Refraction and Shoaling of Nonlinear Internal Waves at the Malin Shelf Break*

Transcription

1 DECEMBER 2003 SMALL 2657 Refraction and Shoaling of Nonlinear Internal Waves at the Malin Shelf Break* JUSTIN SMALL International Pacific Research Center, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, Honolulu, Hawaii (Manuscript received 4 December 2001, in final form 9 May 2003) ABSTRACT This paper applies a numerical model to explain the refraction and shoaling of nonlinear internal waves observed at the Malin slope for a sequence of tidal cycles in the summer of The model is first order in dispersion and second order in nonlinearity and is based on the extended Korteweg de Vries (EKdV) equation. The EKdV model is applied along a set of rays and includes the effect of variable depth. The model predicts that an initial long-wavelength wave, which lies in water of depths between 500 and 900 m, develops into a set of internal solitary waves as it passes across the shelf break and onto the continental shelf, in agreement with observations. The extent of refraction is small because the internal waves are high frequency, and in this case the refraction is not strongly dependent on the nonlinear adjustment to phase speed. In the observations, the internal-wave amplitude, measured in terms of near-surface currents or thermocline displacement, initially grows as the wave crosses the slope and then is capped as the shelf is reached. The numerical experiments suggest that this behavior is due to a particular nonlinear feature of the EKdV equation, which predicts the existence of limiting wave amplitudes. The properties of simulated internal waves that arose from an idealized initial waveform were close to those observed. However, the numerical evolution of waves from a realistic initial condition showed some differences to the observed. It is suggested that these differences are due to neglect of strong nonlinearity and turbulence in the model. 1. Introduction Recently the study of nonlinear and solitary internal waves has received much attention, often motivated by the ability of the waves to disrupt acoustic propagation (Headrick et al. 2000) and sea-drilling operations (Bole et al. 1994), to displace and sometimes transport nutrients (Lamb 1997; Sharples et al. 2001), to contribute to crossshelf exchange (Inall et al. 2001), and to cause ocean turbulence (Inall et al. 2000; Pinkell 2000). Recent studies have focused on the generation of the internal tide and associated nonlinear waves at topography (e.g., Farmer and Armi 1999; Gerkema 1996; Lamb 1994), the evolution of solitary internal waves from long internal waves (e.g., Holloway et al. 1997, 1999; New and Pingree 2000; Lamb and Yan 1996), the detection of internal waves by synthetic aperture radar (SAR; Brandt et al. 1996; Zheng et al. 2001; Small et al. 1999a; and many others see the references * International Pacific Research Center Contribution Number 212 and School of Ocean and Earth Science and Technology Contribution Number Corresponding author address: Dr. Justin Small, International Pacific Research Center, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, 2525 Correa Rd., Honolulu, HI small@hawaii.edu contained online at miyata/ IWavesPublicationList.htm and also see udel.edu), and the detailed structure of the solitary waves (e.g., Choi and Camassa 1999; Stanton and Ostrovsky 1998; Small et al. 1999a). The aim of this paper is to investigate the importance of refraction and shoaling effects on nonlinear internal waves of the type observed in deep water off the Malin slope off the United Kingdom. Oceanographic data are utilized from the Shelf Edge Study (SES) and Shelf Edge Study Acoustic Measurement Experiment (SES- AME) experiments, which took place at the Malin slope and shelf in 1995 and 1996 (see left panel of Fig. 1 for location and positions of instrumentation). The current paper investigates nonlinear internal waves that cross the slope and are associated with an internal tide. Numerical simulations are made to help to interpret the internal tide behavior that was observed from satellite and in situ data in the two experiments. The present study significantly extends that of Small et al. (1999a,b) to include numerical simulations over a sloping bottom, including the effects of second-order nonlinearity and refraction. It is believed to be the first time these effects have been simulated in unison in a realistic application, and it follows on from the idealized studies of Small (2001a,b). This paper focuses on particular packets of high-fre American Meteorological Society

