Blocking and trapping of waves in an inhomogeneous flow

Dynamics o f Atmospheres and Oceans, 20 (1993) 79-106

79

Elsevier Science Publishers B.V., A m s t e r d a m

Instabilities/turbulence/shear layers

Blocking and trapping of waves in an inhomogeneous flow Steven R. Long

,,a Ronald J. Lai b, Norden E. Huang c, Geoffry R. Spedding d

a NASA Goddard Space Flight Center, Wallops Flight Facility, Observational Science Branch, Code 972, Wallops Island, VA 23337, USA b Minerals Management Service, Mail Stop 4310, 381 Elden Street, Herndon, VA 22070, USA c NASA Goddard Space Flight Center, Oceans and Ice Branch, Code 971, Greenbelt, MD 20771, USA d Department of Aerospace Engineering, University of Southern California, Los Angeles, CA 90089-1191, USA (Received 27 May 1992; revised 21 December 1992; accepted 30 December 1992)

ABSTRACT The blocking and trapping of waves and wave packets by inhomogeneous flow fields were studied in a laboratory setting. Evidence for multiple reflections and thus trapping within the current gradient zone near the blocking point are the main results obtained, and lead to the following conclusions: (1) a strong current gradient zone near the blocking point can trap wave energy, which will shift the wavenumbers into the capillary region, to match the microwave wavelength of many remote sensing instruments; (2) this mechanism provides a direct link between strong current gradient zones (bathymetric features in tidal flows, eddies, current boundaries, etc.) and the surface wave structure, which in turn allows instruments such as the Synthetic Aperture Radar (SAR) to form images of bottom features based solely on the small wave conditions; (3) an indication of direct energy transfer from the carrier waves to low-frequency waves was also observed, and resolved by wavelet analysis. The instantaneous spectra from wavelet analysis reveal that a large portion of wave energy transfers directly into the low-frequency band from the carrier waves at the trapping zone. Subharmonics may play a critical role in the energy transfer process, but details are still to be established.

1. I N T R O D U C T I O N

Among all the wave-current interaction phenomena, the most interesting one is blocking, when the wave is stopped in an area where the velocity of the local current is larger in magnitude than that of the wave group but is opposite in direction to it. Blocking as an intriguing dynamic phe-

* C o r r e s p o n d i n g author. 0377-0265/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved

80

R.J. LAI ET AL.

n o m e n o n is still unresolved in detail. Smith (1975) was the first to derive a uniform asymptotic solution for water waves propagating against a co-linear, adverse and slowly varying current. His solution suggests reflection rather than breaking near the blocking area. This result was later confirmed by Basovich and Talanov (1977), and extended by Peregrine and Smith (1979) and Peregrine and Thomas (1979) to include nonlinear and finite-amplitude effects. Badulin et al. (1983) also produced qualitative laboratory results to support double-reflection and blocking based on kinematics only. Recently, Shyu and Phillips (1990) extended the uniform asymptotic solution of blocking to capillary-gravity waves. The reflection caused by non-breaking blocking could have important implications in remote sensing of ocean phenomena, especially for Synthetic Aperture Radar (SAR) image interpretations. The classic wave-current interaction theory stipulates that at blocking the wave number can be compressed only by a factor of four. In the reflection theory, however, the wavenumber can change from the original value to the capillary range. Thus, the compression ratio would become almost open ended. Furthermore, in the classic theory, the wavenumber compression occurs only near the immediate neighborhood of the critical point, whereas in the reflection theory, the wavenumber compression could occur over a large region that spans the initial blocking area and the secondary blocking area, and even beyond. This is in all probability the real mechanism for the SAR image formation of bottom topographies for many geographic locations, as reported by various investigators. Let us take the SAR image of the English Channel, as shown in Fig. l(a) (Fu and Holt, 1982), as an example. Although the electromagnetic wave used in SEASAT-1 SAR can penetrate only a few centimeters under the water surface, the image actually reveals the bathymetric features fairly faithfully, for, in some coastal waters, the local current gradient can be the consequence of the bathymetric variations. Figure l(b) shows the bottom topography and the tidal current as determined from the tide table. Here the spatial tidal current variations are caused by the bathymetric features. The maximum tidal current over the ridges is of the order of 1 m s-l, which is capable of blocking the waves from up to 10 m in length and producing 'trapped' waves of the order of 25 cm in length over a wide trapping zone. These 25 cm waves are the required Bragg wavelength for SEASAT-1 SAR. The gradual change of the grey tone in the image attests to the fact that the Bragg wave exists over a large area. If we do not take into account double-reflection and trapping (this will be discussed below), the longest wave that can be involved in the image formation is about 1 m in length, which is shortened by the four-to-one compression. Waves shorter than 1 m will not last long enough to experience the whole velocity

