DJL equations

The Dubreil-Jacotin-Long (DJL) equation is derived from the steady Euler equations. The result is a single equation for the isopycnal displacement $\eta$. Here are a few cases:

Boussinesq with constant background current $U_{0}$

$\nabla ^{2}\eta +{\frac {N^{2}(z-\eta )}{U_{0}^{2}}}\eta =0$

where $N^{2}(z)=-{\frac {g}{\rho _{0}}}{\frac {d{\bar {\rho (z)}}'}{dz}}$. ${\bar {\rho }}(z)$ is the far upstream density profile, $\rho _{0}$ is a constant reference density ,and $g$ is the acceleration due to gravity.

Boussinesq with non-constant background current $U(z)$

$\nabla ^{2}\eta +{\frac {N^{2}(z-\eta )}{U(z-\eta )^{2}}}\eta +{\frac {U'(z-\eta )}{U(z-\eta )}}\left[1-\left(\eta _{x}^{2}+(1-\eta _{z})^{2}\right)\right]=0$

Non-Boussinesq constant background current $U_{0}$

$\nabla ^{2}\eta +{\frac {N^{2}(z-\eta )}{U_{0}^{2}}}\eta +{\frac {N^{2}(z-\eta )}{2g}}\left[\eta _{x}^{2}+\eta _{z}(\eta _{z}-2)\right]$

where Failed to parse (syntax error): {\displaystyle \[N^2(z) = \frac{-g\bar{\rho}'(z)}{\bar{\rho}(z)}}