The Hasegawa-Wakatani model of plasma turbulence

This repository contains a reference implementations for the Hasegawa-Wakatani model in two dimensions using Python. The purpose is to provide a playground for education and scientific purposes: be it testing numerical or machine learning methods, or building related models quicker, while improving the comparability of results.

Stable, verified parameters and values are published along with this repository.

Installation

Install a pure NumPy version via

pip install hw2d

and to include accelerators (currently supporting: Numba), use the following:

pip install hw2d[accelerators]

Usage

Running python -m hw2d will let you run a hw2d simulation. It exposes the CLI Interface of the code located in src/hw2d/run.py with all parameters available there. Simply run python -m hw2d --help to get a full rundown of all available parameters along with their explanations.

If accelerators like Numba are installed, these will be used automatically. To manually decide which function should be run with which accelerator, simply change the imports in srd/hw2d/model.py to select the appropriate function. Default uses Numba for periodic gradients and the Arakawa scheme, and to be extended in the future.

Example Usage

Example of running a fully converged turbulent simulation:

python -m hw2d --step_size=0.025 --end_time=1000 --grid_pts=512 --c1=1.0 --k0=0.15 --N=3 --nu=5.0e-08 --output_path="test.h5" --buffer_length=100 --snaps=1 --downsample_factor=2 --movie=1 --min_fps=10 --speed=5 --debug=0

The code will run a grid of 512x512 in steps of 0.025 from 0 to 1000 for an adiabatic coefficient of 1.0 and hyperdiffusion of order 3 with a coefficient of 5.0e-08. The resulting data will be saved to "test.h5" after every 1 frame and written in batches of 100 using a 2x downsampled representation (256,256). This file will then be turned into a movie with at least 10 frames per second, running at 5t per second. The entire process will use no debugging.

Full documentation is available at: https://the-rccg.github.io/hw2d/

Reference Methods

The implementation presented here is by no means meant to be the optimal, but an easy to understand starting point to build bigger things upon and serve as a reference for other work. This reference implementation uses:

• Gradients $\left(\partial_x, \partial_y, \nabla \right)$: Central finite difference schemes (2nd order accurate)
• Poisson Bracket $\left([\cdot, \cdot]\right)$: Arakawa Scheme (2nd order accurate, higher order preserving)
• Poisson Solver $\left(\nabla^{-2}\cdot\right)$: Fourier based solver
• Time Integration $\left(\partial_t\right)$: Explicit Runge-Kutta (4th order accurate)

The framework presented here can be easily extended to use alternative implementations.

Contributions encouraged

Pull requests are strongly encouraged.

The simplest way to contribute is running simulations and committing the results to the historical runs archive. This helps in exploring the hyper-parameter space and improving statistical reference values for all.

If you don't know where to start in contributing code, implementing new numerical methods or alternative accelerators make for good first projects!

Code guidelines

All commits are auto-formatted using Black to keep a uniform presentation.

The Hasegawa-Wakatani Model

The HW model describes drift-wave turbulence using two physical fields: the density $n$ and the potential $\phi$ using various gradients on these.

$$ \begin{align} \partial_t n &= c_1 \left( \phi - n \right) - \left[ \phi, n \right] - \kappa_n \partial_y \phi - \nu \nabla^{2N} n \\ \partial_t \Omega &= c_1 \left( \phi - n \right) - \left[ \phi, \Omega \right] - \nu \nabla^{2N} \Omega \\ \Omega &= \nabla^2 \phi \end{align} $$

hw2d_sample.mp4

Dynamics of the different phases

The model produces self-organizing turbulent structures in a three distinct stages: initial self-organization, linear drift waves, and a stable turbulent phase.

For the interesting intermediary phase for the adiabatic coefficient, c1=1, the initial perturbation will start organizing to produce linear drift waves through the $\partial_t \phi$ component. The system transitions into this first linear phase at roughly t=15, saturates at around t=45, and breaks down to transition into the turbulent phase at about t=80. The turbulent phase is visually saturated at around t=125, but physical parameters overshoot and only fall into the long term stable phase at around t=200.

