Taylor’s hypothesis

number. Its value depends on the length scale of the convective system, the rotation rate, and kinematic viscosity. The Taylor number Ta is

Ta = (2ωU)2/	ν H 2			2
				= 4ω2H 4/ν2
		U

where H is depth of fluid, is rotational angular velocity, ν is kinematic viscosity, and U is a typical velocity. If Ta is equal or greater than one, the rotational effects are significant. See Taylor instability for a slightly different form for Ta.

Taylor’s hypothesis See frozen field approximation.

Taylor state A configuration of magnetic field in a conducting, rotating fluid so as to obey the constraint that the azimuthal component of the Lorentz force integrates to zero over the surface of cylinders coaxial with the rotation axis. This can be true for fluid in a magnetostrophic balance, i.e., balance of magnetic, pressure, buoyant, and coriolis forces, which may hold for the Earth’s outer core if viscosity and inertia are both small. Taylor’s constraint holds for a magnetostrophic flow if it is anelastic (i.e., ·(ρu) = 0 where ρ is the density and u the flow velocity) and if the gravitational force has no azimuthal (φ) component (or, at minimum, ρgφ also integrates to zero on the cylinder). The constraint may be written as:

T = s [( × B) × B]φ d = 0

where s is the distance from the rotation axis, B is the magnetic field, and denotes the axial cylinder. Neither inertia nor viscosity are identically zero in the Earth’s outer core, and it has been suggested that the Taylor torque T may be balanced by viscous forces acting on the cylinder at the core-mantle boundary, such as in Braginski’s Model-Z dynamo. However, even in this case in the limit of vanishing viscosity, the above constraint is asymptotically satisfied. The effect of non-zero inertia is that a non-zero Taylor torque is balanced by a term representing acceleration, which leads to torsional oscillations. The net result is a “basic state” satisfying the above constraint with superimposed torsional oscillations.

TCB See coordinate time.

TCG See coordinate time.

TDB See dynamical time.

TDT See dynamical time.

tectonics The study of the large scale movements and deformation of a planetary crust. Planetary crusts are affected by extensional and compressional forces caused by regional or global processes. The lunar maria display compressional features in their centers and extensional features along their edges, caused by subsidence of the dense lava flows which comprise the maria. Mercury is shrinking due to the cooling and solidification of its large iron core, which is creating compressional tectonics on its surface. The icy moons of the outer solar system display extensional tectonics due to the cooling and solidification of the ice in their interiors. Earth’s crust is in a constant state of flux due to the influence of plate tectonics, which causes large segments of the crust to move. There is evidence from satellite magnetometer measurements of former surface tectonic activity on Mars.

tectosphere A layer of rock up to several hundred kilometers thick at the top of the mantle underlying some of the continents, which has been observed by seismology to be distinct (perhaps colder) than the rest of the upper part of the mantle, i.e., as “continental roots”. The tectosphere is thought to be relatively viscous and tightly coupled to the overlying continent, which implies that it is important for determining how the motions of the continent are coupled to those in the underlying mantle. There are different ideas as to how tectosphere may be formed, including simple cooling of relatively immobile mantle underlying a continental plate and the buildup of buoyant by-products of slab subduction.

Telesto Moon of Saturn, also designated SXIII. Discovered by Smith, Reitsema, Larson, and Fountain in 1980, it orbits Saturn in the same orbit as Tethys, but leading it by 60◦. This is one of the two stable Lagrange points in the Saturn–

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tension (cosmic string)

Tethys system. Calypso orbits at the other. Its orbit has a semimajor axis of 2.95 ×105 km. Its size is 17 × 14 × 13 km, but its mass has not been measured. It has a geometric albedo of 0.5 and orbits Saturn once every 1.888 Earth days.

temperature (1.) In thermodynamics, the integrating factor in the first law of thermodynamics:

dE = dU + T dS

where the energy E, the internal energy U, and the entropy S are functions of the state of the equilibrium system, and dE, dU, dS are their functional differentials; a function proportional to the pressure of a hypothetical perfect gas held at constant volume.

(2.) In statistical mechanics, a measure of the translational molecular kinetic energy.

(3.) The degree of hotness measured on a conventional temperature scale.

(Definitions (1.) and (2.) agree for equilibrium systems, as does (3.) for some appropriate range of temperature specific to the device.)

(4.) In equilibrium photon dynamics, the inverse multiplier of E/k (where k = Boltzmann’s constant) in the Plack distribution f = f (E/kT ); the temperature of a black body radiation. In non-equilibrium situations, e.g., astronomical observations, a derived measure which agrees with one of the above definitions for some range of applicability.

