Vlasov example

In this example, we extend the Vlasov-Maxwell input file shown in A first Gkeyll simulation to simulate a more general kinetic plasma. Because the Vlasov-Maxwell system of equations is widely applicable in plasma physics, this example is intended to illustrate some of the functionality of the Vlasov-Maxwell solver a user may desire for production science. For more extensive documentation on all of the available options for Vlasov-Maxwell simulations, we refer the reader to VlasovMaxwell App: Vlasov-Maxwell equations on a Cartesian grid.

This simulation is based on the studies of [Skoutnev2019] and [Juno2020] and concerns the evolution of instabilities driven by counter-streaming beams of plasma. This example demonstrates the flexibility of the Vlasov-Maxwell solver by showing how one extends Vlasov-Maxwell simulations to higher dimensionality, in this case 2x2v. The input file for this example is also a standard performance benchmark for Gkeyll and timings for this input file with varying grid resolution can be found in the gkyl repo in the Benchmarks/ folder.

Physics model and initial conditions

This example solves the Vlasov-Maxwell system of equations

\[\begin{split}\frac{\partial f_s}{\partial t} &+ \mathbf{v}\cdot\nabla f_s + \frac{q_s}{m_s} \left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right)\cdot\nabla_{\mathbf{v}}f = 0, \\ \frac{\partial\mathbf{B}}{\partial t} &+ \nabla\times\mathbf{E} = 0, \\ \epsilon_0\mu_0\frac{\partial\mathbf{E}}{\partial t} &- \nabla\times\mathbf{B} = -\mu_0\mathbf{J},\end{split}\]

in two spatial dimensions and two velocity dimensions (2x2v). For this example, the ions are taken to be a stationary, neutralizing background, and therefore do not contribute to the plasma current \(\mathbf{J}\). The electrons are initialized as two equal density, equal temperature, counter-propagating, Maxwellian beams

\[f_e (x, y, v_x, v_y) = \frac{m_e n_0 }{2 \pi T_e} \left [ \exp \left (- m_e \frac{(v_x)^2 + (v_y - u_d)^2}{2 T_e} \right ) + \exp \left (- m_e \frac{(v_x)^2 + (v_y + u_d)^2}{2 T_e} \right ) \right ].\]

The electromagnetic fields are initialized as a bath of fluctuations in the electric and magnetic fields in the two configuration space dimensions,

\[B_z(t=0)=\sum_{n_x,n_y=-N,-N}^{N,N}\tilde B_{n_x,n_y}\sin \left (\frac{2\pi n_x x}{L_x}+\frac{2\pi n_y y}{L_y}+\tilde \phi_{n_x,n_y} \right ),\]

where \(N=16\) and \(\tilde B_{n_x,n_y}\) and \(\tilde \phi_{n_x,n_y}\) are random amplitudes and phases. The electric fields, \(E_x, E_y\) are initialized similarly. Note that the sum goes from \(-N\) to \(N\) so as to initialize phases from 0 degrees to 180 degrees.

Input file

The full Lua input file (found here) has three components: the App dependencies, the Preamble, and the App. In the App dependencies section, we load the necessary components of Gkeyll to perform a Vlasov-Maxwell simulation, as well as any additional functionality we require:

--------------------------------------------------------------------------------
-- App dependencies
--------------------------------------------------------------------------------
-- Load the Vlasov-Maxwell App
local Plasma = require("App.PlasmaOnCartGrid").VlasovMaxwell()
-- Pseudo-random number generator from SciLua package for initial conditions.
-- Specific rng is the mrg32k3a (multiple recursive generator) of Ecuyer99.
local prng = require "sci.prng"
local rng = prng.mrg32k3a()

The Preamble to set up the initial conditions is:

--------------------------------------------------------------------------------
-- Preamble
--------------------------------------------------------------------------------
-- Constants
permitt = 1.0                               -- Permittivity of free space
permeab = 1.0                               -- Permeability of free space
lightSpeed = 1.0/math.sqrt(permitt*permeab) -- Speed of light
chargeElc = -1.0                            -- Electron charge
massElc = 1.0                               -- Electron mass

-- Initial conditions
-- 1 = Right-going beam; 2 = Left-going beam.
nElc1 = 0.5
nElc2 = 0.5

ud = 0.1                                    -- Drift velocity of beams
uxElc1 = 0.0
uyElc1 = ud
uxElc2 = 0.0
uyElc2 = -ud

R = 0.1                                     -- Ratio of thermal velocity to drift velocity
TElc1 = massElc*(R*ud)^2
TElc2 = massElc*(R*ud)^2
vthElc1 = math.sqrt(TElc1/massElc)
vthElc2 = math.sqrt(TElc2/massElc)

k0_TS = 6.135907273413176                   -- Wavenumber of fastest growing two-stream mode
theta = 90.0/180.0*math.pi                  -- 0 deg is pure Weibel, 90 deg is pure two-stream
kx_TS = k0_TS*math.cos(theta)
ky_TS = k0_TS*math.sin(theta)

k0_Weibel = 2.31012970008316                -- Wavenumber of fastest growing Weibel mode
theta = 0.0/180.0*math.pi                   -- 0 deg is pure Weibel, 90 deg is pure two-stream
kx_Weibel = k0_Weibel*math.cos(theta)
ky_Weibel = k0_Weibel*math.sin(theta)
kx = k0_Weibel
ky = k0_TS/3.0

perturb_n = 1e-8
-- Perturbing the first 16 wave modes with random amplitudes and phases.
-- Note that loop goes from -N to N to sweep all possible phases.
N=16
P={}
for i=-N,N,1 do
   P[i]={}
   for j=-N,N,1 do
      P[i][j]={}
      for k=1,6,1 do
        P[i][j][k]=rng:sample()
      end
   end
end

-- Domain size and number of cells
Lx = 2*math.pi/kx
Ly = 2*math.pi/ky
Nx = 16
Ny = 16
vLimElc = 3*ud                              -- Maximum velocity in velocity space
NvElc = 16

-- Maxwellian in 2x2v
local function maxwellian2D(n, vx, vy, ux, uy, vth)
   local v2 = (vx - ux)^2 + (vy - uy)^2
   return n/(2*math.pi*vth^2)*math.exp(-v2/(2*vth^2))
end

The Preamble defines the constants in the normalization standard outlined in Normalized units for the Vlasov-Maxwell system and sets the parameters and perturbations to the wave modes of interest for the study. Note that because the dimensionality of the simulation is now 2x2v, the normalization of the Maxwellian has correspondingly changed from the 1x1v Langmuir wave simulation described in A first Gkeyll simulation.

The App can be further subdivided into a number of sections

plasmaApp = Plasma.App {
   -----------------------------------------------------------------------------
   -- Common
   -----------------------------------------------------------------------------
   ...

   -----------------------------------------------------------------------------
   -- Species
   -----------------------------------------------------------------------------
   ...

   -----------------------------------------------------------------------------
   -- Fields
   -----------------------------------------------------------------------------
   ...
}
--------------------------------------------------------------------------------
-- Run application
--------------------------------------------------------------------------------
plasmaApp:run()

The Common section of the App defines input parameters which will be utilized by all solvers in the simulation. For example, the configuration space extents and number of configuration space cells (lower, upper, cells), as well as what directions, if any, utilize periodic boundary conditions (periodicDirs), and how to parallelize the simulation (decompCuts,useShared).

