dispersion_MTSI

In this project we will try to write some Python code to solve the dispersion relation and study the MTSI mode in different situations.

The MTSI Dispersion relation

From S. Janhunen et. al, Physics of Plasmas 25, 082308 (2018); doi: 10.1063/1.5033896

We have to solve :

with $$\omega_{pi}$$ and $\omega_{pi}$ the ion and electron plasma frequencies, respectively, $\omega = \omega_r + i \gamma$ the complex pulsation, $k$ the norm of the wave vector $\vec{k} = k_x \vec{e_x} + k_y \vec{e_y} + k_z \vec{e_z}$. The direction $z$ is parallel to the magnetic field $B$, $x$ is in the direction of the electric field $E$, and $y$ is in the direction of the $E x B$ drift. The drift velocity $\vec{v_0} = \frac{E x B}{B^2} = \frac{E}{B} \vec{e_y}$. Lastly, $\Omega_{ce} = \frac{q B}{m_e}$ is the electron cyclotron frequency.

Approche

The usual approache is to

1. Normalise the relation, using the usual $\lambda_{De}$ Debye lengh and the Ions plasma pulsation
2. Fixe the wave vector $\vec{k}$
3. Solve for the complex pulsation $\omega = \omega_r + i \gamma$
4. I iterate step 2. and 3. over the parameter space needed.

Normalised Relation

The wave vectors are normalized by the debye length $$\lambda_{De} = \sqrt{ \frac{\epsilon_0 k_B T_e}{n_e q_e^2}}$$ with $\epsilon_0$ the vacuum permitivity, $k_B$ the Boltzman constant, $T_e$ the electron temperature (in Kelvin), $n_e$ the electron density, and $q_e$ the elementary charge.

The pulsation are normalized by the ion plasma pulsation $$\omega_{pi} = \sqrt{ \frac{n_e q_e^2}{m_i \epsilon_0}}$$ with $m_i$ the ion mass.

The velocities are normalized by $$\lambda_{De} \omega_{pi} = \sqrt{\frac{k_B T_e}{m_i}} = u_B$$ the Bohm speed.

The normalized equation reads

with $\tilde{\omega} = \frac{\omega}{\omega_{pi}}$, $\tilde{k} = k \lambda_{De}$, and $\tilde{v_0} = \frac{v_0}{u_B}$. The normalized equation is not much simpler than the previous version, but now the variables are of the order of unity.