magnetoacoustic wave

or a remnant field. The moon and Mars currently do not have active magnetic fields, but rocks from both bodies indicate that magnetic fields were present in the past. The large Jovian moons also appear to have weak magnetic fields. The Earth’s magnetic field has undergone polarity reversals throughout history, the discovery of which helped advance the hypothesis of sea floor spreading, which eventually led to the theory of plate tectonics. The area of space where a planetary body’s magnetic field interacts with the solar wind is called the magnetosphere.

magnetoacoustic wave Any compressive hydromagnetic wave; frequently called magnetosonic wave. Magnetoacoustic waves are characterized by fluctuations of density and magnetic field strength, and by linear polarization of fluctuations in the velocity and magnetic field vectors. Their phase and group velocities are anisotropic in the sense that they depend on the direction of propagation with respect to the mean magnetic field.

The theory of magnetoacoustic waves can be developed in the framework of either magnetohydrodynamics or kinetic theory. The magnetohydrodynamic approach gives two different kinds of magnetoacoustic wave, conventionally called the fast and slow magnetoacoustic modes. For propagation parallel to the mean magnetic field, the fast (slow) mode propagates at the greater (lesser) of the sound speed CS and Alfvén speed CA; for propagation transverse to the mean magnetic field, the fast mode propagates at the speed

1

C = CA2 + CS2 2 ,

and the slow mode has zero propagation speed. The other qualitative difference between the two modes is that the fluctuations in density and magnetic field strength are correlated positively for the fast mode, negatively for the slow mode. In the limit of strong magnetic field, the slow mode disappears, and the fast mode propagates at CA for all directions; in the limit of weak magnetic field, the slow mode disappears once again, and the fast mode is simply a sound wave. The phase speed of a magnetoacoustic wave, unlike the Alfvén wave, depends on amplitude, so

that magnetoacoustic waves can steepen to form shock waves.

The kinetic-theory approach yields analogs of the fast and slow modes, but additionally gives other, very strongly damped modes. Moreover, in a hot plasma like the solar wind, even the fast and slow modes are subject to strong Landau damping. Indeed, although solar wind fluctuations have often been identified with Alfvén waves, they can rarely be associated with magnetoacoustic waves. Landau damping of magnetoacoustic waves may be an important mechanism of plasma heating in some circumstances. In particular, it may severely restrict the formation of shock waves from magnetoacoustic disturbances. See Alfvén wave, hydromagnetic wave.

magneto-fluid mechanics	See magnetohy-
drodynamics.

magnetogram Graphic representation of solar magnetic field strength and polarity. Magnetograms show a hierarchy of spatial scales in the photospheric magnetic field from sunspots to the general magnetic network.

magnetohydrodynamics (MHD) The study of the dynamics of a conducting fluid in the presence of a magnetic field, under the assumption of perfect, or partial (resistive MHD) locking of the plasma to the field lines. Important in many branches of physics, but in geophysics it is primarily important in the study of planetary magnetospheres, ionospheres, and cores. In these non-relativistic cases, Maxwell’s equations for electrodynamics and the Lorentz field transformation yield the induction equation (Faraday’s law):

∂B = × (V × B) − × (η × B) ∂t

where B is the magnetic field, V is the velocity of the conducting fluid, and η is the magnetic diffusivity of the fluid. The first term on the righthand side represents the effects of fluid flow on the magnetic field, and the second term on the right-hand side represents diffusion of the magnetic field, and in a region where η is constant may be written +η 2B. In the case of a simple conducting fluid in the presence of a magnetic

© 2001 by CRC Press LLC

magnetometer

field, the force due to the magnetic field (the Lorentz force) per unit volume is:

fB = 1 ( × B) × B

µ

where µ is the magnetic permeability.

In a medium characterized by a scalar electrical conductivity σ , the electric current density J in cgs units is given by Ohm’s law

J = σ (E + V×B/c) ,

where E is the electric field, B the magnetic field, and c the (vacuum) speed of light. Many fluids of importance for space physics and astrophysics have essentially infinite electrical conductivity. For these systems to have finite current density,

E + V×B/c = 0 .

Thus, the electric field can be eliminated from the magnetohydrodynamic equations via Faraday’s law. Also, the current density has disappeared from Ohm’s law, and has to be calculated from Ampère’s law (the displacement current is negligible in nonrelativistic magnetohydrodynamics). Thus, for a nonrelativistic inviscid fluid that is a perfect electrical conductor, the Euler equations for magnetohydrodynamics are the equation of continuity

∂ρ + · (ρV) = 0 , ∂t

where ρ is the mass density, the momentum equation

ρ∂ + V· V = − P + fB

∂t

where P is the pressure, and F is the body force per unit volume (due, for example, to gravity), plus some additional condition (such as an energy equation, incompressibility condition, or adiabatic pressure-density relation) required for closure of the fluid-mechanical equations. Faraday’s law completes the magnetohydrodynamic equations.

