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Interesting paper on hydrodynamics, magnetohydrodynamics, and astrophysical plasmas. Solid, in-depth paper.
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HYDRODYNAMICS, MAGNETOHYDRODYNAMICS, AND ASTROPHYSICAL PLASMAS E. N. Parker, Department of Physics, University of Chicago I Introduction Not infrequently one encounters in the literature the assertion that hydrodynamics (HD) does not describe the large-scale bulk motion of a collisionless gas. Then for an ionized gas containing a magnetic field there is the widely held notion that electric current j is the cause of the magnetic field, and hence is the fundamental field variable rather than the magnetic field B itself. This point of view rejects magnetohydrodynamics (MHD) and attempts to work with the current density j and electric field E in treating the dynamics of the field and plasma. But the equations of Newton and Maxwell form the theoretical basis for the dynamics and cannot be expressed in terms of j and E in any tractable form. The equations become nonlinear global integro-differential equations and can be used only in special cases where symmetries suitably constrain the form of the fields. So lacking workable field equations the j,E enthusiasts turn to fantasy, e.g. the idea of an electric circuit analog, and the remarkable idea that the electric field E drives the bulk flow. The standard textbooks on MHD and plasma physics generally provide correct derivations of MHD from Newton and Maxwell in terms of the magnetic field B and the bulk fluid velocity v, but do not directly address the confusion that exists concerning j and E. The purpose of the present writing is to confront those issues, showing that HD follows from conservation of particles, momentum, and kinetic energy regardless of the presence or absence of collisions, and MHD applies to any gas that has no significant electrical insulating properties. II Hydrodynamics Consider the large-scale bulk motion v of a freely moving collisionless gas with number density N. It is not intuitively obvious that the bulk motion is HD in character when one reflects that every particle is moving with constant velocity u in a straight line, completely unaffected by the presence of the other particles. How can there be the familiar HD pushing and shoving of one region of fluid by another? The answer lies in understanding the concept of pressure, representing the momentum density of the thermal motions transported by the thermal motions, regardless of the presence or absence of collisions. An obvious necessary condition for application of HD is that there be enough particles for the local mean particle density N to be statistically well defined. The essential point is that with N statistically well defined, the other mean quantities, e.g. v, Nv, Nv2 , etc. are also statistically well defined. To show the implications of this statistical requirement, treat a bulk flow v with characteristic dynamical scale L. In order that the spatial partial derivatives be well defined, consider differencing the field variables on some small-scale
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l (> 1 in order that N be statistically well defined. The statistical fluctuations are of the order of
where
NN
1Nl3( ) 1/ 2
.
Suppose, then, that we require to be at least as small as l/L, requiring that
.
With l = 10-3 L the requirement becomes > 1015. More sophisticated estimates can be constructed, but the present order of magnitude is sufficient to show that, except in the extreme circumstances of shock front structure, the structure of the terrestrial magnetopause, or the structure of the current sheets arising in rapid reconnection, the statistical requirement is amply fulfilled. For instance, in the flow of solar wind around the terrestrial magnetosphere the dynamical scale is in excess of 103 km and the number density is greater than one ion/cm3, yielding >> 1015. Consider, then, the dynamics of the large-scale bulk flow. Summing over all the
particles in the elemental volume V = l3, indicated by the symbol , we have
N = ,
with
,
where is the velocity of an individual particle. Decompose into the mean bulk velocity vi and the thermal velocity relative to the bulk motion, so that . Obviously
.
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Consider, then, conservation of particles, momentum, and kinetic energy. It is readily shown from Gausss theorem that the time rate of change of the density F of a conserved quantity is equal to the negative divergence of the flux of F. Begin with the fact that the time rate of change of F within a fixed volume W enclosed by the surface S is given by the inflow of F across S. Thus, with the aid of Gausss theorem
t d
3rF r( )W = dS fluxof F( )
S ,
= dr3 fluxof F( )V .
This relation applies to all arbitrary V. Hence the integrands must be equal at every point in space, and it follows that
Ft = fluxof F( ).
Note that the flux of particles is
,
providing the familiar equation
.
For conservation of particles, The momentum density is
,
where M is the mass of an individual particle. The flux of momentum density is
,
where the pressure tensor is defined as
,
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representing the flux of the momentum transported by the thermal motions. This is the pressure, of course, regardless of whether there are, or are not, collisions. Conservation of momentum requires that the time rate of change of the momentum density is equal to the negative divergence of the flux of momentum density, so
.
Using the equation for conservation of particles, this reduces to the familiar Euler equation,
,
i.e Newtons equation of motion, recognizing that the momentum flux is equivalent to a force. If an external force (dynes/cm3) is introduced, the momentum equation becomes
.
