Seguridad en la Red

The mass-ux concept as described in the previous section will be used to nd an expression for the contribution of convection to the grid-mean condensate content,l. The starting point for the derivation is the grid-mean equation for the time evolution of l in the presence of convection, which can be written as

@l @t =A(l) +c;e;GP ; 1 @z@ w0l0 cv (5.9)

where l is the grid-mean condensate content, A(l) represents all transport processes of l except for the convective ux which is represented by the last term on the right hand side. c, e, and GP are the grid average of condensation, evaporation and generation of precipitation.

Making use of the mass-ux concept (see section 5.2.1 and equation (5.5)), the convective ux of cloud condensate can be rewritten (omitting downdraughts for the moment) as,

w0l0

cv =Mu(lu;l); (5.10)

with Mu as the updraught mass-ux, lu the updraught condensate content and l the grid-

mean condensate content.

Since the aim is to derive cloud source/sink terms due to convection, it is assumed that convection is the only active process in the grid box. Hence the only contribution to the grid-mean condensation c is condensation in convective updraughts and therefore c = cu.

Furthermore it is assumed that precipitation is only generated in convective updraughts so thatGP =GP;u. There are several ways of including the evaporation eects of convection on

the grid-mean condensate. T93 identies the compensating subsiding motion as the major inuence on evaporation and hence denes

e=ecv =;

Mu

a dqdzs

!

ma;

whereais the cloud fraction and qs the grid-mean saturation specic humidity; the subscript

5.2. The generation of convective clouds 69 details of the evaporation term for the further derivation of the equations in the context of this chapter. Introducing the assumptions made and substituting (5.10) into (5.9) results in

@l @t ! cv =cu ;GP;u; 1 @z@ h Mu(lu;l) i ;ecv: (5.11)

The steady state equation for the updraught condensate ux can be written as (e.g., Tiedtke, 1989; section 5.2.1)

@

@z(Mulu) =Eul;Dulu+cu;GP;u; (5.12)

whereEu andDu are the entrainment and detrainment rates, cu represents the condensation

and GP;u describes the generation of precipitation in the updraughts. Note, that (5.12)

forms part of most mass-ux convection parametrization schemes, so that the values of the variables above are known after this parametrization is applied in the GCM. Historically the rst term on the right hand side, which describes the entrainment of condensate, is neglected in convection parametrizations. However, it is obvious that it has to be taken into account when applying a mass-ux scheme for convection together with a cloud scheme describing condensate as a prognostic variable, because of the possibility of updraughts penetrating already cloudy areas. The entrainment of the grid-mean value of condensate l implies that it is equally likely for the updraughts to occur in the cloudy area aas it is for them to occur in the cloud-free area (1;a). Introducing (5.12) into (5.11) leads to

@l @t ! cv = Du lu; Eu l+ 1@z@ (Mul);ecv: (5.13)

This equation reveals the interaction of convection with the grid-mean condensate in an interesting way. The rst three terms on the right hand side all represent transport terms whereas the fourth term describes the evaporation. Condensate is produced by condensa- tion in the convective updraughts and enters the grid-mean cloud condensate budget when detrained into the environment. At the same time the condensate in already existing clouds is entrained into the updraughts. The assumption of mass continuity implies compensating downward motion between the updraughts, which will lead to a ux of the grid-mean con- densate. The divergence of that ux contributes to the local change of cloud condensate. This process is described by the third term in (5.13).

Equation (5.13) can be further simplied by using the mass continuity equation for the updraughts (5.6). This leads to the nal form of (5.9) for convective processes as

70 5. The parametrization of cloud generation @l @t ! cv = Du (lu;l) + Mu @z@l ;ecv: (5.14)

Equation (5.14) gives a very similar expression for the inuence of convection on the large scales as for other variables, such as heat and specic humidity, through a detrainment and a subsidence term (see section 5.2.1). However, it diers from the other variables in that the detrainment term is the dominant one in the case of condensate. Together with a denition of the evaporation term, this equation describes unambiguously the connection between grid-mean condensate and its convective source.

Figure 5.1: Schematic of the relevant quantities in the link between a mass-ux convection scheme and

a prognostic cloud scheme for a given model level k. The \half-levels" k;1=2 and k+ 1=2 represent the

boundaries of model level k, for which all uxes are calculated. For the meaning of all other symbols the reader is referred to the text.

A schematic illustration of (5.14) can be found in Figure 5.1 (top panel). The condensate that is produced in the convective updraughts is detrained into the environment. Because of mass continuity, the injected mass (carrying lu) replaces the same amount of mass present

5.2. The generation of convective clouds 71 in the convective updraughts induces a downward motion in the environment, which itself advects grid-mean condensate and whose adiabatic heating eect leads to evaporation of cloud water.