HYDROMAGNETIC WAVES IN THE UPPER ATMOSPHERE AND SOLAR ACTIVITY

Arthur Beiser New York University, New York, N.Y.

In general, disturbances can propagate in a hydromagnetic medium in three modes. The best known of these is the so-called Alfven, or transverse mode, which is a noncompressive disturbance traveling in the direction of the field line. I t may be understood intuitively by comparing the field lines to strings under tension. Here the inertia corresponding to the mass of the string comes from the fact that in a highly conducting gas the material and the field lines move together. Thus a transverse wave in a hydromagnetic medium is like a pulse propagating along a string. The other two modes, commonly called fast and slow, are combinations of gas dynamical effects and effects due purely to the existence of the magnetic field. If we suppose a com- pressive wave traveling in a direction normal to that of the field, since the field and the material move together the field lines will be compressed where the material is compressed. The compression of the field lines leads to an in- crease in the potential energy involved in the compression and so to an in- crease in the velocity of the wave over that of a pure sound wave. In the upper atmosphere, the velocity of an AlfvCn wave is so much greater than the velocity of a sound wave that the wave velocity for the fast mode is the same in all directions. The slow mode, on the other hand, propagates largely along the field lines with some three dimensional spreading. Alfven waves, then, travel along the lines of force of the magnetic field with a characteristic speed depending upon the ambient ion density and the magnetic flux density; fast waves may travel in any direction relative to the field and have a speed greater than the speed of sound but less than the Alfven speed, though in the outer atmosphere the fast and Alfven speeds are essentially identical; and slow waves, which also may travel in any direction but whose speed is less than the sound speed. FIGURE 1 is a plot of V / V , versus S = V J V , , where V is the wave speed, VA the Alfven speed, and V , the sound speed.l In a medium of finite conductivity, the damping of the fast mode is independent of direction of propagation and is the same as that of the Alfven mode which travels along the field lines. The slow mode, however, is rapidly damped when its direction of propagation deviates from that of the field lines.

In a recent paper Francis and Karplus2 have studied the absorption of hy- dromagnetic waves vertically incident on the atmosphere from an altitude of 550 km. Their treatment was restricted to a magnetic dip angle of 60" cor- responding to a magnetic latitude of about 45". They also invoked ionospheric properties corresponding to the daytime a t sunspot maximum. An incident- hydromagnetic wave separates into an ordinary wave whose properties are independent of the direction of the magnetic field and an extraordinary wave. These two modes are coupled by the Hall effect, which is most important near an altitude of 120 km. FIGURE 2 shows the power dissipation versus altitude above the earth's surface for 2 ordinary waves whose frequencies are 25 radians/

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18 Annals New York Academy of Sciences sec. and 1 radianlsec. at normal incidence with their amplitude initially 100 y. Evidently the power dissipation occurs in a relatively narrow a1 titude band. FIGURE 3 shows power dissipation versus altitude for extraordinary waves of 25 and 1 radian/sec. incident frequency, also with a 100 y field amplitude ini- tially. The ionosphere is essentially transparent for frequencies less than 1 radianlsec.

Weak maghetic fteld S= FIGURE 1.

Dessler3 has pointed out that the energy dissipated in the ionosphere by hydromagnetic waves may represent an important source of ionospheric heat- ing. This effect has been invoked by various authors to explain the following phenomena: (1) the lifting of the F region during geomagnetic storms, ( 2 ) the irregular orbital acceleration of satellites, ( 3 ) the sudden disappearance of trapped radiation from the Argus nuclear explosion coincident with a geo- magnetic storm, (4) the X-ray flux observed a t balloon altitudes below the Van Allen belt during geomagnetic storms, and (5) the decrease in intensity at the lower edge of the Van Allen radiation belt during geomagnetic storms.

I t is interesting to note that the curves showing power absorbed versus alti-

Beiser : Hydromagnetic Waves and Solar Activity 19 tude for solar ultraviolet radiation and hydromagnetic waves are remarkably similar above an altitude of about 100 km.

We now turn to possible sources of hydromagnetic waves that may be inci- dent upon the earth.

The sun constantly emits a totally ionized hydrogen plasma, usually called the solar wind, whose flux a t the earth’s orbit is perhaps lOI3 ions/m.2-sec. The kinetic energy of the protons is about 10-15 joule/proton, and the resulting energy flux a t the earth’s orbit is therefore in the vicinity of watts/m?, about of the flux of solar radiant energy.

w - 2 5

I ---*---#-- \ \-‘--a

Ol I I 1 I 1 %

0 I00 200 330 400 5 ALTITUDE (KM.)

FIGURE 2.

