JE17: Solving (Modified) Hasegawa-Wakatani equations
In this note I describe how to solve the Hasegawa-Wakatani system using Gkeyll, both in its original incarnation as well as modified to describe zonal flows.
Contents
The Hasegawa-Wakatani system
The Hasegawa-Wakatani (HW) system is a coupled set of equations for the plasma density and electrostatic potential that describe resistive drift wave turbulence. See, for example, the early papers by Hasegawa and Wakatani [Wakatani1986], [Hasegawa1987]. These can be written as
where
Note that in general
The hyper-diffusive terms are usually added for numerical
stability. However, the DG scheme implemented in Gkeyll works fine
without these, and hence I have set
The HW system contains two limits that can be obtained by setting
with the potential now determined from
As of writing this note, I have not attempted to solve the Hasegawa-Mima equations with Gkeyll, although it should be an easy task.
The modified Hasegawa-Wakatani system
When restricted to 2D the HW system described above does not contain
zonal flows. Numata et. al [Numata2007] describe a simple
modification that allows capturing zonal flows. This is obtained by
observing that in a tokamak edge any potential fluctuations on a flux
surface is neutralized by the parallel electron motion. Define the
zonal and non-zonal component of any variable
respectively. Note that in the reduced 2D geometry, the
These are identical to the standard HW system, except that the adiabatic coupling terms are computed differently.
A note on solving (M)HW systems with Gkeyll
Given the Poisson bracket solver in Gkeyll (see JE12) all that remains to be done is to
write a Lua program that combines two of these to update the LHS of
the (M)HW system. The drift wave term on the RHS (with accumulate
function.
When solving the MHW system the zonal component is computed using a “moment” updater (also used in the Valsov-Poisson solvers) and subtracting it off to compute the nonzonal components needed in the source terms.
The Lua programs are long, but relatively straightforward. See [s215] for an example input file where these are coded up.
In the HW system, with large value of the adiabaticity parameter
Simulations of the Hasegawa-Wakatani system
In this set of simulations the Hasegawa-Wakatani system was initialized with a Gaussian initial number density profile.
with
Note that for
Comparisons (with different adiabaticity parameter) for the vorticity, potential and number density are shown below. The initial Gaussian profiles undergo linear drift-wave instabilities which are eventually taken over by nonlinear effects. Once the simulation becomes nonlinear vortices are generated, driving the system into a turbulent state.
With increasing adiabaticity differences in the structure of the
generated turbulence are clearly visible. In particular, for
Note
As of writing this, I have not performed any statistical analysis of
the simulations. However, it is clear that the turbulence is in a
saturated state. One way to see this is to monitor the time-step as
the simulation progresses. This effectively tracks the

Comparison of vorticity (

Comparison of number density (

Comparison of potential (
Simulations of the modified Hasegawa-Wakatani system
For the MHW system, the simulations were initialized with random noise
for
Vortices rapidly form and the solution goes turbulent, initially
showing similar vortex patterns as in the unmodified HW
system. However, zonal flows soon set in and the turbulent
fluctuations in the electrostatic potential are suppressed. The

Comparison of vorticity (

Comparison of number density (

Comparison of of potential (
References
- Wakatani1986
Masahiro Wakatani and Akira Hasegawa, “A collisional drift wave description of plasma edge turbulence”, Physics of Fluids, 27 (3), 1984.
- Hasegawa1987
Akira Hasegawa and Masahiro Wakatani, “Self-Organization of Electrostatic Turbulence in a Cylindrical Plasma”, Physical Review Letters, 59 (14), 1987.
- Numata2007
Numata, R., Ball, R., & Dewar, R. L, “Bifurcation in electrostatic resistive drift wave turbulence”. Physics of Plasmas, 14 (10), 102312, 2007.