Examining the Validity of Approximations to Fully ThreeDimensional Shallow-Water Acoustic Propagation through
Nonlinear Internal Gravity Waves
Timothy F. Duda
Applied Ocean Physics and Engineering Department
Woods Hole Oceanographic Institution
Woods Hole, MA 02543 USA
Abstract – The anisotropic nature of most oceanic sound speed
structures, which have longer horizontal characteristic scale lengths
than vertical ones, means that horizontal refraction of sound can often
be neglected in computations. Cases of especially strong lateral
sound-speed variation, such as in steep nonlinear internal gravity
waves, where horizontal refraction effects are known to occur, can
often be effectively studied by considering the behavior of
adiabatically propagating vertical normal modes. However, this is not
always appropriate. Here, we describe propagation of low-frequency
(less than 1 kHz) sound on continental shelves in two situations where
mode coupling and horizontal mode refraction occur either
concurrently or in close proximity. These situations are propagation
through crossing or interacting nonlinear wave packets, and
propagation at low horizontal grazing angles into nonlinear internal
wave packets having very short horizontal scale lengths.
I.
INTRODUCTION
Propagating sound at frequencies below 1000 Hz is a useful
tool for remote sensing, detection, and signaling in the ocean,
under the condition that the water depth is greater than a few
wavelengths. This is because the attenuation per unit distance
is low. As the sound travels substantial distance in shallow
water, however, temporally and spatially variable bathymetry,
water column waves and fronts, and sub-bottom structures all
serve to reduce the predictability and temporal stability of
acoustic fields [1]. In addition to mean conditions, fluctuation
properties such as temporal and spatial coherence scales can
also be difficult to predict.
Studies of propagation processes that cause temporal and
spatial variability of the field are typically divided into two
categories: Those that exhibit horizontal deflection out of
vertical planes extending radially from a source (out-of-plane
effects), and thus require a three-dimensional treatment, and
those that do not have out-of-plane effects. Lately, some
treatments of the first situation have used an adiabatic (no
coupling) normal mode approximation, with horizontal
refraction and reflection effects of each mode treated
independently [2]. This is called the vertical-mode/horizontalray method. However, a handful of recent studies have
demonstrated that mode coupling is a very important
component of shallow-water low frequency propagation [e.g.
3-6], so that the adiabatic approximation is not always
appropriate.
Situations where both mode coupling and out-of-plane
effects are important are examined here. The intent is to show
that such situations exist, to begin to examine how they are
characterized, and to show our computational method for
dealing with situations where mode-coupling and out-of-plane
effects occur together and force a fully three-dimensional
treatment. We focus on joint coupling and horizontal effects in
nonlinear internal gravity waves. Another situation where
these effects can occur together would mode-coupling water
column structures (internal waves or fronts) over steeply
sloping bathymetry that would cause out-of plane effects.
Section II describes situations where mode coupling is
expected to occur. Section III shows situations where out-ofplane effects are important. Section IV describes some oceanic
situations where both mode refraction and coupling will occur.
Section V shows some numerical simulation results, and
Section VI is a summary of results.
II. COUPLED-MODE PROPAGATION
Coupled-mode acoustic propagation caused by watercolumn variability is probably common on continental shelves,
although only a few scientific experiments have made detailed
simultaneous measurements of mode arrivals and ocean
internal waves and fronts along the acoustic path for
verification of this [5,7,8]. The amount of energy that is
exchanged between modes is determined by the along-path
sound-speed derivative at depths where mode functions have
significant shape overlap [3,5,9].
A few computational studies of mode coupling caused by
internal waves are have been published in the last few years [4,
6, 10-12], building on earlier work. These papers show that
mode coupling can be major contributor to the high variability
of acoustic fields observed in a few recent experiments [5,8,1314]. Areas of continental shelves with water depths of 60 to
100 m and a water column characterized by sound-speed
contrast of 20 to 30 m/s across a thermocline of 10 to 20 m
thickness are usually found to support large populations of
nonlinear internal gravity waves. Because these waves are
steep they cause mode coupling. Permanent-form or quasipermanent form solutions of the mode-one nonlinear wave
equation can take the form of a single bump pointing away
from the vertical side of the water column having the strongest
density gradient, and toward the side with weaker gradient (e.g.
a near-surface thermocline will support nonlinear solitary
waves of depression) [15]. The stable wave-shape solution of
the KdV equation is well-known: η(x,t) = a sech2 [(x-ct)/L],
where c is wave speed and L is wave half-width. L is
commonly 80 to 100 m, with the entire wave occupying a few
hundred meters. Note that nonlinear internal waves grow
narrower with increasing amplitude, so that large waves are
particularly suited to causing mode coupling. Note that single
fronts can cause mode coupling, and that internal wave
coupling can sometimes be described by the effects of two
fronts, so an internal wave resonance condition that is
prominent in theory may sometimes play a role [e.g. 7], but
does not apply to all coupling situations..
