Energy spectrum of tearing mode turbulence in sheared background field
AAIP/123-QED
Energy spectrum of tearing mode turbulence in sheared background field
Di Hu,
1, 2, a) Amitava Bhattacharjee,
3, 4 and Yi-Min Huang
3, 4 School of Physics, Peking University, Beijing 100871,China. ITER Organization, Route de Vinon sur Verdon, CS 90 046,13067 Saint Paul-lez-Durance, Cedex, France. Department of Astrophysical Sciences, Princeton University, Princeton,New Jersey, 08544, USA Princeton Plasma Physics Laboratory, Princeton University, Princeton,New Jersey, 08540, USA (Dated: 14 October 2018)
The energy spectrum of tearing mode turbulence in a sheared background magneticfield is studied in this work. We consider the scenario where the nonlinear interactionof overlapping large-scale modes excites a broad spectrum of small-scale modes, gen-erating tearing mode turbulence. The spectrum of such turbulence is of interest sinceit is relevant to the small-scale back-reaction on the large-scale field. The turbulencewe discuss here differs from traditional MHD turbulence mainly in two aspects. Oneis the existence of many linearly stable small-scale modes which cause an effectivedamping during energy cascade. The other is the scale-independent anisotropy in-duced by the large-scale modes tilting the sheared background field, as opposed to thescale-dependent anisotropy frequently encountered in traditional critically balancedturbulence theories. Due to these two differences, the energy spectrum deviates froma simple power law and takes the form of a power law multiplied by an exponentialfalloff. Numerical simulations are carried out using visco-resistive MHD equationsto verify our theoretical predictions, and reasonable agreement is found between thenumerical results and our model. a) While visiting at PPPL, Princeton, New Jersey; Electronic mail: hudi [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] J u l . INTRODUCTION The generation of a spectrum of small-scale tearing modes by their large-scale counter-parts is a very relevant issue both in magnetically confined devices such as a reversed-field-pinch (RFP) or a tokamak as well as in space and astrophysical plasmas. For RFPs, theconstant interaction of tearing modes and resistive interchange modes keeps the plasma ina perpetual turbulent state . For tokamaks, the non-linear excitation and overlapping ofa spectrum of tearing modes may break nested flux surfaces and lead to disruption . Inastrophysical plasmas, secondary plasmoid turbulence is found to play a crucial role duringmagnetic reconnection both in kinetic and in resistive MHD investigations.An important aspect of these problems is the back-reaction of small-scale field fluctu-ations on their large-scale counterparts. A well-known example of such back-reaction isthe hyper-resistivity produced in a mean-field theory, which has been a subject of intensivestudies in the past decades . To understand this problem, however, knowledge regard-ing the structure of tearing turbulence spectrum is necessary . Hence, in this paper, wetry to construct a model to describe the structure of tearing-instability-driven turbulencespectrum in a sheared strong magnetic field. While this sheared and strongly magnetizedcase would appear to be most relevant to laboratory plasmas and to space and astrophysicalplasmas characterized by strong guide fields, our approach also provides important qualita-tive insight into more general problems where turbulence is instability-driven due to strongspatial inhomogeneities.Two arguments are commonly invoked when studying the spectrum of MHD turbulence.One is the inertial range argument, which states that there exists a self-similar region in the k space between the energy injection scale and dissipation scale where energy is conservativelytransferred from one scale to another, resulting in a power-law energy spectrum . Theother is the scale-dependent anisotropy which indicates that the ratio between the paralleland perpendicular length scale l (cid:107) /l ⊥ of turbulent eddies depends on l ⊥ . For weak turbulencein a homogeneous magnetic field, three-wave interactions result in no cascade along theparallel direction . Hence, l (cid:107) is independent of l ⊥ , which yields an energy spectrum E (cid:0) k ⊥ , l (cid:107) (cid:1) = E ⊥ ( k ⊥ ) f (cid:0) l (cid:107) (cid:1) ∝ k − ⊥ , where f (cid:0) l (cid:107) (cid:1) is any initial spectrum function of l (cid:107) and k ⊥ ∼ l − ⊥ is the perpendicular wave number. For strong turbulence, assuming no scale-2ependent alignment, the frequently invoked critical balance condition assumes that thenonlinear term and linear term are of the same order, v A /l (cid:107) ∼ v ( l ⊥ ) /l ⊥ , where v A is theAlfv´en speed of the background field and v ( l ⊥ ) is the velocity at a given perpendicular scale l ⊥ . Combining the critical balance assumption with the inertial range argument yieldsthe scale-dependent anisotropy l (cid:107) ∝ l / ⊥ , corresponding to the energy spectrum E ( k ⊥ ) ∝ k − / ⊥ . With scale-dependent alignment, the balance between linear and nonlinear termsbecomes v A /l (cid:107) ∼ v ( l ⊥ ) /v A l ⊥ , leading to the anisotropy relation l (cid:107) ∝ l / ⊥ , and the energyspectrum E ( k ⊥ ) ∝ k − / ⊥ .However, recent development in kinetic turbulence theory has pointed out the possibilitythat stable eigenmodes nonlinearly excited by unstable modes can act as an effective dampingmechanism . This is equally true for tearing turbulence with which we are concernedhere. Unlike the commonly discussed externally driven turbulence in a homogeneous system,instability driven turbulence usually has many stable modes along with a few unstable modeswhich provide the energy for the rest of the spectrum. The effective damping caused by thestable modes interrupt the transfer of energy between scales and thus alter the structureof the spectrum. It may then be expected that the resulting spectrum will deviate fromthe traditional power-law form E ( k ⊥ ) ∝ k β ⊥ and take the form of a power law multipliedby an exponential fall E ( k ⊥ ) ∝ k β ⊥ exp (cid:16) − δk β ⊥ (cid:17) . Here, β , β , δ and β are constantcoefficients. Furthermore, a