Abstract
For a collisionless plasma, the magnetic field enables fluidlike behavior in the directions perpendicular to ; however, fluid behavior along may fail. The magnetic field also introduces an Alfvenwave nature to flows perpendicular to . All Alfven waves are subject to Landau damping, which introduces a flow dissipation (viscosity) in collisionless plasmas. For three magnetized plasmas (the solar wind, the Earth’s magnetosheath, and the Earth’s plasma sheet), shear viscosity by Landau damping, Bohm diffusion, and by Coulomb collisions are investigated. For magnetohydrodynamicturbulence in those three plasmas, integralscale Reynolds numbers are estimated, Kolmogorov dissipation scales are calculated, and Reynoldsnumber scaling is discussed. Strongly anisotropic Kolmogorov and mildly anisotropic Kraichnan turbulences are both considered and the effect of the degree of wavevector anisotropy on quantities such as Reynolds numbers and spectraltransfer rates are calculated. For all three plasmas, Braginskii shear viscosity is much weaker than shear viscosity due to Landau damping, which is somewhat weaker than Bohm diffusion.
The authors wish to thank Pablo Dmitruk, Hans Pecseli, and Dastgeer Shaikh for helpful conversations and to thank Dot Delapp for her help. This work was supported by the NASA Heliospheric Guest Investigator Program, by the NASA Heliospheric Supporting Research and Technology Program, and by the Los Alamos National Laboratory LDRD Program.
I. INTRODUCTION: REYNOLDS NUMBERS, COLLISIONLESS PLASMAS, AND VISCOSITY
II. KOLMOGOROV AND KRAICHNAN TURBULENCES
A. Kraichnan turbulence
B. Kolmogorov turbulence
III. COULOMB SCATTERING
IV. ELECTRON AND ION LANDAU DAMPING
V. BOHM DIFFUSION
VI. RATIOS OF SCALESIZES
A. Dissipation from Landau damping
B. Dissipation from Bohm diffusion
VII. DISCUSSION: THE SOLAR WIND, THE MAGNETOSHEATH, AND THE EARTH’S PLASMA SHEET
A. Turbulence in the solar wind at
B. Turbulence in the Earth’s magnetosheath
C. Turbulence in the Earth’s magnetotail plasma sheet
VIII. SUMMARY
Key Topics
 Turbulent flows
 329.0
 Plasma turbulence
 151.0
 Magnetohydrodynamics
 96.0
 Eddies
 94.0
 Solar wind
 70.0
Figures
A sketch of the three collisionless magnetized plasmas studied herein that contain MHD turbulence: the solar wind, the magnetosheath, and the plasma sheet. The Earth is depicted as a blue sphere. The solarwind plasma (shaded in yellow) impinges on the Earth’s magnetic field from the right. Behind the bow shock (red dashed curve), the shocked solarwind plasma flowing around the magnetosphere is known as the magnetosheath (shaded in green). The magnetosheath is denser, hotter, and has a stronger magnetic field than does the unshocked solarwind plasma. Within the Earth’s magnetotail is a very hot, lowdensity plasma known as the plasma sheet (shaded in purple). In the solar wind and the magnetosheath, no magneticfield lines are indicated. In the magnetosphere, magneticfield lines are drawn in black by the method employed in Ref. 60. Note the large in the turbulent plasma sheet.
A sketch of the three collisionless magnetized plasmas studied herein that contain MHD turbulence: the solar wind, the magnetosheath, and the plasma sheet. The Earth is depicted as a blue sphere. The solarwind plasma (shaded in yellow) impinges on the Earth’s magnetic field from the right. Behind the bow shock (red dashed curve), the shocked solarwind plasma flowing around the magnetosphere is known as the magnetosheath (shaded in green). The magnetosheath is denser, hotter, and has a stronger magnetic field than does the unshocked solarwind plasma. Within the Earth’s magnetotail is a very hot, lowdensity plasma known as the plasma sheet (shaded in purple). In the solar wind and the magnetosheath, no magneticfield lines are indicated. In the magnetosphere, magneticfield lines are drawn in black by the method employed in Ref. 60. Note the large in the turbulent plasma sheet.
A sketch of the regimes of MHD turbulence in space for the solar wind, where parallel and perpendicular are with respect to the direction of the magnetic field . Shaded in purple are regions where the Alfven effect is important and the Kraichnan cascade should hold and shaded in blue are regions where the Alfven effect is negligible and the Kolmogorov cascade should hold. The two regions are separated by the criticalbalance curve (red) where . At perpendicular wavenumbers below the integral scale (thick black vertical dashed curve), the fluctuations are not part of the turbulence: This region is shaded in yellow. At wavenumbers larger than (thin black dashed curve), the fluctuations are too small to be described by MHD: This region is shaded in gray. The top panel is a loglog plot, the bottom panel is a linear plot.
