DJL equations

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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)}.