Constant-$\psi$ Linear Resonant Response Regimes

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

$$G = \frac{[-{\rm i}\,(Q-Q_E)]\,[-{\rm i}\,(Q-Q_E-Q_i)]}{-{\rm i}\,(Q-Q_E-\,Q_e)}.$$ (5.92)

Hence, we deduce that

$$\skew{6}\hat{\mit\Delta} = \frac{2\pi\,{\mit\Gamma}(3/4)}{{\mit\G... ...(Q-Q_E-\,Q_e)]^{3/4}\,[-{\rm i}\,(Q-Q_E)]^{1/4}\,[-{\rm i}\,(Q-Q_E-Q_i)]^{1/4}.$$ (5.93)

This response regime is known as the resistive-inertial regime, because the layer response is dominated by plasma resistivity and ion inertia [2,13,17]. The characteristic layer width is $p_\ast \sim Q^{-1/4}$, which implies that the regime is valid when $P\ll Q^{3/2}$, $Q\gg D^4$, $Q\ll 1$, and $c_\beta\ll Q^{3/4}$.

Suppose that $Q\ll P\,p^2$ and $D^2\,p^2 \ll 1$. It follows that $k=4$, $j=3$, $\nu=1/6$, and $G=P_\varphi$. Hence, we deduce that

$$\skew{6}\hat{\mit\Delta} = \frac{6^{2/3}\,\pi\,{\mit\Gamma}(5/6)}{{\mit\Gamma}(1/6)}\, [-{\rm i}\,\left(Q-Q_E-Q_e\right)]\,P_\varphi^{1/6}.$$ (5.94)

This response regime is known as the viscous-resistive regime, because the layer response is dominated by ion perpendicular viscosity and plasma resistivity [3,11]. The characteristic layer width is $p_\ast \sim P^{-1/6}$, which implies that the regime is valid when $P\gg Q^{3/2}$, $P\gg D^6$, $P\ll Q^{-3}$, and $c_\beta\ll P^{1/2}$.

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

$$G = \frac{-{\rm i}\,(Q-Q_E)}{D^2}.$$ (5.95)

Hence, we deduce that

$$\skew{6}\hat{\mit\Delta}= \frac{\pi\,[-{\rm i}\,\left(Q-Q_E-Q_e\right)]\,[-{\rm i}\,(Q-Q_E)]^{1/2}}{D}.$$ (5.96)

This response regime is known as the semi-collisional regime [9,21]. The characteristic layer width is $p_\ast\sim Q^{-1/2}\,D$, which implies that the regime is valid when $Q\ll D^4$, $P\ll Q^2/D^2$, $Q\ll D$, and $c_\beta\ll Q^{1/2}\,D$.

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

$$G = \frac{P_\perp}{(1+1/\tau)\,D^2}.$$ (5.97)

Hence, we deduce that

$$\skew{6}\hat{\mit\Delta}= \frac{2\pi\,{\mit\Gamma}(3/4)}{{\mit\Ga... .../4)}\,\frac{[-{\rm i}\,(Q-Q_E-Q_e)]\,P_\perp^{1/4}}{(1+1/\tau)^{1/4}\,D^{1/2}}.$$ (5.98)

This response regime is known as the diffusive-resistive regime, because the layer response is dominated by perpendicular energy diffusivity and plasma resistivity [15]. The characteristic layer width is $p_\ast \sim P^{-1/4}\,D^{1/2}$, which implies that the regime is valid when $P\gg Q^2/D^2$, $P\ll D^6$, $P\ll D^2/Q^2$, and $c_\beta\ll P^{1/4}\, D^{3/2}$.