BLOCKING

AND TRAPPING

81

OF WAVES

gradient in their lifetime. On the basis of these considerations, we believe that the existence of the SAR image is evidence of the existence of the trapping zone. Observational evidence to support the theoretical results is still rare. Recently, Lai et al. (1989) and Long et al. (1993) have reported a series of laboratory experiments, in which only continuous wave trains were used; this makes the detection of reflection rather difficult. In this paper, we report a series of supplemental experiments using envelope solitons as probes for reflection. Because the envelope solitons are confined in a finite duration, wavelet analysis instead of the traditional Fourier analysis is used to examine the data. We found tantalizing evidence of double reflection or

ILLUMINATION DIRECTINN ~

0 L

20 km

Fig. 1. (a) Synthetic Aperture Radar (SAR) image of the English Channel obtained from SEASAT-1, with an L-Band SAR. (b) The bottom topographic map of the area imaged by SAR in Fig. l(a), where contours are in meters and current speeds are in meters per second.

82

R.J. L A I E T A L .

(b)

UUNR~HUU=

I I

UU~It~U¢

iI

BELGIUM

B

CONTOURSIN METERS CURRENT SPEEDSINMETERS PER SECOND

2"E 9

~30 'E /

I

/~I'N

I

I

I

i

I

Fig. 1 (continued).

'trapping' for the blocking occurring in inhomogeneous current cases. Before discussing the experiments, we will first summarize the salient theoretical results. 2. T H E O R E T I C A L ANALYSIS

According to the accepted theory, wave motions are governed by the following approximate kinematic and dynamic conservation laws. Kinematically, Ok --

Ot

+

Vn

= 0

(1)

BLOCKING A ND T R A P P I N G O F WAVES

83

and dynamically, ~A

- - + V. [(Cg--}-UIA] =O 0t

(2)

where k is the wavenumber, n is the apparent wave frequency, A is the wave action, defined as E/o- (where E is the wave energy density and o- is the intrinsic wave frequency), Cg is the group velocity and U is the ambient current. Equation (1) is exact; Eqn. (2) is the Wentzel-Kramers-Brillouin method approximation of the action conservation law. Equation (2) is derived with the reflection of the waves neglected; therefore, it is not a uniformly valid solution near the blocking point. Under the steady-state assumption, Eqns. (1) and (2) reduce to n = o- + k" U = constant

(3)

and (Cg +

U)A

= constant

(4)

The critical blocking point then becomes a mathematical singularity by the WKB approximation of the action conservation law: the length of the waves will decrease and the energy of the waves will increase according to Eqns. (3) and (4). If the blocking event were to follow this scenario, the waves should have broken violently before they reached this singular point. Indeed, blocking has been proposed as a principle for the construction of breakwaters (Taylor, 1955). Recent theoretical developments suggest that the WKB approximation of the conservation of action law, which neglected reflection, cannot be uniformly valid in the critical region near the blocking point. As the wave reaches the critical point of blocking, the current can cause the wave to reflect and propagate in the reversed direction without breaking. Badulin et al. (1983) produced qualitative laboratory results to support the doublereflection (or 'trapping') theory. Recently, Shyu and Phillips (1990) established the dynamic aspects by extending the uniform asymptotic solution to the capillary-gravity region. On the basis of the concept of reflection of the waves by an adverse current, Pokazeyev and Rozenberg (1983) and Badulin et al. (1983) derived a simple linear kinematic theory of double-reflection (or 'trapping') for gravity-capillary waves in horizontal current gradient regions. The blocking with reflection as shown by Badulin et al. (1983) from the kinematic conservation law can be summarized as follows. By rewriting Eqn. (3), they obtained

n=(gk +yk3)i/2+k.U

(5 t

84

R.J. LAi ET AL.

(al ."

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,,,<,,,,'"' ,,,,,,,"

N

3 ............

e'~'B 1 /k

_

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**

,°

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,o

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I .........

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4

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Wave Number (rad/cm) (b) A

~

) B2

B1

(.

