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dimensional transmutation

which the dilaton plays the role of a scalar component of gravity and can also be thought of as a variable (in both space and time) Newton’s “constant” (see Brans–Dicke theory). The dilaton couples to matter fields, e.g., Yang–Mills SU(N) fields and Maxwell’s U(1) electromagnetic field. Because of this, the presence of a non-trivial dilaton in general spoils the principle of equivalence. See dilaton.

dilatonic black hole In field theory applied to cosmology, a dilaton is an additional scalar field associated with gravity. A few dilatonic gravity solutions are known in four space-time dimensions which represent black holes with a nontrivial dilaton field. When the latter field couples to the electromagnetic tensor, the black holes must be electrically charged. One exact solution is given by the analog of the spherically symmetric general relativistic black hole with charge, the Reissner–Nordström metric. When angular momentum is present (see Kerr–Newman met- ric) only perturbative solutions are known for either small angular momentum or small electric charge. All the solutions have in common a dilaton which decreases and vanishes at a large distance from the center of the hole. However, near the event horizon the dilaton is nonzero and can possibly affect the scattering of passing radiation.

Other solutions have been found when the dilaton couples to Yang–Mills SU(N) fields giving rise to richer structures. See black hole, dilaton gravity, future/past event horizon, Kerr– Newman metric, Reissner–Nordström metric.

dilution Defined as the total volume of a sample divided by the volume of effluent (contaminant) contained in the sample.

dimension A statement of the number of independent parameters necessary to uniquely define a point in the space under consideration. Everyday experience with space indicates that it is 3-dimensional, hence specifying the x, y, z labels (for instance) of a point uniquely defines the point. If it is wished to specify the locations of two mass points, it is convenient to introduce a 6-dimensional space, giving, for instance, the x, y, z labels of the location of each of the mass points. In special and general rela-

tivity, time is considered a separate dimension, and so events are given by specifying x, y, z, and t (time); thus, spacetime is 4-dimensional. In many mathematical operations, e.g., in integration, the dimension of the space enters explicitly and as a consequence the solution to standard equations depends on the dimension of the space in which the solution is found. It is found effective theoretically to allow noninteger dimensions in those cases. Also, since the ratio of the “volume” to the “surface” depends on the dimension, this concept has been generalized to a “fractal dimension” which is defined in terms of the ratio of these quantities in some suitable sense. See fractal.

dimensional analysis In usual physical descriptions, quantities are assigned units, e.g., centimeter for length, gram for mass, second for time. In dimensional analysis, one constructs a combination of known quantities which has the dimension of the desired answer. Because specific systems have typical values for dimensional quantities, results constructed in this way are usually close to the correctly computed result. Typically dimensional analysis omits factors of order one, or of order π; the results then differ from exactness by less than an order of magnitude. For instance, the typical length associated with a sphere is its radius r (measured in centimeters, say). Its volume by dimensional analysis is then r3 (cm3), whereas the exact value including π and factors of order unity is

4π/3r3 4.18r3 .

 

dimensional transmutation

(Coleman

Weinberg, 1973) In field theory, dimensional transmutation occurs in a first or second order phase transition in an originally massless theory. The classical potential of the massless scalar field ϕ has single minima at ϕ = 0 of a particular shape. Due to the quantum effects, the potential acquires the second minima at ϕc. In turn, the existence of the critical point ϕc leads to spontaneous symmetry breaking (the field is nonzero, even though the underlying theory does not pick out a nonzero value for ϕ: it could have ended up in the other minimum). Additionally, quantum effects can modify the shape of the potential

© 2001 by CRC Press LLC

Dione

at the new minima, so that the field now corresponds to massive particles — this is dimensional transmutation. Such effects are important at the level of cosmic defects in cosmology, and also more fundamentally because, for instance, Newton’s constant of gravity G may have arisen by a similar process from a “simpler” theory with G = 0. See induced gravity, spontaneous symmetry breaking.