The solar wind impinging on the earth’s magnetic field causes it to be con- fined within a cavity in the wind. Estimates of the energy density of the plasma vary, but it seems likely that the plasma and field energy densities are equal at between 6 and 9 earth radii. The problem in three dimensions is too complex for exact solution. Instead, we have made an exact calculation of the cavity shape in 2 dimensions, corresponding to a pair of antiparallel line currents immersed in a plasma ~ t r e a m . ~ The result in the case where the dipole axis is perpendicular to the stream direction is shown in FIGURE 4; note that the neutral points lie somewhat upstream from the dipole axis. The maximum diameter of the cavity a t infinity is twice that a t the axis. Also shown is the cavity surface as calculated on the basis of an approximate pro-

20 Annals New York Academy of Sciences 16

m 0 - 1.2 m

f t u) t

3 -

0 0 I00 2 00 300 400 500

A L T I T U D E (KM.)

FIGURE 3.

FIGURE 4.

Beiser : Hydromagnetic Waves and Solar Activity 2 1

cedure devised by Beard for the three-dimensional case: the agreement is sufficiently good for it to be assumed that Beard's method is probably satis- factory for the 3-dimensional problem. FIGURE 5 shows how the cavity is distorted when the dipole axis is tilted toward the stream by 37", corresponding to the maximum tilt of the geomagnetic axis toward the sun. FIGURE 6 shows the result of an exact calculation of the field configuration within the cavity. (Note that we have not included the thermal effects within the solar wind that will cause the cavity to pinch down at infinity.)

Taking the radius of the cavity nose to be 10 earth radii, its cross-sectional

FIGURE 5.

area is Thus the total power intercepted by the cavity nose is about l O I 4 w. By comparison, the total solar electromagnetic radiation strik- ing the earth is about 2 x 1017 w. The cross-sectional area of the cavity downstream is 4 times greater than that of the cavity nose, so it intercepts 4 times more power. Even so there is not much total energy influx, but its selective absorption may make it more significant than these numbers indicate.

Hydromagnetic waves can be initiated at the cavity wall by two distinct processes: fluctuations in the incident solar wind and instabilities in the cavity wall. In the first case, the impact of irregularities in the solar wind on the geomagnetic field will cause the generation of hydromagnetic waves over the nose of the cavity surface. The result will be large amplitude hydromagnetic waves starting on the cavity surface and propagating inward. The arrival of

sq. m.

22 Annals New York Academy of Sciences these waves at the earth’s surface has been thought5 to result in worldwide sud- den commencement, a notion that has been claimed to give reasonably good agreement with the observed features of sudden commencements (other evi- dence6 casts some doubt on this interpretation). The hydromagnetic waves that leave various portions of the surface will arrive a t a given point on the earth at different times owing to their different paths. Furthermore, the im- pact of an initially plane plasma front on the field does not occur simultane-

FIGURE 6.

ously along the curved geomagnetic surface but occurs first near the sun-earth line. Thus hydromagnetic waves are not generated simultaneously along the nose. These two effects produce a spread in the arrival times, at an observing point on the earth, of signals generated over the entire nose. The resulting delays yield a build-up of the sudden commencement amplitude with a rise time of roughly one to six minutes.

The second source of hydromagnetic waves is, in a way, the most interesting for our purposes. As a general rule, the cavity wall is st able where its center of curvature lies inside the cavity. Hence the nose i s stable; but the regions near the neutral points are not, and the entire downstream part of the wall is

Beiser : Hydromagnetic Waves and Solar Activity 23 not. We are currently studying the propagation of hydromagnetic waves propagating inward from these regions.

In this connection it is interesting to speculate on the possibility of a duct within the upper atmosphere within which hydromagnetic waves may propa- gate unimpeded around the earth. The velocity of hydromagnetic waves in the atmosphere starts at about 227 km./sec. a t an altitude of 200 km., drops to a minimum of perhaps 137 km./sec. at an altitude of 400 km., then increases once more to a maximum of 5,180 km./sec a t an altitude of 3,000 km., and then declines steadily with increasing altitude. Thus refraction effects may cause a wave guide to exist for hydromagnetic waves at an altitude of several hundred km. above the earth.

The work referred to above that has been conducted at New York University received support from the National Science Foundation, the National Aero- nautics and Space Administration, Washington, D. C., and the Advanced Research Projects Agency of the Department of Defense, Washington, D. C.

References

Detailed calculations are in progress on this point.

1. OBAYASHI, T. 1958. Rept. Ionosphere Research Japan. 12: 301. 2. FRANCIS, W. E. & K. KARPLUS. 3. DESSLER, A. J. 1959. J. Geophys. Research. 64: 397. 4. HURLEY, J. 1961. Doctoral thesis. Department of Physics. New York Univ. New

5. DESSLER, A. J., W. E. FRANCIS & E. N. PARKER. 1960. J. Geophys. Research. 66:

6. TKOITSKAYA, V. A. 1961. J. Geophys. Research. 66: 5.

1960. J. Geophys. Research. 66: 3593.

York, N.Y.

2715.