Introduction to Magnetohydrodynamics (MHD) Tony Arber University of Warwick 4th SOLARNET Summer School on Solar MHD and Reconnection
Aim Derivation of MHD equations from conservation laws Quasi-neutrality Validity of MHD MHD equations in different forms MHD waves Alfvén s Frozen Flux Theorem Turbulence Characteristics Shocks Applications of MHD, i.e. all the interesting stuff!, will be in later lectures covering Waves, Reconnection, CMEs and Instabilities etc.
Derivation of MHD Possible to derive MHD from N-body problem to Klimotovich equation Louiville theorem to BBGKY hierarchy Simple fluid dynamics and control volumes First two are useful if you want to study kinetic theory along the way but all kinetics removed by the end Final method followed here so all physics is clear
Ideal MHD Maxwell equation Mass conservation F = ma for fluids Low frequency Maxwell Adiabatic equation for fluids Ideal Ohm s Law for fluids 8 equations with 8 unknowns
Mass Conservation - Continuity F (x) m = Z x+ x x dx F (x + x) x x + x Mass m in cell of width x changes due to rate of mass leaving/entering the cell F (x) Z! @ x+ x dx = F (x) F (x + x) @t x @ @t = lim x!0 F (x) F (x + x) x @ @t + @F (x) @x =0
Mass flux - conservation laws @ @t + @F (x) @x =0 Mass flux per second through cell boundary In 3D this generalizes to F (x, t) = (x, t) v x (x, t) @ + r.( v) =0 @t Z This is true for any conserved quantity so if conserved @U + r.f =0 @t U dx Hence applies to mass density, momentum density and energy density for example.
Convective Derivative In fluid dynamics the relation between total and partial derivatives is Convective derivative: Rate of change of quantity at a point moving with the fluid. Rate of change of quantity at a fixed point in space Often, and frankly for no good reason at all, write D Dt instead of d dt
Adiabatic energy equation If there is no heating/conduction/transport then changes in fluid element s pressure and volume (moving with the fluid) is adiabatic PV = constant Where is ration of specific heats d (PV )=0 dt Moving with a packet of fluid the mass is conserved so V / 1 d dt P =0
Momentum equation - Euler fluid P (x) u x x P (x + x) x x + x Total momentum in cell changes due to pressure gradient @ @t ( u x x) =F (x) F (x + x)+p (x) P (x + x) Now F is momentum flux per second F = u x u x @ @t ( u x)+ @F @x = rp
Momentum equation - Euler fluid Use mass conservation equation to rearrange as @u x @t + u @u x x @x = rp @ux @t + u @u x x = @x rp Since by chain rule du x dt = rp du x (x, t) dt = @u x @t + @x @t @u x @x
For Euler fluid Momentum equation - MHD du dt = rp Force on charged particle in an EM field is F = q(e + v B) how does this change for MHD? Hence total EM force per unit volume on electrons is n e e(e + v B) and for ions (single ionized) is Where n e and n i densities n i e(e + v B) are the electron and ion number
Momentum equation - MHD Hence total EM force per unit volume e(n i n e )E +(en i u i en e u e ) B If the plasma is quasi-neutral (see later) then this is just en(u i u e ) B = j B Where j is the current density. Hence du dt = rp + j B Note jxb is the only change to fluid equations in MHD.
Maxwell equations Not allowed in MHD! Initial condition only Used to update B Low frequency version used to find current density j
Low-frequency Maxwell equations
Displacement current So for low velocities/frequencies we can ignore the displacement current
Quasi-neutrality For a pure hydrogen plasma we have Multiply each by their charge and add to get where σ is the charge density and j is the current density From Ampere's law at low frequencies Hence for low frequency processes this is quasi-nuetrality
MHD Maxwell equation Mass conservation Momentum conservation Low frequency Maxwell Energy conservation 8 equations with 11 unknowns! Need an equation for E
Ohm's Law Equations of motion for ion fluid is Assume quasi-neutrality, subtract electron equation This is called the generalized Ohm's law Note that Ohm's law for a current in a wire (V=IR) when written in terms of current density becomes When fluid is moving this becomes
Magnetohydrodynamics (MHD) Valid for: Low frequency Large scales If η=0 called ideal MHD Missing viscosity, heating, conduction, radiation, gravity, rotation, ionisation etc.
