# Governing Equations¶

MOM6 is a hydrostatic ocean circulation model that time steps either the non-Boussinesq ocean equations (where the flow velocity is divergent: $$\nabla \cdot \mathbf{v} \ne 0$$), or the Boussinesq ocean equations (where velocity is non-divergent: $$\nabla \cdot \mathbf{v} = 0$$). We here display the Boussinesq version since it is most commonly used (as of 2022). We start by casting the equations in geopotentiial coordinates prior to transforming to the generalized vertical coordinates used by MOM6. A more thorough discussion of these equations, and their finite volume realization appropriate for MOM6, can be found in Griffies, Adcroft and Hallberg (2020) [22].

The hydrostatic Boussinesq ocean equations, written using geopotential vertical coordinates, are given by

\begin{split}\begin{align} \rho_o \left[ D_t \mathbf{u} + f \hat{\mathbf{z}} \times \mathbf{u} \right] &= -\rho \, \nabla_z \Phi - \nabla_z p + \rho_o \, \mathbf{\mathcal{F}} &\mbox{horizontal momentum} \\ \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic} \\ \nabla_z \cdotp \mathbf{u} + \partial_{z} w &= 0 &\mbox{continuity} \\ D_t \theta &= \mathbf{\mathcal{N}}_\theta^\gamma - \partial_{z} J_\theta^{(z)} &\mbox{potential or Conservative temp} \\ D_t S &= \mathbf{\mathcal{N}}_S^\gamma - \partial_{z} J_S^{(z)} &\mbox{salinity} \\ \rho &= \rho(S, \theta, z) &\mbox{ equation of state} \\ \mathbf{v} &= \mathbf{u} + \hat{\mathbf{z}} \, w &\mbox{velocity field.} \end{align}\end{split}

The acceleration term, $$\mathbf{\mathcal{F}}$$, in the horizontal momentum equation includes the acceleration due to the divergence of internal frictional stresses as well as from bottom and surface boundary stresses. Other notation is described in Notation for equations.

The prognostic temperature, $$\theta$$, is either potential temperature or Conservative Temperature, depending on the chosen equation of state, and $$S$$ is the salinity. We generally follow the discussion of [43] for how to interpret the prognostic temperature and salinity in ocean models. MOM6 has historically used the Wright (1997) [70] equation of state to compute the in situ density, $$\rho$$. However, there are other options as documented in Equation of State. In the potential temperature and salinity equations, fluxes due to diabatic processes are indicated by $$J^{(z)}$$. Tendencies due to the convergence of fluxes oriented along neutral directions are indicated by $$\mathbf{\mathcal{N}}^\gamma$$, with our implementation of neutral diffusion detailed in Shao et al (2020) [57].

The total or material time derivative operator is given by

\begin{split}\begin{align} D_t &\equiv \partial_{t} + \mathbf{v} \cdotp \nabla \\ &= \partial_{t} + \mathbf{u} \cdotp \nabla_z + w \, \partial_{z}, \end{align}\end{split}

where the second equality explosed the horizontal and vertical terms. Using the non-divergence condition on the three-dimensional velocity allows us to write the material time derivative of an arbitrary scalar field, $$\psi$$, into a flux-form equation

\begin{split}\begin{align} D_t \psi &= ( \partial_{t} + \mathbf{u} \cdotp \nabla) \, \psi \\ &= \partial_{t} \psi + \nabla \cdotp (\mathbf{v} \, \psi) \\ &= \partial_{t} \psi + \nabla_z \cdotp ( \mathbf{u} \, \psi) + \partial_{z} ( w \, \psi). \end{align}\end{split}

