1.5.1. Equations of Motion for the Ocean

We review here the method by which the standard (Boussinesq, incompressible) HPE’s for the ocean written in \(z-\)coordinates are obtained. The non-Boussinesq equations for oceanic motion are:

(1.64)\[\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\boldsymbol{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho } \nabla _{z}p = \vec{\boldsymbol{\mathcal{F}}}_h\]
(1.65)\[\epsilon _{\rm nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} = \epsilon _{\rm nh}\mathcal{F}_{w}\]
(1.66)\[\frac{1}{\rho }\frac{D\rho }{Dt}+ \nabla _{z}\cdot \vec{\mathbf{v}} _{h}+\frac{\partial w}{\partial z} = 0\]
(1.67)\[\rho = \rho (\theta ,S,p)\]
(1.68)\[\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }\]
(1.69)\[\frac{DS}{Dt} = \mathcal{Q}_{s}\]

These equations permit acoustics modes, inertia-gravity waves, non-hydrostatic motions, a geostrophic (Rossby) mode and a thermohaline mode. As written, they cannot be integrated forward consistently - if we step \(\rho\) forward in (1.66), the answer will not be consistent with that obtained by stepping (1.68) and (1.69) and then using (1.67) to yield \(\rho\). It is therefore necessary to manipulate the system as follows. Differentiating the EOS (equation of state) gives:

(1.70)\[\frac{D\rho }{Dt}=\left. \frac{\partial \rho }{\partial \theta }\right| _{S,p}\frac{D\theta }{Dt}+\left. \frac{\partial \rho }{\partial S}\right| _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{Dp}{Dt}\]

Note that \(\frac{\partial \rho }{\partial p}=\frac{1}{c_{s}^{2}}\) is the reciprocal of the sound speed (\(c_{s}\)) squared. Substituting into (1.66) gives:

(1.71)\[\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+ \nabla _{z}\cdot \vec{\mathbf{ v}}+\partial _{z}w\approx 0\]

where we have used an approximation sign to indicate that we have assumed adiabatic motion, dropping the \(\frac{D\theta }{Dt}\) and \(\frac{DS}{Dt}\). Replacing (1.66) with (1.71) yields a system that can be explicitly integrated forward:

(1.72)\[\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\boldsymbol{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho } \nabla _{z}p = \vec{\boldsymbol{\mathcal{F}}}_h\]
(1.73)\[\epsilon _{\rm nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} = \epsilon _{\rm nh}\mathcal{F}_{w}\]
(1.74)\[\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+ \nabla _{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} = 0\]
(1.75)\[\rho = \rho (\theta ,S,p)\]
(1.76)\[\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }\]
(1.77)\[\frac{DS}{Dt} = \mathcal{Q}_{s}\] Compressible z-coordinate equations

Here we linearize the acoustic modes by replacing \(\rho\) with \(\rho _{o}(z)\) wherever it appears in a product (ie. non-linear term) - this is the ‘Boussinesq assumption’. The only term that then retains the full variation in \(\rho\) is the gravitational acceleration:

(1.78)\[\frac{D\vec{\mathbf{v}}_{h}}{Dt}+ f \hat{\boldsymbol{k}} \times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{o}} \nabla _{z}p = \vec{\boldsymbol{\mathcal{F}}}_h\]
(1.79)\[\epsilon _{\rm nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}} \frac{\partial p}{\partial z} = \epsilon _{\rm nh}\mathcal{F}_{w}\]
(1.80)\[\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+ \nabla _{z}\cdot \vec{ \mathbf{v}}_{h}+\frac{\partial w}{\partial z} = 0\]
(1.81)\[\rho = \rho (\theta ,S,p)\]
(1.82)\[\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }\]
(1.83)\[\frac{DS}{Dt} = \mathcal{Q}_{s}\]

These equations still retain acoustic modes. But, because the “compressible” terms are linearized, the pressure equation (1.80) can be integrated implicitly with ease (the time-dependent term appears as a Helmholtz term in the non-hydrostatic pressure equation). These are the truly compressible Boussinesq equations. Note that the EOS must have the same pressure dependency as the linearized pressure term, ie. \(\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}=\frac{1}{c_{s}^{2}}\), for consistency. ‘Anelastic’ z-coordinate equations

The anelastic approximation filters the acoustic mode by removing the time-dependency in the continuity (now pressure) equation (1.80). This could be done simply by noting that \(\frac{Dp}{Dt}\approx -g\rho _{o} \frac{Dz}{Dt}=-g\rho _{o}w\), but this leads to an inconsistency between continuity and EOS. A better solution is to change the dependency on pressure in the EOS by splitting the pressure into a reference function of height and a perturbation:

\[\rho =\rho \left(\theta ,S,p_{o}(z)+\epsilon _{s}p^{\prime } \right)\]

Remembering that the term \(\frac{Dp}{Dt}\) in continuity comes from differentiating the EOS, the continuity equation then becomes:

\[\frac{1}{\rho _{o}c_{s}^{2}}\left( \frac{Dp_{o}}{Dt}+\epsilon _{s}\frac{ Dp^{\prime }}{Dt}\right) + \nabla _{z}\cdot \vec{\mathbf{v}}_{h}+ \frac{\partial w}{\partial z}=0\]

