Aerospace and Electronic Systems Magazine December 2017 - 15
Anderson
The relevance of this to HF radar lies in the fact
that the expansion of the Hamiltonian into a series of
terms of progressively higher order in the canonical
variables enables us to associate specific phenomenology with the individual terms. In the usual expansion, H0 is the energy spectrum as a sum of free
waves, while H1 represents the nonlinear interaction
of two free waves to form a third wave; as the resulting "sum" and "difference" waves do not satisfy the
free wave dispersion relation, this term of the Hamiltonian does not redistribute wave energy among the
wave spectrum components, and it can be removed
by a canonical transformation, simplifying some
computations but perhaps taking us further from the
HF radar perspective. In fact, the "bound" waves
embodied in H1 present a distinctive signature in the
Doppler spectrum, and this has been exploited for cyclone mapping, wave growth studies, current profiling, and other purposes. H2 embodies the interaction
of four waves, which typically manifests itself as the
flux of energy into an under-represented wave by the
collective generosity of three donor waves. It is this
Figure 15.
mechanism that is incorporated in the action-balance
An example of a transient window of opportunity for substantially increased ship detectability during a period of changing wind stress. Epoch here is measured in units of about
equation (ABE) which is widely used in operational
2 min duration.
oceanography to describe the evolution of oceanic
wave fields, but here a cautionary note: as pointed
out by Annenkov and Shrira [16], the ABE as used
in
common
practice
is itself incomplete, not accounting properly
= η,t). One can write the energy density of the wave system
for the nonadiabatic wave field development under rapidly changdirectly in the form
ing conditions. This has significant implications for ship detection
2
2
g
with HF radar, as evident in Figure 15 which shows an example
2
H dz dx
(5)
dx
where ship echo visibility is enhanced by roughly 15 dB under such
z 2
D
circumstances [17]. The three panels in the figure show (a) the temporal evolution of the Doppler spectrum for adiabatic evolution, (b)
The kinematic and dynamic boundary conditions at the surface are
the temporal evolution of the Doppler spectrum for nonadiabatic
expressed by Hamilton's equations,
evolution, and (c) the gain in ship detectability in the latter case.
This is by no means an extreme case, so there is a strong message
∂η δ H
∂ψ
δH
(6)
=
;
=−
for radar task scheduling. The astute radar operator will seize win∂t δψ
∂t
δη
dows of opportunity whenever they arise.
There are several other physical processes that are associated
δH δH
,
with
H3 and H4 and which have observable signatures at HF. Perare functional derivatives, but at this point H is
where
δψ δη
haps the most remarkable are the related phenomena of modualso a function of the velocity potential, ϕ.
lational instability and Fermi-Pasta-Ulam recurrence, which first
Now, ϕ satisfies Laplace's equation,
emerge at H3. Under conditions that prevail in HF radar surveil2
lance,
the primary ocean waves on the sea surface are predicted to
∂φ
(7)
∇ 2φ + 2 = 0
be
unstable
to sideband frequencies to which energy leaks, reduc∂z
ing the amplitude of the primary wave, such as the wave that, in a
with the upper and lower boundary conditions
given instance, might be the one responsible for Bragg scatter to
an HF radar. Experimental observations, such as the one shown in
∂φ ( x, z )
= 0,
φ ( x, z = η ) = ψ ;
z = −D
Figure 16, reveal that a pronounced energy transfer does indeed
(8)
∂z
occur, and that the energy loss can achieve close to the theoretical limit of almost 10 dB. Of course, in this experiment, without
A solution for ϕ, which we might express in the form of a power
a comprehensive deployment of in situ sensors, we cannot assert
series in some small parameter such as wave steepness, can thus be
unequivocally that the Benjamin-Feir instability is responsible for
written in terms of η and ψ. H is now expressible entirely in terms
the observed modulation. In support of the hypothesis, the proof the canonical variables, whose temporal variation is then given
cess is predicted to be cyclic, a manifestation of Fermi-Pasta-Ulan
by Hamilton's equations.
DECEMBER 2017
IEEE A&E SYSTEMS MAGAZINE
13
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