Components of MHD equations in toroidal geometry

Next, we consider the form of the linearized MHD equations in toroidal devices (e.g. tokamak). In these devices, there exist magnetic surfaces. The motion of plasma along the surface and perpendicular to the surface are very different. Thus, it is useful to decompose the perturbed quantities into components lying on the surface and perpendicular to the surface. Following Ref. [3,4], we write the displacement vector and perturbed magnetic field as

$$\ensuremath{\boldsymbol{\xi}}= \xi_{\psi} \frac{\nabla \Psi}{\ver... ...c{(\mathbf{B}_0 \times \nabla \Psi)}{B^2_0} + \xi_b \frac{\mathbf{B}_0}{B^2_0},$$ (64)

and

$$\mathbf{B}_1 = Q_{\psi} \frac{\nabla \Psi}{\vert \nabla \Psi \ver... ...\times \nabla \Psi)}{\vert \nabla \Psi \vert^2} + Q_b \frac{\mathbf{B}_0}{B^2},$$ (65)

where $\Psi = \Psi_{\ensuremath{\operatorname{pol}}} / (2 \pi) + C$ with $\Psi_{\ensuremath{\operatorname{pol}}}$ the poloidal magnetic flux within a magnetic surface and $C$ being an arbitrary constant and (In the process of deriving the eigenmode equation, we do not need the specific definition of $\Psi$. What we need is only that $\nabla \Psi$ is a vector in the direction of $\nabla p_0$ and thus $\nabla \Psi$ is perpendicular to both $\mathbf{B}_0$ and $\mathbf{J}_0$). Taking scalar product of the above two equations with $\nabla \Psi$, $\mathbf{B}_0 \times \nabla \Psi$, and $\mathbf{B}_0$, respectively, we obtain

$$\xi_{\psi} =\ensuremath{\boldsymbol{\xi}} \cdot \nabla \Psi, \xi_... ...\Psi \vert^2} \right), \xi_b =\ensuremath{\boldsymbol{\xi}} \cdot \mathbf{B}_0,$$ (66)

$$Q_{\psi} =\mathbf{B}_1 \cdot \nabla \Psi, Q_s =\mathbf{B}_1 \cdot... ...B}_0 \times \nabla \Psi}{B^2_0} \right), Q_b =\mathbf{B}_1 \cdot \mathbf{B}_0 .$$ (67)

Next, we derive the component equations for the induction equation (37) and momentum equation (35). The derivation is straightforward but tedious. Those who are not interested in these details can skip them and read directly Sec. 3.7 for the final form of the component equations.