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inertial subrange

inelastic collision Collision between two or more bodies in which there is loss of kinetic energy.

inelastic scattering of radiation Scattering in which the wavelength of radiation changes because radiant energy is transferred to the scatterer.

inertia In Newtonian physics, the tendency of a material object to remain at rest, or in a state of uniform motion.

inertia coefﬁcient Also referred to as a mass coefﬁcient; appears in the Morison Equation for description of wave-induced force on a vertical pile or cylinder. Denotes the force that arises due to the acceleration of the ﬂuid around the cylinder. A second term denotes the force due to the square of the instantaneous velocity and includes a drag coefﬁcient.

inertial-convective subrange For wave numbers k well below LB −1, the molecular diffusivity of heat or salt does not inﬂuence the spectrum very much, and so the spectra are similar to the velocity spectrum E(k), which falls off proportional to k−5/3 (inertial-convective subrange). For smaller scales, velocity ﬂuctuations are reduced progressively by viscosity, but the diffusivity of heat or salt is not yet effective (viscous-convective subrange).

inertial coordinate system An unaccelerated coordinate system in which the laws of Newton and the laws of special relativity hold without correction. In the absence of gravity, this coordinate system can be extended to arbitrarily large distances. In the presence of gravity, such a coordinate system can be erected locally, but cannot be extended beyond lengths

corresponding to the typical tidal scale of r =

√

c/ Gρ, where c is the speed of light, G is Newton’s gravitational constant, and ρ is a measure of the matter density.

inertial frequency When waves are long compared with the Rossby radius, the frequency is approximately constant and equal to the Coriolis parameter, f , or twice the Earth’s rotation rate. In this limit, gravity has no effect, so ﬂuid

particles are moving under their own inertia. Thus, f is often called the inertial frequency. The corresponding motion is called the inertial motion or inertial oscillation; the paths are called inertial circles. Likewise the wave with this frequency is known as the inertial wave.

inertial instability The instability that occurs when a parcel of ﬂuid is displaced radially in an axisymmetric vortex with negative (positive) absolute vorticity (planetary vorticity plus relative vorticity) in the northern (southern) hemisphere.

inertial mass The mass that opposes motion. In general, for small accelerations, a = F/m, where m is the inertial mass. The inertial mass is usually contrasted to the passive gravitational mass, which is a factor in the Newtonian force law, and to the active gravitational mass, which generates the gravitational ﬁeld. In Newtonian gravitation, and in general relativity, all these masses are proportional, and are set equal by convention.

inertial oscillation A ﬂuid particle with an initial velocity v but free of force in the Northern Hemisphere will be bent by the inertial Coriolis force of magnitude 2 sin θ to its right, where is the angular velocity of Earth rotation around the North Pole and θ is the local latitude. In the absence of background currents, the particle’s trajectory is a circle with a radius of v/2 sin θ. The particle returns to its original position in 1/(2 sin θ) days (inertial period). Inertial oscillations are often observed in the ocean after strong wind events like hurricanes.

inertial subrange Part of the turbulent kinetic energy spectrum where turbulent kinetic energy is neither produced nor dissipated by molecular diffusion, but only transferred from larger to shorter length scales by initial forces (see also turbulence cascade). Wavenumbers in this part of the spectrum are much larger than the energy containing scales of turbulence and shorter than the Kolmogorov wavenumber kη = /ν3 1/4 at which kinetic energy is dissipated into heat. For sufﬁciently high Reynolds numbers, this part of the spectrum is nearly isotropic and is independent of molecular viscosity. The shape of the energy spectrum (see

inferior conjunction

the ﬁgure) in the inertial subrange is given by Kolmogorov’s “k−5/3” law

(k) = A 2/3k−5/3

where k is the wavenumber and is the dissipation rate of turbulent kinetic energy. The parameter A is a universal constant that is valid for all turbulent ﬂows, and A 0.3 for the streamwise components of (k) and A 1.5 for the cross-stream components.

spectrum		0
spectrum		1				k5/3
		1

turbulence	logΦ	2
turbulence	logΦ	3
normalized		3
normalized		5
		4						dissipation
				inertial subrange					range
					equilibrium range
					equilibrium range

		5	4	3		2	1	0

log (k/kη)

Typical wavenumber spectrum observed in the ocean

plotted against the Kolmogorov wavenumber.

inferior conjunction

See conjunction.

inferior mirage A spurious image of an object formed below its true position by atmospheric refraction when temperature decreases strongly with height. See superior mirage.

inﬁnitesimal canonical transformation In classical mechanics, a canonical transformation in which the change between old canonical variables {qk , pl} with Hamiltonian H(qk , pl , t) and new canonical variables {Qk , Pm} with Hamiltonian K(Qm , Pn , t) is inﬁnitesimal, so that squares of differences can be neglected. A basic form of canonical transformation postulates a generating function F(qk , Pl , t) and

then solves
	pk		=		∂F

				∂qk
	Ql		=		∂F

				∂Pl
	K		=	∂F		+ H

					∂t
Notice that F	=	qk	Pk	(summed on k) generates
	=		Pk			Q	k	= q	k	, Pl = pl
the identity transformation:						Q		= q		, Pl = pl

via the above equation.

