Transcription of Introduction to Magnetohydrodynamics
1 Introduction to MagnetohydrodynamicsNick MurphyHarvard-Smithsonian Center for AstrophysicsJuly 17, 2015To MHD and beyond!IWhat is MHD?IThe equations of MHD and their physical meaningIWaves in MHDIAlfv en wavesISlow magnetosonic wavesIFast magnetosonic wavesIBeyond MHDIE xtensions to MHDIP lasma kinetic theoryIMagnetic reconnectionIFinal thoughtsWhat is MHD?IMagnetohydrodynamics (MHD) couples Maxwell s equationsof electromagnetism with hydrodynamics to describe themacroscopic behavior of conducting fluids such as is important in solar physics, astrophysics, spaceplasma physics, and in laboratory plasma experimentsLeft: The International Thermonuclear Experimental Reactor(ITER.)
2 Currently under construction)Right:Interaction between the solar wind and the Earth smagnetosphereMHD at a glance (SI units)Continuity Equation t+ ( V) = 0 Momentum Equation ( t+V )V=J B pAmpere s law 0J= BFaraday s law B t= EIdeal Ohm s lawE+V B= 0 Divergence constraint B= 0 Adiabatic Energy Equationddt(p )= 0 Definitions:B, magnetic field;V, plasma velocity;J, current density;E, electric field; , mass density;p, plasma pressure; , ratio of specific heats (usually 5/3);t, MHD approximationIAssume the plasma behaves as a fluidIMacroscopic (low frequency, long wavelength) behaviorIAssume that the gyroradius is smallIIgnore the most significant physics advances since 1860.
3 IRelativity (v2 c2)IQuantum mechanicsIDisplacement current in Ampere s lawIAssume the plasma is fully ionizedILimited applicability to weakly ionized plasmas like thephotosphere and chromosphereIAssume collisions are frequent enough that the particledistribution function is MaxwellianINot always true in the solar wind and laboratory plasmasIIdeal MHD assumes no resistivity, viscosity, thermalconduction, or radiative coolingThe continuity equation describes conservation of massV IThe continuity equation written in conservative form is.
4 T+ ( V) = 0 IThe partial derivative / trefers to the change in density ata single point in spaceIThe divergence of the mass flux ( V) says how muchplasma goes in and out of the regionIPut sources and sinks of mass on RHSThe second golden rule of astrophysics The density of wombatstimes the velocity of wombatsgives the flux of wombats. The momentum equation is analogous toma=FIThe momentum equation is ( t+V )V=J B pAdditional forces go on the right hand side ( , gravity).IThe total derivative is given byDDt t+V and represents the derivative you take as you follow a parcelof a static equilibrium:J B= pWhenJ B= 0, the plasma is force-free The pressure gradient force ppushes plasma fromregions of high pressure to low plasma pressureThe Lorentz force term includes two componentsIThe current density is given by the relative drift between ionsand electrons:J=ne(Vi Ve) J Bis analogous toF=qV vector identities and Ampere s law ( 0J= B), werewrite the Lorentz force termJ Bas.
5 J B=(B )B 0 (B22 0)However: the Lorentz force is orthogonal toB, but these twoterms are Lorentz force can be decomposed into two terms withforces othogonal toBusing field line curvatureIThe curvature vector points toward the center of curvatureand gives the rate at which the tangent vector turns:IWe can then write the Lorentz force asJ B Lorentz force= B2 0 magnetic tension (B22 0) magnetic pressure(1)where all terms are orthogonal toB. The operator takesthe gradient only in the direction orthogonal magnetic tension force wants to straighten magneticfield linesIThe tension force is directed radially inward with respect tomagnetic field line curvatureRegions of high magnetic pressure exert a force towardsregions of low magnetic pressureIThe magnetic pressure is given bypB B22 0 The ratio of the plasma pressure to the magnetic pressureis an important dimensionless numberIDefine plasma as plasma pressuremagnetic pressure pB2/2 0 IIf 1 then the magnetic field
6 DominatesISolar coronaIIf 1 then plasma pressure forces dominateISolar interiorIIf 1 then pressure/magnetic forces are both importantISolar chromosphereIParts of the solar wind and interstellar mediumISome laboratory plasma experimentsFaraday s law tells us how the magnetic field varies withtime B t= EBut how do we get the electric field?Ohm s law provides the electric fieldIThe ideal MHD Ohm s law is given byE+V B= 0 IIn ideal MHD, the magnetic field isfrozen-into the plasma. Iftwo parcels of plasma are connected by a magnetic field lineat one time, then they will be connected by a magnetic fieldline at all other resistive MHD, Ohm s law becomesE+V B= Jwhere is the resistivity.
7 Resistivity allows the frozen-incondition to be also include the Hall effect which is important on shortlength scalesWith Ohm s law we can rewrite Faraday s law as theinduction equationIUsing the resistive Ohm s law: B t= (V B) convection+ 0 2B diffusionDiffusion is usually represented by a second order example of resistive diffusion:Thermal conductionIThermal conduction is a common extension to MHDIHeat diffuses much more quickly along magnetic field linesthan orthogonal to themIAnisotropic thermal conduction is a challenge in numericalsimulationsIThe temperature along magnetic field lines is usuallyapproximately constantIExceptions: when localized heating occurs on short timescales,or there are rapid connectivity changesWavesIThere are three primary waves that arise from MHD.
8 IAlfv en waveISlow magnetosonic waveIFast magnetosonic waveIThere are two important speedsIThe sound speed is given byCs p IThe Alfv en speed is given byVA B 0 Alfv en WavesIAlfv en waves propagate at the Alfv en speed,VA B/ 0 IThe restoring force is magnetic tensionIThis is a shear wave with no compression involvedIDisturbances propagate parallel toBSlow and Fast Magnetosonic WavesILeft:The restoring forces for magnetosonic waves propagatingperpendicular toBare given by gas and magnetic pressuregradients. This shows a compressional :The phase velocity of MHD waves as a function ofangle whenBis in thezdirection and is waves are magnetosonic waves propagating alongBHow useful is MHD?
9 IMHD is appropriate for large-scale, low-frequency behaviorIMHD is a good predictor of stabilityINon-MHD effects sometimes stabilize or destabilize..IMHD is often inappropriate when there are non-Maxwelliandistribution functionsIIncluding in collisionless plasmas or when there are energetic,non-thermal particlesIMHD is a reasonable approximation for most solar physicsapplications, but there are many effects beyond MHD that willoften be importantIMHD usually does not usually work well for laboratory plasmasThere are two general approaches to going beyond MHDIE xtended the fluid approximation but includeeffects such asIAnisotropic thermal conductionIDifferent ion and electron temperatures:Tion6=TelectronIDifferent pressures parallel and perpendicular toB.
10 P 6=p IFinite Larmor radius effectsIHall effectIKinetic the fluid approximation and keeptrack of particle distribution intensive but often necessary to includeimportant physical processesISometimes we can take a hybrid between these twoapproachesMagnetic Reconnectionis the breaking and rejoining ofmagnetic field lines in a highly conducting plasmaSolar flares and CMEs are powered by magneticreconnectionIExplosive release of magnetic energyIBidirectional Alfv enic jetsIVery efficient particle accelerationIFlux ropes escape as coronal mass ejections (CMEs)Magnetic reconnection is a fundamental process inlaboratory and astrophysical plasmasIClassical theories based on resistive diffusion predict slowreconnection (weeks to months.)
