sandbox/Antoonvh/internalwacesMG.c

Internal Waves

In a stratified fluid internal waves can exist (also reffered to as gravity waves). An interesting feature of these waves is the so-called dispersion relation between the angle of wave propagation (\theta), stratification strength (N^2) and the freqency of the wave (\omega), according to,

\displaystyle \omega = N^2 \cos(\theta).

Numerical set-up

The Navier-Stokes equantions under Boussinesq approximation are solved on a 256 \times 256 miltigrid. In the centre of the domain an oscillating force exites the internal waves with a freqency corresponding to \theta = 45^o.

#include "grid/multigrid.h"
#include "navier-stokes/centered.h"
#include "tracer.h"

scalar b[];
scalar * tracers = {b};
face vector av[];
double sqN = 1., omega;

b[top]    = neumann (sqN);
b[bottom] = neumann (-sqN);

int main() {
  omega = sqrt(1./2.);
  L0 = 30;
  X0 = Y0 = -L0/2;
  a = av;
  TOLERANCE = 1e-4;
  DT = 0.2/omega;
  N = 256;
  run();
}

The initial stratification is set.

event init (t = 0) {
  foreach()
    b[] = sqN*y;
  boundary ({b});
}

Acceleration

We apply gravity and the localized oscillarory force.

event acceleration (i++) {
  coord del = {0, 1};
  foreach_face() 
    av.x[] = del.x*((b[] + b[-1])/2. +
		    0.1*(sin(omega*t)*((sq(x) + sq(y)) < 1)));
}

Output

We output a .mp4 file showing the evolution of the magnitude of the gradient of the buoyancy field (|\nabla b|).

event output (t += 0.5; t <= 75) {
  scalar grb[];
  foreach() {
    grb[] = 0;
    foreach_dimension()
      grb[] += sq((b[1] - b[-1])/(2*Delta));
    grb[] = sqrt(grb[]);
  }
  output_ppm (grb, file = "grb.mp4", min = 0.8, max = 1.2);
}

Results

The dispersion relation appears to be statisfied.

The next step is to perform this simulation using adaptive grids, See here.