2 Intermediate region plasma model

We consider cylindrical geometry with coordinates $r,\unicode[STIX]{x1D703},z$ and magnetic fields $(0,B_{\unicode[STIX]{x1D703}}(r),B_{Z}(r))$ ; the total field strength is $B$ . The perturbed fields of the tearing mode involve the parallel vector potential, $A_{\Vert }$ , and electrostatic potential, $\unicode[STIX]{x1D711}$ , with frequency $\unicode[STIX]{x1D714}$ and structure $A_{\Vert }\sim A(x)\text{e}^{-\text{i}\unicode[STIX]{x1D714}t+\text{i}k_{z}z-\text{i}m\unicode[STIX]{x1D703}}$ for example. Here $x=r-r_{0}$ , where the resonant surface, $r_{0}$ , is given by $k_{z}B_{Z}(r_{0})-k_{\unicode[STIX]{x1D703}}B_{\unicode[STIX]{x1D703}}(r_{0})=0$ , with $k_{\unicode[STIX]{x1D703}}=m/r$ , so that the wavenumber parallel to the magnetic field is $k_{\Vert }=mx/rL_{s}$ , with $L_{s}$ being the magnetic shear length. In the toroidal geometry of a tokamak, $k_{z}=n/R$ , $L_{s}=Rq/s$ , with $R$ the major radius, $s=r\text{d}(\ln q)/\text{d}r$ and $q$ the safety factor.

The electrons and ions satisfy the linearised, collisionless, kinetic equation:

(2.1) $$\begin{eqnarray}(\unicode[STIX]{x1D714}-k_{\Vert }v_{\Vert })g_{j}=(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{\ast j}^{T})\frac{e_{j}F_{Mj}}{T_{j}}J_{0}\left(\frac{k_{\bot }v_{\bot }}{\unicode[STIX]{x1D714}_{cj}}\right)(\unicode[STIX]{x1D711}-v_{\Vert }A_{\Vert }).\end{eqnarray}$$

Here the perturbed electron distribution, $\unicode[STIX]{x1D6FF}f_{j}$ is written $\unicode[STIX]{x1D6FF}f_{j}=-(e_{j}F_{Mj}/T_{j})\unicode[STIX]{x1D711}+g_{j}$ , where $F_{Mj}$ is the Maxwellian distribution for species $j$ , $v_{\Vert }$ is the parallel velocity and $v_{\bot }$ the perpendicular velocity, while $\unicode[STIX]{x1D714}_{\ast j}^{T}=\unicode[STIX]{x1D714}_{\ast j}(1+\unicode[STIX]{x1D702}_{j}((m_{j}v^{2}/2T_{j})-3/2))$ , with $v^{2}=v_{\Vert }^{2}+v_{\bot }^{2}$ . $J_{0}$ is the Bessel function of zero order with $k_{\bot }$ , where $k_{\bot }^{2}=k_{x}^{2}+m^{2}/r^{2}$ , being the wavenumber perpendicular to the magnetic field, and $\unicode[STIX]{x1D714}_{cj}=e_{j}B/m_{j}$ , with $m_{j}$ the mass of species $j$ , is the cyclotron frequency for species $j$ .

The stability of tearing modes can be investigated from two equations: the vorticity equation and Ampere’s law. The former is obtained by multiplying (2.1) by $e_{j}$ , integrating over velocity space, summing over the two species, $j$ , and applying the quasi-neutrality condition, $\unicode[STIX]{x1D6FF}n_{e}=\unicode[STIX]{x1D6FF}n_{i}$ . The result is:

(2.2) $$\begin{eqnarray}k_{\Vert }J_{\Vert }=\frac{\text{e}^{2}n\unicode[STIX]{x1D711}}{T_{i}}[(1-\unicode[STIX]{x1D6E4}_{0}(b))(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{\ast i})-\unicode[STIX]{x1D714}_{\ast i}\unicode[STIX]{x1D702}_{i}b(\unicode[STIX]{x1D6E4}_{0}(b)-\unicode[STIX]{x1D6E4}_{1}(b))]\equiv \frac{\text{e}^{2}n\unicode[STIX]{x1D711}}{T_{i}}F(b).\end{eqnarray}$$

Here $J_{\Vert }=J_{\Vert ,i}+J_{\Vert ,e}$ is the perturbed parallel current density, $e$ the proton charge and $\unicode[STIX]{x1D6E4}_{l}(b)=I_{l}(b)\text{e}^{-b}$ , where $b=k_{\bot }^{2}T_{i}/(m_{i}\unicode[STIX]{x1D714}_{ci}^{2})$ , with $I_{l}(b)$ the modified Bessel functions of order $l$ . Ampere’s equation is simply:

(2.3) $$\begin{eqnarray}\frac{\text{d}^{2}A_{\Vert }}{\text{d}x^{2}}=-J_{\Vert },\end{eqnarray}$$

where we ignore the small contribution to the left-hand side from $k_{\unicode[STIX]{x1D703}}^{2}$ as $k_{\unicode[STIX]{x1D703}}^{2}x^{2}\ll 1$ . In general, one must consider several regions in $x$ : (i) the innermost layer, $x\sim \unicode[STIX]{x1D6FF}_{e}$ , where $\unicode[STIX]{x1D6FF}_{e}$ is the width of the very narrow region where electron physics such as electron Landau damping or semi-collisional effects play a role in reconnection; (ii) at somewhat larger values of $x$ , namely $x\sim \unicode[STIX]{x1D70C}_{i}>\unicode[STIX]{x1D6FF}_{e}$ , where $\unicode[STIX]{x1D70C}_{i}=v_{Ti}/\unicode[STIX]{x1D714}_{ci}$ , with $v_{Ti}$ the ion thermal velocity, $v_{Ti}=(2T_{i}/m_{i})^{1/2}$ , is the ion Larmor radius, one must retain full finite ion Larmor radius (FLR) effects; (iii) the ‘intermediate region’, $\unicode[STIX]{x1D714}\sim k_{\Vert }v_{Ti}$ , where full ion parallel dynamics operates (ion sound at $x\sim (L_{s}/L_{n})^{1/2}\unicode[STIX]{x1D70C}_{i}$ and ion Landau damping at $x\sim (L_{s}/L_{n})\unicode[STIX]{x1D70C}_{i}$ ); and finally; (iv) the outer ideal MHD region. These regions are shown schematically in figure 1. Solutions of (2.2) and (2.3) must be matched through all these four regions. In earlier work Connor et al. (Reference Connor, Hastie and Zocco2012) considered this problem, but omitted the intermediate region required to ensure $E_{\Vert }\rightarrow 0$ as one approaches the ideal region. We now develop the equations to describe this region.