\\\ Fig. 2(a). Dispersion relationship for waves in the n - k plane, under various current conditions. A m b i e n t adverse currents are as follows: - - , 0 cm s - l ; ...... , - 5 cm s - l ; .... ,-10cms-1; ...... ,-15cms 1; . . . . . . . ,-20cms-1,-"-,-25cms-1;---- - , - 3 0 cm s - l ; , 35 cm s -1. (b). Schematic diagram of wave propagation in the physical plane. Points A, B1 and B2 are the same as in Fig. 2(a).

in which the intrinsic frequency, or, satisfies the dispersion relationship

o = (gk + yk3) '/2

(6)

with y as the kinematic surface tension. The apparent frequency, n, is plotted as a function of k with U as a variable as in Fig. 2(a), in which U varies from zero to - 3 5 cm s-~ with an

BLOCKING AND TRAPPING OF WAVES

85

increment of 5 cm s-1. On the n - k plane, the slope of any curve is the apparent group velocity, or the Doppler-shifted group velocity. Thus, > 0; forward propagation of wave energy On 0-k = 0; blocking (7) < 0; backward propagation of wave energy From kinematic considerations, the existence of a horizontal tangent indicates a blocking condition. Furthermore, the kinematic conservation also dictates that any wave encountering a steady current field with spatial gradients should have a constant apparent frequency throughout; therefore, a horizontal line on the n - k plane also indicates the locus of the wave propagating in a steady but spatially inhomogeneous current field. It has been shown by Long et al. (1993) that, at the air-water interface, the minimum adverse current to cause blocking is 17.58 cm s -1, and the maximum frequency of waves that can be reflected is 2.58 Hz. Schematically, a wave starting from point A, in Fig. 2(a), propagates against an adverse current increasing in its magnitude until reaching point B1, the first blocking point, where the current is about - 2 5 cm s-1, where negative values refer to adverse currents. It is possible to have waves propagate beyond B1 on the n - k plane. In this region, on the n - k plane, however, the slope becomes negative, indicating the backward propagation of the wave energy. As the wave continuously propagates, it will meet a second point of horizontal slope, B2, where the second reflection occurs. Physically, the process of double reflection between B1 and B2 can be shown in Fig. 2(b). As the wave propagates from A to B1, the adverse current encountered increases from - 10 cm s- 1 to - 25 cm s- 1. The wave is reflected at point B1 and propagates backward. This time, the wave starts from a high-velocity region B1 and propagates to a low-velocity region B2, where the current is - 2 0 cm s 1. Even though the propagation direction of the wave and the current are parallel and in the same sense, the relative current which the wave experiences is still negative. The wave will be compressed by the relative negative current and becomes shorter and shorter as the current velocity diminishes, until it reaches B2, where it is reflected again. Beyond B2, the waves becomes predominantly capillary, when no current will be strong enough to block the wave. Between the points B1 and B2, the wave travels back and forth until it escapes. Local energy density, being a scalar quantity, should be high. The action flux, being a vector quantity, however, should be constant. At each reflection, the action flux is conserved, as shown by Smith (1975) and Shyu and Phillips (1990). To emphasize this double-reflection zone with high local energy density, we call this region between B1 and B2 the 'trapping' zone.

86

R.J. LAI ET AL.

In this zone, the wavenumber continues to increase, covering the range from the initial value to that of capillary-gravity waves; the approximation solution, however, can only allow a four-to-one reduction of the wavelength up to the critical point. Important as the possible trapping is, there have never been experimental data to confirm its existence. The experiments by Long et al. (1993), which used a continuous wave train have not fully resolved the questions of trapping and double-reflection, because the probe array failed to detect the return waves when they change their wavelengths as a result of Doppler shift caused by the current. We adopted the wave packet envelope at the suggestion of Dr. Erik Mollo-Christensen because the envelope soliton can maintain its integrity for a long period of propagation; therefore, it can be used as a probe for detecting reflection. 3. E X P E R I M E N T A L P R O C E D U R E S

This series of experiments consists of detection of the double-reflection and trapping with envelope soliton wave packets. The experiments were conducted in the same tank as used by Lai et al. (1989) at NASA Wallops Flight Facility, Wallops Island, Virginia. The test section has 18.3 m length 0.91 m width and 1.22 m height. The water depth is always set at 0.76 m. A more complete description of the research facility has been given by Long (1992). The spatially inhomogeneous current is produced in the tank by a false bottom of a total length 4.88 m located near the central section of the tank. The false bottom and the measurement stations are shown in Fig. 3, and the locations are given in Table 1.