Sound traveling in the same direction as the wave or wave
packet has the highest possible down-range gradient of sound
speed (the gradient responsible for coupling). We call this the
parallel case. For other sound propagation directions, the
sound-speed gradients in the direction of the sound are
stretched by a cosine factor. For situations within 60º of the
parallel case, coupling occurs which is similar to that of the
parallel case, but with reduced sound-speed gradients [3].
Let the angle between the internal wave propagation and
sound propagation directions be φ, so that φ = 0 is the parallel
case. The sound speed gradient in the propagation direction G
and the sound speed gradient in the internal wave direction G0
are related by G0 / G = cosφ . For φ near 90 degrees the
gradient can get very small, leading to adiabatic mode
conditions.
The division between mode-coupling and
adiabaticity in waves scaled to match observations lies
somewhere between 88° and 74°, based on simulations we
have performed with a fully 3D parabolic equation propagation
code. The code is described in Section V.
III. HORIZONTAL SOUND REFRACTION
The horizontal component of sound-speed gradient (or
water-depth gradient) that is normal to sound propagation will
cause horizontal refraction. Using normal modes, a broad-band
sound field can be described by the spectral components
Ψ(r,z,ω) = Σn ψn(r, z,ω) Pn(r,ω), where r is horizontal position,
z is depth, ω is frequency, ψn is vertical shape of normal mode
n, and Pn is mode amplitude. The total sound speed field can
be arrived under adiabatic conditions by tracing rays associated
with each mode using modal group velocities [2]. Clearly,
lateral (out-of-plane) effects of each mode can be computed
and analyzed, as well as the total effect. This is the verticalmode/horizontal-ray method.
Horizontal refraction of sound is currently being investigated
by a number of groups. Badiey et al. [2] show examples of
this effect within typical internal waves. Their results suggest
that sound can be trapped between waves when 90º > |φ| > 82º,
with refraction possibly occurring for lesser |φ| < 82º. It is
possible that refraction and coupling can occur together in the
ocean for very strong waves at φ of 80 to 85 degrees.
Fig. 1. SEASAT SAR image showing crossing internal wave fronts. The
image was taken 31 August 1978 @ 0240 UTC. The typical interval between
waves is about 500 m, and the image width is ~30 km. Data were digitally
processed at CSTARS. © 2005 CSTARS-University of Miami / NASA
IV. INTERNAL WAVE OBSERVATIONS
The results in the previous two sections were obtained for
conditions of isolated internal waves with long, straight wave
crests. Oceanic observations don’t always show waves fitting
that mold, however. Figure 1 shows internal waves on the
continental shelf east of the Middle Atlantic United States.
The image is bright in regions of surface current convergence
on the front side of the internal waves, and dark in smooth
areas of low backscatter caused by diverging surface currents
on the back side of waves. Curved wave crests can be seen.
Note the crossing wave fronts in the center of the image.
Within the crossing waves, sound with low angle |φ| < 60º with
respect to one set of waves (the mode-coupling situation), may
have |φ| > ~80º for the other wave (the refractive situation). In
this zone, the theories in each of the two previous sections can
not be applied because the coupling violates the adiabatic
condition of the normal-mode refraction theory, and the
Fig. 2. Results for 200-Hz propagation through curved internal-wave pair. Depth-averaged intensity times distance from source is shown in dB with an
arbitrary reference. This format eliminates effects of cylindrical spreading. The water depth is 80 m, away from the wave the thermocline with a uniform
gradient lies between 15 and 30 m depth, separating the upper layer with 1520 m/s sound speed from the uniform-gradient lower layer with 1484 m/s sound
speed at its upper limit and 1481 at the seafloor. The waves are sech-squared depression waves with a = 15 and 12 m (upper, lower), L = 77 and 86 m. The
sound from the beam-like source, emitted with y-symmetry, is refracted by the wave pair. Some sound is trapped in the low mode-speed region between the
two waves. Most of the energy is refracted sharply by the lower wave close to the source and then passes through the upper wave because the incident angle
has changed in a way that allows the sound to escape.
refraction can alter mode phases that control coupling [4].
This situation requires a fully three-dimensional propagation
theory or model. There may be some source-receiver
geometries where horizontal refraction may be negligible, but
the complete solution would require mode coupling and
horizontal refraction to be treated together. Results obtained
with a computational model capable of this are shown in the
next section.
V. ACOUSTIC FIELD COMPUTATIONS
Concurrent mode coupling and out-of-plane refraction can
be readily modeled using a Cartesian-coordinate threedimensional parabolic equation solver. Our implementation of
this is described in a recent report [16]. Some results obtained
with it are also shown, with additional results described. Fully
three-dimensional propagation studies of sound propagation in
internal waves have recently been published that uses a code
having cylindrical coordinates. That implementation has
horizontal resolution which decreases at increasing distance
from the source [6,12]. The cylindrical and Cartesian
geometries each have limitations on what can be modeled
effectively.
The code is a simple extension of the classic split-step
Fourier solution used since the mid seventies [17]. Typically,
the acoustic field in a plane is solved for using onedimensional Fourier transforms. The new code simply uses
two-dimensional transforms to compute the field in a volume.
