/**
# The Helmholtz filter

Reading the work of Fodor et al. (2019), I learned about the Helmholtz
filter (Foias et al, 2001): An original field $\phi$, may be filtered
($\phi_f$) with filter length scale $\Delta$, according to:

$$\left(1 - \left(\frac{\Delta}{2\pi}\right)^2\nabla^2\right) \phi_f =
\phi.$$

A neat feature is that $\phi_f$ may be subjected to the same boundary
conditions as $\phi$, such that the filtered field could enherit
crucial properties of the original fields (i.e. assuming a solution
even exists).

We solve the equation with the `Poisson-Helmholtz` solver and visualize
the result with the `output_ppm()` utility.
 */
#include "poisson.h"
#include "utils.h"

#define DELTA 1.0

scalar phi[], phif[];

int Helmholtz_filter (scalar phi, scalar phif, double Delta) {
  double sq_alpha = -sq(Delta/(2.*pi));
  const face vector alphaf[] = {sq_alpha, sq_alpha};
  return poisson (phif, phi, alphaf, unity).i;
}

int main() {
  L0 = 2*pi;
  foreach_dimension()
    periodic (left);
  init_grid (128);
  foreach()
    phi[] = sin(x) * cos(y) + 0.5*noise();
  output_ppm (phi, file = "phi.png", n = 400,
	      min = -1.4, max = 1.4);
  /**
     We start with a noisy field:
     
     ![The original field ($\phi$)](hf/phi.png)

  */
  Helmholtz_filter (phi, phif, DELTA);
  output_ppm (phif, file = "phif.png", n = 400,
	      min = -1.4, max = 1.4);
  /**
     The result for $\phi_f$:

![The filtered field ($\phi_f$)](hf/phif.png)

Looks good..

## A link with diffusion

Recall the Helmholtz filter

$$\left(1 - \left(\frac{\Delta}{2\pi}\right)^2\nabla^2\right) \phi_f =
\phi.$$

One may rewrite it as

$$\left(\frac{\Delta}{2\pi}\right)^2\nabla^2 \phi_f - \phi_f = -\phi, $$
$$\mathrm{or\ as }\rightarrow$$
$$\phi_f - \phi = \left(\frac{\Delta}{2\pi}\right)^2\nabla^2 \phi_f.$$

From which we recognize the structure of the backward-Euler discretization for the diffusion equation:

$$\frac{\phi^{n+1} - \phi^{n}}{\mathrm{d}t} = D\nabla^2\phi^{n+1},$$

where $\phi^{n+1} \leftarrow \phi_f$, $\phi^n \leftarrow \phi$, and $\mathrm{d}t \times D \leftarrow  \left(\frac{\Delta}{2\pi}\right)^2$.   

## References 

Fodor K, Mellado, JP, Wilczek M (2019) *On the role of large-scale
updrafts and downdrafts in deviations form Monin-Obukhov similarity
theory in free convection*. Boundary Layer
Meteorology. [DOI](https://doi.org/10.1007/s10546-019-00454-3)

Foias C, Holm D, Titi E (2001) *The Navier–Stokes-alpha model of fluid
turbulence.* Physica D 152–153:505–519
*/
}