Physical Properties

Numerical values for each frame

The reason why the Hasegawa-Wakatani Model has been the de-facto testing bed for new methods are its statistically stationary properties of the complex turbulent system. The module includes all code needed to generate these values. It goes further, however, and provides reference values along with statistical bounds for the first time for a vast range of values. This allows simple comparison, as well es evaluation of new methods to one reference community built resource.

$$ \begin{align} \Gamma^n &= - \iint{ \mathrm{d}^2x \space \left( n \space \partial_y \phi \right) } \\ \Gamma^c &= c_1 \iint{ \mathrm{d}^2x \space \left(n - \phi \right)^2} \\ E &= \small \frac{1}{2} \normalsize \iint{\mathrm{d}^2 x \space \left(n^2 - \left|\nabla_\bot \phi \right|^2 \right)} \\ U &= \small \frac{1}{2} \normalsize \iint{\mathrm{d}^2 x \space \left(n-\nabla_\bot^2 \phi\right)^2} = \small \frac{1}{2} \normalsize \iint{\mathrm{d}^2 x \space \left(n-\Omega\right)^2} \end{align} $$

Spectral values for each frame

Additionally, some spectral properties are included for more detailed analysis beyond the scalar factors, among these are:

$$ \begin{align} \Gamma^n \small (k_y) \normalsize &= - i k_y \space n \small (k_y) \normalsize \space \phi^* \small (k_y) \normalsize \\ \delta \small (k_y) \normalsize &= - \mathrm{Im}\left( \mathrm{log} \left( n^* \small (k_y) \normalsize \space \phi \small (k_y) \normalsize \right) \right) \\ E^N \small (k_y) \normalsize &= \small \frac{1}{2}\normalsize \big| n \small (k_y) \normalsize \big|^2 \\ E^V \small (k_y) \normalsize &= \small \frac{1}{2}\normalsize \big| k_y \space \phi \small (k_y) \normalsize \big|^2 \end{align} $$

Predictable in- and outflows over time

Finally, due to the definition of the fields as perturbation fields with background density gradients, the system gains and loses energy and enstrophy in a predictable manner over time. The conservation of these are also tested within the continuous integration pipeline. The definitions are given by:

$$ \begin{align} \partial_t E &= \Gamma^N - \Gamma ^c - \mathfrak{D}^E \\ \partial_t U &= \Gamma^N - \mathfrak{D}^U \\ \mathfrak{D}^E &= \quad \iint{ \mathrm{d}^2x \space (n \mathfrak{D^n} - \phi \mathfrak{D}^\phi)} \\ \mathfrak{D}^U &= - \iint{ \mathrm{d}^2x \space (n - \Omega)(\mathfrak{D}^n - \mathfrak{D}^\phi)} \\ with \quad \mathfrak{D}^n \small (x,y) \normalsize &= \nu \nabla^{2N} n \quad and \quad \mathfrak{D}^\phi \small (x,y) \normalsize = \nu \nabla^{2N} \phi \end{align} $$

General notes

It is the common practice across all reference texts to calculate $\int\cdot$ as $\langle \cdot \rangle$ for a unitless box of size one in order to get comparable values for all properties.

Common Issues in Simulating HW2D

Crashing/NaN encountered

within < 10 timesteps

The simulation has exploded in one direction. Most commonly this means that the hyper-diffusion components are too large.

• reduce the hyper diffusion order: N
• reduce the diffusion coefficient: nu
• reduce the initial perturbation: scale

around t=75-125

The timestep is too big in the turbulent phase. CFL criteria are no longer satisfied and simulation crashes.

• reduce: step_size

Chessboard pattern emerges

The energy accumulates at grid scale. Hyper-diffusion component is not large enough to dissipate the energy.

• increase: nu
• increase: N

Physical values deviate from references

The HW2D model can create stable simulations that are under-resolved, through very large hyper-diffusion terms. A higher resolution is needed for this box size.

• increase: grid_pts

If the esntrophy goes way above reference values, however, it can mean that the hyper-diffusion is too small.