Wave Direction

Air

45.7 cm 30.5 cm 45.7 cm

Current

j

False Bottom

,~,

A

A

A

A

A

A

A

A

10

9

8

7 6

5

4

3

I 152cm

1

244cm

Water

J

488 cm Fig. 3. The locations of the false bottom and the measurement stations.

87

BLOCKING AND TRAPPING OF WAVES

TABLE 1 Measurement stations and their fetch Station No.

Fetch (cm)

10 9 8 7 6 5 4 3

191 344 497 580 628 708 861 963

The envelope soliton packet is generated by a programmable wave-maker driven by a signal, ( T - 1260)7r ~T S(t) = a sech 2 1260 c°s 18----0 (8) where the increment of T is 1/(360w 0) s. Thus, the carrier waves are computed with a 1° phase angle resolution, for frequencies of 1.5, 1.75 and 2 Hz. Three amplitude values are used. The combined signal is then converted into an analog voltage to drive the wave paddle. A hyperbolic secant squared function is selected for the envelopes, which ranged from 2.8 to 4.8 m in our experiment, and thus should be considered to be propagating in shallow water in our tank. The hyperbolic secant squared soliton is an exact solution for the KdeV equation (Ablowitz and Segur, 1981). This envelope has similar properties to the hyperbolic secant envelope proposed by Yuen and Lake (1975). In the preliminary tests without current, the solitons used here showed less degeneracy than that used by Yuen and Lake (1975). Digitization of the data commences 20 s before the wave-maker driving signal starts. Each run consists of recording the surface elevation for 120 s from Station 10 to Station 3. As these signals represent transient events, 40 repetitions for each case are run. Between each run, a 5 min quiescent period is allowed for the tank to settle down before the next run starts. The same currents as in the continuous wave series reported by Lai et al. (1989) and Long et al. (1993) are used here. They are 13, 23 and 35 cm s-1 over the top of the false bottom, near the center of the water flow cross-section. 4. RESULTS AND DISCUSSION

4.1. The envelope soliton time series The envelope soliton is a transient signal. The duration of data collection was long enough to cover the event. For each case, 40 repeated tests

88

R.J. LAI ET AL.

were run under identical conditions. The means of each case obtained represent the true ensemble averages. Measurements were made at Stations 3-10 to detect the presence of reflection. Let us first examine the case for no current, as a reference. The soliton packet for this represents the highest driving signal level at 3.75 V, the highest carrier frequency, at 2.0 Hz, with (ak)max = 0.422 at Station 8. As nothing extraordinary occurs here, only the averaged signals at the eight stations are presented in Fig. 4. The soliton passes through each station uneventfully. At each station, both the signal and the reflections from the beaches are identifiable. The reflection of the soliton packet can also be seen from the soliton packet arrival time in Fig. 5. The arrival times of the soliton packets are plotted against distance. Three groups of soliton arrival times can be identified. The primary soliton packets are shown in the lower curve set. The first soliton packets reflected from the beach are shown in the middle curve set, and the second soliton packets reflected from the wave-maker are shown in the upper curve set. The propagation speeds of the soliton packets can also be derived from these curves, and agree well with the linear dispersion relation. Let us next examine a partial blocking case using the same soliton as in the zero current case. Here the soliton packet, with (ak)ma~ = 0.338 at Station 8, is propagating against a current. The maximum current over the false bottom is 23 cm s-1 which is sufficient to block these carrier waves. Figure 6 shows the ensemble averaged signals of all 40 runs. Here the magnitudes of the high-frequency components are drastically reduced by the averaging process compared with the individual realizations to be shown later in the wavelet analysis. The signals standing out are the in-phase part of the solitons, which can now be clearly identified as pulses propagating down channel. At Station 5 (see Fig. 6(f)), the soliton is still clear, but there is a hint of degeneracy of the soliton by splitting. This is probably due to the fact that the current fluctuations have caused the split soliton to have slightly random phases. Beyond Station 5, the only discernible signals are the forerunner soliton of the original packet. As it is of a lower frequency, the forerunner can pass through this current. Next, let us examine the energy density of the soliton packet by plotting the root mean square (r.m.s.) of the signals. In this way, even the out-of phase part of the signal will be represented. The result is given in Fig. 7, which represents the result of first performing the r.m.s, on each realization, followed by the ensemble average of all realizations. The main part of the soliton is clearly shown by the large spikes in the r.m.s, plots. Near the critical blocking area, the energy level shows clearly two occurrences (see Fig. 7(f)) of high values which indicate that the soliton has indeed been

89

BLOCKING AND TRAPPING OF WAVES

1.0

el

0.0 i i

-1.0 1.0

~

;i ,

f

b ,~',j!

