Introduction to Turbulence

Introduction to Turbulence and Heating in the Solar Wind 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. What Is Turbulence? Operational definition for turbulence in plasmas and fluids: turbulence consists of disordered motions spanning a large range of lengthscales and/or timescales. “Energy Cascade” in Hydrodynamic Turbulence Canonical picture: larger eddies break up into smaller eddies ENERGY INPUT LARGE SCALES ENERGY CASCADE SMALL SCALES DISSIPATION OF FLUCTUATION ENERGY Plasma Turbulence Vs. Hydro Turbulence • In plasmas such as the solar wind, turbulence involves electric and magnetic fields as well as velocity fluctuations. • In some cases the basic building blocks of turbulence are not eddies but plasma waves or wave packets. Where Does Turbulence Occur? • Atmosphere (think about your last plane flight). • Oceans. • Sun, solar wind, interstellar medium, intracluster plasmas in clusters of galaxies... What Causes Turbulence? • Instabilities: some source of free energy causes the amplification of fluctuations which become turbulent. Example: convection in stars. • Stirring. Example: stirring cream into coffee. • Requirement: the medium can’t be too viscous. (Stirring a cup of coffee causes turbulence, but stirring a jar of honey does not.) What Does Turbulence Do? • Turbulent diffusion or mixing. Examples: cream in coffee, pollutants in the atmosphere. • Turbulent heating. When small-scale eddies (or wave packets) dissipate, their energy is converted into heat. Example: the solar wind. 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. In Situ Measurements solar wind N T RTN coordinates R spacecraft (E.g., Helios, ACE,Wind, STEREO...) Quantities measured include v, B, E, n, and T Spacecraft Measurements of the Magnetic Field and Velocity A•,•v• '• I I I WAvrs x• So•,Aa I I I I 3537 I bR 0 I • I . m OVR -4 -25 +4 I , 25 -4 5 bN O. 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) Data from the Mariner 5 spacecraft (Belcher & Davis 1971) 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, period is one of the better examplesof the waves whichcorresponds to 4•'p•/B0• = 0.5,wherep• is 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- spondsto a value of D• -• of 6.4 km 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 data (1 per 5.04 rain), scaledin the sameratio as A•,•v• '• I I I WAvrs x• So•,Aa I I I I 3537 I bR 0 I • I m . OVR -4 -25 +4 I , 25 -4 5 bN O. Is This Turbulence? 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) • The velocity and magnetic field in these measurements appear to fluctuate in a random or disordered fashion. 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 • But how do we tell whether there are “velocity requiredvalueof •a seems reasonable. fluctuations spanning a entirely large range of scales,” as in our On other occasionswhen /• = 8•rp•/Bo• is smaller,valuesof • closerto unity wouldbe operational definition of turbulence? 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 expected. Wavesversusdiscontinuities. Figure 2 is an expandedplot of three particular10-rainperiods indicatedon Figure 1, where the crossesare the basic magnetometer data (one reading in approximately4 sec)andthe linesare the plasma data (1 per 5.04 rain), scaledin the sameratio as in Figure 1. On this time scale,the wavesmay be eithergradual(2b)or discontinuous (2a, 2c), with abrupt changeswithin 4 sec.As discussed below, we feel that all three examplesare Alfv•nic, with continuousmagneticfield lines,but with a One way: by examining the power spectrum of the fluctuations. spondsto a value of D• -• of 6.4 km This was determined by the condition that, when this ratio is used for a fixed area plot of v• versusb• for all the data, the stun of the squares of the perpendicular distances from the points to a line of unit slope is minimized. (Mathematicallythis gives D• -• = discontinuity in direction in cases 2a and 2c. Suchabruptchanges occurat a rate of abouti per hour and are enmeshedin more gradualchanges. 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 is significantand probably is due to the anisot- changes that tend to be precededand followedby the ratio of the standard deviations.) The average values of N and N, during this period are 5.4 and 0.4 cm-3, respectively; thus The Magnetic Power Spectrum ~ ) = lim B̃(f Fourier transform of magnetic field power spectrum T !1 P (f ) = lim T !1 Z T /2 ~ e2⇡if t dt B(t) T /2 1 ~ ~ f )i hB̃(f ) · B̃( T • B(t) is the magnetic field vector measured at the spacecraft location. • T is the duration of the measurements considered. (When power spectra are computed using real data, T can not be increased indefinitely; the resulting power spectra are then approximations of the above formulas.) • <...> indicates an average over many such measurements. The Magnetic Power Spectrum ~ ) = lim B̃(f T !1 P (f ) = lim T !1 • • Z T /2 ~ e2⇡if t dt B(t) T /2 1 ~ ~ f )i hB̃(f ) · B̃( T ~ B(f) can be thought of as the part of B that oscillates with frequency f. When turbulence is present, P(f) is non-negligible over a broad range of frequencies. Typically, P(f) has a power-law scaling over frequencies varying by one or more powers of 10. 