p = nkt p ~! " !" /!t + # "u = 0 Assumptions for MHD Fluid picture !du/dt = nq(e + uxb) " #p + other Newton s 2nd law Maxwell s equations

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1 Intro to MHD Newton s 2nd law Maxwell s equations Plasmas as fluids Role of magnetic field and MHD Ideal MHD What do we need to know to understand the sun, solar wind&shocks, magnetospheres? Some material from: Notes from Lysak 4611 Fluid picture Assumptions for MHD Conservation of mass Equation of continuity Forces on fluid Momentum equation Energy equation Often just need equation of state!" /!t + # "u = 0!du/dt = nq(e + uxb " #p + other p = nkt p ~! " Ionized, high density Velocities are slow (v<<c Timescales are slow (t>>ion gyroperiod, plasma period Scale lengths long compared to gyroperiod,mean free path Quasi-neutrality (n e ~n i Ideal - high conductivity

2 Equations of MHD For mass density, center of mass velocity!" /!t + # "u = 0!du/dt = jxb " #p +!g What happens to Maxwell s equations?! B = 0! E = "/# 0! $ B = µ 0 j + 1 c %E/%t 2! $ E = &%B/%t Note p can be anisotropic! Remember slow and n e ~n i! B = 0! E! # B = µ 0 j! # E = $%B/%t Ohm s Law and Frozen-in condition How does magnetic field evolve? Assume currents given by Ohm s law in plasma rest frame and transform to lab frame j =!(E + vxb Compare to 3.55 in our text. Why can we leave out the other terms? m e e 2 n!j/!t = E + vxb " (jxb/en + #p e /e2 n " j/$ j =! (E + vxb E = j/! " vxb!x(j/" # vxb = -$B/$t!x(! % B/µ 0 " #!x(vxb = -$B/$t! " B = µ 0 j # j =! " B/µ 0! " E = $%B/%t Do vector manipulations to get:!b/!t + "x(vxb = #" 2 B where # is 1/µ 0 $ j =!(E + vxb E + vxb = j/! Ideal MHD assume infinite conductivity

3 Ohm s Law and Frozen-in condition Frozen-in condition E + vxb = j/! Ideal MHD assumes infinite conductivity E =!vxb E =!vxb Magnetic field lines move with the plasma!b/!t + "x(vxb = 0 Think of a bundle of field lines and the resulting magnetic flux through a surface The flux through a surface defined by a loop moving with the plasma stays constant. Magnetic field is frozen-into the plasma. What would be effect of a compression of the fluid? Increase in field strength Also can tangle up See posted Russell, Chap.3 pg or our book pg for detailed derivation E =!vxb Warning: Frozen-in condition Use with care Magnetic field lines move with the plasma Very useful construct in many space plasma contexts Structure of magnetized plasmas: Narrow boundaries The simplest picture of magnetized plasmas is a fluid one (MHD. If there is no resistivity, the magnetic flux through a given fluid element stays fixed. This is called the frozen-in condition - E+vxB=0 Most interesting physics is when frozen-in condition is violated and how it happens Parallel electric fields Reconnection Kivelson and Russell, 1995 Results in regions of very different plasma conditions separated by narrow boundaries because plasmas are stuck in a specific magnetic flux tube.

4 Back to evolution of B!B/!t + "x(vxb = #" 2 B where # is 1/µ 0 $ Back to momentum equation!du/dt = jxb " #p +!g Ignore gravity and remember that j =! " B/µ 0 Advective term resistive term!du/dt = ("xbxb/µ 0 # "p advective term V/L diffusive term!/l 2 R M >>1 equivalent to frozen-in For R M <<1, Ratio is V/L/!/L 2 R M =VL/! magnetic Reynolds number Lorentz force term is jxb = ("xbxb/µ 0 More vector stuff to get: jxb = B!B/µ 0!B/!t = "# 2 B B/T ~ "B /L 2 T ~ L 2 /" Example: Typical conductivity on Sun is ~10 mho/m and a sunspot scale size of 10,000 km, the lifetime is the order of 1 year. For Sun, R S! 700,000 km, magnetic decay time is only 5000 years #du/dt = "!p + B!B/µ 0 Equilibrium Example: sunspot Assume we are at equilibrium, du/dt=0!( p + B 2 /2µ 0 = B!B/µ 0 Often rhs is small, then we have pressure balance p + B 2 /2µ 0 = constant Why are sunspots dark? Model magnetic field as vertical in spot

