MHD instabilities and Magnetic Reconnection

Transcription

1 MHD instabilities and Magnetic Reconnection Gunnar Hornig STFC Advanced Summer School 2014

2 Contents Motivation Stability and Instability Kink Instability Tearing Instability Reconnection: History, Literature, Occurrence 2D Reconnection (MHD) 2D Collisionless Reconnection 3D Reconnection Problems

3 Motivation One of the key questions of solar physics (plasma astrophysics) is how the magnetic field interacts with the plasma (e.g. dynamo problem or coronal heating problem). This includes the exchange of energy and momentum between the magnetic field, and the plasma. The latter consists of momentum/energy of the bulk velocity and the random (thermal) velocity. For a high temperature plasma (e.g. the solar corona) the exchange of energy between the magnetic field and the thermal energy seems to be small due to the low value of the resistivity η ~ Τ 2 2µ 0 + r 1 E µ 0 = J E = v j j 2 magnetic energy Poynting flux work done by Lorentz force Ohmic heating Magnetic energy j 2 thermal energy µ v v v j bulk kinetic energy

4 Motivation oth, the resistive and the viscous heating terms are small unless there are strong gradients in the magnetic field (electric currents) or the velocity field. Also if strong gradients in the fields develop the assumptions for the derivation of these terms breaks down (e.g. macroscopic length scale >> mean free path) and the coefficients, and possibly also the form of the terms, may become anomalous. Processes which generate strong gradients are typically instabilities. These instabilities in turn lead to magnetic reconnection.

5 Stability and Instability In all four cases the ball is in equilibrium, but a) is globally stable; b) is an unstable equilibrium (but stable equilibria are close by); c) is an unstable equilibrium (no stable equilibrium exists) d) is a stable equilibrium (for small perturbations) For large perturbations b) behaves like a) and d) like c). MHD stability: Take an MHD equilibrium (e.g. pressure equilibrium, force-free equilibrium,..) and perturb it. Does the amplitude of the perturbation increase (instability) or decrease (stability)? Complication: Other than in the cartoon the space of possible perturbations is infinite dimensional. Typically only a subspace of possible perturbations is tested. This allows only to prove instability but not stability.

6 Linear Stability Linear stability: you assume that all quantities are of the form X(x,t)=X 0 (x)+x 1 (k, )exp(i(k x t)) Here X 0 is the equilibrium quantity and X 1 is a small quantity so that all terms quadratic in X 1 can be neglected. Solving the MHD equations using such an Ansatz yields a dispersion relation = (k), where is a complex function of the wave modes k. Re(): Oscillating solutions (waves) Im(): Damping Im() apple 0 (stability) or exponential growth, Im() > 0 (instability)

7 Non-Linear Stability Define the total energy of a closed system: W (t) = T (t) {z} + V (t) {z} Kinetic energy potential energy In a static equilibrium at t = 0: T (0) = 0, V (0) = V 0 and W 0 = V 0 If we disturb the equilibrium W 0 W = W 0 + W. Define stability: T (t) apple W for any disturbance.

8 Ideal instability: kink instability An ideal instability of flux tubes which occurs at a certain threshold of twist in the flux tube. The threshold for the onset of the instability depends on the details of the considered equilibrium, in particular on the radial profile of the twist, and on the aspect ratio, l/a, where a is the characteristic radius of the configuration (the minor radius in a toroidal system). For a straight, cylindrically symmetric flux tube with fixed (line-tied) ends and uniform twist (the Gold and Hoyle (1960) force-free equilibrium), which is the simplest possible model of a coronal loop, the threshold was numerically determined (Hood and Priest, 1981) to be From T. Török and. Kliem.

9 Resistive instability: tearing instability Initial equilibrium: One dimensional current sheet (Harris sheet) (z) = 0 tanh(z)e x ) j = 0 µ cosh(z) 2 e y. The equilibrium is unstable to long wave length resistive modes. ) Change of topology (not possible for an ideal instability) ) Release of energy stored in the magnetic configuration Growth rate of the instability: γ ~ S -3/5 where S = VA a / η.

10 Tearing instability Courtesy of M Hesse Occurrence of tearing instability in a less symmetric set up: Shearing of an 2D x-type configuration generates a thin current sheet which becomes unstable.

