Turbulent Heating in Goldreich-Sridhar Shear Alfven Cascade

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Turbulent Heating in Goldreich-Sridhar Shear
Alfven Cascade
W Dorland Univ of Maryland with S C
Cowley Imperial Coll London E Quataert UC
Berkeley G W Hammett Princeton Univ
Chandras view of central region of Milky Way in
X-rays
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Project Overview

• Important MHD turbulence exists
• E.g., MRI turns gravitational potential energy
  into magnetic field and electromagnetic
  turbulence via unstable flows. Other examples
  solar wind, scintillations in interstellar medium

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Project Overview

• Important MHD turbulence exists
• E.g., MRI turns gravitational potential energy
  into magnetic field and electromagnetic
  turbulence via unstable flows. Other examples
  solar wind, scintillations in interstellar medium
• Alfvenic turbulence does not directly heat gas

4
Project Overview

• Important MHD turbulence exists
• E.g., MRI turns gravitational potential energy
  into magnetic field and electromagnetic
  turbulence via unstable flows. Other examples
  solar wind, scintillations in interstellar medium
• Alfvenic turbulence does not directly heat gas
• Observable emission dominated by electron
  synchrotron radiation if electron component of
  plasma is sig. heated

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Project Overview

• Important MHD turbulence exists
• E.g., MRI turns gravitational potential energy
  into magnetic field and electromagnetic
  turbulence via unstable flows. Other examples
  solar wind, scintillations in interstellar medium
• Alfvenic turbulence does not directly heat gas
• Observable emission dominated by electron
  synchrotron radiation if electron component of
  plasma is sig. heated
• Observations suggest electron heating is very
  weak, or accretion rate is very low, or both

6
Project Overview

• Important MHD turbulence exists
• E.g., MRI turns gravitational potential energy
  into magnetic field and electromagnetic
  turbulence via unstable flows. Other examples
  solar wind, scintillations in interstellar medium
• Alfvenic turbulence does not directly heat gas
• Observable emission dominated by electron
  synchrotron radiation if electron component of
  plasma is sig. heated
• Observations suggest electron heating is very
  weak, or accretion rate is very low, or both
• Here
• Calculating relative heating of ions and
  electrons from Alfvenic turbulence, using
  first-principles, nonlinear, gyrokinetic
  simulations of the tail of Goldreich-Sridhar
  cascade

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Incompressible MHD Turbulence

• View as nonlinear interactions btw. oppositely
  directed Alfven waves (e.g., Kraichnan 1965)

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Incompressible MHD Turbulence

• View as nonlinear interactions btw. oppositely
  directed Alfven waves (e.g., Kraichnan 1965)
• Consider weak turbulence, where nonlinear time gtgt
  linear time (e.g., Shebalin et al. 1983)

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Incompressible MHD Turbulence

• View as nonlinear interactions btw. oppositely
  directed Alfven waves (e.g., Kraichnan 1965)
• Consider weak turbulence, where nonlinear time gtgt
  linear time (e.g., Shebalin et al. 1983)
• k cannot increase (true for 4-waves as well)

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Incompressible MHD Turbulence

• View as nonlinear interactions btw. oppositely
  directed Alfven waves (e.g., Kraichnan 1965)
• Consider weak turbulence, where nonlinear time gtgt
  linear time (e.g., Shebalin et al. 1983)
• k cannot increase (true for 4-waves as well)
• Turbulence is anisotropic Energy cascades
  perpendicular to local magnetic field

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Strong MHD Turbulence(Goldreich Sridhar 1995)

• Perpendicular cascade becomes more more
  nonlinear

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Strong MHD Turbulence(Goldreich Sridhar 1995)

• Perpendicular cascade becomes more more
  nonlinear
• Hypothesize critical balance linear time
  nonlinear time

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Strong MHD Turbulence(Goldreich Sridhar 1995)

• Perpendicular cascade becomes more more
  nonlinear
• Hypothesize critical balance linear time
  nonlinear time
• Anisotropic Kolmogorov cascade

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Strong MHD Turbulence(Goldreich Sridhar 1995)

• Perpendicular cascade becomes more more
  nonlinear
• Hypothesize critical balance linear time
  nonlinear time
• Anisotropic Kolmogorov cascade
• More more anisotropic on small scales

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Goldreich-Sridhar ? Gyrokinetics
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Nonlinear Gyrokinetic Equations
Gyrokinetic equations describe small scale
turbulence in magnetized plasma written down
from 1978-1982
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Nonlinear Gyrokinetic Equations
Gyrokinetic equations describe small scale
turbulence in magnetized plasma written down
from 1978-1982 Equations describe
self-consistent evolution of 5-dimensional distrib
ution functions (one for each plasma species) in
time
hhs(x, y, z, e, m t)
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Nonlinear Gyrokinetic Equations
Gyrokinetic equations describe small scale
turbulence in magnetized plasma written down
from 1978-1982 Equations describe
self-consistent evolution of 5-dimensional distrib
ution functions (one for each plasma species) in
time
hhs(x, y, z, e, m t)
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Nonlinear Gyrokinetic Equations
Gyrokinetic equations describe small scale
turbulence in magnetized plasma written down
from 1978-1982 Equations describe
self-consistent evolution of 5-dimensional distrib
ution functions (one for each plasma species) in
time
hhs(x, y, z, e, m t)
Plus Maxwells Eqs to get gyro-averaged potential
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Bessel functions represent averaging around
particle gyro-orbit
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Bessel functions represent averaging around
particle gyro-orbit
Reduces dimension of phase space from six to five
at expense of strict locality and removes
extraneous high frequency waves w gt W
excluded
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Project Strategy