Discretizing the flux-form scalar equations means that fluxes transferring scalars between grid cells act in a conservative manner. Consequently, the domain integrated scalar (e.g., total seawater volume, total salt content, total potential enthalpy) is affected only via surface and bottom boundary transport. Such global conservation properties are maintained by MOM6 to within computational roundoff, with this level of precision found to be essential for using MOM6 to study climate. Making use of the flux-form scalar conservation equations brings the model equations to the form

\begin{split}\begin{align} \rho_o \left[ D_t \mathbf{u} + f \hat{\mathbf{z}} \times \mathbf{u} \right] &= -\rho \, \nabla_z \Phi - \nabla_z p + \rho_o \, \mathbf{\mathcal{F}} &\mbox{horizontal momentum} \\ \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic} \\ \nabla_z \cdotp \mathbf{u} + \partial_{z} w &= 0 &\mbox{continuity} \\ \partial_{t} \theta + \nabla_z \cdotp (\mathbf{u} \, \theta) + \partial_{z} (w \, \theta) &= \mathbf{\mathcal{N}}_\theta^\gamma - \partial_{z} J_\theta^{(z)} &\mbox{potential or Conservative temp} \\ \partial_{t} S + \nabla_z \cdotp (\mathbf{u} \, S) + \partial_{z}(w \, S) &= \mathbf{\mathcal{N}}_S^\gamma -\partial_{z} J_S^{(z)} &\mbox{salinity} \\ \rho &= \rho(S, \theta, z) &\mbox{equation of state.} \end{align}\end{split}

## Vector invariant velocity equation¶

MOM6 time steps the horizontal velocity equation in its vector-invariant form. To derive this equation we make use of the following vector identity

\begin{split}\begin{align} D_t \mathbf{u} &= \partial_t \mathbf{u} + \mathbf{v} \cdotp \nabla \mathbf{u} \\ &= \partial_t \mathbf{u} + \mathbf{u} \cdotp \nabla_z \mathbf{u} + w \partial_z \mathbf{u} \\ &= \partial_t \mathbf{u} + \left( \nabla \times \mathbf{u} \right) \times \mathbf{v} + \nabla \left|\mathbf{u}\right|^2/2 \\ &= \partial_t \mathbf{u} + w \, \partial_{z} \mathbf{u} + \zeta \, \hat{\mathbf{z}} \times \mathbf{u} + \nabla_{z} K, \end{align}\end{split}

where we introduced the vertical component to the relative vorticity

(1)\begin{align} \zeta = \hat{\mathbf{z}} \cdot (\nabla \times \mathbf{u}) = \partial_{x}v - \partial_{y} u, \label{eq:relative-vorticity-z} \end{align}

as well as the kinetic energy per mass contained in the horizontal flow

(2)\begin{align} K = (u^{2} + v^{2})/2. \label{eq:kinetic-energy-per-mass} \end{align}

It is just the horizontal kinetic energy per mass that appears when making the hydrostatic approximation, whereas a non-hydrostatic fluid (such as the MITgcm) includes the contribution from vertical motion. With these identities we are led to the MOM6 flux-form equations of motion in geopotential coordinates

\begin{split}\begin{align} \rho_{o} \left[ \partial_t \mathbf{u} + w \, \partial_{z} \mathbf{u} + (f + \zeta) \hat{\mathbf{z}} \times \mathbf{u} \right] &= -\nabla_{z} (p + K) - \rho \, \nabla_{z} \Phi + \rho_{o} \, \mathbf{\mathcal{F}} &\mbox{vector-inv horz velocity} \\ \rho \, \partial_{z} \Phi + \partial_{z} p &= 0 &\mbox{hydrostatic} \\ \nabla_z \cdotp \mathbf{u} + \partial_{z} w &= 0 &\mbox{continuity} \\ \partial_t \theta + \nabla_z \cdotp ( \mathbf{u} \, \theta ) + \partial_z ( w \, \theta ) &= \mathbf{\mathcal{N}}_\theta^\gamma - \partial_{z} J_\theta^{(z)} &\mbox{potential/Cons temp} \\ \partial_t S + \nabla_z \cdotp ( \mathbf{u} \, S ) + \partial_z (w \, S) &= \mathbf{\mathcal{N}}_S^\gamma - \partial_{z} J_S^{(z)} &\mbox{salinity} \\ \rho &= \rho(S, \theta, z) &\mbox{equation of state.} \end{align}\end{split}