11 Magnetic Reconnection A very short history of magnetic reconnection 1946 Giovanelli suggested reconnection as a mechanism for particle acceleration in solar flares Sweet and Parker developed the first quantitative model 1961 Dungey investigated the interaction between a dipole (magnetosphere) and the surrounding (interplanetary) magnetic field which included reconnection Petschek proposed another model in order to overcome the slow reconnection rate of the Sweet-Parker model 1974 J.. Taylor explains the turbulent relaxation of a Reversed Field Pinch by a cascade of reconnection processes which preserve only the total helicity Kadomtsev explains the sawtooth crash (m=1,n=1 mode) in a tokamak by reconnection 1988 Schindler et al. introduced the concept of generalised magnetic reconnection in three dimensions

12 Literature: several thousands publications which refer to magnetic reconnection iskamp, D., Magnetic Reconnection in Plasmas, CUP 2000 Priest and Forbes, Magnetic Reconnection, CUP 2000 Reconnection of Magnetic Fields, Editors: J. irn and E. R. Priest, CUP 2007

13 Reconnection in other systems Hydrodynamics: Reconnection of vorticity ( crosslinking or cut and connect of vortex tubes), e.g. [Kida, S., and M. Takaoka, Vortex Reconnection, Annu. Rev. Fluid Mech. 26, 169 (1994)] Superfluids: Reconnection of quantized vortex elements [e.g. Koplik, J. and Levine, H., Vortex reconnection in superfluid helium, Phys. Rev. Letters 71, 1375, (1993)] Movie by O.N. oratav and R.. Pelz

14 Reconnection in other systems Cosmic Strings: Reconnection of topological defects [e.g. Shellard, E.P.S., Cosmic String interactions, Nucl. Phys. 282, 624, (1987)] Liquid Crystals: Reconnection of topological defects [e.g. Chuang, I., Durrer, R., Turok, N., and Yurke,., Cosmology in the laboratory: Defect Dynamics in Liquid Crystals, Science 251, 1336, (1991)] Knot theory: Surgery of framed knots [e.g. Kauffman, L.H. (1991)Knots and Physics, World Scientific, London] Enzymology: Reconnection (also recombination ) of strands of the DNA [e.g. Sumners D., Untangling DNA, Math. Intell. 12, (1990)]

15 Occurrence of magnetic reconnection: Fusion devices, in particular Tokamaks. Sawtooth oscillations limit confinement (see e.g Kadomtsev, 1987). Reconnection Experiments: Swarthmore Spheromak Experiment (SSX), Princeton Magnetic Reconnection Experiment (MRX), High Temperature Pasma Center Tokyo (TS-3/4), Versatile Toroidal Facility (VTF) at MIT,...

16 Magnetospheres of planets

17 Solar Flares, Coronal Mass Ejections Accretion disks around black holes Magnetars (Soft Gamma Repeaters) A large prominence eruption on the Sun on June 7, 2011 (SDO)

18 What have these systems in common? A primarily ideal, i.e. topology conserving dynamics. A (spontaneous) local non-ideal process, which changes the topology. Characteristics associated with magnetic reconnection: a) generation of an electric field which accelerates particles parallel to b) dissipation of magnetic energy (direct heating) c) acceleration of plasma, i.e. it converts magnetic energy into bulk kinetic energy d) change of magnetic topology Note a)-c) are not properties of magnetic reconnection alone, but occur also in other plasma processes, in particular c) can occur even in an ideal plasma.

19 Role of ideal evolution in reconnection: t E + v =0 t v =0 da = const. for a comoving surface C 2, Alfvén s Theorem (1942): The flux through any comoving surface is conserved. Mathematically this is an application of the Lie-derivative theorem: The magnetic field is transported (or Lie-dragged, Lie-transported, advected) by the flow v. Topology conservation Conservation of flux, Conservation of field lines Conservation of null points Conservation of knots and linkages of field lines C 2 (t)

20 ionless parameter determining the importance o Role of ideal evolution in reconnection: Storage of magnetic energy Generates small scales (typically current sheets) Rm =10000 Rm =10 t (v ) advection term = diusion term R m = (v ) 2 = v 0/l 0 /l 2 0 = l 0v 0,

21 Role of ideal evolution in reconnection: Under ideal evolution a non-trivial topology can store energy trivial topology: ideal relaxation can reduce the magnetic energy to zero. non-trivial topology: (linkage of magnetic flux) ideal relaxation cannot reduce the magnetic energy to zero.

22 Magnetic Helicity: The homogeneous Maxwell s equation yield a balance equation for the helicity density: t A hel. density + ( + E A)= 2 E hel. current hel. source ume yields an expression for the total helicity Integrating over a closed volume (no helicity current across the boundary) yields an expression for the total helicity: d dt V A d 3 x = 2 V E d 3 x, The helicity integral (total helicity) is a measure for the linkage of magnetic flux in the domain. It is invariant under an ideal dynamics.

