Large-scale condensation

Large-scale condensation in an atmospheric general circulation represents the micro-physical that kicks in when an air parcel reaches saturation. Subsequently, the water vapour inside it condenses, forms droplets around condensation nuclei, which grow, become heavy and eventually fall out as precipitation. This process is never actually representable at the resolution of global (or even regional) atmospheric models as typical cloud droplets have a size of micrometers. Atmospheric models therefore rely on large-scale quantities such as specific humidity, pressure and temperature within a given grid cell, even though there might be considerably variability of these quantities within a grid cell if the resolution was higher.

Condensation implementations

Currently implemented are

using SpeedyWeather
subtypes(SpeedyWeather.AbstractCondensation)

2-element Vector{Any}:
 ImplicitCondensation
 NoCondensation

which are described in the following.

Explicit large-scale condensation

We parameterize this process of large-scale condensation when relative humidity in a grid cell reaches saturation and remove the excess humidity quickly (given time integration constraints, see below) and with an implicit (in the time integration sense) latent heat release. Vertically integrating the tendency of specific humidity due to this process is then the large-scale precipitation.

Immediate condensation of humidity $q_i > q^\star$ at time step $i$ given its saturation $q^\star$ humidity calculated from temperature $T_i$ is

\[\begin{aligned} q_{i+1} - q_i &= q^\star(T_i) - q_i \\ T_{i+1} - T_i &= -\frac{L_v}{c_p}( q^\star(T_i) - q_i ) \end{aligned}\]

This condensation is explicit in the time integration sense, meaning that we only use quantities at time step $i$ to calculate the tendency. The latent heat release of that condensation is in the second equation. However, treating this explicitly poses the problem that because the saturation humidity is calculated from the current temperature $T_i$, which is increased due to the latent heat release, the humidity after this time step will be undersaturated.

Implicit large-scale condensation

Ideally, one would want to condense towards the new saturation humidity $q^\star(T_{i+1})$ at $i+1$ so that condensation draws the relative humidity back down to 100% not below it. Taylor expansion at $i$ of the equation above with $q^\star(T_{i+1})$ and $\Delta T = T_{i+1} - T_i$ (and $\Delta q$ similarly) to first order yields

\[q_{i+1} - q_i = q^\star(T_{i+1}) - q_i = q^\star(T_i) + (T_{i+1} - T_i) \frac{\partial q^\star}{\partial T} (T_i) + O(\Delta T^2) - q_i\]

Now we make a linear approximation to the derivative and drop the $O(\Delta T^2)$ term. Inserting the (explicit) latent heat release yields

\[\Delta q = q^\star(T_i) + -\frac{L_v}{c_p} \Delta q \frac{\partial q^\star}{\partial T} (T_i) - q_i\]

And solving for $\Delta q$ yields

\[\left[ 1 + \frac{L_v}{c_p} \frac{\partial q^\star}{\partial T} (T^i) \right] \Delta q = q^\star(T_i) - q_i\]

meaning that the implicit immediate condensation can be formulated as (see also ^{[Frierson2006]})

\[\begin{aligned} q_{i+1} - q_i &= \frac{q^\star(T_i) - q_i}{1 + \frac{L_v}{c_p} \frac{\partial q^\star}{\partial T}(T_i)} \\ T_{i+1} - T_i &= -\frac{L_v}{c_p}( q_{i+1} - q_i ) \end{aligned}\]

With Euler forward time stepping this is great, but with our leapfrog timestepping + RAW filter this is very dispersive (see #445) although the implicit formulation is already much better. We therefore introduce a time step $\Delta t_c$ which makes the implicit condensation not immediate anymore but over several time steps $\Delta t$ of the leapfrogging.

\[\begin{aligned} \delta q = \frac{q_{i+1} - q_i}{\Delta t} &= \frac{q^\star(T_i) - q_i}{ \Delta t_c \left( 1 + \frac{L_v}{c_p} \frac{\partial q^\star}{\partial T}(T_i) \right)} \\ \delta T = \frac{T_{i+1} - T_i}{\Delta t} &= -\frac{L_v}{c_p}( \frac{q_{i+1} - q_i}{\Delta t} ) \end{aligned}\]

For $\Delta t = \Delta t_c$ we have an immediate condensation, for $n = \frac{\Delta t_c}{\Delta t}$ condensation takes place over $n$ time steps. One could tie this time scale for condensation to a physical unit, like 6 hours, but because the time step here is ideally short, but cannot be too short for numerical stability, we tie it here to the time step of the numerical integration. This also means that at higher resolution condensation is more immediate than at low resolution, but the dispersive time integration of this term is in all cases similar (and not much higher at lower resolution).

Large-scale precipitation

The tendencies $\delta q$ in units of kg/kg/s are vertically integrated to diagnose the large-scale precipitation $P$ in units of meters

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

with gravity $g$, water density $\rho$ and time step $\Delta t$. $P$ is therefore interpreted as the amount of precipitation that falls down during the time step $\Delta t$ of the time integration. Note that $\delta q$ is always negative due to the $q > q^\star$ condition for saturation, hence $P$ is positive only. It is then accumulated over several time steps, e.g. over the course of an hour to yield a typical rain rate of mm/h. The water density is taken as reference density of $1000~kg/m^3$

References