Presentation on theme: "MHD vs. kinetic effects in space & solar plasmas"— Presentation transcript:
1MHD vs. kinetic effects in space & solar plasmas David TsiklauriUniversity of SalfordSTFC IntroductorySummer School in Solar & Solar-Terrestrial Physics12 September 2007 Armagh Observatory
2Different types of plasma description / Lecture outline: 1. Fluid / MHD description2. Dynamics of individual particles3. Kinetic (statistical approach)4. Applications (to space plasmas: waves + reconnection)1. Fluid / MHD description:In this approach plasma is described in terms of macro-parameterssuch as density, pressure and velocity of "physically small" fluidvolume elements that contain many plasma particles.Such description is justified if there is a certain ordering of spatial &temporal scales in the physical system. A priori such ordering is notclear from the MHD equations. What is clear is that these equationsare invalid when λ (wavelength) is too small and ω (frequency) istoo large [spatial scale= SS and temporal scales = TS]
3Neglecting charge separation immediately restricts rage of applicability of the MHD with λ >> λD (Debye length)=The assumption that ne= ni i.e. e's & i's move together puts yetanother restriction on TS's of MHD processes, namely, T >> ωci-1 .This inequality follows from the fact that for such fast rotation ascyclotron, e's & i's move differently. (ωci =eB/mic – IC frequency)MHD also neglects tensor nature of thermal pressure:This assumption would clearly hold if collisions play dominant role.However, on TS smaller than collision time this is not the case.In MHD, in addition to λ >> λD , and following ordering should hold(Roberts & Taylor (PRL, 1962 and references therein) :(MHD ordering) (1.1)
4In the situation when ordering between TS and SS of the system is different other types of MHD are used. In particular, if(FLR MHD ordering) (1.2)the pressure tensor is even non-diagonal. In such, cases Finite LarmorRadius (FLR) MHD is used (e.g. Berning & Spatschek PoP 1998):They are closed by some kind of equation of state e.g. isothermal. The gyro-viscous pressure tensor can be found in (Hazeltine & Meiss, Plasma Confinement, 1991).(1.3)
5Other examples of MHD-type equations can be derived based of which terms are significant (i.e. kept) in the generalized Ohm's law.Recall that the simplest form of Ohm's law quoted on previous slideis derived from following consideration:Plasma is a moving conductor and the Ohm's law should be written inthe plasma rest frame For non-relativistic flows the restframe electric field isBecause of quasi-neutrality condition, we require j=j'. Thus, theOhm's law for (moving) plasma is:(1.4)
6Indeed, simple Ohm’s law, Eq. (1 Indeed, simple Ohm’s law, Eq.(1.4), takes into account fluid motion in the magnetic field (appearing as an additional electric field (V × B)/c) but not the effects due to the acceleration, dV /dt , of the plasmafluid volumes (Polygiannakis & Moussas, Plasma Phys. Control. Fus. 43 (2001) 195).In the non-inertial coordinate system co-moving with the flow, the acceleration appears as an inertial force −mdV /dt (where m is the ion mass, since the electrons are much lighter), thus corresponding to an ‘inertially’ exerted electric field −(m/e)(dV /dt).Thus the generalized Ohm’s law must include the total electric field exerted on plasma volume (Landau & Lifshitz Electrodyn. Cont. Med, 1963, p 210):(1.5)
7If we insert dV /dt from the equation of plasma motion we obtain: where n = ρ/m is the number density.Note that Eq.(1.6) is an approximation of more extended expressions, obtained by taking into account both e's & i's equations of motion. However these terms are negligible in the usual MHD limit (e.g. Krall and Trivelpiece 1973).(1.6)In the generalized Ohm’s law (1.6) notice the additional Hall term, (J × B)/enc (for which the plasma model is often called Hall-MHD) and (tensor) pressure gradient, ∇Pth/en, electron inertiaand some other force (e.g. gravity), −F/en, terms.
