Dycore equations and algorithms

This section summarizes the governing equations behind Breeze’s atmospheric dynamics and the anelastic formulation used by AtmosphereModel, following the thermodynamically consistent framework of Pauluis (2008).

We begin with the compressible Navier-Stokes momentum equations and reduce them to an anelastic, conservative form. We then introduce the moist static energy equation and outline the time-discretized pressure correction used to enforce the anelastic constraint.

Compressible momentum equations

Let $ρ$ denote density, $\boldsymbol{u}$ velocity, $p$ pressure, $\boldsymbol{f}$ non-pressure body forces (e.g., Coriolis), and $\boldsymbol{\tau}$ subgrid/viscous stresses. With gravity $- g \hat{\boldsymbol{z}}$, the inviscid compressible equations in flux form are

\[\begin{aligned} &\text{Mass:} && \partial_t ρ + \boldsymbol{\nabla \cdot}\, (ρ \boldsymbol{u}) = 0 ,\\ &\text{Momentum:} && \partial_t(ρ \boldsymbol{u}) + \boldsymbol{\nabla \cdot}\, (ρ \boldsymbol{u} \boldsymbol{u}) + \boldsymbol{\nabla} p = - ρ g \hat{\boldsymbol{z}} + ρ \boldsymbol{f} + \boldsymbol{\nabla \cdot}\, \boldsymbol{\tau} . \end{aligned}\]

Notation $\boldsymbol{\nabla \cdot}\, (ρ \boldsymbol{u} \boldsymbol{u})$ above denotes a vector whose components are $[\boldsymbol{\nabla \cdot}\, (ρ \boldsymbol{u} \boldsymbol{u})]_i = \boldsymbol{\nabla \cdot}\, (ρ u_i \boldsymbol{u})$.

For moist flows we also track total water (vapor + condensates) via

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

where $q^t$ is total specific humidity and $S_q$ accounts for sources/sinks from microphysics and boundary fluxes.

Thermodynamic relations (mixture gas constant $R^m$, heat capacity $c^{pm}$, Exner function, etc.) are summarized in the Thermodynamics section.

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:

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

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 (Pauluis, 2008)

Following Pauluis (2008), Breeze advances a conservative moist static energy density

\[ρᵣ e ≡ ρᵣ c^{pd} θ ,\]

where $c^{p d}$ is the dry-air heat capacity at constant pressure and $θ$ is the (moist) potential temperature. The prognostic equation reads

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

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

Breeze uses an explicit 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 and closures used here

• $ρᵣ(z)$, $pᵣ(z)$: 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.
• $e = c^{pd} \, θ$: Energy variable used for moist static energy in the conservative equation.
• $q^t$: Total specific humidity (vapor + condensates).
• $\phi$: Nonhydrostatic pressure correction potential used by the projection.

See Thermodynamics section for definitions of $R^m(q)$, $c^{pm}(q)$, and $Π$.