A sketch of the regimes of MHD turbulence in space for the solar wind, where parallel and perpendicular are with respect to the direction of the magnetic field . Shaded in purple are regions where the Alfven effect is important and the Kraichnan cascade should hold and shaded in blue are regions where the Alfven effect is negligible and the Kolmogorov cascade should hold. The two regions are separated by the criticalbalance curve (red) where . At perpendicular wavenumbers below the integral scale (thick black vertical dashed curve), the fluctuations are not part of the turbulence: This region is shaded in yellow. At wavenumbers larger than (thin black dashed curve), the fluctuations are too small to be described by MHD: This region is shaded in gray. The top panel is a loglog plot, the bottom panel is a linear plot.
For the typical parameters in Table I, the time scales for turbulence in the solar wind at (top panel), the Earth’s magnetosheath (middle panel), and the Earth’s plasma sheet (bottom panel) are plotted as functions of eddy size . In all three panels the black solid curve is the eddyturnover time for Kolmogorov turbulence, the black dashed curve is the weakened spectraltransfer time for Kraichnan turbulence, the green curve is the Braginskii viscous dissipation time scale, the dark blue curve is the Landaudamping time scale for Kolmogorov turbulence in the reducedMHD regime, the lightblue curve is the Landaudamping time scale for isotropic Kraichnan turbulence, and the red curve is the Bohm diffusion time scale. For the Kraichnan turbulence, is taken for lack of better knowledge.
For the typical parameters in Table I, the time scales for turbulence in the solar wind at (top panel), the Earth’s magnetosheath (middle panel), and the Earth’s plasma sheet (bottom panel) are plotted as functions of eddy size . In all three panels the black solid curve is the eddyturnover time for Kolmogorov turbulence, the black dashed curve is the weakened spectraltransfer time for Kraichnan turbulence, the green curve is the Braginskii viscous dissipation time scale, the dark blue curve is the Landaudamping time scale for Kolmogorov turbulence in the reducedMHD regime, the lightblue curve is the Landaudamping time scale for isotropic Kraichnan turbulence, and the red curve is the Bohm diffusion time scale. For the Kraichnan turbulence, is taken for lack of better knowledge.
For a Maxwellian plasma with , the ion distribution function is plotted (black solid curve) and the electron distribution function is plotted (gray solid curve) as function of . Also plotted as the dashed curves are for ions (black) and for electrons (gray). The position of the peaks of are marked as for ions and for electrons.
For a Maxwellian plasma with , the ion distribution function is plotted (black solid curve) and the electron distribution function is plotted (gray solid curve) as function of . Also plotted as the dashed curves are for ions (black) and for electrons (gray). The position of the peaks of are marked as for ions and for electrons.
For a magnetized plasma with (black curve) and a plasma with (blue curve), the value of in expression (19) is plotted as a function of . The curves are determined from numerical solutions of the linearVlasov–Maxwell equations for the Alfvenwave branch of the plasma dispersion relation. For comparison the value of from Gary and Borovsky (Ref. 100) (for ) is plotted in green, the value of from expression (13b) of Stefant (Ref. 106) is plotted in light blue, and the value of from expression (19b) of Stefant (Ref. 106) (for ) is plotted in purple. The regimes demarked by the red dashed lines apply to the case, where .
For a magnetized plasma with (black curve) and a plasma with (blue curve), the value of in expression (19) is plotted as a function of . The curves are determined from numerical solutions of the linearVlasov–Maxwell equations for the Alfvenwave branch of the plasma dispersion relation. For comparison the value of from Gary and Borovsky (Ref. 100) (for ) is plotted in green, the value of from expression (13b) of Stefant (Ref. 106) is plotted in light blue, and the value of from expression (19b) of Stefant (Ref. 106) (for ) is plotted in purple. The regimes demarked by the red dashed lines apply to the case, where .
A sketch of the omnidirectional energy spectrum of Kolmogorov turbulence with the three characteristic scalesizes (integral scale, Taylor scale, and Kolmogorov dissipation scale) and their relative values in relation to the turbulence Reynolds number . This scaling is only valid if the spectral energy transfer rate is proportional to the local eddyturnover time.
A sketch of the omnidirectional energy spectrum of Kolmogorov turbulence with the three characteristic scalesizes (integral scale, Taylor scale, and Kolmogorov dissipation scale) and their relative values in relation to the turbulence Reynolds number . This scaling is only valid if the spectral energy transfer rate is proportional to the local eddyturnover time.
Using expression (47) with obtained from Fig. 5, the ratio of the Kolmogorov scale for Landau damping to the minimumMHD scale of the plasma is plotted as a function of the ion beta of the plasma. Kolmogorov turbulence in the reducedMHD regime below the criticalbalance curve is assumed in the calculations.
Using expression (47) with obtained from Fig. 5, the ratio of the Kolmogorov scale for Landau damping to the minimumMHD scale of the plasma is plotted as a function of the ion beta of the plasma. Kolmogorov turbulence in the reducedMHD regime below the criticalbalance curve is assumed in the calculations.
