The Parker field model

E. N. Parker has a specific theoretical point of view. He avoids electric fields, E, and currents, j, preferring to work with only the plasma velocity, V, and the magnetic field, B. The rationale is that E can always be derived afterward, if necessary, from E = —V x B and j can be obtained from j = V x B. This approach is basically magnetohydrodynamic (MHD) theory and is widely used by other plasma theorists. It is sometimes referred to as the VB paradigm (Parker, 1996). This approach will be used in the following derivations because it is elegant and simple. Readers may note that in the book about the solar wind written by Parker there is little, if any, mention of electric fields and currents.

Admittedly, this approach is still not the one that is most familiar to readers nor, surprisingly, most workers in magnetospheric plasma physics. Because of the usual classical introduction to electromagnetic theory and experiment, people tend to think in terms of currents and electric fields as fundamental and as the causes of the respective phenomena. However, in plasma physics, especially hydromagnetic (or magnetohydrodynamic or magnetofluid) theory, currents are actually caused by stresses in the plasma and electric fields are caused by relative motion between the plasma and the magnetic field. Thus, the difference is not simply a matter of taste but involves fundamental distinctions between cause and effect.

The Parker model (Parker, 1963) is basically a solar wind model—in fact, the first such model. It is a strictly hydrodynamic model that ignores the magnetic field in so far as it might affect the acceleration of the hot coronal plasma to supersonic speeds followed by its escape from the Sun's strong gravitational field. The cause of the solar wind is solely the internal pressure of the plasma and the gradient in pressure that exerts an outward force able to overcome solar gravity.

The magnetic field is added more or less as a "tracer" in the solar wind flow. Parker knew that the solar wind would be magnetized and that B would play a significant role once the solar wind left the Sun in determining the properties of hydromagnetic waves and in various other perturbations to the steady flow. He specifically investigated eruptive phenomena at the Sun that might cause "blast waves''—that is, shock waves not driven by the injection of fresh solar plasma but able to propagate as large-amplitude waves into the heliosphere. The conclusions of the model regarding the character of the magnetic field are basically correct, since, at large distances, it does not represent a significant energy density or pressure compared with the convective or "ram" pressure of the solar wind. Another useful source of information about the solar wind model and HMF is Hundhausen (1972) that contains early solar wind and magnetic field measurements including attempts to supplement the Parker model in various ways.

It is customary to treat the magnetic field as "frozen-into" collisionless plasma because of the high electrical conductivity. By eliminating any relative motion between the field and plasma with V parallel to B, the electric field vanishes and extremely large currents are avoided. The field travels along with the solar wind (V and B remain parallel) and is transported into space to form the heliospheric magnetic field. In the frame of reference that corotates with the Sun, the solar wind follows a streamline given by rd</dr = v</vr = — fr/vr where f is the angular rotation rate of the Sun. Integration produces < = —f r/vr—recognizable as the expression for an Archimedes spiral. Since B is parallel to V, B</BR = v</vr = —fr/vr = tan <P, the Parker spiral. When the plasma and field vectors are transformed into the inertial/non-corotating frame, the field line is unchanged (according to the special theory of relativity when V ^ c) but the solar wind streamline is radial.

Alternatively, in the inertial frame, a radially flowing solar wind parcel reaches a distance, r = vrt, at time t after leaving the Sun. During that interval, the Sun has rotated counterclockwise as viewed from above the north pole and the (sub-solar) longitude of the solar wind parcel is < = — fit = —fr/vr. Since one end of B is attached to the rotating Sun, the locus of the field line is given by the same equation or B</Br = —f r/vr = tan <P, as above.

Parker Spiral Equation

300 km/s (considered slow wind today). The spirals are magnetic field lines that start out radially and make a large angle to the radial direction by the time they reach 1 AU (the dotted cycle). Arrows added to the field directions indicate their polarity at the Sun. The pluses designate outward-directed (positive) fields and the minuses inward-directed (negative) fields. The field lines divide the circle into two magnetic "sectors". Two of the spirals are the boundaries between the sectors (designated S/B for sector boundary). Adapted from Parker (1963).

300 km/s (considered slow wind today). The spirals are magnetic field lines that start out radially and make a large angle to the radial direction by the time they reach 1 AU (the dotted cycle). Arrows added to the field directions indicate their polarity at the Sun. The pluses designate outward-directed (positive) fields and the minuses inward-directed (negative) fields. The field lines divide the circle into two magnetic "sectors". Two of the spirals are the boundaries between the sectors (designated S/B for sector boundary). Adapted from Parker (1963).

Figure 4.1 shows the radial solar wind velocity vectors (assumed to have a constant speed of 300 km/s) and the spiral magnetic field. Additional features have been added that are discussed later: the polarity of the magnetic field (sunward or outward), the resulting division into magnetic ''sectors'', and the sector boundary (SB) between them.

The above equations can be converted to latitudes other than the equator by substituting fir sin d for Or, where d is the co-latitude measured from the polar axis. Away from the equator, with d — ^/2, the field lines form helices lying on the surface of a cone of half-angle, d (Figure 4.2). Their locus is given by d — 6>0 = constant, 0 = -fir sin 0q/ Vr. The height above the equatorial plane is z — r cos 0q. The distance from the polar axis, p, is simply p — r sin 6>0.