The code shared many features with a code implemented for
optical scattering studies [18-19], with the notable exception
that the new code uses the wide-angle operator [20].
Figure 2 shows depth-averaged intensity from a sound
source at 40-m depth in 80-m of water placed between two
sech-squared solitary waves. Refraction of sound away from
the waves’ centers, which have fast acoustic mode phase
velocities, can be observed. The waves have a prism-like effect
previously described for variable bathymetry [21], with
differential refraction of the modes causing them to diverge.
The beams show refractive effects that are easy to detect when
a sound source is placed in a slow-mode speed zone between
two waves.
Clearly, the presence of a crossing wave or wave packet with
| φ | in the range of zero to 70°, crossing the high-|φ| waves of
the model, would cause mode-coupling. This would have a
strong effect on the resultant field. For example, changing the
mode content in any of the beams would change the effects of
mode stripping on transmission loss versus range curves [22].
When the sound source is not between two waves, it is still
possible for sound to enter waves at |φ| near 90°, but refraction
might be subtle and difficult to detect. The presence of
horizontal refraction in situations like that can be found by
running the code in two ways, then comparing the results.
First, the code can be run with a wave of any geometry. Next,
the code is run with the sound-speed field of the wave at one
azimuth (usually taken to be along the line y=ys, where ys is the
source position) placed at all angles. The second run will not
have refractive effects, so refraction occurring along the line in
the original rum can be detected by comparing the fields in the
vertical plane along the line. The report shows refractive
effects for |φ| ~ 86° for one particular wave choice. Clearly
such a procedure is not required for refraction as pronounced
Detecting mode coupling has proven to be mode difficult
unless the coupling is very pronounced. Performing modefiltering using normal modes found computationally using
readily available software indicates that the code causes mode
energies to oscillate. This may be caused by using incorrect
mode shapes, or by approximations inherent in the split-step
solver. Very strong mode coupling can be verified using this
technique, as shown in the report [16], but weak coupling is
difficult to detect.
Despite the difficulty, the model can be used to study
whether out-of-plane effects and mode coupling can occur
together in a single wave, at appropriate angles |φ| of 80 to 85°.
Figure 3 shows depth-averaged intensity for one three-
Fig. 3. Results for 400-Hz propagation through a single internal wave at the high angle of φ = 81.7º. Depth-integrated intensity times distance from source is
shown, in dB with an arbitrary reference. The center of the internal wave is shown by the line. The wave is a 20-m amplitude, 70-m half-width sech-squared
solitary wave of depression in 80-m water depth, riding on a thermocline as in the Figure 2 simulation. White dashed radial lines from the source are shown for
reference. The y-dimension stretched relative to x. The azimuthal variation of sound energy in the lower part of the figure indicates mode coupling. The zone
of high-intensity at positive x on the source side of the wave indicates horizontal refraction.
dimensional simulation of this type, with |φ| of about 82°. The
high-intensity beams that extend outward from the source in
the lower part of the figure are commonly seen in simulations
of this type. They are caused by azimuthally variable mode
mode coupling and are commonly seen in our simulations.
The azimuthal dependence on coupling arises because the
sound enters the wave at a range that varies with azimuth, so
that phase differences between modes, which control coupling
[22], also vary with azimuth. Horizontal refraction of sound is
evident on the source side of the wave. Out-of-vertical-plane
effects are the only explanation for the observed effect of
intensity increasing with range along the radials that cross the
refracted beam of energy. The wave in the simulation is at
maximum limit of waves observed in a summer 2006
experiment in the Mid-Atlantic Bight east of the Unites States.
VI. SUMMARY
In shallow-water ocean areas populated with nonlinear
internal waves, there are at least two situations where neither
the adiabatic treatment of horizontally refracting sound, nor a
mode-coupling treatment of radially propagating sound are
sufficient at describing the effects of the waves on sound. One
is the intuitively understood case of crossing internal waves,
where one wave (or wave packet) is oriented to propagate with
the sound and cause strong coupling, and the other is oriented
to propagate near normally to the sound, causing string sound
refraction. Satellite images occasionally show crossing wave
behavior in the ocean.
The other situation is near oblique incidence of sound with
internal waves of very high steepness, as illustrated in Figure 3.
The strong horizontal gradients of sound speed normal to the
direction of sound propagation cause the sound to refract.
Likewise, the strong horizontal gradients of sound speed in the
direction of propagation cause mode coupling. These both
occur only in very steep waves; however, waves of sufficient
steepness are occasionally observed. The horizontal effects are
often understood in terms of variations on mode phase
velocities, causing adiabatic modes to refract, but the fact that
modes are not adiabatic complicates the situation and
compromises the accuracy of mode-by-mode calculations.
ACKNOWLEDGMENT
Supporting grants from the Office of Naval Research are
acknowledged. Thanks are given to Prof. Hans Graber of the
University of Miami for providing the satellite image.
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