--------------------------------------------------------------------------------
-- Common
--------------------------------------------------------------------------------
logToFile = true,

tEnd = 50.0,                             -- End time
nFrame = 1,                              -- Number of output frames
lower = {0.0,0.0},                       -- Lower boundary of configuration space
upper = {Lx,Ly},                         -- Upper boundary of configuration space
cells = {Nx,Ny},                         -- Configuration space cells
basis = "serendipity",                   -- One of "serendipity", "maximal-order", or "tensor"
polyOrder = 2,                           -- Polynomial order
timeStepper = "rk3s4",                   -- One of "rk2", "rk3", or "rk3s4"

-- MPI decomposition for configuration space
decompCuts = {1,1},                      -- Cuts in each configuration direction
useShared = true,                        -- If using shared memory

-- Boundary conditions for configuration space
periodicDirs = {1,2},                    -- periodic directions (both x and y)

-- Integrated moment flag, compute integrated quantities 1000 times in simulation
calcIntQuantEvery = 0.001,

The Species section of the App defines the species-specific inputs for the Vlasov-Maxwell simulation within a Plasma.Species table. For example, the velocity space extents and number of velocity space cells (lower, upper, cells), the function which prescribes the initial condition, and the types of diagnostics. More discussion of diagnostic capabilities can be found in VlasovMaxwell App: Vlasov-Maxwell equations on a Cartesian grid.

--------------------------------------------------------------------------------
-- Electrons
--------------------------------------------------------------------------------
elc = Plasma.Species {
  charge = chargeElc, mass = massElc,
  -- Velocity space grid
  lower = {-vLimElc, -vLimElc},
  upper = {vLimElc, vLimElc},
  cells = {NvElc, NvElc},
  -- Initial conditions
  init = function (t, xn)
     local x, y, vx, vy = xn[1], xn[2], xn[3], xn[4]
     local fv = maxwellian2D(nElc1, vx, vy, uxElc1, uyElc1, vthElc1) +
        maxwellian2D(nElc2, vx, vy, uxElc2, uyElc2, vthElc2)
    return fv
  end,
  evolve = true,
  diagnostics = {"M0","M1i","M2ij","M3i","intM0","intM1i","intM2Flow","intM2Thermal"},
},

Note that for this particular simulation the ions are a stationary, neutralizing background that does not contribute to the plasma current, so we only require a species table for the electrons.

The Field section if the final section of the App and specifies the input parameters for the field equation, in this case Maxwell’s equation, in the Plasma.Field table. For example, similar to the Plasma.Species table, the Plasma.Field table contains the initial condition for the electromagnetic field.

--------------------------------------------------------------------------------
-- Field solver
--------------------------------------------------------------------------------
field = Plasma.Field {
  epsilon0 = permitt, mu0 = permeab,
  init = function (t, xn)
     local x, y = xn[1], xn[2]
     local E_x, E_y, B_z = 0.0, 0.0, 0.0
     for i=-N,N,1 do
        for j=-N,N,1 do
           if i~=0 or j~=0 then
              E_x = E_x + perturb_n*P[i][j][1]*math.sin(i*kx*x+j*ky*y+2*math.pi*P[i][j][2])
              E_y = E_y + perturb_n*P[i][j][3]*math.sin(i*kx*x+j*ky*y+2*math.pi*P[i][j][4])
              B_z = B_z + perturb_n*P[i][j][5]*math.sin(i*kx*x+j*ky*y+2*math.pi*P[i][j][6])
           end
        end
     end
     return E_x, E_y, 0.0, 0.0, 0.0, B_z
  end,
  evolve = true,
},

Running the simulation

The input file vm-tsw-2x2v.lua can be run using the gkyl executable

~/gkylsoft/gkyl/bin/gkyl vm-tsw-2x2v.lua

assuming gkyl has been installed in the user’s home directory. When running this simulation, a user should see the following output

Wed Sep 16 2020 11:38:54.000000000
Gkyl built with a4430cbb5d93
Gkyl built on Sep 16 2020 01:25:31
Initializing Vlasov-Maxwell simulation ...
Initializing completed in 1.39731 sec

