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 . Here are a few cases:
Boussinesq with constant background current
where . is the far upstream density profile, is a constant reference density ,and is the acceleration due to gravity.
Boussinesq with non-constant background current
Non-Boussinesq constant background current