Magnetohydrodynamics (MHD)

Outline I. Introduction to Turbulence II. Measurements of Turbulence in the Solar Wind. III. Magnetohydrodynamics (MHD) IV. Alfven Waves, the Origin of the Solar Wind, and Solar Probe Plus V. Introduction to phenomenological theories of MHD turbulence. Magnetohydrodynamics (MHD) • In order to understand the power spectra seen in the solar wind, we need a theoretical framework for analyzing fluctuations on these lengthscales and timescales. • In the solar wind near Earth, phenomena occurring at large length scales (exceeding ≈300 km) and long time scales (e.g., exceeding ≈10 s) can be usefully described within the framework of a fluid theory called magnetohydrodynamics (MHD). • In MHD, the plasma is quasi-neutral, and the displacement current is neglected in Maxwell’s equations (since the fluctuation frequencies are small). Ideal, Adiabatic MHD magnetic field @⇢ = r · (⇢~v ) density @t ◆ ◆ ✓ ✓ ~ · rB ~ B @~v B2 + ⇢ + ~v · r~v = r p + @t 8⇡ 4⇡ ✓ ◆✓ ◆ @ p + ~v · r =0 pressure @t ⇢ ⇣ ⌘ ~ @B ratio of specific heats ~ = r ⇥ ~v ⇥ B @t velocity (The phrase “ideal MHD” means that dissipative terms involving viscosity and resistivity have been neglected. I’ll come back to these terms later.) Ideal, Adiabatic MHD “continuity equation” mass @⇢ conservation = r · (⇢~v ) @t ◆ ◆ ✓ ✓ ~ · rB ~ B @~v B2 + ⇢ + ~v · r~v = r p + @t 8⇡ 4⇡ ✓ ◆✓ ◆ @ p + ~v · r =0 @t ⇢ ⇣ ⌘ ~ @B ~ = r ⇥ ~v ⇥ B @t Ideal, Adiabatic MHD Newton’s 2nd law @⇢ = r · (⇢~v ) @t ◆ ◆ ✓ ✓ ~ · rB ~ B @~v B2 + ⇢ + ~v · r~v = r p + @t 8⇡ 4⇡ ✓ ◆✓ ◆ @ p + ~v · r =0 @t ⇢ ⇣ ⌘ ~ @B ~ = r ⇥ ~v ⇥ B @t Ideal, Adiabatic MHD @⇢ = r · (⇢~v ) @t ◆ ◆ ✓ ✓ ~ · rB ~ B @~v B2 + ⇢ + ~v · r~v = r p + @t 8⇡ 4⇡ ✓ ◆✓ ◆ @ p adiabatic evolution. + ~v · r =0 the specific entropy @t ⇢ [∝ln(p/ργ)] of a fluid ⇣ ⌘ element does not ~ @B ~ change in time. = r ⇥ ~v ⇥ B @t (An alternative, simple approximation is the isothermal approximation, in which p = ρcs2, with the sound speed cs = constant. More generally, this equation is replaced with an energy equation that includes thermal conduction and possibly other heating and cooling mechanisms.) Ideal, Adiabatic MHD @⇢ = r · (⇢~v ) @t ◆ ◆ ✓ ✓ ~ · rB ~ B @~v B2 + ⇢ + ~v · r~v = r p + @t 8⇡ 4⇡ ✓ ◆✓ ◆ @ p + ~v · r =0 Ohm’s Law for @t ⇢ a perfectly ⇣ ⌘ ~ conducting @B ~ = r ⇥ ~v ⇥ B plasma @t Ideal, Adiabatic MHD @⇢ = r · (⇢~v ) @t ◆ ✓ ◆ ✓ ~ · rB ~ B2 B @~v + ~v · r~v = r p + + ⇢ @t 8⇡ 4⇡ ◆✓ ◆ ✓ p @ + ~v · r =0 @t ⇢ ⇣ ⌘ ~ @B ~ = r ⇥ ~v ⇥ B @t • Magnetic forces: magnetic pressure and magnetic tension • Frozen-in Law: magnetic field lines are like threads that are frozen to the plasma and advected by the plasma Waves: small-amplitude oscillations about some equilibrium ~v = ~v0 cos(~k · ~x !t + 0) B0 wavelength λ = 2π/k decreasing λ increasing k k kk energy cascades to small λ, or equivalently to large k k? As an example, let’s look at MHD Waves in “low-beta” plasmas such as the solar corona. b = 8pp ⌧ 1, B2 so magnetic pressure cs = sound speed = B2 greatly exceeds p 8p r gp r B vA = “Alfvén speed” = p 4pr ! b = 2c2s g v2A Plasma Waves at Low Beta Alfven wave fast magnetosonic wave slow magnetosonic wave magnetic pressure thermal pressure magnetic field lines magnetic tension (like a wave w = kk von propagating A a string) undamped w = kvA weakly damped w = kk cs strongly damped Plasma Waves at Low Beta Alfven wave fast magnetosonic wave slow magnetosonic wave magnetic tension magnetic pressure thermal pressure w = kk vA w = kvA p damped vundamped = B/ 4pr ← weakly “Alfven speed” A w = kk cs strongly damped Plasma Waves at Low Beta Alfven wave fast magnetosonic wave slow magnetosonic wave magnetic tension magnetic pressure thermal pressure w = kk vA p vundamped = B/ 4pr A w = kvA weakly damped w = kk cs strongly damped sound speed cs = (γp/ρ)1/2 Plasma Waves at Low Beta Alfven wave fast magnetosonic wave slow magnetosonic wave magnetic tension magnetic pressure thermal pressure w = kk vA w = kvA w = kk cs virtually undamped in collisionless plasmas like the solar wind damped in collisionless plasmas (weakly at β<<1, strongly at β≈1) strongly damped in collisionless plasmas Properties of Alfven Waves (AWs) ~ 0 or anti-parallel to B ~ 0. • Two propagation directions: parallel to B ~ p4⇡⇢0 for AWs propagating in the ⌥B ~ 0 direction, and ⇢ = 0 • ~v = ± B/ ~ 0 |. Also, there are many intervals of time • In the solar wind ⇢/⇢0 ⌧ | B/B ~ p4⇡⇢0 is nearly satisfied, with the sign in which the relation ~v = ± B/ corresponding to propagation of AWs away from the Sun. • For these reasons, and because AWs are the least damped of the largescale plasma waves, AWs or nonlinear AW-like fluctuations likely comprise most of the energy in solar wind turbulence Properties of Alfven Waves (AWs) ~ 0 or anti-parallel to B ~ 0. • Two propagation directions: parallel to B ~ p4⇡⇢0 for AWs propagating in the ⌥B ~ 0 direction, and ⇢ = 0 • ~v = ± B/ ~ 0 |. Also, there are many intervals of time • In the solar wind ⇢/⇢0 ⌧ | B/B ~ p4⇡⇢0 is nearly satisfied, with the sign in which the relation ~v = ± B/ corresponding to propagation of AWs away from the Sun. A•,•v•because WAvrs x• AWs So•,Aa are the least damped 3537 • For these reasons, and of the large'• I I I I I I I I I • I . scale plasma waves, AWs or nonlinear AW-like fluctuations likely comprise most of bthe R 0 energy in solar wind turbulence m OVR -4 -25 +4 I , 25 -4 5 bN O. (Belcher & Davis 1971) 0 vN -4 5 B N B o I ! I 8 I I I 12 I 16 I 20 I I I oN 24 TIME (HRS) correctedfor aberration due to the spacecraft ropy in the pressure.This requires that 4•r motion). The two lower curves on the plot are {•, -- p•_)/Bo • be 0.40. The averageduring proton number density and magnetic field this periodof (2kTo/m•)•%the most probable strength((B) = 5.3 % (N) = 5.4 cm-3).This protonvelocity,was observedto be 47 km/sec, whichcorresponds to 4•'p•/B0• = 0.5,wherep• is period is one of the better examplesof the waves and illustrates their most characteristic features: closecorrelationbetweenb and v, variationsin b comparableto the field strength, and relatively little variation in field strength or density. In this case the average magnetic field is inward along the spiral and the correlation between b and v is positive; when the magnetic field is outward the correlationin periodsof goodwaves is negative.This indicatesoutward propagation (seeequation1). The scaleratio usedfor plotting the magnetic field and velocity variations in Figure i corre- the mean proton pressure. With reasonable valuesof the electronand a pressures and of the pressure anisotropy[Hundhausen et al., 1967],the requiredvalueof •a seemsentirelyreasonable. On other occasionswhen /• = 8•rp•/Bo• is smaller,valuesof • closerto unity wouldbe expected. Wavesversusdiscontinuities. Figure 2 is an expandedplot of three particular10-rainperiods indicatedon Figure 1, wherethe crossesare the basic magnetometer data (one reading in approximately4 sec)andthe linesare the plasma spondsto a value of D• -• of 6.4 km data (1 per 5.04 rain), scaledin the sameratio as This was determined by the condition that, in Figure 1. On this time scale,the wavesmay be when this ratio is used for a fixed area plot eithergradual(2b)or discontinuous (2a, 2c), with of v• versusb• for all the data, the stun of the abrupt changes within 4 sec.As discussed below, squares of the perpendicular distances from we feel that all three examplesare Alfv•nic, the points to a line of unit slope is minimized. with continuousmagneticfield lines,but with a (Mathematicallythis gives D• -• = discontinuity in direction in cases 2a and 2c. the ratio of the standard deviations.) The Suchabruptchanges occurat a rate of abouti per average values of N and N, during this per- hour and are enmeshedin more gradualchanges. iod are 5.4 and 0.4 cm-3, respectively; thus The visual appearanceof the field fluctuations equationi with • = i givesD• -• = 8.2 km is qualitatively different on the time scalesof sec-•/-F.We feel that the discrepancy between Figuresi and 2. With the scaleusedin Figure2, this predictedvalue and the observedvalue of 6.4 the most prominent structuresare the abrupt Properties of Alfven Waves (AWs) ~ 0 or anti-parallel to B ~ 0. • Two propagation directions: parallel to B ~ p4⇡⇢0 for AWs propagating in the ⌥B ~ 0 direction, and ⇢ = 0 • ~v = ± B/ ~ 0 |. Also, there are many intervals of time • In the solar wind ⇢/⇢0 ⌧ | B/B ~ p4⇡⇢0 is nearly satisfied, with the sign in which the relation ~v = ± B/ corresponding to propagation of AWs away from the Sun. • For these reasons, and because AWs are the least damped of the largescale plasma waves, AWs or nonlinear AW-like fluctuations likely comprise most of the energy in solar wind turbulence What role do AWs and AW turbulence plan in the solar wind? One possibility: AW turbulence may be one of the primary mechanisms responsible for generating the solar wind. Let’s take a closer look at how this might work.