Normalization Scheme

The Alfvén speed, which is the typical phase velocity of a compressible-Alfvén wave, is defined

$$V_A =\frac{a}{\tau_A}= \frac{B_z}{\sqrt{\mu_0\,n_0\,m_i}}.$$ (4.23)

[See Equation (3.181).] It is helpful to define the collisionless ion skin-depth,

$$d_i = \left(\frac{m_i}{n_0\,e^{\,2}\,\mu_0}\right)^{1/2}.$$ (4.24)

Note that $d_i=c/\omega_{p\,i}$, where $c$ is the velocity of light in vacuum, and $\omega_{p\,i}$ the ion plasma frequency [2].

Let $l$ be a typical variation lengthscale in the resonant layer. It is convenient to adopt the following normalization scheme that renders all quantities in the drift-MHD fluid equations dimensionless: $\hat{\nabla} = l\,\nabla$, $\hat{t} = t/(l/V_A)$, $\hat{d}_i = d_i/l$, $\hat{\bf E}= {\bf E}/(B_z\,V_A)$, $\hat{\bf j} = {\bf j}/(B_z/\mu_0\,l)$, $\hat{p}=p/(B_z^{\,2}/\mu_0)$, $\hat{p}_0= p_0/(B_z^{\,2}/\mu_0)$, $\hat{\bf V} = {\bf V}/V_A$, $\hat{\bf V}_{E,\ast,i} = {\bf V}_{E,\ast,i}/V_A$, $\hat{V}_\parallel = V_{\parallel\,i}/V_A$, $\hat{\mit\Xi}_\perp = {\mit\Xi}_\perp/(l\,V_A)$, $\hat{\eta}_{\parallel,\perp} =\eta_{\parallel,\perp}/(\mu_0\,l\,V_A)$, and $\hat{\chi}_{\parallel,\perp}= \chi_{\parallel,\perp}/(l\,V_A)$. Equations (4.7)–(4.13) and (4.20)–(4.22) yield the following set of normalized drift-MHD fluid equations:

$$\frac{\partial \hat{\bf V}}{\partial\hat{t}}+\hat{\nabla}\left(\f... ...t{\bf V}_\ast)-\hat{\bf V}_\ast\times (\hat{\nabla}\times \hat{\bf V}_E)\right.$$

$$\left. + (\hat{\nabla}\cdot\hat{\bf V}_E)\,\hat{\bf V}_\ast- (\ha... ...E\times\hat{\bf V}_\ast)\right] +\hat{\nabla}\hat{p} -\hat{\bf j}\times {\bf b}$$

$$-\hat{\mit\Xi}_\perp\hat{\nabla}\cdot\left(\hat{\nabla}\hat{\bf V... ...bf V}_i^\dag - \frac{2}{3}\,\hat{\nabla}\cdot\hat{\bf V}_i\,{\bf I}\right) = 0,$$ (4.25)

$$\hat{\bf E} +\hat{\bf V}\times{\bf b} +\hat{d}_i\left[\hat{\nabla... ...{\bf b}\times \hat{\nabla}\hat{p})+\hat{\eta}_\parallel\,\hat{\bf j}_\parallel,$$ (4.26)

$$\frac{3}{2}\,\frac{\partial\hat{p}}{\partial\hat{t}}+\frac{3}{2}\... ...({\bf b}\cdot\hat{\nabla}\hat{p}) -\hat{\chi}_\perp\,\hat{\nabla}^2\hat{p} = 0,$$ (4.27)

where

$$\hat{\bf V}_E = \hat{\bf E}\times {\bf b},$$ (4.28)

$$\hat{\bf V}_\ast = \hat{d}_i\,{\bf b}\times \hat{\nabla}\hat{p},$$ (4.29)

$$\hat{\bf V} = \hat{\bf V}_E + \hat{V}_\parallel\,{\bf b},$$ (4.30)

$$\hat{\bf V}_i = \hat{\bf V} +\frac{1}{1+\tau}\,\hat{\bf V}_\ast,$$ (4.31)

and

$$\hat{\nabla}\cdot{\bf b} =0,$$ (4.32)

$$\hat{\nabla}\times \hat{\bf E} = - \frac{\partial{\bf b}}{\partial \hat{t}},$$ (4.33)

$$\hat{\bf j} =\hat{\nabla}\times {\bf b}.$$ (4.34)

Here, use has been made of some standard vector identities.