0.0

i

~ 'i

-~

- 1.0

•!

1.0 [

m

0.0

g

-1.0 I. d

h

0.0 " -].o t. I

0

I

120

I

0

I

120

Time (s) Fig. 4. The ensemble averages of surface elevation vs. time (s) from 40 envelope soliton packet runs for zero current for all eight stations, in the following order of stations: (a) 10; (b) 9; (c) 8; (d) 7; (e) 6; (f) 5; (g) 4; (h) 3.

R.J.LAIETAL.

90

u

<

50

_

0 0

~

e#i~mRlm

.......... . Rm~m~Iml~

i1~111~11~JtH n , l ~ l l m l l [ ~ . ~ a l t l l l a 500 1000 Distance From Wave Maker (cm)

Fig. 5. The arrival times of the soliton packets of frequency 2 Hz, for both the signal and its reflections. ~ , High-amplitude waves with no current; - - - - - - , low-amplitude waves with no current; . . . . . . , high-amplitude waves with the lowest adverse current.

split. A further effort to examine the frequency contents of the high-energy-level part of the signal using wavelet analysis will be discussed below. Finally, let us examine a total blocking case. Figures 8 and 9 correspond to Figs. 6 and 7, respectively. In Fig. 9, as in Fig. 7, the result was obtained by first performing the r.m.s, on each realization, followed by the ensemble average of all realizations. The same soliton packet as in the partial blocking case is used here, but now (ak)m~= 0.270 at Station 8, and the maximum current is 35 cm s-1 over the top of the false bottom. The mean signal is shown in Fig. 8. The soliton packet is clearly identifiable at Stations 10-8. After that, the degeneracy becomes very drastic. Beyond Station 5 (Fig. 8(f)), only a low-level trace of the forerunner can be discerned. As it is of much lower frequency, the forerunner can propagate ahead of the main soliton packet through the current gradient zone. The most intriguing results are the r.m.s, energy series as shown in Fig. 9. The energy of the soliton packet at Stations 10-8 is clearly identifiable. At Stations 7 and 6 (Figs. 9(d) and 9(e)), the energy level stayed high for a long period after the initial soliton group passed the probes: from 40 to 90 s

91

BLOCK/NG AND TRAPPING O F WAVES

,.o

al

e i

-1.0

.

1.0

©

..... 7. . . . . . . . . . . . . . .

-1.o 1.0

t._L "

f-]

i

..............

c

> ,D

0.0 -1.0 1.0

0.0

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........................ i,~

~÷

,

,

,i

h~ •

-1.0 I

0

I. . . . . . .

120 0

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120

Time (s) Fig. 6. The ensemble averages of surface elevation vs. time (s) from 40 envelope soliton packet runs for a medium current case (adverse current of 23 cm s-1) for all eight stations, in the same order as in Fig. 4.

92

R.J. LAI E T AL.

0.1

0.0

"

a

'................................................................

0.1

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0.0 0.1

e

b

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0.0

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Time (s) Fig. 7. The ensemble average of the r.m.s, signal for the case shown in Fig. 6. The ensemble average is done on the r.m.s, result from each realization.

BLOCKING

AND

TRAPPING

93

OF WAVES

1.0 "

e

0.0 I

-1.0 1.0

L-

[

b!

f'

i

0.0

,'p0vl Jl~.....

i I

i O

°~..~

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T i m e (s) Fig. 8. The ensemble averages of surface elevation from 40 envelope soliton packet runs for the blocking current case vs. time (current over the top of the false bottom is 35 cm s-~) for all eight stations, in the same order as in Fig. 4.

,qlenlae I~AOl~u~aouo qB!q oql luql s! lSaJZlu! aulna!laud JO "uo.tlanP°JluI ~ql u! passnas!p 8u!ddeJ1 jo a~u~ls!x~ aql ql!A~ 1u~moa:~e m oJe SlZAZI ~!SttOp /u~taUO qB!q Ie3Ol oqJ~ "9 uo.tlelS Joj s 0L o~ glr moaj p u e L uo!lelS :toj •uo!leZ!leaa qa~a m o z j llnsoa "s'tu'z a q l u o pzlelnOle3 s! O~e.lOAe

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, ¢ . , ,_ ~ ~:,,.,W&,,¢.~;,~,~,m~.O,,9~,~,~,~,~.~?¢~,&~."~,

