Introduction to Phenomenological Theories of MHD Turbulence

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. As discussed previously, Alfven-wave turbulence likely comprises the bulk of the energy in solar-wind turbulence. Rather than investigating the full MHD equations, let’s work with a limit of the full MHD equations that captures the physics of non-compressive Alfven waves but neglects compressive motions. Incompressible MHD ⇢ ✓ @~v + ~v · r~v @t ◆ = ✓ B2 r p+ 8⇡ ◆ + 1 ~ ~ + ⇢⌫r2~v B · rB 4⇡ ~ @B ~ ~ + ⌘r2 B = r ⇥ (~v ⇥ B) @t r · ~v = 0 ⇢ = constant kinematic viscosity resistivity Because Alfven waves (AWs) satisfy ∇⋄v = 0, incompressible MHD captures much of the physics of both small-amplitude AWs and AW turbulence. Because the viscous and resistive terms contain ∇2 they dominate for fluctuations with sufficiently small lengthscales. Elsässer Variables, ~a± represent AWs traveling parallel (a ) or ~0 anti-parallel (a+ ) to B ~ = B0 ẑ + B ~ B p vA = B0 / 4⇡⇢ p ~b = B/ ~ 4⇡⇢ ✓ ◆ 1 B2 ⇧= p+ ⇢ 8⇡ (B0 = constant) ~a ± = ~v ± ~b Substitute the above into the MHD eqns and obtain @~a ± @~a ± ⌥ vA = @t @z r⇧ ~a ⌥ · r~a ± + { terms / to ⌫ or ⌘ } Conserved Quantities in Ideal, Incompressible MHD E= Z 3 d x ~ 2 | B| ⇢|~v |2 + 2 8⇡ Hm = ! Z ⇢ = 4 Z ~·B ~ d3 x A d3 x |~a + |2 + |~a |2 magnetic helicity cross helicity Z Z p Z ~ = ~ 0 + ⇡⇢ d3 x |~a + |2 Hc = d3 x ~v · B d3 x ~v · B 2 integral vanishes when there is no average flow along B0 • energy |~a |2 integral measures the difference in energy between AWs moving parallel and anti-parallel to B0. These “quadratic invariants” are conserved in the “ideal” limit, in which the viscosity and resistivity are set to zero. Conserved Quantities and Cascades • MHD turbulence results from “nonlinear interactions” between fluctuations. These interactions are described mathematically by the nonlinear terms in the MHD equations (e.g., z-⋄∇z+). When you neglect viscosity and resistivity, the equations conserve E, Hm, and Hc. The nonlinear terms in the equations thus can’t create or destroy E, Hm, and Hc, but they can “transport” these quantities from large scales to small scales (a “forward cascade”) or from small scales to large scales (an “inverse cascade”). • At sufficiently small scales, dissipation (via viscosity, resistivity, or collisionless wave-particle interactions) truncates a forward energy cascade, leading to turbulent heating of the ambient medium. Forward and “Inverse” Cascades in 3D Incompressible MHD (Frisch et al 1975) • Energy cascades from large scales to small scales. (Large wave packets or eddies break up into smaller wave packets or eddies.) • Magnetic helicity cascades from small scales to large scales. (Helical motions associated with rotation cause the growth of large-scale magnetic fields, i.e., dynamos.) The Inertial Range of Turbulence • Suppose turbulence is stirred/excited at a large scale or “outer scale” L. • Suppose that the turbulence dissipates at a much smaller scale d, the “dissipation scale.” • Lengthscales λ satisfying the inequality d << λ << L are said to be in the “inertial range” of scales. • Fluctuations with wavelengths in the inertial range are insensitive to the details of either the forcing at large scales or the dissipation at small scales. • Systems with different types of large-scale forcing or small-scale dissipation may nevertheless possess similar dynamics and statistical properties in the inertial range (“universality”). Kolmogorov’s Theory of Inertial-Range Scalings in Hydrodynamic Turbulence v = rms amplitude of velocity di↵erence across a spatial separation ⌧c = “cascade time” ⌧c ⇠ /( v ) = “eddy turnover time” ✏ ⇠ ( v )2 /⌧c = “cascade power” ✏ ⇠ ( v )3 / In the “inertial range,” ✏ is independent of . ! v / 1/3 The shearing/ cascading of eddies of size λ is dominated by eddies of similar size: interactions are “local” in scale. “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 Connection to Power Spectra • Let f = kvsolar wind /2⇡, and E(k)dk = P (f )df , where P (f ) (or E(k)) is the frequency (or wavenumber) power spectrum of the velocity fluctuations. (This velocity power spectrum is defined just like the magnetic ~ ! ~v .) power spectrum introduced earlier in the talk, but with B R1 • The total kinetic energy