Dione Moon of Saturn, also designated SIV. It was discovered by Cassini in 1684. Its orbit has an eccentricity of 0.002, an inclination of 0.02◦, a semimajor axis of 3.77 × 105 km, and a precession of 30.85◦ yr−1. Its radius is 560 km, its mass 1.05 × 1021 kg, and its density is 1.43 g cm−3. Its geometric albedo is 0.7, and it orbits Saturn in a synchronous orbit once every 2.737 Earth days.

dip-equator Isocline with inclination I = 0, geomagnetic equator.

dip, magnetic The local angle observed between the horizontal plane and the direction of the Earth’s magnetic field. Also known as magnetic inclination.

dip slip fault A fault upon which the motion on the fault is vertical.

disappearing filament In solar physics, disappearance of a filament/protuberance without the typical emission of electromagnetic radiation related to a flare. The disappearing filament might be observed as a coronal mass ejection. Compared to coronal mass ejections accompanied by a flare, these are normally rather small and slow coronal mass ejections, too slow to drive a shock wave.

discharge coefficient A calibration coefficient employed to relate the flowrate over a weir to the depth (head) on the weir and other geometric properties.

disconnection event In solar physics, reconnection in the tip of a helmet streamer leads to the expulsion of coronal plasma. In contrast to a typical coronal mass ejection, the disconnected magnetic field configuration is open, and the im-

age in the coronograph is a jetor fan-like structure. Observations suggest disconnection events are related to the emergence of new magnetic flux on the sun. The accompanying disturbance of the coronal structure then “squeezes” the opposing fields in the tip of a helmet streamer, eventually causing reconnection and the expulsion of magnetic flux and matter. Because of the open magnetic field structure, the remainders of such disconnection events do not form magnetic clouds and consequently are more difficult to identify in interplanetary space.

discordant redshift Redshift of a galaxy significantly differing from the redshift of other galaxies belonging to an apparently interacting system of galaxies, such as a group. For example, in the case of the Stephan Quintet, a group of five galaxies, four galaxies have redshifts around 6000 km/s, and one has a discordant redshift of only 800 km/s. Several other intriguing examples exist, in which two galaxies with large redshift difference are connected by bright filaments. Galaxies with discordant redshift can be explained as due to casual superposition of foreground and background galaxies, or in some cases (at small angular separation) on gravitational lensing of background galaxies. However, H. Arp and collaborators have suggested the existence of “non-velocity” redshifts. They concluded that redshift is not a reliable distance indicator, questioning the validity of Hubble’s law and especially of quasar distances deduced from their redshift. This is a distinctly non-standard interpretation, at variance with the opinion of most of the astronomical community.

discrete aurora A term used for aurora appearing in well-defined formations, e.g., arcs and ribbons, distinguishing it from the diffuse aurora. Auroral arcs visible to the eye are of this kind. Discrete auroras are caused by electrons of up to 5 to 15 keV, accelerated earthward along magnetic field lines. Often these same field lines also carry upward-directed currents, suggesting that the electrons are carriers of that current, and the accelerating electric field helps achieve the current intensity demanded by the sources of that current, which are more distant. While those sources provide the energy, the ac-

© 2001 by CRC Press LLC

disparition brusque

celeration process occurs mainly within about 8000 km of Earth.

discrimination (seismic) The use of seismology to determine whether a seismic event was caused by the test of an explosive device (such as the relative slip of rock on either side of a fault, while explosions generate waves through a more isotropic compression of the surrounding rock). The difference in the source mechanisms and their relative durations (explosions tend to be very rapid) means that the seismic signals generated differ in the geometrical pattern of the radiation, in the proportion of different types of seismic wave emitted, and in frequency content. Large explosions that can be detected at long distances may be reliably discriminated through study of the seismic signals, although smaller explosions may be more difficult both to detect and to distinguish from earthquakes. Potential complications include the geological setting of the event (which may not be well known) and whether an effort has been made to decouple the explosion from the surrounding rock by setting it off in a cavity.

disk warp A deviation from planarity of the disk of a galaxy, in which the outer parts of a galactic disk gradually tilt. More precisely, one can think of the disk of a galaxy as a sequence of concentric, adjacent rings: If the disk is flat, the rings are coplanar; if the disk is warped, the inclination and the position angle of the intersection line between rings (line of nodes) varies continuously from ring to ring. A disk warp is more frequently observed in the gaseous than in the stellar component of a galaxy. The distribution of the 21-cm hydrogen emission line often reveals a warp outside the boundaries of the optical disk. Small warps are probably present in the outer regions of most disk galaxies (including the galaxy); strong warps, where the disk plane tilts by 40 to 50◦, as in the case of NGC 660, are of rare occurrence.

dislocation climb Dislocation motion when an edge dislocation moves along the direction perpendicular to its slip plane under the action of a shear stress. The climb of dislocation involves transport of matter. For the dislocation line to climb one interatomic distance up or down, a

line of atoms along the edge of the extra-half plane has to be removed or added according to the position of the extra-half plane, which is completed by diffusion.