2 2658 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 FIG. 1. (left) Location of the main experiment area discussed in this paper. SES moorings are marked with diamonds and labeled S700 S140. The thermistor chain survey track UTC 19 Aug 1995 is shown as the thick dashed line. The times of the thermistor measurements of internal waves are marked with asterisks. The leading-wave measurements are labeled T700 and T400 and are defined in the text. The times of thermistor measurements of the following waves in the packet are marked 2 (i.e., second wave), 3, and 4. The internal-wave fronts from the SAR image of the right-hand panel are marked as thick solid lines for 20 Aug (1136 UTC), including the following waves, and as a thick solid line joining asterisks for 21 Aug 1995 (1136 UTC). The feature A is the main wave discussed in the text. Bathymetry contour levels (m) are at every 100-m intervals, starting at 200 m on the right and getting deeper to the left. Left and bottom axes show the latitude and longitude axes, respectively; right and top show the corresponding range (km). (Tick marks outside the plot refer to range and inside the plot refer to longitude or latitude.) (right) ERS-1 SAR image of the Malin shelf edge, 1136 UTC 20 Aug SES bathymetry contours of 180, 500, and 1000 m are marked in black (dashed), from right to left. An internal-wave feature marked A is present on the slope between the 500- and 1000-m contours. Farther onshelf there are sets of internal waves, the first two marked B1 and B2, which may have resulted from generation during the previous tidal cycle. Dark area is the edge of the swath. Also marked are internal-wave interaction zones, as I. Positions of SES moorings are marked as diamonds, denoting S700, S400, S300, S200, and S140 from left to right.

3 DECEMBER 2003 SMALL 2659 quency internal waves that were observed on each tidal cycle between 19 and 21 August 1995 during neap tides [see, e.g., Inall et al. (2001), their Fig. 3a]. Inall et al. (2000) showed that in August 1995 the energy of highfrequency waves was greatest during the neap tide period, and this may be due to the influence of distantly generated internal tides that would not necessarily arrive at the Malin shelf break at the time of greatest local tidal forcing. In fact, the particular internal waves being studied here do not appear to be products of a locally forced internal tide. This is suggested by the depth of water where the waves were first observed ( m, much deeper than the shelf edge around 200-m depth), combined with the onshelf direction of propagation (see Small et al. 1999a). The signature of the internal waves may be seen on the SAR image of 1136 UTC 20 August 1995 [right panel of Fig. 1; a portion of this image was shown in Small et al. (1999a)]. The feature labeled A on the SAR image corresponds to the internal-wave packet that passed the moorings late on the afternoon of 20 August, while those labeled B1 and B2 are speculated to have passed the moorings early on 20 August (and have moved farther onshelf by the time of the image), an M 2 tidal cycle earlier. Further, the tidal flow U, which at the time of observation was at its weakest around m s 1, never approached the phase speed c of the internal waves (measured at between 0.6 and 0.9 m s 1 ), indicating that the Froude number Fr U/c never approached a critical value of unity for hydraulic jump formation (see e.g., Brandt et al. 1996). These preliminary results suggest that the features observed during the time of interest to this paper, August 1995, at the Malin slope during neap tides, were generated elsewhere. Small (2000) suggested that likely generation sites were Rockall Bank, a known generator of internal tides (DeWitt et al. 1986), and/or Anton Dohrn seamount, one of the seamounts that Xing and Davies (1998) demonstrated numerically to have an important influence on the internal tides in the region. A more comprehensive discussion of the possibility of local and distant generation of internal tides at the Malin shelf break is given in Small (2000). The main topic of this present paper is how the waves behave as they reach the continental slope and undergo refraction and shoaling. The paper is organized as follows. Section 2 introduces the numerical model. Section 3 describes numerical simulations of the transformation process, using both idealized and realistic initial conditions, together with adiabatic predictions of solitary-wave behavior. Section 4 then compares the simulations with the measurements from SES and SESAME and discusses the limitations of the model. Last, concluding remarks are presented. 