Calculations in the theory of tearing instability
aa r X i v : . [ phy s i c s . p l a s m - ph ] S e p Calculations in the theory of tearing instability
Stanislav Boldyrev , and Nuno F. Loureiro Department of Physics, University of Wisconsin, Madison, WI 53706, USA Space Science Institute, Boulder, Colorado 80301, USA Plasma Science and Fusion Center, Massachusetts Institute of Technology, Cambridge MA02139, USAE-mail: [email protected]
Abstract.
Recent studies have suggested that the tearing instability may play a significantrole in magnetic turbulence. In this work we review the theory of the magnetohydrodynamictearing instability in the general case of an arbitrary tearing parameter, which is relevant forapplications in turbulence. We discuss a detailed derivation of the results for the standard Harrisprofile and accompany it by the derivation of the results for a lesser known sine-shaped profile.We devote special attention to the exact solution of the inner equation, which is the centralresult in the theory of tearing instability. We also briefly discuss the influence of shear flows ontearing instability in magnetic structures. Our presentation is self-contained; we expect it to beaccessible to researchers in plasma turbulence who are not experts in magnetic reconnection.
1. Introduction
Numerical simulations and analytic models have suggested that magnetic plasma turbulencetends to form anisotropic, sheet-like current structures at small scales [1–10]. Such structures arenot necessarily associated with the dissipation scale of turbulence. Rather, a hierarchy of sheet-like turbulent eddies is formed throughout the whole inertial interval [11–15]. Recently, it hasbeen realized that given large enough Reynolds number, such anisotropic structures may becomeunstable to the tearing mode at scales well above the Kolmogorov-like dissipation scale [16–21].The Reynolds numbers for which such effects become significant are very large ( Re & ), sotheir definitive study is beyond the capabilities of modern computers. Nevertheless, the rapidprogress in in situ measurements of space plasma brings interest to small scales of magneticturbulence, where such effects may be observed, e.g., [22]. It is, therefore, highly desirable todevelop an understanding of the linear tearing theory in magnetic profiles such as those onemight expect to find throughout the inertial range of turbulence, but not necessarily thoseassociated with dissipative current sheets.This brings attention to the two facets of the theory of tearing instability that are nottraditionally covered in textbooks on magnetohydrodynamics or plasma physics. One is thetheory of reconnection beyond the well-known Furth, Killeen & Rosenbluth regime of smalltearing parameter [23]. This regime assumes limited anisotropy of a reconnecting magneticprofile, so it is not applicable to very anisotropic tearing modes relevant for our study. Theother is the theory of tearing instability for the magnetic profiles that are different from thecanonical tanh-like Harris profile [24]. Such a profile assumes that the reconnecting magneticfield is uniform in space except for the region where it reverses its direction. This is, arguably, nota general situation encountered in turbulence, where the magnetic fields strength varies in spacen similar scales both inside and outside the reconnection region. Different magnetic profilesmay lead to different scalings of the corresponding tearing growth rates, e.g., [18, 19, 25–27]. Tothe best of our knowledge, there are currently no texts methodically covering these aspects oftearing instability. Rather, various relevant analytical results are scattered over the literature,e.g., [28–32].In this work we review the derivation of the standard Harris-profile tearing mode, andaccompany it by a parallel derivation of the results for the sine-shaped profile. We devote specialattention to the discussion of the exact solution of the inner equation, for which we use a methoddifferent from those previously adopted in the literature [28–32]. Although our work is mostlydevoted to the tearing instability in magnetic profiles not accompanied by velocity fields, in theend of our presentation we discuss to what extent shear flows, typically present within turbulenteddies, can modify the results. The goal of our work is to give a self-contained presentation ofthe theory of tearing effects that are most relevant for applications to turbulence. We believe itwill be useful for researchers in turbulence who are not necessarily experts in reconnection.
2. Equations for the tearing mode
We assume that the background uniform magnetic field is in the z direction, and the currentsheet thickness, a , and length, l , are measured in the field-perpendicular plane. The currentsheets are strongly anisotropic, a ≪ l . We denote the reconnecting magnetic field, that is, thevariation of the magnetic field across the current sheet, as B . Such structures can be createdin turbulence if their life times are comparable to the Alfv´enic time τ A ∼ l/V A , where V A isthe Alfv´en speed associated with B . They are formed at all scales, the thinner the structurethe more anisotropic it becomes. A theory describing a hierarchy of such magnetic fluctuations,or turbulent eddies, in MHD turbulence suggests that their anisotropy increases as their scaledecreases, a/l ∝ a / [11, 12, 33–36].It has been proposed that at small enough scales the tearing instability of very anisotropiceddies can compete with their Alfv´enic dynamics. This means that below a certain criticalscale the tearing time should become comparable to τ A , so that the turbulence is mediatedby tearing instability [16–18]. Such a picture has received some numerical and observationalsupport [22, 25, 40]. The theory of tearing instability required to describe strongly anisotropiccurrent sheets, goes beyond the simplified FKR theory and generally requires one to analyzestructures that are different from the Harris-type current sheets.It should be acknowledged that the first analysis of the tearing instability in structures formedby MHD turbulence dates back to 1990 [41]. That analysis was based on the Iroshnikov-Kraichnan model of MHD turbulence [42, 43] that treats turbulent fluctuations as essentiallyisotropic (that is, characterized by a single scale) weakly interacting Alfv´en wave packets. Theanisotropy of turbulent fluctuations has therefore not been quantified in [41]. Moreover, theirmodel assumed the presence of a significant velocity shear in the current layer and adopted thetearing-mode growth rate calculated in the shear-modified FKR regime [44–46]. As a result, theapproach of [41] was qualitatively different from that of the recent studies [16–18].In our discussion we do not impose any limitations on the anisotropy of current structures,that is, we assume them to be anisotropic enough to accommodate the fastest growing tearingmode; this assumption is consistent with the model of MHD turbulence [12] adopted in [16, 18].We do, however, make several important simplifications. First, we assume that the backgroundmagnetic field has only one component, B y ( x ). An optional uniform guide field in z -directionmay also be present; it has no effect on the problem. Second, in most of the work we assumethat the configuration is static, that is, there is no background flows. In section 8 we briefly These ideas stem from the observation that if current sheets were allowed to have arbitrarily large aspect ratios,they would be tearing unstable at rates that diverge when the Lundquist number tends to infinity [37–39]. We were not aware of this important early work at the time when our previous studies [16, 18] were published. iscuss the effects of a shear flow, where, similarly to the magnetic field, the shearing velocityfield is assumed to have only one component, v y ( x ). (We refer the reader to, e.g., [47–49] for abroader discussion of the possible effects of background flows and strong outflows). Finally, ouranalysis is limited to the MHD framework.To obtain the equations governing the tearing mode, we follow the standard procedure andrepresent the magnetic field as B ( x, y ) = B f ( x )ˆ y + b ( x, y ), where f ( x ) describes the profile ofthe reconnecting field, see Fig. (1). Its typical scale, the thickness of the reconnection layer, is a .The weak perturbation field can be represented through the magnetic potential b = − ˆ z × ∇ ψ =( ∂ψ/∂y, − ∂ψ/∂x ). We assume that the background velocity is zero. The incompressible velocityperturbation is represented through the stream function v = ( ∂φ/∂y, − ∂φ/∂x ). We will neglectthe effects of viscosity, but will keep the magnetic diffusivity. The magnetic induction andvelocity momentum equations take the form e.g., [2]: ∂ψ∂t = ∂φ∂y B f + η ∇ ψ, (1) ∂∂t ∇ φ = B f ∂∂y ∇ ψ − B f ′′ ∂ψ∂y . (2)We can use the Fourier transform in the y direction, and represent the fluctuating fields as ψ = ˜ ψ ( ξ ) exp( ik y ) exp( γt ) , (3) φ = − i ˜ φ ( ξ ) exp( ik y ) exp( γt ) . (4)In these expressions and in what follows we will use the dimensionless variables ξ = x/a, (5)˜ η = η/ (cid:0) k V A a (cid:1) , (6) λ = γ/ ( k V A ) , (7) ǫ = k a, (8)where v A is the Alfv´en velocity associated with B . The anisotropy parameter ǫ can be arbitrary.In applications to turbulence, however, the most relevant cases are those corresponding to ǫ ≪ ǫ ≪ ǫ ≈ λψ = f φ + η (cid:2) ψ ′′ − ǫ ψ (cid:3) , (9) λ (cid:2) φ ′′ − ǫ φ (cid:3) = − f (cid:0) ψ ′′ − ǫ ψ (cid:1) + f ′′ ψ, (10)where we denote by primes the derivatives with respect to ξ . An additional simplification ofthese equations can be obtained from the following consideration. We assume that η and λ aresmall parameters (this assumption can be verified a posteriori, from the obtained solution). Therange of scales where the terms including these parameters can be neglected will be called the outer region . The range of scales where they become significant will be called the inner region .We will solve equations (9) and (10) in the outer and inner regions separately, and thenasymptotically match the solutions. In the inner region, where as we will see, ξ ≪
1, we have ∂ /∂ξ ≫
1. Due to the smallness of η and λ , the terms in the square brackets can be relevant (cid:1877)(cid:3398)(cid:1853) (cid:1858) ( (cid:1876) ) (cid:2158)(cid:1853) Figure 1.
Sketch of a general profile of the backgroundmagnetic field.only in the inner region, and, therefore, small ǫ terms can always be neglected in the squarebrackets. The equations then take the form λψ − f φ = ηψ ′′ , (11) − f (cid:0) ψ ′′ − ǫ ψ (cid:1) + f ′′ ψ = λφ ′′ . (12)Those are the equations describing the tearing instability in the very anisotropic case ǫ ≪
1, andthey are the main equations we are going to discuss in this work. The right-hand-side terms inthese equations are relevant only in the inner region. In the outer region, they can be neglected.
3. The outer equation
We start with the outer region. We need to solve the equations φ = λf ψ, (13) ψ ′′ = (cid:18) f ′′ f + ǫ (cid:19) ψ, (14)subject to the boundary conditions ψ, φ → ξ → ±∞ (or to the periodic boundary conditionsif the magnetic profile f ( ξ ) is periodic). Before we discuss the solution we note that Eq. (14) isthe Schrodinger equation with zero energy. In general, it does not have a solution correspondingto given boundary conditions. So the solutions should be found separately for ξ > ξ < ξ where this equation is not applicable. The solutions thus foundwill not, therefore, match smoothly, but will have a discontinuity in the derivative (a break) at ξ = 0.We consider two exactly solvable model cases that correspond to particular profiles f ( ξ ) ofthe reconnecting magnetic field. The first case is f ( ξ ) = tanh( ξ ). It is the so-called Harris profile[24]. It corresponds to a magnetic field that value does not change in space except for a regionof width ξ ∼
1, where it reverses its direction. The second solvable case is f ( ξ ) = sin( ξ ) [50]. Inthis case the magnetic field changes its strength and reverses its direction on the same scales.The latter case is arguably more relevant for the structures encountered in turbulence, and it isespecially convenient for numerical studies as it allows for periodic boundary conditions.n the first case, f = tanh( ξ ), the solution of Eq. (14) for the magnetic field is (e.g., [51]): ψ ( ξ ) = Ae − ǫξ (cid:20) ǫ tanh( ξ ) (cid:21) , ξ ≥ , (15) ψ ( − ξ ) = ψ ( ξ ) . (16)The solution for the velocity function φ ( ξ ) is then easily found from Eq. (13). In order tomatch with the inner solution, it is important to know the asymptotic forms of the velocity andmagnetic fields for ξ ≪
1. Taking into account that ǫ ≪
1, one obtains by expanding the tanh( ξ )in Eq. (15) that φ ′ ( ξ ) ∼ − Aλ/ξ + (2 Aλξ ) / (3 ǫ ). The second term can be neglected when ξ ≪ ǫ .The velocity φ ′ can then be formally expressed in this limit as φ ′ ( ξ ) ∼ − λξ ψ (0) . (17)As we will see later, in order to match the magnetic field one can define the tearing parameter∆ ′ = ψ ′ ( ξ ) − ψ ′ ( − ξ ) ψ (0) , ξ > . (18)It is easy to see that in the region ξ ≪
1, the tearing parameter approaches a constant value∆ ′ = 2 ǫ . (19)In the second case, f ( ξ ) = sin( ξ ), the solution periodic in [ − π, π ] is: ψ ( ξ ) = A sin (cid:20)p − ǫ (cid:18) ξ + π √ − ǫ − π (cid:19)(cid:21) , ξ ≥ , (20) ψ ( − ξ ) = ψ ( ξ ) . (21)The derivative of the φ function is then found as φ ′ ∼ − ( Aλ/ξ )( πǫ /
4) + (
Aλξ/ ξ ≪ ǫ , in which case the φ ′ function has the asymptotic behaviorthat is formally identical to that obtained for the first case, φ ′ ∼ − λξ ψ (0) . (22)Indeed, in this case ψ (0) = Aπǫ /
4. In the region ξ ≪
1, the tearing parameter approaches aconstant value ∆ ′ = 8 πǫ . (23)Note the different scaling of this parameter with ǫ as compared to the previous result (19).It is important to check how deeply into the asymptotic region ξ ≪ f ′′ ∼ f ∼ ξ for both tanh- and sine-shaped magnetic profiles.From Eq. (14) we have ψ ′′ ∼ ψ . Then from Eq. (13) we estimate φ ′′ ∼ ( λ/ξ ) ψ . The right-handsides in Eqs. (11, 12) are, therefore, small if η ≪ λ, (24) λ ≪ ξ . (25) . The inner equation In the inner region we need to keep the right hand sides of Eqs. (11) and (12). For that thesecond derivatives of the fields should be large. For instance, we have to assume ψ ′′ ≫ ψ , whichholds for ξ ≪
1. This allows us to simplify Eqs. (11,12) in the following way: λψ − ξφ = ηψ ′′ , (26) − ξψ ′′ = λφ ′′ . (27)By differentiating Eq. (26) twice, we get λψ ′′ = ξφ ′′ + 2 φ ′ + ηψ ′′′′ . We now exclude ψ ′′ and ψ ′′′′ from this equation by using Eq. (27), ψ ′′ = − ( λ/ξ ) φ ′′ . A few lines of algebra allow one to castthe resulting equation in the form λ φ ′′ + (cid:0) ξ φ ′ (cid:1) ′ − λη (cid:20) φ ′′′′ − (cid:26) φ ′′ ξ (cid:27) ′ (cid:21) = 0 . (28)This equation can trivially be integrated once. Also, noting that it contains only derivatives of φ , we may reduce the order by denoting Y ≡ φ ′ . We then get: Y ′′ − ξ Y ′ − ηλ (cid:0) λ + ξ (cid:1) Y = C. (29)Before we analyze this equation further, we note that the first two terms in the left-hand sidecome from the resistive term in the induction equation. Also, one can directly verify that theterm λ in the parentheses would be absent if we used a common approximation treating ψ ( ξ )as a constant in the inner region, the so-called “constant- ψ ” approximation.In order to find the constant of integration C we need to match asymptotically the solution ofthe inner Eq. (29) with the solution of the outer equation. We know that the outer solution existsin ξ ≫ λ , see Eq. (25), which, for ξ < ξ ≫ λ . What asymptotic does thesolution of Eq. (29) have in region (25)? There are two possibilities: Y ∼ exp (cid:8) ξ / √ λη (cid:9) and Y ∼ − Cηλ/ξ . By evaluating different terms in Eq. (29) for these asymptotic solutions, onecan check that they hold for ξ ≫ ηλ , which is less restrictive than Eq. (25).Obviously, the first asymptotic is not the solution we need, since the outer solution does nothave an exponential growth at these scales. We, therefore, are interested in the inner solutionwith the asymptotic Y ∼ − C ηλξ . (30)In order to match this asymptotic expression with the outer solution Y ≡ φ ′ = − ( λ/ξ ) ψ (0),see expressions (17) or (22), we need to choose C = ψ (0) /η .We note that the region where we asymptotically matched the two solutions is λ ≪ ξ ≪ ǫ / in the case of the tanh-profile, and λ ≪ ξ ≪ ǫ / in the case of the sine-profile. Our solution,therefore, makes sense only when λ /ǫ / ≪ , for tanh-shaped profile , (31) λ /ǫ / ≪ , for sine-shaped profile , (32)which, as can be checked when the solution is obtained, are not restrictive conditions.So far, we have matched the solutions for the velocity field, the φ ′ functions. To completethe asymptotic matching of the inner and outer solutions, we now need to match the magneticfields, that is, the ψ ′ function. This can be done in the following way. From Eq. (27) we get forhe inner solution ψ ′′ = − ( λ/ξ ) φ ′′ . Integrating this equation from ξ ≪ −√ λ to ξ ≫ √ λ , whichfor the inner solution is equivalent to integrating from −∞ to ∞ , we obtain − λ + ∞ Z −∞ φ ′′ ξ dξ = + ∞ Z −∞ ψ ′′ dξ, (33)which, recalling that φ ′ ≡ Y , can be rewritten as + ∞ Z −∞ Y ′ ξ dξ = − ψ (0) λ ∆ ′ . (34)This asymptotic matching condition will define the growth rate λ .It is convenient to change the variables in the following way. Let us introduce a function G such that Y = ( ψ (0) /λ ) G , and the independent variable ζ = ξ /λ . Then, the velocity-functionequation Eq. (29) takes the form4 ζG ′′ − G ′ − β (1 + ζ ) G = β , (35)where primes denote the derivatives with respect to ζ , and we have denoted β ≡ λ /η . Thematching condition (34) then takes the form − ∞ Z √ ζ ∂G∂ζ dζ = λ ∆ ′ . (36)We thus need to solve Eq. (35), and then find the tearing growth rate λ from the matchingcondition (36).The analytic theory of tearing mode is essentially based on Eqs. (35) and (36). Theseequations can be solved exactly. Historically, however, a better known case is a simpler case of β ≪
1, the so-called FKR case [23]. The simplifying assumptions going into the FKR derivationare, however, easier to understand if one knows the exact solution of the problem. Here we,therefore, first concentrate on the exact solution.
5. Solution of the inner equation
Here we present the exact solution of the tearing equation (35). This is an inhomogeneousequation, so its solution is a linear combination of a particular solution of the originalinhomogeneous equation (35) and a general solution of the homogeneous equation4 ζG ′′ − G ′ − β (1 + ζ ) G = 0 . (37)The solution of the homogeneous equation (37) has two possible asymptotics, G ∝ exp ( ± βζ/ ζ → ∞ . We obviously need to consider only the solutions behaving as G ∝ exp ( − βζ/ Y ( ξ ), we need to study the solutions of Eq. (37) in more detail. As can be checked directly, atsmall ζ the solution has the following asymptotic behavior G ( ζ ) ∼ a (cid:18) − β ζ + β (cid:26) − β (cid:27) ζ + . . . (cid:19) + b ζ / (cid:18) ζ + . . . (cid:19) , (38)where a and b are two arbitrary parameters. The “ a -part” of the solution, which is a regularfunction at ζ = 0, corresponds to a solution of the homogeneous version of the original velocityquation (29) that is even in ξ , while the singular “ b -part” corresponds to a solution odd in ξ . Equation (29) is symmetric with respect to ξ → − ξ , therefore, each solution of (29) is a linearcombinations of even and odd solutions.One can see from the asymptotic behavior (38) and from Eq. (37) itself that in the oddsolutions, the signs of the first and second derivatives are the same and they do not change onpositive or negative axes. This means that every odd solution of the homogeneous version ofequation (29) diverges at both ξ → ∞ and ξ → −∞ . In the case of even solutions, the sameanalysis can be applied to the function G ( ζ ) exp (cid:0) β ζ/ (cid:1) when β <
1, from which it followsthat every even solution of homogeneous Eq. (29) diverges at both infinity limits as well. Onlyby choosing a particular relation between a and b can one cancel these divergences either atpositive or negative infinity (but not at both).The method that we will use reproduces all the solutions that decline at ζ → ∞ . Therefore,as we will see, our derived expression for G will contain both even and odd parts, but with arigid relation between a and b , to cancel this divergence. We will need to remove such solutionssince, as has been explained, they diverge either at ξ → −∞ or ξ → ∞ . Later, we will use thiscondition to uniquely define the solution.In order to find the general solution of Eq. (35) we use the following method. The tearingequation (35) is defined only for positive ζ . We can, however, consider this equation on thewhole ζ − axis by formally extending the function G to the negative values of the argument. Forthat we define G ( ζ ) = G ( ζ ) θ ( ζ ) + G ( ζ ) (1 − θ ( ζ )) , (39)Where G and G are solutions of Eq. (35) such that G → ζ → + ∞ , and G → ζ → −∞ , and θ ( ζ ) is the Heaviside step function. These solutions at positive and negativearguments are defined up to arbitrary, declining at infinity solutions of the homogeneousequation, therefore, they can always be chosen so that their amplitudes match at the origin, G (0) = G (0). This provides a formal extension of the G function to the negative arguments.Note that we match only the values of functions G and G at ζ = 0, but not their derivatives.If we consider the operator ˆ L = 4 ζ ∂ ∂ζ − ∂∂ζ − β (1 + ζ ) , (40)we can directly verify that ˆ LG = θ ˆ LG + (1 − θ ) ˆ LG = β . (41)Therefore, the extended function (39) satisfies the same equation (35) on the entire real axis.This function declines at ±∞ , therefore, we can Fourier transform equation (35) using thestandard definition G ( ζ ) = 12 π ∞ Z −∞ G ( k ) e ikζ dk. (42)In the Fourier space the equation takes the form (cid:0) k + β (cid:1) G ′ + (cid:0) k − iβ (cid:1) G = 2 πiβ δ ( k ) . (43) This follows from the fact that the velocity function Y ( ξ ) is analytic at ξ = 0. Indeed, the function P = G exp (cid:0) β ζ/ (cid:1) satisfies the equation 4 ζP ′′ = (cid:2) ζβ (cid:3) P ′ + (cid:2) β − β (cid:3) ζP . Its evensolution has the asymptotic behavior P ∼ a + a (cid:0) β / (cid:1) (cid:0) − β (cid:1) ζ at small ζ . So, when β <
1, the functionitself and, due to the equation it satisfies, its first and second derivatives have the same sign at ζ >
0. Therefore,the function P diverges at infinity, which means that G must diverge as well. (cid:1861)(cid:2010) (cid:3398) (cid:1861)(cid:2010) .. . Figure 2.
Contour of integration in the k plane informula (45). Possible branch cuts necessary to defineanalytic continuations of the integrands in (45) are alsoshown.The first-order ordinary differential equation (43) can easily be solved for k < k >
0. Thesolutions are G ± ( k ) = 2 πiA ± (cid:20) k β (cid:21) − " ikβ − ikβ β , (44)where A ± are two complex constants, and ± signs stand for the solutions defined on the positiveand negative real k axes, respectively. The function G ( ζ ) is real, therefore A − = − A ∗ + .The delta-function in the right-hand side of Eq. (43) implies that the function G ( k ) isdiscontinuous on the real k axis at k = 0, with the discontinuity condition A + − A − = 1.In what follows we will simply denote A + = A and A − = − A ∗ , so that the discontinuitycondition reads A + A ∗ = 1. As a result, the function G can be represented as: G ( ζ ) = iA + ∞ Z (cid:20) k β (cid:21) − " ikβ − ikβ β e ikζ dk − iA ∗ + ∞ Z (cid:20) k β (cid:21) − " − ikβ ikβ β e − ikζ dk. (45)In these integrals the branches of the integrands must be chosen so that they coincide at k = 0,and the integration is performed along the positive real line in a complex plane, see Fig.(2). Wenote that the discontinuity condition defines only the real part of the complex coefficient A , butleaves its imaginary part arbitrary. This reflects the fact the solution is not defined uniquely,but only up to an arbitrary solution of the homogeneous equation (37). It is easy to see that the solution of the homogeneous equation is given by the integral G ( ζ ) = A ∞ Z −∞ (cid:20) k β (cid:21) − " ikβ − ikβ β exp( ikζ ) dk, (46)where A is an arbitrary real constant. or practical calculations, it is convenient to modify Eq. (45) further. In the first integral ofEq. (45) we change the variable of integration k = (cid:18) β i (cid:19) q − q + 1 , (47)while in the second integral we choose k = − (cid:18) β i (cid:19) p − p + 1 . (48)Expression (45) now takes the form G ( ζ ) = − A β √ Z − ( q + 1) q q β e β q − q +1 ζ dq − A ∗ β √ Z − ( p + 1) p p β e β p − p +1 ζ dp. (49)The integrals in Eq. (49) look identical. However, they differ by the contours of integrationthat follow from the changes of variables (47) and (48). If in each integral of Eq. (45), thecontours of integration lie along the real axis in the complex k plane, see Fig. (2), then thecorresponding contours in the q and p planes are defined as shown in Fig (3). .. (cid:3398) . qP Figure 3.
Contours of integration in the q and p complexplanes in formula (49).It is easy to see that for ζ >
0, these integrals will not change if we deform the contours tocoincide with the real axis, as shown in Fig (4). This way, one of the contours of integrationhas to go above the branch cut, and the other one below. It is also useful to integrate by partsonce, in order to avoid dealing with too strong a singularity at the origin, G ( ζ ) = Aβ − β − √ Z − q β − ddq n ( q + 1) e β ζ q − q +1 o dq + A ∗ β − β − √ Z − q β − ddq n ( q + 1) e β ζ q − q +1 o dq . (50) .. (cid:3398) . qP Figure 4.
Equivalent contours of integration in the q and p complex planes in formula (50).In the interval (0 ,
1] both integrals are the same, and their sum can be simplified since A + A ∗ = 1. In the interval [ − , A exp [ iπ ( β − /
4] + A ∗ exp [ − iπ ( β − / A is arbitrary, this sum is arbitrary as well (we assume β < ζ > G ( ζ ) = β − β − √ Z q β − ddq n ( q + 1) e β ζ q − q +1 o dq + C Z − | q | β − ddq n ( q + 1) e β ζ q − q +1 o dq, (51)where C is an arbitrary parameter. The first term in this expression is a particular solutionof the inner equation (35). It has been originally derived in [28–30], where a different approachinvolving expansion in the Laguerre polynomials, has been used. The second term, including afree parameter C , represents the solution of the homogeneous equation (37), the zero mode.It is easy to see that the particular solution is an analytic function at ζ = 0, so it describesan even solution of Eq. (29). Also, this solution converges at ζ → ∞ . Therefore, it represents asolution of Eq. (29) that converges at ξ → ±∞ . The zero mode, on the contrary, is non-analyticat ζ = 0, since its second derivative diverges there. It is therefore a combination of odd andeven solutions of homogeneous equation (29). According to what was said in the beginning ofthis section, a solution of the homogeneous equation diverges at either ξ → −∞ or ξ → + ∞ .We thus have to require C = 0, which removes these divergences. The zero mode is thereforeabsent and the solution is given by the first term in expression (51).
6. Tearing rate in the limit β ≪ (the FKR case) Now that we have obtained the general solution for the inner region, we can find the tearing modegrowth rate by substituting this solution into the matching condition (36). Before consideringthe general case, however, we discuss the important limit of β ≪
1, the so-called FKR case [23](we recall that β = λ /η ). In this limit, we can approximate q ( β − / ≈ q − / in Eq. (51).It is possible to show that this approximation is equivalent to the “constant- ψ ” approximationiscussed after Eq. (29), which demonstrates the equivalence of the constant- ψ approximation tothe FKR case. We see that in this case the integral depends on ζ only through the combination βζ . The matching condition (36) now reads Cβ / = λ ∆ ′ , (52)where the constant C is given by the integral C = ∞ Z r x ddx Z q − ddq n ( q + 1) e x q − q +1 o dq dx. (53)One can easily do this integral by changing the order of integrations. The answer is expressedthrough the gamma functions, C = (cid:16) π (cid:17) Γ (cid:0) (cid:1) Γ (cid:0) (cid:1) ≈ . . (54)From Eq. (52) we find the growth rate of the tearing mode in the FKR regime of β ≪ λ = C − / η / ∆ ′ / . (55)In the dimensional form, this expression can be rewritten as: γ = 2 / C − / η / k − / V / A a − (56)for the tanh-shaped magnetic profile, and γ = 8 / π − / C − / η / k − / V / A a − / (57)for the sine-shaped magnetic profile. In order to obtain these results we have substituted theexpressions (19) and (23) for the corresponding parameters ∆ ′ .Two important points should be made about the FKR solution. First, since in this limitthe inner function G ( ζ ) depends on scale only through the combination βζ , this function has acharacteristic length scale, ζ = 1 /β , which is the so-called inner scale . In terms of the ξ variable,this scale is ξ = ( λη ) / ≫ λ . In dimensional form the inner scale x = δ is given by δ = (cid:18) γηa k V A (cid:19) / , (58)where the growth rate γ is given by either (56) or (57) depending on the chosen magnetic profile.Second, we see that the growth rate and the inner scale formally diverge for k →
0. This isobviously an unphysical behavior. It reflects the fact that the approximation β ≪ G ( ζ ) is given by the exact expression (51). We will see that the growth rate doesnot diverge, but reaches a maximal value at a certain small value of k a . . Tearing rate in the general case Here we analyze the general case, the so-called Coppi case [28–30]. We need to evaluate theintegral in the left-hand-side of (36) using the exact expression for the G function given by (51).This can be easily done in the same way as we obtained (54), − ∞ Z √ ζ ∂G∂ζ dζ = − π β / Γ (cid:16) β − (cid:17) Γ (cid:16) β +54 (cid:17) . (59)The growth rate is found from the transcendental equation [29]: − π β / Γ (cid:16) β − (cid:17) Γ (cid:16) β +54 (cid:17) = λ ∆ ′ . (60)We see that the left-hand-side of this expression is positive when β <
1, and, therefore, theinstability is possible only in this case. In the limit of β ≪
1, we recover the results discussedin the previous section. The left-hand-side of Eq. (60) is small in this limit. The low- β approximation, however, breaks down at k →
0. As we will see momentarily, in this limitthe solution corresponds to β →
1. Indeed, equation (60) can be approximated in this case as √ π β / − β = λ ∆ ′ . (61)Recalling now that β ≡ λ /η , we arrive at the equation for the growth rate √ π β / − β = η / ∆ ′ . (62)From the definitions of the dimensionless parameters η and ∆ ′ , we see that as k a →
0, theright-hand-side of this equation diverges. This is possible only if β → λ = η is the equation that defines the tearing growth rate in this case. Inthe dimensional units, this equation gives γ = η / V / A k / a − / , (63)which is termed the Coppi solution in [32]. The remarkable fact is that as k decreases, theCoppi growth rate decreases as well. This is opposite to the behavior of the FKR growth ratethat increases at decreasing k . This means that there must exist a maximal growth rate ofthe tearing instability, γ ∗ , attainable at a certain wave number k ∗ . One can define this criticalwave number as the wave number at which the Coppi growth rate (63) formally matches theFKR growth rate (56) for the tanh-shaped magnetic profile or (57) for the sine-shaped profile.For the tanh-shaped magnetic profile, a simple algebra gives for the critical wavenumber andthe corresponding maximal tearing growth rate: k ∗ = (2 /C ) / η / V − / A a − / , (64) γ ∗ = (2 /C ) / η / V / A a − / . (65)For the sine-shaped profile, the answer is: k ∗ = (8 /πC ) / η / V − / A a − / , (66) γ ∗ = (8 /πC ) / η / V / A a − / . (67)n the general case, the inner solution G ( ζ ) is not a universal function depending only on βζ .However, for β ≈
1, this function approaches its asymptotic behavior G ( ζ ) ∼ /ζ at ζ ≫
1. Thetypical scale (the inner scale) of this solution is therefore ζ = 1, which in terms of the ξ variablereads ξ = λ . In dimensional variables, the inner scale in this case is δ = γak v A , (68)where γ is given by Eq. (63), or by expressions (65) or (67) for the fastest growing modes.
8. Tearing rates in the presence of a shear flow
In turbulent systems, magnetic field fluctuations are accompanied by velocity fluctuations, sothat both magnetic and velocity shears are present in a turbulent eddy, e.g., [12]. It is thereforeuseful to comment on the influence of a velocity shear on the tearing instability. A velocityshear across the current layer is expected to be less intense than the magnetic shear, otherwise,such a layer would be destroyed by the Kelvin-Helmholtz instability, e.g., [2, 25]. In MHDturbulence, fluctuations indeed have excess of the magnetic energy over the kinetic energy; thedifference between the kinetic and magnetic energies, the so-called residual energy, is negativein the inertial interval, e.g., [52–56].We assume that, similarly to the magnetic field, the background velocity has the structure v ( x, y ) = v ( x )ˆ y . In the presence of this velocity field, the dimensionless system ofequations (9), (10) becomes λψ = f φ + η (cid:2) ψ ′′ − ǫ ψ (cid:3) − i ˜ v ψ, (69) λ (cid:2) φ ′′ − ǫ φ (cid:3) = − f (cid:0) ψ ′′ − ǫ ψ (cid:1) + f ′′ ψ − i ˜ v (cid:0) φ ′′ − ǫ φ (cid:1) + i ˜ v ′′ φ, (70)where ˜ v = v ( x ) /V A is the background velocity profile normalized by the Alfv´en speedassociated with the magnetic profile. Similarly to equations (9), (10), the modified equationscan be simplified in the case of small η and λ as: λψ = f φ + ηψ ′′ − i ˜ v ψ, (71) λφ ′′ = − f (cid:0) ψ ′′ − ǫ ψ (cid:1) + f ′′ ψ − i ˜ v (cid:0) φ ′′ − ǫ φ (cid:1) + i ˜ v ′′ φ. (72)In the outer region , we have from Eqs. (71), (72): λψ = f φ − i ˜ v ψ, (73) − f (cid:0) ψ ′′ − ǫ ψ (cid:1) + f ′′ ψ − i ˜ v (cid:0) φ ′′ − ǫ φ (cid:1) + i ˜ v ′′ φ = 0 . (74)A general analysis of the problem is, unfortunately, not very transparent, e.g., [47]. A simplifiedbut quite informative treatment is, however, possible when the velocity profile is similar to thatof the magnetic field [44]. We therefore assume that ˜ v ( x ) = αf ( x ), where − < α < v ( ξ ) /λ = k v ( ξ ) /γ in dimensional units). Since in the outer region v . V A , theDoppler shift dominates, and we can neglect the term containing λ in Eq. (73). Expressing φ from Eq. (73), and substituting it into Eq. (74) one obtains after simple algebra [44]: (cid:0) − α (cid:1) (cid:26) ψ ′′ − ǫ ψ − f ′′ f ψ (cid:27) = 0 . (75)Since α = 1, the magnetic-field outer equation in this case is identical to the outer equationwithout a velocity shear (13). The asymptotic behavior of the outer solution for the velocityeld is, however, different from Eqs. (17, 22), and it can be found from Eq. (73) to the zerothorder in λ/ ˜ v : φ ( ξ ) ∼ iαψ ( ξ ) , ξ ≪ . (76)This expression holds in the asymptotic matching region δ ≪ ξ ≪
1, where δ is the inner scale.In the inner region δ ∼ ξ ≪ λψ + iαξψ = ξφ + ηψ ′′ , (77) λφ ′′ + iαξφ ′′ = − ξψ ′′ . (78)In order to match the inner solution for the velocity field with the outer solution, we derive fromEq. (78): λ + ∞ Z −∞ φ ′′ ξ dξ = − iαφ ′ | + − − ψ ′ | + − , (79)where the integral goes from ξ ≪ − δ to ξ ≫ δ , and we denote by | + − the jumps of the quantitiesacross the inner layer between the indicated limits. We can use the asymptotic form (76) toevaluate the jump of φ ′ . From Eq. (76) we get λ + ∞ Z −∞ φ ′′ ξ dξ ∼ − (cid:0) − α (cid:1) ψ (0)∆ ′ . (80)In order to do the integral in Eq. (80) we need to know the velocity function φ ( ξ ) in the innerregion. To the best of our knowledge, the exact solution of the inner equations (77, 78) isnot available. We may, however, estimate the integral in the left-hand-side of Eq. (80) in thefollowing way. We note that if λ ≫ αδ , then the shear flow does not affect the inner regionof the tearing mode. The inner solution (and the resulting scaling of the growth rate with theLundquist number S = aV A /η and the anisotropy parameter k a ) can, therefore, be qualitativelyaffected by the shear flow only if λ ≪ αδ , which is essentially the FKR limit. We thus assumethis limit in what follows.Similarly to the case without a flow, the shear-modified FKR limit implies the “constant- ψ ” approximation, which reads to the zeroth order in the small parameter λ/ ( αδ ): φ ( ξ ) ∼ iαψ ( ξ ) ∼ const at ξ ≪
1. This solution trivially satisfies Eqs. (77, 78) to the zeroth order.Obviously, the zeroth order solution does not contribute to the integral in (80), so in order toevaluate this integral we need to go to the first order in λ/ ( αδ ). From Eq. (78) we estimate ψ ′′ ∼ − iαφ ′′ and substituting this into Eq. (77), we get λψ (0) + α ξφ = ξφ − iηαφ ′′ , (81)where in the first term in the left-hand-side we have substituted the zeroth order solution. Inthe region ξ ≫ δ , the resistive term is not important, and we obtain the asymptotic form forthe first-order velocity field: φ ∼ λψ (0)(1 − α ) ξ . (82)his expression does contribute to the integral in Eq. (80). It diverges as ξ decreases, until ξ becomes as small as δ , at which scale the resistive term becomes important and φ does not growanymore. We may then estimate the integral in Eq. (80) as λ + ∞ Z −∞ φ ′′ ξ dξ ∼ λ + ∞ Z δ φ ′′ ξ dξ ∼ − λ ψ (0)(1 − α ) δ , (83)where the inner scale δ is, in turn, estimated from balancing the resistive term in Eq. (81) withthe other terms: δ ∼ αη/ (1 − α ). Substituting this into Eq. (83) and then into Eq. (80) wefinally obtain λ ∼ (cid:2) η | α | (cid:0) − α (cid:1) ∆ ′ (cid:3) / . (84)This result coincides with the more detailed derivations performed in [44, 47] up to a numericalcoefficient of order unity.In dimensional variables, this growth rate takes the following form for the tanh-like magneticprofile: γ ∼ κ / η / V / A a − / , (85)while for the sine-like profile we obtain γ ∼ κ / η / V / A a − k − / , (86)where we denote κ ≡ | α | (cid:0) − α (cid:1) <
1. These solutions are shown in Fig. (5), together with theexpressions (56), (57), and (63), corresponding to the cases without a flow. We see that a shearflow changes the scaling of the growth rate in the FKR regime, but not in the Coppi regime. Inparticular, it does not affect the scaling of the fastest growing mode.Note that the growth rate (85) corresponding to the tanh-like profile is degenerate in thatit is independent of the anisotropy parameter k a . This explains why the early analysis of[41] based on the Iroshnikov-Kraichnan theory of MHD turbulence, which assumes isotropy ofthe turbulent fluctuations (i.e, it implies k a ∼ const), formally led to the same scaling of thefastest growing mode, γ ∗ ∼ S − / , as the analysis of [16, 17] for the tanh-like magnetic profile,cf. Eq. (56). In the non-degenerate sine-like case, however, the fastest growing rate scales as γ ∗ ∼ S − / cf. Eq. (57), which is different from γ ∗ ∼ S − / assumed in [41].
9. Conclusions
We have reviewed the derivation of the anisotropic tearing mode by considering in detail twosolvable cases corresponding to the tanh-shaped and sine-shaped magnetic shear profiles. Givenlarge enough anisotropy, the dominating tearing mode has the dimension ∼ /k ∗ and growswith the rate γ ∗ . We see that these parameters depend on the assumed magnetic shear profile,and they are not universal. Their derivation requires one to go beyond the simplified FKRmodel generally covered in textbooks. We have presented an effective method for solving theinner equation for the current layer in the general case. We have also discussed the influence onthe tearing instability of shearing flows that typically accompany magnetic profiles generated byturbulence. Our work provides a detailed and self-contained discussion of the methods requiredfor the study of tearing effects in turbulent systems. Acknowledgments
SB was partly supported by the NSF grant no. PHY-1707272, NASA Grant No.80NSSC18K0646, and by the Vilas Associates Award from the University of Wisconsin -Madison. NFL was supported by the NSF-DOE Partnership in Basic Plasma Science andEngineering, award no. DE-SC0016215 and by the NSF CAREER award no. 1654168. (cid:1853) / (cid:1874) (cid:3002) (cid:1863) (cid:2868) (cid:1853) . . (cid:2018) (cid:2869)(cid:2870) (cid:1845) (cid:2879)(cid:2869)(cid:2870) (cid:1845) (cid:2879)(cid:2869)(cid:2872) .. (cid:1845) (cid:2879)(cid:2869)(cid:2872) (cid:2018) (cid:2879)(cid:2873)(cid:2872) tanh profile (cid:1863) (cid:2868) (cid:1853) . . (cid:1845) (cid:2879)(cid:2869)(cid:2875) .. (cid:1845) (cid:2879)(cid:2869)(cid:2875) (cid:2018) (cid:2879)(cid:2873)(cid:2875) sine profile (cid:2011)(cid:1853) / (cid:1874) (cid:3002) Figure 5.
Sketch of the tearing-mode growth rate as afunction of ( k a ) for the tanh-like magnetic profile (upperpanel) and the sine-like profile (lower panel). The solid linescorrespond to the growth rates without a shear flow. Thedashed lines show the growth rates in the shear-modified FKRregimes. Here S = aV A /η and κ = | α | (cid:0) − α (cid:1) < References [1] Matthaeus W H and Lamkin S L 1986
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