University:
Boston University
Course:
CAS AS 311 | Planetary Physics
Academic year:

2022
Views:

463

Pages:

26
Author:

Melina Hobbs

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 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. (B0 = constant) Conserved Quantities in Ideal, Incompressible MHD energy magnetic helicity cross helicity integral vanishes when there is no average flow along B0 • 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. • MHD turbulence results from “nonlinear interactions” between fluctuations. These Conserved Quantities and Cascades 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- 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 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 Connection to Power Spectra 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. the way that wave packets displace field lines is the key to understanding nonlinear wave-wave interactions An Alfven Wave Packet in 1D 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 Shearing of a wave packet by field-line wandering Maron & Goldreich (2001) Anisotropic energy cascade 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) • 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⊥ (Ng & Bhattacharjee 1997, Goldreich & Sridhar 1997) (Ng & Bhattacharjee 1997, Goldreich & Sridhar 1997) (Ng & Bhattacharjee 1997) D Critically Balanced, Strong AW Turbulence (Higdon 1983; Goldreich & Sridhar 1995) 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.