Effect of In-Plane Shear Flow on the Magnetic Island Coalescence Instability
Jagannath Mahapatra, Arkaprava Bokshi, Rajaraman Ganesh, Abhijit Sen
EEffect of In-Plane Shear Flow on the Magnetic Island CoalescenceInstability
Jagannath Mahapatra , ∗ Arkaprava Bokshi , Rajaraman Ganesh , † and Abhijit Sen Institute for Plasma Research, Gandhinagar, Gujarat-382 428, India and Homi Bhabha National Institute, Mumbai, Maharashtra-400 094, India
Abstract
Using a 2D Viscoresistive Reduced MagnetoHydroDynamic (VR-RMHD) model, the magneticisland coalescence problem is studied in the presence of in-plane, parallel shear flows. Extendingthe analytical work of Waelbroeck et al [33] and Throumoulopoulos et al [34] in the sub-Alfv´enicflow shear regime for Fadeev equilibrium, the super-Alfv´enic regime is studied for the first timenumerically. A wide range of values of shear flow amplitudes and shear scale lengths have beenconsidered to understand the effect of sub-Alfv´enic and super-Alfv´enic flows on the coalescenceinstability and its nonlinear fate. We find that for flow shear length scales greater than the magneticisland size, the maximum reconnection rate decreases monotonically from sub-Alfv´enic to super-Alfv´enic flow speeds. For scale lengths smaller than the island size, the reconnection rate decreasesupto a critical value v c , beyond which, the shear flow is found to destabilize the islands. The valueof v c decreases with decrease in the value of shear flow length scale. Interestingly, for our rangeof parameters, we find suppression of the Kelvin-Helmholtz instability in super-Alfv´enic flows evenwhen the shear scale length is smaller than the island width. Observation of velocity streamlinesshows that the plasma circulation inside the islands has a stabilizing influence in strong shear flowcases. Plasma circulation is also found to be responsible for decrease in upstream velocity, causingless pile-up of magnetic flux on both sides of the reconnection sheet. ∗ [email protected]; [email protected] † [email protected] a r X i v : . [ phy s i c s . p l a s m - ph ] F e b . INTRODUCTION Magnetic reconnection (MR) is a fundamental phenomenon in dynamically evolvingplasma systems that is responsible for relaxing the magnetic field topology by convertingthe magnetic energy into kinetic energy. This phenomenon heats up the plasma [1, 2] andis frequently observed in both laboratory and space plasma. Examples include saw-toothcrash driven by tearing mode instability (TMI) in tokamaks [3], solar flares, coronal massejection, magnetospheric substorms, etc. (see [4] and references there in). The most favor-able location for MR to occur is the thin current sheet that develops at the X -type magneticnull points generated through MHD instabilities, e.g. the TMI [5], plasmoid instability [15]and island coalescence instability [19–30], to name a few. Any change in plasma parametersaround these local X -points is known to affect the entire process of MR. Large number ofnumerical simulations have been conducted to understand the role of various parameters onMR, such as, the effect of non-uniform resistivity [6], presence of strong guide field [20], pres-ence of asymmetric magnetic field and density on the upstream side of reconnection currentsheet [7, 8] and more. Additionally, as the plasma bulk flow is common in fusion plasmaexperiments (e.g., generated indirectly by neutral beam injection or wave heating/currentdriving mechanisms) as well as in space plasma environment (e.g., caused due to plasma jetsfrom astronomical bodies, stellar wind, etc.), it can also affect MR in many important waysby altering the upstream and downstream flow pattern.As is well known, magnetic field can also affect the flow dynamics. For example, thepresence of magnetic field parallel to shear flow with velocity discontinuity generates Kelvin-Helmholtz instability (KHI) if the difference in velocity