The generator for an inﬁnitesimal canonical transformation is then:

F = qkPk + G qk , Pm , t

where is a small parameter (so that 2 terms are ignored). From the previous equations,


pk	=	Pk +		∂G

				∂qk
Ql	=	ql +	∂G
					.
			∂Pl

The terms on the right involve derivatives of G, which are functions of qk , Pl , and t. However, because of the small parameter , G(qk , Pl , t) differs only at ﬁrst order in from G(qk , pl , t) (the same function, evaluated at a slightly different argument). Hence we can write, to ﬁrst order in ,

∂

Pk = pk − ∂qk G ql , pn , t .

This is an inﬁnitesimal canonical transformation, and G is the generator of the inﬁnitesimal canonical transformation. Compared to the general form, it has the great advantage of being explicit. A full nonlinear transformation can be carried out by integrating inﬁnitesimal transformations. See canonical transformation, Hamilton–Jacobi Theory.

inﬂation In cosmology, a period of rapid universal expansion driven by a matter source whose energy density falls off slowly in time or not at all. The simplest example appears in the behavior of the scale a(t) for an isotropic, homogeneous universe (a cosmology modeled by a Robertson–Walker cosmology):

	a	2		πGρ		k
3	˙		=	8	−		+ *
3	a		=	c2	−	a2	+ *

8πGρ

inhomogeneous models (of the universe)

(G = Newton’s constant; c = speed of light; dot indicates proper time derivative) where ρ is energy density associated with “ordinary” matter (thus ρ decreases as the universe expands), −ak2 reﬂects the global topology (k is a parameter,

k= −1, 0, 1); * is the cosmological constant, which does not depend on the size scale, a, of the universe. As the universe expands, * dominates the right-hand side of this equation for

k= 0, −1; if * is large enough it also dominates the solution for late times for k = +1. In these cases at late times, one has

a exp	3 t .
	*

This exponential growth is called inﬂation. As originally posited by Einstein, * is precisely constant, and observationally at present, neither * nor the term ak2 is greater than order

of magnitude of 8πGρc2 . In modern theories of quantum ﬁelds, however, those ﬁelds can produce long-lived states with essentially constant (and very large) energy density, mimicking a *. In 1979 Guth noted that those states can drive long periods of inﬂation in the very early universe, then undergo a transition to more normal matter behavior. (Hence, * → in the transition called “reheating”.) If the inﬂation exceeds 60 e-foldings, then the ratio of ρ to the −a2k term after reheating becomes consistent with an evolution to the current observed state. This explains why the universe appears nearly “ﬂat” (k = 0) now, solving the “ﬂatness problem”. Similarly, inﬂation blows the size of some very small causally connected region up to a size larger than the current observable universe. This has the effect of suppressing large initial inhomogeneities. As similar by-products of the accelerated expansion, all topological defects (and monopoles in particular) formed before the inﬂationary phase are diluted in such a way that their remnant density, instead of overﬁlling the universe, becomes negligible. At the same time, small-scale quantum ﬂuctuations are spread out by this inﬂation. With correct choice of the inﬂation parameters, one can achieve reasonable consistency with the ﬂuctuations necessary to create the observed large scale structure of the universe. This restriction of parameters could be viewed as restricting the physics at very

early times in the universe. With typical such parameter theories, the period of inﬂation occurred during the ﬁrst 10−35 sec after the Big Bang at temperatures corresponding to very high particle energies, 1014 GeV. See also monopole excess problem, Robertson–Walker cosmological models.

infragravity wave A water gravity wave with a period in the range of 20 sec to 5 min.

infrared Referring to that invisible part of the electromagnetic spectrum with radiation of wavelength slightly longer than red-colored visible light.

infrasound “Sound” waves of frequency below 20 Hz (hence inaudible to humans).