Solar scales and times For a coronal plasma with: Density 10 9 cm -3 Temperature 1MK Magnetic field 10 G D = v th! pe ' 1mm r L = m iv th eb mfp = v th i = ' 10 cm ' 10 km c! pi ' 10 km Observational limits Cadence ~ seconds Pixel ~ 100 km
Validity of MHD Assumed quasi-neutrality therefore must be low frequency and speeds << speed of light Assumed scalar pressure therefore collisions must be sufficient to ensure the pressure is isotropic. In practice this means: mean-free-path << scale-lengths of interest collision time << time-scales of interest Larmor radii << scale-lengths of interest However as MHD is just conservation laws plus lowfrequency MHD it tends to be a good first approximation to much of the physics even when some of these conditions are not met.
Eulerian form of MHD equations @ @t = @P @t = r.( v) P r.v @v @t = v.r.(v) 1 r.p + 1 j B @B @t = r (v B) j = 1 µ 0 r B Final equation can be used to eliminate current density so 8 equations in 8 unknowns
Lagrangian form of MHD equations D Dt = DP Dt = Dv Dt = r.v P r.v 1 r.p + 1 j B DB Dt =(B.r)v B(r.v) D j = 1 r B Dt µ 0 Alternatives D Dt = Specific internal energy density P = ( 1) B P r.v = B.rv
Conservative form @ @t = @ v @t = r. r.( v) vv + I(P + B2 2 ) BB @E E @t = r. + P + B2 v 2µ 0 B(v.B) @B @t = r(vb Bv) E = P 1 + v2 2 + B2 2µ 0 The total energy density
Plasma beta A key dimensionless parameter for ideal MHD is the plasma-beta It is the ratio of thermal to magnetic pressure = 2µ 0P B 2 Low beta means dynamics dominated by magnetic field, high beta means standard Euler dynamics more important / c2 s v 2 A
Univorm B field MHD Waves Constant density, pressure Zero initial velocity Apply small perturbation to system Assume initially in stationary equilibrium r.p 0 = j 0 B 0 Simplify to easiest case with 0,P 0, B 0 = B 0 ẑ constant and no equilibrium current or velocity Apply perturbation, e.g. P = P 0 + P 1
MHD Waves Ignore quadratic terms, e.g. P 1 r.v 1 Linear equations so Fourier decompose, e.g. P 1 (r,t)=p 1 exp i(k.r!t) Gives linear set of equations of the form Ā.ū = ū Where ū =(P 1, 1,v 1,B 1 ) Solution requires det Ā Ī =0
Dispersion relation (Fast and slow magnetoacoustic waves) B k α (Alfvén waves) Alfvén speed Sound speed
Alfvén Waves Incompressible no change to density or pressure Group speed is along B does not transfer energy (information) across B fields
Fast magneto-acoustic waves For zero plasma-beta no pressure Compresses the plasma c.f. a sound wave Propagates energy in all directions
Magnetic pressure and tension Magnetic pressure Magnetic tension
Pressure perturbations
Phase and group speeds Phase speeds: : Group speeds
MHD Waves Movie Throwing a pebble into a plasma lake... For low plasma beta v A >> c s transv. velocity density Three types of MHD waves Alfvén waves magnetic tension (ω=v A k װ ) Fast magnetoacoustic waves magnetic with plasma pressure (ω V A k) Slow magnetoacoustic waves magnetic against plasma pressure (ω C S k װ )
MHD Waves Movie Throwing a pebble into a plasma lake... For low plasma betav A >> c s transv. velocity density Three types of MHD waves Alfvén waves magnetic tension (ω=v A k װ ) Fast magnetoacoustic waves magnetic with plasma pressure (ω V A k) Slow magnetoacoustic waves magnetic against plasma pressure (ω C S k װ )
Linear MHD for uniform media 1. The perturbations are waves 2. Waves are dispersionless 3. ω and k are always real 4. Waves are highly anisotropic 5. There are incompressible - Alfvén waves - and compressible - magnetoacoustic modes However, natural plasma systems are usually highly structured and often unstable See later lectures on waves & instabilities
Resistivity Electron-ion collisions dissipate current If we assume the resistivity is constant then @B @t = r (v B)+ µ 0 r 2 B Ratio of advective to diffusive - magnetic Reynolds number R m = µ 0L 0 v 0 R m >> 1 Usually in space physics (10 6-10 12 ).
