- 4.1.1 Updraft Ensemble
- 4.1.2 Downdraft Ensemble
- 4.1.3 Closure
- 4.1.4 Numerical Approximations
- 4.1.5 Deep Convective Tracer Transport

4.1 Deep Convection

The large-scale budget equations distinguish between a cloud and sub-cloud layer where temperature and moisture response to convection in the cloud layer is written in terms of bulk convective fluxes as

for , where is the height of the cloud base. For , where is the surface height, the sub-cloud layer response is written as

where the net vertical mass flux in the convective region, , is comprised of upward, , and downward, , components, and are the large-scale condensation and evaporation rates, , , , , , , are the corresponding values of the dry static energy and specific humidity, and is the cloud base mass flux.

The updraft ensemble is represented as a collection of entraining plumes, each with a characteristic fractional entrainment rate . The moist static energy in each plume is given by

Mass carried upward by the plumes is detrained into the environment in a thin layer at the top of the plume, , where the detrained air is assumed to have the same thermal properties as in the environment (). Plumes with smaller penetrate to larger . The entrainment rate for the plume which detrains at height is then determined by solving (4.6), with lower boundary condition :

(4.7) | |||

(4.8) | |||

(4.9) | |||

(4.10) | |||

(4.11) |

Since the plume is saturated, the detraining air must have , so that

Then, is determined by solving (4.12) iteratively at each .

The top of the shallowest of the convective plumes, is assumed to be no lower than the mid-tropospheric minimum in saturated moist static energy, , ensuring that the cloud top detrainment is confined to the conditionally stable portion of the atmospheric column. All condensation is assumed to occur within the updraft plumes, so that . Each plume is assumed to have the same value for the cloud base mass flux , which is specified below. The vertical distribution of the cloud updraft mass flux is given by

where is the maximum detrainment rate, which occurs for the plume detraining at height , and is the entrainment rate for the updraft that detrains at height . Detrainment is confined to regions where decreases with height, so that the total detrainment for . Above ,

The total entrainment rate is then just given by the change in mass flux and the total detrainment,

The updraft budget equations for dry static energy, water vapor mixing
ratio, moist static energy, and cloud liquid water, , are:

where (4.18) is formed from (4.16) and (4.17) and detraining air has been assumed to be saturated ( and ). It is also assumed that the liquid content of the detrained air is the same as the ensemble mean cloud water ( ). The conversion from cloud water to rain water is given by

following Lord et al. [116], with .

Since , and are given by (4.13-4.15),
and and are environmental profiles, (4.18) can be
solved for , given a lower boundary condition. The lower boundary
condition is obtained by adding a K temperature perturbation to
the dry (and moist) static energy at cloud base, or
at . Below the lifting condensation level (LCL),
and are given by (4.16) and (4.17). Above
the LCL, is reduced by condensation and is increased by
the latent heat of vaporization. In order to obtain to obtain a
saturated updraft at the temperature implied by , we define
as the temperature perturbation in the updraft, then:

Substituting (4.22) and (4.23) into (4.21),

The required updraft quantities are then

With given by (4.28), (4.16) can be solved for , then (4.19) and (4.20) can be solved for and .

The expressions above require both the saturation specific humidity to be

where is the saturation vapor pressure, and its dependence on temperature (in order to maintain saturation as the temperature varies) to be

(4.31) | |||

(4.32) | |||

(4.33) |

The deep convection scheme does not use the same approximation for the saturation vapor pressure as is used in the rest of the model. Instead,

where , , K and K is the freezing point. For this approximation,

We note that the expression for in the code gives

The expressions for in (4.38) and (4.39) are not identical. Also, and .

Downdrafts are assumed to exist whenever there is precipitation production in the updraft ensemble where the downdrafts start at or below the bottom of the updraft detrainment layer. Detrainment from the downdrafts is confined to the sub-cloud layer, where all downdrafts have the same mass flux at the top of the downdraft region. Accordingly, the ensemble downdraft mass flux takes a similar form to (4.13) but includes a ``proportionality factor'' to ensure that the downdraft strength is physically consistent with precipitation availability. This coefficient takes the form

where is the total precipitation in the convective layer and is the rain water evaporation required to maintain the downdraft in a saturated state. This formalism ensures that the downdraft mass flux vanishes in the absence of precipitation, and that evaporation cannot exceed some fraction, , of the precipitation, where = 0.2.