Figure 1. The regions in which different physical processes occur as the distance, $x=r-r_{0}$ , from the resonant surface, $r_{0}$ , increases: (i) electron reconnecting physics at a scale $\unicode[STIX]{x1D6FF}_{e}$ ; (ii) ion finite Larmor radius effects at a scale $\unicode[STIX]{x1D70C}_{i}$ ; (iii) full ion parallel dynamics, namely ion sound at a scale $(L_{s}/L_{n})^{1/2}\unicode[STIX]{x1D70C}_{i}$ and ion Landau damping effects at a scale $(L_{s}/L_{n})\unicode[STIX]{x1D70C}_{i}$ ; and (iv) the outer ideal MHD region.

Expressions for the parallel current densities, $J_{\Vert ,e}$ and $J_{\Vert ,i}$ , valid in the ion sound region and when $b\ll 1$ , are derived from solving the respective kinetic equations for electrons and ions, equation (2.1):

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle J_{\Vert ,e}=-\text{i}\frac{\text{e}^{2}nE_{\Vert }}{k_{\Vert }^{2}T_{e}}(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{\ast e}) & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle J_{\Vert ,i}=-\text{i}\frac{\text{e}^{2}nE_{\Vert }}{k_{\Vert }^{2}T_{i}}\left[(\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{\ast i})(1+\unicode[STIX]{x1D701}_{i}Z(\unicode[STIX]{x1D701}_{i}))-\unicode[STIX]{x1D714}_{\ast i}\unicode[STIX]{x1D702}_{i}\left(\unicode[STIX]{x1D701}_{i}^{2}(1+\unicode[STIX]{x1D701}_{i}Z(\unicode[STIX]{x1D701}_{i}))-\frac{1}{2}\unicode[STIX]{x1D701}_{i}Z(\unicode[STIX]{x1D701}_{i})\right)\right], & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $E_{\Vert }=-\text{i}(k_{\Vert }\unicode[STIX]{x1D711}-\unicode[STIX]{x1D714}A_{\Vert })$ , $\unicode[STIX]{x1D701}_{i}=\unicode[STIX]{x1D714}L_{s}/k_{\unicode[STIX]{x1D703}}xv_{Ti}$ and $Z(\unicode[STIX]{x1D701}_{i})$ is the plasma dispersion function. Making use of the long-wavelength approximation, $b\sim |\unicode[STIX]{x1D70C}_{i}^{2}\text{d}^{2}/\text{d}x^{2}|\ll 1$ , for the ions in (2.2), eliminating $J_{\Vert ,e,i}$ and introducing the dimensionless variable, $X=1/\unicode[STIX]{x1D701}_{i}$ , so that $X=(2L_{n}/L_{s})(\unicode[STIX]{x1D714}_{\ast i}/\unicode[STIX]{x1D714})(x/\unicode[STIX]{x1D70C}_{i})$ , we obtain the following fourth-order system describing the potentials $\unicode[STIX]{x1D711}$ and $A_{\Vert }$ , replacing the former by $\unicode[STIX]{x1D719}=\unicode[STIX]{x1D711}/v_{Ti}$ and denoting the latter by $A$ : (so that $E_{\Vert }\propto (X\unicode[STIX]{x1D719}-A)$ ):

(2.6) $$\begin{eqnarray}X\frac{\text{d}^{2}A}{\text{d}X^{2}}=\unicode[STIX]{x1D6FD}_{i}\left(1-\frac{\unicode[STIX]{x1D714}_{\ast i}(1+\unicode[STIX]{x1D702}_{i})}{\unicode[STIX]{x1D714}}\right)\frac{\text{d}^{2}\unicode[STIX]{x1D719}}{\text{d}X^{2}},\end{eqnarray}$$

and

(2.7) $$\begin{eqnarray}\displaystyle X\frac{\text{d}^{2}A}{\text{d}X^{2}} & = & \displaystyle \frac{\unicode[STIX]{x1D6FD}_{e}}{2}\left(\frac{L_{s}}{L_{n}}\frac{\unicode[STIX]{x1D714}_{\ast i}}{\unicode[STIX]{x1D714}}\right)^{2}\left(\unicode[STIX]{x1D719}-\frac{A}{X}\right)\left[1-\frac{\unicode[STIX]{x1D714}_{\ast e}}{\unicode[STIX]{x1D714}}+\unicode[STIX]{x1D70F}\left(1-\frac{\unicode[STIX]{x1D714}_{\ast i}}{\unicode[STIX]{x1D714}}\right)\left(1+\frac{Z}{X}\right)\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\unicode[STIX]{x1D70F}\frac{\unicode[STIX]{x1D714}_{\ast i}}{\unicode[STIX]{x1D714}}\frac{\unicode[STIX]{x1D702}_{i}}{X^{2}}\left(1+\frac{Z}{X}-\frac{XZ}{2}\right)\right],\end{eqnarray}$$

where $Z=Z(1/X)$ and the $\unicode[STIX]{x1D6FD}_{j}$ are the ratios of plasma to magnetic pressure for each species. These two equations can be combined to yield:

(2.8) $$\begin{eqnarray}\frac{\text{d}^{2}\unicode[STIX]{x1D719}}{\text{d}X^{2}}=Q(X)\left(\unicode[STIX]{x1D719}-\frac{A}{X}\right),\end{eqnarray}$$

where

(2.9) $$\begin{eqnarray}Q(X)=\frac{1}{2}\left(\frac{L_{s}}{L_{n}}\right)^{2}\frac{(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F})^{2}}{(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}\left[\hat{\unicode[STIX]{x1D714}}-1+(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1)\left(1+\frac{Z}{X}\right)+\unicode[STIX]{x1D702}_{i}\left(\frac{1}{X^{2}}+\frac{Z}{X^{3}}-\frac{Z}{2X}\right)\right],\end{eqnarray}$$

with $\hat{\unicode[STIX]{x1D714}}=\unicode[STIX]{x1D714}/\unicode[STIX]{x1D714}_{\ast e}$ . The quantity $Q(X)$ contains the effects of ion sound when $X\ll 1$ and ion Landau damping in the region $X\sim 1$ . Introducing the ion sound approximation,

(2.10) $$\begin{eqnarray}Q(X)\approx \frac{1}{2}\left(\frac{L_{s}}{L_{n}}\right)^{2}\frac{(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F})^{2}}{(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}\left[\hat{\unicode[STIX]{x1D714}}-1-(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})\frac{X^{2}}{2}\right].\end{eqnarray}$$

The validity of the ion finite Larmor radius and ion sound expansions requires these terms both to be small so that the balance of terms in (2.8), with $Q(X)$ given by (2.10), implies $\hat{\unicode[STIX]{x1D714}}-1\sim \unicode[STIX]{x1D702}_{e}\sim L_{n}/L_{s}<1$ .

We shall find it to be sufficient to consider only the ion sound approximation in order to ensure $E_{\Vert }\rightarrow 0$ , the physics of ion Landau damping only playing a conceptual role that justifies the use of the ‘outward carrying energy’ wave boundary conditions, as in the theory of the electron drift wave (Pearlstein & Berk Reference Pearlstein and Berk1969).

Inserting the solution for $\unicode[STIX]{x1D719}$ into Ampere’s law (2.6) and integrating over $X$ through the ion sound region yields the modification to $\unicode[STIX]{x1D6E5}^{\prime }$ arising from the intermediate region (which represents ion sound and, implicitly, ion Landau damping physics and we designate it as such):

(2.11) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }=\frac{1}{A}\int _{-\infty }^{\infty }\text{d}x\frac{\text{d}^{2}A}{\text{d}X^{2}}=\frac{1}{\unicode[STIX]{x1D70C}_{i}}\int _{-\infty }^{\infty }\frac{\text{d}X}{X}\left[1+\frac{(1+\unicode[STIX]{x1D702}_{i})}{\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right]Q(X)\left(\unicode[STIX]{x1D711}-\frac{A}{X}\right),\end{eqnarray}$$

where we have used (2.8).

3 Solution in the intermediate region

At low $\unicode[STIX]{x1D6FD}_{i}$ , equation (2.6) allows us to write $A\equiv \text{const.}$ , since $A_{\Vert }$ is an even function of $x$ , which we choose to be unity: $A=1$ . Introducing the following scaled variables:

(3.1a-c ) $$\begin{eqnarray}y=\left(\frac{\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}L_{s}}{L_{n}}\right)^{1/2}X,\quad \hat{\unicode[STIX]{x1D711}}=\left(\frac{\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}L_{s}}{L_{n}}\right)^{-1/2}\unicode[STIX]{x1D719},\quad a=\frac{\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}L_{s}}{2L_{n}}\frac{(\hat{\unicode[STIX]{x1D714}}-1)}{(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})},\end{eqnarray}$$

equation (2.8) can then be written as

(3.2a,b ) $$\begin{eqnarray}\frac{\text{d}^{2}}{\text{d}y^{2}}\hat{\unicode[STIX]{x1D711}}+\left(\frac{y^{2}}{4}-a\right)\hat{\unicode[STIX]{x1D711}}=R(y),\quad R(y)=-\frac{a}{y}+\frac{y}{4}.\end{eqnarray}$$

The validity of the ion FLR and ion sound expansions means that (3.2) is only valid on the domain $(L_{s}/L_{n})^{1/2}>y>(L_{n}/L_{s})^{1/2}$ , requiring $(L_{n}/L_{s})\ll 1$ .

To solve (3.2) we adopt two different techniques, corresponding to the two components of the inhomogeneous term, $R(y)$ . For the first term, $R(y)\rightarrow R_{1}(y)=-a/y$ , we employ a Fourier transform for its solution, $\hat{\unicode[STIX]{x1D711}}_{1}$ . For the second one, $R(y)\rightarrow R_{2}(y)=y/4$ , we generalise a technique due to Basu & Coppi (Reference Basu and Coppi1977), involving an integral representation of its solution, $\hat{\unicode[STIX]{x1D711}}_{2}$ .

Introducing the normalisations (3.1), the contribution to (2.11) from the ion sound term in $Q(X)$ becomes:

(3.3) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }=\frac{\unicode[STIX]{x1D6FD}_{i}(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}{\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}\int _{0}^{\infty }\,\text{d}y[1-y\hat{\unicode[STIX]{x1D711}}(y)].\end{eqnarray}$$

(i) Solution $\hat{\unicode[STIX]{x1D711}}_{1}(y)$

Fourier transforming equation (3.2) with $R(y)\rightarrow R_{1}(y)=-a/y$ , we obtain an equation for $\tilde{\unicode[STIX]{x1D711}}_{1}(k)$ :

(3.4) $$\begin{eqnarray}\tilde{\unicode[STIX]{x1D711}}_{1}(k)=\int _{-\infty }^{\infty }\text{d}k\exp (\text{i}ky)\hat{\unicode[STIX]{x1D711}}_{1}(y),\end{eqnarray}$$

namely

(3.5) $$\begin{eqnarray}\frac{1}{4}\frac{\text{d}^{2}}{\text{d}k^{2}}\tilde{\unicode[STIX]{x1D711}}_{1}+\left(\frac{k^{2}}{4}+a\right)\tilde{\unicode[STIX]{x1D711}}_{1}=\text{i}\unicode[STIX]{x03C0}a\,\text{sgn}(k),\end{eqnarray}$$

where the transform of the right-hand side is given by Pegoraro & Schep (Reference Pegoraro and Schep1986), utilising the theory for Fourier transforms of generalised functions (Gel’fand & Shilov Reference Gel’fand and Shilov1964).