across the discontinuity layer isgreater than twice the Alfv´enic velocity ( v A ) [9, 37]. In the past, most of the research workon MR in the presence of shear flows have used two kinds of MHD equilibrium: the Harriscurrent sheet equilibrium [10] and the Fadeev or magnetic current island equilibrium [17](used interchangeably throughout this work). The Harris type equilibrium has a long thincurrent sheet which separates two regions of anti-parallel magnetic field. In the absenceof shear flow, when the current sheet becomes unstable to TMI, it breaks down into largemagnetic islands through MR. However, in the presence of in-plane parallel [11, 12, 14, 15]or anti-parallel [12, 13] shear flows, the rate of island formation, equivalently the reconnec-tion rate, reduces monotonically with increase of shear flow amplitude upto a critical value,2fter which the TM mode transits to the KH mode. This critical shear flow amplitude for aresistive MHD model is v A . Inclusion of Hall physics reduces the critical flow amplitude tosub-Alfv´enic values [16] enabling the KHI and TMI to couple. However, as stronger shearflows suppress the growth and size of islands, it becomes difficult to study the TM generatedmagnetic islands in the presence of Alfv´enic and super-Alfv´enic flows. One of the drawbacksof using the Harris type equilibrium is that the initiation of TMI driven MR requires eitheran external driving force (e.g. Newton’s challenge problem [18]) or the current sheet aspectratio needs to be very large (thin and long current sheet). Moreover, in natural reconnectingsystems, thin current sheets develop dynamically at the X -points. However, pre-formed cur-rent sheet structure of Harris equilibrium is less suitable to address reconnection physics (seeRef. [20] for further details). Interestingly, Fadeev equilibrium [17] has such features inher-ently built into it. As Fadeev equilibrium contains a 1D chain of current filaments separatedby X -type magnetic null points, in the presence of finite dissipation, the force of attractionbetween the parallel current strands brings them closer leading to the formation of a thinreconnection current sheet at the X -point and drives the MR. This self-driven mechanismfor MR is also one of the advantages of this equilibrium [20]. Mutually attracting magneticislands finally coalesce to form a bigger island.In the past, several attempts have been made to understand the island coalescence prob-lem. For example, stability analysis of a 2D island configuration has been studied analyt-ically by Finn and Kaw [27]. To understand the observed fast MR time scale, 2D islandcoalescence problem has been studied numerically using resistive MHD [21–24], Hall-MHD[28, 29], kinetic [25, 26], ten moment two-fluid [20] and hybrid [30] simulation models. Inthe above mentioned body of work, the role of several external parameters such as systemsize, guide field strength on the properties of MR have been addressed. However, the effectof shear flow on island coalescence has not been studied so far. Furthermore, Fadeev equilib-rium has well developed current islands to couple with both sub-Alfv´enic and super-Alfv´enicshear flows. Previously, several analytical studies have reported the shear flow effects on theKelvin-Stuart [27] island configuration (same as Fadeev equilibrium). Throumoulopouloset. al. [34] have constructed a class of magnetic island equilibrium in presence of shearflows by solving Grad-Shafranov equation. Considering only sub-Alfv´enic shear flows thatare relevant to fusion experiments, they have shown stabilization of island equilibrium andformation of pressure islands. Analytical work by Waelbroeck et. al. [33], using a resistive3HD model, verifies modest influence on the stability of magnetic islands when subjectedto a sub-Alfv´enic shear flow. The aim of our present work is to understand the effect ofshear flow on the island