The diagonal terms of represent kinetic energy in each of the three directions, each a conserved quantity. So the tensor equation for represents energy conservation. As already noted, the flux of momentum density is
,
and the flux of this is
The heat flow tensor is
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,
representing the flux of pij transported by the thermal motions. The time rate of change of the momentum flux density is the negative divergence of the total flux of momentum flux density, which can be reduced to
To explore this relation, consider a freely moving gas and neglect temperature variations, so that
One dimensional adiabatic compression is described by
where is taken to be a constant. For two dimensional compression,
For isotropic three dimensional compression,
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The effective viscosity of a collisionless gas, represented by the off diagonal terms of pij. initially grows linearly with time as particles from more distant regions pass across the velocity gradients. These effects follow directly from terms in with . So the dynamics is HD, with some special features as a consequence of the unlimited motion of the individual particles across the gradients in the bulk velocity. It is a simple matter to include the effects of interparticle collisions by introducing linear scattering terms. Thus, in the simplest case, evolution of the thermal anisotropy can be described by equations of the form
where h represents the mean scattering time. The off diagonal terms, representing viscosity, can also be constructed, showing that the effective viscosity is steady in time when the excursions of the individual particles are limited by the collisions. For collision dominated gas we have , and the equation for pressure variation reduces to the familiar
,
where represents the ratio of specific heats and K is the thermal conductivity. In summary, the large-scale bulk motion of a collisionless gas satisfies dynamical equations of the same form as the classical fluid, except that the classical scalar pressure p must be represented by a tensor because the thermal motions need not be isotropic. This general conclusion is hardly surprising because matter, momentum, and kinetic energy are conserved by the moving gas, regardless of the presence or absence of collisions, which themselves conserve matter, momentum, and energy. Going back to the question as to how the net effect of particles moving in straight lines with constant speed can provide HD motions, consider the situation in which a
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moving region of gas bumps into a stationary region. In the presence of collisions the momentum of the moving region is transferred to the particles in the stationary region, setting that region in motion. In the absence of collisions the moving gas penetrates freely into the region of stationary gas, thereby increasing the net momentum in the initially stationary region, just as if there had been collisions. The differences come out in the equation for , where the interpenetration in the absence of collisions appears as a free diffusion effect somewhat of the nature of viscosity. III Magnetohydrodynamics The concept of MHD is that the magnetic field is transported bodily with the bulk motion v of the fluid, so that
.
Consider a gas with enough free electrons and ions that it cannot support any significant electric field in its own frame of reference, where in the moving frame is related to the electric field E and magnetic field B in the laboratory frame by the nonrelativistic Lorentz transformation
.
If does not differ significantly from zero, it follows that
.
Substituting this into the Faraday induction equation gives the familiar MHD equation
.
Only if the gas has significant electrical insulating properties can MHD be avoided. Note that E plays no significant dynamical role, because
.
That is to say, the electric stresses are negligible compared to the magnetic stresses to second order in v/c. They do not drive the fluid motions as is sometimes imagined. [For further explorations of the role of E see V. M. Vasyliunas, Geophys. Res. Letters, 28, 2177-2180 (2001); Ann. Geophys. 23, 1347-1354 (2005)] Note, too, that E depends upon the frame of reference in which the calculation is executed. It would be a curious world indeed if fluid accelerations depended on the frame of reference chosen for the
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calculation. It should be mentioned in passing that j also plays no dynamical role because the moving conduction electrons have no significant momentum or kinetic energy. The condition that is the basis for the MHD condition of transport of the magnetic field B with the bulk motion v of the fluid. The Poynting vector
represents the flux of electromagnetic energy. With the electric field it is readily shown that
,
where is the fluid velocity perpendicular to the magnetic field. The quantity
represents the magnetic enthalpy density, from which it follows that the physical energy of the magnetic field is transported bodily with the fluid velocity. It is on this basis that the field lines are said to be transported bodily with the fluid motion. Consider the not uncommon belief that the electric current j is the cause of the magnetic field B. In any gas in the real physical world, the gas or plasma has at least some slight resistivity, so j is continually driven by a weak , pulling energy out of the magnetic field. Thus, if there were no significant fluid motion, the field would slowly die away because of the necessity to maintain the currents required by Amperes law. Now the concept of cause and effect is well defined in physics, with the cause supplying the energy and momentum responsible for the effect. Therefore B causes j, and not vice versa. Then consider the role of the neglected E , arising from the small resistivity of the highly conducting gas. The Faraday induction equation becomes
.
For a collision dominated plasma write j= E. It follows from Amperes law that
,
so that
,
with
cm2/sec.