recent resistive MHD simulation concerning plasmoid-mediatedturbulence in a sheared magnetic field has found discrepancy from the scale-dependentanisotropy picture and produced an approximately scale-independent anisotropy l (cid:107) ∝ l ⊥ in strong turbulence when the magnitude of the magnetic field perturbation is comparablewith that of the background field . These results raise doubt regarding the validity of thestandard inertial range picture as well as that of scale-dependent anisotropy for tearing modeturbulence in a magnetically sheared system.In the light of the discussion above, in this paper we revisit the problem of the spectrumof tearing mode turbulence. On one hand, the presence of large-scale perturbations in asheared guide field is found to introduce a scale-independent anisotropy in the small-scaleeddies. On the other hand, we find significant effective damping of the turbulence calculatedfrom linear stability of high k ⊥ modes, wherein the effective damping scales as k p ⊥ , with p = 6 / / k ⊥ than that of classical dissipation, which generally3cales as k ⊥ . We provide an analytical model for turbulence under such scale-independentanisotropy and effective damping. Based on this model, the modified spectrum will beobtained by considering the local energy budget in k space. This analytical spectrum willthen be compared with resistive MHD simulation. Reasonable agreement is found betweenanalytical predictions and numerical results.The rest of the paper is arranged as follows. In Section II, the system of interest willbe described and the basic resistive MHD equations will be introduced. In Section III, thetheoretical model regarding the damped tearing turbulence and the modified turbulencespectrum will be discussed. This new spectrum will be checked with simulation results inSection IV, and spectral properties as well as structure functions of the turbulence will bediscussed. The turbulence anisotropy will be studied analytically as well as numerically.Furthermore, this scale-independent anisotropy will be checked for strong turbulence cases.Discussions on the implication of this new form of spectrum to future studies and a conclusionwill be presented in Section V. II. SYSTEM OF INTEREST
We will consider the standard compressible MHD equations with viscosity and resistivityincluded, as follows: ∂∂t ρ + ∇ · ( ρ v ) = 0 , (1) ∂∂t ( ρ v ) + ρ ( v · ∇ ) v + v [ ∇ · ( ρ v )] = −∇ (cid:18) p + B (cid:19) + ( B · ∇ ) B + ν ∇ ( ρ v ) , (2) ∂∂t p + ∇ · ( p v ) = − ( γ A − p ∇ · v , (3) ∂∂t B = ∇ × ( v × B − η J ) . (4)Here, Eq. (1) is the continuity equation, Eq. (2) is the equation of motion, Eq. (3) representsthe equation of state, and Eq. (4) is the Ohm’s law. Here ρ is the plasma density, v isthe velocity, B is the total magnetic field, J is the current density, and p is the pressure.The vacuum permeability µ has been absorbed into ρ and J . Furthermore, γ A = 5 / ν and η stand for classic viscosity andresistivity respectively.In this study, we will consider a simple slab system with coordinates ( x, y, z ), and theboundary conditions are assumed to be periodic at all sides. The sizes of the system in x , y , z directions are X , Y and Z respectively, and the geometric center of the system is chosento be ( x, y, z ) = (0 , , B arethe following: B x = 0 , B y = B y (0) cos (cid:18) πX x (cid:19) , B z = (cid:113) B − B y . (5)Here, B y (0) and B are constants to be specified later. The system is initially in force-free,with the pressure assumed to be constant and set to unity. The corresponding initial currentprofile is J z = − B y (0) 2 πX sin (cid:18) πX x (cid:19) . (6)An artificial constant electric field along z direction is implemented to sustain the initialcurrent profile against resistive diffusion. Although the assumed global geometry is simple,it is sufficient to capture the fundamental physical process of the dynamics of small-scaletearing fluctuations. The qualitative features of the theory are not expected to change inmore realistic global geometry.We define a “safety factor” q ≡ Y B z ZB y (7)and the “rotational transform” µ ≡ q (8)analogous to that of a tokamak. The corresponding q profile is then a function of x . Inthe region x ∈ [ − . , . q profile is shown in Fig. 1, with X = 2, Y = 4, Z = 20, B = 10, B y (0) = 1 .
5, and the corresponding minimum safety factor is given by q (0) = 1 . /
1, 3 / / IG. 1. The analogous safety factor profile in region x ∈ [ − . , .
5] for initial background magneticfield, with X = 2, Y = 4, Z = 20, B = 10 and B y (0) = 1 .
5. The safety factor tends to infinitynear x = ± . B y being zero there. As the turbulence grows in strength, it will have a back-reaction on the mean backgroundfield, leading to self-consistent evolution of the latter. The mean current profile will tendto relax under turbulence spreading , and it is observed that substantial profile flatteningwould occur over time after the turbulence has been fully established. Ultimately, therelaxation would reach a point where there is no free energy available, and the tearingturbulence would then gradually decay away. However, it will be shown in Section IV Athat the characteristic time scale of such decay is much longer than the slowest nonlinearturnover time of eddies, thus the turbulence can be viewed as having attained a quasi-steady-state before decay occurs.
III. ANALYTICAL MODEL FOR TEARING TURBULENCE
The structure of tearing turbulence spectrum will be discussed analytically in this section.Three quantities are needed in order to obtain the spectrum of tearing turbulence in asheared guide field. The first is the effective damping rate caused by small-scale linearlystable modes, the second is the anisotropy property of the tearing turbulence, and the thirdis the local energy transfer in the k space . We will treat the effective damping and6nisotropy property in Section III A and III B respectively, then substitute these results intothe local energy transfer equation in Section III C to obtain the turbulence spectrum. InSection III A, we will first justify the use of linear stability theory in considering the effectivedamping, then provide the k ⊥ scaling of growth rate and further obtain the effective dampingrate for inviscid and viscous limit in Eq. (21)-(23). In Section III B, we will investigatethe scale-dependence of turbulence anisotropy by considering the ratio between the parallelwave number dispersion ∆ k (cid:107) ∼ l − (cid:107) as defined in Eq. (26) and the perpendicular wave number k ⊥ ∼ l − ⊥ . The result is given in Eq. (34) and Eq. (40) for unperturbed and perturbed shearedguide field respectively. Finally, in Section III C, we combine the aforementioned results withthe local energy budget in Eq. (41) and the forward energy transfer rate in Eq. (44) to obtainthe spectrum shape shown in Eq. (46). A. Effective damping caused by linearly stable modes
