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Governing equations

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. See the Anelastic dynamics page for details.

  • CompressibleDynamics: Solves the fully compressible Euler equations with prognostic density. Retains acoustic waves and supports split-explicit time integration for efficiency. See the Compressible dynamics page for details.

Both formulations share the same compressible Navier-Stokes equations as a starting point, with the anelastic formulation obtained as a special case through linearization.

Compressible Navier-Stokes 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 compressible equations in flux form are

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

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

Thermodynamic equation

In addition to mass and momentum, Breeze advances a thermodynamic prognostic variable $χ$ in conservative (flux) form:

\[\partial_t χ + \boldsymbol{\nabla \cdot}\, (χ \boldsymbol{u}) = Π(ρ, χ, \ldots) \, \boldsymbol{\nabla \cdot \, u} + S_χ ,\]

where $Π$ is a formulation-specific compression source and $S_χ$ represents diabatic and diffusive sources.

Breeze supports two concrete choices for $χ$:

  • Liquid-ice potential temperature density ($χ = ρ θ^{li}$): The potential temperature is materially conserved under adiabatic motion, so $Π = 0$. This is the simplest thermodynamic formulation.

  • Static energy density ($χ = ρ e$): The moist static energy $e = c^{pm} T + g z - \mathscr{L}^l_r q^l - \mathscr{L}^i_r q^i$ includes gravitational potential energy and latent heat. In this case $Π \neq 0$ and encodes the pressure work term.

The thermodynamic equation couples to the momentum equation through the equation of state (below) and buoyancy.

Moisture transport

For moist flows we 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.

Equation of state

Pressure is related to density and temperature through the ideal gas law for moist air:

\[p = ρ R^m T ,\]

where $R^m = (1 - q^t) R^d + q^v R^v$ is the mixture gas constant.

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

Symbols and notation

Core variables

  • $ρ$: Density (prognostic for compressible; reference $ρᵣ(z)$ for anelastic)
  • $\boldsymbol{u} = (u, v, w)$: Velocity
  • $\boldsymbol{m} = ρ \boldsymbol{u}$: Momentum
  • $p$: Pressure
  • $T$: Temperature
  • $θ$: Potential temperature
  • $χ$: Thermodynamic prognostic variable ($ρθ$ or $ρe$)

Moisture

  • $q^t$: Total specific humidity (vapor + condensates)
  • $q^v, q^l, q^i$: Vapor, liquid, and ice mass fractions
  • $R^m$: Mixture gas constant
  • $c^{pm}$: Mixture heat capacity at constant pressure

Stresses and forces

  • $\boldsymbol{\tau}$: 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 $\boldsymbol{\nabla\cdot}\, \boldsymbol{\mathcal{T}}$.
  • $\boldsymbol{f}$: Non-pressure body forces (Coriolis)

Thermodynamic closures

  • $Π = (p / p_0)^{R^m / c^{pm}}$: Exner function
  • $\mathbb{C}^{ac} = \sqrt{γ^m R^m T}$: Acoustic sound speed, where $γ^m = c^{pm} / c^{vm}$
  • $b$: Buoyancy

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