:Author: Ammar Hakim


JE9: Tunneling through an electron-cyclotron cutoff layer
=========================================================

In this note I study the propagation of an radio-frequency (RF) wave
into a electron-cyclotron cutoff layer. As in the plasma beach
problem, the ions are assumed to be stationary and are not
evolved. The plasma is initialized with a uniform density and is
threaded with a non-uniform static transverse field. This static field
exerts a Lorentz force on the electrons but is not evolved or included
in the electric field update equations. What this means physically is
that the static field is assumed to be created from a set of external
coils and hence its curl is zero.

The domain is one-dimensional, :math:`0 <x < 0.14`, with open boundary
conditions on either end. The plasma density is set to
:math:`1\times10^{17}`/m :math:`^3`. An electromagnetic wave is driven
by a current applied at the center of the last cell, i.e.,

.. math::

  J_y(t) = J_0\sin(2\pi f_d t)\thinspace
  \sin^2\big(0.5\pi \min(1, t/t_0)\big)

where :math:`f_d = 15\times 10^9` Hz is the drive frequency and
:math:`t_0=10/f_d` is the time at which the source turns on
completely. The second factor in the above equation ramps up the
source slowly so as to avoid exciting small scale oscillations and
extraneous modes. The static magnetic field is set to

.. math::

  B_z(x) = B_0\frac{R_0+x_c}{R_0+x}

where :math:`B_0 = 0.536` Tesla, :math:`R_0 = 5\times10^{-3}` and
:math:`x_c = 0.04` m is the location at which the electron cyclotron
frequency matches the drive frequency. The electron temperature is set
to :math:`1\times10^{-2}` eV, i.e. the electrons are cold.

The problem was run on a 200 cell grid and run to :math:`1.4` ns. The
results of :math:`E_x` and :math:`E_y` are plotted below at different
time frames. The results show that the EM wave suffers a cutoff at
:math:`x_c` and tunnels into the cyclotron layer. It is also seen that
the electrostatic field develops a sharp spike around the cutoff layer
as the wave number becomes infinite there.

.. figure:: s72-Ey.png
  :width: 100%
  :align: center

  The electric field (:math:`E_y`) of the EM wave at different
  times. The black dashed line shows the location of the cyclotron
  cutoff. The wave tunnels through the electron cyclotron resonance
  layer, forming a distinct standing wave pattern late in time. The
  simulation input file is at :doc:`s72
  <../../sims/s72/s72-cyclotron-cutoff>`.

.. figure:: s72-Ex.png
  :width: 100%
  :align: center

  The electrostatic field (:math:`E_x`) at different times. The black
  dashed line shows the location of the cyclotron cutoff. A very sharp
  spike develops at the cutoff location as the wave number becomes
  infinite.

A simulation was performed with the same parameters but with 400 cells
and was run to 50 ns. The electrostatic field component is shown
below.

.. figure:: s73-Ex-inset.png
  :width: 100%
  :align: center

  The electrostatic field (:math:`E_x`) at :math:`t=50` ns. The black
  dashed line shows the location of the cyclotron cutoff. The plot
  shows the sharp spike formed due to the wave number becoming
  infinite. The inset plot is a zoom to show the electrostatic field
  around the resonance layer. The simulation input file is at
  :doc:`s73 <../../sims/s73/s73-cyclotron-cutoff>`.

Conclusions
-----------

In this simulation the propagation of a wave into an electron
cyclotron resonance layer is shown. The EM wave suffers a cutoff at
the resonance layer but tunnels through. The electrostatic field shows
a sharp spike due to the wave number becoming infinite at the
resonance layer. However, the finite size of the grid means that the
spike can only be resolved to the smallest grid size. Even though the
linear theory predicts unlimited growth of the wave number, when the
field amplitude gets large enough the plasma will become non-linear
and the linear theory is no longer valid. The simulations show a
characteristic feature of cyclotron cutoff layers: sharp gradients in
the electrostatic fields and a sudden change in the electromagnetic
wave amplitude.