Boussinesq approximation
For a compressible fluid, the conservation of mass can be written as:
\[{\text{D}_t}{\rho_{\text{tot}}}+ {\rho_{\text{tot}}}{\bnabla}\cdot {\boldsymbol{u}}= 0 \Leftrightarrow
{\partial}_t {\rho_{\text{tot}}}+ {\bnabla}\cdot({\rho_{\text{tot}}}{\boldsymbol{u}}) = 0.\]
The Navier-Stokes equation is
\[{\rho_{\text{tot}}}{\text{D}_t}{\boldsymbol{u}}= -{\bnabla}\tilde p + {\rho_{\text{tot}}}{\boldsymbol{g}}+ {\rho_{\text{tot}}}\nu {\bnabla}^2 {\boldsymbol{u}}.\]
The density is decomposed in 3 parts:
\({\rho_{\text{tot}}}= \rho_0 +
\tilde\rho(z) + \rho'\), where the time average
density is \(\bar\rho(z) = \rho_0 + \tilde\rho(z)\).
The Boussinesq approximation consists in replacing \({\rho_{\text{tot}}}\) by
\(\rho_0\) everywhere except in the buoyancy term.
\[{\text{D}_t}{\boldsymbol{u}}= -{\bnabla}\frac{\tilde p}{\rho_0}
+ \frac{{\rho_{\text{tot}}}{\boldsymbol{g}}}{\rho_0}
+ \nu {\bnabla}^2 {\boldsymbol{u}}.
\]
and the mass conservation becomes \({\bnabla}\cdot {\boldsymbol{u}}=
0\).
We can subtract to this equation its static solution obtained for
\({\boldsymbol{u}}= 0\):
\[0 = -{\bnabla}\frac{\tilde p_s}{\rho_0} + \frac{\bar\rho {\boldsymbol{g}}}{\rho_0}\]
which gives
\[{\text{D}_t}{\boldsymbol{u}}= -{\bnabla}p + {\boldsymbol{b}}+ \nu {\bnabla}^2 {\boldsymbol{u}},\]
where \(p\) is the rescaled dynamic pressure and \({\boldsymbol{b}}\) is the buoyancy.
The conservation of internal energy and mass of salt can be rewritten as
\[{\text{D}_t}{\rho_{\text{tot}}}= \kappa {\bnabla}^2 {\rho_{\text{tot}}},\]
which leads with some assumption to
\[{\text{D}_t}b + N^2 u_z = \kappa {\bnabla}^2 b,\]
where \(N = \sqrt{d_z \bar b_{\text{tot}}}\) is the Brunt-Väisälä frequency.
Linear inviscid equations and eigenmodes
The linear inviscid equations are
\[\begin{split}
\begin{align}
{\partial_t}{\boldsymbol{u}} & = -{\bnabla}p + b e_z,\\
{\partial_t}b & = - N^2 u_z.
\end{align}
\end{split}\]
We want to obtain a wave equation, i.e. a linear differential equation for only
one variable.
To be able to eliminate the pressure, we know that we can take the divergence of
the velocity equation to obtain the linear Poisson equation:
\[ \bnabla^2 p = \p_z b \]
To eliminate the velocity, we can take the time derivative of the buoyancy
equation:
\[ {\partial_{tt}} b = - N^2 ( - \p_z p + b ). \]
\[ {\partial_{tt}}\bnabla^2 b = - N^2 ( - \p_z \bnabla^2 p + \bnabla^2b ). \]
\[ {\partial_{tt}}\bnabla^2 b = - N^2 \bnabla_h^2 b. \]
Taking the time and space Fourier transforms, we obtain the dispersion relation:
\[ \omega^2 = N^2 \left(\frac{k_h}{k}\right)^2 = N^2 \sin^2\theta,\]
where \(\theta\) is the angle between the wave vector and the verticale direction.
Few consequences of this uncommon dispersion relation:
Rotational - toroidal modes
We can also take the vertical componant of the rotational of the velocity equation
to get the equation for the vertical vorticity \(\p_t \omega_z = 0\).