Alfvén s theorem Rate of change of flux through a surface moving with fluid d dt Z S n.b ds = = Z n @B I @t ds v B.dl l Z r (E + v B).n ds S S Magnetic flux through a surface moving with the fluid is conserved with ideal MHD Ohm s law, i.e. no resistivity Often stated as- the flux is frozen in to the fluid
Line Conservation x(x + X,t) x(x + X, 0) X x x(x,t) x(x, 0) = X Consider two points which move with the fluid x i = @x i @X j D Dt x = @u i @X j X j X j = @u i @x k @x k @X j X j =( x.r)u
Line Conservation -2 Equation for evolution of the vector between two points D Dt Also for ideal MHD D Dt x =( x.r)u B Hence if we choose x to be along the magnetic field at t = 0 then it will remain aligned with the magnetic field. Two points moving with the fluid which are initially on the same field-line remain on the same field line in ideal MHD Reconnection not possible in ideal MHD = B.r v
Shown that Implies B and Cauchy Solution x x i = @x i @X j B i = @x i Bj 0 @X j 0 satisfy the same equation hence X j Where superscript zero refers to initial values B i = @x i @X j B 0 j 0 Cauchy solution 0 = = @(x 1,x 2,x 3 ) @(X 1,X 2,X 3 ) B i = @x i @X j B 0 j
MHD based on Cauchy B i = @x i @X j B 0 j 0 = = @(x 1,x 2,x 3 ) @(X 1,X 2,X 3 ) P = const Dv Dt = 1 r.p + 1 µ 0 (r B) B dx dt = v Only need to know position of fluid elements and initial conditions for full MHD solution
Non-ideal terms in MHD Ideal MHD is a set of conservation laws Non-ideal terms are dissipative and entropy producing Resistivity Viscosity Radiation transport Thermal conduction
Non-ideal MHD Resistivity Coriolis Gravity Other Thermal Conduction Radiation Ohmic heating Other
MHD equations Turbulence @U @t + r.f (U) =RHS ln(e k ) Energy input RHS = driver Inertial Range RHS = 0 ln(k) Dissipation RHS = viscous
Kolmogorov 1941 (K41) Energy transfer rate through inertial range is constant " local dissipation rate Fluctuations in flow moves energy between scales Eddie turn over time l l v l Energy transfer from scale l to smaller scales (l) v2 l l = v3 l l
K41-2 (l) v2 l l = v3 l l but (l) =" v l l 1/3 " 1/3 Energy associated with flow lv 2 l E l " 2/3 l 5/3 More commonly E k " 2/3 k 5/3 Ignores anisotropy of MHD S 3 (l) =< (u(x + l) u(x)) 3 >= 4 5 "l
Iroshmikov-Kraichnan (IK) Alfvén waves interact weakly and pass by in time Eddie turn-over time is l v l Time-scale for energy transfer is now l v A l l v l v A v l Repeat K41 but now get E k ("v A ) 1/2 k 3/2
Goldreich-Sridah (G-S) IK model is isotropic G-S extended with critical balance k k v A k? v k Turbulence then only really in k? E k k 5/3 Parallel spectrum, i.e. Alfvén wave, passively fixed by critical balance due to perpendicular cascades This is for strong turbulence when B v B
MHD Characteristics Sets of ideal MHD equations can be written as All equations sets of this types share the same properties they express conservation laws can be decomposed into waves non-linear solutions can form shocks satisfy L1 contraction, TVD constraints
Characteristics is called the Jabobian matrix For linear systems can show that Jacobian matrix is a function of equilibria only, e.g. function of p 0 but not p 1
Properties of the Jacobian Left and right eigenvectors/eigenvalues are real Diagonalisable:
Characteristic waves This example is for linear equations with constant A But A = R R 1 so R 1 A = R 1 @ @t w t R 1 U +. @ @x R 1 U =0 w + = = x 1 Λ. 0 with w R U w is called the characteristic field
Riemann problems is diagonal so all equations decouple i.e. characteristics w i propagate with speed In MHD the characteristic speeds are v x,v x ± c f,v x ± v A,v x ± c s Solution in terms of original variables U This analysis forms the basis of Riemann decomposition used for treating shocks, e.g. Riemann codes in numerical analysis
Temperature T = T (x c s t) Basic Shocks c 2 s = P/ x Without dissipation any 1D traveling pulse will eventually, i.e. in finite time, form a singular gradient. These are shocks and the differentially form of MHD is not valid. Also formed by sudden release of energy, e.g. flare, or supersonic flows.
Rankine-Hugoniot relations U U L U R S(t) x x l x r Integrate equations from x l to x r across moving discontinuity S(t)
Jump Conditions Use Let x l and x r tend to S(t) and use conservative form to get Rankine-Hugoniot conditions for a discontinuity moving at speed v s All equations must satisfy these relations with the same v s
The End