Nonhydrostatic model

The NonhydrostaticModel solves the incompressible Navier-Stokes equations under the Boussinesq approximation and an arbitrary number of tracer conservation equations. Physics associated with individual terms in the momentum and tracer conservation equations –- the background rotation rate of the equation's reference frame, gravitational effects associated with buoyant tracers under the Boussinesq approximation, generalized stresses and tracer fluxes associated with viscous and diffusive physics, and arbitrary "forcing functions" –- are determined by the whims of the user.

The momentum conservation equation

The equations governing the conservation of momentum in a rotating fluid, including buoyancy via the Boussinesq approximation and including the averaged effects of surface gravity waves at the top of the domain via the Craik-Leibovich approximation are

\[ \begin{align} \partial_t \boldsymbol{v} & = - \left ( \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \right ) \boldsymbol{v} - \left ( \boldsymbol{V} \boldsymbol{\cdot} \boldsymbol{\nabla} \right ) \boldsymbol{v} - \left ( \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \right ) \boldsymbol{V} \nonumber \\ & \qquad - \left ( \boldsymbol{f} - \boldsymbol{\nabla} \times \boldsymbol{u}^S \right ) \times \boldsymbol{v} - \boldsymbol{\nabla} p + b \boldsymbol{\hat g} - \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau} + \partial_t \boldsymbol{v}^S + \boldsymbol{F_v} \, , \label{eq:momentum} \end{align}\]

where $b \boldsymbol{\hat g}$ the is the buoyancy (a vector whose default direction is upward), $\boldsymbol{\tau}$ is the kinematic stress tensor, $\boldsymbol{F_v}$ denotes an internal forcing of the velocity field $\boldsymbol{v}$, $p$ is the kinematic pressure, $\boldsymbol{u}^S$ is the 'Stokes drift' velocity field associated with surface gravity waves, and $\boldsymbol{f}$ is the Coriolis parameter, or the background vorticity associated with the specified rate of rotation of the frame of reference.

The terms that appear on the right-hand side of the momentum conservation equation are (in order):

  • momentum advection: $\left ( \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \right ) \boldsymbol{v}$,
  • advection of resolved momentum by the background velocity field $\boldsymbol{V}$: $\left ( \boldsymbol{V} \boldsymbol{\cdot} \boldsymbol{\nabla} \right ) \boldsymbol{v}$,
  • advection of background momentum by resolved velocity: $\left ( \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \right ) \boldsymbol{V}$,
  • Coriolis: $\boldsymbol{f} \times \boldsymbol{v}$,
  • the effective background rotation rate due to surface waves: $\left ( \boldsymbol{\nabla} \times \boldsymbol{u}^S \right ) \times \boldsymbol{v}$,
  • kinematic pressure gradient: $\boldsymbol{\nabla} p$,
  • buoyant acceleration: $b$,
  • vertical unit vector (pointing to the direction opposite to gravity): $\boldsymbol{\hat g}$,
  • molecular or turbulence viscous stress: $\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{\tau}$,
  • a source of momentum due to forcing or damping of surface waves: $\partial_t \boldsymbol{v}^S$, and
  • an arbitrary internal source of momentum: $\boldsymbol{F_v}$.

The tracer conservation equation

The conservation law for tracers in Oceananigans.jl is

\[ \begin{align} \partial_t c = - \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} c - \boldsymbol{V} \boldsymbol{\cdot} \boldsymbol{\nabla} c - \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} C - \boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{q}_c + F_c \, , \label{eq:tracer} \end{align}\]

where $\boldsymbol{q}_c$ is the diffusive flux of $c$ and $F_c$ is an arbitrary source term. Oceananigans.jl permits arbitrary tracers and thus an arbitrary number of tracer equations to be solved simultaneously with the momentum equations.

From left to right, the terms that appear on the right-hand side of the tracer conservation equation are

  • tracer advection: $\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} c$,
  • tracer advection by the background velocity field, $\boldsymbol{V}$: $\boldsymbol{V} \boldsymbol{\cdot} \boldsymbol{\nabla} c$,
  • advection of the background tracer field, $C$, by the resolved velocity field: $\boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} C$,
  • molecular or turbulent diffusion: $\boldsymbol{\nabla} \boldsymbol{\cdot} \boldsymbol{q}_c$, and
  • an arbitrary internal source of tracer: $F_c$.

The following subsections provide more details on the possible forms that each individual term in the momentum and tracer equations can take in Oceananigans.jl.