2. The numerical model of nonlinear internal wave refraction The model employs second-order nonlinear extended Korteweg de Vries (EKdV) theory (Holloway et al. 1999) to describe the wave evolution and a ray method to describe the refraction. The present model modifies the EKdV ray model of Small (2001a,b) to handle realistic ocean stratification and bathymetry. A normalmode approach is taken as described in the appendix. Only the dominant mode is chosen for analysis, and here the first mode is selected in accordance with the observations described in Inall et al. (2000) and Small et al. (1999a). The EKdV ray model is given by 2 t c 0(x) x (x) x 1(x) x (x) xxx (x) S(x) 0, (1) where (x, t) is a waveform, (approximating the maximum displacement found in the water column), t is time, x is range along the ray of interest, and subscripts denote differentiation. The variable coefficients of (1) [c 0 (x), (x), 1 (x), (x), see appendix] are here termed the environmental parameters, and (x) and S(x) are the shoaling and spreading terms (see Small 2001a,b). The numerical scheme of the model was described in Small (2001a,b), and model testing was reported in Small (2000), which verified that in idealized situations the model predictions were accurate in comparison with analytical EKdV solutions. The effect of rotation was also briefly investigated by considering the rotation-modified EKdV equation (Holloway et al. 1999), which includes rotation to first order. In fact it turned out that in this situation the effect of rotation on the internal-wave amplitudes was a very small reduction in amplitude. Small (2000) gives some discussion of this point and notes that the observed phase speeds suggest that rotation is not significantly affecting the internal waves because of the relatively high frequency of the waves. It is acknowledged that a complete description of the development of the internal tide from its long linear form to the high-frequency disintegration may involve the use of a governing equation with full rotation, such as developed by New and Pingree (2000). To compute exact environmental parameters at each water depth, knowledge of the stratification and background current shear is required. Data for these quantities were taken from the Land Ocean Interaction Study (LOIS) SES CD-ROM, available from the British Oceanographic Data Centre at the address given in the acknowledgements. The background density stratification of the Malin shelf/slope environment was dominated by a strong seasonal thermocline that extended from near the surface to 50 m (Fig. 2a). The background buoyancy frequency N(z) given by 1 * 2 N (z) g, (2) z where g is gravitational acceleration and 0 and *(z) are a reference and a profile of background potential density, respectively, is a measure of the stratification. 0

4 2660 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 FIG. 2. Density stratification, Aug 1995: (a) potential density profile to 1000-m depth and (b) buoyancy frequency to 150-m depth. The peak buoyancy frequency was around 10 cph centered around 30-m depth (Fig. 2b). In this paper there will be no discussion of how changes in stratification with range affect internal-wave paths. This is because reliable density profiles over a complete tidal cycle were only available at one location (see the appendix). Further, current shear effects will not be included (the small effect of current shear in the slope current on the internal wave modes is also reported in the appendix). Hence the investigation presented here will show solely the effects of changing water depth on internal-wave propagation. As it turns out, the model predictions capture many of the observed properties of the internal waves, thus partly justifying this approach. The environmental parameters were computed using the relationships given in the appendix. These parameters are plotted in Fig. 3 as a function of water depth and are listed in Table 1. The linear phase speed increases fairly slowly with depth (Fig. 3a) and is the first indication that the refraction will be limited over the slope. The reason is that the normal-mode solution (discussed in the appendix) assumes the initial waveform has a frequency that lies in the band of f K K N, where f is the Coriolis frequency, consistent with the statement made above that we are assuming the waves do not fall under the influence of rotation. Because the internal-wave period along the thermocline is around h, f has a period of 14.4 h, and the value of N FIG. 3. Variation of (a) linear phase speed c 0, (b) dispersive coefficient, (c) nonlinear coefficient, and (d) second-order nonlinear coefficient 1 as a function of water depth. The coefficients and 1 are in units of 10 2 s 1 and 10 4 m 1 s 1 respectively.

