5.2 The generation of convective clouds
5.2.1 The mass-ux concept
Before deriving the source terms from convection for cloud condensate and cloud fraction it is necessary to review the basic ideas of the mass-ux concept as they are applied in many convection parametrization schemes today (e.g., Arakawa and Schubert, 1974; Tiedtke, 1989; Gregory and Rowntree, 1990).
A generic conservation equation for a scalar in an incompressible uid averaged over a model grid volume can be written as (e.g. Stull, 1988, Tiedtke, 1989)
@ @t +
V
h rh +w@ @z =; 1 @zw@ 0 0 +S ; (5.1)where
V
h is the horizontal wind speed, wis the vertical velocity, andS is the body sourceterm for respectively. The overbar denotes the grid-box average and the prime denotes the deviation from that average such that at any point in the grid volume
= + 0:
When interpreted as a grid-box average equation for a large-scale model the left-hand side of (5.1) represents the terms resolved by the model while the right hand side contains terms acting on the subgrid scale which need to be parametrized (see section 2.2).
In an atmosphere with active convection the large-scale (grid-scale in a model) vertical motion,w, is the average of the mean ascent in active cumulus clouds and the mean motion, which can be ascending or descending, in the environment. With the assumption that active convection occupies a fraction of the horizontal area, the mass-uxes in cumulus clouds, Mc, and in the environment, Me, can be dened as
Mc = wc
Me = (1;)we:
Here, both wc and we are dened as the perturbation vertical velocities with respect to the
area-averaged motion, i.e.,
wc = wc;tot;w
66 5. The parametrization of cloud generation Furthermore with the denition of an active convection area and its environment the average value of can be written as
= c+ (1;) e: (5.2)
If it is now assumed that above cloud base the vertical eddy transport is entirely achieved by active cumulus convection; then after some algebra
0w0 =( w
; w) =(1;)( c; e)(wc;we): (5.3)
With the above denition of wc and we and the ususal assumption that w0 = 0 it easily
follows that wc+ (1;)we = 0; and hence, we =; wc 1;:
Equation (5.3) can now be rewritten as
0w0 =w
c( c; e) = Mc( c; e):
Substituting e from (5.2) yields
0w0 = Mc
1;
( c; ): (5.4)
If the averaging area is much larger than the scale of individual convective draughts, which normally are not much larger than a few kilometers, it is reasonable to assume that 1.
Using this assumption in (5.4) leads to the nal approximate form of (5.3) as 0w0 =M
c( c; ): (5.5)
With (5.5) the subgrid-scale ux of the quantity has been expressed as the product of the cloud-scale mass ux and the dierence of the values of inside and outside the cumulus elements. When reintroducing (5.5) into (5.1) it becomes obvious that in order to close the system of equations it is necessary to derive expressions for the convective mass ux, Mc, and the ux of inside the convective elements, Mc c. This requires a model for the
5.2. The generation of convective clouds 67 ensemble of cumulus elements. Examples for such models are given by Arakawa (1969), Ooyama (1971), Yanai et al. (1973), Arakawa and Schubert (1974) and Tiedtke (1989) and only a brief summary will be provided here. For simplicity it will be assumed that there are no convective-scale downdraughts present in the averaging area, so that the net convective mass ux is achieved only by updraughts, i.e., Mc = Mu and c = u, where subscript u
stands for updraught quantities.
The balance equation for mass and quantity in a steady state for an individual convective updraught can be written as
1
@M@zi =i;i
and
1
@M@zi u;i =i ;i u;i+S( u;i);D( u;i);
where i and i represent the rates of mass entrainment into and detrainment out of the
convective updraught. S( u;i) and D( u;i) represent sources and sinks of u;i within the
updraught, such as condensation in the case of representing heat or humidity, or the generation of precipitation if represents cloud condensate carried inside the updraughts. A further simplication of the problem often used in cumulus parametrization can be achieved by summing over all convective elements. This leads to the so-called bulk cumulus model equations as @Mu @z =E;D (5.6) and @Mu u @z =E ;D u+S( u);D( u); (5.7)
where E = and D = . Introducing (5.5), (5.6) and (5.7) into (5.1) and making the additional assumption that sources and sinks for can only occur in convective updraughts (i.e., S =S( u) and D =D( u)), (5.1) simplies to
@ @t +
V
h rh +w@ @z = D( u; ) + Mu @@z : (5.8)This form of the large-scale evolution equation for in a convective atmosphere is usually interpreted as describing the inuence of convection on the large-scale atmosphere through
68 5. The parametrization of cloud generation the detrainment of the updraught values of into the environment and through advection of due to compensating downward motion between convective updraughts. The concept and the equations outlined here will be used in the subsequent sections to show that if the convection process is expressed using the mass-ux concept, there exists a unique and unambiguous coupling to cloud formation.