coalescence problem for a wide range of values of flow amplitude( v ) from sub-Alfvenic to super-Alfvenic and for a wide range of velocity shear scale length( a v ). Using VR-RMHD model and very high resolution computer simulations, we have in-vestigated Fadeev equilibrium in the presence of a tan-hyperbolic flow profile (shear flow issymmetric and anti-parallel on both sides of magnetic islands). We find that in line withprevious studies, the sub-Alfv´enic flows have negligible effect on coalescence instability asthe time required to attain the peak value as well the magnitude of reconnection rate isclose to that in the absence of shear flow. However, for super-Alfv´enic shear flows and irre-spective of shear length scales (compared to magnetic island width), we show the existenceof coalescence instability and suppression of the magnetohydrodynamic Kelvin-Helmholtzinstability (MHD-KHI). We also present extensive results on its quasi-linear and non-linearfate for the entire range of parameters addressed here.The rest of our paper is organized as follows: the VR-RMHD model and the BOUT++numerical framework used for its study are discussed in Section. II. Then, a series of bench-mark results for the resistive island coalescence instability in the absence of in-plane floware discussed in Section. III. In Section. IV, the dynamics of magnetic island - flow shearsystem for different v and a v values are discussed. Finally, our conclusions and potentialfuture work have been discussed in Section. V. II. NUMERICAL SETUP
We solve the VR-RMHD model in a 2D Cartesian geometry (see Ref. [21] and referencesthere in) using the BOUT++ framework [35, 36]. This model assumes plasma as an in-compressible magnetized fluid i.e. ∇ · u = 0. This assumption also reduces the full MHDequations to the vorticity ( ω )-vector potential (Ψ) formalism. The governing equations of4he ω -Ψ formalism read as [21] ∇ · u = 0 , ∇ · B = 0 (1) ∂ω y ∂t = [ ϕ, ω y ] − [Ψ , J y ] + ˆ ν ∇ ω y (2) ∂ Ψ ∂t = [ ϕ, Ψ] + ˆ η ∇ Ψ (3)In the above equations, the out-of-plane or y − component of vorticity ω = ˆ y · (cid:16) (cid:126) ∇ × u (cid:17) = −∇ ⊥ ϕ ; u x = − ∂ z ϕ , u z = ∂ x ϕ , where u (= u x ˆ x + u z ˆ z ) is the in-plane (“poloidal”) velocity ofthe plasma and ϕ is the corresponding stream function. Similarly, the out-of-plane currentdensity J y = ˆ y · (cid:16) (cid:126) ∇ × B (cid:17) = −∇ ⊥ Ψ; B x = − ∂ z Ψ, B z = ∂ x Ψ, where B (= B x ˆ x + B z ˆ z )is the in-plane magnetic field and Ψ is the corresponding magnetic vector potential. Theabove equations are normalized as follows: length L to system length L x , velocity to Alfv´envelocity v A = B/ √ µ ρ and time to the Alfv´enic time t A = L/v A . Here, normalized density ofplasma ρ = 1 and magnetic permeability µ = 1. The normalized viscosity ˆ ν and resistivityˆ η are defined as ˆ ν = ν/ ( Lv A ) and ˆ η = η/ ( µ Lv A ), where ν is the kinematic viscosity and η isa constant resistivity. The main variables ω y and Ψ are normalized as ω y L/v A and Ψ / ( BL ).The Poisson bracket is defined as [ f, g ] z,x = ( ∂ z f )( ∂ x g ) − ( ∂ x f )( ∂ z g ).Equations (2) and (3) are solved in a 2D uniform Cartesian grid (0 ≤ x ≤ L x and0 ≤ z ≤ L z ) where L x = L z . For the rest of the paper, we use dimensionless, normalizedquantities unless otherwise specified. Initial equilibrium current density J y , vorticity ω y and perturbed vector potential Ψ profiles are[21], J y = 1 − (cid:15) a B (cid:104) cosh (cid:16) x − L x / a B (cid:17) + (cid:15) cos (cid:16) za B (cid:17)(cid:105) = ⇒ Ψ = − a B ln (cid:20) cosh (cid:18) x − L x / a B (cid:19) + (cid:15) cos (cid:18) za B (cid:19)(cid:21) ω y = v a v (cid:104) cosh (cid:16) x − L x / a v (cid:17)(cid:105) Ψ = A cos( z ) sin( πx/L x ) (4)Here, the parameter (cid:15) = 0 . a B = 1 / π determines the sim-ulation domain size as L x = L z = 4 πa B = 2, Ψ is the perturbed vector potential with5mplitude A = 0 .
