# Governing Equations¶

The Boussinesq hydrostatic equations of motion in height coordinates are

$\begin{split}\begin{eqnarray} D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \times \boldsymbol{u} + \frac{\rho}{\rho_o} \boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\mathcal{F}} &\mbox{ momentum} \\ \rho \, \frac{\partial \Phi}{\partial z} + \frac{\partial p}{\partial z} &= 0 &\mbox{ hydrostatic} \\ \boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \frac{\partial w}{\partial z} &= 0 &\mbox{ thickness} \\ D_t \theta &= \boldsymbol{\mathcal{N}}_\theta^\gamma - \frac{\partial J_\theta^{(z)}}{\partial z} &\mbox{ potential temp} \\ D_t S &= \boldsymbol{\mathcal{N}}_S^\gamma - \frac{\partial J_S^{(z)}}{\partial z} &\mbox{ salinity} \\ \rho &= \rho(S, \theta, z) &\mbox{ equation of state.} \end{eqnarray}\end{split}$

where notation is described in Notation for equations, $$\boldsymbol{\mathcal{F}}$$ represents the accelerations due to the divergence of stresses including those provided through boundary interactions.

The prognostic thermodynamic variables are potential temperature, $$\theta$$, and salinity $$S$$, which are related to in situ density $$\rho$$ through the  equation of state. In the potential temperature and salinity equations, fluxes due to diabatic, vertically oriented processes are indicated by $$J^{(z)}$$. The tendency due to the convergence of fluxes oriented along neutral directions is indicated by $$\boldsymbol{\mathcal{N}}^\gamma$$. Our implementation of this neutral diffusion parameterization is detailed in Shao et al. (personal comm.)

The total derivative is

$\begin{split}\begin{eqnarray} D_t & \equiv \frac{\partial}{\partial t} + \boldsymbol{v} \cdotp \boldsymbol{\nabla} \\ &= \frac{\partial}{\partial t} + \boldsymbol{u} \cdotp \boldsymbol{\nabla}_z + w \frac{\partial}{\partial z}. \end{eqnarray}\end{split}$

The non-divergence of flow allows a total derivative to be re-written in flux form:

$\begin{split}\begin{eqnarray} D_t \theta &= \frac{\partial}{\partial t} + \boldsymbol{\nabla} \cdotp ( \boldsymbol{v} \theta ) \\ &= \frac{\partial}{\partial t} + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \frac{\partial ( w \theta )}{\partial z}. \end{eqnarray}\end{split}$

The above equations of motion can thus be written as:

$\begin{split}\begin{eqnarray} D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \times \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\mathcal{F}} &\mbox{ momentum}\\ \rho \, \frac{\partial \Phi}{\partial z} + \frac{\partial p}{\partial z} &= 0 &\mbox{ hydrostatic} \\ \boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \frac{\partial w}{\partial z} &= 0 &\mbox{ thickness} \\ \frac{\partial \theta}{\partial t} + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \frac{\partial ( w \theta )}{\partial z} &= \boldsymbol{\mathcal{N}}_\theta^\gamma - \frac{\partial J_\theta^{(z)}}{\partial z} &\mbox{ potential temp} \\ \frac{\partial S}{\partial t} + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} S ) + \frac{\partial ( w S )}{\partial z} &= \boldsymbol{\mathcal{N}}_S^\gamma - \frac{\partial J_S^{(z)}}{\partial z} &\mbox{ salinity} \\ \rho &= \rho(S, \theta, z) &\mbox{ equation of state.} \end{eqnarray}\end{split}$

## Vector Invariant Equations¶

MOM6 solves the momentum equations written in vector-invariant form.

A vector identity allows the total derivative of velocity to be written in the vector-invariant form:

$\begin{split}\begin{eqnarray} D_t \boldsymbol{u} &= \partial_t \boldsymbol{u} + \boldsymbol{v} \cdotp \boldsymbol{\nabla} \boldsymbol{u} \\ &= \partial_t \boldsymbol{u} + \boldsymbol{u} \cdotp \boldsymbol{\nabla}_z \boldsymbol{u} + w \partial_z \boldsymbol{u} \\ &= \partial_t \boldsymbol{u} + \left( \boldsymbol{\nabla} \times \boldsymbol{u} \right) \times \boldsymbol{v} + \boldsymbol{\nabla} \underbrace{\frac{1}{2} \left|\boldsymbol{u}\right|^2}_{\equiv K} . \end{eqnarray}\end{split}$

The flux-form equations of motion in height coordinates can thus be written succinctly as:

$\begin{split}\begin{eqnarray} \partial_t \boldsymbol{u} + \left( f \widehat{\boldsymbol{k}} + \boldsymbol{\nabla} \times \boldsymbol{u} \right) \times \boldsymbol{v} + \boldsymbol{\nabla} K + \frac{\rho}{\rho_o} \boldsymbol{\nabla} \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla} p &= \boldsymbol{\mathcal{F}} &\mbox{ momentum} \\ \boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \partial_z w &= 0 &\mbox{ thickness} \\ \partial_t \theta + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \partial_z ( w \theta ) &= \boldsymbol{\mathcal{N}}_\theta^\gamma - \frac{\partial J_\theta^{(z)}}{\partial z} &\mbox{ potential temp} \\ \partial_t S + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} S ) + \partial_z ( w S ) &= \boldsymbol{\mathcal{N}}_S^\gamma - \frac{\partial J_S^{(z)}}{\partial z} &\mbox{ salinity} \\ \rho &= \rho(S, \theta, z) &\mbox{ equation of state} \end{eqnarray}\end{split}$

where the horizontal momentum equations and vertical hydrostatic balance equation have been written as a single three-dimensional equation.