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176

BLOCKING A N D T R A P P I N O O F WAVES

95

Fig. 10. Wavelet analysis of a single realization for the case of zero current at Station 8 (see also Fig. ll(a) for scaling). The figure presents the color-coded modulus and the corresponding energy level, in perspective view.

has a longer duration at the upstream Station 7 than at the downstream Station 6. The wave signal cannot be blocked at Station 7, for the current is not strong enough there; blocking can occur only at Station 6 and beyond. One possible explanation is that the soliton is reflected at the blocking area (Station 6), and Station 7 is actually located in the trapping zone. As discussed above, our probes used here cannot discriminate the direction of the wave propagation. The distance between Stations 7 and 6, unfortunately, is too short for the soliton packets to separate clearly. The local high energy density coupled with the extended durations offers the tantalizing suggestion that double-reflection and trapping have indeed occurred. Additional evidence can also be cited from experiments using continuous waves, which showed a capillary wave radiating from the blocking point against the current, as reported by Long et al. (1993). F u r t h e r experiments are planned, with modification of the false bottom to produce a much gentler current gradient over a longer transitional zone, and with an optical sensor similar to the one used by J~ihne and R i e m e r

96 (a)

(b)

Fig. 11.

R.J. LAI E T A I

BLOCKING A N D T R A P P I N G OF WAVES

(c)

(d)

Fig. 11 (continued).

97

98

R.J. LAI E T AL.

(1990), which can detect the wave propagation directions. Because of the limitations of the present experimental conditions, the available data cannot be treated as a definitive proof of trapping, but they can serve as a good start.

4.2. Wavelet analysis results Because the envelope solitons are transient events, we cannot use the standard Fourier analysis to examine the frequency contents of their energy. We decided to use the newly developed wavelet analysis (see, e.g. Farge, 1992) to probe further into the frequency constituents of the soliton during its propagation, blocking and trapping. The wavelet analysis program has been developed at the University of Southern California (USC) over the last 3 years (see Spedding et al., 1992; Dallard and Spedding, 1993). The one-dimensional program we used is based on a Morlet mother wavelet (Morlet, 1981, 1983; Grossman and Morlet, 1984), which is a Gaussian modulated sine and cosine wave packet. Through translation and dilation, we can examine the local properties of the signal in great detail. More information on the USC wavelet analysis program has been given by Spedding et al. (1992). To examine the dynamics of wave-current interaction processes near the critical blocking point, we present the wavelet analysis results in Figs. 10-13. Figure 10 presents a perspective view of the wavelet modulus, giving frequency vs. time information, with the energy levels being color coded. Figures l l ( a ) - l l ( d ) represent data taken at Station 8-5 for zero current cases. The last two series (Figs. 12 and 13) are for the corresponding stations near the critical points (Stations 5, 6, 7 and 8) in the partial and total blocking cases discussed in the previous section. Each figure of the series (Figs. 11-13) giving the wavelet analysis results consists of three panels. The top one is the modulus of the wavelet analysis. It is closely associated with the energy level of the signal contoured in the renormalized color scale as a function of time on the horizontal axis, and frequency on the vertical axis. The second panel is the phase angle of the

Fig. 11. Wavelet analysis result for the case of zero current for Stations 8, 7, 6 and 5, with the modulus given in the top panel, the phase in the middle panel and the time series data in the bottom panel. The edge effects of a wavelet of finite size are shown in the curve noticeable near the left and right bottom edges of the modulus and argument (phase) panels. This marks the boundary below which the result is unreliable. The horizontal axis shows time; the vertical axis shows frequency. (a) Station 8; (b) Station 7; (c) Station 6; (d) Station 5.

BLOCKING A N D T R A P P I N G OF WAVES

(a)

(b)

Fig. 12. Same as in Fig. 11, but for the case of medium current with the maximum current over the top of the false bottom at 23 cm s - 1.

100 (c)

(d)

Fig. 12 (continued).

