Microphysics

Source terms

Sources from cloud microphysics $\mathcal{S}$ represent the transfer of mass between different water categories such as cloud water, cloud ice or precipitation, as well as the latent heat release due to phase changes. The model supports three different cloud microphysics and precipitation representations:

• equilibrium cloud formation coupled with a 0-moment microphysics scheme,
• nonequilibrium cloud formation coupled with a 1-moment microphysics scheme representing both liquid and ice phase precipitation,
• nonequilibrium cloud formation coupled with a 2-moment microphysics scheme representing liquid phase precipitation.

The equilibrium 0-moment option does not introduce any new variables to the state vector. The cloud condensate and phase partitioning are diagnosed using saturation adjustment and the 0-moment microphysics provides a sink on total water due to precipitation. Precipitation is immediately removed from the computational domain. The nonequilibrium 1-moment option expands the state vector by four microphysics tracers: cloud liquid water, cloud ice, rain and snow $(q_{liq}, q_{ice}, q_{rai}, q_{sno})$. The nonequilibrium 2-moment option expands the state vector by four microphysics tracers: cloud liquid water and droplet number concentration, rain water and drop number concentration: $(q_{liq}, N_{liq}, q_{rai}, N_{rai})$.

All microphysics mass tracers are part of the working fluid and are defined as a ratio of the tracer mass over the mass of the working fluid. The different cloud and precipitation source terms are provided by CloudMicrophysics.jl library and are defined as the change of mass normalized by the mass of the working fluid. See the CloudMicrophysics.jl docs for more details.

Considering the transition from $x \rightarrow y$ where $x$ and $y$ can be any of the microphysics tracers

\[\mathcal{S}_{x \rightarrow y} := \frac{\frac{dm_x}{dt}}{m_{dry} + m_{vap} + m_{liq} + m_{ice} + m_{rai} + m_{sno}}\]

If $\mathcal{S}_{x \rightarrow y}$ is a sink of $q_{tot}$ from the 0-moment scheme it has a corresponding sink on density and energy:

\[\frac{d}{dt} \rho = \frac{d}{dt} \rho q_{tot} = \rho \mathcal{S}_{x \rightarrow y}\]

\[\frac{d}{dt} \rho e = \rho \mathcal{S}_{x \rightarrow y} (I_{y} + \Phi)\]

where $I_{y}$ is the internal energy of the $y$ phase.

In nonequilibrium cloud formation and the 1-moment and 2-moment schemes, since all microphysics tracers are part of the working fluid, microphysics sources do not introduce corresponding sources/sinks of total water, density or total energy.

Sedimentation

All microphysics tracers sediment with a bulk (group) sedimentation velocity parameterized via CloudMicrophysics.jl. Sedimentation is done implicitly through a first-order upwinding scheme. Because all tracers are part of the working fluid, their sedimentation results in sedimentation terms for density and total energy.

Stability and positivity

Microphysics tracers should remain positive throughout the simulation. The numerics of the model however, may result in errors that lead to the spurious formation of small negative numbers. Most common causes of those errors are:

• spurious oscillations caused by the high order horizontal transport scheme,
• time integration of microphysics sources at time-step that is longer than the stability limit,
• use of hyperdiffusion.

Our strategy is to minimize the untoward effects of those errors.

Limiters

All microphysics source terms are individually limited by the available mass of the source tracer $x$. We typically set $lim_x = \frac{q_x}{a \; dt}$ where $a > 1$. The limiter caps positive tendencies (sources) and negative tendencies (sinks) independently:

\[\text{tendency\_limiter}(\mathcal{S}, lim_{pos}, lim_{neg}) = \begin{cases} \min(\mathcal{S}, lim_{pos}), & \mathcal{S} \geq 0 \\ -\min(-\mathcal{S}, lim_{neg}), & \mathcal{S} < 0 \end{cases}\]

If the source tendency exceeds the available tracer $lim_{pos}$, it is capped at the available amount. Similarly, sink tendencies are capped by $lim_{neg}$. Below figure illustrates the behavior of the limiter for positive and negative force with the positive bound set to 5 and the negative bound set to 2.

Hyperdiffusion

Hyperdiffusion ($\nabla^4$ operator) is a tendency applied in order to remove noise buildup at the small scales and improve the model stability. It's more selective than standard diffusion operator, and applies the damping only at the smallest scales of the simulation without degrading the sharp features of the modeled tracers.

