Magnetohydrodynamics Basic MHD

MagnetohydrodynamicsConservative form of MHD equationsCovection and diffusionFrozen-in field linesMagnetohydrostatic equilibriumMagnetic field-aligned currentsAlfvén wavesQuasi-neutral hybrid approachBasic MHD(isotropic pressure)(or energy equation)Relevant Maxwell’sequations; displacementcurrent neglected( )Note that in collisionless plasmas and may need to beintroduced to the ideal MHD’s Ohm’s lawbefore1

In case of the diffusion becomes very slowand the evolution of B is completely determinedby the plasma flow (field is frozen-in to the plasma)convection equationThe measure of the relative strengths of convectionand diffusion is the magnetic Reynolds number R mLet the characteristic spatial and temporal scales be&and the diffusion timeThe order of magnitude estimates for the terms of the induction equation areand the magnetic Reynolds number is given byThis is analogous to the Reynolds number in hydrodynamicsIn fully ionized plasmas R m is often very large. E.g. in the solar windat 1 AU it is 10 16 – 10 17 . This means that during the 150 million kmtravel from the Sun the field diffuses about 1 km! Very ideal MHD:viscosityExample: Diffusion of 1-dimensional current sheetInitially:in the frame co-moving with the plasma (V = 0)has the solutionTotal magnetic flux remains constant = 0,but the magnetic energy densitydecreases with timeNowOhmic heating, or Joule heating4

Example: Conductivity and diffusivity in the SunPhotosphere• partially ionized, collisions with neutrals cause resisitivity: Photospheric granulesObservations of the evolution of magnetic structuressuggest a factor of 200 larger diffusivity, i.e., smaller R mExplanation: Turbulence contributes to the diffusivity:This is an empirical estimate,nobody knows how to calculate it!Corona• above 2000 km the atmosphere becomes fully ionizedSpitzer’s formula for the effective electron collision time numerical estimate for the conductivityEstimating the diffusivity becomesFor coronal temperatureHome exercise: Estimate R m in the solar wind close to the Earth5

Frozen-in field linesA concept introduced by Hannes Alfvén but laterdenounced by himself, because in the Maxwelliansense the field lines do not have physical identity.It is a very useful tool, when applied carefully.Assume ideal MHD and consider two plasma elementsjoined at time t by a magnetic field line and separated byIn time dt the elements move distances udt andIn ”Introduction to plasma physics” it was shown that the plasma elementsare on a common field line also at time t + dt In ideal MHD two plasma elements that are on a common field line remain ona common field line. In this sense it is safe to consider ”moving” field lines.Another way to express the freezing is to showthat the magnetic flux through a closed loopdefined by plasma elements is constantdl x VDtCC’dlSS’BClosed contour C defined by plasma elements at time tThe same (perhaps deformed) contour C’ at time t + tArc element dl moves in time t the distance Vt andsweeps out an area dl x VtConsider a closed volume defined by surfaces S, S’ andthe surface swept by dl x Vt when dl is integratedalong the closed contour C.As ·B = 0, the magnetic flux through the closed surfacemust vanish at time t + t , i.e,Calculate d/dt when the contour C moves with the fluid:6

The integrand vanishes, ifThus in ideal MHD the flux through a closed contour defined byplasma elements is constant, i.e., plasma and the field move togetherThe critical assumption was the ideal MHD Ohm’s law. This requires that theExB drift is faster than magnetic drifts (i.e., large scale convection dominates).As the magnetic drifts lead to the separation of electron and ion motions (J),the first correction to Ohm’s law in collisionless plasma is the Hall termso-called Hall MHDIt is a straightforward exercise to show that in Hall MHD the magnetic fieldis frozen-in to the electron motion andWhen two ideal MHD plasmas with different magnetic field orientations flowagainst each other, magnetic reconnection can take place. Magneticreconnection can break the frozen-in condition in an explosive way andlead to rapid particle accelerationExample of reconnection:a solar flare7

Magnetohydrostatic equilibriumAssume scalar pressureandi.e., B and J are vector fields onconstant pressure surfacesWrite the magnetic force as:Magnetohydrostatic equilibriumafter elimination of the currentAssume scalar pressure andnegligible (BB)Plasma beta:magneticpressuremagneticstress and torsionThe current perpendicular to B:Diamagnetic current: this is the way how B reacts on the presenceof plasma in order to reach magnetohydrostatic equilibrium.This macroscopic current is a sum of drift currents and of the magnetizationcurrent due to inhomogeneous distribution of elementary magnetic moments(particles on Larmor motion)denseplasmatotalcurrenttenuousplasmaIn magnetized plasmas pressure and temperature can be anisotropicDefine parallel and perpendicular pressures asUsing these we can derive themacroscopic currents fromthe single particle drift motionsandwhere8