R.J. LAI E T A]

B L O C K I N G A ND T R A P P I N G OF WAVES

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signal as a function of frequency and time (the argument of the wavelet analysis). It gives the phase behavior during the processes of blocking and trapping. This phase information is normally not available from Fourier techniques, and is generally wrapped around at two pi when available from other techniques. Both these results also contain a curve near the bottom corners (left and right) which mark the boundary below which the results are unreliable, as a result of edge effects of a finite wavelet. With this information, the time series of the signal can be reconstituted exactly. The last panel is the corresponding original time series data. Two interesting points are worthy of further discussion in these figures: the degeneracy of the soliton when the soliton packet propagates against the current, and the increase in energy density noted in the low-frequency range. Figures l l ( a ) - l l ( d ) are for the case with no current. At Station 8, the time series shows a single large packet of the soliton around 12 s, its first reflection around 50 s, and the second reflection around 90 s. The reflections here are caused by the beaches, which are not perfect absorbers of the wave energy. The movement of the soliton closely adheres to the linear dispersion relation as shown in Fig. 5. The main part of the soliton shows clearly in the modulus panel at the frequency of the carrier. There is, however, a frequency redistribution within the soliton, which shows a trend of increasing frequency components at the front. This frequency distribution indicates that within the soliton packet a sorting of frequency has already taken place, so that the lower-frequency components are in the front part of the packet. When the current is zero, the reflected solitons are discernible as local energy highs. Overall, there is very low energy in the low-frequency (frequency less than 1 Hz) components. The energy level is so low that the phase function, defined as the ratio of the imaginary and real parts of the wavelet transforms, cannot be well determined at some locations. The program we used will leave the space blank if the phase cannot be defined unequivocally. The characteristics of the zero-current cases are well-defined packets with only low-frequency components near the soliton envelope. Figures 12(a)-12(d) are for the partial blocking case. The soliton passes the probe and leaves some high-frequency noise, similar to the normal event for a packet of waves passing through the fixed probe. At Station 8, the soliton arrives later, as a result of the Doppler shift. At Station 5, over the top of the false bottom, the soliton signal passes the probe some 30 s after the digitization started. The soliton then degenerates and splits into two: the first part has a slightly lower frequency than the second part. That the degeneracy of the soliton was possible was discussed by Yuen and Lake (1975), but the adverse currents have certainly accelerated the degeneracy process. Beyond Station 5, there is no longer any signal for the second

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soliton, but the first soliton, with a lower-frequency carrier, is able to propagate through the current. Its packets reflected by the beaches are also visible at the 65 and 90 s marks. The Doppler shift can delay the arrival of the initial packet, but the second reflection should have the effects of both the positive and the negative shifts cancelled; therefore, its arrival time is identical to that in the case of zero current. Figures 13(a)-13(d) are for the blocking case. The packet seems to have degenerated into multiple solitons. There is a great increase of carrier wave amplitudes at Station 7 and 6, where the velocity has the greatest gradient. At Station 5, the current reaches 35 cm s-l; no wave can pass this critical point. All the energy after the blocking appears in the low-frequency range. Other than the fact that we can see soliton degeneration in the modulus panel, which we have discussed in great detail in the previous section, the steady increase in the low-frequency energy is intriguing. We even have some energy below 0.5 Hz. This energy exists all the time, even before the soliton has been generated. Most of the increase in energy intensity, however, is confined to low-frequency components after the trapping occurs. Wavelet analysis clearly shows the importance of the low-frequency components in the wave-current interaction. Although details are still to be explored, some speculations will be given here. The most important effects of the low-frequency components are their ability to absorb energy and play a pivotal role in the action balance. The low-frequency components can be the subharmonics of the carriers. If these subharmonics can receive energy directly from the wave motion during the strong wave-current interaction, they can provide the missing action density flux as discussed by Long et al. (1993). The existence of the subharmonics can be seen from the spectra presented in Long et al. (1993). The effects of the subharmonics, as discussed by Longuet-Higgins (1978) for gravity waves of finite amplitude but small ak, can be identified with the Benjamin-Feir instabilities. On inspection of the wave spectra for continuous waves in the zero current case, the many sidebands are typical (see, for example, Fig. 9 of Long et al. (1993)), and are indications of the existence of subharmonics as discussed by Longuet-Higgins (1978). The mechanism by which the current-wave interactions transfer energy into these subharmonic components remains to be proved. Another possible explanation for the importance of low-frequency components is that the topographic trapped waves are the results of subharmonics as discussed by Long et al. (1993). However, the details of the generation of the topographic trapped modes by the subharmonics need to be studied more carefully before a definite answer can be given. Another unknown is the relative importance of the subharmonics in the envelope soliton packets.

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(a)

(b)

Fig. 13. Same as in Fig. 11, but for the case of blocking current with the maximum current over the top of the false bottom at 35 cm s - 1.

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(d)

Fig. 13 (continued).

