Models

All the equations are dimensionless.

Nomenclature:

• n : density
• φ : electric potential
• Ω = Δφ : vorticity
• κ : density gradient
• C : adiabatic parameter
• ν : viscosity
• D : diffusivity
• Xₖ : X in the Fourier space
• X̅ = 1/Ly ∫_y X(x,y) dy : zonal average
• X̃ = X - X̅ : fluctuations
• [X,Y] = ∂ˣX∂ʸY - ∂ʸX∂ˣY : Poisson bracket
• X* : conjugate of X

hasegawa_wakatani_pspectral_2d

Pseudo-spectral method (periodic boundaries) of the Hasekawa-Wakatani model in 2D.

We used the 2/3 rule for the zero padding to avoid aliasing effects. These equations are expressed in the Fourier space. $\frac{\partial {\varphi }_k}{\partial t}=-\frac{C_k}{k^2}({\varphi }_k-n_k)-{\nu }_k{\varphi }_k+\frac{1}{k^2}[\varphi ,\mathrm{\Delta }\varphi {]}_k\frac{\partial n_k}{\partial t}=C_k({\varphi }_k-n_k)-i\kappa k_y{\varphi }_k-D_k{n}_k-[\varphi ,n{]}_k$ where $[\varphi ,\mathrm{\Delta }\varphi {]}_k=\mathrm{RFFT}2(\mathrm{iRFFT}2(i{k}_x{\phi }_k)\mathrm{iRFFT}2(-i{k}_y{k}^2{\phi }_k)-\mathrm{iRFFT}2(i{k}_y{\phi }_k)\mathrm{iRFFT}2(-i{k}_x{k}^2{\phi }_k))$

\	$C_k$	${\nu }_k$	$D_k$
$k_y\ne 0$	$C$	${\nu }_x\ast k_x^2+{\nu }_y\ast k_y^2$	$D_x\ast k_x^2+D_y\ast k_y^2$
$k_y=0$ (zonal flows)	$0$	${\nu }_z$	$D_z$

The initial state is initialised in the Fourier space.

hasegawa_wakatani_pspectral_2d:
  grid_size: 682
  domain: 50
  atol: 1.0e-10
  rtol: 1.0e-10
  νx: 1.0e-4
  νy: 1.0e-4
  νz: 1.0e-5
  Dx: 1.0e-4
  Dx: 1.0e-4
  Dz: 1.0e-5
  tf: 500
  video_length: 10

pspectral_2D.mp4

Runtime (GPU RTX 3060) ≈ 1h20

hasegawa_wakatani_pspectral_1d

Pseudo-spectral method of the single poloidal mode of Hasekawa-Wakatani.

It is simply a wrapper of hasegawa_wakatani_pspectral_2d where the grid_size in the y direction is 4. The domain in the y direction is 2*pi/ky where ky is the most unstable wave number (maximizing the growth rate γ) computed numerically. See also hasegawa_wakatani_findiff_1d.

hasegawa_wakatani_pspectral_1d:
  grid_size: 682
  domain: 50
  νx: 1.0e-2
  νy: 1.0e-4
  νz: 1.0e-5
  Dx: 1.0e-4
  Dy: 1.0e-4
  Dz: 1.0e-5
  tf: 300
  atol: 1.0e-10
  rtol: 1.0e-10
  video_length: 30
  video_fps: 100

We increased νx because the loss of νy/modes in the y direction

hasegawa_wakatani_findiff_1d

Finite difference method of the single poloidal mode model.

In this model, only one poloidal mode is kept. Therefore, we can express n, φ and Ω this way: $X=\bar{X}+X_k{e}^{i{k}_{y}y}+X_k^{\ast }{e}^{-i{k}_{y}y}$ where ky is a parameter, the single poloidal mode, the most unstable wave number. The equations become: $\frac{\partial {\mathrm{\Omega }}_k}{\partial t}=i{k}_y({\varphi }_k\frac{\partial \bar{\mathrm{\Omega }}}{\partial x}-{\mathrm{\Omega }}_k\frac{\partial \bar{\varphi }}{\partial x})+C({\varphi }_k-n_k)+{\nu }_x\frac{{\partial }^2{\mathrm{\Omega }}_k}{\partial x^2}-{\nu }_y{k}_y^2{\mathrm{\Omega }}_k$

$$\frac{\partial n_k}{\partial t}=i{k}_y((\frac{\partial \bar{n}}{\partial x}-\kappa ){\varphi }_k-{\mathrm{\Omega }}_k\frac{\partial \bar{\varphi }}{\partial x})+C({\varphi }_k-n_k)+D_x\frac{{\partial }^2{n}_k}{\partial x^2}-D_y{k}_y^2{n}_k$$

$$\frac{\partial \bar{\mathrm{\Omega }}}{\partial t}=2k_y\frac{\partial }{\partial x}\mathrm{Im}{\varphi }_k^{\ast }{\mathrm{\Omega }}_k-{\nu }_z\bar{\mathrm{\Omega }}\frac{\partial \bar{n}}{\partial t}=2k_y\frac{\partial }{\partial x}\mathrm{Im}{\varphi }_k^{\ast }{n}_k-D_z\bar{n}$$