We consider the effective damping under the assumption of weak nonlinearity, that is, thenonlinear interaction is assumed to be sufficiently weak that it does not change the linearouter region solution. Hence, we can still use linear theory to consider the mode structure,and the effective damping rate can be estimated from the negative linear growth rate.The justification of using the linear growth rate to estimate effective damping may beformulated more precisely as follows. The effective island width w for a given Fouriercomponent of the magnetic perturbation ˜ B k ( x, y, z ) = ˜ B (0) k ( x ) exp ( ik y y − ik z z ) has thefollowing dependence on mode numbers and the magnetic perturbation strength: w ∼ ( − ψ/ Ψ (cid:48)(cid:48) s ) / ∼ (cid:32) ˜ B x B z L s k y (cid:33) / , (9)where ψ is the perturbed oblique flux ψ ≡ ˜ A · (cid:126)h, (cid:126)h ≡ (cid:126)e z + ( k z /k y ) (cid:126)e y . (10)Here, (cid:126)h is the oblique direction defined by given k y and k z . Also, ˜ B x is the x component of thecorresponding magnetic perturbation and Ψ (cid:48)(cid:48) s is the second order derivative of backgroundoblique flux taken at the resonant surface. Furthermore, L s ≡ Zq/s is the magnetic shearlength and s ≡ Y q (cid:48) /q is the magnetic shear. In the inviscid limit, the tearing layer width7cales as x η ∼ (cid:18) ηv A L s k y (cid:19) / (∆ (cid:48) ) / , (11)where ∆ (cid:48) ≡ ψ (cid:48) s /ψ s (cid:12)(cid:12) + − is the tearing stability index; the minus and plus signs here denote theleft and the right side of the resonant surface. Alternatively, in the viscous regime we have x η ∼ (cid:18) ηv A L s k y (cid:19) / P / m , (12)where the magnetic Prandtl number P m ≡ ν/η . The following two factors justify the useof linear stability analysis. First, the perturbation amplitudes of high- k modes are ordersof magnitude smaller than that of low k modes, thus the effective width of a high- k islandwill also be much smaller than that of a low- k island. Second, the effective island widthwill shrink faster than the tearing layer width for increasing k , as the power dependence on k for the former is greater than that of the latter. Simple estimation using the turbulencespectrum obtained later in Section IV indicates that, in our case of weak turbulence, theisland width will be smaller than the tearing layer width when k ⊥ ≥
25. Furthermore,the contribution from hyper-resistivity is also ignored since it is proportional to the drivenmode width to the fourth power, making its contribution less important for very small-scalemodes. We now examine the linear growth rate of the small-scale modes. The ideal linear eigen-equation for slab geometry can be written as : ∂ x ψ = (cid:18) k + F (cid:48)(cid:48) F (cid:19) ψ. (13)Here, F ≡ B · k , and k is the wave number perpendicular to the oblique direction (cid:126)h . Itshould be noted that we have k ⊥ (cid:39) k due to k (cid:107) (cid:28) k ⊥ as a result of the localized small-scalemode structure. For straight tearing modes with k z = 0, F (cid:48)(cid:48) /F remains finite even at theresonant surface where F = 0. If k ⊥ (cid:29) F (cid:48)(cid:48) /F , then the eigen-structure has the followingform near resonant surface x = x s : ψ (cid:39) ψ s exp ( − k ⊥ | x − x s | ) . (14)Hence, for high k modes which are linearly stable, we have:∆ (cid:48) ≡ ψ (cid:48) s ψ s (cid:12)(cid:12)(cid:12) + − (cid:39) − k ⊥ . (15)8or oblique modes, there is a logarithmic singularity in the derivative of the ideal solutionsince F (cid:48)(cid:48) /F is singular near the resonant surface . However, the contribution of thislogarithmic singularity to ψ (cid:48) is even in parity near the resonant surface, thus does notcontribute to ∆ (cid:48) . Hence, the ∆ (cid:48) of high- k oblique modes should have the same form asthat of straight modes as shown in Eq. (15). Numerical solution of Eq. (13) confirms thisstatement .The linear growth rate for oblique tearing modes in the inviscid limit is given by γ = η / (∆ (cid:48) ) / (cid:0) k ⊥ B (cid:48) ys (cid:1) / ρ − / , (16)while in the viscous regime we have γ = η / P − / m ∆ (cid:48) (cid:0) k ⊥ B (cid:48) ys (cid:1) / ρ − / . (17)Here, B (cid:48) ys is the x gradient of B y taken at resonance x s . The stable eigenmodes satisfyingthese dispersion relations are similar in mode structure and parity to the unstable modesthat drive the turbulence. Equations (16) and (17) give the following k ⊥ dependence for γ : γ ∝ − η / k / ⊥ (18)in the inviscid limit and γ ∝ − η / P − / m k / ⊥ (19)in the viscous regime.As has been mentioned in Section II, the background magnetic field is constantly evolvingthroughout the time-evolution of turbulence, hence we need to track the evolution of B (cid:48) ys numerically as the turbulence evolves. We define the following characteristic length scale of B y variation: λ ≡ B y (0) B (cid:48) ys . (20)Thus, the effective damping in k space can be written as:[ ∂ t E ( k )] damping = 2 γE ( k ) = − DS − p/ ( k ⊥ λ ) p E ( k ) , (21)with p = 6 / p = 4 / E ( k ⊥ ) = v ( k ⊥ ) /k ⊥ is the “energy density” in k ⊥ space. We consider a priori the equipartition ofmagnetic and kinetic energy for medium to high k ⊥ . (We will check the validity of this9ssumption a posteriori ). The Lundquist number S is defined as S ≡ τ ∗ η /τ A , with τ ∗ η ≡ λ /η and τ A ≡ Z/v A , while v A is the Alfv´en speed corresponding to the guide field. Furthermore, D is the effective damping coefficient with dimension of 1 /t . Combining Eq. (16) or Eq. (17)with Eq. (21), we obtain D = 1 . (cid:18) Zλ (cid:19) / (cid:18) B y B z (cid:19) / τ − A (22)in the inviscid limit and D = 2 (cid:18) Zλ (cid:19) / (cid:18) B y B z (cid:19) / τ − A P − / m (23)in the viscous regime. The damping rate given in Eq.˙(21) has a weaker dependence on k ⊥ than the classical dissipation does, making the distinction between the inertial range andthe dissipation range hard to define. Thus, the present physical situation, in which dampingappears to be important at all scales, does not permit a strict delineation of an inertial rangein tearing turbulence. B. Scale-independent anisotropy in sheared background field
