Dycore equations and algorithms

This section summarizes the governing equations behind Breeze's atmospheric dynamics used by AtmosphereModel. Breeze supports two dynamical formulations:

• AnelasticDynamics: Filters acoustic waves by linearizing about a hydrostatic reference state, following the thermodynamically consistent framework of Pauluis (2008). Suitable for most large-eddy simulations and mesoscale applications.

• CompressibleDynamics: Solves the fully compressible Euler equations with prognostic density. Retains acoustic waves and is useful for validation, acoustic studies, and problems where full compressibility is important.

We begin with the compressible Navier-Stokes momentum equations and reduce them to an anelastic, conservative form. We then describe the fully compressible formulation, and finally 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 $\boldsymbol{\tau}$ the kinematic (per-mass) subgrid/viscous stresses. We denote the corresponding dynamic (per-volume) stresses by $\boldsymbol{\mathcal{T}} = ρ \, \boldsymbol{\tau}$. With gravity $- g \hat{\boldsymbol{z}}$, the inviscid compressible equations in flux form are

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

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:

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

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.

Compressible dynamics

Breeze also supports fully compressible dynamics via CompressibleDynamics, which directly time-steps the density without an anelastic constraint. This formulation retains acoustic waves and is suitable for problems where compressibility effects are important, such as:

• Acoustic wave propagation
• Shock-resolving simulations
• Validation against fully compressible reference solutions

Prognostic equations

The compressible formulation advances density $ρ$ as a prognostic variable alongside momentum:

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

Equation of state

Pressure is computed from the ideal gas law using prognostic density and potential temperature:

\[p = ρ R^m T\]

where $ρ$ is density, $R^m$ is the mixture gas constant, and $T$ is temperature.

For dry air, this simplifies to the classic Poisson equation relating pressure, density, and potential temperature.

Time stepping

CompressibleDynamics uses fully explicit time integration without a pressure correction step. This permits acoustic waves to propagate and requires a CFL condition based on the sound speed:

\[Δt < \frac{Δx}{c_s}, \quad \text{where} \quad c_s = \sqrt{\frac{c^{pm}}{c^{vm}} R^m T}\]

is the adiabatic sound speed (approximately 340 m/s at standard conditions).

Comparison with anelastic dynamics

Time discretization and pressure correction (Anelastic)

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 and closures used here

Anelastic dynamics

• $ρᵣ(z)$, $pᵣ(z)$: 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.

Compressible dynamics

• $ρ$: Prognostic density field.
• $p$: Pressure computed from equation of state.
• $θ$: Potential temperature.
• $c_s$: Sound speed.

Diffusion and turbulence closure notation

• $\boldsymbol{\tau}$: Kinematic (per-mass) subgrid/viscous stress tensor returned by Oceananigans closures.
• $\boldsymbol{\mathcal{T}} = ρ \, \boldsymbol{\tau}$: Dynamic (per-volume) stress used in the momentum equation; Breeze computes flux divergences as $\boldsymbol{\nabla\cdot}\, \boldsymbol{\mathcal{T}}$.

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