The ratio of the first term to the second term on the right hand side of the induction equation is given by the well known magnetic Reynolds number,
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in terms of the characteristic dynamical scale L and velocity v. In the modest dimensions of the terrestrial laboratory it is not easy to make NR large compared to one, whereas in astrophysical plasmas NR is general 106 or more as a consequence of large L and v. Cold planetary atmospheres and interstellar clouds are exceptions, of course. For large NR the principal effect is the bulk transport of B with the fluid motion v, i.e. simple MHD. IV Partially Ionized Gases Consider briefly the partially ionized gas, reputed to lie outside MHD because it does not include the Hall effect. We treat the problem here in terms of the electron fluid, the ion fluid, and the neutral gas fluid, assuming the ions to be singly ionized. Denote the number density of the neutral gas by N and the number density of the ions and of the electrons by n. Then let v = mean bulk velocity of neutral gas w = mean bulk velocity of ions u = mean bulk velocity of electrons i = ion-neutral collision time e= electron-neutral collision time = ion-electron collision time p = pressure of neutral gas The equation of motion for the neutral gas is
,
where we use a simple frictional drag to represent the effects of collisions between the electron, ion, and neutral fluids. Consider a slightly ionized gas, n
10
.
The algebraic task is to eliminate u and w from these equations so as to obtain E and the induction equation for B. The current density is given by j=ne(w-u) so that Amperes law provides the relation
.
At this point we neglect the electron and ion inertia and sum the two eqns. of motion, yielding
Hence, for the neutral atoms
,
which is the usual MHD momentum equation. It also follows that
where
It follows that
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E = - v Bc B( )B[ ]B4ncQ +
M/ i m / e4neQ B( ) B
+c
4ne2M/?i( ) m/?e( )
Q +m
B
Define
Hall coefficient,
Pedersen coefficient,
Ohms coefficient,
Write b = B/B, where B is some characteristic magnetic field strength, so that
The Faraday induction equation becomes
.
This is the usual resistive MHD eqn. with the two extra terms in the curly brackets, representing the Pedersen resistivity and the Hall effect, respectively. In terms of the non-dimensional Lorentz force
,
the induction equation can be written
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.
The magnetic energy equation, formed from the scalar product of the induction equation with b, can be written
The right hand side represents the dissipation of magnetic energy. The term in square brackets on the left hand side represents the circulation of magnetic energy. The essential point for the present discussion is that the terms in in the induction equation all involve second derivatives over space, i.e. , whereas the MHD term for bulk transport of the field involves only first order derivatives, . Hence, for large-scale dynamics the Hall, Pedersen, and Ohmic terms are only small corrections to the basic MHD bulk transport term. We emphasize again that MHD does not treat the structure of shock fronts, thin current sheets, and magnetopauses, but it universally applies to the large-scale hydrodynamic flow of a partially ionized gas provided only that there are enough electrons and ions that the gas is a good conductor of electricity, i.e. has no significant electrical insulating properties, so that the electric field
in the moving frame of the gas can be neglected to a first approximation. [For further discussion, see E. N. Parker, J. Geophys. Res. 101, 10587-10625, (1996), and Conversations on Electric and Magnetic Fields in the Cosmos. pp 102-107, Princeton University Press, Princeton, (2007)] The foregoing calculation can be carried through including the ion and electron pressures and inertias, along with applied forces Li, Le per unit mass. The result is
The additional three bracketed terms on the right hand side of the induction equation represent such things as the thermoelectric effect, the Biermann battery effect, the
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Eddington-Sweet effect, etc. which are negligible under ordinary large-scale circumstances in astrophysical settings. V Compatibility of Newton and Ampere Another concern expressed by the proponents of j and E has been whether Amperes law is properly satisfied by the electric currents required for MHD and produced by the Newtonian motions of the electrons and ions. This is a nonproblem, of course, because Amperes law and Newtons equations of motion are both fundamental laws of nature. If they were incompatible in some way, then one or both of them would not be a fundamental law of nature. However, it is instructive to see how Newton and Ampere fit together. So consider a collisionless plasma, made up of equal numbers of electrons and singly charged ions. Calculate the electron and ion motions using the guiding center approximation, and sum over the particle motions to obtain the electric current. Then substitute the current into Amperes law and see what it tells us. As a matter of definition write
,
where u is the electric drift velocity of a particle subjected to the mutually perpendicular electric field E and magnetic field B. We assume that any initial electric field parallel to B is not long lived in large-scale astrophysical plasmas, even if it is sometimes interesting in the laboratory. It is readily shown that the perpendicular E is equal to , for which the Faraday induction equation yields
,
and the Poynting vector is
.
So we are dealing with MHD. The magnetic field moves bodily with the velocity u. Consider a particle of mass M and charge e with velocity w. The velocity along the magnetic field B is denoted by wpara and the perpendicular component by wperp. The motion of the guiding center is readily shown to be
Note that the motion of the guiding center along the field is described by
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dvdt
para
= w perp22B4 B B B ( )B[ ]{ }
in the absence of any significant electric field parallel to B. Define the pressures perpendicular and parallel to the magnetic field in terms of the sum over unit volume of the particle kinetic energies,
.
Summing over the individual particle trajectories of the electrons and the ions [See E. N. Parker, Phys. Rev. 107, 924 (1957)] it can be shown that the current density is
.
Substituting this result for j into Amperes law leads to
NM dudt = p +B28?
+
B ( )B[ ] perp4 1+
pperp ppara?2 /4
This is the Newtonian momentum equation for the overall electric drift velocity u perpendicular to the magnetic field. The term represents the transverse
force exerted by the tension as a consequence of the curvature of the field. The additional term involving represents the net centrifugal force of the thermal motions along those curved field lines. So our calculation shows that Amperes law is automatically satisfied if the bulk velocity u is described by Newtons equation of motion. The fundamental laws are mutually compatible, and we need never worry whether Amperes law is properly satisfied by the Newtonian motion of the individual particles. VI Equivalent Electric Circuits It was proposed many years ago [See H. Alfven and P. Carlquist, Solar Physics, 1, 220 228 (1967)] that the flow of electric currents in an MHD system can be described by the familiar laws of electric circuits, with the magnetic field energy represented by an equivalent inductance. This reduces the mathematical problem from the solution of partial differential equations to the much simpler solution of ordinary differential equations. This enormous simplification has been seized upon to offer solutions to a variety of magnetospheric and solar dynamical phenomena, e.g. flares, eruptive prominences, etc. No derivation of the electric circuit analog from basic principles has been presented, the seductive powers of the idea are evidently sufficient for their
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acceptance. Unfortunately there are two fundamental differences between MHD and the electric circuit analog. First, the electric circuit has a fixed connectivity or pattern. The current is confined to wires and the interconnections of the wires are fixed. Second, the wires are fixed in the laboratory reference frame so that the wires experience the electric fields in that frame. In contrast, in the swirling plasma it is Amperes law that dictates the current flow patterns and interconnections, which may vary with time as the field is deformed in the plasma flow. Equally important is the fact that the electric currents flow in the frame of reference of the moving plasma, in which reference frame there is nothing more than the insignificant electric field . So there are none of the interesting inductive effects that arise in the electric circuit analog. For instance, sudden blocking of the current produces no immense electromotive forces and charged particle acceleration, contrary to what has been claimed. In fact the sudden blocking of the current path causes the current to reroute or reconnect itself so as to avoid the blockage. This is accompanied by a sudden freeing of the magnetic field at the blockage, upsetting the magnetic stress balance and initiating Alfven waves that set the fluid in motion, thereby converting the free magnetic energy into kinetic energy of fluid motion rather than into powerful electric fields. An isolated twisted flux tube affords a convenient example. The twisted tube consists of a longitudinal field B0 in the z-direction filling the entire space and an azimuthal field Baz circling the z-axis and vanishing beyond some finite radius R. This isolated twisted flux bundle carries no net current, but there is a current flowing one way along the axis and a return current flowing back at a larger radius where the field cuts off. Imagine that the electric circuit closes across end plates at each distant end of the twisted tube. The electric circuit analog would be a closed circuit with a net current I flowing in one direction along a distributed inductance L per unit length such that LI2 /2 is equal to the magnetic energy per unit length. Suppose then, that the twisted tube is suddenly chopped through with a sheet of electrical insulation of small thickness D (
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motion. The magnetic energy is quickly transformed into fluid motion by the passage of the torsional wave fronts, whose passing unwinds the initial twist in the flux bundle, leaving behind a rotating fluid in which Baz = 0. The electric current flow I is closed on each side of D by the radial currents in the departing waves. VII Conclusion The foregoing demonstrations of the hydrodynamic behavior of collisionless fluids and the magnetohydrodynamic behavior of any and all fluids lacking electrical insulating properties shows that the large-scale dynamics of a conducting fluid, derived from Newton and Maxwell, reduces quite generally to MHD. The large-scale dynamics of a magnetized plasma is described in terms of the magnetic field B transported with the velocity v the plasma. The electric current j and the electric field E are related to B and v, of course, and easily calculated once B and v are known. Unfortunately j and E, playing no direct role in the dynamics, are too distant from B and v to serve as useful general proxies for pursuing MHD.