C O M M E N T ON T H E P A P E R ' S O L U T I O N OF O N E - F L U I D

M O D E L E Q U A T I O N S W I T H S H O R T R A N G E R E T A R D I N G

M A G N E T I C F O R C E S F O R T H E Q U I E T S O L A R W I N D '

BY C U P E R M A N AND H A R T E N

AARON BARNES Space Science Diwrsion, NASA-Ames Research Center, Moffett Field, Calif., U.S.A.

(Received 27 May, 1975)

Cuperman and Harten (1974) recently studied the effect of coronal magnetic fields on the flow of the solar w.~nd by calculating one-fluid models, using the assumption that the radial magnetic force varies according to a power law in heliocentric distance r. They concluded that short range retarding magnetic forces would account for the steep- ness of the observed density profile near the Sun, which cannot be explained by con- ventional spherically symmetric flow models. In this note it is shown that although magnetic forces probably do in fact steepen the coronal density profile, the magnetic- force model of Cuperman and Harten implies an unrealistically strong interplanetary magnetic field, and therefore cannot be regarded as a demonstration of the desired conclusion.

First, consider qualitatively how coronal magnetic fields may affect solar wind flow. One expects the solar wind to flow away from the Sun along open magnetic flux tubes, which on the average are vertical but whose field lines diverge away from the vertical direction with increasing altitude. The tension in the field decreases as the lines diverge, so that the net contribution to the magnetic force by tension has a downward vertical component. The contribution to the vertical force from magnetic pressure may be directed either upward or downward, depending on how the fluid pressure varies from field line to field line. On balance one expects net downward magnetic force to be more likely than net upward force, although net upward force is certainly possible.

If it is assumed that the net magnetic force is downward, or 'retarding', this force is locally equivalent to an increase in gravity. Since the lower corona is nearly in hydro- static equilibrium, the effect of a downward magnetic force is to decrease the scale height of the plasma, i.e., to steepen the plasma density profile in comparison with the case of no magnetic force. This conclusion is supported by numerous calculations (e.g., Parker, 1964; Pneuman and Kopp, 1970; Durney and Pneuman, 1975). Thus, it is not surprising that Cuperman and Harten found that a power-law radial retarding force steepened the density profile near the base. The question that remains is whether their theory gives a model of magnetic acceleration that is consistent with the observed quiet solar wind.

The Cuperman and Harten model is based on one-fluid radial equations of motion

Astrophysics and Space Science 40 (1976) 35-38. All Rights Reserved Copyright �9 1976 by D. Reidel Publishing Company, Dordrecht-Holland

36 A. BARNES

nv,r 2 = const. (1)

GMomn 1 mnv,-~rdV" + ~rrd (nkT) + r 2 c (] x B)r = 0, (2)

dT 11 d 3nv,k-~r - v ,kT 2 r 2 dr r2Km~ ~ r = 0. (3)

The notation is the same as Cuperman and Harten, except that Gaussian units are used for the magnetic term, and the radial velocity is written as v,. The ' frozen-in' magnetic field condition implies that v and B are everywhere parallel (neglecting solar rotation). Now B cannot be purely radial (a radial magnetic field produces no j x-B force), so that v must have nonradial components. Thus, Equations (1)-(3) are based on very strong symmetry assumptions. For example the continuity equation is

l ~ r 1 [ ~ 0 ~ ] r2 (nv,r 2) + ~ (nvo sin 0) + (nvo) = 0. (4)

The q~ term can be eliminated by assuming n and v o to be independent of ~b. The 0 term cannot be eliminated by assumptions of symmetry, but can be made to vanish in the equatorial plane by assuming that (8/~O)(nvo)= 0 when 0 = •/2. The 0 term can vanish for general 0 only if Vo =-O.

Similarly, the radial component of the momentum equation is

Or, Vo OV, vo ev~ v~ + v~ v,-~r + - + r ~'0 r sin 0 ~q~ r

1 ~ (nkT) GMo 1 - mn ~r - r -----T- +" mnc (j x B),. (5)

The only way this equation can be made to reduce to Equation (2) is to take v~= vo = O, in which case B~ = Bo = 0, and (j x B), = 0. However, one might assume the centrifugal force terms to be small in comparison with v, 8v~/Or, in which case Equation (2) is approximately recovered if ~v,/~O = 8v~/~q3 = O.

Similar considerations apply to the equation div B = 0. Altogether, given the fact that v and B must have nonradial components, Equations (1)-(3) are approximately correct only if (1) the analysis is restricted to the equatorial plane (or else vo--Bo = 0

everywhere), (2) v,, vo, vr and B~, Bo, B~ are independent of 0 and ~b, and (3) centri- fugal force terms are neglected in the momentum equation.

