Magnetohydrodynamics. Basic MHD

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1 Magnetohydrodynamics Conservative form of MHD equations Covection and diffusion Frozen-in field lines Magnetohydrostatic equilibrium Magnetic field-aligned currents Alfvén waves Quasi-neutral hybrid approach Basic MHD (isotropic pressure) (or energy equation) Relevant Maxwell s equations; displacement current neglected ( ) Note that in collisionless plasmas and may need to be introduced to the ideal MHD s Ohm s law before 1

2 Conservation of mass density: Navier-Stokes equation(s): Some hydrodynamics Energy equation: polytropic index viscosity, if these are called Euler equation(s) In case we can write these in the conservation form is the momentum density is a tensor with components kinetic energy density primitive variables conserved variables thermal energy density MHD: add Ampère s force In MHD we neglect the displacement current, thus Now the energy density contains also the magnetic energy density 2

3 Ideal MHD: Faraday s law reduces to or using conserved variables: Thus we have found 8 equations in the conservation form for 8 conserved variables Convection (actually advection) and diffusion Take curl of the MHD Ohm s law and apply Faraday s law Thereafter use Ampère s law and the divergence of B to get the induction equation for the magnetic field (Note that has been assumed constant) Assume that plasma does not move diffusion equation: diffusion coefficient: If the resistivity is finite, the magnetic field diffuses into the plasma to remove local magnetic inhomogeneities, e.g., curves in the field, etc. Let L B the characteristic scale of magnetic inhomogeneities. The solution is where the characteristic diffusion time is 3

4 In case of the diffusion becomes very slow and the evolution of B is completely determined by the plasma flow (field is frozen-in to the plasma) convection equation The measure of the relative strengths of convection and diffusion is the magnetic Reynolds number R m Let the characteristic spatial and temporal scales be & and the diffusion time The order of magnitude estimates for the terms of the induction equation are and the magnetic Reynolds number is given by This is analogous to the Reynolds number in hydrodynamics In fully ionized plasmas R m is often very large. E.g. in the solar wind at 1 AU it is This means that during the 150 million km travel from the Sun the field diffuses about 1 km! Very ideal MHD: viscosity Example: Diffusion of 1-dimensional current sheet Initially: in the frame co-moving with the plasma (V = 0) has the solution Total magnetic flux remains constant = 0, but the magnetic energy density decreases with time Now Ohmic heating, or Joule heating 4

5 Example: Conductivity and diffusivity in the Sun Photosphere partially ionized, collisions with neutrals cause resisitivity: Photospheric granules Observations of the evolution of magnetic structures suggest a factor of 200 larger diffusivity, i.e., smaller R m Explanation: Turbulence contributes to the diffusivity: This is an empirical estimate, nobody knows how to calculate it! Corona above 2000 km the atmosphere becomes fully ionized Spitzer s formula for the effective electron collision time numerical estimate for the conductivity Estimating the diffusivity becomes For coronal temperature Home exercise: Estimate R m in the solar wind close to the Earth 5

6 Frozen-in field lines A concept introduced by Hannes Alfvén but later denounced by himself, because in the Maxwellian sense the field lines do not have physical identity. It is a very useful tool, when applied carefully. Assume ideal MHD and consider two plasma elements joined at time t by a magnetic field line and separated by In time dt the elements move distances udt and In Introduction to plasma physics it was shown that the plasma elements are on a common field line also at time t + dt In ideal MHD two plasma elements that are on a common field line remain on a common field line. In this sense it is safe to consider moving field lines. Another way to express the freezing is to show that the magnetic flux through a closed loop defined by plasma elements is constant dl x VDt C C dl S S B Closed contour C defined by plasma elements at time t The same (perhaps deformed) contour C at time t + t Arc element dl moves in time t the distance Vt and sweeps out an area dl x Vt Consider a closed volume defined by surfaces S, S and the surface swept by dl x Vt when dl is integrated along the closed contour C. As B = 0, the magnetic flux through the closed surface must vanish at time t + t, i.e, Calculate d/dt when the contour C moves with the fluid: 6

7 The integrand vanishes, if Thus in ideal MHD the flux through a closed contour defined by plasma elements is constant, i.e., plasma and the field move together The critical assumption was the ideal MHD Ohm s law. This requires that the ExB 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 term so-called Hall MHD It is a straightforward exercise to show that in Hall MHD the magnetic field is frozen-in to the electron motion and When two ideal MHD plasmas with different magnetic field orientations flow against each other, magnetic reconnection can take place. Magnetic reconnection can break the frozen-in condition in an explosive way and lead to rapid particle acceleration Example of reconnection: a solar flare 7

8 Magnetohydrostatic equilibrium Assume scalar pressure and i.e., B and J are vector fields on constant pressure surfaces Write the magnetic force as: Magnetohydrostatic equilibrium after elimination of the current Assume scalar pressure and negligible (BB) Plasma beta: magnetic pressure magnetic stress and torsion The current perpendicular to B: Diamagnetic current: this is the way how B reacts on the presence of plasma in order to reach magnetohydrostatic equilibrium. This macroscopic current is a sum of drift currents and of the magnetization current due to inhomogeneous distribution of elementary magnetic moments (particles on Larmor motion) dense plasma total current tenuous plasma In magnetized plasmas pressure and temperature can be anisotropic Define parallel and perpendicular pressures as Using these we can derive the macroscopic currents from the single particle drift motions and where 8

