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Euler equations

sometimes called sand seas since they represent vast regions of sand. Most ergs are located in desert basins downwind from terrain which is experiencing high amounts of erosion. Various types of dunes occur within the ergs. Ergs are found on Earth and around the north polar cap on Mars.

Ernst equation (1967) The complex partial differential equation

ReEFE = E · E

in a Euclidean 3-space introduced by F.J. Ernst. The axially symmetric solutions E = E(x1, x2) represent the metric potentials of stationary, axisymmetric vacuum gravitational ﬁelds. A second form

¯ − 1 = 2¯ ·

(ξξ )Fξ ξ ξ ξ

is obtained by the substitution E = (ξ −1)/(ξ + 1). The Ernst equation has been found to describe additionally various physical systems, such as colliding plane waves in general relativity, monopoles in SU(2) gauge theory, and states of Heisenberg ferromagnets. The symmetries of the equation and solution generating techniques have been extensively studied.

erosion Rock is destroyed by weathering. Mechanical weathering breaks up rock into small particles which can be transported hydrologically. In chemical weathering the rock is dissolved by water. In either case, the removal of the rock is erosion.

ERP Earth rotation parameters.

Ertel potential vorticity	See potential vor-
ticity.

eruption Volcanos are subject to periodic eruptions during which molten rock (magma) and/or volcanic ash ﬂows to the Earth’s surface. Eruptions can occur for months between periods of quiescence.

eruptive prominence Solar prominences can become activated and exhibit several types of large-scale motion. The prominence may grow and become brighter with a corresponding in-

crease in helical motion or ﬂow along the prominence axis. This type of activation can lead to an eruption, especially if it exceeds a height of 50,000 km. In this case, it rises as an erupting prominence and eventually disappears. The eruption of a quiescent prominence (disparition brusque) is a slow process lasting several hours. The eruption of an active region prominence is much more rapid taking about 30 min.

escape velocity In Newtonian physics, under gravitational interactions, the gravitational ﬁeld contributes negatively to the total energy of a particle. The escape velocity is that speed giving a zero total energy (kinetic + gravitational potential). A particle moving without dissipation at or exceeding the escape velocity will escape to inﬁnity.

eskers Long narrow ridges of sand and gravel deposited in the middle of glaciers. They have stream-like shapes and are believed to form from meltwater streams ﬂowing in tunnels beneath a melting glacier. Sinuous ridges in the high latitude regions of Mars have been suggested to be eskers.

estuary A sheltered region that lies inshore of a sea and is subject to tidal action. Generally has a reduced salinity, compared to the adjoining sea, as a result of river and other freshwater inputs.

ET See ephemeris time (ET).

eternal black hole A black hole that was not formed in the collapse of matter, but was present ab initio as a stable topological structure in spacetime.

Eudoxos of Cnidus Greek geometer and astronomer, born in Cnidus, Asia Minor approximately 408 B.C., died about 353 B.C. Eudoxos established the foundation of geometrical principles developed by Euclid, and applied the subject to the study of the moon and the planets.

Euler equations The equations of motion for an inviscid ﬂuid. Equivalent to the Navier– Stokes equations if viscosity is taken as zero.

Eulerian

Represents Newton’s 2nd Law for an inviscid ﬂuid.

Eulerian A term used to denote a description of ﬂuid behavior which involves description of ﬂuid ﬂow parameters (particularly velocity) at ﬁxed points in space. The alternative is a Lagrangian description, which essentially involves describing the behavior of selected ﬂuid particles as they move through space.

Eulerian coordinates In hydrodynamics, physical parameters such as pressure, ﬂuid velocity, and density can be expressed as functions of positions in space and time; thus, a coordinate system ﬁxed to an external reference frame, in which physical phenomena, for instance hydrodynamical ﬂow, move through the hydrodynamical grid. Named after Leonhard Euler (1707– 1783). See Lagrangian coordinates.

Eulerian representation Description of a phenomenon relative to a framework ﬁxed in space. Measurements in moorings are typical applications of Eulerian-type observations. See Lagrangian representation.

Eulerian velocity Velocity measured from a ﬁxed point or set of points (i.e., by a moored current meter). See also Lagrangian velocity.

Euler, Leonhard Mathematician (1707– 1783). Made contributions in the areas of algebra, theory of equations, trigonometry, analytic geometry, calculus, calculus of variations, number theory, and complex numbers. He also made contributions to astronomy, mechanics, optics, and acoustics.