The conservation of magnetic flux requires that J* BR • dA — constant where A is the element of area on the surface of a sphere enclosing the Sun. In terms of the solid angle (w), J* BR • dA — J* BRr2dw so that BRr2 = constant or BR ~ 1/r2. It follows from the above equation for B^/BR that B^ ~ 1 /r. Hence, the magnetic field magnitude is given by B — BR(1 + (fir sind/Vr)2)1/2 .

Parkers Magnetic Field

Figure 4.2. The Parker model in the solar meridional plane. The view is perpendicular to the solar equator and rotation axis. Several field lines are shown originating at different solar latitudes. The solar wind velocity vectors are radial. The field lines are helices lying on cones with half-angles equal to the source latitudes. At a given radial distance, the fields are tightly spiraled in the equator and radial over the pole.

Figure 4.2. The Parker model in the solar meridional plane. The view is perpendicular to the solar equator and rotation axis. Several field lines are shown originating at different solar latitudes. The solar wind velocity vectors are radial. The field lines are helices lying on cones with half-angles equal to the source latitudes. At a given radial distance, the fields are tightly spiraled in the equator and radial over the pole.

The remaining field component, BN, is necessarily zero because B||V in the corotating frame and, hence, in the inertial frame. It is occasionally incorrectly stated that Parker assumed BN = 0; however, it actually follows from the basic assumption that V is radial at the Sun.

The Parker model is a steady-state model; however, it has also proven useful in many applications involving time variations in V and B. For example, there has been recent scientific interest in time variations in BN at the Sun associated with motions of the footpoints of the magnetic field lines that accompany convective motions of the solar plasma such as in granules or super-granules. This possibility was discussed first by Jokipii and Parker (1969).

With this information as background, we are ready to describe the global properties of the HMF during minimum solar activity. In reviewing the field measurements, the only feature of the solar field that is needed is the largest scale dipole component that can be conveniently described in terms of two opposed magnetic poles. As with other planetary and stellar dipoles, the solar dipole is tilted relative to the Sun's axis of rotation. As the Sun rotates, the magnetic equator wobbles up and down in an inertial frame and fields with opposite polarities (outward or inward) are customarily observed each solar rotation and in opposite hemispheres. This approach provides an opportunity to compare the Parker model with observations. Section 4.3 will introduce complexity associated with magnetic field and solar wind structures that are also present during solar minimum.

4.2.2 Br and open flux

In the Parker model the heliospheric magnetic field is derived from the radial field component at the Sun. BR is caused by currents inside the Sun and between the photosphere and the corona but not by currents in the solar wind. Currents in a steady solar wind give rise instead to B<. Therefore, it is reasonable to begin our discussion with observations of BR, especially since that component contains implicit information about the solar magnetic field.

In-ecliptic observations extending over many missions and years have documented the essential correctness of the Parker magnetic field model. However, Ulysses observations have clarified important aspects of the Parker model, especially the importance of the magnetic field at the solar wind source.

Images of the solar corona typically show a deviation of the magnetic field lines in the polar cap from being strictly radial but diverging much like those from the pole of a bar magnet. However, this divergence was traditionally attributed to excess plasma pressure in the polar caps rather than to the effect of the magnetic field.

In spite of the presumed dominance of the plasma pressure in the corona, early in-ecliptic measurements showed that the magnetic field energy density, and consequently the magnetic pressure, equaled or exceeded the plasma pressure when both were extrapolated back to the corona (Davis, 1966). This possibility was considered by Parker (1963) but not enough was known when he formulated his theory to justify any but the simplest assumptions. The early in-ecliptic measurements near the orbit of Earth showed that the energy density of the solar wind (nMV2/2) exceeded the magnetic energy density (B2/^) and internal plasma energy density (3nkT/2, where k is the Boltzmann constant and T is temperature) by a factor of approximately 100. However, the magnetic field at the Sun is radial and with BR ~ r—2, BR = 3.5nT at 1 AU (216 solar radii) grows by (216)2 and becomes 1.6 x 105 nT = 1.6 gauss at the Sun. B2/^ is then increased to 0.21 erg/cm3. On the other hand, conservation of mass implies that n - r—2 so that nMVR/2 = 64 x 10—10 erg/cm3 at 1 AU (n = 5cm3, VR = 420km/s) and becomes 3 x 10—4 erg/cm3 at the Sun. The magnetic energy density is 700 times larger and dominates the energy density of the solar wind. This conclusion led to models that included the effect of the magnetic field at the solar wind source. Ulysses observations have provided convincing evidence that clearly show the vital role of the magnetic field in the source region.

Ulysses provided the first opportunity to study the dependence of BR on helio-graphic latitude. Prior studies of the polarity (inward/outward) of the interplanetary magnetic field by in-ecliptic spacecraft clearly indicated that the fields were associated with the Sun's global magnetic field (i.e., with the solar magnetic dipole and the polar cap fields). That suggested that the fields should be stronger at high latitudes just as for a dipole field.

However, the initial Ulysses measurements near solar minimum unexpectedly showed that r2BR was «3nT (AU)2 independent of latitude without increasing toward the poles (Smith and Balogh, 1995). Panel (a) of Figure 4.3 shows this parameter was constant in the south and north hemispheres above latitudes of ±20°. Panel (b) of the figure compares the Ulysses measurements with simultaneous

Ulysses Heliograph¡c Latitude ■80 -50 | -40 -20 0 20 40 60 60

Ti-H-1-1-1-1—H

Mil -t—j—1-1-1-h-r

+1 0

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