The scale dependence of the ratio between the parallel and the perpendicular length scalesof eddies is of great interest since it directly affects the nonlinear turnover rate and thusfurther influences the forward energy cascade rate of turbulence. The nonlinear turnoverrate for MHD turbulence can be modeled as :1 τ nl (cid:39) v ( k ⊥ ) l ⊥ l (cid:107) v A . (24)Here, v ( k ⊥ ) represents kinetic perturbation at k ⊥ scale.For weak turbulence generated by oppositely propagating Alfv´en waves with straightbackground field lines, the three-wave interaction preserves the k (cid:107) space structure of thebeating waves, thus preventing any energy cascade along the direction parallel to the back-ground magnetic field . A simple way to see this is by considering the resonant conditionof wave number and frequency for three-wave interaction . We have: k + k = k , ω ± + ω ∓ = ω ± . (25)Here, ω + = v A k (cid:107) and ω − = − v A k (cid:107) represent the angular frequencies of the forward andthe backward propagating Alfv´en waves, respectively. The oppositely propagating waves10ndicate that either k (cid:107) or k (cid:107) must be zero to satisfy both resonance conditions for the wavenumber and the frequency. Hence, there is no cascade of energy along k (cid:107) and the nonlinearturnover rate scales as τ nl ∝ v ( k ⊥ ) l − ⊥ as a result.On the other hand, for a spectrum of modes in a sheared guide field, the parallel lengthscale l (cid:107) (cid:39) / ∆ k (cid:107) , where ∆ k (cid:107) is the dispersion in parallel wave number, and perpendicularlength scale l ⊥ (cid:39) /k ⊥ , where k ⊥ is the perpendicular wave number. The dispersion inparallel wave number, ∆ k (cid:107) , is defined as (cid:0) ∆ k (cid:107) (cid:1) ≡ (cid:10) k (cid:107) (cid:11) k ⊥ ,x − (cid:10) k (cid:107) (cid:11) k ⊥ ,x . (26)Here, (cid:104) f (cid:105) k ⊥ ,x represents averaging quantity f over k (cid:107) for a given k ⊥ and across the ( y, z )plane for a given x . Averaging over the ( y, z ) plane is necessary because the small-scale modestructures are very localized and we are looking at the spectrum at a specific x . Withinthe framework of weak turbulence theory in a strong guide field where the average fieldis assumed to be unperturbed, we will find a similar independence between l (cid:107) and l ⊥ inthe turbulence spectrum, i.e., ∆ k (cid:107) ∝ k ⊥ , although the physical mechanism is somewhatdifferent from that described above. However, it can be seen that the inclusion of a finitelarge-scale perturbation will introduce an additional relationship between ∆ k (cid:107) and k ⊥ inthe spectrum, so long as we have ˜ B L L s k ⊥ /B z (cid:29)
1, where ˜ B L is the random large-scaleperturbation, L s is the shear length of background guide field, and B z is the guide field alongthe ignorable direction. It is important to note that k ⊥ in this criterion is the perpendicularwave number of the small-scale modes rather than the large-scale perturbation. Thus, theleft-hand-side of the aforementioned criterion should not be confused with the Kubo numberof the large-scale perturbation, defined as the ratio between the nonlinear and linear terms κ ≡ (cid:16) ˜ B/B (cid:17) / (cid:0) l (cid:107) /l ⊥ (cid:1) .We assume the perturbation has the following form: ˜ B k ( x, y, z ) = ˜ B (0) k ( x ) exp ( ik y y − ik z z ),where m ≡ Y k y / π and n ≡ Zk z / π . For the unperturbed background field, we have: k (cid:107) = 2 π (cid:18) B y B mY − B z B nZ (cid:19) = k y YZ B z B (cid:16) µ − nm (cid:17) = − k y ∆ xL s B z B , µ ≡ /q. (27)Again, L s ≡ Zq/s , q ≡ Y B z /ZB y , and s ≡ Y q (cid:48) /q . We repeat for emphasis that B y and B z here do not contain the contribution of large-scale perturbation. The length ∆ x ≡ x − x s represents the distance to the resonant surface for a given m/n .It will be shown later on in Section IV A that the characteristic length scale of turbulencestrength envelope is much larger than 1 /k ⊥ in cases we are interested in, thus ψ s can be11ssumed to be independent of ∆ x for a given x . Then Eq. (14) yields (cid:104) f (cid:105) k ⊥ ,x = (cid:42) (cid:82) ∞−∞ exp ( − k ⊥ | ∆ x | ) f d ∆ x (cid:82) ∞−∞ exp ( − k ⊥ | ∆ x | ) d ∆ x (cid:43) x . (28)Note that here we integrate over ∆ x instead of k (cid:107) because dk (cid:107) ∝ d ∆ x so long as k y ∝ k ⊥ .For the denominator, we have: (cid:90) ∞−∞ exp ( − k ⊥ | ∆ x | ) d ∆ x = e k ⊥ ∆ x k ⊥ (cid:12)(cid:12)(cid:12) −∞ − e − k ⊥ ∆ x k ⊥ (cid:12)(cid:12)(cid:12) ∞ = 1 k ⊥ . (29)Thus we obtain: (cid:10) k (cid:107) (cid:11) k ⊥ = 0 , (cid:0) ∆ k (cid:107) (cid:1) = k ⊥ (cid:90) ∞−∞ exp ( − k ⊥ | ∆ x | ) (cid:18) k y ∆ xL s B z B (cid:19) d ∆ x. (30)Because the localized mode structure also implies that all the small-scale modes which canbe “seen” from x have similar µ , we can approximately write: k ⊥ = B z B k y + B y B k z , (31) Zk z Y k y (cid:39) µ ( x ) . (32)Therefore, we obtain k y (cid:39) Z Z + Y µ ( x ) B B z k ⊥ . (33)Substituting the above relationship into Eq. (30), the parallel length scale l (cid:107) for small scaleperturbations is found to be independent of k ⊥ l − (cid:107) = (cid:0) ∆ k (cid:107) (cid:1) (cid:39) Z Z + Y µ ( x ) L s ∝ k ⊥ . (34)This is similar to the weak turbulence limit discussed in Ref. [12] and Ref. [13], although theunderlying physics is quite different.Now, let us consider the effect of a large-scale perturbation on the small-scale anisotropy.We consider the summation of several large-scale modes as a random magnetic perturbationstrong enough to twist the field “seen” by the small-scale modes. Let ˜ B L be the perturbationcomponent in ( y, z ) plane. Thus, the parallel wave number for each mode is now: k (cid:107) = − k y ∆ xL s B z B + ˜ B Ly B k y − ˜ B Lz B k z . (35)Recalling Eq. (32), for given x , we have: k (cid:107) (cid:39) − B z B k y ∆ xL s + ˜ B Ly B (cid:34) − YZ µ ( x ) ˜ B Lz ˜ B Ly (cid:35) k y . (36)12or simplicity, we define the following parameters: T ≡ B z B , U ≡ ˜ B Ly B (cid:34) − YZ µ ( x ) ˜ B Lz ˜ B Ly (cid:35) . (37)An important feature of the latter parameter is that the contribution from the large-scaleperturbation vanishes upon taking the ( y, z ) plane average since U vanishes under suchspatial average, although U does not.Carrying out the same method used above, we also obtain (cid:10) k (cid:107) (cid:11) k ⊥ ,x = (cid:28) U B B z k ⊥ (cid:29) x = 0 , (38)as well as (cid:10) k (cid:107) (cid:11) k ⊥ ,x (cid:39) (cid:42) k ⊥ B B z (cid:90) ∞−∞ e − k ⊥ ∆ x (cid:34) T (cid:18) k ⊥ L s (cid:19) ∆ x − T U k ⊥ L s ∆ x + U k ⊥ (cid:35) d ∆ x (cid:43) x = (cid:28) B B z (cid:18) T L s + U k ⊥ (cid:19)(cid:29) x . (39)Thus, so long as 2 U k ⊥ L s /T (cid:29)
1, we have∆ k (cid:107) = (cid:115)(cid:28) B B z (cid:18) T L s + U k ⊥ (cid:19)(cid:29) x (cid:39) B B z (cid:113) (cid:104) U (cid:105) x k ⊥ ∝ k ⊥ , (40)resulting in a scale-independent anisotropy l (cid:107) /l ⊥ ∝ l ⊥ . Here, we emphasize that U/T canstill be small for the condition 2 U k ⊥ L s /T (cid:29) L s k ⊥ ,with k ⊥ being the wave number of small-scale modes. C. Local energy budget in k ⊥ space The impact of non-negligible dissipation on the structure of the spectrum has been studiedby considering the local energy budget in the k space . We will follow this methodol-ogy here, albeit in the context of turbulence with scale-independent anisotropy instead ofturbulence that is critically balanced one, as discussed in Section III B.Under the local interaction assumption, the local energy budget in the k ⊥ -space naturallyarises from considerations of the effective damping and classical resistive diffusion, : − DS − p/ ( k ⊥ λ ) p E ( k ⊥ ) − τ ∗− η ( k ⊥ λ ) E ( k ⊥ ) = dT ( k ⊥ ) dk ⊥ , (41)13here τ ∗ η ≡ λ /η , T ( k ⊥ ) is the energy forward transfer rate at scale k ⊥ , and E ( k ⊥ ) = v ( k ⊥ ) /k ⊥ is the energy density in the k ⊥ space. The energy budget Eq. (41) can be solvedto yield the energy spectrum if T ( k ⊥ ) can be written as a function of E ( k ⊥ ). The traditionalscaling for MHD turbulence without scale-dependent alignment indicates that T ( k ⊥ ) = v ( k ⊥ ) τ nl (cid:39) v ( k ⊥ ) l ⊥ l (cid:107) v A . (42)Here, v ( k ⊥ ) represents the kinetic perturbation at k ⊥ scale, and v A is the Alfv´en speedmeasured with the guide field. Due to the equipartition of kinetic and magnetic energy, T ( k ⊥ ) also represents the forward cascade of magnetic energy as v ( k ⊥ ) = v A (cid:16) ˜ B ( k ⊥ ) /B (cid:17) with ˜ B ( k ⊥ ) as the magnetic perturbation at k ⊥ scale.Using the scale-independent anisotropy discussed before, the forward transfer rate is now T ( k ⊥ ) = v ( k ⊥ ) l ⊥ αv A , α ≡ l ⊥ l (cid:107) . (43)Here, α is a constant characterizing the scale-independent anisotropy, the value of whichwill be extracted from numerical simulations of tearing turbulence. We follow the closuretechnique used in Refs. [21] - [23], and write the forward energy transfer rate T ( k ⊥ ) = v ( k ⊥ ) k ⊥ ( αv A ) − = E ( k ⊥ ) k ⊥ ( αv A ) − v ( k ⊥ ) = E ( k ⊥ ) (cid:15) / k / ⊥ ( αv A ) − / . (44)Here, we have used the closure v ( k ⊥ ) (cid:39) (cid:15) / k − / ⊥ ( αv A ) / . This closure technique effec-tively builds the inertial power-law behavior into the energy spectrum as an asymptote in thelow-damping limit. Therefore, as can be seen later in this section, the spectrum approachesa simple power law when the effective damping vanishes.Substituting Eq. (44) into Eq. (41), we obtain a linear first order ordinary differentialequation for the energy spectrum: ddk ⊥ E ( k ⊥ ) k ⊥ = − E ( k ⊥ ) − (cid:104) DS − p/ ( k ⊥ λ ) p − / + 2 τ ∗− η ( k ⊥ λ ) / (cid:105) × E ( k ⊥ ) (cid:15) − / ( αv A ) / λ / . (45)The tearing turbulence spectrum with linear stabilities act as effective damping is then: E ( k ⊥ ) ∼ ( αv A ) / (cid:15) / k − / ⊥ exp (cid:34) − DS − p/ (cid:15) − / λ / ( αv A ) / p − ( k ⊥ λ ) p − / (cid:35) × exp (cid:34) − τ ∗ η (cid:15) − / λ / ( αv A ) / / k ⊥ λ ) / (cid:35) , (46)14ith p = 6 / p = 4 / D is given by Eq. (22) and Eq. (23). From Eq. (46), it can be seen thatthe primary impact of effective damping is an exponential multiplier on the original powerlaw. In the limit of small effective damping, the spectrum recovers the simple power-lawbehavior predicted by the assumption of an inertial range. Also, the power-law behavior inthe no damping limit tends to be k − / ⊥ due to the scale independent anisotropy l (cid:107) ∝ l ⊥ . IV. SIMULATIONS
In this section, the analytical result from Section III will be tested against resistive MHDsimulation using the same set of equations (1) - (4) described in Section II. Specifically,weare concerned with the structure of the energy spectrum E ( k ⊥ ) and the dependence of k ⊥ /k (cid:107) on k ⊥ . We will first examine the strong guide field case where B y /B z ∼ O (10 − )and ˜ B L /B z ∼ O (10 − ). We will then consider the case of comparable guide field where B y /B z ∼ O (1) and the large-scale perturbed field is only one order of magnitude smallerthan the guide field ˜ B L /B z ∼ O (10 − ). In the latter case, the Kubo number for thelarge-scale perturbation is κ = (cid:16) ˜ B L /B (cid:17) (cid:0) L (cid:107) /L ⊥ (cid:1) ≥
1, corresponding to the regime whereturbulent shearing is comparable with parallel propagation. Here, the Kubo number κ isequivalent to the χ used by Goldreich and Sridhar in Ref. [17]. Numerical observation ofthe magnetic shear length indicates we have 2 ˜ B L L s k ⊥ /B z > O (10 ) when k ⊥ >
