DJL equations

The Dubreil-Jacotin-Long (DJL) equation is derived from the steady, incompressible Euler equations. The result is a single, non-linear 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 $N^2(z) = -g\frac{\bar{\rho}'(z)}{\rho_0}$. $\bar{\rho}(z)$ 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 $U(z)$

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

Again, $N^2(z) = -g\frac{\bar{\rho}'(z)}{\rho_0}$.

Non-Boussinesq constant background current $U_0$

$\nabla^2\eta + \frac{N^2(z-\eta)}{U_0^2}\eta +\frac{N^2(z-\eta)}{2g}\left[\eta_x^2 +\eta_z(\eta_z-2)\right] =0$

where $N^2(z) = -g\frac{\bar{\rho}'(z)}{\bar{\rho}(z)}$.