in velocity fluctuations per unit mass is 0.5 0 E(k)dk. • The mean square velocity fluctuation at lengthscale ( v )2 ' • If v / 1/3 / k1 1/3 Z ⌘ k1 1 is given by 2k1 0.5k1 E(k) dk ⇠ k1 E(k1 ) , then E(k) / k P (f ) / f 5/3 , and 5/3 . • We saw earlier in this talk that this type of scaling is seen in magneticfield measurements. A similar scaling is also seen in velocity fluctuation measurements, although the exponent appears to be somewhat smaller than 5/3 (Podesta et al 2007). Alfven-Wave Turbulence • Wave propagation adds an additional complication. • Here, I’m going to walk you through some difficult physics, and try to convey some important ideas through diagrams rather than equations. • These ideas are useful and have been influential in the field, but represent a highly idealized viewpoint that misses some physics and is not universally accepted. • This is challenging material the first time you see it, but these notes will hopefully serve as a useful introduction, and one that you can build upon with further study if you wish to learn more. Nonlinear terms - the basis of turbulence No nonlinear terms linear waves. Small nonlinear terms fluctuations are still wavelike, but waves interact (“weak turbulence” or “wave turbulence”). Large nonlinear terms strong turbulence, fluctuations are no longer wave-like. a± a± vA = a · a± t z Note that the nonlinear terms vanish unless a+ and a are both nonzero. Nonlinear interactions result from “collisions between oppositely directed wave packets” (Iroshnikov 1963, Kraichnan 1965). a + (vA ẑ + a+ ) · a = t If a+ = 0, the a waves follow the background field B0 ẑ. When a+ = 0 and a+ vA , the a waves approximately follow the field lines corresponding to B0 ẑ and the part of B associated with the a+ waves (Maron & Goldreich 2001). the way that wave packets displace field lines is the key to understanding nonlinear wave-wave interactions Alfven wave packet, with δB ⊥ B0 δB An Alfven Wave Packet in 1D B0 perturbed magnetic field line if v = -b = -δB/(4πρ)1/2, it is an a wave packet that moves to the right phase velocity An “incoming” a+ wave packet from the right would follow the perturbed field line, moving to the left and down. An Alfven Wave Packet in 3D !B B0 field lines If v = b, then a+ = 0 and this is an a right without distortion. wave packet that propagates to the An ”incoming” a+ wave packet approaching from the right would follow the perturbed field lines, moving left and down in the plane of the cube nearest to you and moving to the left and up in the plane of the cube farthest from you. BEFORE COLLISION: ! ! field line B0 a! wave packet phase velocity a+ wave packet DURING COLLISION: each wave packet follows the field lines of the other wave packet AFTER COLLISION: wave packets have passed through each other and have been sheared a+ wave packet a! wave packet Shearing of a wave packet by field-line wandering Maron & Goldreich (2001) Anisotropic energy cascade BEFORE COLLISION: ! ! field line B0 a! wave packet phase velocity a+ wave packet DURING COLLISION: each wave packet follows the field lines of the other wave packet AFTER COLLISION: wave packets have passed through each other and have been sheared a+ wave packet a! wave packet In weak turbulence, neither wave packet is changed appreciably during a single “collision,” so, e.g., the right and left sides the “incoming” a+ wave packet are affected in almost exactly the same way by the collision. This means that the structure of the wave packet along the field line is altered only very weakly (at 2nd order).You thus get small-scale structure transverse to the magnetic field, but not along the magnetic field. (Large perpendicular wave numbers, not large parallel wave numbers.) (Shebalin, Matthaeus, & Montgomery 1983, Ng & Bhattacharjee 1997, Goldreich & Sridhar 1997) Anisotropic Cascade (Shebalin, Montgomery, & Matthaeus 1983) B • As energy cascades to smaller scales, you can think of wave packets breaking up into smaller wave packets. • During this process, the length λ|| of a wave packet measured parallel to B remains constant, but the length λ⊥ measured perpendicular to B gets smaller. • Fluctuations with small λ⊥ end up being very anisotropic, with λ|| >> λ⊥ λ|| λ⊥ B λ|| λ|| λ⊥ λ⊥ a- wave packet a+ wave