Reduced Neoclasssical Drift-MHD Model

The Alfvén speed, $V_A$, and the collisionless ion skin-depth, $d_i$, are defined in Equations (4.23) and (4.24), respectively. Let $l$ be a typical variational lengthscale in the inner region. It is convenient to adopt the following normalization scheme that renders all quantities in the neoclassical drift-MHD fluid equations dimensionless: $\hat{\nabla} = l\,\nabla$, $\hat{t} = t/(l/V_A)$, $\hat{d}_i = d_i/l$, $\hat{\bf E}= {\bf E}/(B_z\,V_A)$, $\hat{\bf j} = {\bf j}/(B_z/\mu_0\,l)$, $\hat{j}_{bs} = j_{bs}/(B_z/\mu_0\,l)$, $\hat{j}_{nc\,e,i} = j_{nc\,e,i}/(B_z/\mu_0\,l)$, $\hat{p}=p/(B_z^{\,2}/\mu_0)$, $\hat{p}_0= p_0/(B_z^{\,2}/\mu_0)$, $\hat{\bf V} = {\bf V}/V_A$, $\hat{\bf V}_{\ast,i,E} = {\bf V}_{\ast,i,E}/V_A$, $\hat{V}_\parallel = V_{\parallel\,i}/V_A$, $\hat{x}= (r-r_s)/l$, $\hat{\mit\Xi}_{\theta}= l/(V_A\,\tau_\theta)$, $\hat{\mit\Xi}_\perp = {\mit\Xi}_\perp/(l\,V_A)$, $\hat{\eta}_{\parallel,\perp,\parallel\,e}=\eta_{\parallel,\perp,\parallel\,e}/(\mu_0\,l\,V_A)$, $\hat{\chi}_{\parallel,\perp}= \chi_{\parallel,\perp}/(l\,V_A)$. Here, $V_{\parallel\,i}$ is the ion parallel fluid velocity [see Equation (2.321)]. Equations (11.1)–(11.3) yield the following set of normalized neoclassical drift-MHD fluid equations:

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

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

$$+\hat{\mit\Xi}_\theta\left(\hat{\bf V}_i-\alpha_\theta\,\frac{1}{... ...bf V}_i^\dag - \frac{2}{3}\,\hat{\nabla}\cdot\hat{\bf V}_i\,{\bf I}\right) = 0,$$ (11.16)

$$\hat{\bf E} +\hat{\bf V}\times {\bf b} +\hat{d}_i\left[\hat{\nabl... ...abla}\hat{p})+\hat{\eta}_\parallel\,(\hat{j}_\parallel -\hat{j}_{bs})\,{\bf b},$$ (11.17)

$$\frac{3}{2}\,\frac{\partial\hat{p}}{\partial\hat{t}}+\frac{3}{2}\... ...hat{j}_{bs}+\frac{\hat{E}_\parallel}{\hat{\eta}_\parallel}\right){\bf b}\right]$$

$$-\frac{\tau'}{1+\tau'}\,\hat{d}_i\,\hat{p}_0\,\hat{\nabla}\cdot\l... ...({\bf b}\cdot\hat{\nabla}\hat{p}) -\hat{\chi}_\perp\,\hat{\nabla}^2\hat{p} = 0,$$ (11.18)

where

$$\hat{j}_{bs} = -\alpha_{bs}\,\frac{\partial\hat{p}}{\partial\hat{x}},$$ (11.19)

$$\hat{j}_{nc\,e,i} = -\alpha_{nc\,e,i}\,\frac{\partial\hat{p}}{\partial\hat{x}},$$ (11.20)

$$\hat{\bf V}_E = \hat{\bf E}\times {\bf b},$$ (11.21)

$$\hat{\bf V}_\ast = \hat{d}_i\,{\bf b}\times \hat{\nabla}\hat{p},$$ (11.22)

$$\hat{\bf V} = \hat{\bf V}_E + \hat{V}_\parallel\,{\bf b},$$ (11.23)

$$\hat{\bf V}_i = \hat{\bf V} + \frac{1}{1+\tau}\,\hat{\bf V}_\ast.$$ (11.24)

