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On shear viscosity and the Reynolds number of magnetohydrodynamic turbulence in collisionless magnetized plasmas: Coulomb collisions, Landau damping, and Bohm diffusion
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10.1063/1.3155134
/content/aip/journal/pop/16/8/10.1063/1.3155134
http://aip.metastore.ingenta.com/content/aip/journal/pop/16/8/10.1063/1.3155134

Figures

Image of FIG. 1.
FIG. 1.

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 solar-wind plasma (shaded in yellow) impinges on the Earth’s magnetic field from the right. Behind the bow shock (red dashed curve), the shocked solar-wind 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 solar-wind plasma. Within the Earth’s magnetotail is a very hot, low-density plasma known as the plasma sheet (shaded in purple). In the solar wind and the magnetosheath, no magnetic-field lines are indicated. In the magnetosphere, magnetic-field lines are drawn in black by the method employed in Ref. 60. Note the large in the turbulent plasma sheet.

Image of FIG. 2.
FIG. 2.

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 critical-balance 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 log-log plot, the bottom panel is a linear plot.

Image of FIG. 3.
FIG. 3.

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 eddy-turnover time for Kolmogorov turbulence, the black dashed curve is the weakened spectral-transfer time for Kraichnan turbulence, the green curve is the Braginskii viscous dissipation time scale, the dark blue curve is the Landau-damping time scale for Kolmogorov turbulence in the reduced-MHD regime, the light-blue curve is the Landau-damping 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.

Image of FIG. 4.
FIG. 4.

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.

Image of FIG. 5.
FIG. 5.

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 linear-Vlasov–Maxwell equations for the Alfven-wave 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 .

Image of FIG. 6.
FIG. 6.

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 eddy-turnover time.

Image of FIG. 7.
FIG. 7.

Using expression (47) with obtained from Fig. 5, the ratio of the Kolmogorov scale for Landau damping to the minimum-MHD scale of the plasma is plotted as a function of the ion beta of the plasma. Kolmogorov turbulence in the reduced-MHD regime below the critical-balance curve is assumed in the calculations.

Image of FIG. 8.
FIG. 8.

For Kolmogorov turbulence (top panel) and Kraichnan turbulence (bottom panel) in the solar wind at , 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 solar wind to the large-eddy scalesize . The vertical axis extends from the proton gyroperiod upward. The horizontal dashed line denotes the age of the solar-wind plasma at (about ). All parameters come from Table I and is taken.

Image of FIG. 9.
FIG. 9.

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 large-eddy 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.

Image of FIG. 10.
FIG. 10.

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 large-eddy scalesize . The vertical axis extends from the proton gyroperiod upward. The horizontal dashed line denotes the approximate age of the plasma-sheet plasma [about (Ref. 175)]. All parameters come from Table I and is taken.

Image of FIG. 11.
FIG. 11.

Using time-resolution measurements of the solar-wind velocity from the ACE spacecraft for the year 2001, the logarithm of the ratio of the solar-wind fluctuations is binned. To create the distribution plotted in red, the solar-wind bulk velocity is subtracted off the velocity measurement parcel by parcel (see text) and to create the distribution plotted in blue the solar-wind bulk velocity is subtracted off the measurements using a running average. The dashed curve is the distribution of isotropically distributed vectors.

Tables

Generic image for table
Table I.

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.

Generic image for table
Table II.

Mechanisms that can dissipate MHD fluctuations in a plasma to act as a “viscosity.” Note that the last source listed (plasma-wave diffusion) may be one of the mechanisms underlying Bohm diffusion.

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/content/aip/journal/pop/16/8/10.1063/1.3155134
2009-08-18
2014-04-20
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752b84549af89a08dbdd7fdb8b9568b5 journal.articlezxybnytfddd
Scitation: On shear viscosity and the Reynolds number of magnetohydrodynamic turbulence in collisionless magnetized plasmas: Coulomb collisions, Landau damping, and Bohm diffusion
http://aip.metastore.ingenta.com/content/aip/journal/pop/16/8/10.1063/1.3155134
10.1063/1.3155134
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