23 Role of ideal evolution in reconnection: Allows to build up magnetic energy in the field Allows to store magnetic energy in complex magnetic topology

24 2D Reconnection 2D -> = x(x,y) ex + y(x,y) ey; N= η J = Jz(x,y) ez E + v = N; N = v E + w = 0 with w = v v w = E, E = 0; E w =0 2 The reconnection occurs only at an x-point of the magnetic field. The source term of the magnetic helicity density vanishes, E =0. One can map the plasma flow v to an ideal flow w which transports the magnetic field and becomes singular at the x-point. Note the process is only locally stationary, globally the field is always time-dependent.

25 2D Reconnection Since for a generic null is linear in x, w is of the type x/x 2 along the inflow direction, hence it has a 1/x singularity at the origin. Thus a cross-section of magnetic flux is transported in a finite time onto the null-line. T = 0 T dt = 0 x 0 dt/dx dx = 0 x 0 1/w x dx x 2 0/2 This gives the impression that the flux is cut and reconnected at the z-axis. Rate of reconnected flux: d rec dt = d dt nda = E z dz (= w x y dz)

26 2D Reconnection: First simple models of two-dimensional steady state reconnection have been set up by Sweet and Parker (1957/8) and later Petcheck (1964) on the basis of conservation of mass and flux in the framework of resistive MHD. They provide a scaling of the Alfvén Mach number (the ratio of inflow plasma velocity to inflow Alfvén velocity) in terms of the Lundquist number. L e is the global scale length, v Ae = e / µ 0 is the Alfvén speed in the inflow region M Ae = v e v Ae = E v Ae e = absolute reconnection rate maximum inflow of flux = relative reconnection rate, The electric field at the x-point is an absolute measure of the reconnected flux. d 2 rec dt dz = E z oth definitions are equivalent if the magnetic field does not vary along the inflow channel.

27 2D Reconnection: The models provide a scaling of the Alfvén Mach number in terms of the Lundquist number: S = µ 0 L e v Ae / M Ae = v e v Ae = E v Ae e = 1 S Sweet-Parker S=108 = ln S Petschek S=10 8 = Due n rates to the of the high order Lundquist of numbers =01 yield in the time Solar scales Corona for global ( energy 12 ) the releas scaling of the reconnection rate with S is important. Reconnection rates of the order of MAe = 0.1 yield time scales for global energy release and magnetic reconfiguration that are consistent with those seen in solar flares and magnetospheric substorms (Miller et al., 1997). In this respect Petschek s model was a significant improvement to Sweet &Parker as it provides higher reconnection rates. The term fast reconnection is used to describe reconnection rates which scale like Petschek reconnection or faster.

28 2D Reconnection: Collisionless Collisionless: There are not enough collisions between electrons and ions to explain the reconnective electric field at the x-point as caused by resistivity (E η j). Generalised Ohm s law: m e ne 2 j t + (j v+ vj 1 ne jj = E + v 1 ne j + 1 ne P e j Electron inertial length: de=c/ωpe on which the electr ( pe = 4ne 2 /m e ) Schematic of the multiscale structure of the dissipation region during anti-parallel reconnection. Electron (ion) dissipation region in white (gray) with scale size c/ωpe (c/ωpi). Electron (ion) flows in long (short) dashed lines. Inplane currents marked with solid dark lines and associated out-of-plane magnetic quadrupole field in gray (From Drake & Shay, Collisionless Reconnection in ``Reconnection of magnetic fields'', CUP 2007)

29 2D Reconnection: Collisionless Data from a PIC simulation of anti-parallel reconnection with mi/me =100, Ti/Tp = 12.0 and c = 20.0 showing: (a) the current Jz and in-plane magnetic field lines; (b) the self-generated out-of-plane field z (c) the ion outflow velocity vx; (d) the electron outflow velocity vxe; and (e) the Hall electric field Ey. Noticeable are the distinct spatial scales of the electron and ion motion, the substantial value of z which is the signature of the standing whistler and the strong Hall electric field Ey, which maps the magnetic separatrix. The overall reconnection geometry reflects the open outflow model of Petschek rather than the elongated current layers of Sweet-Parker (From Drake & Shay, Collisionless Reconnection in ``Reconnection of magnetic fields'', CUP 2006).

