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 1D (SETUP == 1)

In this section (configuration #01) we present numerical results of the dispersion relation of whistler waves and we compare it with the above analitical formula, in a manner similar to Huba 2003. 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 Harris current sheet.


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 9, 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