The magnetization isSumming up J G , J C and J M we get the total currentMagnetohydrostatic equilibrium in anisotropic plasma is described byIn time dependent problems the polarization current must be addedForce-free fields a.k.a.magnetic field-aligned currentsIn case of the pressure gradientis negligible and the equilibrium isi.e., no Ampère’s forceFlux-rope:The field and the currentin the same directionis a non-linear equationIf and are two solutions, does not need to be another solutionNow is constant along BIf is constant in all directions, the equationbecomes linear, the Helmholtz equationNote that potential fieldsare trivially force-free9

Force-free fields as flux-ropes in spaceLoops in theSolar coronaCoronal Mass Ejection(CME)Interplanetary Coronal Mass Ejection(ICME)Example: Linear force-free model of a coronal arcadeConstruct a model that• looks like an arc in the xz-plane• extends uniformly in the y-directionThe form of the Helmholtz equationsinusoidal inx-directionsuggests a trial of the form:with the conditionvanishesat large z10

The solutionlooks like:Arcs in the xz-planeProjection of field lines in thexy-plane are straigth linesparallel to each otherCase of so weak current that we can neglect it ( = 0):Potential field (no distortion due to the current)Solve the two-dimensional Laplace equationLook for a separable solutionMagnetic field lines are now arcsThis is fulfilled bywithout the shear in the y-directionGrad – Shafranov equationOften a 3D structure isessentially 2D(at least locally)e.g. the linear force-free arcade aboveor an ICMEConsider translational symmetryB uniform in z directionAssume balance btw. magneticand pressure forcesThe z component reduces to11

REGIONAL GROWTH PARTNERSHIPHamden Middle School would be tooclose to the Dixwell Avenue clinic• Typically need 2,000 square feet for afully developed clinic; less is neededfor a school-based model with only anurse practitioner• The recreation facilities currentlyin the former middle school wouldcomplement a clinicBoys & Girls ClubOn November 11, 2006 the project teammet with Jeff Starcher, Executive Directorof the Boys and Girls Club of SoutheasternConnecticut. The following summarizesthe key points of the meeting.• Interest in a Boys and Girls Clubneeds to come from the community• A club could be up and running veryquickly• Boys and Girls Club is very interestedin the former middle school site,particularly the gymnasium, cafeteria,and shop classrooms• Possible for a Boys and Girls Club tooccupy only a portion of the buildingand partner with other tenants (e.g.,artists, theater groups, Hamden Parksand Recreation Department)• The diversity of the neighborhood isideal for a Boys and Girls ClubThe Boys and Girls Club of SoutheasternConnecticut is extremely interested in thepotential created by the former HamdenMiddle School redevelopment.Jeff Starcher feels that a Boys and GirlsClub could serve the neighborhood, aswell as the broader community. Becausethe organization is community-based andcommunity-driven, the impetus for a clubmust come from a coalition of communitymembers who would like to furtherpursue a Boys and Girls.If this type of coalition is formed, RGPcould work to bring the Boys and GirlsClub and the community together andfacilitate the creation of a neighborhoodclub.Whitney Players Theatre CompanyOn January 17, 2007 the project teammet with Cindy Simell-Devoe of the WhitneyPlayers Theatre Company, an amateurtheater company located in Hamden.The following summarizes the key pointsof the meeting.• Desperate need for theater and performancespace in Hamden• Many small theater/dance/musicteachers are looking to rent space toteach classes year-round• The most difficult times of year forspace are October, March, and May• Space could also be useful for smallscaleretail shops for local artists12Research/Interviews

Some general properties of force-free fieldsProof: As B vanishes faster than r –2 at large r, we can writethat is zero for force-free fieldsProof: ExerciseThus if there is FAC in a finite volume, it must be anchored to perpendicularcurrents at the boundary of the volume (e.g. magnetosphere-ionospherecoupling through FACs)Magnetospheric and ionospheric currents are coupled through field-alignedcurrents (FAC)Solar energyMagnetosphere(transports and stores energy)FACIonosphere(dissipate energy)“The ionosphere and magnetosphere are coupled in so many differentways that nearly every magnetospheric process bears on the ionospherein some way and every ionospheric process on the magnetosphere”Wolf, 197413