Euler pole Euler’s theorem states that a movement on a sphere may be represented as a rotation about an axis. This holds not only for the movement of a point on a sphere but also the movement of a continuous undeformable patch on a sphere. This turns out to be important for the theory of plate tectonics, which models the surface of the planet as a set of rigid plates in relative motion. Therefore, an axis may be found for each plate relative to some given reference frame. These axes are known as Euler poles. In particular, by considering the motion of one

plate relative to another plate, one may ﬁnd the pole of relative rotation. If two plates meet at a mid-ocean ridge, then the ridge will often be fractured by transform faults oriented perpendicular to a line running toward the pole of rotation between the two plates. Because velocity about the pole depends on distance from the pole, the velocity between adjoining plates will vary along the boundary between the plates unless the boundary happens to lie on a cylinder centered on the pole. This means that, for example, subduction may be much faster at one end of a trench than the other.

Euler potentials Two scalar functions of position (α,β) used for describing a magnetic ﬁeld B and satisfying B = α × β. Their use is equivalent to that of a vector potential A = α β (of no particular gauge, non-covariant) and is useful because the points of any magnetic ﬁeld line share the same value of α and β. As long as the ﬁeld inside Earth is excluded, (α,β) in the Earth’s vicinity are usually unique to one ﬁeld line (not valid in more general geometries) and this allows using them in mapping ﬁeld lines and as variables in the theory of plasma convection by electric ﬁelds in the magnetosphere. Field line motion can also be described by Euler potentials. In toroidal geometries, (α,β) are generally multiply valued.

euphotic depth In oceanography, the depth to which signiﬁcant phytoplankton photosynthesis can take place; typically taken to be the depth at which photosynthetically available radiation falls to 1% of its value just below the surface [m].

euphotic zone In oceanography, the water layer in which signiﬁcant phytoplankton photosynthesis can take place; typically taken to be the layer down to which photosynthetically available radiation falls to 1% of its value just below the sea surface.

euphotic zone midpoint In oceanography, the layer at which photosynthetically available radiation falls to 10% of its value just below the sea surface.

evolutionary track

Europa Moon of Jupiter, also designated JII. Discovered by Galileo in 1610, it is one of the four Galilean satellites. Its orbit has an eccentricity of 0.009, an inclination of 0.47◦, a precession of 12.0◦ yr−1, and a semimajor axis of

6.71 × 105 km. Its radius is 1569 km, its mass 4.75 × 1022 kg, and its density 2.94 g cm−3. It

has a geometric albedo of 0.64, and orbits Jupiter once every 3.551 Earth days. Europa’s surface is covered with ice, and it appears to have liquid water under the ice.

eustatic sea-level The worldwide change of sea level elevation with time. Speciﬁcally, eustatic sea-level describes global sea level variations due to absolute changes in the total quantity of water present in the oceans. These changes are distinct from redistributions of water from circulation, tides, and density changes. Diastrophic eustatism is a change of sea level resulting from variation in capacity of the ocean basins, through the deeping of basins or raising of continents. Glacial eustatism refers to changes in sea level produced by withdrawal or return of water to the oceans.

eutrophic water Water with high phytoplankton biomass; chlorophyll a concentration exceeds 10 mg m−3.

evaporation (E) The process by which water moves from the liquid state to the vapor state follows Fick’s ﬁrst law and can be written in ﬁ- nite difference form as E = KEua(es − ea), which shows that E increases with increasing wind speed (ua) and with the difference between the vapor pressure of the evaporating surface (es) and the overlying air (ea); KE is a coefﬁcient that reﬂects the efﬁciency of vertical transport of water vapor by turbulent eddies of the wind. Sublimation is the process by which water moves from the solid state (snow and ice) to the vapor state.

evapotranspiration (ET ) The sum of all processes by which water changes phase to vapor and is returned to the atmosphere, including sublimation from snow and ice, evaporation from lakes and ponds, and transpiration by vegetation. Actual evapotranspiration (AET) is the amount of evapotranspiration that actually oc-

curs under given climatic and soil-moisture conditions. Potential evapotranspiration (PET) is the amount of evapotranspiration that would occur under given climatic conditions if soil moisture was not limiting. Moisture deﬁcit is PETAET and is the amount of irrigated water added to maximize crop yield.

evening cloud (Mars) Evening clouds of Mars appear near the evening limb or terminator in spring to summer of the northern hemisphere. Similar to morning clouds, evening clouds show a diurnal variation. In their most active period from late spring to mid summer, evening clouds appear in early to mid afternoon, increasing their brightness as they approach the evening limb. They form the equatorial cloud belt together with morning clouds. See morning cloud, af- ternoon cloud.

event horizon A null hypersurface in spacetime separating observable events from those that an observer can never see. A uniformly accelerated observer in Minkowski space-time experiences an event horizon. An absolute event horizon in a weakly asymptotically simple space-time is the boundary of the causal past of future null inﬁnity. A black hole is a domain of space-time separated from null inﬁnity by an absolute event horizon, which is the surface of the region surrounding a black hole where the gravitational ﬁeld is strong enough to prevent the escape of anything, even light. It is a oneway membrane in the sense that no light signal can be transmitted across it from the interior to the external world. See black hole.

