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Chapter 5. Convection
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Chapter 1. Overview
Chapter 2. Radiation
Chapter 3. Turbulent diffusion and interactions with the surface
Chapter 4. Subgrid-scale orographic drag
Chapter 5. Convection
Chapter 6. Clouds and large-scale precipitation
Chapter 7. Land suface parametrization
Chapter 8. Methane oxidation
Chapter 9. Climatological data
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5.9 Discretization of the model equations
The flux divergence in the large-scale budget equations (5.1) and in the cloud equations (5.3) and (5.12) are approximated by centred finite differences as
|
(5.47) |
The definition of the large-scale variables at half levels pose a problem, when the half-level values defined by linear interpolation of full-level values very noisy profiles evolve in time particularly with regard to humidity. Much smoother profiles are obtained when the half-level values are determined by downward extrapolation from the next full level above along a cloud-ascent through that level:
|
(5.48) |
where
is the saturation moist static energy. Using an extrapolation
like (5.48) for calculating
the downward transports is also more consistent with the calculation of
the updraughts where cloud air is transported upwards through level 
with the thermal
state below that level and equally with the downdraughts which depend only
on values of s and q above that level. Similarly, because
of (5.47) the downward transport of environmental
air through the same level accounts now only for thermal properties above
that level. The choice of a moist adiabat for extrapolation is dictated
by the property of the moist static energy which is, by convection in the
absence of downdraughts, only changed through the fluxes of moist static
energy
|
(5.49) |
As the lines of the saturation moist static energy
through point
and the updraught moist static energy are almost parallel,
apart from entrainment effects, the difference 
is little affected by the vertical discretization.For horizontal winds, values at model half levels are set to those on the full model level below.
The ascent in the updraughts is obtained by vertical integration of (5.3). Starting at the surface the condensation level (equal to the lowest half-level which is saturated or supersaturated and where buoyancy is greater than
) is determined from an adiabatic ascent. The cloud profile
above cloud base is determined layer by layer by first doing a dry adiabatic
ascent with entrainment and detrainment included and then adjusting temperature
and moisture towards a saturated state, taking into account condensation
and freezing processes. The buoyancy of the parcel is calculated taking
into account the effects of cloud and precipitation water loading i.e.
|
(5.50) |
Special care has to be taken in the discretization of (5.8) because of overshooting effects. A centred differencing scheme is used so that
|
(5.51) |
Finally, we mention that for numerical reasons the environmental air must not be convectively unstably stratified:
|
(5.52) |
In fact, one of the forecasts with the ECMWF global model became numerically unstable when (5.23) was not imposed.
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05.04.2002 |
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