EM 1110-2-1100 (Part II)
(Change 1) 31 July 2003
effects would begin to be significant. This region of saturated energy densities is termed the equilibrium
range of the spectrum.
(9) Kitaigorodskii (1962) extended the similarity arguments of Phillips to distinct regions throughout
the entire spectrum where different mechanisms might be of dominant importance. Pierson and Moskowitz
(1964) followed the dimensional arguments of Phillips and supplemented these arguments, with relationships
derived from measurements at sea. They extended the form of Phillips spectrum to the classical Pierson-
Moskowitz spectrum
&4
α g 2 f &5
f
exp 0.74
E(f) '
(II-2-32)
fu
(2 π)4
where
speed)
(10) Based on these concepts of spectral wave growth due to wind inputs via Miles-Phillips mechanisms
and a universal limiting form for spectral densities, first-generation (1G) wave models in the United States
were born (Inoue 1967, Bunting 1970). It should be pointed out here that the first model of this type was
actually developed in France (Gelci, Cazale, and Vassel 1957); however, that model did not incorporate the
limiting Pierson-Moskowitz spectral form as did models in the United States. In these models, it was
recognized that waves in nature are not only made up of an infinite (continuous) sum of infinitesimal wave
components at different frequencies but that each frequency component is made up of an infinite (continuous)
sum of wave components travelling in different directions. Thus, when waves travel outward from a storm,
a single "wave train" moving in one direction does not emerge. Instead, directional wave spectra spread out
in different directions and disperse due to differing group velocities associated with different frequencies.
This behavior cannot be modeled properly in parametric (significant wave height) models and understanding
of this behavior formed the basic motivation to model all wave components in a spectrum individually. The
term discrete-spectral model has since been employed to describe models that include calculations of each
separate (frequency-direction) wave component. The equation governing the energy balance in such models
is sometimes termed the radiative transfer equation and can be written as
' &cG L E(f, θ, x, y, t) % j S(f, θ, x, y, t)k
K
ME(f, θ, x, y, t)
P
(II-2-33)
P
Mt
k'1
where
E(f,θ,x,y,t) = spectral energy density as a function of frequency (f), propagation direction (θ), two
horizontal spatial coordinates (x and y) and time (t)
S(f,θ,x,y,t)k = the kth source term, which exists in the same five dimensions as the energy density
The first term on the right side of this equation represents the effects of wave propagation on the wave field.
The second term represents the effects of all processes that add energy to or remove energy from a particular
frequency and direction component at a fixed point at a given time.
(11) In the late 1960's evidence of spectral behavior began to emerge which suggested that the
equilibrium range in wave spectra did not have a universal value for α. Instead, it was observed that α varied
II-2-40
Meteorology and Wave Climate
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