Starting main loop of Vlasov-Maxwell simulation ...
Step 0 at time 0. Time step 0.0360652. Completed 0%
0123456789 Step   139 at time 5.01307. Time step 0.0360652. Completed 10%
01234

The full screen output can be found here, which includes performance details for the simulation. This example was run with a single core of a 10th gen Intel i9 (Comet Lake) processor. Increasing the resolution to \(32^2 \times 32^2\) and now running the simulation using all 10 cores of the Intel i9 using

~/gkylsoft/openmpi/bin/mpirun -n 10 ~/gkylsoft/gkyl/bin/gkyl vm-tsw-2x2v.lua

we obtain the following performance with useShared=true and the installed MPI from the Gkeyll build.

Postprocessing

The output of this simulation is the following set of files:

  • Distribution functions: vm-tsw-2x2v_elc_#.bp.
  • Electromagnetic fields: vm-tsw-2x2v_field_#.bp.
  • Velocity moments: vm-tsw-2x2v_elc_M0_#.bp, vm-tsw-2x2v_elc_M1i_#.bp, vm-tsw-2x2v_elc_M2ij_#.bp, and vm-tsw-2x2v_elc_M3i_#.bp.
  • Field energy: vm-tsw-2x2v_fieldEnergy.bp.
  • Volume integrated moments: vm-tsw-2x2v_elc_intM0.bp, vm-tsw-2x2v_elc_intM1i.bp, vm-tsw-2x2v_elc_intM2Flow.bp, and vm-tsw-2x2v_elc_intM2Thermal.bp.

Snapshots (frames) are labeled by the number # at the end of the file name, while volume integrated diagnostics that are computed as a time-series, such as the field energy, are written out as a single file. Since nFrame=1 in the input file, the only frames that are output are 0, corresponding to the initial condition, and 1, corresponding to the end of the simulation.

Since this simulation has two configuration space dimensions, postgkyl creates pcolor plots when run from the command line with pgkyl. We can compare the initial condition and final state of the magnetic field, \(B_z\), (of the \(32^2 \times 32^2\) simulation) in two separate figures with the pgkyl command:

pgkyl vm-tsw-2x2v_field_0.bp vm-tsw-2x2v_field_1.bp interp sel --comp 5 plot -b --fix-aspect
../_images/vm-tsw-2x2v_field.png

\(B_z\) magnetic field at \(t=0\) (left) and \(t=50 \omega_{pe}^{-1}\), the end of the simulation (right).

The default postgkyl colorbar is sequential and useful for visualizing data such as distribution functions, which will vary from 0 (zero phase space density/no particles) to some number (corresponding to a local increase in phase space density). However, we can see that the colorbar for the magnetic field varies between roughly equal positive and negative numbers, and thus a diverging colormap may yield a more useful representation of the data. In addition, we can utilize the flexibility of the interpolate command to interpolate the discontinuous Galerkin data onto an even finer mesh

pgkyl vm-tsw-2x2v_field_0.bp vm-tsw-2x2v_field_1.bp interp -i 6 sel --comp 5 plot -b --fix-aspect --diverging --xlabel '$x (d_e) $' --ylabel '$y (d_e) $'
../_images/vm-tsw-2x2v_field_finer_interp.png

\(B_z\) magnetic field at \(t=0\) (left) and \(t=50 \omega_{pe}^{-1}\), the end of the simulation (right), now with a diverging colorbar, finer interpolation, and labels.

where we have now added labels with the normalized units in \(x\) and \(y\). Note that the default interpolation level for polynomial order 2 is 3 (polyOrder + 1).

We can likewise visualize diagnostic moments such as the first velocity moment elc_M1i

pgkyl vm-tsw-2x2v_elc_M1i_1.bp interp -i 6 plot --fix-aspect --diverging --xlabel '$x (d_e) $' --ylabel '$y (d_e) $'
../_images/vm-tsw-2x2v_elc_M1i.png

\(\mathrm{M}1_x\) first velocity moment (left) and \(\mathrm{M}1_y\) first velocity moment (right) at \(t=50 \omega_{pe}^{-1}\), the end of the simulation.