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5. CONCLUSIONS

Based on our present study using envelope solitons as probes, we have found additional evidence to support the existence of double-reflection and trapping. The solitons in adverse currents degenerate at a much faster pace, as a result of the increasing energy density caused by the currents. There is clear indication of direct energy transfer from the carrier waves to low-frequency waves, as dictated by the existing theory. Subharmonics may be the critical elements in the processes, but details are still to be established. ACKNOWLEDGEMENTS

This work has been supported by the following sources: the Fluid Mechanics and Physical Oceanography Programs of the Office of Naval Research (R.J.L., N.E.H, G.R.S.) and the N A S A Oceanic Processes SRT Program (N.E.H, S.R.L.). The authors wish to express their appreciation for this generous support. REFERENCES Ablowitz, M.J. and Segur, H., 1981. Solitons and the Inverse Scattering Transform. SIAM, Philadelphia, PA, 425 pp. Badulin, S.T., Pokazeyev, K.V. and Rozenberg, A.D., 1983. A laboratory study of the transformation of regular gravity-capillary waves on inhomogeneous flows. Izv. Atmos. Ocean Phys., 19: 782-787. Basovich, A.Y. and Talanov, V.L., 1977. Transformation of short surface waves on inhomogeneous currents. Izv. Atmos. Ocean. Phys., 13: 514-518. Dallard, T. and Spedding, G.R., 1993. Two-dimensional wavelet transforms: generalisation of the Hardy space and application to experimental studies. Eur. J. Mech. B. Fluids, 12: 104-124. Farge, M., 1992. Wavelet transforms and their applications to turbulence. Annu. Rev. Fluid Mech., 24: 395-457. Fu, L.L. and Holt, B., 1982. SEASAT Views Oceans and Sea Ice with Synthetic-Aperture Radar. JPL Publ. 81-120. Jet Propulsion Laboratory, Pasadena, CA. Grossman, A. and Morlet, J., 1984. Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15: 723-736. J~ihne, B. and Riemer, K.S., 1990. Two-dimensional wave number spectra of small-scale water waves. J. Geophys. Res., 95:11531-11546. Lai, R.J., Long, S.R. and Huang, N.E., 1989. Laboratory studies of wave-current interaction: kinematics of the strong interaction. J. Geophys. Res., 94: 16201-16214. Long, S.R., 1992. NASA Wallops Flight Facility: Air-Sea Interaction Research Facility. NASA Tech. Ref. Publ. RP-1277, 36 pp. Long, S.R., Lai, R.J. and Huang, N.E., 1993. Laboratory studies of wave-current interaction: dynamics of the strong interaction. J. Geophys. Res. (submitted). Longuet-Higgins, M.S., 1978. The instabilities of gravity waves of finite amplitude in deep water: II Subharmonics. Proc. R. Soc. London, Ser. A. 360: 489-505.

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Morlet, J., 1981. Sampling theory and wave propagation. Proc. 51st Annu. Meet. Soc. Explor. Geophys., Los Angeles. Morlet, J., 1983. Sampling theory and wave propagation. In: C.H. Chen (Editor), NATO ASI, Vol. F1, Issues on Acoustic Signal/Image Processing and Recognition. SpringerVerlag, Berlin. Peregrine, D.H. and Smith, R., 1979. Nonlinear effects upon waves near caustics. Philos. Trans. R. Soc. London, Ser. A, 293: 341-370. Peregrine, D.H. and Thomas, G.P., 1979. Finite-amplitude deep-water waves on currents. Philos. Trans. R. Soc. London, Ser. A, 293: 371-390. Pokazeyev, K.V. and Rozenberg, A.D., 1983. Laboratory studies of regular gravity-capillary waves in currents. Izv. Atmos. Ocean Phys., 23: 429-435. Shyu, J.H. and Phillips, O.M., 1990. The blockage of gravity and capillary waves by longer waves and currents. J. Fluid Mech., 217: 115-141. Smith, R., 1975. The reflection of short gravity waves on a non-uniform current. Proc. Cambridge Philos. Soc., 78: 517-525. Spedding, G.R., Dillard, T. and Browand, F.K., 1992. The design and application of 2D wavelet transforms for fluid turbulence data analysis. Proc. Int. Conf. 'Wavelets and Turbulence', Princeton, NJ. Taylor, G.I., 1955. The action of a surface current as a breakwater. Proc. R. Soc. London, Ser. A, 231: 466-478. Yuen, H.C. and Lake, B.M., 1975. Nonlinear deep water waves: theory and experiment. Phys. Fluids, 18: 956-975.