The linear, collisional, constant-*ψ* drift-tearing mode is analyzed for different regimes of the plasma-*β*, ion-skin-depth parameter space with an unreduced, extended-magnetohydrodynamic model. New dispersion relations are found at moderate plasma *β* and previous drift-tearing results are classified as applicable at small plasma *β*.

## I. INTRODUCTION

Experimental, fusion-plasma discharges typically operate in regimes away from ideal-magnetohydrodynamic (MHD) stability boundaries. The ideal-MHD modes that exist outside these boundaries, which are unable to modify the magnetic topology, are often deleterious to confinement and can lead to a rapid loss of the plasma stored energy. Analysis with a resistive-MHD model shows a second class of modes is possible. These resistive-MHD modes are a combination of macroscopic ideal-MHD behavior through-out most of the plasma volume and boundary-layer dynamics, where resistivity is important near a resonant magnetic-flux surface, a surface where the mode structure and the magnetic topology are aligned in poloidal and toroidal periodic variation. Although the plasma dynamics associated with these resistive modes are usually less violent than ideal modes, finite resistivity allows for modification of the magnetic topology. For example, magnetic islands formed from saturated resistive-tearing modes can enhance energy and particle transport from the plasma core to the edge via large field-aligned transport.

The tearing instability^{1} is one such multi-scale mode: a combination of macroscopic structure, the ideal-MHD response through-out most of the plasma volume; and microscopic structure, the boundary-layer physics near the resonant surface which minimally includes resistive MHD. Ideal-MHD flows advect magnetic flux to the resonant surface where a large, localized current sheet is formed. This leads to slow growth on a hybrid-time scale that is a combination of the ideal Alfvén time and the time scale of the pertinent boundary-layer physics. With a resistive-MHD model, the current-sheet size is determined by the magnitude of the plasma resistivity: smaller resistivity results in a more localized layer. In high-temperature fusion plasmas, which have very small resistivity, the boundary-layer width can approach the ion gyroradius where ion finite-Larmour-radius (FLR) and electron-ion-fluid-decoupling effects become important. When the more mobile electron fluid is decoupled from the ion fluid near the layer, it can more effectively transport flux into the layer and thus destabilize the mode (increase the growth rate). Alternatively, when the fluids are decoupled and drift in opposed directions within the resonant flux surface, the sheared relative motion can stabilize the mode (reduce the growth rate). A sufficient model to capture these FLR effects to first order is extended-MHD with Braginskii-like closures.^{2–5} The zeroth-order FLR plasma drift, the **E** × **B** drift, causes the electron and ion fluids to drift with the same velocity and thus are not stabilizing. The first-order FLR drifts have an orientation that is dependent on the sign of the charge of the species and thus are stabilizing. With respect to influence on the tearing mode, the most studied first-order FLR drift is the fluid diamagnetic drift^{6} but stabilizing effects are also attributed to drifts proportional to the gradient and curvature of the magnetic field.^{7}

A previous parametric regime analysis of the tearing mode without drift effects is given by Ahedo and Ramos.^{8} They characterize small-Δ^{′} tearing-mode parameter space by seven regimes as illustrated schematically in Fig. 1. A single-fluid model (resistive MHD) describes the dynamics in parameter-space-region PR1 as first discovered by Furth *et al*.^{1} In PR5, at very small values of *β* and large ion-skin depth (*d _{i}*), the semicollisional description of Drake and Lee is valid.

^{9}Without drift-effects, the semicollisional description is valid when

*β*is smaller than the square of the tearing skin depth normalized by the mode wavenumber (ignoring some factors of order unity). Thus, even for a large tearing skin depth of 1 cm, the validity constraint is still approximately $\beta \u226a10\u22124$ with a mode wavelength of 1 m and it is unlikely this regime is relevant to tokamak discharges. At moderate values of the plasma-

*β*parameter and

*d*(thus moderate values of the ion gyroradius, $\rho i\u223c\beta di$), the tearing dispersion relation from electron-MHD

_{i}^{10}is recovered (PR3). Mirnov

*et al*. derive a unified dispersion relation for PR4 which limits to that found in PR3 and PR5.

^{11}They describe the decoupling effects of the mode as mediated through interaction with the kinetic-Alfvén wave in PR3 and the whistler wave in PR5. The dispersion relations for the remaining transitional regimes in this parameter space (PR2 and PR6) are derived by Ahedo and Ramos.

^{8}There is no known solution in PR0.

Our results largely follow the parameter-space characterization of Ref. 8, however our calculations include diamagnetic and magnetic-field-gradient drift contributions. In a sense, the main concept of our study is to add a third dimension out of the page of Fig. 1 that corresponds to the drift frequency. In Sec. II, we describe the extended-MHD model and our small-Δ^{′}, large-guide-field assumptions. The extended-MHD equations are linearized and reduced to a system of two second-order equations in Secs. III and IV. Although gyroviscosity is included within our model equations, our final solutions do not include it in its full, unreduced form as discussed in Sec. IV A. Our notation allows us to express two approximations to ion gyroviscosity in our final solutions. With the first approximation, the contributions from ion gyroviscosity are neglected, and, in the second approximation, we use the standard ion gyroviscous cancellation (where ion gyroviscosity exactly cancels with the diamagnetic drift term in the parallel vorticity equation, an approximation originally from Ref. 6).

Our intention is to clarify the relevant regimes to fusion plasmas and to provide a benchmark for extended-MHD drift-tearing computations which use an unreduced-MHD model. As such, our study differs from much of the prior work in that we do not start with a reduced-MHD model, but rather we apply tearing ordering to the full extended-MHD equations. Our main dispersion relation results are derived in Sec. V for drift tearing in PR1 through PR5. We recover the result of Coppi at small values of the plasma-*β* parameter in the single-fluid regime (PR1, Refs. 6 and 12) and the result of Drake and Lee in the semicollisional regime (PR5, Ref. 9). New dispersion relations are found in PR2 through PR4.

## II. MODEL EQUATIONS AND ORDERINGS

With an unreduced-MHD model, the plasma fluid is described by a continuity equation

for the plasma density (*n*) evolution, a center-of-mass momentum equation

for the bulk-plasma velocity (**v**), and an energy equation

for the plasma temperature (*T _{α}*). The subscript indicates either the ion or electron species,

*m*is a species' mass, and Γ is the adiabatic index. The plasma is assumed to be an ideal gas and thus the species pressure (

_{α}*p*; $p=\u2211p\alpha $) is given by the ideal-gas law,

_{α}*p*=

_{α}*nT*. As appropriate for low-frequency plasma dynamics, we assume quasi-neutrality (

_{α}*n*≃

_{e}*n*for an ion charge state of unity) and drop the displacement-current term in Ampere's law (

_{i}*μ*

_{0}

_{ }

**J**= ∇×

**B**, where

*μ*

_{0}is permeability of free space), which provides a relation between the magnetic field (

**B**) and the current density (

**J**=

*ne*(

**v**

_{i}−

**v**

_{e}) where

*e*is the electron charge). These approximations analytically eliminate both light and Langmuir waves. The electron momentum equation is used as an expression for the electric field (

**E**)

commonly referred to as the generalized Ohm's law (*m _{e}* is the electron mass and

*η*is the electrical resistivity caused by electron-ion collisions). Faraday's law ($\u2202B/\u2202t=\u2212\u2207\xd7E$) in conjunction with Eq. (4) produces the induction equation, which describes the evolution of the magnetic field. This system of equations is considered to be a two-fluid model when the Hall term (

**J**×

**B**/

*ne*) is retained as the magnetic field is then advected by the electron flow (

**v**

_{e}=

**v**

_{i}−

**J**/

*ne*) instead of bulk-flow advection from the

**v**×

**B**term.

These equations require closure expressions for the stress tensors ($\Pi \alpha $) and heat fluxes (**q**_{α}). We use the Braginskii-like^{3–5} “cross” terms (first-order FLR terms) as the closure: gyroviscosity

and cross-heat flux

where *q _{α}* is a species' charge. The rate-of-strain tensor (

**W**

_{α}) is defined as $W\alpha =\u2207v\alpha +\u2207v\alpha T\u2212(2/3)I\u2207\xb7v\alpha $. This choice of closure neglects the perpendicular and parallel (to

**B**) closure terms and additional contributions to the gyroviscous stress;

^{3,13}however, the retained terms are commonly included in state-of-the-art extended-MHD codes and have contributions that enter the model equations on the same order as the diamagnetic-drift terms.

To further estimate the importance of the cross-closure terms, consider flows on the order of the sound speed, $cs=\Gamma (Ti+Te)/mi$, which for comparable species' temperatures is on the same order as the ion thermal speed, $vT\alpha =T\alpha /m\alpha $. The ion gyroviscous term then scales as *ρ _{i}*/

*L*relative to the ∇

*p*term in the momentum equation, Eq. (2), whereas the electron gyroviscous term scales as $me/mi(\rho e/L)$ relative to the ∇

*p*term in the generalized Ohm's law, Eq. (4). Here, $\rho \alpha =vT\alpha /\omega c\alpha $ is the gyroradius, where $\omega c\alpha =q\alpha B/m\alpha $ is the gyrofrequency and

_{e}*L*is a characteristic gradient length scale. Furthermore, the ratio of the electron to ion gyroradius is the square root of the mass ratio, $me/mi$. Thus, if the ion gyroviscous term is significant and the first-order ion-FLR model remains valid, $\rho i/L\u2272O(1)$, then the electron gyroviscous term is expected to be smaller than other terms in the generalized Ohm's law by at least the mass ratio. As such, we neglect contributions from electron gyroviscosity in our equations. Next, consider the cross-heat flux terms relative to $p\alpha \u2207\xb7v\alpha $ in Eq. (3). The ion cross-heat flux scales as $\rho i/L$, but the electron cross-heat flux scales as $mi/me(\rho e/L)$. Thus, if the ion cross-heat flux is significant ($\rho i/L\u2272O(1)$) then the electron cross-heat flux enters the equations on the same order and must be retained.

For the purposes of our study, the tearing instability is generated from an imposed $z\u0302$-oriented current sheet in a Cartesian slab. There are distant conducting walls at $x=\xb1\u221e$, the *ŷ* and $z\u0302$ directions are infinite, and the $z\u0302$ direction is symmetric. The tearing mode drive is fueled by free energy from the global configuration but growth of the mode is limited by the small-scale physics that breaks the frozen-flux theorem within the tearing boundary layer. As this boundary-layer physics is the focus of our study, the slab configuration is locally analogous to a toroidal configuration without curvature contributions where $x\u0302$ is a radial (flux) coordinate, *ŷ* is approximately a cross-field coordinate and $z\u0302$ is approximately a parallel-field coordinate. We decompose all fields into imposed, *x*-dependent, background fields (“0” subscript) and periodic-in-*ŷ*, perturbation fields (tilde), e.g., $B=B0(x)+B\u0303(x)\u2009exp\u2009(iky+\gamma t)$. Here, $k=ky\u0302$ is the perturbation wavenumber and *γ* is the complex growth rate. The radial ($x\u0302$) component of all background vector fields is zero. Perturbation vector fields and wavenumber use a magnetic-coordinate system where the *ŷ* and $z\u0302$ components are expressed as parallel-to and perpendicular-to the magnetic field.

In our subsequent analysis, we ignore the effects of flow shear but retain the effect of advection by bulk background flows. We impose orderings appropriate for the tearing boundary layer: (1) the equilibrium magnetic-shear-length scale (*L _{s}*) is comparable to the inverse wavelength, $kLs\u223cO(1)$; (2) a moderately large guide- to shear-magnetic-field ratio, such that $\u03f5B=Bz(x=0)/By(x=\u221e)\u223cO(\u03f53/4)$; (3) a small tearing skin depth (

*δ*), $kx\u223ck\delta \u223cO(\u03f5)$; (4) slow dynamics, $\omega \tau A\u223cO(\u03f53/2)$; and (5) slowly varying profiles within the layer, e.g. $\delta n0\u2032/n0\u223cO(\u03f5)$. Here,

*ϵ*is a small parameter ($\u03f5\u226a1$) and $\tau A\u22121=kvA=kB0/min0\mu 0$ is the Alfvén time.

We assume that the mode is collisional and thus that electron-inertia term is small relative to resistivity. In order to clarify the regime of validity for the collisional tearing mode, we assume that electron inertia is dominated by the contribution from current density and that advection in the electron inertia term, which is of the same order as electron gyroviscosity, is small. Thus, after linearization

The relative magnitude of resistivity compared to electron inertia classifies the tearing mode as collisionless ($de2\gamma \u226b\eta /\mu 0$) or collisional ($de2\gamma \u226a\eta /\mu 0$), where $d\alpha $ is a species' skin depth ($m\alpha /\mu 0ne2$). The assumption that the current density contribution dominates is justified a-posteriori in PR3 through PR5, as the magnetic field perturbation dominates in these regimes. Additionally, as $de=me/midi$, these large-*d _{i}* regimes (where the growth rate is also typically enhanced by fluid decoupling) are the most likely to be collisionless.

These assumptions, with the exception of the assumption of a collisional mode, are consistent with the expected conditions for core tearing in a high-temperature tokamak discharge. Our analysis can accommodate very small values of *β* ($cs2/vA2$), however we assume the growth rate is subsonic ($\gamma 2\u226ak2cs2$). Ahedo and Ramos show that when this assumption is violated without drift effects, the eigenfunction structure is modified but the growth rate is unchanged.^{14} In the following discussion, we use two normalizations: the hat which indicates normalization by Alfvén time/velocity and characteristic field strengths ($\omega \u0302=\omega \tau A$, $L\u0302=kL,\u2009v\u0302=v/vA(x=0),\u2009B\u0302=B/B0(x=0)$, $n\u0302=n/n0(x=0)$, and $p\u0302=p/vA2min0(x=0)$) and the overbar which is a tearing specific normalization introduced in Sec. IV.

## III. LINEARIZED EQUATIONS

Following convention, we define $\xi \u0302=\gamma \u0302v\u0302x$ as the displacement vector and

consistent with Ref. 8, where $\lambda =\mu 0J\xb7B/B2$ and $k\u0302\u2225=k\xb7B0/kB0$. After linearization and applying the assumptions of Sec. II, the radial induction equation becomes

where the Lundquist number is defined as $S=vA\mu 0/k\u22a5\eta $. The left side of this equation is a term representing the rate-of-change of $B\u0302x$. The notation $\gamma \u0302i=\gamma \u0302+ik\u0302\xb7v\u03020$, and $\gamma \u0302e=\gamma \u0302+ik\u0302\xb7v\u0302e0\u2243\gamma \u0302i\u2212i\omega \u0302*$ gathers the advective and temporal-derivative contributions into a single term. The terms on the right side of Eq. (9) result from the **v** × **B**, Hall, and resistive terms, respectively. Contributions from the ∇*p _{e}* term vanish as

and the equilibrium gradients of *T _{e}* and

*n*are oriented in the $x\u0302$ direction. Other than ignoring flow shear and applying our ordering to the resistive term, Eq. (9) is exact.

The location where **k**⋅ **B**_{0} = 0 is the resonant magnetic-flux surface. Away from the resonant surface the contribution from the **v** × **B** term dominates and all other terms may be neglected. When fluid decoupling and/or drift effects are significant, the Hall term dominates near the resonant surface. At the resonant surface the **v** × **B** and Hall terms vanish and thus the resistive contributions must be retained. Our calculations assume the resonant surface is located at *x* = 0. The standard treatment of these equations is to apply a boundary-layer analysis, where the ideal-MHD equations describe the solution in the outer region (away from the resonant surface), and the full model is used in the inner layer near the resonant surface. These solutions are matched using the discontinuity in the logarithmic derivative of the perturbed radial magnetic field of the outer solution (Δ^{′})

where the prime indicates a partial derivative with respect to *x*. With a resistive-MHD model and without drift-effects, an equilibrium is tearing unstable (*γ* > 0) if Δ*′* > 0;^{1} thus Δ^{′} is both a matching and stability parameter. We assume that $\Delta \u0302\u2032\u223cO(1)$ (equivalent to $kLs\u223cO(1)$ in a slab) and thus $Bx\u2032\u0302\u223cB\u0302x$, as follows from Eq. (11). Expanding $B\u0302x$ at *x* = 0,

and noting that $x\u0302\u223cO(\u03f5)$ allows us to treat $B\u0302x$ as a constant—an assumption known as the constant-*ψ* approximation. Derivatives of other perturbed fields are assumed to raise the relative size of the field by $\u03f5\u22121$, e.g., $\u03f52\xi \u2033\u0302\u223c\xi \u0302$ and $\u03f5Bx\u2033\u0302\u223cB\u0302x$. This approximation results from the large, localized gradients of perturbed fields within the boundary layer. Consider, for example, that the reconnecting inflows of the tearing mode produce a displacement vector that changes sign across the boundary layer.

After linearization, the parallel induction equation becomes

where $\omega *\alpha $ is a species diamagnetic-drift frequency ($kp\alpha 0\u2032/n0eB0$), $\omega *$ is the total diamagnetic-drift frequency ($\omega *i+\omega *e$), $\omega *n$ is the density-gradient drift $(kT0n0\u2032/n0eB0)$, and $\u2207\u22a5=\u2207\u2212ik\u2225b\u0302$. The first two pairs of terms on the right side are the contributions from the **v** × **B** and Hall terms, respectively. The terms involving $n\u0302$ and $p\u0302e$ result from the ∇*p _{e}* term and the last term is the effect of resistivity.

The components of the linearized momentum equation are

and

The perpendicular and parallel components (Eqs. (15) and (16)) are used to construct an expression for $\u2207\u0302\xb7v\u0302$. The first terms on the right side of Eqs. (14) and (16) are drift contributions from **J** × **B**.

The linearized continuity, ion-energy and electron-energy equations are

and

respectively. To allow for a systematic study of the effect of different advective models, we introduce the *σ _{pe}* and $\gamma \u0302pe$ notation. If the advective term uses the bulk flow then $\gamma \u0302pe=\gamma \u0302i$ and

*σ*= 0, whereas advection by the electron flow leads to $\gamma \u0302pe=\gamma \u0302e$ and

_{pe}*σ*= 1. To compute the linearized cross heat-flux contributions we first expand the heat-flux vector as

_{pe}Noting that $J0\xb7\u2207f0,\u2009B0\xb7\u2207f0$, $B0\xb7\u2207B0$, and $\u2207f0\xd7\u2207g0$ vanish for our slab configuration, we may assume the coefficients of these terms are equilibrium quantities during linearization. After linearization and ordering (specifically, we drop terms where $\omega \u0302*\u226bk\u0302\u2225\lambda \u03020d\u0302i$), we find

where

and

Again we introduce *σ _{qα}* as a marker with value $\sigma qi=\u2212\sigma qe=(5/2)(\Gamma \u22121)/\Gamma $ when the cross heat flux is included in the model and $\sigma q\alpha =0$ when it is not. Equations (18), (19), and (21) may be combined to produce expressions for $p\u0302=p\u0302i+p\u0302e$ and $p\u0302e$. Thus

and

where $c\u0302sqe2=Cqec\u0302se2,\u2009c\u0302sq2=Cqec\u0302se2+Cqic\u0302si2$, $c\u0302sp2=c\u0302spe2+c\u0302spi2$,

and *E _{t}* =

*E*+

_{i}*E*.

_{e}## IV. SYSTEM OF EQUATIONS

We next algebraically reduce Eqs. (9), (13)–(17), (25), and (26) from a system of eight equations to a system of five. These five equations use $B\u0302x,\u2009\u2207\u0302\xb7v\u0302$, $Q\u0302,\u2009\xi \u0302$, and $v\u0302\u2225$ as primary variables. Two of these are unmodified from the system of eight: the radial induction equation, Eq. (9), and the parallel velocity equation, Eq. (16). One is slightly modified: the parallel induction equation provides an expression for $Q\u0302$ after $n\u0302$ and $p\u0302$ are eliminate. And two new equations are derived: an expression for $\u2207\u0302\xb7v\u0302$ and a parallel vorticity equation which governs $\xi \u0302$.

After multiplying by $\gamma \u0302i$ and substituting Eq. (25) for $p\u0302$,

The inertial contributions ($\gamma \u0302i2\u2207\u0302\xb7v\u0302$) are dropped as they are small compared to the $c\u0302sp2\u2207\u0302\xb7v\u0302$ term from $p\u0302$ in Eq. (25). Without drift and FLR effects only the first and third terms on the right side contribute to $\u2207\u0302\xb7v\u0302$. The second term on the right side is a drift-like term from $v\u0303\xb7\u2207p$ and $v\u0303\xb7\u2207n$ and the remaining terms are contributions from electron advection (∼*C _{pe}*), cross heat flux ($\u223ccsq2$), and ion gyroviscosity.

After eliminating $B\u0302\u2225,\u2009n\u0302$ and $p\u0302$ from the parallel induction equation, Eq. (13), we find

where

and

Without drift effects, all contributions from ∇*p _{e}* and ∇

*n*vanish (the latter of these results from the 1/

*ne*factors in Ohm's law). In particular, these contributions lead to the

*A*,

*E*,

_{n}*C*, and $c\u0302sq2$ factors in Eq. (34).

_{pe}The only unused equation from our original system of eight is the radial momentum equation, Eq. (14). To find an expression for $p\u0302\u2032$, we take the derivative of Eq. (32)

Again, we ignore the inertial term ($\gamma \u0302i2\u2207\u0302\xb7v\u0302$). Substituting into Eq. (14) and applying the tearing ordering

Without drift and FLR effects, this equation becomes the standard form of the parallel vorticity equation, $\gamma \u0302i\gamma \u0302\xi \u2033\u0302\u2243\u2212ik\u0302\u2225Bx\u2033\u0302$.

We now have a system of five equations: Eqs. (9), (16), (33), (34), and (38). The discussion of the tearing-ordered contributions from ion gyroviscosity is deferred until Sec. IV A. Without these contributions, compressibility and parallel flows only couple to this system through the parallel induction equation, Eq. (34). Thus, in the single-fluid regime where the Hall effect and ion gyroviscosity may be ignored, only two equations, the radial induction and parallel vorticity equations, are required to find the dispersion relation.

### A. Considerations of ion gyroviscosity

With tearing-ordered gyroviscous contributions, the compressibility equation (Eq. (33)) becomes

and the parallel-momentum equation (Eq. (16)) becomes

where *σ _{gv}* is a marker for ion gyroviscosity (set to unity when gyroviscosity is included and otherwise zero), the modified ion-gyroviscous growth rate is

and $\gamma \u0302E\xd7B$ is the Doppler-shifted growth rate. The tearing-ordered ion-gyroviscous contributions to parallel-vorticity equation (Eq. (38)) are

The $i\omega \u0302*i\gamma \u0302\xi \u2033\u0302$ term produces the standard gyroviscous cancellation and cancels the advective diamagnetic drift. However, as there are many additional terms in this equation, this cancellation is inexact. The $i\omega \u0302*c\u0302si2\gamma \u0302\xi \u2033\u0302/\Gamma $ term is the result of a drift proportional to the gradient of the magnetic field as previously discussed in detail for tearing in a cylindrical pinch configuration^{7} (it has been re-characterized in terms of *ω*_{*} through equilibrium force balance). Combining Eqs. (38) and (42) and again applying the tearing ordering gives

The last two terms on the right side of Eq. (43), which are present only with ion gyroviscosity, raise the differential order of the system of equations. Without these gyroviscous contributions, compressibility and parallel flow terms can be eliminated algebraically from the parallel induction equation, Eq. (34), by substituting Eqs. (33) and (16). We do not presently have a solution to the system of equations with ion gyroviscosity, and thus we proceed without the full contributions.

Prior work typically includes only the standard gyroviscous cancellation as a model of ion gyroviscosity. Although we can not justify this approximation from a tearing-ordered-equations stand point, we retain the $\gamma \u0302gvi$ terms as is in order to facilitate comparison. The two relevant limits are then without gyroviscosity ($\gamma \u0302gvi\u2192\gamma \u0302i$), and with the exact gyroviscous cancellation ($\gamma \u0302gvi\u2192\gamma \u0302E\xd7B$).

### B. Tearing normalized system of equations

Without ion gyroviscosity, compressibility, and parallel flow can be eliminated algebraically. Substituting Eqs. (16) and (33) into Eq. (34), we find

where

and

Equations (9), (43) (with *σ _{gv}* = 0), and (44) now comprise our system of equations for $B\u0302x,\u2009Q\u0302$, and $\xi \u0302$. The first two terms on the right side of Eq. (44) are the contributions from the Hall term and resistive diffusion, respectively; the remaining terms result from a combination of compressibility, parallel flows, ∇

*p*contributions, inertia and the

_{e}**v**×

**B**term. Compressibility and parallel flows contribute the $k\u0302\u22252$ and $\omega \u0302*$ terms on the right side of Eq. (44) as well as the $\gamma \u0302i/c\u0302sp2$ terms in the $\tau \xaf$ factors. The ∇

*p*term in Ohm's law contributes the $\omega \u0302*n/c\u0302s2$ terms in the $\tau \xaf$ factors.

_{e}With the constant-*ψ* approximation, where $B\u0302x$ is assumed constant within the small tearing layer, Eq. (43) is used to eliminate $Bx\u2033\u0302$; which results in a system of two coupled equations for $Q\u0302$ and $\xi \u0302$. We use a tearing normalization for these equations similar to Ref. 8 with the dimensionless variables

and the dimensionless parameters

With this normalization, $\sigma \xaf$ is the ion skin depth, *d _{i}*, normalized to the tearing skin depth, $\delta =(S\gamma \u0302)\u22121/2$. Validity of a first-order FLR model requires

*ρ*<

_{i}*δ*. A good rule of thumb for plasmas with comparable ion and electron temperatures is to use the ion sound gyroradius,

*ρ*=

_{s}*c*/

_{s}*ω*, and require $\rho s/\delta =c\u0302s\sigma \xaf=\beta \sigma \xaf<1$. After expanding $k\u2225$ and retaining only the leading order term in

_{ci}*x*, $k\u2225\u2032x$,

and

compose the system of second-order coupled equations.

Equation (51), a combination of the radial-induction and parallel-vorticity equations, governs the ion dynamics and is composed of the contribution from resistivity on the left side, and the contributions from the **v** × **B**, Hall, and inertial terms, respectively, on the right side. In the single-fluid limit where the Hall term ($x\xaf2Q\xaf$) can be ignored, this equation alone governs the bulk-flow-mediated mode dynamics. Equation (52), a combination of the parallel-induction and parallel-vorticity equations, governs the electron dynamics. The left side of this equation is the contribution from diffusion of the parallel field and third term on the right side is the contribution from the Hall term ($k\u0302\u2225d\u0302iBx\u2033\u0302$). The $\tau \xaf$ parameters scale as *β*^{−1} and are typically important only at small values of *β*. The dominant *β*^{−1} contributions result from the gradient of the electron pressure in Ohm's law (terms involving $\omega \u0302*n$) and perpendicular compressibility (otherwise). There are other contributions to $\tau \xafQ$ and $\tau \xaf\xi $ from the parallel-field inertia and the **v** × **B** term, respectively, however these term are unimportant from a practical perspective. The first and last terms on the right side are also contributions from perpendicular compressibility and are important in the moderate-*β* transition regime (PR2).

## V. DRIFT-TEARING DISPERSION RELATIONS BY PARAMETRIC REGIME

Once solutions for $Q\xaf$ and $\xi \xaf$ are found, the dispersion relation may be computed by integrating the radial induction equation (Eq. (9)) and applying the boundary condition $B\u0303r\u2032(\xb1\u221e)=0$. The resulting equation is

where we have defined *D* for notational convenience. The right side of this expression is the contribution from resistivity, thus the integrand of left side of this expression is the ideal radial Ohm's law. As resistivity is only significant in the layer, proper matching of the inner and outer region solutions ensures the integrand vanishes outside the layer and the integral converges.

We next derive the dispersion relation in the various parametric regimes as summarized in Table I. We begin in the single-fluid regime (PR1) with $\tau \xafQ\u226a1$ (near PR2) and work our way clockwise around Fig. 1. We do not address PR6 which was solved numerically in Ref. 8. We finish again in the single-fluid regime (PR1) with $\tau \xafQ\u226b\sigma \xaf2$ (near PR6) where we recover the drift-tearing result of Ref. 6.

. | PR1a . | PR2 . | PR3 . | PR4 . | PR5 . | PR1b . |
---|---|---|---|---|---|---|

Regime | $\sigma \xaf2\u226a1$ | $\sigma \xaf2\u223c1$ | $\sigma \xaf2\u226b1$ | $\sigma \xaf2\u226b1$ | $\sigma \xaf2\u226b1$ | $\tau \xafQ\u226b\sigma \xaf2$ |

boundary | and | and | and | and | and | and |

$\tau \xafQ\u226a1$ | $\tau \xafQ\u226a1$ | $\tau \xafQ\u226a\sigma \xaf$ | $\tau \xafQ\u223c\sigma \xaf$ | $\sigma \xaf\u226a\tau \xafQ\u226a\sigma \xaf2$ | $\tau \xafQ\u226b1$ | |

or | ||||||

$\tau \xafQ\u226a\sigma \xaf$ | ||||||

Dominant field | $\xi \xaf$ | $\xi \xaf$ and $Q\xaf$ | $Q\xaf$ | $Q\xaf$ | $Q\xaf$ | $\xi \xaf$ and $Q\xaf$ |

B diffusion _{x} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

$B\u2225$ diffusion | ✓ | ✓ | ✓ | |||

Hall decoupling | ✓ | ✓ | ✓ | ✓ | ||

∇⋅ v decoupling | ✓ | ✓ | ✓ | |||

No drift reference | 1 | 8 | 10 | 11 | 9 | 1 |

Drift reference | new | new | new | new | 9 | 6 |

Hall drift | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

∇p drift _{e} | ✓ | ✓ | ✓ | |||

∇⋅ v drift | ✓ | ✓ | ✓ | ✓ |

. | PR1a . | PR2 . | PR3 . | PR4 . | PR5 . | PR1b . |
---|---|---|---|---|---|---|

Regime | $\sigma \xaf2\u226a1$ | $\sigma \xaf2\u223c1$ | $\sigma \xaf2\u226b1$ | $\sigma \xaf2\u226b1$ | $\sigma \xaf2\u226b1$ | $\tau \xafQ\u226b\sigma \xaf2$ |

boundary | and | and | and | and | and | and |

$\tau \xafQ\u226a1$ | $\tau \xafQ\u226a1$ | $\tau \xafQ\u226a\sigma \xaf$ | $\tau \xafQ\u223c\sigma \xaf$ | $\sigma \xaf\u226a\tau \xafQ\u226a\sigma \xaf2$ | $\tau \xafQ\u226b1$ | |

or | ||||||

$\tau \xafQ\u226a\sigma \xaf$ | ||||||

Dominant field | $\xi \xaf$ | $\xi \xaf$ and $Q\xaf$ | $Q\xaf$ | $Q\xaf$ | $Q\xaf$ | $\xi \xaf$ and $Q\xaf$ |

B diffusion _{x} | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

$B\u2225$ diffusion | ✓ | ✓ | ✓ | |||

Hall decoupling | ✓ | ✓ | ✓ | ✓ | ||

∇⋅ v decoupling | ✓ | ✓ | ✓ | |||

No drift reference | 1 | 8 | 10 | 11 | 9 | 1 |

Drift reference | new | new | new | new | 9 | 6 |

Hall drift | ✓ | ✓ | ✓ | ✓ | ✓ | ✓ |

∇p drift _{e} | ✓ | ✓ | ✓ | |||

∇⋅ v drift | ✓ | ✓ | ✓ | ✓ |

### A. PR1a

We use PR1a as a notation for the upper left quadrant of Fig. 1 where

Examination of the system of tearing equations (Eqs. (51) and (52)) shows $Q\xaf\u226a\xi \xaf$. Thus, the electron equation (Eq. (52)) may be ignored and the governing equation is simply

The solution for $\xi \xaf$ can be expressed in terms of the parabolic cylinder function

as $\xi \xaf=R\xaf\u22121/4U(0,R\xaf\u22121/4x\xaf)$. Integrating Eq. (53), the drift dispersion relation is

where $\gamma \u0302MHD$ is the single-fluid growth rate without drift effects

### B. PR2

Regime PR2 is the transition at moderate *β* between the single-fluid regime, PR1, and the electron-MHD regime, PR3. Here, we assume $\Lambda \xaf\u223cx\xaf\u223c1$, $\xi \xaf\u223cQ\xaf$ and

Thus, the system of tearing equations becomes

and

Following the method outlined in Ref. 8 for the solution of a similar system of equations (where $R\xaf\u21921$ and $\Lambda \xaf\u21920$), we transform this system of equations into two independent parabolic cylinder equations

where $V\xafi=\xi \xaf+aiQ\xaf$ and *i* = 1, 2. This transformation requires

and

The solution for each $V\xafi$ is $V\xafi=\lambda i1/4CiU(0,\lambda i1/4x\xaf)$. Integrating Eq. (53) to find the dispersion relation gives

This may be expressed in a more explicit form as $D=2\Gamma (34)2f2(\sigma \xaf,R\xaf,\Lambda \xaf)$, where

The limits of this expression under the same approximations as PR1a and PR3 are consistent with the dispersion relations found in these regimes. Consider the limit where $\sigma \xaf2\u226a1$, in this case $f2(\sigma \xaf,R\xaf,\Lambda \xaf)\u2192(1+\Lambda \xafR\xaf/4)R\xaf1/4$. With the additional limit $\Lambda \xaf\u226a1$ (as is the case in PR1a), $f2(\sigma \xaf,R\xaf,\Lambda \xaf)\u2192R\xaf1/4$ and we recover Eq. (57). In the limit where $\sigma \xaf2\u226b1$, $f2(\sigma \xaf,R\xaf,\Lambda \xaf)\u2192\sigma \xaf\u22121/2$. As we shall see in Subsec. V C, this limit is the dispersion relation found in the electron-MHD regime, PR3.

### C. PR3

In the electron-MHD regime, the resistive diffusion of $B\u2225$ balances the Hall term in the parallel induction equation, and the parallel-vorticity equation is not needed. The orderings of this regime are a small tearing layer and large $B\u2225,\u2009x\xaf\u22121\u223c\sigma \xaf1/2\u223cQ\xaf$, small ion displacement, $\xi \xaf\u223c\sigma \xaf\u22123/2$, and large *d _{i}*, $\sigma \xaf2\u226b1$, such that $\Lambda \xaf\u226a\sigma \xaf2$, $\tau \xaf\xi \u226a\sigma \xaf3,\u2009\tau \xafB\u226a\sigma \xaf3/2$ and $\tau \xafQ\u226a\sigma \xaf$. After substituting the ordered electron equation, Eq. (52), into the ordered ion equation, Eq. (51), the governing equation in this regime is

The solution to this equation is $Q\xaf=\sigma \xafU(0,\sigma \xafx\xaf)$. Integrating Eq. (53), we find $D=2\Gamma (34)2\sigma \xaf\u22121/2$ (the limit of *D* from PR2 when $\sigma \xaf2\u226b1$) and the dispersion relation is then

In this regime, the growth rate scales as $di1/2S\u22121/2$ and the mode simply rotates at the electron drift frequency; there is no drift stabilization. This result is not particularly surprising, as the mode is mediated purely by the electron fluid through the induction equation. Contributions from ion compressibility, parallel ion flows, and ion vorticity do not play a role.

### D. PR4

The PR4 regime is the transition between the $B\u2225$-diffusion (PR3) and the semicollisional (PR5) regimes. The orderings of this regime are similar to PR3; a small tearing layer with a large $B\u2225$, $x\xaf\u22121\u223c\sigma \xaf1/2\u223cQ\xaf$, small ion displacement, $\xi \xaf\u223c\sigma \xaf\u22123/2$, however $\tau \xafQ$ is comparable to the normalized ion skin depth which is large, $\sigma \xaf2\u226b1$, such that $\Lambda \xaf\u226a\sigma \xaf2,\u2009\tau \xaf\xi \u226a\sigma \xaf3$, $\tau \xafB\u223c\sigma \xaf3/2$ and $\tau \xafQ\u223c\sigma \xaf$. Thus, the $\tau \xafQ$ and $\tau \xafB$ contributions must both be retained in Eq. (52), and the system of tearing equations becomes

and

These equations may be combined into a single non-homogeneous parabolic cylinder equation for $Q\xaf$

The solution to this equation up to a constant of integration, following the method outlined in Ref. 15, is

with the constraint

where $a=\tau \xafQ/2\sigma \xaf$. The constant of integration (either *A*_{+} or *A*_{−}) is found by matching the layer equations with the outer solution (in practice, requiring that the integral of Eq. (53) converges), which provides the additional condition $A+=\u2212i\sigma \xaf/2$. Integrating Eq. (53) determines the dispersion relation as

Drift effects modify the dispersion relation through the left side, in particular, the drift modified growth rate $\gamma \u0302e$ and the drift effects contained in $\tau \xafQ$ and $\sigma \xaf$. In the limit where $\tau \xafQ\u226a\sigma \xaf$, the left side of Eq. (75) becomes $\gamma \u0302e\Gamma (34)2/2\pi $, consistent with the dispersion relation of PR3. In the opposite limit, where $\tau \xafQ\u226b\sigma \xaf$ the left side of the equation becomes $\gamma \u0302e\tau \xafQ/2\sigma \xaf$, which is consistent with the dispersion relation found in Sec. V E for the semicollisional regime, PR5. Although $\tau \xafB$, which scales similarly in magnitude to $\tau \xafQ$, affects the eigenfunction, it does not modify the growth rate. In the limit of PR3, both $\tau \xafB$ and $\tau \xafQ$ are small and thus the results are consistent. In the limit of PR5, where $\tau \xafB$ is again expected to be large, $\tau \xafB$ contributes an even parity term to the eigenfunction and thus again does not contribute to the dispersion relation after integration of Eq. (53).

### E. PR5

The orderings in the semicollisional regime are similar to PR3 and PR4, with a large $B\u2225,\u2009x\xaf\u22121\u223c\sigma \xaf1/2\u223cQ\xaf$, and small ion displacement, $\xi \xaf\u223c\sigma \xaf\u22123/2$. However, in this regime the $\tau \xaf$ terms are larger than normalized ion skin depth (but not too large), $\sigma \xaf2\u226b1$, such that $\Lambda \xaf\u226a\sigma \xaf2,\u2009\tau \xaf\xi \u226a\sigma \xaf3$, and $\sigma \xaf\u226a\tau \xafQ\u223c\tau \xafB\u226a\sigma \xaf2$. The diffusion of $B\u2225$ may be neglected and the Hall term in Eq. (52) is balanced by the $\tau \xafQ$ and $\tau \xafB$ terms. After substitution of the ordered electron equation, Eq. (52), into the ordered ion equation, Eq. (51), the governing equation for this regime is

The solution is algebraic

The dispersion relation, found by integrating Eq. (53), is then

The growth rate scales as $\rho s2/3S\u22121/3$ and drift effects are contained on the left side of Eq. (78). The eigenfunction, $Q\xaf$, only contains even-parity contributions from $\tau \xafB$, which vanish during the integration of Eq. (53) and thus do not contribute to the dispersion relation.

This is the two-fluid drift-regime first described by Drake and Lee.^{9} Simplifying this expression further by assuming *β* ≪ 1 (which defines this regime), *σ _{pe}* =

*σ*= −

_{qi}*σ*= 1, we find

_{qe}When the electron temperature is much larger than the ion, *f _{Ti}* = 0 and

*f*= 1, the standard two-thirds, one-third dispersion relation is attained. Our results are not identical to the kinetic analysis of Drake and Lee. However, our present fluid model does not include ion and electron gyroviscosity or heat-flux contributions to the frictional force. The inclusion of cross heat flux cancels contributions from the pure density-gradient drifts in the dispersion relation, as with

_{Te}*σ*= 0 but

_{qα}*σ*= 1, one instead finds

_{pe}### F. PR1b

This regime is the low-*β*, drift limit of the single-fluid regime. With the corresponding orderings, $1\u226a\tau \xafQ\u223c\tau \xaf\xi $, $\sigma \xaf2\u226a\tau \xafQ\u223c\tau \xaf\xi $ and $\tau \xafB\u223c\Lambda \xaf\u223c1$, the $\tau \xafQ$ and $\tau \xaf\xi $ terms balance in the electron equation, Eq. (52), and thus $\xi \xaf\u223cQ\xaf$ as $\tau \xafQQ\xaf=\u2212\tau \xaf\xi \xi \xaf$. Substituting this balance into Eq. (51), the governing equation becomes

The solution is $\xi \xaf=R\xaf\u22121/4M\u22123/4U(0,R\xaf\u22121/4M1/4x\xaf)$, where $M=1\u2212\tau \xaf\xi /\tau \xafQ$. Assuming $\beta \u226a1$, and thus $M\u2243\gamma \u0302e/\gamma \u0302i$, integration of Eq. (53) gives the dispersion relation

With an exact gyroviscous cancellation, $\gamma \u0302gvi=\gamma \u0302E\xd7B$, this is the standard drift-tearing dispersion relation as found by Coppi.^{6}

## VI. CONCLUSION

This work is an analytic investigation of the collisional, constant-*ψ*, drift-tearing mode with an unreduced, extended-MHD model. Our new analytic results describe the experimentally relevant, moderate-*β* portion of the drift-tearing phase space. We emphasize that our definition of moderate *β* encompasses the values that are pertinent for a fusion reactor (*β* ∼ 1%−25%). Our results cannot be directly applied to tokamak discharges, as we do not retain the effects of ion gyroviscosity and plasma shaping and curvature. Instead, the ultimate benefits of this work are to map the drift-tearing mode behaviour through the different regimes and to provide an analytic result against which extended-MHD, boundary-layer-dynamics computations can be verified. As the tearing-layer dynamics result from the balance of otherwise small terms, this verification is a novel way to test an extended-MHD implementation.

## ACKNOWLEDGMENTS

The authors would like to thank Carl Sovinec and Chris Hegna for stimulating discussions and Carl Sovinec for comments on a draft of the text. This material is based on work supported by U.S. Department of Energy, Office of Science, Office of Fusion Energy Sciences under Award Nos. DE-FC02-06ER54875 and DE-FG02-08ER54972.