Note that elc_M1i has two components due to the fact that this simulation has two velocity dimensions, and both components are visualized when this pgkyl command is utilized. The left plot is the \(v_x\) velocity moment and the right plot is the \(v_y\) velocity moment. Further details on the diagnostics available and their definitions can be found in VlasovMaxwell App: Vlasov-Maxwell equations on a Cartesian grid.

We can also visualize the distribution function from this simulation. However, for this simulation the distribution function if four-dimensional, two configuration space and two velocity space dimensions. Postgkyl offers a number of options for down-selecting the data to be more amenable to visualizing. For example, we can read-in a subset of the data and visualize the distribution function in velocity space \(v_x-v_y\) in the lower left corner of the domain

pgkyl vm-tsw-2x2v_elc_1.bp --z0 0 --z1 0 interp -i 6 sel --z0 0.0 --z1 0.0 plot --xlabel '$v_x (v_{th_e}) $' --ylabel '$v_y (v_{th_e}) $' --vmin 0.0
../_images/vm-tsw-2x2v_elc_vxvy.png

Note that the immediate --z0 0 --z1 0 tells postgkyl to read in only the first \(x\) and \(y\) configuration space grid cells (while still reading in all of velocity space). Because we are then interpolating the data onto a finer mesh, the data is still four dimensional so we pass the abbreviated select command sel to finally down-select to the lower-left corner of the configuration space domain. These selective read-in commands are vital for very large arrays where the cost in memory and CPU time can be quite large to read-in and manipulate the data structure of interest.

Alternatively, if we do want to read-in the whole array, we can perform other manipulations to the distribution function such we can still easily visualize the data. For example, we can use the integrate command to integrate the distribution function over \(x\) and \(v_x\) to produce a \(y-v_y\) plot of the electron distribution function.

pgkyl vm-tsw-2x2v_elc_1.bp interp integrate 0,2 plot --xlabel '$y (d_e) $' --ylabel '$v_y (v_{th_e}) $' --vmin 0.0
../_images/vm-tsw-2x2v_elc_yvy.png

Finally, since we performed this simulation at two different resolutions, and interesting diagnostic to look at is a comparison of integrated quantities between the two simulations. For ease of plotting we have moved the data from the two simulations to two different folders, res1 (\(16^2 \times 16^2\)) and res2 (\(32^2 \times 32^2\)). Here, we are being agnostic on what a user might have named these two different simulations and labeling them ourselves with postgkyl.

pgkyl res1/*fieldEnergy.bp -l '$16^2 \times 16^2$' res2/*fieldEnergy.bp -l '$32^2 \times 32^2$' select --comp 5 plot --logy --xlabel '$t (\omega_{pe}^{-1})$' --ylabel '$\int B_z^2$' -f0
../_images/vm-tsw-2x2v_fieldEnergy.png

Integrated magnetic field energy, \(|B_z|^2\), plotted as a function of time comparing the lower resolution calculation, \(16^2 \times 16^2\) (blue), and higher resolution calculation, \(32^2 \times 32^2\) (orange).

References

[Skoutnev2019]Skoutnev, V., Hakim, A., Juno, J., & TenBarge, J. M. (2019). “Temperature-Dependent Saturation of Weibel-Type Instabilities in Counter-streaming Plasmas”, Astrophysical Journal Letters, 872, (2). https://doi.org/10.3847%2F2041-8213%2Fab0556
[Juno2020]Juno, J., Swisdak, M. M., TenBarge. J. M., Skoutnev, V., & Hakim, A. “Noise-induced magnetic field saturation in kinetic simulations”, Journal of Plasma Physics, 86, (4). https://doi.org/10.1017/S0022377820000707