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.

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 instability1 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 drift6 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 (di), 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 β104 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 di (thus moderate values of the ion gyroradius, ρiβdi), the tearing dispersion relation from electron-MHD10 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.

FIG. 1.

Tearing mode parameter space in terms of normalized β, τ¯Q1, and ion skin depth, σ¯, (as originally defined in Ref. 8). The diagram uses a small parameter value of 0.04 to determine regime boundaries.

FIG. 1.

Tearing mode parameter space in terms of normalized β, τ¯Q1, and ion skin depth, σ¯, (as originally defined in Ref. 8). The diagram uses a small parameter value of 0.04 to determine regime boundaries.

Close modal

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.

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

nt=·nv,
(1)

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

mindvdt=J×Bp·Πi,
(2)

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

nΓ1dαTαdt=pα·vα·qα,
(3)

for the plasma temperature (Tα). The subscript indicates either the ion or electron species, mα is a species' mass, and Γ is the adiabatic index. The plasma is assumed to be an ideal gas and thus the species pressure (pα; p=pα) is given by the ideal-gas law, pα = nTα. As appropriate for low-frequency plasma dynamics, we assume quasi-neutrality (ne ≃ ni for an ion charge state of unity) and drop the displacement-current term in Ampere's law (μ0J = ∇×B, where μ0 is permeability of free space), which provides a relation between the magnetic field (B) and the current density (J = ne(vi − ve) 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)

E=v×B+J×Bnepene·Πene+ηJmeedevedt,
(4)

commonly referred to as the generalized Ohm's law (me is the electron mass and η is the electrical resistivity caused by electron-ion collisions). Faraday's law (B/t=×E) 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 (ve = vi − J/ne) instead of bulk-flow advection from the v × B term.

These equations require closure expressions for the stress tensors (Πα) and heat fluxes (qα). We use the Braginskii-like3–5 “cross” terms (first-order FLR terms) as the closure: gyroviscosity

Πα=mαpα4qαB[ b̂×Wα·(I+3b̂b̂)(I+3b̂b̂)·Wα×b̂ ],
(5)

and cross-heat flux

q=5pα2qαB0b̂×Tα,
(6)

where qα is a species' charge. The rate-of-strain tensor (Wα) is defined as Wα=vα+vαT(2/3)I·vα. 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=Γ(Ti+Te)/mi, which for comparable species' temperatures is on the same order as the ion thermal speed, vTα=Tα/mα. The ion gyroviscous term then scales as ρi/L relative to the ∇p term in the momentum equation, Eq. (2), whereas the electron gyroviscous term scales as me/mi(ρe/L) relative to the ∇pe term in the generalized Ohm's law, Eq. (4). Here, ρα=vTα/ωcα is the gyroradius, where ωcα=qαB/mα is the gyrofrequency and 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, ρi/LO(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 pα·vα in Eq. (3). The ion cross-heat flux scales as ρi/L, but the electron cross-heat flux scales as mi/me(ρe/L). Thus, if the ion cross-heat flux is significant (ρi/LO(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 ẑ-oriented current sheet in a Cartesian slab. There are distant conducting walls at x=±, the ŷ and ẑ directions are infinite, and the ẑ 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 x̂ is a radial (flux) coordinate, ŷ is approximately a cross-field coordinate and ẑ 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)+B̃(x)exp(iky+γt). Here, k=kŷ is the perturbation wavenumber and γ is the complex growth rate. The radial (x̂) component of all background vector fields is zero. Perturbation vector fields and wavenumber use a magnetic-coordinate system where the ŷ and ẑ 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 (Ls) is comparable to the inverse wavelength, kLsO(1); (2) a moderately large guide- to shear-magnetic-field ratio, such that ϵB=Bz(x=0)/By(x=)O(ϵ3/4); (3) a small tearing skin depth (δ), kxkδO(ϵ); (4) slow dynamics, ωτAO(ϵ3/2); and (5) slowly varying profiles within the layer, e.g. δn0/n0O(ϵ). Here, ϵ is a small parameter (ϵ1) and τA1=kvA=kB0/min0μ0 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

meedevedtmee(ve0·+t)ṽemenee2J̃t.
(7)

The relative magnitude of resistivity compared to electron inertia classifies the tearing mode as collisionless (de2γη/μ0) or collisional (de2γη/μ0), where dα is a species' skin depth (mα/μ0ne2). 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-di 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 (γ2k2cs2). 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 (ω̂=ωτA, L̂=kL,v̂=v/vA(x=0),B̂=B/B0(x=0), n̂=n/n0(x=0), and p̂=p/vA2min0(x=0)) and the overbar which is a tearing specific normalization introduced in Sec. IV.

Following convention, we define ξ̂=γ̂v̂x as the displacement vector and

Q̂=k̂2B̂ik̂Bx̂+iλ̂0B̂x,
(8)

consistent with Ref. 8, where λ=μ0J·B/B2 and k̂=k·B0/kB0. After linearization and applying the assumptions of Sec. II, the radial induction equation becomes

γ̂eB̂x=ik̂γ̂ξ̂+k̂d̂iQ̂+S1Bx̂,
(9)

where the Lundquist number is defined as S=vAμ0/kη. The left side of this equation is a term representing the rate-of-change of B̂x. The notation γ̂i=γ̂+ik̂·v̂0, and γ̂e=γ̂+ik̂·v̂e0γ̂iiω̂* 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 ∇pe term vanish as

x̂·×pene=x̂·[(kBTene)×n]
(10)

and the equilibrium gradients of Te and n are oriented in the x̂ direction. Other than ignoring flow shear and applying our ordering to the resistive term, Eq. (9) is exact.

The location where k⋅ B0 = 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)

Δ=limϵ0B̃x(x)|x=ϵx=ϵB̃x(0),
(11)

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 Δ̂O(1) (equivalent to kLsO(1) in a slab) and thus Bx̂B̂x, as follows from Eq. (11). Expanding B̂x at x = 0,

B̂x=B̂x(0)+Bx̂(0)x̂+...,
(12)

and noting that x̂O(ϵ) allows us to treat B̂x 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 ϵ1, e.g., ϵ2ξ̂ξ̂ and ϵBx̂B̂x. 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

γ̂eB̂=̂·v̂+ω̂*γ̂ξ̂d̂i(iω̂*+iω̂*nΓĉs2)Q̂+k̂d̂i[Bx̂+ik̂B̂B̂x]iω̂*in̂iω̂*nΓĉs2p̂e+S1B̂,
(13)

where ω*α is a species diamagnetic-drift frequency (kpα0/n0eB0), ω* is the total diamagnetic-drift frequency (ω*i+ω*e), ω*n is the density-gradient drift (kT0n0/n0eB0), and =ikb̂. The first two pairs of terms on the right side are the contributions from the v × B and Hall terms, respectively. The terms involving n̂ and p̂e result from the ∇pe term and the last term is the effect of resistivity.

The components of the linearized momentum equation are

γ̂iγ̂ξ̂=ω̂*d̂iB̂+ik̂B̂xB̂p̂(̂·Π̂)x,
(14)
γ̂iv̂=iQ̂ip̂(̂·Π̂),
(15)

and

γ̂iv̂=ω̂*d̂iB̂xik̂p̂(̂·Π̂).
(16)

The perpendicular and parallel components (Eqs. (15) and (16)) are used to construct an expression for ̂·v̂. 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

γ̂in̂=ω̂*nΓĉs2γ̂ξ̂d̂î·v̂,
(17)
γ̂ip̂i=ω̂*iγ̂ξ̂d̂iĉsi2̂·v̂(Γ1)̂·q̂i,
(18)

and

γ̂pep̂e=ω̂*eγ̂ξ̂d̂iĉse2̂·v̂+σpe(iω*eΓfTeiω*n)(Q̂iλ̂0B̂x)σpeĉse2(iω̂*+ik̂λ̂0d̂i)n̂(Γ1)̂·q̂e,
(19)

respectively. To allow for a systematic study of the effect of different advective models, we introduce the σpe and γ̂pe notation. If the advective term uses the bulk flow then γ̂pe=γ̂i and σpe = 0, whereas advection by the electron flow leads to γ̂pe=γ̂e and σpe = 1. To compute the linearized cross heat-flux contributions we first expand the heat-flux vector as

·qα=·[5pα2qαBb̂×Tα]=5pα2nqαB2[μ0J·(pαnnpα)2λB·(pαnnpα)]+5pαnqαB4[pαn(B×n)B×pα]·B·B+5pα2n2qαB2B·(pα×n).
(20)

Noting that J0·f0,B0·f0, B0·B0, and f0×g0 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 ω̂*k̂λ̂0d̂i), we find

(Γ1)̂·q̂α=iω̂*q1αĉsα2Γn̂iω̂*qαp̂α(γ̂αiω̂*qα)ĉsα2Cqα(Q̂iλ̂0B̂x+2ik̂Bx̂),
(21)

where

Cqα=σqαiω̂*αfTαiω̂*nγ̂αiω̂*qα,
(22)
iω̂*q1α=σqα(Γiω̂*α+ĉsα2iω̂*),
(23)

and

iω̂*qα=σqαfTα(Γiω̂*n+ĉs2iω̂*).
(24)

Again we introduce σ as a marker with value σqi=σqe=(5/2)(Γ1)/Γ when the cross heat flux is included in the model and σqα=0 when it is not. Equations (18), (19), and (21) may be combined to produce expressions for p̂=p̂i+p̂e and p̂e. Thus

p̂=Etγ̂ξ̂d̂iĉsp2γ̂î·v̂+(Cpe+ĉsq2)Q̂(Cpe+ĉsq2)iλ̂0B̂x+2ĉsq2ik̂Bx̂,
(25)

and

p̂e=Eeγ̂ξ̂d̂iĉspe2γ̂î·v̂+(Cpe+ĉsqe2)Q̂(Cpe+ĉsqe2)iλ̂0B̂x+2ĉsqe2ik̂Bx̂,
(26)

where ĉsqe2=Cqeĉse2,ĉsq2=Cqeĉse2+Cqiĉsi2, ĉsp2=ĉspe2+ĉspi2,

ĉspi2=ĉsi2γ̂iiω̂*q1i/Γγ̂iiω̂*qi,
(27)
ĉspe2=ĉse2γ̂peiω̂*q1e/Γγ̂peiω̂*qe,
(28)
Cpe=σpeiω̂*eΓfTeiω̂*nγ̂peiω̂*qe,
(29)
Ei=(γ̂iiω̂*qi)1(ω̂*ifTiiω̂*nω̂*q1iγ̂i),
(30)
Ee=(γ̂peiω̂*qe)1(ω̂*efTeiω̂*nω̂*q1eγ̂iω̂*nfTeΓσpeiω̂*γ̂i),
(31)

and Et = Ei + Ee.

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 B̂x,̂·v̂, Q̂,ξ̂, and v̂ 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 Q̂ after n̂ and p̂ are eliminate. And two new equations are derived: an expression for ̂·v̂ and a parallel vorticity equation which governs ξ̂.

Equations (15) and (16) are combined to provide an expression for ̂·v̂,

p̂=γ̂î·v̂γ̂iγ̂ξ̂Q̂+ik̂ω̂*d̂iB̂r+i(̂·Π̂gv)+ik̂(̂·Π̂gv).
(32)

After multiplying by γ̂i and substituting Eq. (25) for p̂,

ĉsp2̂·v̂=γ̂i2γ̂ξ̂γ̂iEtγ̂ξ̂d̂i+γ̂i(1+Cpe+ĉsq2)Q̂γ̂i(k̂ω̂*d̂iλ̂0+Cpe+ĉsq2)iλ̂0B̂x+2γ̂iĉsq2ik̂Bx̂iγ̂i[ (̂·Π̂gv)+k̂(̂·Π̂gv) ].
(33)

The inertial contributions (γ̂i2̂·v̂) are dropped as they are small compared to the ĉsp2̂·v̂ term from p̂ in Eq. (25). Without drift and FLR effects only the first and third terms on the right side contribute to ̂·v̂. The second term on the right side is a drift-like term from ṽ·p and ṽ·n and the remaining terms are contributions from electron advection (∼Cpe), cross heat flux (csq2), and ion gyroviscosity.

After eliminating B̂,n̂ and p̂ from the parallel induction equation, Eq. (13), we find

(γ̂iiω̂*)Q̂=(A1)̂·v̂+ik̂v̂+k̂d̂iBx̂+(ω̂*+iω̂*nΓĉs2En)γ̂ξ̂d̂i[iω̂*+iω̂*nΓĉs2(1+Cpe+ĉsqe2)]Q̂+iω̂*nΓĉs2(Cpe+ĉsqe2)iλ̂0B̂x+S1Q̂,
(34)

where

A=iω̂*iγ̂i+Γĉspe2ĉs2iω̂*nγ̂i,
(35)

and

En=Ee+ω̂*iγ̂i.
(36)

Without drift effects, all contributions from ∇pe 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, En, Cpe, and ĉsq2 factors in Eq. (34).

The only unused equation from our original system of eight is the radial momentum equation, Eq. (14). To find an expression for p̂, we take the derivative of Eq. (32)

p̂=γ̂iγ̂ξ̂ω̂*nΓĉs2γ̂iγ̂ξ̂d̂i+ω̂*d̂iQ̂Q̂+ik̂ω̂*d̂iBx̂+(ω̂*d̂i+ω̂*2k̂λ̂0d̂i2)iλ̂0B̂x+i(̂·Π̂gv)+ik̂(̂·Π̂gv)+iλ̂0(̂·Π̂gv).
(37)

Again, we ignore the inertial term (γ̂i2̂·v̂). Substituting into Eq. (14) and applying the tearing ordering

γ̂iγ̂ξ̂=2k̂λ̂0Q̂ik̂Bx̂+ω̂*nΓĉs2γ̂iγ̂ξ̂d̂i2iω̂*d̂iλ̂0B̂x(̂·Π̂)ri(̂·Π̂gv)ik̂(̂·Π̂gv)iλ̂0(̂·Π̂gv).
(38)

Without drift and FLR effects, this equation becomes the standard form of the parallel vorticity equation, γ̂iγ̂ξ̂ik̂Bx̂.

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.

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

ĉsp2̂·v̂=γ̂iEtγ̂ξ̂d̂i+γ̂i(1+Cpe+ĉsq2)Q̂γ̂i(k̂ω̂*d̂iλ̂0+Cpe+ĉsq2)iλ̂0B̂x+2γ̂iĉsq2ik̂Bx̂+γ̂iγ̂gviγ̂ξ̂σgviγ̂i2ĉsi2Γd̂iγ̂ξ̂,
(39)

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

γ̂gviv̂=ω̂*d̂iB̂x+ik̂ĉsp2γ̂î·v̂ik̂γ̂i(Cpe+ĉsq2)Q̂+ik̂Etγ̂ξ̂d̂iσgvĉsi2Γλ̂0d̂iγ̂ξ̂σgvĉsi2Γd̂ik̂γ̂ξ̂,
(40)

where σgv is a marker for ion gyroviscosity (set to unity when gyroviscosity is included and otherwise zero), the modified ion-gyroviscous growth rate is

γ̂gvi=γ̂E×B+iω̂*iσgv(iω̂*iiω̂*ĉsi2Γ),
(41)

and γ̂E×B is the Doppler-shifted growth rate. The tearing-ordered ion-gyroviscous contributions to parallel-vorticity equation (Eq. (38)) are

(̂·Π̂)ri(̂·Π̂gv)ik̂(̂·Π̂gv)iλ̂0(̂·Π̂gv)=iω̂*ω̂*id̂î·v̂2iω̂*(ω̂*i+ω̂*ĉsi2Γ)γ̂ξ̂d̂iiĉsi2Γd̂i((̂·v̂)+ik̂v̂)(iω̂*i+iω̂*ĉsi2Γ)γ̂ξ̂.
(42)

The iω̂*iγ̂ξ̂ 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 iω̂*ĉsi2γ̂ξ̂/Γ 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 configuration7 (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

γ̂gviγ̂ξ̂=2k̂λ̂0Q̂ik̂Bx̂2iω̂*d̂iλ̂0B̂xiσgvĉsi2Γd̂i((̂·v̂)+ik̂v̂).
(43)

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 γ̂gvi terms as is in order to facilitate comparison. The two relevant limits are then without gyroviscosity (γ̂gviγ̂i), and with the exact gyroviscous cancellation (γ̂gviγ̂E×B).

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

τ̂QQ̂=k̂d̂iBx̂+Sg1Q̂k̂2γ̂gviQ̂+(τ̂Bk̂ω̂*γ̂gvid̂iλ̂0)iλ̂0B̂x+τ̂ξγ̂ξ̂d̂i,
(44)

where

τ̂Q=γ̂i+iω̂*nΓĉs2(1+Cpe+ĉsqe2)γ̂iĉsp2(1+Cpe+ĉsq2)(A1),
(45)
τ̂B=iω̂*nΓĉs2(Cpe+ĉsqe2)+γ̂i(Cpe+ĉsq2)(A1)ĉsp2,
(46)

and

τ̂ξ=ω̂*+iω̂*nΓĉs2En(A1)ĉsp2γ̂iEt.
(47)

Equations (9), (43) (with σgv = 0), and (44) now comprise our system of equations for B̂x,Q̂, and ξ̂. 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, ∇pe contributions, inertia and the v × B term. Compressibility and parallel flows contribute the k̂2 and ω̂* terms on the right side of Eq. (44) as well as the γ̂i/ĉsp2 terms in the τ¯ factors. The ∇pe term in Ohm's law contributes the ω̂*n/ĉs2 terms in the τ¯ factors.

With the constant-ψ approximation, where B̂x is assumed constant within the small tearing layer, Eq. (43) is used to eliminate Bx̂; which results in a system of two coupled equations for Q̂ and ξ̂. We use a tearing normalization for these equations similar to Ref. 8 with the dimensionless variables

x¯=x̂d̂0,ξ¯=ik̂d̂0γ̂ξ̂B̂r(0)γ̂e,Q¯=k̂d̂0d̂iQ̂B̂r(0)γ̂e,
(48)

and the dimensionless parameters

d̂0=(γ̂E×Bk̂2S)1/4,σ¯2=γ̂E×B2d̂i2k̂2d̂04=γ̂E×Bd̂i2S,R¯=γ̂gviγ̂E×B,Λ¯=iω̂*γ̂eγ̂E×Bγ̂gvi,
(49)
τ¯Q=γ̂E×B(k̂d̂0)2τ̂Q,τ¯ξ=iγ̂E×B(k̂d̂0)2τ̂ξ,andτ¯B=iγ̂E×Bd̂iγ̂ed̂0(τ̂B+2iω̂*).
(50)

With this normalization, σ¯ is the ion skin depth, di, normalized to the tearing skin depth, δ=(Sγ̂)1/2. Validity of a first-order FLR model requires ρi < δ. A good rule of thumb for plasmas with comparable ion and electron temperatures is to use the ion sound gyroradius, ρs = cs/ωci, and require ρs/δ=ĉsσ¯=βσ¯<1. After expanding k and retaining only the leading order term in x, kx,

R¯2ξ¯x¯2=x¯2(ξ¯+Q¯)x¯,
(51)

and

2Q¯x¯2=(R¯1x¯2+τ¯Q)Q¯+R¯σ¯22ξ¯x¯2+τ¯ξξ¯τ¯B+Λ¯x¯
(52)

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¯2Q¯) 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 (k̂d̂iBx̂). The τ¯ 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 ω̂*n) and perpendicular compressibility (otherwise). There are other contributions to τ¯Q and τ¯ξ 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).

Once solutions for Q¯ and ξ¯ are found, the dispersion relation may be computed by integrating the radial induction equation (Eq. (9)) and applying the boundary condition B̃r(±)=0. The resulting equation is

D=dx¯(1x¯ξ¯x¯Q¯)=k̂1/2Δ̂γ̂eγ̂E×B1/4S3/4,
(53)

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 τ¯Q1 (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 τ¯Qσ¯2 (near PR6) where we recover the drift-tearing result of Ref. 6.

TABLE I.

A summary of the parametric regime boundaries, significant terms and fields, and prior references if applicable.

PR1aPR2PR3PR4PR5PR1b
Regime σ¯21 σ¯21 σ¯21 σ¯21 σ¯21 τ¯Qσ¯2 
boundary and and and and and and 
 τ¯Q1 τ¯Q1 τ¯Qσ¯ τ¯Qσ¯ σ¯τ¯Qσ¯2 τ¯Q1 
  or     
  τ¯Qσ¯     
Dominant field ξ¯ ξ¯ and Q¯ Q¯ Q¯ Q¯ ξ¯ and Q¯ 
Bx diffusion ✓ ✓ ✓ ✓ ✓ ✓ 
B diffusion  ✓ ✓ ✓   
Hall decoupling  ✓ ✓ ✓ ✓  
∇⋅ v decoupling  ✓  ✓ ✓  
No drift reference 1  8  10  11  9  1  
Drift reference new new new new 9  6  
Hall drift ✓ ✓ ✓ ✓ ✓ ✓ 
pe drift    ✓ ✓ ✓ 
∇⋅ v drift  ✓  ✓ ✓ ✓ 
PR1aPR2PR3PR4PR5PR1b
Regime σ¯21 σ¯21 σ¯21 σ¯21 σ¯21 τ¯Qσ¯2 
boundary and and and and and and 
 τ¯Q1 τ¯Q1 τ¯Qσ¯ τ¯Qσ¯ σ¯τ¯Qσ¯2 τ¯Q1 
  or     
  τ¯Qσ¯     
Dominant field ξ¯ ξ¯ and Q¯ Q¯ Q¯ Q¯ ξ¯ and Q¯ 
Bx diffusion ✓ ✓ ✓ ✓ ✓ ✓ 
B diffusion  ✓ ✓ ✓   
Hall decoupling  ✓ ✓ ✓ ✓  
∇⋅ v decoupling  ✓  ✓ ✓  
No drift reference 1  8  10  11  9  1  
Drift reference new new new new 9  6  
Hall drift ✓ ✓ ✓ ✓ ✓ ✓ 
pe drift    ✓ ✓ ✓ 
∇⋅ v drift  ✓  ✓ ✓ ✓ 

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

τ¯Q,τ¯ξ,τ¯B,σ¯2,Λ¯1x¯ξ¯.
(54)

Examination of the system of tearing equations (Eqs. (51) and (52)) shows Q¯ξ¯. Thus, the electron equation (Eq. (52)) may be ignored and the governing equation is simply

R¯ξ¯=x¯2ξ¯x¯.
(55)

The solution for ξ¯ can be expressed in terms of the parabolic cylinder function

U(0,x¯)=x¯201dμ(1μ2)1/4exp[μx¯22],
(56)

as ξ¯=R¯1/4U(0,R¯1/4x¯). Integrating Eq. (53), the drift dispersion relation is

γ̂eγ̂gvi1/4=γ̂MHD5/4,
(57)

where γ̂MHD is the single-fluid growth rate without drift effects

γ̂MHD=S3/5(Δ̂2Γ(34)2)4/5k̂2/5.
(58)

Regime PR2 is the transition at moderate β between the single-fluid regime, PR1, and the electron-MHD regime, PR3. Here, we assume Λ¯x¯1, ξ¯Q¯ and

τ¯Q,τ¯ξ,τ¯B1orτ¯Q,τ¯ξ,τ¯Bσ¯.
(59)

Thus, the system of tearing equations becomes

Q¯=R¯1x¯2Q¯+R¯σ¯2ξ¯+Λ¯x¯,
(60)

and

R¯ξ¯=x¯2(Q¯+ξ¯)x¯.
(61)

Following the method outlined in Ref. 8 for the solution of a similar system of equations (where R¯1 and Λ¯0), we transform this system of equations into two independent parabolic cylinder equations

λi1V¯i=x¯2V¯iCix¯,
(62)

where V¯i=ξ¯+aiQ¯ and i = 1, 2. This transformation requires

λi=R¯1+σ¯2ai,
(63)
ai=12±121+4R¯σ¯2,
(64)

and

Ci=1Λ¯R¯(ai1).
(65)

The solution for each V¯i is V¯i=λi1/4CiU(0,λi1/4x¯). Integrating Eq. (53) to find the dispersion relation gives

D=2πΓ(34)Γ(14)[C1a1λ114C2a2λ214a1a2].
(66)

This may be expressed in a more explicit form as D=2Γ(34)2f2(σ¯,R¯,Λ¯), where

f2(σ¯,R¯,Λ¯)=i=1,212[1Λ¯R¯2((1)i1+4R¯σ¯21)][1+(1)i(1+4R¯σ¯2)12]×[R¯1+σ¯22+(1)iσ¯σ¯24+R¯1]14.
(67)

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 σ¯21, in this case f2(σ¯,R¯,Λ¯)(1+Λ¯R¯/4)R¯1/4. With the additional limit Λ¯1 (as is the case in PR1a), f2(σ¯,R¯,Λ¯)R¯1/4 and we recover Eq. (57). In the limit where σ¯21, f2(σ¯,R¯,Λ¯)σ¯1/2. As we shall see in Subsec. V C, this limit is the dispersion relation found in the electron-MHD regime, PR3.

In the electron-MHD regime, the resistive diffusion of B 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,x¯1σ¯1/2Q¯, small ion displacement, ξ¯σ¯3/2, and large di, σ¯21, such that Λ¯σ¯2, τ¯ξσ¯3,τ¯Bσ¯3/2 and τ¯Qσ¯. After substituting the ordered electron equation, Eq. (52), into the ordered ion equation, Eq. (51), the governing equation in this regime is

Q¯=σ¯2x¯2Q¯σ¯2x¯.
(68)

The solution to this equation is Q¯=σ¯U(0,σ¯x¯). Integrating Eq. (53), we find D=2Γ(34)2σ¯1/2 (the limit of D from PR2 when σ¯21) and the dispersion relation is then

γ̂e=S1/2(d̂ik̂)1/2Δ̂2Γ(34)2.
(69)

In this regime, the growth rate scales as di1/2S1/2 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.

The PR4 regime is the transition between the B-diffusion (PR3) and the semicollisional (PR5) regimes. The orderings of this regime are similar to PR3; a small tearing layer with a large B, x¯1σ¯1/2Q¯, small ion displacement, ξ¯σ¯3/2, however τ¯Q is comparable to the normalized ion skin depth which is large, σ¯21, such that Λ¯σ¯2,τ¯ξσ¯3, τ¯Bσ¯3/2 and τ¯Qσ¯. Thus, the τ¯Q and τ¯B contributions must both be retained in Eq. (52), and the system of tearing equations becomes

Q¯=τ¯QQ¯+R¯σ¯2ξ¯τ¯B,
(70)

and

R¯σ¯2ξ¯=σ¯2x¯2Q¯σ¯2x¯.
(71)

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

Q¯=τ¯QQ¯+σ¯2x¯2Q¯σ¯2x¯τ¯B.
(72)

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

Q¯(x¯)=A+0exp(ikx¯σ¯2)U(a,k)U(a,0)dk+A0exp(ikx¯σ¯2)U(a,k)U(a,0)dk,
(73)

with the constraint

A+A=iσ¯3/22+τ¯B2σ¯U(a,0)U(a,0),
(74)

where a=τ¯Q/2σ¯. 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+=iσ¯/2. Integrating Eq. (53) determines the dispersion relation as

γ̂eΓ[(3+τ¯Q/σ¯)/4]Γ[(1+τ¯Q/σ¯)/4]=S1/2k̂d̂iΔ̂2π.
(75)

Drift effects modify the dispersion relation through the left side, in particular, the drift modified growth rate γ̂e and the drift effects contained in τ¯Q and σ¯. In the limit where τ¯Qσ¯, the left side of Eq. (75) becomes γ̂eΓ(34)2/2π, consistent with the dispersion relation of PR3. In the opposite limit, where τ¯Qσ¯ the left side of the equation becomes γ̂eτ¯Q/2σ¯, which is consistent with the dispersion relation found in Sec. V E for the semicollisional regime, PR5. Although τ¯B, which scales similarly in magnitude to τ¯Q, affects the eigenfunction, it does not modify the growth rate. In the limit of PR3, both τ¯B and τ¯Q are small and thus the results are consistent. In the limit of PR5, where τ¯B is again expected to be large, τ¯B contributes an even parity term to the eigenfunction and thus again does not contribute to the dispersion relation after integration of Eq. (53).

The orderings in the semicollisional regime are similar to PR3 and PR4, with a large B,x¯1σ¯1/2Q¯, and small ion displacement, ξ¯σ¯3/2. However, in this regime the τ¯ terms are larger than normalized ion skin depth (but not too large), σ¯21, such that Λ¯σ¯2,τ¯ξσ¯3, and σ¯τ¯Qτ¯Bσ¯2. The diffusion of B may be neglected and the Hall term in Eq. (52) is balanced by the τ¯Q and τ¯B terms. After substitution of the ordered electron equation, Eq. (52), into the ordered ion equation, Eq. (51), the governing equation for this regime is

0=τ¯QQ¯+σ¯2x¯2Q¯σ¯2x¯τ¯B.
(76)

The solution is algebraic

Q¯=σ¯2x¯+τ¯Bσ¯2x¯2+τ¯Q.
(77)

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

γ̂eτ̂Q1/2=S1/2k̂Δ̂πd̂i.
(78)

The growth rate scales as ρs2/3S1/3 and drift effects are contained on the left side of Eq. (78). The eigenfunction, Q¯, only contains even-parity contributions from τ¯B, 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), σpe = σqi= −σqe = 1, we find

γ̂e2/3(γ̂E×Bγ̂ifTiγ̂e+fTeγ̂i)1/3=Sg1/3(k̂Δ̂πĉsd̂i)2/3.
(79)

When the electron temperature is much larger than the ion, fTi = 0 and fTe = 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 σ = 0 but σpe = 1, one instead finds

γ̂e2/3((γ̂E×BfTeΓiω̂*n)(γ̂E×B+fTiΓiω̂*n)γ̂e)1/3=Sg1/3(k̂Δ̂πĉsd̂i)2/3.
(80)

This regime is the low-β, drift limit of the single-fluid regime. With the corresponding orderings, 1τ¯Qτ¯ξ, σ¯2τ¯Qτ¯ξ and τ¯BΛ¯1, the τ¯Q and τ¯ξ terms balance in the electron equation, Eq. (52), and thus ξ¯Q¯ as τ¯QQ¯=τ¯ξξ¯. Substituting this balance into Eq. (51), the governing equation becomes

R¯ξ¯=x¯2(1τ¯ξτ¯Q)ξ¯x¯.
(81)

The solution is ξ¯=R¯1/4M3/4U(0,R¯1/4M1/4x¯), where M=1τ¯ξ/τ¯Q. Assuming β1, and thus Mγ̂e/γ̂i, integration of Eq. (53) gives the dispersion relation

γ̂MHD5/4=γ̂e3/4γ̂i1/4γ̂gvi1/4.
(82)

With an exact gyroviscous cancellation, γ̂gvi=γ̂E×B, this is the standard drift-tearing dispersion relation as found by Coppi.6 

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.

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.

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