9 The magnetization is Summing up J G, J C and J M we get the total current Magnetohydrostatic equilibrium in anisotropic plasma is described by In time dependent problems the polarization current must be added Force-free fields a.k.a. magnetic field-aligned currents In case of the pressure gradient is negligible and the equilibrium is i.e., no Ampère s force Flux-rope: The field and the current in the same direction is a non-linear equation If and are two solutions, does not need to be another solution Now is constant along B If is constant in all directions, the equation becomes linear, the Helmholtz equation Note that potential fields are trivially force-free 9

10 Force-free fields as flux-ropes in space Loops in the Solar corona Coronal Mass Ejection (CME) Interplanetary Coronal Mass Ejection (ICME) Example: Linear force-free model of a coronal arcade Construct a model that looks like an arc in the xz-plane extends uniformly in the y-direction The form of the Helmholtz equation sinusoidal in x-direction suggests a trial of the form: with the condition vanishes at large z 10

11 The solution looks like: Arcs in the xz-plane Projection of field lines in the xy-plane are straigth lines parallel to each other Case of so weak current that we can neglect it ( = 0): Potential field (no distortion due to the current) Solve the two-dimensional Laplace equation Look for a separable solution Magnetic field lines are now arcs This is fulfilled by without the shear in the y-direction Grad Shafranov equation Often a 3D structure is essentially 2D (at least locally) e.g. the linear force-free arcade above or an ICME Consider translational symmetry B uniform in z direction Assume balance btw. magnetic and pressure forces The z component reduces to 11

12 Now also These are valid if Total pressure the Grad-Shafranov equation The GS equation is not force-free. For a force-free configuration we can set P = 0 While non-linear, it is a scalar eq. Examples of using the Grad-Shafranov method in studies of the structure of ICMEs, see, Isavnin et al., Solar Phys., 273, , 2011 DOI: /s z Black arrows: Magnetic field measured by a spacecraft during the passage of an ICME Contours: Reconstruction of the ICME assuming that GS equation is valid 12

13 Some general properties of force-free fields Proof: As B vanishes faster than r 2 at large r, we can write that is zero for force-free fields Proof: Exercise Thus if there is FAC in a finite volume, it must be anchored to perpendicular currents at the boundary of the volume (e.g. magnetosphere-ionosphere coupling through FACs) Magnetospheric and ionospheric currents are coupled through field-aligned currents (FAC) Solar energy Magnetosphere (transports and stores energy) FAC Ionosphere (dissipate energy) The ionosphere and magnetosphere are coupled in so many different ways that nearly every magnetospheric process bears on the ionosphere in some way and every ionospheric process on the magnetosphere Wolf,

14 Recall: Poynting s theorem EM energy is dissipated via currents J in a medium. The rate of work is defined J E no energy dissipation! F v q E v J E work performed by the EM field energy flux through the surface of V change of energy in V Poynting vector (W/m 2 ) giving the flux of EM energy EJ > 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 magnetopause opened by reconnection acts as a generator (Poynting flux into the magnetosphere) Dayside reconnection is a load ( J E > 0 ) and does not create current Solar wind pressure maintains J CF and the magnetopause current 14

15 Currents in the magnetosphere In large scale (MHD) magnetospheric currents are source-free and obey Ampère s law Thus, 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 B Recall 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 magnetosphere A simple MHD generator Plasma flows across the magnetic field F = q (u x B) e down, i + up Let the electrodes be coupled through a load current loop, where the current J through the plasma points upward Now Ampère s force J x B decelerates the plasma flow, i.e., the energy to the newly created J (and thus B) in the circuit is taken from the kinetic energy. Kinetic energy EM energy: Dynamo! The setting can be reversed by driving a current through the plasma: J x B accelerates the plasma plasma motor 15

16 Boundary layer generator Note: This part of the figure (dayside magnetopause) is out of the plane One scenario how an MHD generator may drive field-aligned current from the LLBL: Plasma penetrates to the LLBL through the dayside magnetopause (e.g. by reconnection. The charge separation ( electrodes of the previous example) is consistent with E = V x B. The FAC ion the inside part closes via ionospheric currents (the so-called Region 1 current system) FAC from the magnetosphere We start from the (MHD) current continuity equation where The divergence of the static total perpendicular current is [ Note: ( M) = 0 ] In the isotropic case this reduces to The polarization current can have also a divergence Integrating the continuity equation along a field line we get: = b ( V) is the vorticity Sources of FACs: pressure gradients & time-dependent 16

