Large internal solitary waves have been observed by Shuttle SIR-A and SIR-B at locations in the Andaman Sea and the New York Bight. Satellite imagery and oceanographic measurements are used in conjunction with hydrodynamic interaction and electromagnetic scattering models to estimate the expected SAR image intensity modulations associated with the internal waves. There is reasonable agreement between the predicted and observed internal wave signatures

Apel, J. R.

Gasparovic, R. F.

Thompson, D. R.

English

NASA Technical Reports Server

N87-17147
HYDRODYNAMICS OF INTERNAL SOLITONS AND A COMPARISON OF
SIR-A AND SIR-B DATA WITH OCEAN MEASUREMENTS
J. R. Apel, R. F. Gasparovic and D. R. Thompson
The Johns Hopkins University
Applied Physics Laboratory
Laurel, Maryland
ABSTRACT
Large internal solitary waves have been observed by Shuttle SlR-A
and SIR-B at locations in the Andaman Sea and the New York Bight. Satellite
imagery and oceanographic measurements are used in conjunction with hydro-
dynamic interaction and electromagnetic scattering models to estimate the
expected SAR image intensity modulations associated with the internal waves.
There is reasonable agreement between the predicted and observed internal wave
signatures.
I. ANDAMANSEA SOLITONS
The Andaman Sea is located along the eastern side of the Indian
Ocean between the Malay Peninsula and the Andaman and Nicobar Islands. It is
a maritime regime containing internal solitary waves of extraordinary ampli-
tude, wavelength and speed that exhibit signatures via surface roughness
variations that are visible to the eye, to cameras, to scanning imagers, and
to imaging radars. Internal solitary waves, or solitons, are generally waves
of depression, more or less isolated from their fellows, that propagate over
great distances with very little change in shape. Several satellite images
have revealed that the Andaman Sea is rich in such excitations. The joint
US-USSR space mission, Apollo-Soyuz, returned color photographs that showed
narrow stripes of rough water in the sun glint region. These were interpreted
by Apel (1979) as surface signatures of large internal waves, presumably
produced by semidiurnal tidal flow over underwater topography; the inferred
group speeds were approximately 2.5 m/s, as derived from their interpacket
separation. In addition, examinations of visible wavelength imagery from the
scanner on the Defense Meteorological Satellite Program (DMSP) spacecraft
revealed similar surface signatures on several dates. Synthetic aperture
radar images from the Shuttle Imaging Radar-A (SIR-A) on November Ii, 1981,
also contain several internal wave packets in the vicinity of the Andaman and
Nicobar Islands (Cimino et al., 1982; Ford et al., 1983; Apel et al., 1986).
Figure I is a section of the SIR-A radar image of that date, having
dimensions of 51.2 by 51.2 km, with the white marks along the lower border
being l-s time tics (or a distance .of 7.14 km). North is along the direction
shown, at an angle of approximately 38° from the lower edge. It is a high
resolution (40-m), geographically corrected representation of the surface wave
roughness variations and has been selected for further analysis because of the
qualities that the internal waves possess. The objectives of the analysis are:
(a) to use the historical density data in conjunction with the known nonlinear
properties of internal solitons to estimate the amplitudes, currents, strain
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rates, and propagation speeds from the SIR-A image; (b) to estimate wind speed
and direction from the image; and (c) to apply theories of radar backscatter
modulation to the deduced hydrodynamic and wind properties, in order to cal-
culate the radar signatures. The signatures will then be compared with image
intensity modulations.
The internal solitons are revealed mostly by dark (smooth) regions,
with the associated rough regions not clearly visible, probably due to the
large, 43 ° earth-incidence angle of the synthetic aperture radar. The approx-
imately semicircular dark region near right center is thought to be an area of
rainfall toward which the local winds are blowing, as revealed by subtle wind
row streaks overlying portions of the waves; the wind direction is nearly from
the west. The wind speed is estimated to be about 3 to 4 m/s, based on the
general level of backscatter and the visibility of the waves, but the error in
this estimate is easily ±i00 percent.
The hydrodynamic properties are estimated from the image and from
the historical density data by the application of a "modulated cnoidal wave"
model of the solitary waves (Apel and Gonzales, 1983). In the Korteweg-deVries
equation describing such solitons, a Mode-i solitary wave train can be written
