DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

energy grade line A visual representation of
the energy in a flow. Indicates the sum of the
velocity head, V 2 /2g, elevation, and pressure
head, p/γ, for the flow, where V is the flow
speed,g is the acceleration of gravity,p is pressure,
and γ is the unit weight (weight per unit
volume) of the fluid.
energy-momentum relations — special
relativity In special relativity physical laws
must be the same in reference systems moving
uniformly with respect to each other; that
is, they must be invariant under Lorentz Transformations.
In special relativity time is not an
absolute variable, and therefore special relativity
is mathematically described through a fourdimensional
spacetime with time as the first coordinate
in addition to the three spatial coordinates.
In order for the Lorentz electromagnetic
force to be incorporated into a law of mechanics
that is invariant under Lorentz transformations,
and for the mechanics law to also reduce
to Newton’s law at low velocities, a fourcomponent
relativistic momentum vector is defined
such that the first component equals the
energy of a given particle and the other three
components equal the momentum components
oftheparticle. Sincethefour-vectorscalarproduct
is invariant under Lorentz transformations,
one obtains the relativistic relationship between
the energy and the momentum of a particle by
calculating the four-vector scalar product of the
four-vector momentum with itself.
energy per unit length (cosmic string) In
the framework of a cosmological model with
the generation of topological defects, a cosmic
string is an approximation of a vacuum vortex
defect in terms of a line-like structure, confined
to a two-dimensional world sheet. For a complete
macroscopic description (as opposed to
microscopic, in terms of relevant fields like the
Higgs field and other microscopic fields coupled
to it) we need to know quantities such as the
string tension T and the energy per unit length
U (often denoted µ in the literature). For a microscopic
model, specified by its Lagrangian,
we can compute its energy-momentum tensor
T µν by standard methods. Given the cylindrical
symmetry of the string configuration, the energy
© 2001 by CRC Press LLC
per unit length is calculated as
U = 2π rdrT tt ,
entropy
where T tt is the time-time component of the
energy-momentum tensor. See equation of state
(cosmic string), Goto–Nambu string, tension
(cosmic string), wiggle (cosmic string).
enthalpy An extensive thermodynamic potential
H given by
H = U − PV ,
where U is the internal energy, P is the pressure,
and V is the volume of the system. The
change of the enthalpy is the maximum work
that can be extracted from a reversible closed
system at constant P . For a reversible process
at constant S and P , work stored as enthalpy can
be recovered completely.
entrainment Jets and plumes, moving
through fluid at rest, have the tendency to entrain
ambient fluid into the flow. The rate ∂Q/∂x
[m 2 s −1 ], at which the volume flow Q [m 3 s −1 ]
increases per unit distance x [m], is called the entrainment
rate. The most customary parameterization
of the entrainment rate is ∂Q/∂x = ERu
(Morton, 1959), where R and u are circumference
and velocity of the flow Q. The nondimensional
proportionality factor E is called
the entrainment coefficient, which increases as
the gradient Richardson number Ri decreases.
This implies that entrainment is more efficient
for large velocity differences and small density
differences. Often entrainment involves the
transport of one substance in another, such as
suspended particles in a current, or parcels of
moist air in dry winds.
entropy That thermodynamic potential defined
by the exact differential
dS = dQ/T ,
where dQ is a reversible transfer of heat in a
system, and T is the temperature at which the
transfer occurs. Entropy is conceptually associated
with disorder; the greater the entropy the
less ordered energy is available.
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