• Mimic larger scale turbulence with Langevin
  antenna
• Allows arbitrarily chosen spectrum of driven
  wavenumbers

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Project Strategy

• Mimic larger scale turbulence with Langevin
  antenna
• Allows arbitrarily chosen spectrum of driven
  wavenumbers
• Amplitude at a given k determined by Langevin
  equation e.g, frequency wk kz vA with
  decorrelation time t

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Project Strategy

• Mimic larger scale turbulence with Langevin
  antenna
• Allows arbitrarily chosen spectrum of driven
  wavenumbers
• Amplitude at a given k determined by Langevin
  equation e.g, frequency wk kz vA with
  decorrelation time t
• Check that GK code recovers MHD (GS) results

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Project Strategy

• Mimic larger scale turbulence with Langevin
  antenna
• Allows arbitrarily chosen spectrum of driven
  wavenumbers
• Amplitude at a given k determined by Langevin
  equation e.g, frequency wk kz vA with
  decorrelation time t
• Check that GK code recovers MHD (GS) results
• Check that damping is correctly recovered, esp.
  with resolution of typical nonlinear simulation

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Project Strategy

• Mimic larger scale turbulence with Langevin
  antenna
• Allows arbitrarily chosen spectrum of driven
  wavenumbers
• Amplitude at a given k determined by Langevin
  equation e.g, frequency wk kz vA with
  decorrelation time t
• Check that GK code recovers MHD (GS) results
• Check that damping is correctly recovered, esp.
  with resolution of typical nonlinear simulation
• Do simulations for a range of b and Ti/Te and a
  range of spatial scales

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GK Code Recovers MHD (GS) Results

Max kx, ky
Non-white noise Stirring DwNL w0 wAlfven
k4 hyperviscosity scaled by shearing rate

• Simulate turbulence in a box gtgt ri, negligible
  gyro effects.
• Reproduces MHD results (Goldreich-Sridhar)
• Kolmogorov power spectrum and anisotropy

b 8 (800). xyzEnergy(Vpar/V )
50501001220
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GK Code Recovers Kinetic Damping

• Use antenna with chirped frequency to excite wave
• Lorentzian response gives real frequency and
  kinetic damping rate

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GK Code Recovers Kinetic Damping

• Weakly damped modes at long wavelengths
• Strongly damped modes at short wavelengths
• This is the essential wave-particle interaction

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GK Code Recovers Kinetic Waves

• Shear Alfven waves at long wavelength
• Transition to kinetic Alfven waves as ions are
  demagnetized
• GK code agrees well with linear theory

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GK Code Recovers Kinetic Damping

• Weakly damped modes at long wavelengths
• Strongly damped modes at short wavelengths
• This is the essential wave-particle interaction
  that determines turbulent heating rates

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Numerical Distortions Small

• Turbulence runs use implicitness in space and
  time for parallel dynamics
• Distortions of linear waves small even for fully
  implicit dynamics, typical time step

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Wide Range of Wavelengths Resolved

• Turbulence runs require simultaneous resolution
  of wide range of scales
• Little distortion of waves introduced by parallel
  grid
• (Perpendicular dynamics are treated
  pseudo-spectrally)

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Weak Collisions Dont Distort Dynamics

• Velocity-space diffusion needed to prevent
  velocity-space echoes, severe distortion of
  dynamics
• GK code uses realistic pitch-angle scattering
  collision operator
• Wide range of collisionalities for which dynamics
  are essentially collisionless

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Summary of Progress to Date

• Massively parallel nonlinear gyrokinetic code
  developed (GS2)

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Summary of Progress to Date

• Massively parallel nonlinear gyrokinetic code
  developed (GS2)
• GS2 recovers nonlinear MHD results at long
  wavelengths

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Summary of Progress to Date

• Massively parallel nonlinear gyrokinetic code
  developed (GS2)
• GS2 recovers nonlinear MHD results at long
  wavelengths
• GS2 accurately describes Landau and transit-time
  (Barnes) damping, over wide range of beta and
  temperature ratios

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Summary of Progress to Date

• Massively parallel nonlinear gyrokinetic code
  developed (GS2)
• GS2 recovers nonlinear MHD results at long
  wavelengths
• GS2 accurately describes Landau and transit-time
  (Barnes) damping, over wide range of beta and
  temperature ratios
• Ready to finish project as soon as a few
  additional diagnostics are installed and tested.