This can be cast in the canonical form for parabolic cylinder functions by introducing $\unicode[STIX]{x1D705}=2k$ . The solution of (3.5) can then be written as:

(3.6) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D711}}_{1}(\unicode[STIX]{x1D705},a) & = & \displaystyle AE(-a,\unicode[STIX]{x1D705})+BE^{\ast }(-a,\unicode[STIX]{x1D705})\nonumber\\ \displaystyle & & \displaystyle +\,\text{i}\unicode[STIX]{x03C0}a\frac{E(-a,\unicode[STIX]{x1D705})}{W}\int _{0}^{\unicode[STIX]{x1D705}}\text{d}\unicode[STIX]{x1D705}^{\prime }E^{\ast }(-a,\unicode[STIX]{x1D705}^{\prime })\,\text{sgn}(\unicode[STIX]{x1D705}^{\prime })\nonumber\\ \displaystyle & & \displaystyle -\,\text{i}\unicode[STIX]{x03C0}a\frac{E^{\ast }(-a,\unicode[STIX]{x1D705})}{W}\int _{0}^{\unicode[STIX]{x1D705}}\text{d}\unicode[STIX]{x1D705}^{\prime }E(-a,\unicode[STIX]{x1D705}^{\prime })\,\text{sgn}(\unicode[STIX]{x1D705}^{\prime }),\end{eqnarray}$$

where $E(-a,\unicode[STIX]{x1D705})$ and $E^{\ast }(-a,\unicode[STIX]{x1D705})$ are the wave-like parabolic cylinder functions (Abramowitz & Stegun Reference Abramowitz and Stegun1972), the Wronskian, $W=-2i$ and $A$ and $B$ are constants to be determined by applying appropriate boundary conditions.

The acceptable solution in configuration space (i.e.  $x$ -space) is a wave carrying energy outwards from the resonant layer, eventually decaying due to ion Landau damping (Pearlstein & Berk Reference Pearlstein and Berk1969). Since the electron drift wave is a ‘backwards’ wave with group velocity opposed to its phase velocity, this translates into suppressing the wave solution with positive phase velocity in $x$ -space. In $k$ -space this implies supressing the one with an apparent negative phase velocity, so that

(3.7) $$\begin{eqnarray}B=\frac{\text{i}\unicode[STIX]{x03C0}a}{W}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D705}^{\prime }E(-a,\unicode[STIX]{x1D705}^{\prime })\text{sgn}(\unicode[STIX]{x1D705}^{\prime }).\end{eqnarray}$$

We also require $\hat{\unicode[STIX]{x1D711}}_{1}(\unicode[STIX]{x1D705},-a)$ to be odd in $\unicode[STIX]{x1D705}$ with $\hat{\unicode[STIX]{x1D711}}_{1}(0,a)=0$ . This requires

(3.8) $$\begin{eqnarray}A=-\frac{\text{i}\unicode[STIX]{x03C0}aE^{\ast }(-a,0)}{WE(-a,0)}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D705}^{\prime }E(-a,\unicode[STIX]{x1D705}^{\prime })\text{sgn}(\unicode[STIX]{x1D705}^{\prime }).\end{eqnarray}$$

Thus

(3.9) $$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D711}}_{1}(\unicode[STIX]{x1D705},a) & = & \displaystyle \frac{\text{i}\unicode[STIX]{x03C0}a}{W}\left[E^{\ast }(-a,\unicode[STIX]{x1D705})\int _{\unicode[STIX]{x1D705}}^{\infty }\text{d}\unicode[STIX]{x1D705}^{\prime }E(-a,\unicode[STIX]{x1D705}^{\prime })\text{sgn}(\unicode[STIX]{x1D705}^{\prime })\right.\nonumber\\ \displaystyle & & \displaystyle -\,\frac{E^{\ast }(-a,0)}{E(-a,0)}E(-a,\unicode[STIX]{x1D705})\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D705}^{\prime }E(-a,\unicode[STIX]{x1D705}^{\prime })\text{sgn}(\unicode[STIX]{x1D705}^{\prime })\nonumber\\ \displaystyle & & \displaystyle +\left.E(-a,\unicode[STIX]{x1D705})\int _{0}^{\unicode[STIX]{x1D705}}\text{d}\unicode[STIX]{x1D705}^{\prime }E^{\ast }(-a,\unicode[STIX]{x1D705}^{\prime })\text{sgn}(\unicode[STIX]{x1D705}^{\prime })\right].\end{eqnarray}$$

(ii) Solution $\hat{\unicode[STIX]{x1D711}}_{2}(y)$

Introducing $z=iy^{2}/4$ and $\hat{\unicode[STIX]{x1D711}}_{2}(y)=u\exp (-z)$ into (3.2) with $R(y)\rightarrow R_{2}(y)=y/4$ , we obtain the equation:

(3.10) $$\begin{eqnarray}z\frac{\text{d}^{2}u}{\text{d}z^{2}}+\left(\frac{1}{2}-2z\right)\,\frac{\text{d}u}{\text{d}z}+\left(-\frac{1}{2}+\text{i}a\right)u=-\frac{\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}z^{1/2}\text{e}^{z}}{2}\end{eqnarray}$$

for $u(z)$ . Generalising the technique of Basu & Coppi (Reference Basu and Coppi1977), we introduce the transform:

(3.11) $$\begin{eqnarray}u(z)=z^{\unicode[STIX]{x1D6FD}}\int _{C}\text{d}tv(t)\text{e}^{zt},\end{eqnarray}$$

where the contour $C$ remains to be chosen for convenience. Inserting the substitution (3.11) in (3.10) leads to

