Governing equationsΒΆ
The Boussinesq hydrostatic equations of motion in height coordinates are
\[\begin{split}D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \frac{\rho}{\rho_o} \boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdotp \boldsymbol{\underline{\tau}} , \\
\rho \partial_z \Phi + \partial_z p &= 0 , \\
\boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \partial_z w &= 0 , \\
D_t \theta &= \boldsymbol{\nabla} \cdotp \boldsymbol{Q}_\theta , \\
D_t S &= \boldsymbol{\nabla} \cdotp \boldsymbol{Q}_S , \\
\rho &= \rho(S, \theta, z) ,\end{split}\]
where notation is described in Notation for equations. \(\boldsymbol{\underline{\tau}}\) is the stress tensori and \(\boldsymbol{Q}_\theta\) and \(\boldsymbol{Q}_S\) are fluxes of heat and salt respectively.
The total derivative is
\[\begin{split}D_t & \equiv \partial_t + \boldsymbol{v} \cdotp \boldsymbol{\nabla} \\
&= \partial_t + \boldsymbol{u} \cdotp \boldsymbol{\nabla}_z + w \partial_z .\end{split}\]
The non-divergence of flow allows a total derivative to be re-written in flux form:
\[\begin{split}D_t \theta &= \partial_t + \boldsymbol{\nabla} \cdotp ( \boldsymbol{v} \theta ) \\
&= \partial_t + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \partial_z ( w \theta ) .\end{split}\]
The above equations of motion can thus be written as:
\[\begin{split}D_t \boldsymbol{u} + f \widehat{\boldsymbol{k}} \wedge \boldsymbol{u} + \frac{\rho}{\rho_o}\boldsymbol{\nabla}_z \Phi + \frac{1}{\rho_o} \boldsymbol{\nabla}_z p &= \boldsymbol{\nabla} \cdotp \boldsymbol{\underline{\tau}} ,\\
\rho \partial_z \Phi + \partial_z p &= 0 ,\\
\boldsymbol{\nabla}_z \cdotp \boldsymbol{u} + \partial_z w &= 0 ,\\
\partial_t \theta + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} \theta ) + \partial_z ( w \theta ) &= \boldsymbol{\nabla} \cdotp \boldsymbol{Q}_\theta ,\\
\partial_t S + \boldsymbol{\nabla}_z \cdotp ( \boldsymbol{u} S ) + \partial_z ( w S ) &= \nabla \cdotp \boldsymbol{Q}_S ,\\
\rho &= \rho(S, \theta, z) .\end{split}\]