Magnetohydrodynamics has been applied broadly to magnetized systems of space physics and astrophysics. However, many of these systems are of such low density that the mean-free

paths and/or times for Coulomb collisions are longer than typical macroscopic length and/or time scales for the system as a whole. In such cases, at least in principle, a kinetic-theory rather than fluid-mechanical description ought to be used. But plasma kinetic theories introduce immense complexity, and have not proved tractable for most situations. Fortunately, experience in investigations of the solar wind and its interaction with planetary and cometary obstacles has shown that MHD models can give remarkably good (though not perfect) agreement with observations. This may be so partly because, absent collisions, charged particles tend to be tied to magnetic field lines, resulting in a fluid-like behavior at least for motion transverse to the magnetic field.

Moreover, moments of the kinetic equation give continuity equation and momentum equation identical to their fluid counterparts except for the fact that the scalar pressure is replaced by a stress tensor. Turbulent fluctuations can also influence transport properties and enforce fluid-like behavior. On the other hand, predictions from kinetic theory can and do depart from the MHD description in important ways, particularly for transport phenomena, instability, and dissipation.

magnetometer An instrument measuring magnetic fields. Until the middle of the 20th century, most such instruments depended on pivoted magnetized needles or needles suspended from torsion heads. Compass needles measured the direction of the horizontal component (the magnetic declination, the angle between it and true north), dip needles measured the magnetic inclination or dip angle between the vector and the horizontal component, and the frequency of oscillation of a compass needle allowed the field strength to be determined. Alternatively, induction coils, rotated rapidly around a horizontal or vertical axis, produced an induced e.m.f. which also gave field components.

Induction coils have also been used aboard spinning Earth satellites, but most spaceborne instruments nowadays use fluxgate magnetometers, also widely used for geophysical prospecting from airplanes and for scientific observations on the ground. These are not absolute but require calibration, and for precision work there-

© 2001 by CRC Press LLC

magnetopause

fore alkali vapor instruments are often used, generally in conjunction with fluxgates. These are absolute instruments, observing the intensity of the magnetic field by the frequency at which a swept-frequency radio signal causes enhanced absorption of a light beam in a glass cell filled with alkali vapor. Rubidium or strontium vapor is generally used, and the beam is emitted from the same element, in a narrow frequency range. The mechanism is based on optical pumping of energy sub-states of these atoms.

Other instruments (used mainly on the ground) include the proton precession magnetometer, on which the widespread technique of nuclear magnetic resonance in medicine is based. The Overhauser effect magnetometer is related to this but it is more precise, and it can be used as an alternative to the alkali vapor instrument in geomagnetic survey satellites.

magnetopause The boundary of the magnetosphere, separating plasma attached to the Earth from that flowing with the solar wind. The magnetopause is defined by the surface on which the pressure of the solar wind is balanced by that of the Earth’s magnetic field. The “nose” of the magnetopause, on the sunward side of the Earth is 15 Earth radii away, on average. As the pressure of the solar wind changes, the magnetopause shrinks or expands accordingly. The idea of a magnetopause has been around since about 1930 when Chapman and Ferraro proposed a theory that explained geomagnetic storms as interactions of the Earth’s magnetosphere with plasma clouds ejected from the sun. The magnetopause was first discovered in 1961 by NASA’s Explorer 12 spacecraft.

magnetosonic wave	See magnetoacoustic
wave.

magnetosphere The planetary region of space where a body’s magnetic field can be detected. As the solar wind plasma embedded with interplanetary magnetic field (IMF) flows around the planet, it interacts with the Earth’s magnetic field and confines it to a cavity called the magnetosphere. Since the solar wind is supersonic, a shock known as Bow shock is formed on the sunward side of the magnetosphere. The solar wind then flows across the bow shock and

its speed is reduced from supersonic to subsonic. The solar wind ahead is deflected at a boundary between the magnetosphere and the solar wind known as the magnetopause. The subsolar point on the magnetopause is about 10 Earth radii from the center of the Earth. The bow shock is about 3 Earth radii from the sunward side of the magnetopause at the subsolar point. The region between the bow shock and the magnetopause is called the magnetosheath.