where

$$\bar{\mathrm{\Omega }}=(\frac{{\partial }^2}{\partial x^2}-k_y^2)\bar{\varphi }{\mathrm{\Omega }}_k=\frac{{\partial }^2}{\partial x^2}{\varphi }_k$$

and (from the dispersion relation, see)

$$\begin{array}{rlrl}{k}_y& =\underset{k_y}{\mathrm{argmax}}\gamma (\mathbf{k}){|}_{k_x=0}& \gamma (\mathbf{k})& =-A+\sqrt{\frac{H+G}{2}}\\ H& =\sqrt{G²+\left(C\kappa \frac{k_y}{k²}\right)²}& A& =\frac{1}{2}(Dk²+C+(\frac{C}{k²}+\nu k²))\\ G& =B²+\left(\frac{C}{k}\right)²& B& =\frac{1}{2}(Dk²+C-(\frac{C}{k²}+\nu k²))\end{array}$$

hasegawa_wakatani_findiff_1d:
  grid_size: 682
  domain: 50
  νx: 1.0e-2
  νy: 1.0e-4
  νz: 1.0e-5
  Dx: 1.0e-4
  Dy: 1.0e-4
  Dz: 1.0e-5
  acc: 10
  tf: 300
  atol: 1.0e-10
  rtol: 1.0e-10
  video_length: 30
  video_fps: 100

Periodic

Note the result is similar to hasegawa_wakatani_pspectral_1d because it is the same problem solved with 2 different methods.

Dirichlet

We simply add this line as a parameter in the .yaml file:

  boundary: dirichlet

Force

The goal is to create a natural density gradient. In this configuration:

• we set κ=0
• we add a source term S in the equation of n̅ wich represents the particle injection from the center. Its distribution a gaussian centered on the left hand side of the domain. We also add a small well on the right hand side to act like the loss of the particle at the edges. The equation becomes: $\frac{\partial \bar{n}}{\partial t}=2k_y\frac{\partial }{\partial x}\mathrm{Im}{\varphi }_k^{\ast }{n}_k-(D_z+\mathrm{well})\bar{n}+S$
• the boundary conditions are:

side	n̅, Ω̅	nₖ, Ωₖ
left	neumann	dirichlet
right	dirichlet	dirichlet

  κ: 0
  ky: 1.3
  tf: 2000000
  boundary: force 0.001

where 0.001 is the amplitude of the source term and we increased the simulation time tf.

hasegawa_wakatani_findiff_2d

Finite difference method of the Hasegawa-Wakatani model in 2D (centered grid).

This is much slower than hasegawa_wakatani_pspectral_2d

$$\frac{\partial \mathrm{\Delta }\varphi }{\partial t}=-[\varphi ,\mathrm{\Delta }\varphi ]+C(\tilde{\varphi }-\tilde{n})+\nu {\mathrm{\Delta }}^2\tilde{\varphi }-{\nu }_z\mathrm{\Delta }\bar{\varphi }\frac{\partial n}{\partial t}=-[\varphi ,n]+C(\tilde{\varphi }-\tilde{n})+D\mathrm{\Delta }\tilde{n}-D_z\bar{n}-\kappa \frac{\partial \varphi }{\partial y}$$

We compute φ by solving Ω=Δφ at each time step.

hasegawa_wakatani_findiff_2d:
  grid_size: 800
  domain: 50
  D: 1.0e-3
  Dz: 1.0e-5
  ν: 1.0e-3
  νz: 1.0e-5
  acc: 6
  atol: 1.0e-10
  rtol: 1.0e-6
  tf: 500
  video_length: 10
  video_fps: 60

Periodic

findiff_2d_periodic.mp4

Dirichlet

  boundary: dirichlet

The initial state is a gaussian in the density field n.

findiff_2d.mp4

runtime (GPU RTX 3060) ≈ 90h

hasegawa_mima_pspectral_2d

Pseudo-spectral method of the Hasegawa-Mima model. This is the abiabatic limit of the Hasegawa-Wakatani model: when C→∞, we have n=ϕ.

The equation is expressed in the Fourier space. $\frac{\partial {\varphi }_k}{\partial t}=-\frac{[\varphi ,\varphi -\mathrm{\Delta }\varphi {]}_k+i{k}_y\kappa {\varphi }_k}{1+k^2}-\nu k^2{\varphi }_k+\mathrm{forcing}\nu k^2={\nu }_z\text{if}{k}_y=0$ Where the forcing term is a gaussian centered in kx=0 and ky=force_ky with an amplitude force_amplitude and a random phase at each step.

hasegawa_mima_pspectral_2D:
  grid_size: 682
  ν: 1.0e-4
  force_ky: 1
  force_amplitude: 5.0e-3
  atol: 1.0e-10
  rtol: 1.0e-6
  tf: 500

mima_pspectral_2d.mp4

hasegawa_wakatani_pspectral_3d

Similar to hasegawa_wakatani_pspectral_2d but $k^2=k_x^2+k_y^2+k_z^2$

hasegawa_wakatani_pspectral_3d:
  grid_size: 128
  domain: 50
  ν: 2.0e-2
  νz: 1.0e-5
  D: 1.0e-2
  Dz: 1.0e-5
  atol: 1.0e-10
  rtol: 1.0e-10
  tf: 300

pspectral_3d.mp4