Recall: Poynting’s theoremEM energy is dissipated via currents J in a medium.The rate of work is definedJ E no energy dissipation!F v q E v J Ework performedby the EM fieldenergy flux throughthe surface of Vchange ofenergy in VPoynting vector(W/m 2 ) giving theflux of EM energyEJ > 0: magnetic energy converted into kinetic energy (e.g. reconnection)JE < 0: kinetic energy converted into magnetic energy (dynamo)How are magnetospheric currents produced?E Here ( J · E < 0 ), i.e., the magnetopauseopened by reconnection acts as a generator(Poynting flux into the magnetosphere)Dayside reconnectionis a load ( J · E > 0 ) anddoes not create currentSolar wind pressuremaintains J CFand the magnetopausecurrent14

Currents in the magnetosphereIn large scale (MHD) magnetospheric currents are source-freeand obey Ampère’s lawThus, if we know B everywhere, we can calculate the currents(in MHD J is a secondary quantity!).However, it is not possible to determine B everywhere,as variable plasma currents produce variable BRecall the electric current in (anisotropic) magnetohydrostatic equilbirum:In addition, a time-dependent electric field drives polarization current:These are the main macroscopic currents ( B) in the magnetosphereA simple MHD generatorPlasma flows across the magnetic fieldF = q (u x B) e – down, i + upLet the electrodes be coupled througha load current loop, where the currentJ through the plasma points upwardNow Ampère’s force J x B deceleratesthe plasma flow, i.e., the energy to thenewly created J (and thus B) in thecircuit is taken from the kinetic energy.Kinetic energy EM energy: Dynamo!The setting can be reversed by driving a currentthrough the plasma: J x B accelerates the plasma plasma motor15

Boundary layer generatorNote: This part of the figure(dayside magnetopause)is out of the planeOne scenario how an MHD generatormay drive field-aligned current fromthe LLBL:Plasma penetrates to the LLBL throughthe dayside magnetopause (e.g. byreconnection.The charge separation (“electrodes”of the previous example) is consistentwith E = – V x B .The FAC ion the inside part closes viaionospheric currents (the so-calledRegion 1 current system)FAC from the magnetosphereWe start from the (MHD) current continuity equationwhereThe divergence of the static total perpendicular current is[ Note: · ( M) = 0 ]In the isotropic case this reduces toThe polarizationcurrent can havealso a divergenceIntegrating the continuity equation along a field line we get: = b · ( V)is the vorticitySources of FACs:pressure gradients &time-dependent 16

FAC from the ionosphereWrite the ionospheric () current as a sum ofPedersen and Hall currents and integrate thecontinuity equation along a field line:Recall:Approximate the RHS integral asR1R1where P = h P and H = h areheight-integrated Pedersen and Hallconducitivities and h the thickness ofthe conductive ionosphere (~ the E-layer)The sources and sinks of FACs in the ionosphereare divergences of the Pedersen current andgradients of the Hall conductivityR2R2R1: Region 1, poleward FACR2: Region 2, equatorward FAC17

Current closureThe closure of the current systems is a non-trivial problem.Let’s make a simple exercise based on continuity equationLet the system be isotropic. The magnetospheric current is(the last term contains inertial currents)Integrate this along a field line from one ionospheric end to the otherwe neglect thisAssuming the magnetic field north-south symmetric, this reduces toThis must be the same as the FAC from the ionosphere and we get anequation for ionosphere-magnetosphere couplingOne more view on M-I coupling18

Gauge transformation:Magnetic helicityA is the vector potentialHelicity is gauge-independent onlyif the field extends over all spaceand vanishes sufficiently rapidlyHelicity is a conserved quantity if• the field is confined within a closed surface S on which• the field is in a perfectly conducting medium for whichTo prove this, note that fromwe getto within a gauge transformationFor finite magnetic field configurationshelicity is well defined if and only ifon the boundaryon SCalculate a littleand both B and V are ^ n on Sthus on S19

Example: Helicity of two flux tubes linked togetherCalculateFor thin flux tubesis approximately normal to the cross section S(note: here S is not the boundary of V !)Helicity of tube 1:ThusSimilarlyIf the tubes are woundN times around each otherComplex flux-tubes on the SunHelicity is a measure of structural complexity of the magnetic field20