The parameterization is closed, i.e., the cloud base mass fluxes are
determined, as a function of the rate at which the cumulus consume
convective available potential energy (CAPE). Since the large-scale
temperature and moisture changes in both the cloud and sub-cloud layer
are linearly proportional to the cloud base updraft mass flux (*e.g.* see
eq. 4.2 - 4.5), the CAPE change due to convective
activity can be written as

where is the CAPE consumption rate per unit cloud base mass flux. The closure condition is that the CAPE is consumed at an exponential rate by cumulus convection with characteristic adjustment time scale s:

The quantities , , , , are defined on layer interfaces, while , , are defined on layer midpoints. , , , are required on both midpoints and interfaces and the interface values are determined from the midpoint values as

All of the differencing within the deep convection is in height coordinates. The differences are naturally taken as

(4.44) |

where and represent values on the upper and lower interfaces, respectively for layer . The convention elsewhere in this note (and elsewhere in the code) is . Therefore, we avoid using the compact notation, except for height, and define

(4.45) |

so that corresponds to the variable dz(k) in the deep convection code.

Although differences are in height coordinates, the equations are cast in flux form and the tendencies are computed in units . The expected units are recovered at the end by multiplying by .

The environmental profiles at midpoints are

(4.46) | |||

(4.47) | |||

(4.48) | |||

(4.49) | |||

(4.50) | |||

(4.51) |

The environmental profiles at interfaces of , , , and are determined using (4.43) if is large enough.

(4.52) |

For and the condition is

(4.53) |

Interface values of are not needed and interface values of are given by

(4.54) |

The unitless updraft mass flux (scaled by the inverse of the cloud base mass flux) is given by differencing (4.13) as

(4.55) |

with the boundary condition that . The entrainment and detrainment are calculated using

(4.56) | |||

(4.57) | |||

(4.58) |

Note that and differ only by the value of .

The updraft moist static energy is determined by differencing (4.18)

with , where is the layer of maximum .

Once is determined, the lifting condensation level is found by
differencing (4.16) and (4.17) similarly to
(4.18):

The detrainment of is given by not by , since detrainment occurs at the environmental value of . The detrainment of is given by , even though the updraft is not yet saturated. The LCL will usually occur below , the level at which detrainment begins, but this is not guaranteed.

The lower boundary conditions,
and
, are determined from the first midpoint values in the plume,
rather than from the interface values of and . The solution of
(4.61) and (4.62) continues upward until the
updraft is saturated according to the condition

(4.63) | |||

(4.64) |

The condensation (in units of m) is determined by a centered differencing of (4.16):

(4.65) |

(4.66) |

The rain production (in units of m) and condensed liquid are then determined by differencing (4.19) as

(4.67) |

and (4.20) as

(4.68) |

Then

(4.69) | |||

(4.70) | |||

(4.71) |

We assume the updrafts and downdrafts are described by a steady state mass continuity equation for a ``bulk'' updraft or downdraft

The subscript is used to denote the updraft () or downdraft () quantity. here is the mass flux in units of Pa/s defined at the layer interfaces, is the mixing ratio of the updraft or downdraft. is the mixing ratio of the quantity in the environment (that part of the grid volume not occupied by the up and downdrafts). and are the entrainment and detrainment rates (units of s) for the up- and down-drafts. Updrafts are allowed to entrain or detrain in any layer. Downdrafts are assumed to entrain only, and all of the mass is assumed to be deposited into the surface layer.

Equation 4.72 is first solved for up and downdraft mixing ratios and , assuming the environmental mixing ratio is the same as the gridbox averaged mixing ratio .

Given the up- and down-draft mixing ratios, the mass continuity equation used to solve for the gridbox averaged mixing ratio is

These equations are solved for in subroutine `CONVTRAN`. There
are a few numerical details employed in `CONVTRAN` that are worth
mentioning here as well.

- mixing quantities needed at interfaces are calculated using the geometric mean of the layer mean values.
- simple first order upstream biased finite differences are used to solve 4.72 and 4.73.
- fluxes calculated at the interfaces are constrained so that the
resulting mixing ratios are positive definite.
*This means that this parameterization is not suitable for moving mixing ratios of quantities meant to represent perturbations of a trace constituent about a mean value*(in which case the quantity can meaningfully take on positive and negative mixing ratios). The algorithm can be modified in a straightforward fashion to remove this constraint, and provide meaningful transport of perturbation quantities if necessary.*the reader is warned however that there are other places in the model code where similar modifications are required because the model assumes that all mixing ratios should be positive definite quantities*.