01. For all our simulations, we have set ˆ η = ˆ ν = 10 − . In the limitof (cid:15) = 0, Fadeev equilibrium reduces to the Harris current sheet of shear width a B . Fig.6a shows spatial profiles of J y (left panel colormap) and Ψ (left panel contour) and ω y (right panel colormap, for a v = 2 a B case). Periodic and Dirichlet boundary conditions arerespectively implemented along the z and x directions. In BOUT++, we have used FFTbased calculations along z direction and finite difference based calculations along x direction.Value of J y , Ψ , ω y and ϕ at x = 0 and 2 is zero (Dirichlet boundary). All the runs arecarried out for two different grid sizes d x = 4 × − , d z = 2 × − ( N x = 512 , N z = 1024)and d x = 2 × − , d z = 10 − (corresponding N x = 1024 , N z = 2048) As given in Refs.[22, 23], the width of the reconnection current sheet (say δ ) generated during island coa-lescence ∼ ˆ η / when ˆ η ≥ − . In our case, ˆ η = 1 × − (corresponding δ (cid:39) . z -directional grid size is sufficient enough to resolve the reconnection current sheet.Equations (2)-(3) are solved using the BOUT++ framework (http://github.com/boutproject) which was originally developed for tokamak edge turbulence studies. The BOUT++ frame-work uses a finite difference technique based 3D nonlinear solver. Recent developments andthe use of “Object Oriented Programming (OOP)” concepts of C++ language have enabledthe users to solve arbitrary number of coupled, general partial differential equations [35, 36].In solving our set of VR-RMHD equations, we have used an energy conserving Arakawabracket method for calculating the nonlinear terms and the implicit CVODE time solverfrom the SUNDIALS [41] package. For all the runs presented here, the time solver takes atime step of 0.05 t A .To check the correctness of the newly implemented model, we first test for energy con-servation. In ideal-RMHD, total energy ( E ideal = 1 / (cid:82) d τ ( |∇ ⊥ ϕ | + |∇ ⊥ Ψ | )) of the systemremains conserved. Taking J y (see Eq. 4) as the initial profile and ˆ η = ˆ ν = 0, E ideal ( t ) asa function of time is plotted in Fig. 1. Constant E ideal ( t ) values at our presently workinggrid sizes (1024 × × τ = d x d z and the operator ∇ ⊥ = ˆ x∂ x + ˆ z∂ z . One of the importantparameters to investigate is the reconnection rate. In literature [21], this reconnection rate( M ) is calculated from the reconnection flux ψ r as M = ∂ t ψ r ( ψ r = Ψ X − Ψ O , with Ψ X and Ψ O the magnetic flux at X -point and O-point, respectively). In the absence of in-plane6IG. 1: Total energy E ideal ( t ) in ideal MHD limit for 5 different grid sizes.shear flows, the line joining the O-points always aligns itself with the line x = 1 (see forinstance Fig. 2), hence it becomes easy to calculate M from the time derivative of ψ r .However, in the presence of shear flows, the line joining the O-points gets twisted because offlow dynamics and it becomes numerically expensive to compute the reconnection rate fromreconnection flux. Reconnection rate can also be calculated by measuring the reconnectionelectric field E y at the X -point using Eq. 3, as E y = − ηJ y | X [19, 21]. In the absence ofshear flows, the nonlinear term (first term in right side of Eq. 3) has negligible contributionfor the calculation of E y inside the reconnecting current sheet [21]. In the presence of shearflows, this also found to be valid. Hence, in this work, the reconnection rate is calculatedas M = E y = − ηJ y | X for our benchmark studies in the absence of shear flows as well asstudies with shear flows. III. ISLAND COALESCENCE WITHOUT SHEAR FLOWS: BENCHMARK
For the benchmark purpose, we have considered the resistive island coalescence problem[21–23] without any shear flow. Benchmark study uses the initial profile of equilibrium J y and perturbed Ψ (as given by Eq. 4) with v = 0. We have used the grid size d z = 5 × − ,d x = 1 × − . Six different resistivity values have been considered between 5 × − and2 × − (see Fig. 3b). Here our length scale L = 2, whereas it is L = 1 in Ref. [21] becauseof their quarter domain simulation; our time scale is therefore twice of that in Ref. [21]. In7 a)(b)(c)(d) FIG. 2: Fadeev equilibrium evolution in the absence of shear flow. Left panel shows J y (colormap), Ψ (contours) and right panel shows ω y (colormap), velocity (streamlines) atvarious time points t : (a) 0 . t A (b) 2 . t A (c) 3 . t A and (d) 5 . t A .the absence of in-plane shear flow, the time evolution of a Fadeev equilibrium is shown inFig. 