(3.12) $$\begin{eqnarray}\displaystyle & & \displaystyle z^{\unicode[STIX]{x1D6FD}-1}\int _{C}\text{d}tv(t)\text{e}^{zt}\left\{z^{2}(t^{2}-2t)+z\left[\left(2\unicode[STIX]{x1D6FD}+\frac{1}{2}\right)(t-1)+\text{i}a\right]+\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FD}-1)+\unicode[STIX]{x1D6FD}/2\right\}\nonumber\\ \displaystyle & & \displaystyle \quad =-\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}z^{1/2}\text{e}^{z}/2.\end{eqnarray}$$

Choosing $\unicode[STIX]{x1D6FD}=1/2$ , which eliminates the term independent of $z$ on the right-hand side bracket, equation (3.12) simplifies to

(3.13) $$\begin{eqnarray}\int _{C}\text{d}tv(t)\text{e}^{zt}\left\{z^{2}(t^{2}-2t)+z\left[\frac{3}{2}(t-1)+\text{i}a\right]\right\}=-\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}\text{e}^{z}/2.\end{eqnarray}$$

Integrating by parts in $t$ ,

(3.14) $$\begin{eqnarray}\displaystyle & & \displaystyle [v(t)\text{e}^{zt}(t^{2}-2t)]_{t_{1}}^{t_{2}}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\int _{C}\text{d}t\text{e}^{zt}\left\{\frac{\text{d}}{\text{d}t}[((t^{2}-2t))v(t)]-\left[\frac{3}{2}(t-1)+\text{i}a\right]v(t)\right\}=-\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}\text{e}^{z}/2,\qquad\end{eqnarray}$$

where $t_{1}$ and $t_{2}$ are the endpoints of contour $C$ and are to be chosen later for convenience.

Equation (3.14) is solved if

(3.15) $$\begin{eqnarray}\frac{\text{d}}{\text{d}t}[((t^{2}-2t))v(t)]-\left[\frac{3}{2}(t-1)+\text{i}a\right]v(t)=0\end{eqnarray}$$

and

(3.16) $$\begin{eqnarray}[v(t)\text{e}^{zt}(t^{2}-2t)]_{t_{1}}^{t_{2}}=-\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}\text{e}^{z}/2.\end{eqnarray}$$

Equation (3.15) is satisfied by:

(3.17) $$\begin{eqnarray}v(t)=\frac{{\hat{c}}}{t^{1/4+\text{i}a/2}(2-t)^{1/4-\text{i}a/2}}\end{eqnarray}$$

and (3.16) by choosing $t_{1}=0,t_{2}=1$ and the constant ${\hat{c}}=\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}/2$ . Inserting solution (3.17) into (3.11) for $u(z)$ , determines $\hat{\unicode[STIX]{x1D711}}_{2}(y)=u(z)\exp (-z)$ with $z=iy^{2}/4$ .

4 The $\unicode[STIX]{x1D6E5}^{\prime }$ -integral

Inserting the solution $\hat{\unicode[STIX]{x1D711}}=\hat{\unicode[STIX]{x1D711}}_{1}+\hat{\unicode[STIX]{x1D711}}_{2}$ into (3.3) and integrating over $y$ through the ion sound region yields $\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }$ , which we can express as

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }=\unicode[STIX]{x1D6E5}^{\prime (1)}+\unicode[STIX]{x1D6E5}^{\prime (2)},\end{eqnarray}$$

where

(4.2) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}^{\prime (1)}=\frac{\unicode[STIX]{x1D6FD}_{i}(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}{\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}\int _{0}^{\infty }\,\text{d}y[1-y\hat{\unicode[STIX]{x1D711}}_{1}(y)]\end{eqnarray}$$

and

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}^{\prime (2)}=\frac{\unicode[STIX]{x1D6FD}_{i}(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}{\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}\int _{0}^{\infty }\text{d}y[1-y\hat{\unicode[STIX]{x1D711}}_{2}(y)].\end{eqnarray}$$

To evaluate $\unicode[STIX]{x1D6E5}^{\prime (1)}$ we introduce the Fourier transform for $\hat{\unicode[STIX]{x1D711}}_{1}(y,a),\tilde{\unicode[STIX]{x1D711}}_{1}(\unicode[STIX]{x1D705},a)$ , perform the integration over $y$ , using the resulting delta function, $\unicode[STIX]{x1D6FF}(k)$ , to reduce the integral to:

(4.4) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}^{\prime (1)}=\frac{\unicode[STIX]{x1D6FD}_{i}(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}{2\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}\text{i}\,\frac{\text{d}}{\text{d}k}\tilde{\unicode[STIX]{x1D711}}_{1}(k,a)_{k=0}.\end{eqnarray}$$

From (3.9) we have the result:

(4.5) $$\begin{eqnarray}\text{i}\frac{\text{d}}{\text{d}k}\tilde{\unicode[STIX]{x1D711}}_{1}(k,a)_{k=0}=-\frac{2a\unicode[STIX]{x03C0}}{E(-a,0)}\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D705}^{\prime }E(-a,\unicode[STIX]{x1D705}^{\prime })\text{sgn}(\unicode[STIX]{x1D705}^{\prime }),\quad a>0.\end{eqnarray}$$

As shown in the appendix, equation (A 10), the integral over $\unicode[STIX]{x1D705}^{\prime }$ can be performed analytically to yield:

(4.6) $$\begin{eqnarray}\text{i}\frac{\text{d}}{\text{d}k}\tilde{\unicode[STIX]{x1D711}}_{1}(k,a)_{k=0}=-\frac{\displaystyle 2\unicode[STIX]{x03C0}\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}a\unicode[STIX]{x1D6E4}\left(\frac{3}{4}-\frac{\text{i}a}{2}\right)}{\displaystyle \unicode[STIX]{x1D6E4}\left(\frac{5}{4}-\frac{\text{i}a}{2}\right)}F\left(\frac{1}{2},\frac{1}{4}-\frac{\text{i}a}{2};\frac{5}{4}-\frac{\text{i}a}{2};-1\right).\end{eqnarray}$$

