Pressure perturbation equation

[ealucy]

Hello,
I'm working with the dynamic pressure perturbation equation (Laplacian of p = -roh*((dudx)**2+(dvdy)**2+(dwdz)**2)). At small scales, what boundary conditions, mxLoop, and tolerance would you reccomend?

[miniufo]

Hi, you can use latex here. So the equation looks like:

$$\nabla^2 p = - \rho \left[\left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial v}{\partial y}\right)^2 + \left(\frac{\partial w}{\partial z}\right)^2\right]$$

For mxLoop and tolerance, both determine the end of iteration. So larger mxLoop or smaller tolerance will give you a more converged result (this case is certainly convergent). You could practice with different parameters to see if the final results change a lot or not. If the difference is invisible, then it could be very close to the solution. For extreme case, you could use mxLoop=1 and compare the result with mxLoop=10, you will see very large difference.

For boundary conditions (BCs), it is a bit complex.

• If your domain is global, then just use periodic in the zonal and extend in the meridional;
• For regitional case, the solutions heavily depend on your BCs. See here for a discussion. Physically, the Laplacian operator is a smooth operator. So the forcing (on the right-hand-side of equation) outside the domain will also influence the solution inside, right through the BCs. So without setting the forcing outside the domain, you cannot uniquely determine the solution. But there are a lot of papers talking about solving Poission equation in a limited region. All of them focus on how to specify the BCs. You can search them on google.
• But if you could specified the "observed" pressure anomaly at boundaries, then you can get the solution close to what you expect.

Just remember that the solution has two parts: one induced by BCs, and the other induced by forcings on the r.h.s. The one induced by BCs could be understood as that induced by forcings outside the region.

[ealucy]

So mxLoop=1 is an extreme and likely not producing a converged result?

[miniufo]

mxLoop=1 means you only iterate one time and then break the loop of iteration. If the equation is exactly Poisson-type, it should converge.