inherent optical property (IOP) In oceanography, any optical quantity that depends only on the properties of the water and is independent of the ambient light ﬁeld; examples include the absorption coefﬁcient, the scattering coefﬁcient, and the beam attenuation coefﬁcient; apparent optical properties become inherent optical properties if the radiance distribution is asymptotic.

inhomogeneous models (of the universe)

Cosmological models that do not obey the cosmological principle. Several such models have been derived from Einstein’s equations, starting as early as 1933 with the solutions found by Lemaître (often called the “Tolman–Bondi” model) and McVittie. The research in this direction was partly motivated by the intellectual challenge to explore general relativity and to go beyond the very simple Robertson– Walker cosmological models, but also by the problems offered by observational astronomy. The latter include the creation and evolution of the large-scale matter-distribution (in particular the voids), the recently discovered anisotropies in the microwave background radiation, also gravitational lenses that cannot exist in the Robertson–Walker models. The more interesting effects predicted by inhomogeneous models are:

1. The Big Bang is, in general, not a single event in spacetime, but a process extended in

initial condition

time. Different regions of the universe may be of different age.

2. The curvature index k of the Robertson– Walker models is, in general, not a constant, but a function of position. Hence, some parts of the universe may go on expanding forever, while some regions may collapse in a ﬁnite time to the ﬁnal singularity.

+(m) m−,, where m is the mass of the star. The index , may vary for different mass ranges, but it is always positive, implying that high mass stars are formed less frequently than low mass stars. According to E.E. Salpeter, +(m) m−2.35, for all masses. From this law, we expect that for one star of 20 solar masses (M ) ≈ 1000 stars of 1M are formed. According to G.E.

3.The spatially homogeneous and isoMiller and J.M. Scalo, the IMF valid for the solar

tropic models (see homogeneity, isotropy) are unstable with respect to the formation of condensations and voids. Hence, the formation of clusters of galaxies and voids is a natural phenomenon rather than a problem.

4. An arbitrarily small electric charge will prevent the collapse to the ﬁnal singularity. This also applies to the initial singularity — the model with charge, extended backward in time, has no Big Bang. The charge may be spread over all matter (then it has to be sufﬁciently small compared to the matter density) or concentrated in a small volume; the result holds in both cases. Reference: A. Krasinski,´ Inhomogeneous Cosmological Models. Cambridge University Press, 1997.

initial condition A boundary condition applied to hyperbolic systems at an initial instant of time; or for ordinary differential equations (one independent variable) applied at one end of the domain. For more complicated situations, for instance, in theories like general relativity one may specify the state of the gravitational ﬁeld and its derivative (which must also satisfy some consistency conditions) at one instant of time. These are the initial conditions. One obtains the later behavior by integrating forward in time.

initial data In general relativity, see con- straint equations.

initial mass function (IMF) The distribution of newly formed stars as a function of mass. The initial mass function is estimated from the photometric and spectroscopic properties of stars in open clusters and associations of stars. Ideally, the IMF can be measured counting the stars of each spectral type in an association of stars so young that the shortest-lived massive stars are still in the main sequence. The initial mass function is usually assumed to be of the form

neighborhood can be approximated as +(m)

m−1.4, for 0.1 < m

, +(m)

m−2.5,

for 1

3.5

, for

10M

, and +(m)

m−

10M . This law predicts fewer high mass

stars (m 10M ) for a given number of solar

mass stars than Salpeter’s law.

injection boundary

A line in the nightside

equatorial magnetosphere along which dispersionless plasma injections appear to be generated.

inner core The Earth’s inner core has a radius of 1215 km, and is solid. The inner core is primarily iron and its radius is growing as the Earth cools. As the outer core solidiﬁes to join the inner core, elements dissolved in the outer core are exsolved. The heat of fusion from the solidiﬁcation of the inner core provides a temperature gradient, and convection, and the ascent of these light exsolved components are major sources of energy to drive the geodynamo.

inner radiation belt A region of trapped protons in the Earth’s magnetosphere, typically crossing the equator at a geocentric distance of 1.2 to 1.8 RE (Earth radii), with energies around 5 to 50 MeV. The inner belt is dense enough to cause radiation damage to satellites that pass through it, gradually degrading their solar cells. Manned space ﬂights stay below the belt.

The 1958 discovery of the inner belt was the ﬁrst major achievement of Earth satellites. The belt seems to originate in the neutron albedo, in secondary neutrons from the collisions between cosmic ray ions and nuclei in the high atmosphere. The trapping of its particles seems very stable, enabling them to accumulate over many years. The planet Saturn also seems to have an inner belt, with the neutrons coming from cosmic ray collisions with the planet’s rings.

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