2, similar to Fig. 2 of Ref. [21]. As discussed earlier, Fadeev equilibrium is an idealMHD equilibrium, hence when perturbed with finite resistivity, the plasma is able to breakthe frozen-in condition and cross the X -point [27]. This allows the attraction force betweenthe current filaments to become dominant. The typical perturbation profile Ψ , having amaxima of magnetic flux at the X -point, further accelerates the instability. This movement8f islands towards the X -point is shown in Fig. 2a. Fig. 2b shows that the coalescenceprocess has started and the reconnection sheet has formed at the X -point. In Fig 2c, thecurrent sheet is fully developed and the reconnection rate is at its maximum. After this,the magnetic flux piles up on both sides of the current sheet causing a slow down of thecoalescence process, hence reconnection rate. The reconnection rate at these time framesare marked in Fig. 3a. In Fig. 1 of Ref [21], the reconnection rate for ˆ η = 2 × − attainsa peak value at t (cid:39) t A . Likewise in our case, as seen in Fig. 3a, E y peaks around t (cid:39) . t A (recall that our t A is twice that defined in Ref. [21]).Furthermore, to verify the dependence of reconnection rate on resistivity, we have plotted (a) (b) FIG. 3: Left panel shows the reconnection rate vs. time plot for ˆ ν = ˆ η = 10 − . Points (a),(b), (c) and (d) correspond to those shown in Fig. 2. Right panel shows the maximumreconnection rate vs. normalized resistivity.the maximum reconnection rate versus different ˆ η values in Fig. 3b. As reported in Ref.[21], the reconnection rate scales as ∝ ˆ η / (Sweet-Parker scaling) for resistivity lower thana critical value [31] (here, for ˆ η ≤ − ). This clearly verifies the correctness of our solver. IV. ISLAND COALESCENCE WITH SHEAR FLOWS
We now turn on the shear flow through an initial vorticity profile as given in Eq. 4.The shear flow profile and simulation parameters are chosen such that in the absence ofcurrent filaments ( (cid:15) → v = 1 . v A and a v = a B / m = 3 ( m is the z -directional/periodicmode number) mode is the most unstable mode followed by the m = 2 for both HD-KHI and MHD-KHI (see Fig. 4); in the nonlinear regime other modes with m > m = 2 like structure may beexpected to be strongly affected by both linear and nonlinear KHI and vice-versa.Effects of in-plane shear flow in the island coalescence problem are studied by changingFIG. 4: Power spectrum plot of u x ( x = 1 , k z , t ) for MHD-KHI for a v = 0 . a B and v = 1 . v A .three important parameters in the vorticity (or velocity) profile: (1) flow shear strength v (2) shear width a v and (3) the direction of shear flow parallel or anti-parallel to the magneticfield. Parameters used for these simulation are given in Section II.10 . Varying flow shear amplitude v At first, we study the effect of shear flow with different amplitude values, keeping theshear length scale fixed at a v = 2 a B (shear flow length scale is larger than the island size,see Fig. 7a). As discussed in literature [24, 25], the x -directional width, a I , of FadeevFIG. 5: Time evolution of the reconnection electric field ( E y ) at the X -point for a v = 2 a B ,with v varied between 0 . v A to 1 . v A .equilibrium island system is decided by the parameter (cid:15) as a I = 2 √ (cid:15) (1 − (cid:15)/ a B . For (cid:15) (cid:39) . a B (cid:39) a I (for comparison , see Fig. 7a). Here, throughout this work, the velocityshear width a v is compared with a B . In this case study, v is varied keeping the flow shearwidth fixed at a v = 2 a B . In Fig. 5, the time variation of the reconnection electric field isshown for different v values. From this figure, it can be observed that for lower values ofshear flow amplitude ( v (cid:46) . E y vs. time matches with the no-shear flow ( v = 0)curve (see Fig. 3a). For these v values, three distinct phases can be identified: (a) Thereconnection rate first increases slowly (sub-exponential increase in E y ) upto the time (cid:39) t A ,slowly displacing the islands from their initial positions towards the X -point. (b) The linearphase of the coalescence instability continues upto (cid:39) . t A when the peak reconnection rateis achieved; in this phase both islands accelerate towards the X -point (exponential increasein E y ) resulting in the thinning of the reconnecting current sheet. (c) This motion causes themagnetic flux to pile up on both sides of the X -point resulting in a slowing down of islandmotion and decrease in the E y value. The islands bounce back and forth several times and11 a)(b)(c)(d)(e) FIG. 6: Left panel showing J y (colormap), A y (contours) and right panel showing ω (colormap), velocity (streamlines) at times t = (a) 0 (b) 2 t A (c) 4 t A (d) 8 t A and (e) 15 t A for a v = 2 a B and v = 1 . v A . Width of streamlines represent flow magnitude.finally the coalescence