I want to simulate density current in the sea or river by vertical 2D model. How should I solve this one? Is it adequate to solve Euler equation and advection-diffusion equation? Then, how should I treat free surface ??

Regards.

Have you heard of the shallow water equation?

IN this case you solve for the thickness (height) of the layer and for an incompressible flow. This allows you to simulate the surface waves (gravity waves) in the sea, atmosphere, etc...

IS the 2D vertical the (z,x) plane, where z is the vertical (direction of the gravity is -z) and x is the latteral dimension (horizon e.g.).

Then basically you solve one equation for the velocity in x and one equation for h (the vertical thickness of the fluid).

If this is what you are looking for, then I'll give you some references.

Cheers, Patrick

Well, I think the shallow water equation is not a good choice for solving such a problem. There may be some misunderstanding to whom read your statement. The shallow water equation does not take into account the vertical acceleration, so hydrostatic pressure distribution along vertical direction is assumed. Therefore it is difficult to simulate density current which propagates x-, z- direction using it (x-, y- direction is possible since it is horizontal 2-D model).