Here $F(a,b;c;z)$ is the hypergeometric function and $\unicode[STIX]{x1D6E4}(z)$ the gamma function (Abramowitz & Stegun Reference Abramowitz and Stegun1972). Thus:

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}^{\prime (1)}=-\frac{\unicode[STIX]{x1D6FD}_{i}(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}{2\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}I_{0}(a),\end{eqnarray}$$

with

(4.8) $$\begin{eqnarray}I_{0}(a)=-\frac{\displaystyle 2\unicode[STIX]{x03C0}\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}a\unicode[STIX]{x1D6E4}\left(\frac{3}{4}-\frac{\text{i}a}{2}\right)}{\displaystyle \unicode[STIX]{x1D6E4}\left(\frac{5}{4}-\frac{\text{i}a}{2}\right)}F\left(\frac{1}{2},\frac{1}{4}-\frac{\text{i}a}{2};\frac{5}{4}-\frac{\text{i}a}{2};-1\right).\end{eqnarray}$$

$I_{0}(a)$ , not to be confused with the modified Bessel function, is displayed in figure 2.

Figure 2. The real and imaginary parts of $I_{0}(a)$ as a function of the parameter $a=(L_{s}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}/2L_{n})(\hat{\unicode[STIX]{x1D714}}-1)/(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})$ .

Inserting $\hat{\unicode[STIX]{x1D711}}_{2}$ into integral (4.3) yields

(4.9) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6E5}^{\prime (2)}=\frac{\unicode[STIX]{x1D6FD}_{i}(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}{\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \quad \times \,\int _{0}^{\infty }\text{d}y\left[1-\frac{\text{i}y^{2}}{4}\int _{0}^{1}\text{d}t\exp \left(-\frac{\text{i}y^{2}t}{4}\right)(1-t)^{-1/4-\text{i}a/2}(1+t)^{-1/4+\text{i}a/2}\right].\end{eqnarray}$$

Using the identity

(4.10) $$\begin{eqnarray}1=\exp \left(-\frac{\text{i}y^{2}}{4}\right)+\frac{\text{i}y^{2}}{4}\int _{0}^{1}\text{d}t\exp \left(-\frac{\text{i}y^{2}t}{4}\right)\end{eqnarray}$$

we can rewrite (4.9) as

(4.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}^{\prime (2)} & = & \displaystyle \frac{\unicode[STIX]{x1D6FD}_{i}\left(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i}\right)}{\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}\left\{\int _{0}^{\infty }\text{d}y\exp \left(-\frac{\text{i}y^{2}}{4}\right)\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\int _{0}^{\infty }\text{d}y\frac{\text{i}y^{2}}{4}\int _{0}^{1}\text{d}t\exp \left(-\frac{\text{i}y^{2}t}{4}\right)[1-(1-t)^{-1/4-\text{i}a/2}(1+t)^{-1/4+\text{i}a/2}]\right\}.\qquad\end{eqnarray}$$

Now the integration over $y$ can be readily performed to yield:

(4.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}^{\prime (2)} & = & \displaystyle \sqrt{\unicode[STIX]{x03C0}}\text{e}^{-\text{i}\unicode[STIX]{x03C0}/4}\frac{\unicode[STIX]{x1D6FD}_{i}(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}{\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}\nonumber\\ \displaystyle & & \displaystyle \times \,\left\{1+\frac{1}{2}\int _{0}^{1}\text{d}t\,t^{-3/2}[1-(1-t)^{-1/4-\text{i}a/2}(1+t)^{-1/4+\text{i}a/2}]\right\}.\end{eqnarray}$$

Thus, we require the integral

(4.13) $$\begin{eqnarray}\hat{I} (a)=\frac{1}{2}\int _{0}^{1}\text{d}t\,t^{-3/2}[1-(1-t)^{-1/4-\text{i}a/2}(1+t)^{-1/4+\text{i}a/2}].\end{eqnarray}$$

This can be evaluated analytically, as shown in (B 9) of appendix B:

(4.14) $$\begin{eqnarray}\hat{I} (a)=-1+\frac{\displaystyle \sqrt{(2\unicode[STIX]{x03C0})}\unicode[STIX]{x1D6E4}\left(\frac{3}{4}-\frac{\text{i}a}{2}\right)}{\displaystyle \unicode[STIX]{x1D6E4}\left(\frac{1}{4}-\frac{\text{i}a}{2}\right)}.\end{eqnarray}$$

Figure 3. The real and imaginary parts of $I_{1}(a)$ as a function of the parameter $a=(L_{s}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}/2L_{n})(\hat{\unicode[STIX]{x1D714}}-1)/(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})$ .

Figure 4. The real and imaginary parts of $I(a)=I_{0}(a)+I_{1}(a)$ as a function of the parameter $a=(L_{s}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}/2L_{n})(\hat{\unicode[STIX]{x1D714}}-1)/(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})$ .

Thus, from (4.7) and (4.12)–(4.14) we obtain:

(4.15) $$\begin{eqnarray}\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }=\frac{\unicode[STIX]{x1D6FD}_{i}(\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}+1+\unicode[STIX]{x1D702}_{i})}{2\unicode[STIX]{x1D70C}_{i}}\left(\frac{L_{s}}{L_{n}\hat{\unicode[STIX]{x1D714}}\unicode[STIX]{x1D70F}}\right)^{1/2}(I_{0}(a)+I_{1}(a)),\end{eqnarray}$$

where

(4.16) $$\begin{eqnarray}I_{1}(a)=\sqrt{2}\unicode[STIX]{x03C0}\text{e}^{-\text{i}\unicode[STIX]{x03C0}/4}\frac{\displaystyle \unicode[STIX]{x1D6E4}\left(\frac{3}{4}-\frac{\text{i}a}{2}\right)}{\displaystyle \unicode[STIX]{x1D6E4}\left(\frac{1}{4}-\frac{\text{i}a}{2}\right)},\end{eqnarray}$$

which is evaluated numerically and displayed in figure 3. The combination

(4.17) $$\begin{eqnarray}I(a)=I_{0}(a)+I_{1}(a)\end{eqnarray}$$

is plotted in figure 4. When $a=0$ , the expression (4.15) reduces to the analytic result for the real part of $\unicode[STIX]{x1D6E5}^{\prime }$ quoted in Coppi et al. (Reference Coppi, Mark, Sugiyama and Bertin1979) for the case $\unicode[STIX]{x1D702}_{e}=0$ .

