DJL equations: Difference between revisions
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The Dubreil-Jacotin-Long (DJL) equation is derived from the steady Euler equations. The result is a single equation for the isopycnal displacement <math>\eta</math>. Here are a few cases: | |||
== Boussinesq with constant background velocity <math>U_0</math> == | == Boussinesq with constant background velocity <math>U_0</math> == | ||
<math>\nabla^2 \eta + \frac{N^2(z-\eta)}{U_0^2} = 0</math> | <math>\nabla^2 \eta + \frac{N^2(z-\eta)}{U_0^2}\eta = 0</math> | ||
where <math> N^2(z) = -\frac{g}{ | 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> | |||
Revision as of 10:34, 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
Failed to parse (syntax error): {\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}