Stochastic Acceleration in Turbulence:

Stochastic Accelerationin Turbulence:

L. A. Fisk

University of Michigan

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

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

0.5

2006.02005.52005.02004.5

Slope of : energy spectrum

( ) ~ j E E ( ) E

10 -2

10 -1

10 0

10 1

10 2

10 3 TS 351DOY

( ≥1) Ion Z intensities 40 - 53 keV 85 - 139 220 - 550 1050 - 2000

: 2005/321 Data through

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:€

f ∝ v−5

€

j ∝T −1.5

€

U ∝T−2

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€

∂U∂t

+∇ ⋅ST = −∂

∂Tδu ⋅∇PT( )

€

PT =2

3TU

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.

€

∂ET∂t

+∇ ⋅ TST( ) +∂

∂TTδu ⋅∇PT( ) = δu ⋅∇PT

€

ET = TU

€

TST

€

PT = 2TU /3

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.

€

∂ET∂t

+∇ ⋅ TST( ) +∂

∂TTδu ⋅∇PT( ) = δu ⋅∇PT

0 0

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 €

∂∂T

2

3T 2δu ⋅∇U

⎛

⎝ ⎜

⎞

⎠ ⎟= 0

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.