8The relative importance of the Hall term can be examined if writing (1.6) as (e.g. see also Parks Physics of Space Plasmas. An Introd. 1991, p 278):(1.7)whereis the ‘total’ electric field without the Hall term;is the electron gyrofrequency;is the frequency of charge collisions (assuming that the conductivity is mainly due to electrons);Then (1.7) shows that the Hall current, J × B/B, becomes dominantover the ‘usual’ electric field-aligned current, J, if ωce >> ν which isvalid for the limiting case of strong magnetic fields or of rarecharge collisions.
9Indeed, many astrophysical problems involve nearly collisionless plasmas (e.g. the solar wind), while confined plasma experiments involve plasmas embedded in strong magnetic fields.Therefore, the Hall current in those cases is expected to be as much or even more important than the electric field-aligned current.Let us make simple estimate for solar corona:ωce = 1.76 ×107 B[G]= 1.76 ×109 rad s-1 (for 100 Gauss)ν = (4.8 × 10-10)2 × 2. × 109/(9.1 × × 6. × 1016) ≈ 10 s-1 . Thusωce / ν ≈ 108 >> 1.At the same time we should be aware that the importance of the Hallterm applies only on small scales, e.g. magnetic reconnection.For details see Bhattacharjee, Annu. Rev. Astron. Astrophys , 365; Ma & Bhattacharjee (1996) Priest & Forbes (2000); and Birn et al., J. Geophys. Res., 106, 3715 (2001).
102. Dynamics of individual particles In this approach plasma is described in terms of the dynamics ofindividual particles.Plasma has natural tendency to disperse. This is due to the fact thatplasma particles move randomly in every direction (thermal motion).Unless there is a restraining force, plasma will disperse & cease to exist.One possibility is that particle collisions, which naturally tend todeflect them from their ran-away trajectories could play role of therestraining force.However, in many plasmas, particularly space plasmas, collisions aretoo rare, i.e. the collision frequency ν << other characteristicfrequencies of the system.In such cases magnetic field, which acts via Lorentz force is far moreimportant than collisions as the restraining force.
11The Lorentz force acting on a charge is: If we only consider action of B-field on an electron (E-field is usuallyignored because it is of the order of V2/c2 <<1), then Newton's 2ndlaw gives:(2.1)which after defining electron cyclotron frequency vector,gives(2.2)In principle Eq.(2.2) can be solved (details ine.g. R.O. Dendy, Plasma Dynamics, chapter 2)and thus particle trajectory can be determined.In a uniform magnetic field the path of anelectron is helical as shown in the figure →The helix is produced by uniform circularmotion about a point that moves withconstant speed parallel to the magnetic field.
12The electron position can be written as: which implicitly defines the guiding centre (GC) position (see fig.→).The first term on RHS describes the parallel dynamics, while the second clearly describes perpendicular dynamicsis the mean position of the electron if the rapid variation (rotation)is averaged out. This means position and also (i.e. thedrift velocity of the GC) often contains all the information required,which is useful when one wants to follow path of over atimescale >> ωce-1.
