Convection

Convection is the atmospheric process of rising motion because of positively buoyant air parcels compared to its surroundings. In hydrostatic models like the primitive equation model in SpeedyWeather.jl convection has to be parameterized as the vertical velocity is not a prognostic variable that depends on vertical stability but rather diagnosed to satisfy horizontal divergence. Convection can be shallow and non-precipitating denoting that buoyant air masses can rise but do not reach saturation until they reach a level of zero buoyancy. But convection can also be deep denoting that saturation has been reached during ascent whereby the latent heat release from condensation provides additional energy for further ascent. Deep convection is therefore usually also precipitating as the condensed humidity forms cloud droplets that eventually fall down as convective precipitation. See also Large-scale condensation in comparison.

Convection implementations

Currently implemented are

using SpeedyWeather
subtypes(SpeedyWeather.AbstractConvection)

3-element Vector{Any}:
 DryBettsMiller
 NoConvection
 SimplifiedBettsMiller

which are described in the following.

Simplified Betts-Miller convection

We follow the simplification of the Betts-Miller convection scheme ^{[Betts1986][BettsMiller1986]} as studied by Frierson, 2007 ^{[Frierson2007]}. The central idea of this scheme is to represent the effect of convection as an adjustment towards a (pseudo-) moist adiabat reference profile and its associated humidity profile. Meaning that conceptually for every vertical column in the atmosphere we

1. Diagnose the vertical temperature and humidity profile of the environment relative to the adiabat up to the level of zero buoyancy.
2. Decide whether convection should take place and whether it is deep (precipitating) or shallow (non-precipitating).
3. Relax temperature and humidity towards (corrected) profiles from 1.

Reference profiles

The dry adiabat is

\[T = T_0 (\frac{p}{p_0})^\frac{R}{c_p}\]

The temperature $T$ of an air parcel at pressure $p$ is determined by the temperature $T_0$ it had at pressure $p_0$ (this can be surface but it does not have to be) and the gas constant for dry air $R = 287.04 J/K/kg$ and the heat capacity $c_p = 1004.64 J/K/kg$. The pseudo adiabat follows the dry adiabat until saturation is reached (the lifting condensation level, often abbreviated to LCL), $q > q^\star$. Then it follows the pseudoadiabatic lapse rate

\[\Gamma = -\frac{dT}{dz} = \frac{g}{c_p}\left( \frac{ 1 + \frac{q^\star L_v }{(1-q^\star)^2 R_d T_v}}{ 1 + \frac{q^\star L_v^2}{(1-q^\star)^2 c_p R_v T^2}}\right)\]

with gravity $g$, heat capacity $c_p$, the saturation specific humidity of the parcel $q^\star$ (which is its specific humidity given that it has already reached saturation), latent heat of vaporization $L_v$, dry gas constant $R_d$, water vapour gas constant $R_v$, and Virtual temperature $T_v$. Starting with a temperature $T$ and humidity $q = q^\star$ at the lifting condensation level temperature aloft changes with $dT = -\frac{d\Phi}{c_p}(...)$ between two layers separated $d\Phi$ in geopotential $\Phi$ apart. On that new layer, $q^\star$ is recalculated as well as the virtual temperature $T_v = T(1 + \mu q^\star)$. $\mu$ is derived from the ratio of dry to vapour gas constants see Virtual temperature. Note that the pseudoadiabatic ascent is independent of the environmental temperature and humidity and function of temperature and humidity of the parcel only (although that one starts with surface temperature and humidity from the environment). Solely the level of zero buoyancy is determined by comparing the parcel's virtual temperature $T_v$ to the virtual temperature of the environment $T_{v,e}$ at that level. Level of zero buoyancy is reached when $T_v = T_{v,e}$ but continues for $T_v > T_{v,e}$ which means that the parcel is still buoyant. Note that the virtual temperature includes the effect that humidity has on its density.

The (absolute) temperature a lifted parcel has during ascent (following its pseudoadiabat, dry and/ or moist, until reaching the level of zero buoyancy) is then taken as the reference temperature profile $T_{ref}$ that the Betts-Miller convective parameterization relaxes towards as a first guess (with a following adjustment as discussed below). The humidity profile is taken as $q_{ref} = RH_{SBM}T_{ref}$ with a parameter $RH_{SBM}$ (default $RH_{SBM} = 0.7$) of the scheme (Simplified Betts-Miller, SBM) that determines a constant relative humidity of the reference profile.