as the product of an eigen function, W(z), giving the relative amplitude vari-
ation with depth, z, and a horizontal amplitude function, A(s,t), describing
propagation along a horizontal arc, s(x,y), in the x,y plane. Thus, we write
for the amplitude, _,
_7(x,y,z,t) = W.(.Z) A(x,y,t) . (l)
The equations obeyed by the separation functions W and A are
d2 W [ N2(z) ]dz z + k 2 °: 2 1 W = O,
w(o) = W(-H) = O, (2)
(where the Ocean bottom is at the depth -H), and the K-dV equation,
-- +Co +_' _-:--v + aA- = O.
Ot Os
(3)
Here, _ = _(k£) is the dispersion equation relating wave angular frequency, _,
to linear wave number, k£; the quantities, N, y, and = are environmental par-
ameters termed the Brunt-V_is_l_ (or buoyancy) frequency, the dispersion
coefficient, and the nonlinear coefficient, respectively. These may all be
calculated from a knowledge of the variation of density with depth.
92
A useful approximate solution to the K-dV equation describing a
boundedwave packet may be written as a modulated cnoidal function, wherein a
propagating envelope function, no(S-Ct), moving at speed, c, is assumed to
multiply the periodic cnoid, just as a modulated cosine maybe used to describe
a propagating linear wave packet. In this fashion, the varying amplitudes and
degrees of nonlinearity existing within a group of waves such as those in
Fig. i maybe modeled without invoking the full apparatus of inverse scattering
theory in the analysis. Algebraic relationships exist between the stretched
wavelength, %s, and k_, m, no, and c, and can be used to estimate the amplitude
and speed if the other quantities can be evaluated. These relationships are
xs = • - _® (4)
7r12 7r12 '
_o _o
m - , (5)
½ 7k_ c- Co
CO/o
= ce + co
3
(6)
In Eq. 4, _ = 2_/k_ is the small-amplitude, linear wavelength and K(m) is the
complete elliptic integral of the first kind, which for m=0 reduces to _/2,
while for m + i, K(m) + m. In Eq. 6, the speed c is given in terms of the
long-wavelength speed co for which k_0. The speed of a solitary wave is in-
creased by the nonlinear amplitude-dependent term in Eq. 6; thus, larger
solitons travel faster than smaller ones, all else being equal.
Using these quantities, the modulated cnoidal wave solution can be
written as
n(x,y,z,t) = - W(Z)no (s - ct)
× cn_ [V2(kt.s-oJt + _)].
(7)
To apply Eqs. 4 to 7 to the case at hand, the image of Fig. 1 is
used to evaluate (a) X_ from the wavelength at the rear of the wave packet, and
(b) the stretched wavelength, %s, from the spacings of the other waves
throughout the packet. The ratio Xs/X_ is used to calculate the values of
K(m) and m from Eq. 4; from Eq. 5 one then derives no.
Having obtained the wave amplitudes, the known hydrodynamics of the
waves may be used to obtain the currents, u and w, and the horizontal strain
rate, au/as, as well as the other desired properties.
93
In order to evaluate the equations numerically, functional forms of
W(z) and its derivative are required; these are obtained via numerical solu-
tions to Eq. 2. This in turn requires a vertical profile of N2(z) in terms
of water density, p:
N2(Z) = 1 "g" dp (8)
p dz
A temperature profile was obtained from Osborne et al. for the Andaman Sea
(Osborne and Butch, 1980) and used in this calculation.
Figure 2 illustrates the estimated surface current and surface
strain rates versus distance, s - ct, throughout approximately 40 km of the
packet--1Thes peak currents are about 1.1 m/s, the strain rates are 1.3 xl0 -3 and peak amplitudes (not shown) are approximately 60 m -
reasonable values in light of the measurements made in the Andaman to date
(Osborne and Burch, 1980).
Figure 3 shows a comparison between the measured intensity modula-
tion along the cut through the SlR-A image of Fig. i and the predicted modula-
tion from our wave-current interaction model. One can see that the predicted
modulation depth agrees fairly well with that measured from the synthetic
aperture radar image. The intensity minima occur at nearly the same position
and the spatial behavior of the predicted and measured curves is in reasonable
agreement, although the theory underestimates the observations by about a
factor of two.
II. NEW YORK BIGHT
During Orbit 91 (Ii October 1984) of the SIR-B mission on the
shuttle Challenger, a synthetic aperture radar image of oceanic internal waves
was obtained simultaneously with ship and current-meter measurements of the
waves. These observations were part of the second phase of the Synthetic
Aperture Radar Signature Experiment (SARSEX), a major experiment sponsored by
the Office of Naval Research to investigate the basic hydrodynamic and
electromagnetic phenomena responsible for internal wave signatures in SAR
images and to test theories for predicting such signatures. The SIR-B image