For Kolmogorov turbulence (top panel) and Kraichnan turbulence (bottom panel) in the solar wind at , the time scales for eddyturnover (black line), Bohm diffusion (red line), Landau damping (blue line), and Braginskii shear viscosity (green line) are plotted. The horizontal axis extends from of the solar wind to the largeeddy scalesize . The vertical axis extends from the proton gyroperiod upward. The horizontal dashed line denotes the age of the solarwind plasma at (about ). All parameters come from Table I and is taken.
For Kolmogorov turbulence (top panel) and Kraichnan turbulence (bottom panel) in the solar wind at , the time scales for eddyturnover (black line), Bohm diffusion (red line), Landau damping (blue line), and Braginskii shear viscosity (green line) are plotted. The horizontal axis extends from of the solar wind to the largeeddy scalesize . The vertical axis extends from the proton gyroperiod upward. The horizontal dashed line denotes the age of the solarwind plasma at (about ). All parameters come from Table I and is taken.
For Kolmogorov turbulence (top panel) and Kraichnan turbulence (bottom panel) in the Earth’s magnetosheath, the time scales for eddy turnover (black line), Bohm diffusion (red line), Landau damping (blue line), and Braginskii shear viscosity (green line) are plotted. The horizontal axis extends from of the magnetosheath plasma to the largeeddy scalesize . The vertical axis extends from the proton gyroperiod upward. The horizontal dashed line denotes the approximate age of the magnetosheath plasma (about of flow time). All parameters come from Table I and is taken.
For Kolmogorov turbulence (top panel) and Kraichnan turbulence (bottom panel) in the Earth’s magnetosheath, the time scales for eddy turnover (black line), Bohm diffusion (red line), Landau damping (blue line), and Braginskii shear viscosity (green line) are plotted. The horizontal axis extends from of the magnetosheath plasma to the largeeddy scalesize . The vertical axis extends from the proton gyroperiod upward. The horizontal dashed line denotes the approximate age of the magnetosheath plasma (about of flow time). All parameters come from Table I and is taken.
For Kolmogorov turbulence (top panel) and Kraichnan turbulence (bottom panel) in the Earth’s magnetotail plasma sheet, the time scales for eddy turnover (black line), Bohm diffusion (red line), Landau damping (blue line), and Braginskii shear viscosity (green line) are plotted. The horizontal axis extends from of the plasma sheet to the largeeddy scalesize . The vertical axis extends from the proton gyroperiod upward. The horizontal dashed line denotes the approximate age of the plasmasheet plasma [about (Ref. 175)]. All parameters come from Table I and is taken.
For Kolmogorov turbulence (top panel) and Kraichnan turbulence (bottom panel) in the Earth’s magnetotail plasma sheet, the time scales for eddy turnover (black line), Bohm diffusion (red line), Landau damping (blue line), and Braginskii shear viscosity (green line) are plotted. The horizontal axis extends from of the plasma sheet to the largeeddy scalesize . The vertical axis extends from the proton gyroperiod upward. The horizontal dashed line denotes the approximate age of the plasmasheet plasma [about (Ref. 175)]. All parameters come from Table I and is taken.
Using timeresolution measurements of the solarwind velocity from the ACE spacecraft for the year 2001, the logarithm of the ratio of the solarwind fluctuations is binned. To create the distribution plotted in red, the solarwind bulk velocity is subtracted off the velocity measurement parcel by parcel (see text) and to create the distribution plotted in blue the solarwind bulk velocity is subtracted off the measurements using a running average. The dashed curve is the distribution of isotropically distributed vectors.
Using timeresolution measurements of the solarwind velocity from the ACE spacecraft for the year 2001, the logarithm of the ratio of the solarwind fluctuations is binned. To create the distribution plotted in red, the solarwind bulk velocity is subtracted off the velocity measurement parcel by parcel (see text) and to create the distribution plotted in blue the solarwind bulk velocity is subtracted off the measurements using a running average. The dashed curve is the distribution of isotropically distributed vectors.
Tables
Typical values of relevant parameters for the solar wind at , for the Earth’s magnetosheath, and for the Earth’s magnetotail plasma sheet (cf. Refs. 59, 81, and 145). The factor is the (unknown) wavevector anisotropy of the Kraichnan turbulence.
Typical values of relevant parameters for the solar wind at , for the Earth’s magnetosheath, and for the Earth’s magnetotail plasma sheet (cf. Refs. 59, 81, and 145). The factor is the (unknown) wavevector anisotropy of the Kraichnan turbulence.
Mechanisms that can dissipate MHD fluctuations in a plasma to act as a “viscosity.” Note that the last source listed (plasmawave diffusion) may be one of the mechanisms underlying Bohm diffusion.
Mechanisms that can dissipate MHD fluctuations in a plasma to act as a “viscosity.” Note that the last source listed (plasmawave diffusion) may be one of the mechanisms underlying Bohm diffusion.
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