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Stochastic Acceleration in Turbulence:

Stochastic Acceleration in Turbulence:. L. A. Fisk University of Michigan. Evolution of low-energy ion energy spectrum across TS. Increasing intensities j(40 keV) ≈10 3 f/u Spectral fluctuations markedly decreased by 2005.1

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Stochastic Acceleration in Turbulence:

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  1. Stochastic Accelerationin Turbulence: L. A. Fisk University of Michigan

  2. Evolution of low-energy ion energy spectrum across TS • Increasing intensities j(40 keV) ≈103 f/u • Spectral fluctuations markedly decreased by 2005.1 • Spectrum of protons 40-4000 keV evolving to single power-law with index =1.5

  3. Different Expressions for the Observed Power Law • When expressed as a distribution function in velocity space: • When expressed as differential intensity [T is kinetic energy per nucleon]: • When expressed as differential number density:

  4. The Transport Equations • In the frame of the solar wind with random convective motions. • Continuity equation is • ST is the differential stream; term on right is the change in differential number density due to the turbulence doing work on the particles. • The differential pressure is

  5. Transport Equations • Multiply equation by kinetic energy T and manipulate, • The differential energy density is • The spatial transport of energy is • The third term is the flow of energy in energy space, with • The term on the right is the work done on the particles by the turbulent motions.

  6. Equilibrium • In a steady state: • Time derivative must be zero • Flow of energy into and out of the core must be zero. • The work done on the particles by the turbulence must be balanced by the flow of energy [heat flux] into and out of the volume. • The particles receive energy from the turbulence and do an equal amount of work on the turbulence, and the heat flux distributes the energy. 0 0

  7. Equilibrium Spectrum • The flow of energy into and out of the core must be zero, which requires that • Which is satisfied by UT‑2 as is observed

  8. Summary Statements • The tail is formed by a cascade in energy, in which the tail particles are in equilibrium receiving and performing an equal amount of work on the turbulence. • The tail pressure is proportional to the pressure in the core particles; the proportionality constant depends on the amplitude of the fluctuations in pressure. • The tails start where the particles are sufficiently mobile to undergo stochastic acceleration. The tails stop when the particle gyro-radii exceed the scale size of the turbulence. • The proportionality of the tail and core particles can be used to specify the composition of the tail particles.

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