Hyperdiffusion is a higher order derivative operator, and as a result does not guarantee positivity. The user has a choice to opt-in certain microphysics tracers to use hyperdiffusion. By default hyperdiffusion is applied to total water and cloud tracers, but not precipitating tracers. The magnitude of hyperdiffusion acting on precipitation tracers can be changed by adjusting the free parameter tracer_hyperdiffusion_factor.

Diffusion

ClimaAtmos provides different horizontal and vertical diffusion schemes that can be used to improve model stability and reduce the negative numbers and spurious oscillations.

Horizontal diffusion tendency is based on either the Smagorinsky-Lilly model [1] or the Anisotropic Minimum-Dissipation model (AMD) [2] and is applied explicitly.

Vertical diffusion tendency can be based on either of the above models, or computed as a decaying with height function that is capped at some value above the tropopause. Vertical diffusion can be applied implicitly. When using the decay with height options (VerticalDiffusion or DecayWithHeight), similar to hyperdiffusion, diffusion is applied to total water and cloud tracers. The magnitude of diffusion acting on precipitation tracers can be scaled using the tracer_vertical_diffusion_factor. There is no such scaling applied when using the Smagorinsky-Lilly or AMD models.

Non-negativity constraints

Often, the diffusion and limiters described above are not enough to ensure positivity of the microphysics tracers. ClimaAtmos supports three additional constraints that can be used to enforce non-negativity of the microphysics tracers. This is controlled by the tracer_nonnegativity_method in the AtmosWater struct. The availalble options are:

• TracerNonnegativityElementConstraint: This option enforces non-negativity by consistently ensuring that the mass of the tracer is conserved within the element. It uses the Limiters.compute_bounds! and Limiters.apply_limiter! functions to redistribute the mass of the tracer within the element such that the tracer concentration is non-negative and bounded by the maximum value in the element. Effectively, this method borrows mass from the neighboring nodes within the element to fill the negative holes. This method is conservative and does not introduce any source/sink of total water mass.

• TracerNonnegativityVaporConstraint: This option enforces non-negativity by borrowing mass from the water vapor. If a microphysics tracer $q_x$ becomes negative at a given node, it is set to zero. Since the total water content $q_{tot}$ is conserved during this operation, and $q_{tot} = q_{vap} + \sum q_x$, setting a negative $q_x$ to zero implicitly decreases $q_{vap}$.

  \[q_x = \max(0, q_x)\]

  This method is applied instantaneously at the end of each time step (or stage). It preserves the total water mass but redistributes it between phases. It should be used with caution as it can lead to negative water vapor if the negative hole in $q_x$ is large, although usually the negative values are small and there is plenty of water vapor available.

• TracerNonnegativityVaporTendency: This option is similar to TracerNonnegativityVaporConstraint in that it borrows mass from water vapor, but it does so via a tendency term rather than an instantaneous adjustment. It computes a tendency that tends to restore the tracer to zero over the timestep $\Delta t$. The tendency is limited by the available water vapor $q_{vap}$ to avoid creating negative vapor.

  \[\frac{\partial q_x}{\partial t} = \dots + \mathcal{S}_{fixer}\]

  where $\mathcal{S}_{fixer}$ is positive if $q_x < 0$. This method is less aggressive than the instantaneous constraint and integrates the correction into the time stepping scheme.

Aerosol Activation for 2-Moment Microphysics

Aerosol activation uses functions from the CloudMicrophysics.jl library, based on the Abdul-Razzak and Ghan (ARG) parameterization. ARG predicts the number of activated cloud droplets assuming a parcel of clear air rising adiabatically. This formulation is traditionally applied only at cloud base, where the maximum supersaturation typically occurs.

To enable ARG to be used locally (i.e., without explicitly identifying cloud base), CloudMicrophysics.jl implements a modified equation for the maximum supersaturation that accounts for the presence of pre-existing liquid and ice particles. This allows activation to be applied inside clouds. To ensure that activation occurs only where physically appropriate, we apply additional clipping logic:

• If the predicted maximum supersaturation is less than the local supersaturation (i.e., supersaturation is decreasing), aerosol activation is not applied.
• If the predicted number of activated droplets is less than the existing local cloud droplet number concentration, activation is also suppressed.

This ensures that droplet activation occurs only in physically meaningful regions—typically near cloud base—even though the activation routine can be applied throughout the domain.