulus updraft in a layer of thickness z leads to mixing of Mcdz of environmental air with an equal amount of cloudy air. K-F

assumed that all of the negatively buoyant mixtures (f>fc) will be rejected from the updraft immediately while positively buoyant mixtures will be incorporated into the updraft. Let P(f) be the pdf of mixing fractions. Then: fc E 2o M u fP( f )df 0 1 D 20 M u (1 f ) P( f )df fc This assumes that negatively buoyant air detrains back to the environment without requiring it to descend to a level of nuetral bouyancy first). Emanuel: Mixtures are all combinations of environement air and undiluted cloud-base air. Each mixture ascends(positively buoyant)/descends (negatively buoyant), typically without further mixing to a level of nuetral buoyancy where it detrains. Shallow convection: Including decent to a nuetral buoyancy level (with evaporation of cloud water) before final detrainment requires gives rise to cooling associated with evaporatively driven downdrafts in the upper levels of cumulus cloud systems noted as a diagnostic requirement by Cho(1977)

Closure and Triggering Triggering: It is frequently observed that moist convection does not occur even when there is a positive amount of CAPE. Processes which overcome convective inhibition must also occur. Closure: The simple cloud models used in mass flux schemes do not fully determine the mass flux. Typically an additional constraint is needed to close the formulation. The closure problem is currently still poorly constrained by theory. Closure Schemes In Use Moisture convergence (Kuo, 1974- for deep precipitating convection) Quasi-equilibrium [Arakawa and Schubert, 1974 and descendants (RAS, Z-M, Zhang&Mu, 2005)] Prognostic mass-flux closures (Pan & Randall, 1998;Scinocca&McFarlane, 2004) Closures based on boundary-layer forcing (Emanuel&Zivkovic-Rothman, 1998; Bretherton et al., 2003)

Emanuel& Zivkoc-Rothman(1998): Bretherton, McCaa, & Grenier, MWR, 2003: Z-M scheme: all plumes have the same base mass flux Mb Mb exp[ ( z zb )]d {exp[ ( z )( z zb )] 1} /( z zb ) M u max 0 max 0 ( z ) max ( z* ) max ; z z* Closure based on CAPE depletion: z LNB CAPE g ((T ) v und

Tv ) /T vdz Tv T (1 .608q l ) zb [(CAPE ) / t ]c M b F CAPE / a M b CAPE /( a F ) Prognostic closure: M b CAPE M b t d M b 0.