Evershed effect The strong horizontal outﬂow observed in the penumbra of sunspots, extending several sunspot diameters from the umbral edge. This outﬂow has speeds of 2 to 6 kms−1 as detected in photospheric and chromospheric lines. The Evershed ﬂow is concentrated within dark penumbral ﬁlaments that are thought to be surface manifestations of convective rolls aligned with nearly horizontal ﬁeld in the penumbra.

evolutionary track As a star ages and moves off the zero age main sequence, its effective temperature (Teff ) and luminosity (L) change.

exact solution

The path of a star in the Teff , L plane (the Hertzsprung–Russell, HR, diagram) as those physical parameters change is called the star’s evolutionary track. This track is primarily a function of the star’s initial mass. Collections of these tracks derived from theoretical stellar evolution models are often plotted on an HR diagram along with data from a collection of stars to estimate initial masses and ages.

exact solution (of Einstein’s equations) A solution of Einstein’s equations, i.e., a metric. The term exact solution is used only when it is necessary to stress that the metric was derived from Einstein’s equations by strict mathematical procedures. The opposites of exact are an approximate (i.e., a perturbative solution) and a numerical solution. This opposition is irrelevant for several purposes; for example, all observational tests of the relativity theory necessarily involve approximations, and numerical calculations are often performed on top of exact solutions to obtain quantitative results. However, some theoretical problems (among them cosmic censorship and the existence of singularities) have been extensively studied within the framework of exact solutions.

excitation temperature In a given object, the constituent atoms and molecules are in quantum mechanically excited states, and the ratio of their numbers in the ground state to the excited state is characterized by an excitation temperature, such that

Nu = gl e−hν/kTex

Nl gu

where Nu and Nl are the number per volume in the upper and lower states, respectively, and gl and gu are the statistical weights. The excitation temperature Tex characterizes this distribution for a given upper and lower state. If the atoms and molecules are in thermodynamic equilibrium (collision dominated), then the kinetic temperature, which describes a distribution in speed of the atoms and molecules, is equal to the excitation temperature.

exosphere The outermost part of the Earth’s atmosphere, beginning at about 500 to 1000 km above the surface. It is characterized by den-

sities so low that air molecules can escape into outer space.

exotic terrane Continental deformation is partly achieved through folding, which preserves the juxtaposition of stratigraphic layers, but also through faulting. A terrane is a block of continent bounded by faults with a stratigraphy that is internally coherent but distinct from that of surrounding blocks. Mountain chains may be composed of several such blocks. A “suspect terrane” is a terrane whose geology appears to be unrelated to the surrounding rock. If it can be shown (through paleomagnetic methods, for example) that the rock in the terrane originated far away from the surrounding rock, then the terrane is known as an exotic terrane. This may happen if part of a continent is sliced away by a long strike-slip fault (such as the San Andreas fault) and is later attached to the continent or another continent elsewhere, or if an island arc or other oceanic landform collides with and is engulfed by a continent.

expansion The mathematical quantity (a scalar) that determines the rate of change of volume of a given small portion of a continuous medium (see also acceleration, kinematical invariants, rotation, shear). An observer placed in a medium in which acceleration, rotation, and shear all vanish while expansion is nonzero would see every point of the medium recede from (or collapse toward) him/her with the velocity v proportional to its distance from him/her, l, so that v = 13 θl. The factor θ is the expansion (θ > 0 for receding motion, θ < 0 for collapse). (In astronomy, H = 13 θ is called the Hubble parameter.) A medium with the abovementioned properties is used as a model of the universe in the Friedmann–Lemaître cosmological models. In these models, the expansion is uniform, i.e., at any given time it is the same at all points of space and can only vary with time. In general (in particular in inhomogeneous models of the universe) the expansion can be different at every point of space. In a perfect ﬂuid or dust such a nonuniform expansion must be accompanied by shear.

expansion of the universe The recession of distant galaxies from our galaxy. The evidence