34 Roberto Bruno and Vincenzo Carbone Magnetic Power Spectra in the Solar Wind from these observations (Bavassano et al., 1982b; Denskat and Neubauer, 1983). In Figure 23 we re-propose similar observations taken by Helios 2 during its primary mission to the Sun. trace of magnetic fied spectral matrix 7 10 0.3AU f -0.89 Is This Turbulence? -1 6 10 f 0.7AU 5 2 power density [nT /Hz] 10 -5/3 Yes - the magnetic fluctuations measured by the spacecraft span a broad range of timescales. -1.06 -1.07 0.9AU 4 10 -1.72 3 10 -1.67 2 Similar spectra are observed at all locations explored by spacecraft in the solar wind. -1.70 10 1 10 (Bruno & Carbone 2005) 0 10 -5 10 -4 -3 10 10 -2 10 -1 10 frequency [Hz] Figure 23: Power density spectra of magnetic field fluctuations observed by Helios 2 between 0.3 and 1 AU within the trailing edge of the same corotating stream shown in Figure 17, during the first mission to the Sun in 1976. The spectral break (blue dot) shown by each spectrum, moves to lower and lower frequency as the heliocentric distance increases. These power density spectra were obtained from the trace of the spectral matrix of magnetic field fluctuations, and belong to the same corotating stream observed by Helios 2 on day 49, at a heliocentric distance of 0.9 AU, on day 75 at 0.7 AU and, finally, on day 104 at 0.3 AU. All the spectra are characterized by two distinct spectral slopes: about 1 within low frequencies and about a Kolmogorov like spectrum at higher frequencies. These two regimes are clearly separated by a knee in the spectrum often referred to as “frequency break”. As the wind expands, the frequency break moves to lower and lower frequencies so that larger and larger scales become part of the Kolmogorov-like turbulence spectrum, i.e., of what we will indicate as “inertial range” (see discussion at the end of the previous section). Thus, the power spectrum of solar wind fluctuations is not solely function of frequency f , i.e., P (f ), but it also depends on heliocentric distance r, i.e., Living Reviews in Solar Physics http://www.livingreviews.org/lrsp-2005-4 What Causes the Time Variation Seen in Spacecraft Measurements? vphase • Consider a traveling plasma wave (more on plasma waves soon). Imagine you’re viewing this wave as you move away from the Sun at the same velocity as the solar wind. From your perspective, the time variation of the magnetic field is the result of the wave pattern moving past you at the wave phase speed relative to the solar-wind plasma, vphase, which is typically ≈30 km/s in the solar wind near Earth. • If the wavenumber of the wave is k, the angular frequency of the magnetic field in this case is kvphase. This frequency characterizes the “intrinsic time variations” of the magnetic field in the solar wind frame. But What if You Measure B Using a “Stationary” Spacecraft That Does Not Move with the Solar Wind? vsolar-wind vphase satellite • Near Earth, the speed at which the solar wind flows past a satellite is highly supersonic, typically >10vphase, where vphase is the phase speed in the plasma rest frame. • “Taylor’s Frozen-in Flow Hypothesis”: Time variation measured by a spacecraft results primarily from the advection of spatially varying quantities past the spacecraft at high speed. The “intrinsic time variation” in the solar-wind frame has almost no effect on the measurements. The spacecraft would see almost the same thing if the fields were static in the solar-wind frame. • If a wave with wavevector k is advected past the spacecraft (i.e., δB eik⋅x), and the wave is static in the solar-wind frame, the spacecraft measures a magnetic oscillation with angular frequency ω = 2πf = k⋅vsolar-wind. • Frequencies measured by a spacecraft thus tell us about k (spatial structure) rather than the intrinsic time variation that would be seen in the plasma rest frame. ∝ 34 Roberto Bruno and Vincenzo Carbone Taylor’s Frozen-in Flow Hypothesis from these observations (Bavassano et al., 1982b; Denskat and Neubauer, 1983). In Figure 23 we re-propose similar observations taken by Helios 2 during its primary mission to the Sun. trace of magnetic fied spectral matrix 7 10 0.3AU f -0.89 -1 6 10 f 0.7AU 5 2 power density [nT /Hz] 10 -5/3 The frequency spectra measured by satellites correspond to wavenumber spectra in the solar-wind frame. -1.06 -1.07 0.9AU 4 10 -1.72 3 10 -1.67 2 -1.70 10 1 10 (Bruno & Carbone 2005) 0 10 -5 10 -4 -3 10 10 -2 10 -1 10 frequency [Hz] Figure 23: Power density spectra of magnetic field fluctuations observed by Helios 2 between 0.3 and 1 AU within the trailing edge of the same corotating stream shown in Figure 17, during the first mission to the Sun in 1976. The spectral break (blue dot) shown by each spectrum, moves to lower and lower frequency as the heliocentric distance increases. These power density spectra were obtained from the trace of the spectral matrix of magnetic field fluctuations, and belong to the same corotating stream observed by Helios 2 on day 49, at a heliocentric distance of 0.9 AU, on day 75 at 0.7 AU and, finally, on day 104 at 0.3 AU. All the spectra are characterized by two distinct spectral slopes: about 1 within low frequencies and about a Kolmogorov like spectrum at higher frequencies. These two regimes are clearly separated by a knee in the spectrum often referred to as “frequency break”. As the wind expands, the frequency break moves to lower and lower frequencies so that larger and larger scales become part of the Kolmogorov-like turbulence spectrum, i.e., of what we will indicate as “inertial range” (see discussion at the end of the previous section). Thus, the power spectrum of solar wind fluctuations is not solely function of frequency f , i.e., P (f ), but it also depends on heliocentric distance r, i.e., Living Reviews in Solar Physics http://www.livingreviews.org/lrsp-2005-4 Question: in the data below, the v and B fluctuations are highly correlated --- what doesWAvrs this mean? We’ll come back to this... A•,•v• x• So•,Aa '• I I I I I I I I bR 0 I • I 3537 . m OVR -4 -25 +4 I , 25 -4 5 bN O. 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) Data from the Mariner 5 spacecraft (Belcher & Davis 1971) 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, period is one of the better examplesof the waves whichcorresponds to 4•'p•/B0• = 0.5,wherep• is 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