First-guess relaxation

With the Reference profiles $T_{ref}, q_{ref}$ obtained, we relax the actual environmental temperature $T$ and specific humidity $q$ in the column

\[\begin{aligned} \delta q &= - \frac{q - q_{ref}}{\tau_{SBM}} \\ \delta T &= - \frac{T - T_{ref}}{\tau_{SBM}} \end{aligned}\]

with the second parameter of the parameterization, the time scale $\tau_{SBM}$. Note that because this is a first-guess relaxation, these tendencies are not actually the resulting tendencies from this scheme. Those will be calculated in Corrected relaxation.

Note that above the level of zero buoyancy no relaxation takes place $\delta T = \delta q = 0$, or, equivalently $T = T_{ref}$, $q = q_{ref}$ there. Vertically integration from surface $p_0$ to level of zero buoyancy in pressure coordinates $p_{LZB}$ yields

\[\begin{aligned} P_q &= - \int_{p_0}^{p_{LZB}} \delta q \frac{dp}{g} \\ P_T &= \int_{p_0}^{p_{LZB}} \frac{c_p}{L_v} \delta T \frac{dp}{g} \end{aligned}\]

$P_q$ is the precipitation in units of $kg / m^2 / s$ due to drying (as a consequence of the humidity tendency) and $P_T$ is the precipitation in the same units due to warming (as resulting from temperature tendencies). Note that they are the vertically difference between current profiles and the references profiles, so if $P_q > 0$ this would mean that a convective adjustment to $q_{ref}$ would release humidity from the column through condensation, but $P_q$ can also be negative. Consequently similar for $P_T$.

Convective criteria

We now distinguish three cases

1. Deep convection when $P_T > 0$ and $P_q > 0$
2. Shallow convection when $P_T > 0$ and $P_q <= 0$
3. No convection for $P_T <= 0$.

Note that to evaluate these cases it is not necessary to divide by $\tau_{SBM}$ in the first-guess relaxation, neither are the $1/g$ and $\tfrac{c_p}{g L_v}$ necessary to multiply during the vertical integration as all are positive constants. While this changes the units of $P_T, P_q$ they can be reused in the following.

Deep convection

Following Frierson, 2007 ^{[Frierson2007]} in order to conserve enthalpy we correct the reference profile for temperature $T_{ref} \to T_{ref, 2}$ so that $P_T = P_q$.

\[T_{ref, 2} = T_{ref} + \frac{1}{\Delta p c_p} \int_{p_0}^{p_{LZB}} c_p (T - T_{ref}) + L_v (q - q_{ref}) dp\]

$\Delta p$ is the pressure difference $p_{LZB} - p_0$. The terms inside the integral are rearranged compared to Frierson, 2007 to show that the vertical integral in First-guess relaxation really only has to be computed once.

Shallow convection

In the following we describe the "qref" scheme from Frierson, 2007 which corrects reference profiles for both temperature and humidity to guarantee that $P_q = 0$, i.e. no precipitation during convection. In that sense, shallow convection is non-precipitating. Although shallow convection is supposed to be shallow we do not change the height of the convection and keep using the $p_{LZB}$ determined during the calculation of the Reference profiles.

\[\begin{aligned} \Delta q &= \int_{p_0}^{p_{LZB}} q - q_{ref} dp \\ Q_{ref} &= \int_{p_0}^{p_{LZB}} -q_{ref} dp \\ f_q &= 1 - \frac{\Delta q}{Q_ref} \\ q_{ref, 2} &= f_q q_{ref} \\ \Delta T &= \frac{1}{\Delta p} \int_{p_0}^{p_{LZB}} -(T - T_{ref}) dp \\ T_{ref,2} &= T_{ref} - \Delta T \end{aligned}\]

Corrected relaxation

After the reference profiles have been corrected in Deep convection and Shallow convection we actually calculate tendencies from

\[\begin{aligned} \delta q &= - \frac{q - q_{ref, 2}}{\tau_{SBM}} \\ \delta T &= - \frac{T - T_{ref, 2}}{\tau_{SBM}} \end{aligned}\]

with $\tau_{SBM} = 2h$ as default.

Convective precipitation

The convective precipitation $P$ results then from the vertical integration of the $\delta q$ tendencies, similar to Large-scale precipitation.

\[P = -\int \frac{\Delta t}{g \rho} \delta q dp\]

In the shallow convection case $P=0$ due to the correction even though in the first guess relaxation $P<0$ was possible, but for deep convection $P>0$ by definition.

Dry convection

In the primitive equation model with humidity the Betts-Miller convection scheme as described above is defined. Without humidity, a dry version reduces to the Shallow convection case. The two different shallow convection schemes in Frierson 2007^{[Frierson2007]}, the "shallower" shallow convection scheme and the "qref" (as implemented here in Shallow convection) in that case also reduce to the same formulation. The dry Betts-Miller convection scheme is the default in the primitive equation model without humidity.

References