30 Reconnection geometry: 2D: (x,y) and v(x,y) are in one plane. The diffusion region (gray area) is bounded in the plane. 2.5D: and v are functions of (x,y) but has a component out of the plane. The diffusion region is bounded in (x,y) but not in z. 3D: and v are functions of (x,y,z). Also the diffusion region is bounded in all 3 dimensions Historically most work on reconnection has been done using the (much more simple) 2D and 2.5D case. However, while this approximation is justified for Tokamaks and other technical plasmas which have an inbuilt symmetry, in astrophysical plasmas reconnection is inherently 3D.

31 Reconnection canflux onlyand occur thetopology: plasma isno non-ideal. Conservation of magnetic fieldifline reconnection. Conservation of magnetic flux and field line topology: No reconnection. Reconnection can only occur if the plasma is non-ideal. E + vis reconnection. = N (e.g. N = j Conservation of magnetic flux and field line plasma topology: No c flux andreconnection field line topology: No reconnection. vs Slippage : if the Reconnection can only occur non-ideal. vno (e.g. =N = j) N Reconnection only occur thefield plasma is non-ideal. Conservation of can magnetic flux ifand topology: reconnection. E + line v t = N ur if the plasma is non-ideal. xample where the transport velocity the magnetic flux wis differs from velocity vj) in + a Reconnection can only occur of if the dynamics non-ideal. E +v the = plasma Nif (e.g. plasma N = Nv= plasma region is v plasma = is ideal N (v = w). The blue and on-ideal region (gray) while outside the non-ideal the Reconnection can only occur if the non-ideal. t E + v = N (e.g. N = j) vw 0 N blue cross === for w =inv the d cross sections with(e.g. the plasma velocity. The section always remains E + v are moving = N N = j) if N v + t t eal domain whileohms the red the non-ideal vinduction region. = equation N lawcross section crosses if N = v + t E +vw v = N = N (e.g. N = = 0 for w = vj) v xamples which satisfy t (w ) 0: and v smooth: No reconnection a) N = v += but slippage. t if N = v + t w = 0 for w = v v if N = v + Ideal plasma dynamics: Fora) N = vb) and v t = = 0Reconnection N N+= v and + v singular: (2D). w for w = v v v smooth: No reconnection but slippage. t t +v w = a)0 NE for w = v+= v0and v smooth:if NoNreconnection = v but slippage. = v + c) N = v + Reconnection (3D). t b) N = v + and v singular: Reconnection (2D). the evolution ideal. called slippage (ofreconnection plasma w.r.t.but the slippage. field). a) Nis=still v +This is and v smooth: 1 No Flux E=+velocity. v = transport P (n) are w= v evelocities w is called a flux transport non-unique. b) N v + and v singular: Reconnection (2D). w = 0 for w = v v e n d v smooth:c)no reconnection but slippage. Nb) =N v=v+ (3D). + Reconnection and v singular: Reconnection (2D). t E.g Hall MHD: MHD: Reconnection c) N(2D). =+ v Reconnection + Reconnection (3D). d vhall singular: c) N = v (3D). 1 a) N = v + and v smooth: No reconnection but1slippage. E+v = j w = ve = v j en en econnectionb)(3d). N = v + and v singular: Reconnection (2D). many other cases under constraints: e.g. 2D dynamics with = 0: occurs in an the topology m of (3D). the field changes. If this Forc) N = v + Reconnection j e + called v reconnection. = globally (vj + jv)... diffusion. isolated region only E it is If it occurs it is + called 2 ne t

32 Reconnection: E + v = N with N = v + implies either a) E 0 and (E Φ). This is 3D- (or 0 or E 0) reconnection or 2.5D (guide field) reconnection b) E =0 and E 0 where =0. This is 2D (or x-point or E =0) reconnection. Remark: Note that these are conditions on the evolution of the electromagnetic field - they are not restricted to MHD. Two cases of reconnection: vanishing or non-vanishing helicity source term (E ).

33 3D Reconnection: Stationary E+ v x = N; E = N 0; stationary: E= - Φ. = NR ds n id + (x 0,y 0 ) id + v = R N N n id. + v n id. = R id. + v id. =0 v = v n id. + v id. General property: No direct coupling between external flow (M Ae ) and absolute reconnection rate. Non-ideal solutions shows counter rotating flows above and below the reconnection region. The asicbasic non-ideal 3D reconnection solution issolutions inherentare 3Dnot and well nonperiodic 3D is not a simple approximation of any approximated by 2 or 2.5D reconnection solutions. 2/2.5 D solution.