taken at 05:03:08 EDT (Fig. 4) shows several packets of tidally generated
internal waves, low to moderate wind fronts, and the location of the research
vessel Cape. The geographical area extends from about i00 km south of Long
Island to immediately northeast of the Hudson Canyon, near 40=N, 72°15'W, a
region that had been studied by Apel and coworkers (Apel et al., 1975a,b)
during observational programs on internal waves on the continental shelf.
In situ measurements were made of horizontal water velocity com-
ponents from current meters at depths of 15 and 67 m, water column temperature
and density profiles, 200-kHz downward-looking echo sounder internal wave pro-
files, surface wave height spectra, and wind speed. The major properties of
the internal wave field can be estimated when these measurements are used in
conjunction with the type of internal wave model used above, with certain
modifications. From the in situ measurements and a solution of the internal
94
wave equation, we find that _o = 9 m, L = 250 m, and c = 0.4 m/s. At the
surface, the estimated maximum induced current is uo = 0.15 m/s, and the maxi-
mum surface current gradient, or strain rate, is approximately 0.004 s-I
(Gasparovic et al., 1986).
A more extensive observational program of some nonlinear internal
waves was carried out four weeks earlier during Phase I of SARSEX. This effort
was structured to quantify the dominant processes contributing to the SARsig-
natures of internal waves: (i) the internal wave hydrodynamics, especially the
surface current strain rates; (ii) modifications to the surface wave spectrum
induced by the currents; (iii) the radar backscatter from the modulated surface
waves; and (iv) the formation of SARimages from the backscatter modulations
(Gasparovic et al., 1985).
A theory describing the perturbation of the surface wave height
spectrum by the internal wave surface currents has been used to calculate the
expected backscatter modulations associated with the internal waves (Hughes,
1978). This theory assumes that the energy input from the wind acts to
restore the surface wave height spectrum S(k, x , t), which is perturbed by the
presence of the internal wave currents u(x,t), to its equilibrium value
So(k). In particular, it is assumedthat the fractional change in the wave
action spectrum
N(k, x, t) = p_ S(k, x, t) (9)
(where p is the water density) is proportional to the departure of the
spectrum from its equilibrium value No(k). Mathematically, this implies
)iv dt - +S,.VxN+ k. VkN
B(k)
- (N - No)
No (10)
where
(ii)
and
(12)
95
In these equations, the indices i, j = i, 2 denote horizontal coordinates, and
repeated indices are summed. The function B(k) - the so-called wave relaxa-
tion rate - determines the effectiveness of both wind and dissipation in
restoring equilibrium (Hughes, 1978). There is at present considerable uncer-
tainty about its exact functional form, although there is general agreement
that it increases with wave number and wind speed. We made the calculations
that follow by using Hughes' expression for B(k) which is consistent with
field observations.
To relate the internal wave-induced change in surface wave spectrum
to the SAR image intensity modulation, we make use of the fact that ocean
backscatter at the SIR-B radar wavelength Xo = 23.5 cm and incidence angle
e is dominated by Bragg scattering (Thompsonand Gasparovic, 1986; Thompson,
1986). In this limit, the normalized radar cross section a is proportional
to the surface wave spectrum of the Bragg waves, that is, those waves travel-
ing in the radar "look" direction with a wavelength XB = Xo/2(sin e). The
fractional change in cross section from its meanvalue aN is given by
A___ = N(ka, x, t) - No(kB)
O"0 N0(kB )
(13)
where kB is the Bragg wave number. When the internal wave features propagate
in the radar look direction as in Fig. 4, the SAR image intensity variations
are essentially equivalent to the cross-sectional variations given by Eq. 13.
Image intensity variations for some of the features of the internal
waves in Fig. 4 are shown in Fig. 5, which was taken through the wave group
along the vertical line A-A'; it shows nine distinct waves in the packet with
peak-to-peak intensity variations ranging from 2 to 12 dB. The average spacing
between waves in this packet is approximately 380 m. There is another group
of waves near the center of the image along the vertical l{ne B-B' and the
four waves at the front of this packet have peak-to-peak intensity variations
of 3 to I0 dB.
We have made predictions of the SAR image intensity modulations for
these internal waves using the above theory, with internal wave and surface
parameters obtained from the in situ observations. Maximum internal wave