17 FAC from the ionosphere Write the ionospheric () current as a sum of Pedersen and Hall currents and integrate the continuity equation along a field line: Recall: Approximate the RHS integral as R1 R1 where P = h P and H = h are height-integrated Pedersen and Hall conducitivities and h the thickness of the conductive ionosphere (~ the E-layer) The sources and sinks of FACs in the ionosphere are divergences of the Pedersen current and gradients of the Hall conductivity R2 R2 R1: Region 1, poleward FAC R2: Region 2, equatorward FAC 17

18 Current closure The closure of the current systems is a non-trivial problem. Let s make a simple exercise based on continuity equation Let 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 other we neglect this Assuming the magnetic field north-south symmetric, this reduces to This must be the same as the FAC from the ionosphere and we get an equation for ionosphere-magnetosphere coupling One more view on M-I coupling 18

19 Gauge transformation: Magnetic helicity A is the vector potential Helicity is gauge-independent only if the field extends over all space and vanishes sufficiently rapidly Helicity 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 which To prove this, note that from we get to within a gauge transformation For finite magnetic field configurations helicity is well defined if and only if on the boundary on S Calculate a little and both B and V are ^ n on S thus on S 19

20 Example: Helicity of two flux tubes linked together Calculate For thin flux tubes is approximately normal to the cross section S (note: here S is not the boundary of V!) Helicity of tube 1: Thus Similarly If the tubes are wound N times around each other Complex flux-tubes on the Sun Helicity is a measure of structural complexity of the magnetic field 20

21 Woltjer s theorem For a perfectly conducting plasma in a closed volume V 0 the integal is invariant and the minimum energy state is a linear force-free field Proof: Invariance was already shown. To find the minimum energy state consider Small perturbations: on S and added transform to surface integral that vanishes because Thus if and only if Magnetohydrodynamic waves V 1 Dispersion equation for MHD waves Alfvén wave modes MHD is a fluid theory and there are similar wave modes as in ordinary fluid theory (hydrodynamics). In hydrodynamics the restoring forces for perturbations are the pressure gradient and gravity. Also in MHD the pressure force leads to acoustic fluctuations, whereas Ampère s force (JxB) leads to an entirely new class of wave modes, called Alfvén (or MHD) waves. As the displacement current is neglected in MHD, there are no electromagnetic waves of classical electrodynamics. Of course EM waves can propagate through MHD plasma (e.g. light, radio waves, etc.) and even interact with the plasma particles, but that is beyond the MHD approximation. 21

22 Nature, vol 150, (1942) Dispersion equation for ideal MHD waves eliminate P eliminate J eliminate E We are left with 7 scalar equations for 7 unknowns ( m0, V, B) Consider small perturbations and linearize 22

23 () () Find an equation for V 1. Start by taking the time derivative of () () Insert () and () and introduce the Alfvén velocity as a vector Look for plane wave solutions Using a few times we have the dispersion equation for the waves in ideal MHD Alfvén wave modes Propagation perpendicular to the magnetic field: Now and the dispersion equation reduces to clearly And we have found the magnetosonic wave Making 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 condition This mode has many names in the literature: - Compressional Alfvén wave - Fast Alfvén wave - Fast MHD wave 23

24 Propagation parallel to the magnetic field: Now the dispersion equation reduces to Two different solutions (modes) 1) V 1 B 0 k the sound wave 2) V 1 B 0 k V 1 This mode is called - Alfvén wave or - shear Alfvén wave Propagation at an arbitrary angle e z B 0 k e y e x Dispersion equation Coeff. of V 1y shear Alfvén wave From the determinant of the remaining equations: Fast (+) and slow ( ) Alfvén/MHD waves 24

25 v A > v s fast fast magnetosonic slow sound wave sound wave magnetosonic v A < v s Wave normal surfaces: phase velocity as function of fast fast slow Beyond MHD: Quasi-neutral hybrid approach MHD is not an appropriate description if the physical scale sizes of the phenonomenon to be studied become comparable to the gyroradii of 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 approach In QN approach electrons are still considered as a fluid (small gyroradii) but ions are described as (macro) particles Quasi-neutrality requires that we are still in the plasma domain (larger scales than De ) 25

26 As electrons and ions are separated, Ohm s law is important. Recall the generalized Ohm s law (similar to Hall MHD) 0 Inclusion of pressure term requires assumptions of adiabatic/isothermal behavior of electron fluid & T e Note: The QN equations cannot be casted to conservation form 26

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

28 Beyond MHD: Kinetic Alfvén waves At short wavelengths kinetic effects start to modify the Alfvén waves. Easier than to immediately go to Vlasov theory, is to look for kinetic corrections to the MHD dispersion equation For relatively large beta e.g., in the solar wind, at magnetospheric boundaries, magnetotail plasma sheet the mode is called oblique kinetic Alfvén wave and it has the phase velocity For smaller beta ( ), e.g., above the auroral region and in the outer magnetosphere, the electron thermal speed is smaller than the Alfvén speed electron inertial becomes important inertial (kinetic) Alfvén wave or shear kinetic Alfvén wave Kinetic 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 result Full Vlasov treatment leads to the dispersion equation modified Bessel function of the first kind For details, see Lysak and Lotko, J. Geophys. Res., 101, ,

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