34 3D Reconnection: Reconnection rate 0= E dl = E dl + E r dr E r dr L R 1 R 2 E dl =2 E r dr =2 v z dr L The rate of mismatching of flux is given by the difference of the plasma velocity above and below the reconnection region. d mag dt = L E dl = R w in w out dr

35 3D Reconnection: Stationary Superimpose an ideal solution (x-type flow): Region I: Ideal evolution Region II: Slippage (separation bounded) Region III: Reconnection (separation exponential) Flux through region III = E dl.

36 E 0 Reconnection: Stationary v id >v n id v id v n id v id <v n id

37 E 0 Reconnection Reconnection rate = E dl along the field line where it is maximal External reconnection rate = voltage drop between the stagnation points of the ideal flows = magnetic flux per unit time entering (or leaving) the reconnection region Internal reconnection rate = sum of the voltage dierence between the stagnation points and the reconnection line = magnetic flux per unit time reconnecting within the flux tube bounding the diusion region only d rec dt = E dl = d ext rec dt + d int rec dt

38 E 0 Reconnection: Time-dependent

39 E 0 Reconnection There are no pairwise reconnecting field lines. For processes bounded in time reconnecting flux tubes exist which show a twist consistent with the helicity production.

40 E 0 Reconnection: Null points Process depends on the direction of the electric field at the null point. Two basic cases: Electric field aligned with the spine axis Electric field tangent to the fan plane Generic (hyperbolic) 3D null point. Fan: blue, Spine: red. Fan planes of two null points intersect in a separator.

41 E 0 Reconnection: Null points Spine aligned electric field (electric current) Fan aligned electric field

42 Reconnection Cascades Dundee braiding experiment: Start with a force-free field modelled on the pigtail braid with total helicity H=0. Let the field relax (resistive t r(v ) = Taylor hypothesis: the end state of a turbulent relaxation should be a linear force-free field with H=0 => = 1 ez Result: We found a non-linear force-free field (with H=0) in contradiction to Taylor s hypothesis. Wilmot-Smith, A.L., Pontin, D.I., Hornig, G., A&A. 516 A5 (2010) Pontin, D.I., Wilmot-Smith, A.L., Hornig, G., Galsgaard, K, A&A, 525, A57 (2011)

43 Formation of a turbulent cascade of reconnection in a braided magnetic field:

44 3D Reconnection: Summary Three-dimensional reconnection is structurally different from 2 or 2.5 dimensional reconnection. The reconnection rate is given by the integral over the parallel electric field along a certain field line. The reconnection rate is not directly related to inflow or outflow velocities. The flux undergoing reconnection is restricted to thin flux layers intersecting in the non-ideal region. In particular we have no 1-1 correspondence of reconnecting field lines. Instead we have flux surfaces which are mapped onto each other. For a reconnection process which is limited in time, these flux surfaces are closed, i.e. they form perfectly reconnecting flux tubes. The counter rotating flows induce a twist in these tubes, which is consistent with the helicity production of the process.

45 Common Misconceptions about Reconnection: Magnetic reconnection heats the plasma The Ohmic heating due to the reconnection event itself is small due to the small size of dissipation region and a low resistivity. The major part of the magnetic energy released is converted into kinetic energy. The heating usually occurs away from the reconnection region proper via a number of non-ideal plasma processes (shocks, waves, adiabatic heating, viscous heating). Magnetic reconnection occurs at an X-line The notion of an X-line (a magnetic field line which has in a plane perpendicular to it an X-type field) is a notion deduced from 2D (or 2.5D) models. There is no distinguished X-line in a generic 3D magnetic field, instead there are whole regions of magnetic flux which satisfy this criterium. Moreover reconnection can also occur at an 0-line. Magnetic reconnection occurs only along separators A separator is a field line formed by the intersection the fan planes of two null points. It is, for instance, not present in Tokamaks but still we have reconnection in these devices.

46 Common Misconceptions about Reconnection: Magnetic reconnection is associated with fast (Alfvénic) flows Mach numbers of are typically found in stationary 2D simulations. However, it is known that reconnection in reality is 3D and time-dependent. There is no necessity for time-dependent processes to have high flow velocities.

47 Some open questions: How does collisionless reconnection work in 3D configurations? Is there a generic dissipation mechanism? How common is magnetic reconnection? What is the spectrum of reconnection? Is reconnection at magnetic null points spontaneous or driven? How does reconnection work in relativistic systems? Observational consequences?

48 What is the problem? Complex three-dimensional geometry Coupling of micro (kinetic) physics and large scale dynamics (cross-scale coupling) Spatial resolution in simulations is too low by many orders of magnitude. global length scale electron-inertial length = 108 m 10 1 m = global length scale 109 gyro e ects = 108 m 10 2 m = 1010 (Typical resolution of a 3D simulation: 300)