Normalized form of eigenmodes equations

Denote the strength of the equilibrium magnetic field at the magnetic axis by $B_a$, the mass density at the magnetic axis by $\rho_a$, and the major radius of the magnetic axis by $R_a$. Define a characteristic speed $V_A \equiv B_a / \sqrt{\rho_a {\textmu}_0}$, which is the Alfvén speed at the magnetic axis. Using the Alfvén speed, we define a characteristic frequency $\omega_A \equiv V_A / R_a$. Multiplying the matrix equation (134) by $2{\textmu}_0 / B_a^2$ gives

$$\left(\begin{array}{cc} \overline{E}_{11} & \overline{E}_{12}\\ \... ...}\right) \left(\begin{array}{c} \overline{P}_1\\ \xi_{\psi} \end{array}\right),$$ (154)

with new quantities defined as follows: $\overline{P}_1 \equiv P_1 / [B_a^2 / (2{\textmu}_0)]$, $\overline{E}_{11} \equiv E_{11} \frac{2{\textmu}_0}{B_a^2}$, $\overline{E}_{12} \equiv E_{12} \frac{2{\textmu}_0}{B_a^2}$, $\overline{F}_{12} \equiv F_{12} \frac{2{\textmu}_0}{B_a^2}$, $\overline{E}_{21} \equiv E_{21} \frac{2{\textmu}_0}{B_a^2}$, $\overline{E}_{22} \equiv E_{22} \frac{2{\textmu}_0}{B_a^2}$, and $\overline{F}_{22} \equiv F_{22} \frac{2{\textmu}_0}{B_a^2}$. Using the equations (135), (136), (140), (137), (138), and (142), the expression of $\overline{E}_{11}$, $\overline{E}_{12}$, $\overline{E}_{21}$, $\overline{E}_{22}$, $\overline{F}_{12}$, and $\overline{E}_{22}$ are written respectively as

$$\overline{E}_{11} = - \frac{2 \overline{\omega}^2 \overline{\rho}... ...eft( \frac{\vert \nabla \Psi \vert^2}{B^2_0} \mathbf{B}_0 \cdot \nabla \right),$$ (155)

$$\overline{E}_{12} = - 2 \kappa_s \gamma \overline{p}_0,$$ (156)

$$\overline{E}_{21} = \kappa_s \frac{4}{B_a^2},$$ (157)

$$\overline{E}_{22} = \frac{2 B_0^2 / B_a^2 + \gamma \overline{p}_0... ...athbf{B}_0 \cdot \nabla \left( \frac{\mathbf{B}_0 \cdot \nabla}{B^2_0} \right),$$ (158)

$$\overline{F}_{12} = \frac{2}{B_a^2} {\textmu}_0 \sigma \mathbf{B}... ...la \Psi \vert^2}{B^2_0} S \right) + 2 \kappa_s \frac{d \overline{p}_0}{d \Psi},$$ (159)

$$\overline{F}_{22} = - \frac{4}{B_a^2} \frac{\kappa_{\psi}}{\vert \nabla \Psi \vert^2},$$ (160)

where $\overline{\omega} = \omega / \omega_A$, $\overline{\rho}_0 = \rho_0 / \rho_a$, $\overline{p}_0 = p_0 / [B_a^2 / (2{\textmu}_0)]$. Next, consider re-normalizing the matrix equation (145). Multiply the first equation of matrix equation (145) by $2{\textmu}_0 / B_a^2$, giving

$$\nabla \Psi \cdot \nabla \left(\begin{array}{c} \overline{P}_1\\ ... ...rray}{c} \xi_s\\ \nabla \cdot \ensuremath{\boldsymbol{\xi}} \end{array}\right),$$ (161)

with the new matrix elements defined as follows: $\overline{C}_{12} \equiv C_{12} \frac{2{\textmu}_0}{B_a^2}$, $\overline{D}_{11} \equiv D_{11} \frac{2{\textmu}_0}{B_a^2}$, and $\overline{D}_{12} \equiv D_{12} \frac{2{\textmu}_0}{B_a^2}$. (Note that, although the second equation of matrix equation (161) uses $\overline{P}_1$, instead of $P_1$, as a variable , it is actually identical with the second equation of matrix equation (145) because the $\overline{P}_1$ term is multiplied by zero.) Using Eqs. (147), (148) (149) , we obtain

$$\overline{C}_{12} = \frac{2}{R_a^2} \overline{\omega}^2 \overline... ...abla \Psi \vert^2 S}{B^2_0} + 2 \kappa_{\psi} \frac{d \overline{p}_0}{d \Psi} .$$ (162)

$$\overline{D}_{11} = \frac{2}{B_a^2} (\vert \nabla \Psi \vert^2 S ... ...mu}_0 \sigma) \frac{\vert \nabla \Psi \vert^2}{B^2_0} \mathbf{B}_0 \cdot \nabla$$ (163)

$$\overline{D}_{12} = 2 \gamma \overline{p}_0 \kappa_{\psi} .$$ (164)

Note that, after the normalization, all the coefficients of the resulting equations are of the order $10^0$, thus, are suitable for accurate numerical calculation. Also note that, for typical tokamak plasmas, the normalization factor $2{\textmu}_0 / B_a^2$ is of the order $10^{- 6}$, which is six order away from $10^0$. Therefore the normalization performed here is necessary for accurate numerical calculation. [If the normalizing factor is two (or less) order from $10^0$, then, from my experience, it is usually not necessary to perform additional normalization for the purpose of optimizing the numerical accuracy, i.e., the original units system has provided a reasonable normalization. Of course, suitable re-normalization will be of benefit to developing a clear physical insight into the problem in question.]