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In this subsection the source terms for cloud fraction and condensate derived above will be compared to those used in T93 (see Chapter 3). First an analysis of the dierent behaviour of some crucial quantities related to the condensation/cloud formation process will be carried out. After that some simple examples of numerically explicit solutions will be presented to highlight shortcomings in the original parametrization and show the improved physical consistency of the terms derived here.

Prediction of cloud fraction

First the production terms of cloud fraction for the two schemes will be compared. The cloud fraction source term used in T93 is

a=;(1;a

t₎ qs

qts;qt

: (5.31)

The dierence between this and (5.29) can be expressed by the ratio (a)T93

(a)new = 2(1;at)

(5.32) It is immediately apparent, that the original scheme produces larger changes in a than the new one for a given change inqs. The dierence increases with initial cloud fraction. For the

special case that qs =;(qts;qt), the original scheme would always predict cloud fraction

one, whereas the scheme derived from the distribution would yield cloud fractions smaller than one, consistent with the fact that a part of the grid box has specic humidities lower than the grid average.

Environment humidity

Although never explicitely parametrized, the choice of source terms for a and l has impli- cations for the evolution of specic humidity in the cloud-free part of the grid box, qe. The

behaviour of the two parametrizations compared here might provide some insight into the physical soundness of the schemes. The starting point of the investigation is the expression of grid-mean specic humidity as

5.3. The generation of non-convective clouds 81

q=aqs+ (1;a)qe; (5.33)

as used before.

The change in q can then be written in nite dierence form as

q=aqs+ (1;a)qe+ a(qs;qe) + a(qs;qe): (5.34)

Note that the last term on the right hand side of equation (5.34) is of second order and will be retained following accuracy arguments made above. Assuming the only change in specic humidity in the grid box is due to the condensation process itself, q =;l and acan be

replaced with the terms derived above for the two cloud schemes and (5.34) can be solved for qe. For the T93 scheme this gives

aqs+ aqs =aqs+ (1;a)qe;qs+ aqs;aqe;

which nally yields

qe = ₍₁ qs ;a;a)

: (5.35)

Hence, in this scheme, the change in the specic humidity in the cloud-free environment is proportional to the change in saturation specic humidity, implying a reduction in qe in the

case of condensation (qs < 0). The factor of proportionality is a function of the cloud

fraction at the end of a timestep. As will be shown below, this has serious implication for the relative humidity of the cloud-free part. Since the cloud fraction is always positive and smaller than one the change in qe is always larger than that in qs implying a drying out of

the cloud-free air in relative humidity terms, which will be particularly strong for large cloud fractions. It is worthwhile noting that the only eect of the second-order term is that a is retained in the denominator of (5.35), which does not change the general behaviour of the scheme in changing qe.

The behaviour of the new scheme diers signicantly from the original. Again q and a are substituted to give

aqs+ 12aqs=aqs+ (1;a)qe;(1;a)

qs

2 + aqs;aqe;

82 5. The parametrization of cloud generation

qe = ₂qs: (5.36)

In this scheme the change in qe is again proportional to that in qs, implying a drying out

of the environment in absolute humidity terms. However, the change in qe is always half of

that in qs independent of cloud fraction, implying an increase in the relative humidity. The

reason for the drying out is immediately apparent from Figure 5.3. By producing clouds in part of the grid box, the denition of qe itself changes due to a change in the cloud-free area.

The new qe is now an average over the much drier area marked by (1;a) which contains

those points of the distribution which are too dry to contribute to the condensation.

Simple explicit solutions

The prognostic equations for cloud fraction and cloud condensate, i.e., (3.1) and (3.2) for the original T93 scheme and (5.29) and (5.30) for the scheme derived here can be solved numerically in an idealized framework. It is assumed that in a given volume, the saturation specic humidity is reduced in every timestep by an amount suciently small to ensure stable and accurate numerical solutions. This forcing inqs could be achieved for instance by

some continuous uplift. The only process assumed to occur in the volume is cloud formation. The equations for cloud fraction and condensate will be solved together with the respective humidity equation, which in this idealized setup reduces to

q =;l;

Note that by imposing the change in qs, qs;forc, the need for an explicit temperature

equation does not arise. However, the latent heat release due to condensation processes does counteract the imposed forcing and needs to be taken into account. This is achieved in the following way. The net change in qs over one timestep can be written as

qs = qs;forc+ qs;cond; (5.37)

where qs;forc is the prescribed forcing and qs;cond is the change in qs due to condensation

heating. Since the imposed changes in qs, and hence in temperature, are assumed to be

small, qs at time t+ 1 can be written with a high degree of accuracy as

qs(Tt+1) qs(T t_{) +} dqs dT(Tt+1 ;T t_{): (5.38)}

5.3. The generation of non-convective clouds 83 Using the rst law of thermodynamics (5.38) can be rewritten for the condensation contri- bution as qs;cond =; L cpdqdTsqcond: (5.39) Using qconda t_qs;

which is true for both sets of equations used here, (5.39) can be substituted into (5.37) to give the nal value for the forcing in qs in the presence of condensation as

qs qs;forc 1 +a_{t Lcp}dqs dT : (5.40) 0 100 200 300 400 500 600 700 800 900 1000 Time step 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 CC orig CC newls RH orig RH newls RHe orig RHe newls

Figure 5.4: Time evolution of cloud cover (solid), grid-mean relative humidity (dashed), and relative

humidity in the cloud free part of the grid box (dot-dashed) for the original T93 scheme (thin lines) and the source terms proposed here (thick lines). The solutions are for an idealized setup with initial conditions of

qs = 10g kg ;1, q = 8 g kg ;1, a= 0,l= 0, and a forcing of q s = ;0:01 gkg

;1. The latent heating

eects of condensation are accounted for.

Figure 5.4 shows the evolution of cloud cover, grid-mean relative humidity,RH, and relative humidity in the cloud free part, RHe, for integrations with the original scheme (thin lines)

84 5. The parametrization of cloud generation qs = 10 g kg ;1, q = 8 g kg ;1, T = 285:65K, p = 900 hPa, a = 0, and l = 0. q s;forc is set to ;0:01 gkg ;1.

As expected from (5.32), the original scheme increases cloud fraction faster than the new formulation initially. However, as was obvious from (5.35) this occurs at the expense of an unphysical behaviour of the environmental value of specic and therefore relative humidity. This feature becomes obvious in a dramatic way around time step 450. Here, the cloud fraction as predicted by the original scheme becomes larger than the grid-mean relative humidity. Since

RH =a+ (1;a)RHe;

this implies the impossible situation of negative values of relative humidity in the cloud-free part of the grid box. This is an example how ad-hoc choices for sources and sinks of cloud fraction and condensate can lead to an unphysical behaviour of the parametrization, if not checked for physical soundness. Note also that the original scheme would never give a cloud fraction equal to one. The new scheme does not show any of these problems. Although slower, cloud fraction evolves in balance with both the grid-mean and environmental values of relative humidity and all three reach the value of one within the same timestep.