13Guiding Centre approximation. In general, when B=B(r,t) then particle dynamics is rather complicated. However, when L >> rL,e and T >> ωce-1 (L and T are spatial and times scales of the magnetic field variation), the Guiding Centre approximation apples.In this approach the helical trajectory of a particle in magnetic field is approximated by a smooth drift motion of the GC as depicted in this figure:This speeds up numericse.g. Genot et al. (2004)Ann. Geophys., 6, 2081
14Let’s look at relation between some characteristic frequencies: electron collision frequency νe, plasma frequency ,and cyclotron frequency for solar corona and fusionplasmas.Parameters can be taken from e.g.NRL plasma formulary:Fusion plasmas n=1014cm-3 T=103eVSolar corona n=109cm-3 T=102eV(note 1eV=11600K)HzHzHzHere n [cm-3]; T [eV]; B in [G] andCoulomb logarithm (for electrons) is
15B. The Larmor radius is comparable to Debye radius: fce Hzfp HzHzFusion2.8x101110111.4x105Corona2.8x1083x10853enTwo importantconclusions followfrom these estimates:1. For the both cases fce/ νe ≈ few x106 which means that between every collision electrons rotate millions of times around magnetic field line. Thus for solar coronal and fusion plasma magnetic field plays far more important role as a restraining force than collisions.2. For the both cases fce/ fe ≈ 1 which means that. This coincidence is responsible for a great degree of complexity in the plasma behaviour.A. Mathematically this makes the dispersion relations become difficult to treat.B. The Larmor radius is comparable to Debye radius:
163. Kinetic (statistical approach) In plasma kinetics instead of studying dynamics of individual particles,without loss of generality, system is ascribed distribution function (DF)fα(r,p,t) which is the probability of finding species α, in intervals(r,r+dr) and (p,p+dp) at t=t. Hence, normalization condition should be:Using this microscopic distribution function, macroscopic (i.e. hydro-dynamic) quantities can be defined as n-th order moments :(3.1) 0-th order(3.2) 1-st orderi.e. 0-th order gives number density,1st order gives definition of the hydrodynamic velocity Vα.Here we will consider non-relativistic case i.e. pi=mvi,Also the following notation is used throughout:
172-nd order moment allows to define the pressure tensor pij: (3.3)Vlasov equationLet us derive equation which governs dynamics of the distributionfunction – with the latter, we can construct all the quantities we need.If plasma is rarefied (particle collisions can be ignored) and particlesare not created or annihilated, then all we need to require is that the DFdoes not change in time:which is the same as:(3.4)
18In principle any force can be put in Eq. (3 In principle any force can be put in Eq.(3.4), but for plasma, which is acollection of charged particles, and the strongest force acting is of EMnature, it has to be the Lorentz force:(3.5)When E(r,t) and B(r,t) EM-fields are determined from the Maxwell'sequations in which instead of charge and current densities ρq and jthe following equations are used: and ,then eq.(3.5) is called Vlasov's equation with self-consistent EM fields.Self-consistent in the sense that, Eq.(3.5) provides DF, fα, whichchanges under the effect of EM-fields. In turn, change in DF meansre-distribution of charges, i.e. change in EM-fields; or in one line:EM-fields affect DF, which in turn affects EM-fields in aself-consistent way.
19We have seen how taking different order moments of the DF gives different macroscopic quantities. Now, we will show how taking different order moments of the Vlasov's equation gives differentconservation laws (mass, momentum, energy, etc.).Let us take 0-th order moment of Eq.(3.5):=0which after multiplying by mα and noting that ρα= mα nα we obtainEq.(3.6) is mass conservation (continuity) equation for species α. Familiar single fluid MHD version of Eq.(3.6) can be obtained by:(3.6)mass, charge, current densities, hydrodynamic velocity
204. Applications (to space plasmas: waves + reconnection) Previously we discussed charged particle dynamics. In principle,plasma dynamics can be described by solving the equation of motionfor each individual particle (initially distributed by e.g. Maxwelldistribution), supplemented by the Maxwell's equations in whichcharge and current densities are determined self-consistently i.e. bysumming the spatially and temporally changing distribution of plasmacharged particles.This type of approach is called Particle-In-Cell (PIC) simulation.Typically 100s of millions of particles are used, i.e. above mentionedequations are solved for each of the 100s of millions of particles!
21This may sound complicated, but it is still better than tackling Vlasov's equations in 6D (3V, 3D) space: consider memory constraints:if one doubles the resolution then need 26=64 times more RAM!PIC has shortcomings too: it is hard to properly resolve high velocitytails of the distribution function: if you typically have 100 particles percell at v = vth than for v >> vth, there are only few particles left (poorstatistics). Also, PIC data is usually quite noisy – needs smoothing.In PIC simulation EM-fields are defined on some spatial grid(i.e. are discrete variables), while particle positions are continuous.Particles start to move under Lorentz force, hence charge distributionchanges; this changes EM-fields (which are calculated usingMaxwell's equations in which charge and current densities aredetermined self-consistently), and so on.
