Plasma displacement vector $\xi$

In dealing with the linear case of MHD theory, it is convenient to introduce the plasma displacement vector $\ensuremath{\boldsymbol{\xi}}$, which is defined through the following equation

$$\mathbf{u}_1 = \frac{\partial \ensuremath{\boldsymbol{\xi}}}{\partial t} .$$ (27)

Using the definition of $\ensuremath{\boldsymbol{\xi}}$ and the fact that the equilibrium quantities are independent of time, the linearized induction equation (25) is written

$$\frac{\partial \mathbf{B}_1}{\partial t} = \frac{\partial}{\partial t} [\nabla \times (\ensuremath{\boldsymbol{\xi}} \times \mathbf{B}_0)] .$$ (28)

Similarly, the equation for the perturbed pressure [Eq. (23)] is written

$$\frac{\partial p_1}{\partial t} = \frac{\partial}{\partial t} [-\... ...i}} \cdot \nabla p_0 - \gamma p_0 \nabla \cdot \ensuremath{\boldsymbol{\xi}}] .$$ (29)

In terms of the displacement vector, the linearized momentum equation (24) is written

$$\rho_0 \frac{\partial^2 \ensuremath{\boldsymbol{\xi}}}{\partial t... ...thbf{B}_0 +{\textmu}_0^{- 1} (\nabla \times \mathbf{B}_0) \times \mathbf{B}_1 .$$ (30)

Equations (28), (29), and (30) constitute a closed system for $\mathbf{B}_1$, $p_1$, and $\ensuremath{\boldsymbol{\xi}}$.