Now consider what this symmetry means for the magnetic field. The magnetic force density is

1 (j x B ) , = 1 c ~ [(curl B) x B], =

1

- 8 ~ r 2 ~r [ r ~ ( M + B ~ ) ] =

_ 1 e (rZB~)" (6) 8~r z ~r

COMMENT ON THE PAPER BY CUPERMAN AND HARTEN 37

If the magnetic force is retarding,

1 (] x B), FB(r) >10. nc

(7)

Combining (6) and (7), and integrating, we obtain r

B~(r) = - - B~(Ro) + -.~ 02n(o)Fn(o) do. (8)

R|

Equation (8) shows that B• decreases no faster than r-1 (as can be seen directly from (6) and (7)), and that

r { f ;,2 B• >~ 8~ 02n(o)F.(o) d 0 >

R|

P

fo2F.(o ) . > de~ , (9)

R|

since n is a decreasing function of r. Inequality (9) gives a lower bound on Bj. at (say) r = 2Ro.

2RQ 2n

B.I.(nR@) > ( ~ - ~ Q H ( 2 R o ) f g 2 F B ( r ) d r } l / 2 ; (10)

R|

so that

2RQ

(2nn(2R| i/a f r%(r) ,,2 R|

since B• decreases no faster than r - i Inequality (11) implies that BI is unacceptably large far from the Sun. To illustrate this point, take model V of Cuperman and Harten, which they consider to give the best coronal densities. In that model

FB(r) = 1.2 R---~e - - dynes,

n(2Ro) = 1.36 x 106 cm -3.

Using these values in inequality (10) we find that B•174 G; and (11) then implies that B• > 10-3 G = 100y at the orbit of the Earth, grossly inconsistent with the observed Bl ~ 37. In order to reduce the implied B• by the necessary factor of > 30, FB(1 R| would have to be reduced by a factor of >900 thereby reducing n(1 Ro) by a factor ,-~ 10. Thus the Cuperman and Harten model requires either an unacceptably high magnetic field far from the Sun or an unacceptably low density at the coronal base.

38 A. BARNES

It might be argued that the region of significant magnetic force is limited to r< 3 Ro, so that the assumption of magnetic field symmetry far from the Sun could be relaxed, allowing B j_ to fall to reasonable values without otherwise affecting the model. How- ever, it is difficult to imagine why the strong symmetry assumed near the Sun should not hold for larger heliocentric distances as well. Thus, the large value of B• far from the Sun appears to be an inevitable consequence of the symmetry of the Cuperman and Harten model.

The Cuperman and Harten model therefore replaces the density problem of spheri- cally symmetric models with a choice between a density problem and a magnetic field problem. This dilemma illustrates that magnetic effects canot be adequately represented in models which impose too much symmetry, a point which has already been empha- sized in somewhat different contexts by Suess and Nerney (1973) and by Gussenhoven and Carovillano (1973). Even in problems that admit the most symmetry possible, it seems to be necessary to deal explicitly with nonradial components of the flow (Barnes, 1974). The most plausible models of magnetic effects on coronal flow abandon many assumptions of symmetry and solve the equations of motion in coordinates in which one set of coordinate lines follows the magnetic field lines (Parker, 1964; Pneuman and Kopp, 1970; Durney and Pneuman, 1975; and many others).

Note added in proof. In their reply to this note, Cuperman and Harten (1975) point out that the magnetic force of their models is terminated at r/Re = 5. This fact does not alter the objections raised above; in fact inequality (11), which is the basis of our argument, is independent of the behavior of FB(r) beyond r = 2 Re I The laws of mag- netohydrodynamics, under the symmetry constraints needed by Cuperman and Harten, rigorously imply an unacceptably large interplanetary magnetic field in their model. To state that the magnetic field strength appears 'explicitly only in the non- radial equations of motion which are not required in this paper' does not justify ignoring what the laws of physics imply that field strength to be.

References

Barnes, A.: 1974, Astrophys. J. 188, 645. Cuperrnan, S. and Harten, A.: 1974, Astrophys. Space Sci. 27, 383. Cuperman, S. and Harten, A.: 1975, Astrophys. Space Sci. 40, 111. Durney, B. and Pneuman, G. W. : 1975 Solar Phys. 40, 461. Gussenhoven, M. S. and Carovillano, R. L.: 1973, Solar Phys. 29, 233. Parker, E. N.: 1964, Astrophys. J. 139, 690. Pneuman, G. W. and Kopp, R. A.: 1970, SolarPhys. 13, 176. Suess, S. T. and Nerney, S. F.: 1973, Astrophys. Jr. 184, 17.