5 DECEMBER 2003 SMALL 2661 TABLE 1. Variation of EKdV coefficients with water depth. Also included is the depth at which the first mode has its maximum value, referred to as the scale depth, and the maximum internal-wave amplitude c predicted by EKdV theory. The large value of c at 800- m depth is probably due to a weakness of EKdV when applied at this large dpeth: this value does not affect the present simulations where the wave amplitude is O(10 m). The coefficients are defined in the appendix. Water depth (m) Phase speed c 0 (m s 1 ) (10 2 s 1 ) (m 3 s 1 ) (10 4 m 1 s 1 ) c (m) Scale depth H (m) in the thermocline is around 10 cph (see Fig. 2b), the inequality seems to be justified. When the frequency of the internal waves is much greater than f, the phase speed is only weakly dependent on water depth, as seen in Fig. 3a. The dispersive coefficient increases rapidly with water depth (Fig. 3b), while the nonlinear coefficient changes little and is always negative (Fig. 3c). This is typical of situations in which the stratification is concentrated in the upper part of the water column, and it may be compared with the two-layer case with upperand lower-layer thicknesses h 1 and h 2, respectively, where (h 1 h 2 )/h 1 h 2 (Small 2001a). Consequently in the two-layer case with h 1 h 2, is also negative. [When 0, KdV theory predicts that solitary internal waves must depress the interface (Ostrovsky and Stepanyants 1989), as observed in this experiment.] The second-order coefficient 1 (Fig. 3d) smoothly varies between and m 1 s 1, showing that 1 can take both signs in real density stratification as opposed to being always negative in two layers [Pelinovskii et al. (2000): however, they also found that 1 can be positive in a three-layer ocean]. TABLE 2. Speed across ground calculated and direction from mooring triangle 1 (S300, S200, S140). This includes the mean value and upper and lower bounds calculated using the uncertainties in arrival times due to the instrument sampling interval. Time of wave packet 19 Aug, A.M. 19 Aug, P.M. 20 Aug, A.M. 20 Aug, P.M. 21 Aug, A.M. 21 Aug, P.M. Speed across ground (cm s 1 ) Mean Lower Upper Propagation direction ( True) Mean Lower Upper TABLE 3. Speed across ground calculated and direction from mooring triangle 2 (S700, S300, S200). Time of wave packet 19 Aug, A.M. 19 Aug, P.M. 20 Aug, A.M. 20 Aug, P.M. 21 Aug, A.M. 21 Aug, P.M. Speed across ground (cm s 1 ) Mean Lower Upper Propagation direction ( True) Mean Lower Upper The environmental parameters were calculated for 140, 200 m, and then every 100 m of water depth to 1000 m: for points in between these depths the coefficients were found by linear interpolation. The spreading component [S(x)] is computed as discussed in Small (2001b; section 2b), and the variable depth term (x) is as given by Small [2001a, his (12) and (9)]. Water depths at each location along each ray were found from bilinear interpolation of the SES bathymetry (data from the LOIS SES CD). It should be noted that Doppler shifting of internal waves by the barotropic tidal current will not be considered, for both observations and model. As discussed in Small et al. (1999a), the barotropic tidal and slope currents were weak during the period of interest and were typically smaller than 0.1 m s 1. This value is less than the typical uncertainty in measured phase speed (Tables 2 4), and so the barotropic currents are neglected. 3. Simulation of the extent of refraction of the internal waves In this section a simulation of the extent of refraction of the internal waves is made. To this end the initial condition is chosen to be the same waveform on each ray, with an initial wavefront approximating the position of the internal wave A in Fig. 1. Throughout this paper a convention is made that the amplitude of an internal wave refers to the absolute value of the peak-to-trough displacement. In all cases the internal waves are of depression, and in most cases the peak-to-trough ampli- TABLE 4. Additional estimates of speed across ground and direction. The calculations marked * were derived from arrival times from the towed thermistor chain measurements and the S700 mooring. Those marked and # were derived from two positions on the wavefront in the SAR image and the S700 mooring for 20 Aug and 21 Aug. Time of wave packet 19 Aug, A.M.* 20 Aug, P.M. 21 Aug, P.M. # Speed across ground (cm s 1 ) Mean Lower Upper Propagation direction ( True) Mean Lower Upper