process completes. Hence, for lower shear flow strengths (comparedto v A ) and larger shear scale length (compared to a B ), in-plane flows negligibly affect the12verall coalescence process.For v ≥ . v A , the slowly growing phase of E y starts showing oscillations driven bystronger shear flows trying to peel off the islands and altering the magnetic field profile inthe vicinity of the X -point. In this phase, as v is increased, the magnitude of oscillationin E y increases. In Fig. 5, one can observe that even when the strength of shear flowis super-Alfv´enic ( v = 1 . , . first phase , the valueof E y continues to increase in the second phase . This indicates the survival as well as thecontinuation of current island coalescence in super-Alfv´enic shear flows when the initial shearflow scale length a v = 2 a B . Another interesting point to notice is the decrease in magneticflux pile-up as shear flow amplitude increases. Upto v = 0 . v A , one can notice bouncing ofislands after initial merging (decrease in E y value after peak reconnection, see Fig 5). As v increases further, there is no clear sign of the second peak in E y following the first maximawith the E y decreasing continuously. This indicates that with strong shear flows, the rateof flux pile up reduces, causing a slowing down of merging.In Fig. 6, the time evolution of islands is shown for v = 1 . v A . Fig. 6a shows the initialprofiles. The effect of shear flow can be seen in Fig. 6b where the islands get displaced alongthe x -direction. Also, the vorticity profile has now changed from a single sheet to an m = 2( m is the poloidal/z-directional mode number) like profile, similar to the initial J y profile.The velocity streamline shows plasma circulation inside the current islands. The flow speedinside the current island is much less than the outer shear flow magnitude With v > v A [37],the MHD-KHI tries to destabilize the m = 1 mode (which is the initial seed perturbationΨ ). But the rotational flow inside the islands (see Fig. 6b-6c) stabilizes them against theshear flow and suppresses the KHI, even though the v > v A condition is satisfied. Thesestabilized magnetic islands eventually become susceptible to the coalescence instability. Thecirculation is the potential cause of suppression of upstream flow (flow into the reconnectionsheet) causing less pile-up of magnetic flux and smaller reconnection rate. After islandmerging, a large island survives with rotation of plasma column. B. Varying velocity shear scale length a v To see the effect of velocity shear length, we take a v = 2 a B , a B , a B / , a B /
4, and foreach a v value, v is scanned over a range of values between 0 . v A and 1 . v A . For comparison,13 a) (b) FIG. 7: (a) Equilibrium vorticity profiles and J y profile at z = 0 .
5. (b) Reconnectionelectric field E y vs. shear flow strength for different shear flow scale length for two set ofgrid sizes.we have plotted the shear width of the initial vorticity profiles (see Eq. 4) along with theinitial J y ( x, z = 0 .
5) profile at the location where the width of island is maximum. In Fig.7a, both the velocity shear width and island width almost matches for a v = a B (since islandwidth a I (cid:39) a B ). One may expect a strong influence of shear flow on the island dynamicsfor shear width smaller than island width. In this section, we have discussed the evolutionof islands in presence of shear flow when a v = 0 . a B .Fig. 7b shows the variation in the peak values of E y for the full parametric scan over v and a v . For a v = 2 a B , the peak E y value decreases monotonically with the increase of v ,as discussed in the previous section. As E y ∝ − J y , E y attains its maximum value when thecurrent density in the reconnection current sheet is minimum (negatively maximum). As a v = 2 a B , current islands undergo coalescence process without much distortion. Higher v values induce stronger plasma circulation inside the islands and this in-turn decreases theupstream velocity of plasma into the reconnection sheet causing a monotonic decrease inpeak E y value. However, for a v ≤ a B , shear flow is trying to destabilize the islands. For a v = a B , . a B and 0 . a B , the peak E y first decreases upto a critical value of v , we callit v c ; these values are 0.9 v A , 0.7 v A and 0.5 v A respectively. Up to v c , the shear flow isnot strong enough and in these cases, the peak E y is the reconnection rate driven by thecoalescence process. For v > v c , the shear flow becomes stronger and tries to peel off theislands. This peeling also changes the current distribution near the X -point, generating very14hin current sheets which we are measuring as