surface strain rates of 0.002 to 0.005 s-I were assumed, consistent with
intrapacket variations encountered during the Phase I observations. The
large-scale brightness variations in Fig. 4 are indicative of the presence of
wind speeds greater than the value i m/s recorded at the ship location which
is in a dark region of the image. For the wave packet along A-A', we assumed
a wind speed of 2 m/s along line B-B'; the actual wind speed is presumed to
lie between these two values. With a wind speed of 2 m/s, the calculated
peak-to-peak SAR image intensity modulations range from 2 to 7 dB for maximum
surface strain rates of 0.002 and 0.005 s-1, respectively. These calculated
modulations agree with the observations along lines A-A' and B-B' to within 3
to 5 dB, or within a factor of 2 to 3. This result is consistent with initial
findings for the Phase I SAR aircraft L-band images, for which more extensive
comparisons between theory and observations are possible.
96
ACKNOWLEDGEMENTS
The authors appreciate the SIR-A and SIR-B data support by the Jet
Propulsion Laboratory. This work was supported by U.S. Naval Sea Systems
Commandunder Contract N00024-56-C-5301.
REFERENCES
J. R. Apel, H. M. Byrne, J. R. Proni, and R. L. Charnell, J. Geophys. Res. 80,
865 (1975).
J. R. Apel, J. R. Proni, H. M. Byrne, and R. L. Sellers, Geophys. Res. Lett.
2, 128 (1975).
J. R. Apel, "Observations of Internal Wave Surface Signatures in ASTP Photo-
graphs," Apollo-Soyuz Test Pro_ect II, F. EI-Baz and D. M. Warner, eds. NASA
SP-412 (1979).
J. R. Apel and F. I. Gonzalez, J. Geophys. Res. 88, 4459 (1983).
J. R. Apel, D. R. Thompson, D. G. Tilley, and P. Van Dyke, "Hydrodynamics and
Radar Signatures of Internal Solitons in the Andaman Sea," Johns Hopkins APL
Technical Diget, 6, 330 (1986).
J. B. Cimino and C. Elachi, "Shuttle Imaging Radar-A (SIR-A) Experiment,"
NASA/JPL 82-77, pp. 5-53 (1982).
J. P. Ford, J. B. Cimino, and C. Elachi, "Space Shuttle Columbia Views the
World with Imaging Radar: The SIR-A Experiment," NASA/JPL 82-95, pp. 144-145
(1983).
R. F. Gasparovic, J. R. Apel, D. R. Thompson, and J. S. Tochko, "A Comparison
of SIR-B Synthetic Aperture Radar Data with Ocean Internal Wave Measurements,"
Science, 232, 1529 (1986).
B. A. Hughes, J. Geophys. Res. 83, 455 (1978).
A. R. Osborne and T. L. Butch, "Internal Solitons in the Andaman Sea," Science
208, _51 (1980).
SARSEX Experiment Team, "SARSEX Interim Report" (Report STD-R-1200, Johns
Hopkins University Applied Physics Laboratory, Laurel, MD, May 1985).
D. R. Thompson and R. F. Gasparovic, Nature (London) 320, 345 (1986).
D. R. Thompson, "Intensity Modulations in Synthetic Aperture Radar Images of
Ocean Surface Currents and the Wave/Current Interaction Process," Johns
Hopkins APL Technical Digest, 6, 346 (1986).
97
ORIGINAL PAGE IS
OF POOR QUALITY
Figure I. L-band radar image from SIR-A
taken November ii, 1981, near
the Andaman Islands, showing a
packet of 6-kilometer-long
solitons and what is thought
to be a rain squall. Dimen-
sions are 51.2 by 51.2
kilometers.
98
12
g_ o.8
O.
_ _ 0.4
N 0
I I I
_A
_ 1.0
c- 0
_ _ 0
i.- O.
_ 0
u _
_ "-" -1.0
U _
_m 0
]
10
] T
20 30 40
Distance from front of packet, s - ct (kilometers)
Figure 2. (a) Calculated horizontal surface currents and
(b) surface strain rates for cnoidal wave
packet. The front of the packet is at left.
99
M
O
D
U
L
A
T
I
O
N
2.0
1.5
1.0
0.5
I I I
* tJa.lI dl
•I: ^ n:.,_',:
,;, ,,, q
J Itll II I
I'
Ij
I a ILa lJ I
I_ I IL" Ii
'i l {
I I I
0 10000 20000 30000 40000
RANGE (M)
Figure 3. Comparison of observed (dashed curve) and calculated
(solid curve) radar backscatter power modulation from
rectangle of Fig. i. The phase of the calculated modula-
tion is reproduced quite well, but the amplitudes are
generally underestimated by factors of 2 or more. The
overall agreement is considered reasonable.
I00
ORIGINAL PAGE IS
OF POOR QUALITY
Figure 4. SAR image of oceanic internal wave features
from orbit 96 of the SIR-B mission. The bright
linear features crossing lines A-A' and B-B'
are due to increased surface wave roughness
within the internal wave packets, which are
propagating toward the top of the image (313 °
true heading). A surface wind of I m s-1 ,
blowing toward 190 ° true heading, was recorded
at the ship position. The radar incidence
angle at the center of the image is 20 °.
i01
O¢-
10
0
-10 -
0
Figure 5.
I
Wave 9
-.S
I I I
8 7 4 3
I I I
1 2 3 4 5
Distance (kin)
SAR image intensity traces through Fig. 4 along line A-A'.
Trace has been averaged over 625 m in the azimuth direction
and then smoothed 62.5 m in the range direction.
102


Hydrodynamics of internal solitons and a comparison of SIR-A and SIR-B data with ocean measurements

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