Nonconstant-$\psi$ Linear Resonant Response Regimes

Suppose that $Q\gg P\,p^2$ and $D^2\,p^2 \ll 1$. It follows that $k=0$, $j=1$, $\nu=1/2$, and

$$G = [-{\rm i}\,(Q-Q_E)]\,[-{\rm i}\,(Q-Q_E-Q_i)].$$ (5.103)

Hence, we deduce that

$$\skew{6}\hat{\mit\Delta} = -\frac{\pi}{[-{\rm i}\,(Q-Q_E)]^{1/2}\,[-{\rm i}\,(Q-Q_E-Q_i)]^{1/2}}.$$ (5.104)

This response regime is known as the inertial regime, because the layer response is dominated by ion inertia [2,13]. Note that the plasma response in the inertial regime is equivalent to that of two closely-spaced shear-Alfvén resonances that straddle the rational surface [4]. In fact, it is easily demonstrated that in real space,

$$\skew{6}\hat{\mit\Delta} = \int_{-\infty}^\infty \frac{dX}{(Q-Q_E)\,(Q-Q_E-Q_i)-X^{\,2}},$$ (5.105)

which suggests that the resonances lie at $X=\pm \sqrt{(Q-Q_E)\,(Q-Q_E-Q_i)}$. The characteristic layer width is $p_\ast \sim Q^{-1}$, which implies that the regime is valid when $P\ll Q^3$, $Q\gg D$, $Q\gg 1$, and $c_\beta\ll 1$.

Suppose that $Q\ll P\,p^2$ and $D^2\,p^2 \ll 1$. It follows that $k=2$, $j=2$, $\nu=1/4$, and

$$G = [-{\rm i}\,(Q-Q_E-Q_e)]\,P_\varphi.$$ (5.106)

Hence, we deduce that

$$\skew{6}\hat{\mit\Delta} =- \frac{\pi}{2}\,\frac{{\mit\Gamma}(1/4)}{{\mit\Gamma}(3/4)}\, [-{\rm i}\,(Q-Q_E-Q_e)]^{-1/4}\,P_\varphi^{-1/4}.$$ (5.107)

This response regime is known as the viscous-inertial regime, because the layer response is dominated by ion perpendicular viscosity and ion inertia [13]. The characteristic layer width is $p_\ast \sim Q^{-1/4}\,P^{-1/4}$, which implies that the regime is valid when $P\gg Q^3$, $P\gg D^4/Q$, $P\gg Q^{-3}$, and $c_\beta\ll Q^{-3/4}\,P^{1/4}$.

Suppose, finally, that $Q\ll P\,p^2$ and $D^2\,p^2\gg 1$. It follows that $k=0$, $j=1$, $\nu=1/2$, and

$$G = \frac{[-{\rm i}\,(Q-Q_E-Q_e)]\,P_\perp}{(1+1/\tau)\,D^2}.$$ (5.108)

Hence, we deduce that

$$\skew{6}\hat{\mit\Delta}= -\frac{\pi\,(1+1/\tau)^{1/2}\,D}{[-{\rm i}\,(Q-Q_E-\,Q_e)]^{1/2}\,P_\perp^{1/2}}.$$ (5.109)

This response regime is known as the diffusive-inertial regime, because the layer response is dominated by perpendicular energy diffusivity and ion inertia [15]. The characteristic layer width is $p_\ast\sim Q^{-1/2}\,P^{-1/2}\,D$, which implies that the regime is valid when $Q\ll D$, $P\ll D^4/Q$, $P\gg D^2/Q^2$, and $c_\beta\ll Q^{-3/2}\,P^{-1/2}\,D^3$.