Woltjer’s theoremFor a perfectly conducting plasma in a closed volume V 0 the integalis invariant and the minimum energy state is a linear force-free fieldProof: Invariance was already shown.To find the minimum energy state considerSmall perturbations:on S andaddedtransform to surface integralthat vanishes becauseThusif and only ifMagnetohydrodynamic wavesV 1• Dispersion equation for MHD waves• Alfvén wave modesMHD is a fluid theory and there are similar wave modes as in ordinary fluidtheory (hydrodynamics). In hydrodynamics the restoring forces for perturbationsare the pressure gradient and gravity. Also in MHD the pressure force leads toacoustic fluctuations, whereas Ampère’s force (JxB) leads to an entirely newclass of wave modes, called Alfvén (or MHD) waves.As the displacement current is neglected in MHD, there are noelectromagnetic waves of classical electrodynamics. Of course EM waves canpropagate through MHD plasma (e.g. light, radio waves, etc.) and eveninteract with the plasma particles, but that is beyond the MHD approximation.21

Nature, vol 150, 405-406 (1942)Dispersion equation for ideal MHD waveseliminate Peliminate Jeliminate EWe are left with 7 scalar equations for 7 unknowns ( m0 , V, B)Consider smallperturbationsand linearize22

()()Find an equation for V 1 . Start by taking the time derivative of ()()Insert () and () and introduce the Alfvén velocity as a vectorLook for plane wave solutionsUsinga few times we havethe dispersion equation for the waves in ideal MHDAlfvén wave modesPropagation perpendicular to the magnetic field:Nowand the dispersion equation reduces toclearlyAnd we have found the magnetosonic waveMaking a plane wave assumption also for B 1 (very reasonable, why?) (i.e., B 1 || B 0 )The wave electric field follows from the frozen-in conditionThis mode has many names in the literature:- Compressional Alfvén wave- Fast Alfvén wave- Fast MHD wave23

Propagation parallel to the magnetic field:Now the dispersion equation reduces toTwo different solutions (modes)1) V 1 || B 0 || k the sound wave2) V 1 B 0 || k V 1This mode is called- Alfvén wave or- shear Alfvén wavePropagation at an arbitrary angleke xe zB 0e yDispersionequation Coeff. of V 1y shear Alfvén waveFrom the determinant of the remaining equations:Fast (+) and slow (–) Alfvén/MHD waves24

v A > v sfastfastmagnetosonicslowsound wavesound wavemagnetosonicv A < v sWave normal surfaces:phase velocity asfunction of fastfastslowBeyond MHD:Quasi-neutral hybrid approachMHD is not an appropriate description if the physical scale sizes of thephenonomenon to be studied become comparable to the gyroradiiof the particles we are interested in, e.g.• details of shocks (at the end of the course)• relatively small or nonmagnetic objects in the solar wind (Mercury, Venus, Mars)What options do we have?• Vlasov theory• Quasi-neutral hybrid approachIn QN approach electrons are still considered as a fluid (small gyroradii) butions are described as (macro) particlesQuasi-neutrality requires that we are still in the plasma domain(larger scales than De )25

As electrons and ions are separated, Ohm’s law is important.Recall the generalized Ohm’s law(similar to Hall MHD)0Inclusion of pressure termrequires assumptions ofadiabatic/isothermal behaviorof electron fluid & T eNote: The QN equations cannotbe casted to conservation form26

Quasi-neutral hybrid simulations of Titan’s interaction with the magnetosphereof Saturn during a flyby of the Cassini spacecraft 26 December 2005From Ilkka Sillanpää’s PhD thesis, 2008.Escape of O+ and H+ from the atmosphere of Venus.Using the quasi-neutral hybrid simulationFrom Riku Järvinen’s PhD thesis, 2010.27

Beyond MHD: Kinetic Alfvén wavesAt short wavelengths kinetic effects start to modify the Alfvén waves.Easier than to immediately go to Vlasov theory, is to look for kineticcorrections to the MHD dispersion equationFor relatively large betae.g., in the solar wind, at magnetospheric boundaries, magnetotail plasma sheetthe mode is called oblique kinetic Alfvén wave and it has the phase velocityFor smaller beta ( ),e.g., above the auroral region and in the outer magnetosphere, the electronthermal speed is smaller than the Alfvén speed electron inertial becomes importantinertial (kinetic) Alfvén wave orshear kinetic Alfvén waveKinetic Alfvén waves• carry field-aligned currents (important in M-I coupling)• are affected by Landau damping (although small as long as is small),recall the Vlasov theory resultFull Vlasov treatment leads to the dispersion equationmodified Bessel functionof the first kindFor details, see Lysak and Lotko, J. Geophys. Res., 101, 5085-5094, 199628