20 forboth of the above cases. The numerical algorithm will follow that of Ref. [6] and Ref. [35].Five-point finite difference scheme is used to calculate derivatives, and trapezoidal leapfrogis used for time stepping scheme. The resistivity is set to be η = 1 × − , and the magneticPrandtl number is P m ≡ ν/η = 1, and thus we are in the viscous regime discussed above. Inthe numerical scheme, an additional artificial fourth-order dissipation is also implementedto damp small-scale fluctuations at grid size. This should not be confused with the realhyper-dissipation self-consistently generated by the nonlinear terms .15 a) (b)(c) (d)FIG. 2. The perturbed energy profile for different times. Most of the perturbed energy locateswithin x ∈ [ − . , . A. Strong guide field case
In this subsection, we will compare the tearing turbulence in a strong guide field withour previous theoretical model. Let X = 2. Y = 4, Z = 20, B y (0) = 1 . B = 10.At the beginning of the simulation, small initial perturbations with harmonics m/n = 3 / / / x ∈ [ − . , . y - z plane, providing us the sum of perturbationenergy over the whole spectrum, (cid:104) ˜ v (cid:105) and (cid:68) ˜ B (cid:69) , given by (cid:68) ˜ B (cid:69) = (cid:10) B (cid:11) − (cid:104) B (cid:105) , (47) (cid:10) ˜ v (cid:11) = (cid:10) v (cid:11) − (cid:104) v (cid:105) . (48)These perturbation energies as functions of x are plotted in Fig. 2 for different times. Initially,the dynamics is dominated by a few large-scale unstable modes, and the envelope of theirmode structure determines the perturbation energy profile, as can be seen from Fig. 2 (a) andFig. 2 (b). Later, the initial islands overlap with each other and generate a large spectrum ofsmall-scale modes, and the perturbation energy profile becomes smooth in the core region,as seen in Fig. 2 (c) and Fig. 2 (d). By the time the tearing turbulence enters quasi-steadystate, both the magnetic and kinetic energy perturbations are confined within the region x ∈ [ − . , . k modes actually agrees rather well with the equipartition assumption.The alignment of the turbulent eddies to the local mean field is also of interest. That is,we wish to know whether or not the eddies have elongated structure along the mean fielddirection, as would be expected from highly magnetized MHD turbulence. Here, we lookat the local property of magnetic perturbation for a given x position, and perform Fourierdecomposition along y and z direction for all components of magnetic field. We take thezeroth order harmonic as the local mean field for the given x position, while all the otherharmonics correspond to modes with various scales. The alignment of those modes to thedirection of local mean field line can be represented by looking at k · B . We once againwrite k · B = B y k y − B z k z = k y YZ (cid:16) µ − nm (cid:17) B z . (49)17 a) (b)FIG. 3. The alignment of turbulent eddies to local mean field line at x = 0. The color hererepresents the logarithm of perturbed energy density of (a) magnetic perturbation, and (b) kineticperturbation. The black dashed lines are the contours of k · B . It can be seen that the turbulenceis highly anisotropic, and tend to align with the direction of strong mean field. The red dashedlines represent the contours of k ⊥ in ( m, n ) plane. Such alignment of small-scale tearing modes can then be checked by looking at the distri-bution of the 2-D perturbed energy spectrum (cid:12)(cid:12)(cid:12) ˜ B k (cid:12)(cid:12)(cid:12) and | ˜ v k | in ( m, n ) space. The result for x = 0 is shown in Fig. 3, where the logarithm of the perturbed energy is plotted as a functionof mode number m and n . The black dashed line represents the contour of k · B , with theone originating from the (0 ,
0) point corresponding to k · B = 0. It can be seen that theenergy spectrum strongly aligns with the local mean field, indicating a strongly anisotropicstructure. It is noteworthy that, for a magnetically sheared system, this localization in the k space directly corresponds to the localization of mode structures near their respective res-onant surfaces in configuration space. Due to this localized mode structure, the amplitudeof the mode decreases rapidly as we move away from its resonant surface. Thus, only modeswhich are near resonance (corresponds to low k (cid:107) ) can be seen from the spectrum shown inFig. 3, resulting in observed localization in the k space. This is especially true for high- k modes. The red dashed lines represent the contours of k ⊥ . The strong alignment behavior ofsmall-scale perturbation in the presence of strong guide field indicates that we have k (cid:107) (cid:28) k ⊥ ,and consequently k y + k z = k ⊥ + k (cid:107) (cid:39) k ⊥ . This confirms our previous assumptions.With the alignment of eddies known, we now look at how this highly anisotropic turbu-lence establishes itself. From Fig. 2 (c) and (d), it can be seen that the turbulence strength18 IG. 4. The time evolution of the tearing turbulence energy spectrum. At t = 0, there are onlyseveral unstable modes. These large scale modes gradually excite a spectrum of small-scale modesby interacting with each other. At a later time, the tearing turbulence reaches a quasi-steady state. is rather flat in the central region, this implies that we can use the local spectrum for agiven x to represent the evolution of global tearing turbulence. Here, we choose to look atthe spectrum evolution at x = 0. The energy density in k ⊥ space E ( k ⊥ ) can be obtained byintegrating over the red dashed lines in Fig. 3. The spectrum of E ( k ⊥ ) for several differenttimes is presented in Fig. 4. The logarithm of magnetic energy perturbation is shown asa function of the logarithm of the perpendicular scale k ⊥ . It can be seen that at t = 0there are only several large-scale unstable modes. Then the interaction of these large-scalemodes gradually stir up small-scale modes. At a later time, the tearing turbulence reaches aquasi-steady state as can be seen in Fig. 4. The structure of E ( k ⊥ ) spectrum changes verylittle from t = 63 . t = 109 . τ nl ∼ t = 85 .
6. It can be seen that for high- k modesthe kinetic and magnetic energy are approximately the same, while at the largest scale thereis a departure from equipartition. The departure does not significantly impact our theoreti-cal analysis in Section III C, since we are primarily concerned with small-scale modes which19 IG. 5. Comparison between kinetic and magnetic spectrum for t = 85 .