If the time- and space-scales of the motions of interest are longer than those of acoustic modes, then \(\frac{Dp^{\prime }}{Dt}\ll \frac{Dp_{o}}{Dt}, \nabla \cdot \vec{\mathbf{v}}_{h}\) in the continuity equations and \(\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{ Dp^{\prime }}{Dt}\ll \left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{Dp_{o}}{Dt}\) in the EOS (1.70). Thus we set \(\epsilon_{s}=0\), removing the dependency on \(p^{\prime }\) in the continuity equation and EOS. Expanding \(\frac{Dp_{o}(z)}{Dt}=-g\rho _{o}w\) then leads to the anelastic continuity equation:

(1.84)\[\nabla _{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}- \frac{g}{c_{s}^{2}}w = 0\]

A slightly different route leads to the quasi-Boussinesq continuity equation where we use the scaling \(\frac{\partial \rho ^{\prime }}{\partial t}+ \nabla _{3}\cdot \rho ^{\prime }\vec{\mathbf{v}}\ll \nabla _{3}\cdot \rho _{o}\vec{\mathbf{v}}\) yielding:

(1.85)\[\nabla _{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{ \partial \left( \rho _{o}w\right) }{\partial z} = 0\]

Equations (1.84) and (1.85) are in fact the same equation if:

\[\frac{1}{\rho _{o}}\frac{\partial \rho _{o}}{\partial z} = -\frac{g}{c_{s}^{2}}\]

Again, note that if \(\rho _{o}\) is evaluated from prescribed \(\theta _{o}\) and \(S_{o}\) profiles, then the EOS dependency on \(p_{o}\) and the term \(\frac{g}{c_{s}^{2}}\) in continuity should be referred to those same profiles. The full set of ‘quasi-Boussinesq’ or ‘anelastic’ equations for the ocean are then:

(1.86)\[\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\boldsymbol{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{o}} \nabla _{z}p = \vec{\boldsymbol{\mathcal{F}}}_h\]
(1.87)\[\epsilon _{\rm nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}} \frac{\partial p}{\partial z} = \epsilon _{\rm nh}\mathcal{F}_{w}\]
(1.88)\[\nabla _{z}\cdot \vec{\mathbf{v}}_{h}+\frac{1}{\rho _{o}}\frac{ \partial \left( \rho _{o}w\right) }{\partial z} = 0\]
(1.89)\[\rho = \rho \left(\theta ,S,p_{o}(z) \right)\]
(1.90)\[\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }\]
(1.91)\[\frac{DS}{Dt} = \mathcal{Q}_{s}\] Incompressible z-coordinate equations

Here, the objective is to drop the depth dependence of \(\rho _{o}\) and so, technically, to also remove the dependence of \(\rho\) on \(p_{o}\). This would yield the “truly” incompressible Boussinesq equations:

(1.92)\[\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\boldsymbol{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{c}} \nabla _{z}p = \vec{\boldsymbol{\mathcal{F}}}_h\]
(1.93)\[\epsilon _{\rm nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}} \frac{\partial p}{\partial z} = \epsilon _{\rm nh}\mathcal{F}_{w}\]
(1.94)\[\nabla _{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} = 0\]
(1.95)\[\rho = \rho (\theta ,S)\]
(1.96)\[\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }\]
(1.97)\[\frac{DS}{Dt} = \mathcal{Q}_{s}\]

where \(\rho _{c}\) is a constant reference density of water. Compressible non-divergent equations

The above “incompressible” equations are incompressible in both the flow and the density. In many oceanic applications, however, it is important to retain compressibility effects in the density. To do this we must split the density thus:

\[\rho =\rho _{o}+\rho ^{\prime }\]

We then assert that variations with depth of \(\rho _{o}\) are unimportant while the compressible effects in \(\rho ^{\prime }\) are:

\[\rho _{o}=\rho _{c}\]
\[\rho ^{\prime }=\rho (\theta ,S,p_{o}(z))-\rho _{o}\]

This then yields what we can call the semi-compressible Boussinesq equations:

(1.98)\[\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\boldsymbol{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{c}} \nabla _{z}p^{\prime } = \vec{\boldsymbol{\mathcal{F}}}_h\]
(1.99)\[\epsilon _{\rm nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho _{c}}\frac{\partial p^{\prime }}{\partial z} = \epsilon _{\rm nh}\mathcal{F}_{w}\]
(1.100)\[\nabla _{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} = 0\]
(1.101)\[\rho ^{\prime } = \rho (\theta ,S,p_{o}(z))-\rho _{c}\]
(1.102)\[\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }\]
(1.103)\[\frac{DS}{Dt} = \mathcal{Q}_{s}\]

Note that the hydrostatic pressure of the resting fluid, including that associated with \(\rho _{c}\), is subtracted out since it has no effect on the dynamics.

Though necessary, the assumptions that go into these equations are messy since we essentially assume a different EOS for the reference density and the perturbation density. Nevertheless, it is the hydrostatic (\(\epsilon_{\rm nh}=0\)) form of these equations that are used throughout the ocean modeling community and referred to as the primitive equations (HPE’s).