If the object radiates like a true black body, all of these temperatures are equivalent; however, a perfect black body in nature is rare. A perfect black body is thermalized, that is, the atoms or molecules of the object are in perfect thermodynamic equilibrium. Most stars have a radiation distribution with wavelength that closely but not exactly approximates that of a black body. The sun, for example, has a radiation temperature that approximates a black body with TB = 6300 K; however, its effective temperature, which compares the total output power to that of a black body, is better approximated by a Teff = 5800 K, and the color temperature of the sun which compares the energy output ratio over two wavelength intervals is Tc = 6500 K. See excitation temperature, effective temperature, blackbody temperature, brightness temperature, color temperature.

temperature inversion An increase of atmospheric temperature with altitude. Under this condition the typical lapse rate is reversed, and great stability is created, which strongly damps vertical motions and vertical turbulent transport. Wind shear will exist between the top and the bottom of the temperature inversion layer. A temperature inversion acts as a ceiling, preventing further upward convection, and is generally the limit for cloud development. Marked and persistent inversions occur at lower levels, with subsiding air in major anti-cyclonic cells, such as the Azores high-pressure zone and cold anticyclones over continents. Temperature inversion arises for different reasons, such as frontal inversion, subsidence inversion, trade-wind inversion, radiation inversion, advective inversion, turbulence inversion, and stratospheric inversion. See density inversion.

temperature variance dissipation rate

Measure of the rate at which gradient fluctuations of temperature are smoothed out by turbulence. It is defined by

χT = 2κT ( T )2

where κT is the thermal diffusivity, T is the temperature, z is the vertical coordinate, and denotes the gradient operator. The temperature variance dissipation rate can be regarded as the thermal equivalent of the dissipation rate of velocity fluctuations !.

In oceanic or atmospheric studies, generally only one component of the temperature gradient is measured, and χT is estimated from

χT 2λκT	∂T		2
			.
	∂z

The factor λ is a scaling parameter that reflects the level isotropy. For a completely isotropic turbulence field λ = 3.

tension (cosmic string)	For a stringlike topo-
logical defect, the quantity
T = −2π	rdrT zz ,

where r is the radial coordinate in a cylindrical reference frame aligned with the string, and T µν

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tension head (ψ)

is the energy-momentum tensor computed from the microscopic field configuration, in the particular case of a straight string. For an ordinary relativistic string we have T tt + T zz = 0; hence we recover the equation of state for a structureless Goto–Nambu string, where U (the energy per unit length) and T are constant and equal to each other. For more general vortex-forming field theories, the corresponding cosmic string model will be characterized by variable tension and energy per unit length which, in a generic state, will be related by an inequality of the form T ≤ U. See duality in elastic string models, energy per unit length (cosmic string), equation of state (cosmic string), Goto–Nambu string.

tension head (ψ) There is tension on pore water in the unsaturated zone because water is held to the mineral grains by surface-tension forces. It is conventional to measure pressure (p) relative to atmospheric pressure, thus in the unsaturated zone p < 0 and ψ > 0. Negative pressure is often called suction or tension, and ψ is called tension head when p < 0. Tension head is also known as capillary-pressure head and moisture potential. Tension in the unsaturated zone generally varies from about −0.35 m at field capacity to −3,100 m, where water is adsorbed by mineral grains directly from the air. The units are length (height) in meters.

tensor A generalization of the concepts of vector and of one-forms as operators on functions and vectors.

A rank-2 contravariant tensor T is a linear sum of operators of the form

T = A B(·, ·) ;

where A, B are vectors, and the notation indicates that the vector A acts on the function that is the first argument producing a number, and the vector B acts on the function that is the second argument producing a number, and the results of these operations is multiplied (ordinary multiplication). For instance

T (g, h) = A(g)B(h) ;

where A acts on g, and B acts on h. Since A, B can be thought of as fields defined at least in a region of space, and the arguments f, g are

scalar functions of position, the tensor, and the result of this operation, are functions of position.

If we have a set of basis contravariant vectors, then the general rank 2 contravariant vector can be expressed:

T = T ij ei ej .

Higher rank contravariant tensors are formed by repeatedly multiplying vectors using .

A rank-2 covariant tensor is constructed similarly to act on a pair of vectors; the result is the ordinary product of the results for each factor. A general covariant rank-2 tensor can be written

as

g = gij σ i σ j ,

where σ i constitutes a set of basis one-forms. Higher rank covariant tensors, or tensors of

mixed covariant/contravariant rank can be created by repeated use of . See summation convention.

terminal velocity For an object falling through a normal fluid (e.g., air or water) the retarding force increases with the velocity of the object. Hence, there is a speed at which the retarding force equals the weight of the body less any buoyant forces. At this speed the body will not accelerate to fall faster; this is the terminal velocity.

terminator The line on a planet or other solar system object, between sunlit and night sides of the planet, the sunrise/sunset line. Extended to bodies in other solar systems.