packet • v ? = rms velocity di↵erence across a distance ? in the plane perpen~ = velocity fluctuation of wave packets of ? size ? . dicular to B • The contribution of one of these wave packets to the local value of ~v · r~v ~ 0 .) is ⇠ ( v ? )2 / ? . (For AWs, ~v ? B • Assumption: wave packets of size ? are sheared primarily by wave packets of similar size (interactions are “local” in scale). • A collision between two counter-propagating wave packets lasts a time t ⇠ k /vA . • A single collision between wave packets changes the velocity in each wave packet by an amount ⇠ t ⇥ ( v ? )2 / ? • The fractional change in the velocity in each wave packet is ⇠ t⇥( v v ? ? )2 / ? ⇠ k vA v ? ? (Ng & Bhattacharjee 1997, Goldreich & Sridhar 1997) B λ|| λ|| λ⊥ λ⊥ a- wave packet a+ wave packet • The fractional change in the velocity in each wave packet is ⇠ • In weak turbulence, t⇥( v v ? )2 / ? ? ⇠ k v vA ? ? ⌧ 1, while in strong turbulence & 1. • In weak turbulence, the e↵ects of successive collisions add incoherently, like a random walk. The cumulative fractional change in a wave packet’s velocity after N collisions is thus ⇠ N 1/2 . In order for the wave packet’s energy to cascade to smaller scales, this cumulative fractional change must be ⇠ 1. 2 wave packet collisions in order to cause • This means that it takes N ⇠ a wave packet’s energy to cascade. • The cascade time is therefore ⌧c ⇠ N k /vA ⇠ 2 k /vA . (Ng & Bhattacharjee 1997, Goldreich & Sridhar 1997) B λ|| λ|| λ⊥ λ⊥ a- wave packet a+ wave packet • The cascade time is therefore ⌧c ⇠ N = k v ? /( ? vA ), we obtain ⌧c ⇠ k /vA vA k ⇠ 2 k /vA . Recalling that 2 ? 2 v ? • The cascade power ✏ is v 2 ? /⌧c , or ✏⇠ • Noting that ✏ is independent of is constant, we obtain v ? / ? 1/2 ? . v4? k vA 2? within the inertial range, and that (Ng & Bhattacharjee 1997) k 1/2 • Substituting this scaling into the expression for , we find that / ? . At sufficiently small scales, will increase to ⇠ 1, and the turbulence will become strong! a wave packet D a+ wave packet • after colliding wave packets have inter-penetrated by a distance D satisfying the relation D v ? ⇥ ⇠1 vA ? the leading edge of each wave packet will have been substantially sheared/altered relative to the trailing edge. The parallel length of the wave packet therefore satisfies k . D, or equivalently . 1. • In weak turbulence, ⌧ 1 but grows to ⇠ 1 as ? decreases. Once reaches a value ⇠ 1 (strong turbulence), remains ⇠ 1, the state of “critical balance.” (Higdon 1983; Goldreich & Sridhar 1995) Critically Balanced, Strong AW Turbulence (Higdon 1983; Goldreich & Sridhar 1995) • In critical balance, = k vA ⇥ v ? ? ⇠1 and the linear time scale k /vA is comparable to the nonlinear time scale ? / v ? at each perpendicular scale ? , and the turbulence is said to be “strong.” • the energy cascade obeys the same arguments as hydrodynamic turbulence: ⌧c ⇠ ? / v ? and ✏ ⇠ v 2 ? /⌧c ⇠ v 3 ? / ? . • Since the cascade power ✏ is independent of 1/3 v ?/ ? . • the condition ⇠( k /vA ) ⇥( v ? / ?) ? in the inertial range, ⇠ 1 then implies that k / 2/3 ? . Some Open Questions • What is the origin of Alfven waves propagating towards the Sun in the solar-wind rest frame? (Reflection? Velocity-shear instabilities?) • At small scales, where MHD no longer applies, what is the proper description of solar-wind turbulence? • Which microphysical processes are responsible for the dissipation of solar wind turbulence at small scales? (Cyclotron heating? Landau damping? Stochastic heating? Magnetic reconnection?) • What are the processes at the Sun that launch waves into the solar wind? Which types of waves are launched, and what are the power spectra of these waves at the base of the solar atmosphere? Summary • Turbulence is measured at all locations that spacecraft have explored in the solar wind. • Alfven-wave turbulence likely accounts for most of the energy in solar-wind turbulence. • Alfven waves may provide the energy required to power the solar wind, and Alfven-wave turbulence may be the mechanism that allows the wave energy to dissipate and heat the solar-wind plasma. • Solar Probe Plus will provide the in situ measurements needed to determine the mechanisms that heat and accelerate the solar wind within the solar-wind acceleration region.