23Particle-in-cell simulations of circularly polarized Alfven wave (AW) phase mixing: What is Phase Mixing?Phase Mixing is a mechanism originally proposed by Heyvaerts &Priest (1983) that suggests that AW dissipation in plasmas withinhomogeneity across the magnetic field is greatly enhanced:Classical (resistivity) dissipation:Phase Mixing (enhanced) dissip.:All previous Phase Mixing studies were performed in MHDapproximation (Heyvaerts & Priest 1983; Nocera et al. 1986; Parker1991; Nakariakov et al. 1997; DeMoortel et al. 2000; Botha et al.2000; Tsiklauri et al. 2001, 2002, 2003; Hood et al. 2002; Tsiklauri &Nakariakov 2002).
24The problem with this, of course, is that MHD approximation will eventually break down: First when transverse scale in a wavefront will reach ion gyro-radius, ri, and then the electron one, re.Hence we decided to perform Particle-in-cell i.e. kinetic simulationsof circularly polarized AW phase mixing for the first time.The results are published in two papers:Tsiklauri D. , J.-I. Sakai, S. Saito,Astron. Astrophys., 435, 1105, (2005).New J. Phys., 7, 79, (2005).What was new to expect?Ability to study wave—particle interactions!How “dissipation”(collisionless) is modifiedin the kinetic regime!What happens to individualspecies (electrons and ions)?
25Key facts about PIC simulations of Alfven wave phase mixing: We use 2D 3V fully relativistic, electromagnetic, PIC code.System size Lx=5000Δ by Ly=200Δ cells.Each cell has 100 electrons and ions. Total of 478 x 106 particles!Plasma density is enhanced by a factor of 4 in the middle of simulation domain:Temperatures and therm. speeds of e,i are varied so that ptot=const.Mass ratio used: mi / me = 16.Alfven wave phase mixing takes place.Dissipation of Alfven waves is greatly enhanced due to wave-particle interactions (as shown in the following slides).
27Developedstage ofPhase-Mixingt = 55ωci.NotPhase-MixedAWcomps.By and EzPhase-MixedAWcomponentsBz and EyN.L.generatedBxN.L.generatedelectrondensity
28Simulation results: Evidence of electron acceleration Electric field that accelerates electronsElectr. Phase Space (Vx vs x) and (Vx vs y)
29What about particle distribution functions? Vx, Vy, Vz –distributionfunctions ofelectrons(top row)and ions(bottom row)at t = 0(dottedcurves) andt = 55/ωci(solid curves).ElectronacclerationIon wave-broadening
30What about AW amplitude decay law? Two snapshots of the AW Bz(x, y = 148)component at t = 54.69/ωci (solid line)and t = 46.87/ωci (dotted line).The dashed line represents fit0.056 exp[− (x/1250)3].This nicely reproducesHeyvaerts & Priest (1983) result!i.e. our kinetic simulations recoveredMHD AW amplitude decay law:Also, measured AW speed (by the twosnapshots) corresponds to the bumpsin the electron distribution function!