At the magnetopause, the solar wind pressure outside is balanced by the magnetic field pressure inside the magnetosphere. As the solar wind sweeps past the Earth, the magnetic field lines from the polar cap are pulled toward the nightside to form a geomagnetic tail. This tail can be observed as far as 1000 Earth radii, as the combined pressure of the field and plasma prevent its closing on the night side. Moreover, the polar cap magnetic field lines do not close resulting in a thin sheet in the equatorial plane known as the neutral sheet across which an abrupt field reversal occurs. The region where magnetic field lines from sub-auroral latitudes tend to close is known as plasma sheet of thickness about 1 Earth radii. The plasma sheet consists of energetic particles in the energy range of about 1 to 10 kev and is the source of radiation belt particles.

magnetospheric substorms Apart from quiet variations in the Earth’s magnetic field like solar quiet (S q) and lunar variations resulting from the generation of ionospheric currents by solar and lunar tides, other observed variations are far from constant and without any periodicity. Such variations result from the interaction of solar wind with the geomagnetic field of the Earth and are denoted by D. These are called magnetospheric storms and have characteristic times of minutes to days. The magnetospheric storm consists of frequently occurring small storms called magnetospheric substorms.

The intensity of the storm is given in terms of the AE (auroral electrojet) index and of the substorm by the Dst index. Each substorm is associated with injection of energetic protons of few tens of kev in the Van Allen belts. When frequent injection occurs, a proton belt known as ring current is formed whose strength is given in terms of Dst. At first there is a sudden com-

© 2001 by CRC Press LLC

magnitude

mencement (SC) in the geomagnetic field, an impulsive increase in the H component with a risetime of a few minutes with an amplitude of several nano teslas observed over the entire Earth with a spread in arrival time of less than 1 min. The increase in H during SC is maintained for a few hours known as the initial phase until a large H decrease of the main phase begins, resulting from the ring current buildup. The fi- nal recovery phase, of typically a day or more, is faster at first and then slower, and results from a decrease in ring current to prestorm values. The effects of the magnetospheric storms and substorms in the ionosphere are called ionospheric storms.

magnetostratigraphy The use of measurements of the magnetization of strata for absolute or relative dating purposes. Sedimentary rock containing magnetized particles that tend to orient with the Earth’s field upon or soon after deposition will thus make a record of the geomagnetic field: similarly, hot rock containing minerals with a high magnetic susceptibility will record the geomagnetic field at the time at which they are cooled to below their Curie temperature. Hence, strata that was deposited or cooled at a particular site at different times may exhibit differently oriented magnetizations, as the Earth’s field changes with time and also the location of the strata may have moved or rotated with respect to the Earth’s rotation axis. Since the pattern of reversals of the Earth’s field is fairly well known for several epochs, a good record of the variation of the magnetization at a particular site may sometimes be used to date the rocks.

magnetostrophic A dynamic state of a rotating conducting fluid which is threaded by a magnetic field. The leading order force balance is between the Lorentz (magnetic) force, the Coriolis force, the pressure gradient, and buoyancy. Other terms, such as that associated with inertia or viscosity, are deemed much smaller. This is the force balance commonly assumed to hold for the Earth’s core, as the viscosity of the core fluid is thought to be very low. There are several types of magnetostrophic waves including MAC (magnetic-archimedean-coriolis) waves which involve the archimedean force (i.e., buoyancy)

and MC (magnetic-coriolis) waves that do not. A magnetostrophic flow should generally satisfy Taylor’s condition that the azimuthal component of the Lorentz force integrates to zero on axial cylinders, except in certain circumstances where the gravitational force has an azimuthal component. See Taylor state.

magnetotail The long stretched out nightside of the magnetosphere, the region in which substorms begin. It ranges from about 8 Earth radii nightwards of the Earth and has been observed out to 220 Earth radii. The magnetotail is created by the interaction of the solar wind with the Earth’s magnetosphere. The solar wind particles are deflected around the Earth by the bow shock, dragging the nightside magnetosphere out into a long tail. It is estimated that the magnetotail may extend to over 1000 Earth radii from the Earth.

magnitude In astronomy, the system of logarithmic measurement of brightness of stars was invented by the ancient Greeks. The Greeks had six equal divisions from the brightest star to the faintest observable by eye, the brightest being first magnitude and the faintest sixth magnitude. The human eye is sensitive to logarithmic changes in brightness, and so this scheme was logarithmic in nature. The modern definition of magnitude takes the difference√in one magnitude of brightness being equal to 5 100. The magnitude system is related to the brightness of an object, such that

m = −2.5 log(F) + K

where F is the observed flux in units of erg s−1 cm−2. The quantity m is the apparent magnitude. Flux can be written as a function of the intrinsic luminosity of the star such that

F = L 4πR2

where L is the intrinsic luminosity and R is the distance to the star. The difference in two magnitudes is then

m1 − m2	= 2.5 log	F1		.
			F2

The constant K is defined by assigning magnitudes to a number of standard stars, which are

© 2001 by CRC Press LLC