Finally, Maxwell's equations yield

$$\hat{\nabla}\cdot{\bf b} =0,$$ (11.25)

$$\hat{\nabla}\times \hat{\bf E} = - \frac{\partial{\bf b}}{\partial \hat{t}},$$ (11.26)

$$\hat{\bf j} =\hat{\nabla}\times {\bf b}.$$ (11.27)

As before (see Section 4.4), all quantities in the inner region are assumed to be functions of $\hat{x}$, $\zeta=m\,\theta-n\,\varphi$, and $\hat{t}$ only. Here, $\theta$ and $\varphi$ are the poloidal and toroidal angles, respectively, whereas $m$ and $n$ are the poloidal and toroidal mode numbers, respectively, of the tearing mode. (See Sections 3.2 and 3.3.) We can write

$$\hat{p} = \hat{p}_0 +\delta p^{(1)},$$ (11.28)

$${\bf b} = \left(1+\delta b^{(1)}\right){\bf n} + \hat{\nabla}\times\left(\psi^{(1)}\,{\bf n}\right),$$ (11.29)

$$\hat{\bf E} =\hat{\nabla}\phi^{(1)}+\left(\hat{E}_\parallel^{(2)} - \frac{\pa... ...}\times\left( \frac{\partial^{(1)}\chi^{(1)}}{\partial\hat{t}}\,{\bf n}\right),$$ (11.30)

$$\hat{\bf V}_E = \hat{\nabla}\phi^{(1)}\times {\bf b} -\hat{\nabla}\left(\frac{\partial^{(1)}\chi^{(1)}}{\partial\hat{t}}\right),$$ (11.31)

$$\hat{\bf V}_\ast = \hat{d}_i\,{\bf b}\times \hat{\nabla}\delta p^{(1)},$$ (11.32)

$$\hat{\bf V} = \hat{\bf V}_E + \hat{V}_\parallel^{(1)}\,{\bf b} +\hat{\nabla}{\mit\Upsilon}^{(2)},$$ (11.33)

$$\hat{\bf V}_i = \hat{\bf V} +\frac{1}{1+\tau}\,\hat{\bf V}_\ast,$$ (11.34)

where ${\bf n}$ is defined in Equation (4.37), $\hat{E}_\parallel^{(2)}$ is the (normalized) constant inductive component of the parallel electric field that maintains the equilibrium parallel current density in the inner region against ohmic decay, and $\hat{\nabla}^2\chi^{(1)} =\delta b^{(1)}$. The superscript $(1)$ indicates a quantity that is first order in our ordering scheme. Zeroth order terms are left without superscripts, whereas second order terms are given the superscript (2).

Evaluating the normalized neoclassical drift-MHD fluid equations, (11.16)–(11.18), up to second order, we obtain

$$\hat{\nabla}\left(\delta p^{(1)}+ \delta b^{(1)}\right)+\left(\fr... ...l^{(1)},\phi^{(1)}\right] +\left[\psi^{(1)},\delta b^{(1)}\right]\right){\bf n}$$

$$+\hat{\nabla}\left(\frac{\partial^{(1)}\phi^{(1)}}{\partial\hat{t... ... b^{(1)}\,\delta b^{(1)}+\frac{2\,\epsilon^{(1)}\,\epsilon^{(1)}}{q_s^2}\right)$$

$$-\hat{\nabla}^2\phi^{(1)}\,\hat{\nabla}\phi^{(1)}+\frac{\hat{d}_i... ...2\delta p^{(1)}\,\hat{\nabla}\phi^{(1)}\right)+ J^{(1)}\,\hat{\nabla}\psi^{(1)}$$

$$+\hat{\mit\Xi}_\theta^{(1)}\left(\frac{\epsilon^{(1)}}{q_s}\,\hat... ...t) {\bf e}_\theta - {\mit\Xi}_{\perp}^{(1)}\,\hat{\nabla}^{2}{\bf V}_i^{(1)}=0,$$ (11.35)

$$\hat{d}_i\,\hat{\nabla}\left(\delta p^{(1)}+ \delta b^{(1)}\right)$$

$$+\left(\hat{E}_\parallel^{(2)}-\frac{\partial^{(1)}\psi^{(1)}}{\p... ...(1)}\,\alpha_{bs}\,\frac{\partial\delta p^{(1)}}{\partial\hat{x}}\right){\bf n}$$