oscillations in the temporal evolution of E y .Hence, here the peak E y value is not because of coalescence driven reconnection, althoughthe islands coalesce after a long time (see Fig. 8). In Fig. 8a, 8b and 8c, one can clearlynotice the peeling off effect of shear flow on the magnetic islands. However in Fig. 8d, at16 t A , vorticity patches have been formed with flow circulation coinciding with the magneticislands. This confirms the stabilizing effect of circulation on the islands. In Fig. 8e, at 37 t A ,these surviving islands coalesce to form a large single island, as in the case of a v > a B .Comparison of peak E y vs. v in Fig. 7b at two different grid size is also plotted. For v ≤ v c , the peak E y is same for both lower and higher resolution. This implies, for thesecases, the current sheet at the X -point is well resolved by lower and higher resolution.However, for v > v c , the shear flow generates very thin current sheets by destabilizingcurrent islands. The lower resolution is not enough to resolve these thin current sheets.This explains the data points for lower and higher resolution are matching for v ≤ v c butnot matching for v > v c .As the MHD-KHI gets destabilized in anti-parallel magnetic field configuration [9], shearflows anti-parallel to magnetic field is also tested for Fadeev equilibrium. As reported inthe literature for TMI cases [13], we found no difference in the results compared to parallelconfiguration discussed above One explanation for this could be that the stabilizing role ofthe flow induced plasma circulation inside the islands is independent of direction of shearflow and the KHI gets suppressed for the range of v and a v discussed here. Higher values of v may destabilize the current island and generate KHI. Then one can observe the differencein growth rate of KHI in parallel and anti-parallel configuration. V. SUMMARY
In the present work, using a 2D VR-RMHD model implemented in the BOUT++ frame-work, we have carried out a systematic study of the effect of in-plane shear flow on the islandcoalescence problem. Our results in the absence of in-plane shear flow is in very good agree-ment with previously reported work for our set of parameters. We have applied in-planeshear flows, both parallel and anti-parallel to the magnetic fields. We have calculated thepeak reconnection electric field ( E y ) at the X -point for different v values keeping a v = 2 a B .To see the effect of shear length scale, we have calculated E y for four different value of a v .15 a)(b)(c)(d)(e) FIG. 8: Left panel shows J y (colormap), A y (contours) and right panel shows ω (colormap), velocity (streamlines) at time t = (a) 2 . t A (b) 4 . t A (c) 10 . t A (d) 16 . t A and (e) 37 . t A for a v = a B / v = 1 . v A .The main findings are as follows:1. For shear scale length larger than island width, the peak E y or reconnection rate16ecreases as v increases.2. Increase in the value of v also reduces magnetic flux pile-up as the plasma rotationflow inside the islands induced by shear flow reduces the upstream velocity.3. For the range of a v and v values considered here, the island coalescence instabilitydominates. For v > v A , the KHI tries to destabilize the islands, but after set up ofplasma circulation inside the islands4. Anti-parallel shear flows have the same effect on the current islands as the parallelshear flows (hence not shown).Present study is confined to a single uniform resistivity value for which the plasma is pre-dominantly collisional. Hence the two fluid effects (Hall physics) and kinetic effects (FLReffects) are safely ignored. In case of TMI, with slab geometry, several authors have reporteda quadrupolar out-of-plane magnetic (say B || ) field induced by out-of-plane shear flow [38].Hall physics also generates quadrupolar B || because of Hall electric field [29]. Hence, strongout-of-plane flows distort the Hall-induced B || and generate secondary islands [39]. Also, inthe past, for 3D cylindrical geometry, the effect of axial and helical flows on resistive TMIhave been shown to be important [40]. Hence, we believe it would be very interesting tostudy the effect of out-of-plane flows and helical flows including Hall physics and kineticeffects on 2D island structure as well as 3D flux tubes. These problems will be addressed infuture. ACKNOWLEDGMENT
The simulations are performed on the Antya cluster at the Institute for Plasma Research(IPR). We would like to thank Dr N. Bisai, IPR for his valuable inputs.17
ATA AVAILABILITY
The data that support the findings of this study are available from the correspondingauthor upon reasonable request.
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