6. It can be seen that theequipartition of energy is reasonably satisfied for high- k ⊥ modes, while there is some departurefrom equipartition for large-scale modes. The perturbed magnetic energy is approximately oneorder of magnitude larger than the kinetic energy for the largest mode. are linearly stable rather than the unstable large-scale modes. As a side note, the fact thatthe magnetic perturbation is one order of magnitude larger than the kinetic perturbationfor largest scale modes is also consistent with the observation in Fig. 2, as the total magneticperturbation energy is also one order of magnitude larger than the total kinetic perturbationenergy.The next important property we are interested in is the structure function of the turbu-lence, which provides us information regarding the scale dependence of its anisotropy andthus has significant impact on the energy transfer rate and consequently the turbulencespectrum. We follow the procedure detailed in Ref. [6] and Ref. [36], and define the followingtwo-point structure functions: F k (cid:0) l (cid:107) , l ⊥ (cid:1) ≡ (cid:10) | v ( ζ + l ) − v ( ζ ) | (cid:11) , (50) F m (cid:0) l (cid:107) , l ⊥ (cid:1) ≡ (cid:10) | B ( ζ + l ) − B ( ζ ) | (cid:11) . (51)Here, ζ = ( x, y, z ) is the position of a random point in the configuration space, and l is arandom vector. Thus, ζ + l and ζ define a random pair of points in configuration space. Thebracket (cid:104) f (cid:105) here indicates an ensemble average over a large number of random pairs. Due20o the strong localization of mode structure demonstrated in Fig. 3, we look at a 2D versionof the structure function in our study. That is, we take the random pairs within the y - z plane for a given x instead of considering the full 3D space. The parallel and perpendicularcomponent of l is defined by the local mean field direction, which is calculated by averagingthe magnetic field at two points. We average over 10 random pairs of points, and obtainthe structure function for both kinetic and magnetic perturbation as functions of l (cid:107) and l ⊥ .The contours of this structure function in (cid:0) l (cid:107) , l ⊥ (cid:1) then reflect the anisotropy of the eddy atdifferent l ⊥ scales.To extract this anisotropy information, we search for the intersection of a given contourof F k (cid:0) l (cid:107) , l ⊥ (cid:1) or F m (cid:0) l (cid:107) , l ⊥ (cid:1) with the l (cid:107) and l ⊥ axis respectively. Thus we can obtain a pairof l (cid:107) and l ⊥ for a given contour of F k (cid:0) l (cid:107) , l ⊥ (cid:1) or F m (cid:0) l (cid:107) , l ⊥ (cid:1) , the ratio of which represents theanisotropy at a given scale. A scan of these l (cid:107) and l ⊥ pairs then shows the scale dependenceof turbulence anisotropy. The anisotropy thus obtained is plotted in Fig. 6, with two scalings k (cid:107) ∝ k ⊥ and k (cid:107) ∝ k / ⊥ plotted as black dashed lines. It can be seen that the anisotropicbehavior of simulation result largely agrees with our analytical model and is mostly scale-independent. There is some discrepancy between the length scale of kinetic and magneticperturbations, which might be the consequence of their different distribution width in ( k y , k z )space as can be seen in Fig. 3. The ratio between parallel and perpendicular length scaleultimately deviates from the scale-independent scaling at the very small scale where classicaldissipation kicks in. Lastly, it is observed that l (cid:107) is two orders of magnitude larger than l ⊥ ,thus we hereby take α ∼ − as a reasonable estimation. This estimation also agrees withour prediction by Eq. (40) since we also have ˜ B L /B z ∼ O (10 − ).With the characteristic structure known, we can finally check our analytical model givenby Eq. (46) against the simulation results. The magnetic perturbation spectrum for tearingturbulence is shown in Fig. 7 for t = 85 . x = 0. The simulation result is comparedwith three analytical models: our damped turbulence model as shown in Eq. (46), a sim-ple power law E ( k ⊥ ) ∝ k β ⊥ as a result of the traditional inertial range argument, and thespectrum produced by Eq. (46) if only the resistivity is included as damping. From numer-ical observation, we estimate the characteristic length scale to be λ (cid:39)
10 near the centralflattened region where the resonant surfaces of the concerned modes lie. Here λ can belarger than X since it only serves as an indication of the local magnetic field gradient. Theonly free parameter in Eq. (46) is then the energy injection rate (cid:15) , which will be used to21 − − r ⊥ ∼ k − ⊥ r k ∼ k − k k k ∼ k ⊥ k k ∼ k / ⊥ x = 0 . t =77.6 v B FIG. 6. The anisotropy of tearing turbulence for various scale l ⊥ . Both the kinetic and magneticperturbation is shown. The scaling of both k (cid:107) ∝ k ⊥ and k (cid:107) ∝ k / ⊥ are shown as black dashed lines.It can be seen that the turbulence anisotropy is scale independent k (cid:107) ∝ k ⊥ for most of the scales. fit the simulation result. On the other hand, the power index β in the simple power lawwill also be used as a free parameter to fit the numerical result. The fitting exercise yields (cid:15) (cid:39) . × − and β (cid:39) − .
0, with fixed parameters λ = 10 and α = 0 .
01. The fitted curvesare shown in Fig. 7. It can be seen that our analytical model is in better agreement with thesimulation result than either the simple power law or the spectrum obtained by assumingthat it is determined by the effect of resistivity only. It is noteworthy that although thefinal decay of the turbulence spectrum is due to the influence of resistivity, the actual curvedeviates from the inertial range curve due to the presence of effective damping. While thisdeviation might suggest that there exists an inertial range with a steeper slope represented,for instance, by the blue dashed line, this is not the case since the behavior seen is causedby slow exponential decay and cannot be represented accurately by a power law.
B. Weaker guide field case
The strong guide field case has been investigated in the previous subsection. Reasonableagreement has been found between the simulation result and our theoretical prediction22
IG. 7. The spectrum of tearing turbulence for t = 85 . x = 0, compared with three analyticalmodels. The black solid line represents the simulation result, the red dashed line is a fitting ofEq. (46) using only the energy injection rate (cid:15) as a free parameter, the blue dashed line correspondsto a simple power law E m ( k ⊥ ) ∝ k − . , and the green dashed line is the spectrum produced byEq. (46) if we only consider the resistive damping. The simulation result agrees rather well withour analytical prediction. obtained in Section III. The magnitude of the perturbation has been found to be two orderof magnitude smaller than the guide field. However, we are also interested in cases where theguide field is weaker, and the large-scale Kubo number κ = (cid:16) ˜ B L /B (cid:17) (cid:0) L (cid:107) /L ⊥ (cid:1) (cid:39)
1. Again, κ here is equivalent to the χ used in Ref. [17]. Note that, in this case of stronger turbulence,the perturbed field is still smaller than the guide field, although the Kubo number mayexceed unity due to anisotropy.The initial magnetic fields are now B y (0) = 1 . B = 2 .
