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 velocity $U_{0}$

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

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

Boussinesq with non-constant background current $U_{(}z)$

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla^2 \eta + \frac{N^2(z-\eta)}{U(z-\eta)^2}\eta + \frac{U'(z-\eta)}{U(z-\eta)\left(1- \left(\eta_x2 +(1-\eta_z)^2\right)\right) = 0}