$$+\hat{\nabla}\left[{\mit\Upsilon}^{(2)}-\hat{\eta}_\perp^{(1)}\,(... ... b^{(1)}\,\delta b^{(1)}+\frac{2\,\epsilon^{(1)}\,\epsilon^{(1)}}{q_s^2}\right)$$

$$+\hat{d}_i\,J^{(1)}\,\hat{\nabla}\psi^{(1)}-2\,\delta b^{(1)}\,\hat{\nabla}\phi^{(1)}=0,$$ (11.36)

$$\frac{3}{2}\,\frac{\partial^{(1)}\delta p^{(1)}}{\partial\hat{t}}... ...{V}_\parallel^{(1)},\psi^{(1)}\right]+\hat{\nabla}^2{\mit\Upsilon}^{(2)}\right)$$

$$+\frac{5}{2}\,\lambda\,\hat{p}_0\,\hat{d}_i\,[\delta p^{(1)},\del... ...{(1)}}\left[\frac{\partial^{(1)}\psi^{(1)}}{\partial \hat{t}},\psi^{(1)}\right]$$

$$-\hat{\chi}_\parallel^{(-1)}\left[\left[\delta p^{(1)},\psi^{(1)}\right],\psi^{(1)}\right]-\hat{\chi}_\perp^{(1)}\,\hat{\nabla}^2\delta p^{(1)}=0,$$ (11.37)

where $J^{(1)}$ is defined in Equation (4.49), $[A,B]\equiv \hat{\nabla}A\times \hat{\nabla}B\cdot{\bf n}$, and

$$\alpha_{nc} = \frac{\tau'}{1+\tau'}\,\alpha_{bs} + \frac{2}{5}\,\frac{\tau'}{1+\tau'}\,\alpha_{nc\,e} + \frac{1}{1+\tau'}\,\alpha_{nc\,i},$$ (11.38)

$$\alpha_\parallel = \frac{\tau'}{1+\tau'}\left(1+\frac{2}{5}\,\frac{\hat{\eta}_\parallel}{\hat{\eta}_{\parallel\,e}}\right).$$ (11.39)

To first order, Equations (11.35) and (11.36) again give the equilibrium force balance constraint

$$\delta b^{(1)} = -\delta p ^{(1)}.$$ (11.40)

(See Section 4.4.) The scalar product of Equation (11.35) with ${\bf n}$ yields

$$\frac{\partial^{(1)}\hat{V}_\parallel^{(1)}}{\partial\hat{t}} =\left[\phi^{(1)},\hat{V}_\parallel^{(1)}\right] -\left[\delta p^{(1)},\psi^{(1)}\right]$$ (11.41)

$$\phantom{=}-\hat{\mit\Xi}_\theta^{(1)}\,\frac{\epsilon^{(1)}}{q_s... ...ght] \right) +\hat{\mit\Xi}_\perp^{(1)}\,\hat{\nabla}^2\hat{V}_\parallel^{(1)}.$$

The scalar product of Equation (11.36) with ${\bf n}$ gives

$$\frac{\partial^{(1)}\psi^{(1)}}{\partial \hat{t}}= \left[\phi^{(1... ..._{bs}\,\frac{\delta p^{(1)}}{\partial\hat{x}}\right) + \hat{E}_\parallel^{(2)}.$$ (11.42)

The scalar product of the curl of Equation (11.35) with ${\bf n}$ yields

$$\frac{\partial^{(1)}\, \hat{\nabla}^2\phi^{(1)}}{\partial\hat{t}} = \left[\phi^{(1)}, \hat{\nabla}^2\phi^{(1)}\right]+\frac{\hat{d}... ...ta p ^{(1)}\right] +\left[\hat{\nabla}^2\delta p^{(1)},\phi^{(1)}\right]\right)$$

$$\phantom{=} +\left[J^{(1)},\psi^{(1)}\right]+ \hat{\mit\Xi}_\thet... ...ft(1-\alpha_\theta\,\frac{\eta_i}{1+\eta_i}\right)\delta p^{(1)}\right] \right)$$

$$\phantom{=}+ \hat{\mit\Xi}_\perp^{(1)}\,\hat{\nabla}^4\left( \phi^{(1)}-\frac{\hat{d}_i}{1+\tau}\,\delta p^{(1)}\right).$$ (11.43)

