eqn_motion_ocn.rst

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:

$$\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p = \vec{\mathbf{\mathcal{F}}}$$

$$\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} = \epsilon _{nh}\mathcal{F}_{w}$$

$$\frac{1}{\rho }\frac{D\rho }{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}} _{h}+\frac{\partial w}{\partial z} = 0$$

ρ = ρ(θ, S, p)

$$\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }$$

$$\frac{DS}{Dt} = \mathcal{Q}_{s}$$

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 ρ forward in eq-zns-cont, the answer will not be consistent with that obtained by stepping eq-zns-heat and eq-zns-salt and then using eq-zns-eos to yield ρ. It is therefore necessary to manipulate the system as follows. Differentiating the EOS (equation of state) gives:

$$\frac{D\rho }{Dt}=\left. \frac{\partial \rho }{\partial \theta }\right| _{S,p}\frac{D\theta }{Dt}+\left. \frac{\partial \rho }{\partial S}\right| _{\theta ,p}\frac{DS}{Dt}+\left. \frac{\partial \rho }{\partial p}\right| _{\theta ,S}\frac{Dp}{Dt}$$

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

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

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 eq-zns-cont with eq-zns-pressure yields a system that can be explicitly integrated forward:

$$\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho }\mathbf{\nabla }_{z}p = \vec{\mathbf{\mathcal{F}}}$$

$$\epsilon _{nh}\frac{Dw}{Dt}+g+\frac{1}{\rho }\frac{\partial p}{\partial z} = \epsilon _{nh}\mathcal{F}_{w}$$

$$\frac{1}{\rho c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} = 0$$

ρ = ρ(θ, S, p)

$$\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }$$

$$\frac{DS}{Dt} = \mathcal{Q}_{s}$$

Compressible z-coordinate equations

Here we linearize the acoustic modes by replacing ρ with ρ_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 ρ is the gravitational acceleration:

$$\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p = \vec{\mathbf{\mathcal{F}}}$$

$$\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}} \frac{\partial p}{\partial z} = \epsilon _{nh}\mathcal{F}_{w}$$

$$\frac{1}{\rho _{o}c_{s}^{2}}\frac{Dp}{Dt}+\mathbf{\nabla }_{z}\cdot \vec{ \mathbf{v}}_{h}+\frac{\partial w}{\partial z} = 0$$

ρ = ρ(θ, S, p)

$$\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }$$

$$\frac{DS}{Dt} = \mathcal{Q}_{s}$$

These equations still retain acoustic modes. But, because the “compressible” terms are linearized, the pressure equation eq-zcb-cont 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 eq-zcb-cont. 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:

ρ = ρ(θ, S, p_o(z) + ϵ_sp^′)

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

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

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

$$\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z}- \frac{g}{c_{s}^{2}}w = 0$$

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

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

Equations eq-za-cont1 and eq-za-cont2 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 ρ_o is evaluated from prescribed θ_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:

$$\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{o}}\mathbf{\nabla }_{z}p = \vec{\mathbf{\mathcal{F}}}$$

$$\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{o}}+\frac{1}{\rho _{o}} \frac{\partial p}{\partial z} = \epsilon _{nh}\mathcal{F}_{w}$$

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

ρ = ρ(θ, S, p_o(z))

$$\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }$$

$$\frac{DS}{Dt} = \mathcal{Q}_{s}$$

Incompressible z-coordinate equations

Here, the objective is to drop the depth dependence of ρ_o and so, technically, to also remove the dependence of ρ on p_o. This would yield the “truly” incompressible Boussinesq equations:

$$\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p = \vec{\mathbf{\mathcal{F}}}$$

$$\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho }{\rho _{c}}+\frac{1}{\rho _{c}} \frac{\partial p}{\partial z} = \epsilon _{nh}\mathcal{F}_{w}$$

$$\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} = 0$$

ρ = ρ(θ, S)

$$\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }$$

$$\frac{DS}{Dt} = \mathcal{Q}_{s}$$

where ρ_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:

ρ = ρ_o + ρ^′

We then assert that variations with depth of ρ_o are unimportant while the compressible effects in ρ^′ are:

ρ_o = ρ_c

ρ^′ = ρ(θ, S, p_o(z)) − ρ_o

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

$$\frac{D\vec{\mathbf{v}}_{h}}{Dt}+f\hat{\mathbf{k}}\times \vec{\mathbf{v}} _{h}+\frac{1}{\rho _{c}}\mathbf{\nabla }_{z}p^{\prime } = \vec{\mathbf{ \mathcal{F}}}$$

$$\epsilon _{nh}\frac{Dw}{Dt}+\frac{g\rho ^{\prime }}{\rho _{c}}+\frac{1}{\rho _{c}}\frac{\partial p^{\prime }}{\partial z} = \epsilon _{nh}\mathcal{F}_{w}$$

$$\mathbf{\nabla }_{z}\cdot \vec{\mathbf{v}}_{h}+\frac{\partial w}{\partial z} = 0$$

ρ^′ = ρ(θ, S, p_o(z)) − ρ_c

$$\frac{D\theta }{Dt} = \mathcal{Q}_{\theta }$$

$$\frac{DS}{Dt} = \mathcal{Q}_{s}$$

Note that the hydrostatic pressure of the resting fluid, including that associated with ρ_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 (ϵ_nh = 0) form of these equations that are used throughout the ocean modeling community and referred to as the primitive equations (HPE’s).