5 The effective $\unicode[STIX]{x1D6E5}^{\prime }$ for stability

The result (4.15) for $\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }$ , the modification to $\unicode[STIX]{x1D6E5}^{\prime }$ due to the ion sound/ion Landau damping physics, can be substituted into expressions in the literature for the drift tearing mode growth rate at finite ion temperature which have ignored the presence of these effects.

For example, the semi-collisional tearing mode with large ion Larmor orbits in the absence of the ion sound intermediate layer has a growth rate $\hat{\unicode[STIX]{x1D6FE}}$ given by $\text{Im}(\unicode[STIX]{x1D6FF}\hat{\unicode[STIX]{x1D714}})$ , where $\hat{\unicode[STIX]{x1D714}}=\hat{\unicode[STIX]{x1D714}}_{0}+\unicode[STIX]{x1D6FF}\hat{\unicode[STIX]{x1D714}}$ , and $\unicode[STIX]{x1D6FF}\hat{\unicode[STIX]{x1D714}}$ satisfies the dispersion relation (Connor et al. Reference Connor, Hastie and Zocco2012):

(5.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}\hat{\unicode[STIX]{x1D714}}\sim -\frac{\unicode[STIX]{x1D6FF}_{sc}\hat{\unicode[STIX]{x1D714}}_{0}^{1/2}}{\unicode[STIX]{x03C0}\hat{\unicode[STIX]{x1D6FD}}}\text{e}^{-\text{i}\unicode[STIX]{x03C0}/4}(\unicode[STIX]{x1D6E5}^{\prime }-\unicode[STIX]{x1D6E5}_{\text{crit}}^{\prime }).\end{eqnarray}$$

Here $\hat{\unicode[STIX]{x1D714}}_{0}=(1+0.74\unicode[STIX]{x1D702}_{e})$ , $\unicode[STIX]{x1D6FF}_{sc}=(\unicode[STIX]{x1D708}_{e}\unicode[STIX]{x1D70C}_{e}L_{s}^{2}/k_{\unicode[STIX]{x1D703}}v_{Te}L_{n})^{1/2}$ , with $\unicode[STIX]{x1D708}_{e}$ the electron collision frequency, $\unicode[STIX]{x1D70C}_{e}$ the electron Larmor radius and $v_{Te}$ the electron thermal velocity, is the semi-collisional width ( $\unicode[STIX]{x1D6FF}_{sc}\ll \unicode[STIX]{x1D70C}_{i}$ ) and

(5.2) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{\text{crit}}^{\prime } & = & \displaystyle \frac{\sqrt{\unicode[STIX]{x03C0}}\hat{\unicode[STIX]{x1D6FD}}}{\unicode[STIX]{x1D70C}_{i}}\frac{(\hat{\unicode[STIX]{x1D714}}_{0}-1)^{2}}{\hat{\unicode[STIX]{x1D714}}_{0}}\left(1+\hat{\unicode[STIX]{x1D714}}_{0}\unicode[STIX]{x1D70F}-\frac{\unicode[STIX]{x1D702}_{i}}{2}\right)\ln (\unicode[STIX]{x1D70C}_{i}/(\unicode[STIX]{x1D6FF}_{sc}\hat{\unicode[STIX]{x1D714}}_{0}^{1/2}))\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x03C0}\hat{\unicode[STIX]{x1D6FD}}}{\unicode[STIX]{x1D70C}_{i}}\hat{\unicode[STIX]{x1D714}}_{0}(\hat{\unicode[STIX]{x1D714}}_{0}-1)\overline{I},\end{eqnarray}$$

with $\hat{\unicode[STIX]{x1D6FD}}=\unicode[STIX]{x1D6FD}_{e}(L_{s}/L_{n})^{2}/2$ .  $\overline{I}(\hat{\unicode[STIX]{x1D714}}_{0}(\unicode[STIX]{x1D702}_{e}),\unicode[STIX]{x1D702}_{i,}\unicode[STIX]{x1D70F})$ is an integral that is negative and numerical evaluation shows that it is well represented by $\overline{I}\cong -1.3\unicode[STIX]{x1D702}_{e}^{1/2}$ (Connor et al. Reference Connor, Hastie and Zocco2012).Footnote ¹ The first contribution to $\unicode[STIX]{x1D6E5}_{\text{crit}}^{\prime }$ in (5.2) is due to finite ion Larmor radius effects (FLR), while the second term represents diamagnetic effects. The most sensitive dependence of $\unicode[STIX]{x1D6E5}_{\text{crit}}^{\prime }$ is on $\unicode[STIX]{x1D702}_{e}$ .

To introduce the effect of the intermediate ion sound layer, we replace $\unicode[STIX]{x1D6E5}^{\prime }$ in (5.1) by $(\unicode[STIX]{x1D6E5}^{\prime }-\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime })$ . Using (4.15) and (5.1), with the result for $I(a)$ displayed in figure 4, we show $\unicode[STIX]{x1D6E5}_{\text{eff}}^{\prime }$ , the critical value of $\unicode[STIX]{x1D6E5}^{\prime }$ (i.e. the value of $\unicode[STIX]{x1D6E5}^{\prime }$ for which the growth rate, as calculated from (5.1), vanishes) as a function of $\unicode[STIX]{x1D702}_{e}$ in figure 5, for typical values of the other parameters ( $\hat{\unicode[STIX]{x1D6FD}}=0.05$ , $\unicode[STIX]{x1D6FF}_{sc}/\unicode[STIX]{x1D70C}_{i}=10^{-3},\unicode[STIX]{x1D70F}=1,\unicode[STIX]{x1D702}_{i}=0,L_{n}/L_{s}=0.1$ ). Because $\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }$ is complex, both its real and imaginary parts contribute to the growth rate in (5.1). We see that the contribution from the ion sound/ion Landau damping layer is less important than those arising from $\unicode[STIX]{x1D6E5}_{\text{crit}}^{\prime }$ , representing the FLR and diamagnetic terms, except near $\unicode[STIX]{x1D702}_{e}=0$ where it remains finite. There it provides a persistent stabilising contribution, unlike the others, which tend to zero.