31Ok, so what we've learned: if one drives Alfven (IC) waves (0.3ωci) in plasma with transverse density inhomogeneity then E|| is generated ...What is the essential physics? or a minimal model which can do the job?It turns out [see Tsiklauri, New J. Phys. 9, 262 (2007)] that a two fluidmodel (which allows for electron and ion separate dynamics) can do it!Here mi/me=262, takes 4 days on 1 CPU – equivalent PIC simulationwould have taken 4 month on 64 CPUs!E||=100Vm-1
32Magnetic reconnection is one of important possible ways of Magnetic reconnection during collisionless, stressed, X-point collapse using Particle-in-Cell simulationMagnetic reconnection is oneof important possible ways ofconverting magnetic fieldenergy into heat and acceleratedplasma particles.Main problem in plasmaheating (solar corona,Tokamak) is thatSpitzer resistivity is ~ T -3/2,i.e. more heating =plasma starts to behaveas a superconductor.resistive (collisional)or collisionlessspontaneousorforcedsteadytimedependentreconnectionMain aspects of reconnectioncan be classified as:
33Resistive reconnection of all said types is very well studied (e. g Resistive reconnection of all said types is very well studied (e.g. Priest& Forbes book, CUP 2000), although there are some open questions,particularly in 3D. Collisionless reconnection is a relatively recentdevelopment (e.g. Birn & Priest book, CUP 2007, chapter 3.1, Fig.3.1)The key question is which termin the generalized Ohm's law isbreaking the frozen-in condition?Each term has different spatialscales associated with it:For electron inertia – c/ωpe – electron skin-depth;For the Hall term – c/ωpi – ion skin-depth;For the pressure tensor – rLi – ion Larmor radius;
34Validation of our PIC (Particle-In-Cell) code by reproduction of GEM challenge resultsGeneral Advice:Always try to reproduce previous results when using a new code orusing an old code for a new application!GEM result[Pritchett, JGR, 106, 3783, (2001)]Our resultTime evolution of the reconnected magnetic flux difference, Δψ
35Magnetic reconnection during collisionless, stressed, X-point collapse using Particle-in-Cell simulationFor details see Tsiklauri & Haruki, PoP (accepted)Solar flare modelOur equivalent numerical simulation(Hirayama 1974)a = 1a > 1Aschwanden “Physics ofthe Solar corona An Introduction”Priest & Forbes “Magnetic ReconnectionMHD Theory & Applications”
36The modelMagnetic field:Parameters:Lx = Ly = 400D (lD = 1D)wpe Dt = 0.05N = 1.6 million e-i pairs (n0 = 100 / cell)L = 200Dc/wpe = 10Dmi / me = 100vte / c = 0.1wce/wpe = 1.0 (for B = B0 )Imposed current:b = 0.02vd / c = (a = 1.2)Boundary conditions (this is crucial):E and B fields – No flux trough boundaryParticles Reflection
37Generation of out-of-plane electric field Out-of-plane electric field in the X-point (magnetic null) vs time. Thisfield is a measure of magnetic reconnection.wpet = 250 corresponds to 1.25 (Alfven times)
38Current sheet generation Time evolutionof spatialdistribution oftotal currentjz in the X-Yplane.Note the currentpeaks at the sametime as Ez(on previousslide)wpet = 0100Y / (c/wpe)X / (c/wpe)170250jz / j0Max (jz / j0) = 16
39Generation of Quadruple out-of-plane magnetic field Time evolution Bz in the X-Y planeSuch field isregarded asevidence forHall effectphysics –separationof electronand ion flow.[e.g. Birn, et al.,JGR, 106, 3715(2001);Uzdensky &Kulsrud, PoP,13, 2305 (2006)]wpet = 0100Y / (c/wpe)X / (c/wpe)170250Bz / B0
40Visualization of magnetic reconnection by tracing dynamics of individual magnetic field lines:| B | / B0 =1.55, 1.60,1.65, 1.70yx
41Particle acceleration in the current sheet The local electron energy spectrum (distribution function) near the current sheet at t = 0 (dashed curve) and t = 250 (solid curve) for α= 1.20.Fit (solid straight line) tothe high energy partof the spectrum shows aclear power law:In the vicinity of X-typeregion in the Earth'smagneto-tail observationsshow power law index isbetween -4.8 and -5.3[Oieroset, et al., PRL 89, , (2002)]
42Separation of electron and ion flow in the current sheet Electron inflow (a) concentrated along the separatrices. They deflect from the current sheet on the scale of electron skin depth, with the electron outflow speeds being ≈ the external Alfven speed 0.13c.Ion inflow (b) startsto deflect from thecurrent sheet onion skin depthscale [10(c/ωpe)].Outflow speeds ≈0.03c.Task: read chapter 3.1from Birn & Priest 2007book and comparethese to their Fig. 3.1(slide 33)
43Energetics of the reconnection process One of the main problems insolar physics is the time scaleof energy release during solarflares:Normal resistive time is:1015 Alfven times (108 yr)Flares typical time is:Alfven times!In our simulation of x-pointcollapse up to 20% of initialmagnetic energy isreleased in justone Alfven time!