PLUTO Test Problems  4.4-patch2
Functions
init.c File Reference

Whistler Waves. More...

Functions

void VectorRotate (double *v, int s)
 

Detailed Description

The dispersion relation for whistler waves is given by (Huba, 2003):

\[ \omega = \frac{k^2_x B}{e n_e} \]

where $k_x = 2\pi m/L_x$ is the wave number along the x direction, $ L_x $ is the domain lenght in the horizontal direction, $ e $ and $ n_e $ are the electron charge and number density B is the magnetic field strength.

Whistler waves 1 and 2D (SETUP == 1)

In configurations #01 and #02 we test the propagation of whistler waves in 1D and 2D, respectively, and the results can be compared with the analytical dispersion relation, in a manner similar to Huba 2003. In 2D, the setup is rotated by an amount proportional to the ratio between the vertical and horizontal axis extents. The magnetic field is in the x-direction, $\vec{B} = B_0 \hvec{e}_x $, and the system is perturbed with $ \delta B_y = +\delta B \cos(2\pi m x/Lx) $ and $ \delta B_z = -\delta B \sin(2\pi m x/Lx) $, where $ m = 1, 2, ..., $ is the mode number, $ B_0 = 100 $ G and $ \delta B = 10^{-3} $ G. We also set $ \rho = en_e = 1 $.

We show in Fig. 1 the temporal evolution of the perpendicular component of the magnetic field ( $B_y$) and the its fourier composition. In Fig. 2 we show the comparison between the analytical and numerical values of the whistler wave frequency as a function of $ m $.

whistler_waves_fig1.png
Fig. 1 - Temporal evolution of the perpendicular component of the magnetic field (B_y) at x_0 = 0.5 (left), and its fourier decomposition (real amplitude) as a function of frequency for the run with mode number m = 6.


whistler_waves_fig2.png
Fig. 2 - Pseudocolor rendering of the current density (color) with magnetic field lines at t = 57 for the Hall MHD simulation of the Whistler waves.


Whistler waves 2D (SETUP == 2)

In this section (configuration #02) we test the correct propagation of the whistler waves in 2D, in a manner similar to ViganĂ² et al., 2012. The initial magnetic field is $B_x = B_0 + B_1 \cos(k_x y) \cos(k_x x)$, $B_y = B_1 \sin(k_x y) \sin(k_x x)$ and $ B_z = \sqrt{2} B_1 \sin(k_x y) \cos(k_x x)$, where $ B_0 = 10^3 $ and $B_1 = 1$. Again we set $ \rho = en_e = 1 $.

The equations above admit wave solution travelling along $ x $ with speed

\[ v_w = -\frac{c}{4 e n_0} \frac{\sqrt{2} }{L_y} m B_0 \]

where $ L_y$ is the domain length in the vertical direction, and again $ m $ is the number of wave modes.

In Fig. 3 we show the result of our simulation ( $ B_y $) at $ t = 0 $. We checked the scaling of the whistler speed with $ m $, shown in Fig. 4.

whistler_waves_fig3.png
Fig. 3 - Pseudocolor rendering of B_y at t = 0 for the simulation of the 2D whistler waves defined by the above equations (positive values in black and negative values in red).


whistler_waves_fig4.png
Fig. 4 - The whistler speed for different values of m (red circles) compared with the analitical values from the expression of v_w
Author
E. Striani (edoar.nosp@m.do.s.nosp@m.trian.nosp@m.i@ia.nosp@m.ps.in.nosp@m.af.i.nosp@m.t)
A. Mignone (migno.nosp@m.ne@p.nosp@m.h.uni.nosp@m.to.i.nosp@m.t)
B. Vaidya (bharg.nosp@m.av.v.nosp@m.aidya.nosp@m.@uni.nosp@m.to.it)
Date
May 17, 2017

References:

Function Documentation

◆ VectorRotate()

void VectorRotate ( double *  v,
int  s 
)

Rotate a vector <v[0], v[1], v[2]> based on the domain aspect ratio. s = 1 use normal rotation s = -1 use inverse rotation