Figure 5. The effective critical value of $\unicode[STIX]{x1D6E5}^{\prime }\unicode[STIX]{x1D70C}_{i}$ , $\unicode[STIX]{x1D6E5}_{\text{eff}}^{\prime }\unicode[STIX]{x1D70C}_{i}$ (full (blue) line), as a function of $\unicode[STIX]{x1D702}_{e}$ for typical values of the other parameters: $\hat{\unicode[STIX]{x1D6FD}}=0.05$ , $\unicode[STIX]{x1D6FF}_{sc}/\unicode[STIX]{x1D70C}_{i}=10^{-3}$ , $\unicode[STIX]{x1D70F}=1$ , $\unicode[STIX]{x1D702}_{i}=0$ , $L_{n}/L_{s}=0.1$ . $\unicode[STIX]{x1D6E5}_{\text{crit}}^{\prime }$ is the contribution from FLR and diamagnetic effects (long dashed (brown) line), while $(\text{Re}[\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }]-\text{Im}[\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }])$ is the contribution from the real and imaginary parts of the ion Landau damping terms (dashed (green) line).

6 Discussion and conclusions

We have investigated the impact of ion sound and ion Landau damping on the effective tearing mode stability parameter, $\unicode[STIX]{x1D6E5}^{\prime }$ , in a hot tokamak plasma. These effects are essential in ensuring the parallel electric field approaches zero as one attempts to match solutions of the inner resonant layer equations to those of the external ideal MHD model. Because the tearing mode frequency is effectively determined by the resonant layer physics and depends on $\unicode[STIX]{x1D702}_{e}$ , it transpires that the effect is significant even for weak electron temperature gradients. In this situation one can introduce an intermediate region, between that where the key resonant layer physics lies and the ideal MHD region, where electron drift wave physics dominates (Coppi et al. Reference Coppi, Mark, Sugiyama and Bertin1979). Indeed, by recalling the ideas of Pearlstein & Berk (Reference Pearlstein and Berk1969) one does not need to explicitly consider the ion Landau damping region, replacing it by the boundary condition that the drift waves must only carry energy outwards. Matching solutions through this layer provides a link between the true $\unicode[STIX]{x1D6E5}^{\prime }$ and the effective one seen by the inner layer.

The effect of the intermediate layer is parametrised by the quantity $a=(\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}L_{s}/2\unicode[STIX]{x1D714}_{\ast e}L_{n})((\unicode[STIX]{x1D714}-\unicode[STIX]{x1D714}_{\ast e})/(\unicode[STIX]{x1D714}\unicode[STIX]{x1D70F}+(1+\unicode[STIX]{x1D702}_{i})\unicode[STIX]{x1D714}_{\ast e}))$ . Recalling that a good fit to the drift tearing mode frequency is $\hat{\unicode[STIX]{x1D714}}_{0}=(1+0.74\unicode[STIX]{x1D702}_{e})$ (Connor et al. Reference Connor, Hastie and Zocco2012), we see that $a$ is indeed essentially a function of $\unicode[STIX]{x1D702}_{e}$ and $L_{s}/L_{n}$ . However, the validity of the theory requires the electron temperature gradient to be rather weak, with $\unicode[STIX]{x1D702}_{e}\sim L_{n}/L_{s}$ , i.e.  $L_{T_{e}}\sim L_{s}$ , so that $a\sim 0(1)$ (Coppi et al. Reference Coppi, Mark, Sugiyama and Bertin1979). Although the effects of the intermediate region increase as $\unicode[STIX]{x1D702}_{e}$ increases, a valid treatment for $\unicode[STIX]{x1D702}_{e}\sim 0(1)$ would require a numerical calculation involving the full effects of ion Larmor radius and ion Landau damping, as the approximations inherent in the simple electron drift wave model, namely the long-wavelength ion Larmor radius and the ion sound expansion, no longer hold.

Given these constraints on $\unicode[STIX]{x1D702}_{e}$ , we have applied the results to a realistic hot tokamak plasma model, although in cylindrical geometry, where the ion Larmor orbit exceeds the reconnecting layer width, which itself is described by semi-collisional electron physics (Cowley et al. Reference Cowley, Kulsrud and Hahm1986; Connor et al. Reference Connor, Hastie and Zocco2012) and compared the stabilising effects from the ion sound/ion Landau damping physics with other stabilising effects, as a function of $\unicode[STIX]{x1D702}_{e}$ . The sum of these effects then leads to a critical value, $\unicode[STIX]{x1D6E5}_{\text{eff}}^{\prime }$ for tearing instability. Although $\unicode[STIX]{x1D6E5}_{\text{ILD}}^{\prime }$ is generally a less important stabilising term than the FLR and diamagnetic contributions in $\unicode[STIX]{x1D6E5}_{\text{crit}}^{\prime }$ , it persists as $\unicode[STIX]{x1D702}_{e}\rightarrow 0$ , unlike the others.

The calculation in this paper is only strictly valid for a collisionless plasma slab in a sheared magnetic field geometry, as it only ignores possible curvature, collisional and trapped particle effects. However, these are weak in the range of $x\,(\unicode[STIX]{x1D70C}_{i}<x<\unicode[STIX]{x1D70C}_{i}L_{s}/L_{n})$ that is relevant for the ion dynamics involved in the calculation; it is therefore a meaningful calculation for more general geometries.