• increase: nu

References

The region between the adiabatic and hydrodynamic limit is defined at c_1=1. For this dynamic and a box size of k0=0.15, a minimum grid size of 512x512 is needed at a dt=0.025. To generate a stable simulation with hyper-diffusion (N=3) requires a value of nu=5e-08.

Reference Step Sizes

Minimum step sizes for the system can be evaluated by setting hyper-diffusion to zero N=0 and nu=0 and running to about age=200 to reach into the turbulent steady-state regime.

integrator	c1	Box Size	grid_pts	max dt	converged?
rk4	1.0	0.15	1024x1024	0.025	True
rk4	1.0	0.15	512x512	0.025	True
rk4	1.0	0.15	256x256	0.05	False
rk4	1.0	0.15	128x128	0.05	False
rk4	1.0	0.15	64x64	0.05	False
rk4	1.0	0.15	32x32	0.05	False

However, steady-state turbulent stages are reached later for various c1 values. Given the different values for the adiabatic coefficient, different step_sizes dtare required. Below is a table of stable and converged simulation parameters to use.

For a box size of 0.15 and using an rk4 scheme as an integrator, here are the values:

c1	grid_pts	max dt	nu	age	$\Gamma_n\pm\delta\Gamma_n$
0.01	512x512	0.005	5e-09	750+	$2.79\pm0.80$
0.1	512x512	0.01	5e-09	200+	$1.51\pm0.21$
0.25	512x512	0.015	5e-09	200+	$1.17\pm0.13$
0.5	512x512	0.02	5e-09	200+	$0.89\pm0.09$
0.75	512x512	0.025	5e-09	200+	$0.72\pm0.06$
0.9	512x512	0.025	5e-09	200+	$0.65\pm0.05$
1.0	512x512	0.025	5e-09	250+	$0.60\pm0.05$
1.1	512x512	0.025	5e-09	250+	$0.58\pm0.04$
1.25	512x512	0.025	5e-09	250+	$0.54\pm0.04$
1.5	512x512	0.025	5e-09	250+	$0.46\pm0.04$
2.0	512x512	0.025	5e-09	300+	$0.38\pm0.03$
2.5	512x512	0.02	5e-09	400+	$0.31\pm0.03$
3.0	512x512	0.02	5e-09	500+	$0.26\pm0.02$
4.0	512x512	0.015	5e-09	650+	$0.18\pm0.02$
5.0	512x512	0.01	5e-09	800+	$0.13\pm0.02$

Reference Timetraces

Sample traces are given for 512x512, dt=0.05, c1=1, N=3, and nu=5e-08. Note that the statistical nature does mean single simulations can deviate for quite some time from the statistical mean.

Reference Values

Reference values are averaged over 25 runs starting from well within the turbulent steady-state t=300 with the standard deviation across the simulations denoted by $\pm$. Each run to t=1,000 at 512x512 and dt=0.025 requires roughly 125GB (3 million floats/frame for 3 fields over 40,000 frames per simulation), meaning the summary contains information for 10TB of data. This does not include the hypterparameter stabilization tests. As a result, it is practically unfeasible to supply this data.

Metric	HW2D Data	Stegmeir	Camargo	HW	Zeiler
****	512x512	Stegmeir et al.	Camargo et al.	In Stegmeir	Zeiler et al.
$\Gamma_n$	$0.60 \pm 0.01$	$0.64$	$0.73$	$0.61$	$0.8$
$\delta\Gamma_n$	$0.05 \pm 0.01$	$n/a$	$n/a$	$n/a$	$n/a$
$\Gamma_c$	$0.60 \pm 0.01$	$n/a$	$0.72$	$n/a$	$n/a$
$\delta\Gamma_n$	$0.03 \pm 0.01$	$n/a$	$n/a$	$n/a$	$n/a$
$E$	$3.78 \pm 0.07$	$3.97$	$4.4$	$3.82$	$6.1$
$\delta E$	$0.29 \pm 0.03$	$0.26$	$0.16$	$0.26$	$0.51$
$U$	$13.2 \pm 0.91$	$n/a$	$12.8$	$n/a$	$n/a$
$\delta U$	$0.68 \pm 0.08$	$n/a$	$1.66$	$n/a$	$n/a$