dislocation creep A continuous, usually slow deformation of solid crystalline materials resulting from the motion (glide, climb, and cross slip) of dislocations under the action of shear stress. The resistance to dislocation motion includes: (1) intrinsic resistance (the Peierls stress); (2) impurities, and (3) mutual interaction of dislocations. In general, the largest resistance force controls the rate of deformation. Dislocation creep is characterized by (1) a non-linear dependence of strain rate on stress (n = 3 −5, n is stress sensitivity of creep rate at the steady-state stage); (2) crystals deform on specific crystallographic planes along certain orientations so that it can produce strong lattice preferred orientation; (3) significant transient creep is expected.

dislocation energy (self-energy of dislocation)

The energy stored in a solid by the existence of a unit length of dislocation line. The magnitude of dislocation energy depends on the nature of particular dislocations such as the type and configuration. For example, the Peierls energy is the energy associated with the Peierls dislocation, which can be divided into two portions: the elastic energy stored in the two half-crystals, and the misfit energy (core energy) associated with the distorted bond.

dislocation glide The manner of dislocation motion when a dislocation moves on its slip plane under the action of a shear stress. Glide of dislocation involves no transport of matter by diffusion: atomic bonds are shifted, but there is no need for atoms to be brought in or evacuated. The resistance to the dislocation glide is intrinsic, mainly from the Peierls stress.

disparition brusque The sudden disappearance, within the space of a few hours, of a solar filament. This disappearance begins with a slow rising motion at a few kms−1 and is typically associated with a brightening in X-rays and occasionally by the appearance of Hα flare ribbons.

© 2001 by CRC Press LLC

dispersion

dispersion A phenomenon in which wave velocity (phase velocity, group velocity) changes with its wavelength. In seismology, for a layered structure, phase velocity becomes closer to S-wave velocity of the lower and upper layers for longer and shorter wavelengths, respectively. Phenomena in which phase velocity increases and decreases with increasing wavelength are respectively referred to as normal dispersion and reverse dispersion. (This is the case for visible light in glass and is the usual case in seismology.) In seismology, curves representing the relation between surface wave velocity and its wavelength are called dispersion curves, from which the velocity structure of the crust and the mantle can be estimated.

dispersionless injection A sudden rise in the intensity of energetic ions in the Earth’s nightside equatorial magnetosphere, in general at or beyond synchronous orbit, occurring simultaneously over a wide range of energies. It is widely held that such particles must have been accelerated locally because if their acceleration occurred some distance away, the faster ones would have arrived first.

dispersion measure (DM) The integral along the line of sight distant source of the electron number density.

The pulse arrival time for two different frequencies f2, f1is related by

2t = e2/ (2πmec) 2 f −2 DM .

dispersive Tending to spread out or scatter; having phase and group velocities that depend on wavelength. Used to describe both physical and numerical processes.

displacement vector In a Euclidean space, the difference vector between position vectors to two points. The displacement vector is often thought of as the difference in position of a particular object at two different times.

dissipation In thermodynamics, the conversion of ordered mechanical energy into heat. In computational science, deliberately added to differential equations to suppress short wave-

length oscillations that appear in finite representations of differential equations, but have no analog in the differential equations themselves.

dissipation of fields One of the basic concepts of magnetohydrodynamics. In the case of a finite conductivity, the temporal change of the magnetic flux in a plasma can be written as

∂B = c2 2B ∂t 4πσ

with B being the magnetic flux, σ the conductivity, and c the speed of light. Formally, this is equivalent to a heat-conduction equation, thus by analogy we can interpret the equation as describing the temporal change of magnetic field strength while the magnetic field lines are transported away by a process that depends on conductivity: the field dissipates. Note that while the magnetic flux through a given plane stays constant, the magnetic energy decreases because the field-generating currents are associated with ohmic losses. Magnetic field dissipation seems to be important in reconnection. See reconnection.

Aside from the conductivity, the temporal scale for field dissipation depends on the spatial scale of the field. With τ being the characteristic time scale during which the magnetic field decreases to 1/e and L being the characteristic spatial scale of the field, the dissipation time can be approximated as

τ ≈ 4πσ L2 . c2

Thus, the dissipation depends on the square of the characteristic scale length of the field: Smaller fields dissipate faster than larger ones. Thus in a turbulent medium, such as the photosphere, where the field lines are shuffled around and therefore a polarity pattern on very small spatial scales results, the field dissipates rather quickly. Or in other words, turbulence can accelerate magnetic field dissipation.

Note that for infinite conductivity the dissipation time becomes infinite as well, leading to frozen-in fields.

dissipation of temperature variance For scales smaller than the Batchelor scale temperature fluctuations T become extinguished by the

© 2001 by CRC Press LLC

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