Terrestrial Coordinate Time (TCG) Terrestrial Dynamical Time (TDT) has been deemed by the International Astronomical Union (IAU) to require supplementation with another standard because the progress of TDT depends, in a sense, on the gravitational potential on the geoid. The IAU, therefore, established in 1991 a time standard representing what an SI clock would measure in a coordinate system, such that the barycenter of the Earth was stationary in this nearly inertial system, as was the clock, but the clock was so far removed from the Earth (but not the sun and planets) that it suffered no effect of the Earth’s gravity or rotation. That time is T CG =

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Tethys

T T +LG ·(JD−2443144.5)·86400 sec, where LG = 6.96929 · 10−10 by definition as of 1992,

JD stands for the Julian Date in TDT, and TT stands for Terrestrial Time, which is essentially the same as TDT. Presumably, the “constant”, LG, is subject to revision when and if the potential on the geoid is redetermined. For details, see IERS Technical Note 13, The IERS Standards

(Ed.: D.D. McCarthy) (U.S. Naval Observatory, Washington, 1992).

Terrestrial Dynamical Time (TDT) In 1977, Dynamical Time was introduced in two forms, TDT and Barycentric Dynamical Time (TDB). The difference in these two consists of periodic terms due to general relativity and does not exceed about 1.7 milliseconds in any year. TDT is used for the determination of the orbits of objects orbiting the Earth. It advances at the same rate as International Atomic Time (TAI), being equal, for all practical purposes, to TAI + 32.184 s. Both TAI and TDT contain relativistic effects of the Earth’s motion around the sun, which reflect the fact that any reference frame centered on Earth’s center is not inertial. Any of TDT, TAI, and TCG are thus subject to further special relativistic corrections in their relationship to any barycentric time standard, such as TCB. For details, see IERS Technical Note 21, The IERS Conventions (Ed.: D.D. McCarthy) (U.S. Naval Observatory, Washington, 1996). See dynamical time.

terrestrial heat flow Heat flux from the interior of the Earth. It usually means the conductive heat flux measured at the surface of the Earth, and it is usually called heat flow or heat flow density. The present global average is about 0.06 W/m2.

terrestrial planets Those planets that are similar to the Earth in their characteristics. There are four terrestrial planets in our solar system: Mercury, Venus, Earth, and Mars. Characteristics of the terrestrial planets include small size (< 13,000 km in diameter), high density (> 3000 kg/m3, indicating rocky compositions), close to the sun (< 1.5 astronomical units), few or no moons, and no rings. These characteristics differ considerably from those of the Jovian planets, which dominate the outer solar sys-

tem. Although Pluto is similar to the terrestrial planets with its small size, one moon, and lack of rings, its large distance from the sun and its lower density (about 2000 kg/m3, indicating an icy composition) place it in a category of its own, where it shares characteristics with many of the large icy bodies in the outer solar system. The terrestrial planets are sometimes also called the inner planets, since they are located in the inner part of the solar system.

Terrestrial Time (TT) In 1991, the International Astronomical Union (IAU) defined an “ideal form of Terrestrial Dynamical Time”, and designated Terrestrial Time. This definition appears to be primarily intended to extend the scope of TDT off the geoid, by imposing certain restrictions on relativistic coordinate systems. It is used in dynamical theories of solar system motions, and it could take into consideration the need to bring dynamical measures of time (from the motions of spacecraft orbiting the Earth) into accord with atomic time. For practical purposes, it is identical with Terrestrial Dynamical Time. For details, see IERS Technical Note 13, The IERS Standards (Ed.: D.D. McCarthy) (U.S. Naval Observatory, Washington, 1992).

tesla A unit of magnetic flux equal to one weber per square meter.

Tethys Moon of Saturn, also designated SIII. Discovered by Cassini in 1684. Its surface shows a crater, Odysseus, with a radius of 200 km. This is some 40% of the radius of Tethys and is probably at the limit of what the moon could sustain without breakup. The crater is now flattened and conforms to Tethys’ spherical shape. There is also a large valley, Ithaca Chasma, which is 2000 km long and extends 3/4 of the way around Tethys. It is some 100 km wide and 3 to 5 km deep. Tethys’ orbit has an eccentricity of 0, an inclination of 1.86◦, a precession of 72.25◦ yr−1, and a semimajor axis of

2.95 × 105 km. Its radius is 530 km, its mass 7.55 × 1020 kg, and its density 1.21 g cm−3. It

has a geometric albedo of 0.9, and orbits Saturn once every 1.888 Earth days.

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