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 [69] 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.