Linearized ideal MHD equation

Next, consider the linearized version of the ideal MHD equations. Use $\mathbf{u}_0$, $\mathbf{B}_0$, $p_0$, and $\rho_0$ to denote the equilibrium fluid velocity, magnetic field, plasma pressure, and mass density, respectively. Use $\mathbf{u}_1$, $\mathbf{B}_1$, $p_1$, and $\rho_1$ to denote the perturbed fluid velocity, magnetic field, plasma pressure, and mass density, respectively. Consider only the case that there is no equilibrium flow, i.e., $\mathbf{u}_0 = 0$. From Eq. (21), the linearized equation for the time evolution of the perturbed pressure is written as

$$\frac{\partial p_1}{\partial t} = - \gamma p_0 \nabla \cdot \mathbf{u}_1 -\mathbf{u}_1 \cdot \nabla p_0 .$$ (23)

The linearized momentum equation is

$$\rho_0 \frac{\partial \mathbf{u}_1}{\partial t} = - \nabla p_1 +{... ...thbf{B}_0 +{\textmu}_0^{- 1} (\nabla \times \mathbf{B}_0) \times \mathbf{B}_1 .$$ (24)

The linearized induction equation is

$$\frac{\partial \mathbf{B}_1}{\partial t} = \nabla \times (\mathbf{u}_1 \times \mathbf{B}_0) .$$ (25)

These three equations constitute a closed system for $\mathbf{u}_1$, $\mathbf{B}_1$, and $p_1$. Note that the linearized equation for the perturbed mass density $\rho_1$

$$\frac{\partial \rho_1}{\partial t} = - \rho_0 \nabla \cdot \mathbf{u}_1 -\mathbf{u}_1 \cdot \nabla \rho_0$$ (26)

is not needed when solving the system of equations (23)-(25) because $\rho_1$ does not appear in equations (23)-(25).