5. To maintain a similarinitial safety factor profile with the one shown in Fig. 1, the system size is now X = 2, Y = 4 and Z = 4. We are mainly concerned with the anisotropic behavior and the energyspectrum of the turbulence, and we wish to determine whether or not the distinctive featuresexhibited in our weak turbulence simulation persist in this stronger turbulence case.We first examine the anisotropy. Again, we look at the contours of structure functionsfor both the kinetic and magnetic perturbation as described in Section IV A, and we use thesame technique detailed there to extract the turbulence anisotropy for different scales. The23 − − r ⊥ ∼ k − ⊥ − r k ∼ k − k k k ∼ k ⊥ k k ∼ k / ⊥ x = 0 . t =59.6 v B FIG. 8. The scale-independent anisotropy for a weaker guide field case. Although the turbulenceis now less anisotropic than in the strong guide field case, the fundamental scale-independentanisotropy remains the same comparing with what is shown in Fig. 6. x position is chosen at x = 0, and time t = 59 .
6, when the turbulence has already reachedthe quasi-steady state. The anisotropy is shown in Fig.8, with the two scalings k (cid:107) ∝ k ⊥ and k (cid:107) ∝ k / ⊥ plotted as black dashed lines. It can be seen that this stronger turbulence casestill follows the scale-independent anisotropy behavior described in Section III B and onlydeviates from it at very small scales. In fact, the scale-independent anisotropy is even bettercompared to that shown in Fig. 6, possibly due to a stronger large-scale perturbation.We then look at the structure of the turbulence spectrum. The magnetic perturbationspectrum for x = 0 and t = 59 . E ( k ⊥ ) ∝ k − . ⊥ and the spectral form predicted by ourmodel as described by Eq. (46), with fixed parameters λ = 10 and α = 0 .
05. The fittingresult returns (cid:15) = 8 . × − . Reasonable agreement is again found between the numericalresult and our prediction, with a gradual departure from the original k − / scaling well beforeentering the resistive dissipation scale.A noteworthy feature of this stronger turbulence case is that the Kubo numbers for thelargest perturbations exceed unity. From Fig. 8, it can be seen that the scale-independentanisotropy is approximately l (cid:107) /l ⊥ (cid:39)
20. At the same time, numerical observation from Fig. 924
IG. 9. The spectrum of tearing turbulence for t = 59 . x = 0, compared with two analyticalmodel. As is shown in Fig. 7, the simulation result agrees rather well with our analytical prediction,suggesting that our analytical model works well even for not-so-weak turbulence. indicates the largest scale perturbation has ˜ B L /B ∼ O (10 − ). Hence, for the large-scaleperturbations, we have κ (cid:39)
2, while for smaller scale perturbation the Kubo number steadilydecreases as the perturbation strength decreases. This is different from the critical balancescenario where the Kubo number remains on the order of unity across all scales. Thisdeviation from critical balance is very similar to that discussed by Huang et al. in Ref. [6].Thus, we conclude that our analysis can also be applied to the case where the nonlinearmixing is stronger than the linear parallel propagation, such as those reported in plasmoidturbulence, where the critical balance condition was frequently assumed to be true.
V. DISCUSSION AND CONCLUSION
Instability driven tearing turbulence in sheared magnetic field is studied in this work. Theturbulence consists of several large-scale unstable modes and a broad spectrum of small-scalelinearly stable modes which are excited by their large-scale counterparts. It is found thatthe linearly stable modes will act as an effective damping mechanism which has a weakerdependence on k ⊥ than classical dissipation. For inviscid and viscous regimes, the depen-dence scales as k / ⊥ and k / ⊥ respectively. The weak dependence indicates that this damping25echanism will manifest itself long before turbulence eddies reach the resistive or viscousdissipation scales. Consequently, a well-defined inertial range cannot be identified, anddamping must be considered at all scales. Furthermore, we argue that the tilting of shearedbackground field by large-scale perturbations will impose a scale-independent anisotropy forsmall-scale modes. This anisotropic behavior then determines the scale dependence of theforward energy cascade rate.With the knowledge of effective damping rate and energy cascade rate at hand, thestructure of this damped turbulence can be obtained by considering local energy budget inthe k ⊥ space. The key idea is that the difference of energy forward transfer rate between thetwo ends of any interval in the k ⊥ space corresponds to the damping within that interval.The resulting spectrum features a power law multiplied by an exponential falloff, as opposedto the pure power-law spectrum obtained by using the standard inertial range argument.The above analytical result is checked against visco-resistive MHD simulations. Theturbulence is found to be highly anisotropic and tends to align with the strong local mean-field direction. The two-point structure functions are calculated to investigate anisotropicproperty at different scales, and a scale-independent anisotropy is found, confirming our l (cid:107) ∝ l ⊥ argument. Furthermore, the equipartition between kinetic and magnetic energy isfound to be valid for the turbulence in question. The numerical result appears to agree wellwith our analytical model based on effective damping.The behavior of a stronger turbulence, where the Kubo number exceeds unity for certainscales, is also investigated. We find that the scale-independent anisotropy and the energyspectrum continue to hold for the stronger turbulence case, indicating that our analysisremains applicable even for the scenario where the perpendicular turbulence shearing isstronger than the parallel propagation.With this knowledge regarding spectrum structure, the next step would be consideringthe back-reaction of small-scale turbulence on large scales. This involves a sum of quadraticform of perturbed quantities over the whole spectrum, which requires knowledge regardingthe form of spectrum given by our study here. An example is the small-scale spreading ofmean field described by hyper-resistivity, as has been studied in Ref. [3] and Ref. [4]. Ouranalysis here provides a solid basis for future study along these lines. These studies are leftto future work. 26 cknowledgments The authors thank P. H. Diamond, X.-G. Wang, H.-S. Xie and L. Shi for fruitful discus-sion. This work is partially supported by the National Natural Science Foundation of Chinaunder Grant No. 11261140326 and the China Scholarship Council. A. Bhattacharjee andY.-M. Huang acknowledge support from NSF Grants AGS-1338944 and AGS-1460169, andDOE Grant de-sc0016470. Simulations were performed with supercomputers at the NationalEnergy Research Scientific Computing Center. D. Hu publishes this paper while working inITER Organization. ITER is a Nuclear Facility INB-174. The views and opinions expressedherein do not necessarily reflect those of the ITER Organization.
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