# 1.Derive the dispersion relation for low frequency plasma waves at parallel propagation in ideal MHD.

We assume $\mathbf{k} = k_\parallel \hat{z}$, and $\mathbf{B}_0 = B_0 \hat{z}$. Give the linearization of ideal MHD that:

that we have:

then we can obtain that:

and:

combine the formula (3) and (4), we can get:

we can get tow solutions, one is from the z component that:

and another one is:

# 2.Redo the above analysis by keeping the Hall term in the generalized Ohm’s law.

Like the procedure of Ideal-MHD, from the Maxwell equations we can obtain that:

and Generalized Ohm’s law:

so that:

with:

to get the $\mathbf{J_1}$:

put it into the momentum equation that: $\omega \rho_0 \mathbf{u_1} = i \mathbf{J}_1 \times \mathbf{B_0} + p_1 \mathbf{k}$, we have:

with the $\rho_0 = n_0 (m_i+m_e) \approx n_0 m_i$ and $p_0 = n_0 k_B T$, the z component demonstrate that:

rewrite the x, y component into the form of below:

to have a solution, the coefficient determinant must be zero, it means that:

with the $\Omega_i = \frac{e B_0}{m_i}$, $\lambda_i = \sqrt{\frac{m_i}{\mu_0 n_0 e^2}}$, we can derive that:

it can be reduced to a simpler form by consider the factorization of the trem that can’t be zero.

# 3.Solve the dispersion relation s obtained in Problem 2 numerically. Plot the dispersion relations and discuss the difference between the ideal and Hall MHD dispersion relations.

the figure of the dispersion is:

the difference between the ideal MHD and Hall MHD dispersion relations is the x, y component contribute a Fast Magnetic wave which is coincident with the Alfven wave at the low frequency but the Ideal MHD has only the Alfven wave term. It has three positive solution roots for the dispersion equations that we can draw three curve to illustrate it. Besides this, we can also find a resonant phenomenon when the $\mathbf{k}$ is sufficient large. The Red dotted line is the false root because of the general division operation in solving process.

the code to plot the dispersion has been presented below: