DJL equations: Difference between revisions

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<math>\nabla^2 \eta + \frac{N^2(z-\eta)}{U_0^2}\eta = 0</math>
<math>\nabla^2 \eta + \frac{N^2(z-\eta)}{U_0^2}\eta = 0</math>


where <math> N^2(z) = -\frac{g}{\rho_0}\frac{d\bar{\rho'(z)}}{dz}</math>.  <math>\bar{\rho}{z} </math> is the far upstream density profile and <math>g</math> is the acceleration due to gravity.   
where <math> N^2(z) = -\frac{g}{\rho_0}\frac{d\bar{\rho'(z)}}{dz}</math>.  <math>\bar{\rho}(z)</math> is the far upstream density profile and <math>g</math> is the acceleration due to gravity.   


== Boussinesq with non-constant background current <math>U_(z)</math> ==
== Boussinesq with non-constant background current <math>U_(z)</math> ==


<math>\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</math>
<math>\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</math>

Revision as of 11:35, 7 July 2011

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

Boussinesq with constant background velocity

where . is the far upstream density profile and is the acceleration due to gravity.

Boussinesq with non-constant background current