Anelastic dynamics

AnelasticDynamics filters acoustic waves by linearizing about a hydrostatic reference state, following the thermodynamically consistent framework of Pauluis (2008). This formulation is suitable for most large-eddy simulations and mesoscale applications.

Anelastic approximation

To filter acoustic waves while retaining compressibility effects in buoyancy and thermodynamics, we linearize about a hydrostatic, horizontally uniform reference state $(pᵣ(z), ρᵣ(z))$ with constant reference potential temperature $θᵣ$. The key assumptions are

• Small Mach number and small relative density perturbations except in buoyancy.
• Hydrostatic reference balance: $\partial_z pᵣ = -ρᵣ g$.
• Mass flux divergence constraint: $\boldsymbol{\nabla \cdot}\, (ρᵣ\,\boldsymbol{u}) = 0$.

Define the specific volume of moist air and its reference value as

\[α = \frac{R^m T}{pᵣ} , \qquad αᵣ = \frac{R^d θᵣ}{pᵣ} ,\]

where $R^m$ is the mixture gas constant and $R^{d}$ is the dry-air gas constant. The buoyancy appearing in the vertical momentum is

\[b ≡ g \frac{α - αᵣ}{αᵣ} .\]

Conservative anelastic system

With $ρᵣ(z)$ fixed by the reference state, the prognostic equations advanced in Breeze are written in conservative form for the $ρᵣ$-weighted fields:

• Continuity (constraint):

\[\boldsymbol{\nabla \cdot}\, (ρᵣ \boldsymbol{u}) = 0 .\]

• Momentum:

\[\partial_t(ρᵣ \boldsymbol{u}) + \boldsymbol{\nabla \cdot}\, (ρᵣ \boldsymbol{u} \boldsymbol{u}) = - ρᵣ \boldsymbol{\nabla} \phi + ρᵣ \, b \hat{\boldsymbol{z}} + ρᵣ \boldsymbol{f} + \boldsymbol{\nabla \cdot}\, \boldsymbol{\mathcal{T}} ,\]

where $\phi$ is a nonhydrostatic pressure correction potential defined by the projection step (see below). Pressure is decomposed as $p = pᵣ(z) + p_h'(x, y, z, t) + p_n$, where $p_h'$ is a hydrostatic anomaly (obeying $\partial_z p_h' = -ρᵣ b$) and $p_n$ is the nonhydrostatic component responsible for enforcing the anelastic constraint. In the discrete formulation used here, $\phi$ coincides with the pressure correction variable.

• Total water:

\[\partial_t(ρᵣ q^t) + \boldsymbol{\nabla \cdot}\, (ρᵣ q^t \boldsymbol{u}) = S_q .\]

Moist static energy

Breeze advances a conservative moist static energy density

\[ρᵣ e ≡ ρᵣ \left ( c^{pm} T + g z - \mathscr{L}^l_r q^l - \mathscr{L}^i_r q^i \right ),\]

where $c^{p m}$ is the mixture heat capacity, $T$ is temperature, $g$ is gravitational acceleration, $z$ is height, $\mathscr{L}^l_r$ is the latent heat of condensation (vapor to liquid) at the energy reference temperature, and $\mathscr{L}^i_r$ is the latent heat of deposition (vapor to ice) at the energy reference temperature.

According to Pauluis (2008), the moist static energy obeys

\[\partial_t(ρᵣ e) + \boldsymbol{\nabla \cdot}\, (ρᵣ e \boldsymbol{u}) = ρᵣ w b + S_e ,\]

with vertical velocity $w$, buoyancy $b$ as above, and $S_e$ including microphysical and external energy sources/sinks. The $ρᵣ w b$ term is the buoyancy flux that links the energy and momentum budgets in the anelastic limit.

Thermodynamic closures needed for $R^m$, $c^{pm}$ and the Exner function $Π = (pᵣ / p_0)^{R^m / c^{pm}}$ are given in Thermodynamics section.

Time discretization and pressure correction

The anelastic formulation uses a multi-stage time integrator for advection, Coriolis, buoyancy, forcing, and tracer terms, coupled with a projection step to enforce the anelastic constraint at each substep. Denote the predicted momentum by $\widetilde{(ρᵣ \boldsymbol{u})}$. The projection is

1. Solve the variable-coefficient Poisson problem for the pressure correction potential $\phi$:

   \[\boldsymbol{\nabla \cdot}\, \big( ρᵣ \, \boldsymbol{\nabla} \phi \big) = \frac{1}{Δt} \, \boldsymbol{\nabla \cdot}\, \widetilde{(ρᵣ \boldsymbol{u})} ,\]

   with periodic lateral boundaries and homogeneous Neumann boundary conditions in $z$.

2. Update momentum to enforce $\boldsymbol{\nabla \cdot}\, (ρᵣ \boldsymbol{u}^{n+1}) = 0$:

   \[ρᵣ \boldsymbol{u}^{n+1} = \widetilde{(ρᵣ \boldsymbol{u})} - Δt \, ρᵣ \boldsymbol{\nabla} \phi .\]

In Breeze this projection is implemented as a Fourier-tridiagonal solve in the vertical with variable $ρᵣ(z)$, aligning with the hydrostatic reference state. The hydrostatic pressure anomaly $p_h'$ can be obtained diagnostically by vertical integration of buoyancy and used when desired to separate hydrostatic and nonhydrostatic pressure effects.

Symbols

• $ρᵣ(z)$, $pᵣ(z)$: Reference density and pressure satisfying hydrostatic balance for a constant $θᵣ$.
• $α = R^m T / pᵣ$, $αᵣ = R^d θᵣ / pᵣ$: Specific volume and its reference value.
• $b = g (α - αᵣ) / αᵣ$: Buoyancy.
• $e$: Moist static energy.
• $q^t$: Total specific humidity (vapor + condensates).
• $\phi$: Nonhydrostatic pressure correction potential used by the projection.