Finally, the scalar product of the curl of Equation (11.36) with ${\bf n}$ gives

$$\hat{\nabla}^2{\mit\Upsilon}^{(2)} = 2\left[\delta p^{(1)}, \phi^{(1)}\right] +\hat{d}_i\left[J^{(1)}, \psi^{(1)}\right].$$ (11.44)

The previous equation can be combined with Equations (11.37) and (11.40) to produce

$$\frac{\partial^{(1)}\delta p^{(1)}}{\partial \hat{t}} = \left[\phi^{(1)},\delta p^{(1)}\right] -c_\beta^{\,2}\left[\hat... ...t[\alpha_{nc}\,\frac{\partial\delta p^{(1)}}{\partial\hat{x}},\psi^{(1)}\right]$$ (11.45)

$$\phantom{=} -c_\beta^{\,2}\,\hat{d}_i\left[\frac{\alpha_{\paralle... ...2}{3}\,(1-c_\beta^2)\,\hat{\chi}_\perp^{(1)}\,\hat{\nabla}^{\,2}\delta p^{(1)},$$

where $c_\beta$ is defined in Equations (4.65) and (4.66).

Our final reduced neoclassical drift-MHD model takes the form [7,8,10]

$$\frac{\partial\psi}{\partial\hat{t}} = \left[\phi,\psi\right] +\hat{d}_i\,\frac{\tau}{1+\tau}\left[\de... ...lpha_{bs}\,\frac{\partial\delta p}{\partial\hat{x}}\right) + \hat{E}_\parallel,$$ (11.46)

$$\frac{\partial\delta p}{\partial \hat{t}} = \left[\phi,\delta p\right] -c_\beta^{\,2}\left[\hat{V}_{\parall... ...el}}{\hat{\eta}_{\parallel}}\,\frac{\partial\psi}{\partial\hat{t}}, \psi\right]$$

$$\phantom{=} +\frac{2}{3}\,(1-c_\beta^2)\,\hat{\chi}_\parallel\lef... ...ht] + \frac{2}{3}\,(1-c_\beta^2)\,\hat{\chi}_\perp\,\hat{\nabla}^{\,2}\delta p,$$ (11.47)

$$\frac{\partial U}{\partial \hat{t}} = \left[\phi, U\right]+\frac{\hat{d}_i}{2\,(1+\tau)} \left(\hat{\... ...right]+\left[U,\delta p\right] +\left[\hat{\nabla}^2\delta p,\phi\right]\right)$$

$$\phantom{=} +\left[J,\psi\right]+ \hat{\mit\Xi}_\theta\,\frac{\pa... ...au}\left(1-\alpha_\theta\,\frac{\eta_i}{1+\eta_i}\right)\delta p\right] \right)$$

$$\phantom{=}+ \hat{\mit\Xi}_\perp\,\hat{\nabla}^4\left( \phi-\frac{\hat{d}_i}{1+\tau}\,\delta p\right),$$ (11.48)

$$\frac{\partial\hat{V}_\parallel}{\partial\hat{t}} =\left[\phi,\hat{V}_\parallel\right] -\left[\delta p,\psi\right]-... ...au}\left(1-\alpha_\theta\,\frac{\eta_i}{1+\eta_i}\right)\delta p\right] \right)$$

$$\phantom{=}+\hat{\mit\Xi}_\perp\,\hat{\nabla}^2\hat{V}_\parallel,$$ (11.49)

where

$$J = -\frac{2\,\epsilon_s}{q_s}\,+\hat{\nabla}^2\psi,$$ (11.50)

$$U =\hat{\nabla}^2 \phi.$$ (11.51)

Here, we have suppressed the ordering superscripts. If we compare Equations (11.46)–(11.51), to our previous reduced non-neoclassical drift-MHD equations, (4.67)–(4.74), then we can see that the former set of equations contain many additional terms. The additional term involving the parameter $\alpha_{bs}$ in Equation (11.46) is due to the bootstrap current. The additional terms involving the parameters $\alpha_{nc}$ and $\alpha_\parallel$ in Equation (11.47) are due to neoclassical parallel momentum and heat fluxes. Finally, the additional terms involving the parameter ${\mit\Xi}_\theta$ in Equations (11.48) and (11.49) are due to neoclassical poloidal flow damping.