5 Plasma beta Measure of relative importance of magnetic pressure and plasma pressure! = p/ p B = p /(B 2 /µ 0 p B =4x10-13 B 2 Pa, where B in nt. ( 1 Pa = 1 N/m 2, and atmospheric pressure is 10 5 Pa Solar wind beta example from Lepri et al., Astrophysical Journal 674 ( doi: / Example of plasma and magnetic field data from the Ulysses SWICS, SWOOPS, and MAG instruments. (a The magnetic flux measuring in situ, (b plasma beta, (c solar wind proton speed, (d oxygen charge state ratio, and (e solar wind proton density. For solar wind, n~ 10 cm -3, T~ 10 ev and B~ 10 nt,! = 0.4. Earth s auroral zone, n~ 1 cm -3, T~ 1 kev, B~10 µt.! = 4x10-6 In plasma sheet, n n~ 1 cm -3, T~ 1 kev, B~10 nt, what is!? Other examples: coronal hole!~10-3 Is this a low beta or high beta plasma? Back to momentum equation!du/dt = "#p + B #B/µ 0 " #B 2 /2µ 0 B #B/µ 0 " #B 2 /2µ 0 = # (BB/µ 0 " B 2 I/2µ 0 Magnetic stress tensor: B 2 /2µ 0 is isotropic pressure term B 2 /µ 0 is magnetic tension!du/dt = "#(p + B 2 /2µ 0 + B #B/µ 0!du/dt = "#(p + p B + B #B/µ 0

6 Example: force on plasma Example: force on plasma B = (2y,1,0 j =!xb/µ 0 = (0,0,"2 /µ 0 jxb = (2/µ 0,"4y /µ 0,0 B = (2y,1,0 alternate approach Use jxb = B!B/µ 0 B!B = (2,0,0!B 2 = (0,8y,0 B!B/µ 0 = (2 /µ 0,"4 y /µ 0,0 Plasma slingshot to right Important in flares, cmes, tail reconnection Example: force on plasma Example: force on plasma j =!xb /µ 0 = (0,0,2bxe "x 2 /µ 0 jxb = (2bxe "2x 2 /µ 0,0,0 B = (0,be!x 2,0 B = (0,be!x 2,0 alternate approach Use jxb = B!B/µ 0 B!B/µ = 0 "!B 2 /2µ 0 = (2bxe "2x 2 /µ 0,0,0 Plasma pushed away from high B region

7 Start with MHD equations!" /!t + # "u = 0!du/dt = "#p + #xb/µ 0!B/!t + "x(vxb = 0 usual linearization B = B 0 + b,! =! 0 +! 1, p = p 0 + p 1,!" 1 /!t + # u = 0!u/!t = $#p 1 + (#xbxb 0 /µ 0!p 1 /!t = c s 2!" 1 /!t!b/!t = #x(uxb 0 Plasma waves Need energy equation, use adiabatic dp/dt = c s 2 d!/dt c s = "p! Assume no initial velocity, uniform medium, no gravity Fourier transform! 1 =!e i( k r r "#t,b = be i( k r r "#t,etc. $ % ik r and & /&t % "i#!" 1 /!t + # u = 0!u/!t = $#p 1 + (#xbxb 0 /µ 0!p 1 /!t = c s 2!" 1 /!t!b/!t = #x(uxb 0 r!i"# 1 + i# 0 k u = 0!i"# 0 u =!ik r p 1 + (ik r xbxb 0 /µ 0!i"p 1 =!i"c s 2 # 1!i"b = i r k x(uxb 0 do lots of algebra to get Plasma waves Fourier transform! 2 u = c s 2 k(k u + 1 µ 0 (ikx(ikx(uxb 0 xb 0! 1 =!e i( k r r "#t,b = be i( k r r "#t,etc. $ % ik r and & /&t % "i# General MHD Dispersion Relation We have plasma velocity, u Phase velocity, v ph ="/k Wavevector k is perepndicular to phase fronts! 2 u = c s 2 k(k u + 1 µ 0 (ikx(ikx(uxb 0 xb 0 Start with something familiar - sound wave. Set B 0 =0! 2 u = c s 2 k(k u dot with k! 2 = c s 2 k 2 phase velocity is! /k = ±c s group velocity is "! /"k = c s k Group velocity, v gr= #"/#k Longitudinal (u along k, compressive