DICTIONARY OF GEOPHYSICS, ASTROPHYSICS, and ASTRONOMY

inferior conjunction
the figure) in the inertial subrange is given by
Kolmogorov’s “k −5/3 ”law
(k) =Aɛ 2/3 k −5/3
where k is the wavenumber and ɛ 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.
normalized turbulence spectrum
log Φ
1
2
3
4
5
0
5
k 5/3
inertial subrange
equilibrium range
4 3 2 1
log (k/k )
η
dissipation
range
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
{q k ,pl} with Hamiltonian H(q k ,pl ,t)
and new canonical variables {Q k ,Pm} with
Hamiltonian K(Q m ,Pn,t)is infinitesimal, so
that squares of differences can be neglected.
A basic form of canonical transformation postulates
a generating function F(q k ,Pl,t) and
© 2001 by CRC Press LLC
242
0
then solves
pk = ∂F
∂qk Q l = ∂F
∂Pl
K = ∂F
∂t +H
Notice that F =qkPk (summed on k) generates
the identity transformation: Qk =qk ,Pl=pl
via the above equation.
The generator for an infinitesimal canonical
transformation is then:
F=q k
Pk+ɛG q k
,Pm,t
where ɛ is a small parameter (so that ɛ2 terms
are ignored). From the previous equations,
pk = Pk +ɛ ∂G
∂qk Q l = q l +ɛ ∂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 ɛ,G(qk ,Pl,t)
differs only at first order in ɛ from G(qk ,pl,t)
(the same function, evaluated at a slightly different
argument). Hence we can write, to first
order in ɛ,
Pk =pk −ɛ ∂
∂q k
G q l
,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):
3
˙a
a
2
= 8πGρ
c 2
k
− +
a2