6 2662 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 FIG. 4. Initial conditions for the internal-wave simulations applied to each ray. Here the drop in the thermocline of D 30 m occurs over m either side of the 6-km-wide bucket : (a) complete domain, and (b) zoom on right-hand edge of feature. tude corresponds to the depression from the mean background level (unless specified otherwise). a. Idealized simulation In this case the initial internal wave is an idealized representation of the observations shown in Small et al. (1999a) that are summarized later in Fig. 10. The thermistor chain data of the internal-wave packet at the position T700 (see left-hand panel of Fig. 1 for location) show that the thermocline drops m over a period of about 5 min, equivalently 200 m. The idealized initial waveform has hence been chosen to be a smooth drop FIG. 5.(Continued) (b) Evolution and transformation of the waveform along the ray marked in black and white on (a). The amplitudes are incremented for each plot by 30 m. The water depth where the lead solitary wave is located is annotated at right (m). Initial waveform is bottom plot. Range axis is x ct, where c is the leadingwave speed. in the thermocline from the resting level. Various initial amplitudes of the drop (hereinafter called D) between 15 and 40 m have been tried to see if they evolve into similar features. The thermocline was assumed to be level behind the jump for a distance of 6 km (chosen to be similar to the typical observed wave packet length). At the end of this drop a restoration to the zero level was made to satisfy the model boundary condi- FIG. 5. (a) Results of nonlinear refraction model with initial peak trough amplitude of 30 m. Thick white lines show rays: thin white lines show wave fronts plotted every 1hofpropagation: grayscale contours are depth (m). Black and white striped line is the ray analyzed in (b). Mooring positions are shown as asterisks. Overlaid in thick black are the wave fronts from the SAR image of 20 Aug (Fig. 1) with the wave front A labeled in white and B1 and B2 labeled in black. Bottom and left axes indicate range in kilometers (together with the inner tick marks); top and right axes indicate the longitude and latitude, respectively (together with outer tick marks).

7 DECEMBER 2003 SMALL 2663 FIG. 6. (a) Evolution of wave amplitude and phase speed along the ray marked in black and white in Fig. 5a. Amplitudes are shown as asterisks. The analytical solitary-wave phase speed for these amplitudes and water depths is shown as diamonds. The ray-model phase speed is shown as a solid line. The figure is for nonlinear EKdV model with initial amplitude 30 m. (b) Bathymetry along ray. tions. The condition is illustrated in Fig. 4a. Note that this idealized initial condition is used to determine the general trend in wave refraction and evolution: more precise simulations involving real data as initial conditions are considered in section 3c. The drop in the thermocline was modeled with a tanh function with half-length scale L of 100 m (corresponding to a full extent of the front of m; see Fig. 4b). It should be noted here that the evolution was not very sensitive to the initial feature width: simulations with half-widths of 50 m and then of 200 m gave maximum differences of 2 m in amplitude (in the deep water, negligible in shallow water) and of 0.05 m s 1 in phase speed (again occurring in the deep water). This suggests that the model quickly adjusts to the natural waveform of the numerical system, and so the following simulations were all made with the same value of L 100 m. The idealized waveforms shown in Fig. 4 were applied to a set of rays perpendicular to an initial wave front. The initial wave front is a circular arc approximation to the wave front A in Fig. 1 and is shown in Fig. 5a (the deepest wave front). Figure 5a also shows the ray diagram for an initial amplitude of D 30 m with the wave front traces from Fig. 1 plotted for comparison. The extent of refraction is small for reasons discussed below. For the ray highlighted in Fig. 5a, the evolution of the waveform is shown in Fig. 5b. An undular bore of three four oscillations rapidly develops out of the initial form. Quickly the first three waves disperse and separate into solitary waveforms (by water depths of m). When the lead wave has reached 443-m depth it has an amplitude of 50 m and is separated from the second wave by 1300 m. By the time the lead wave reaches 213-m depth the first and second waves have amplitudes of 50 and 40 m, respectively, and have not dispersed further. As the waves move onto the shelf, the lead waves remain separated by m and the first two waves have similar amplitudes of 40 m. The third wave is considerably weaker and lags the second wave by 2 km by the end of the run. Figure 6a summarizes how the amplitude of the lead wave varies with range and water depth. A significant difference between linear and nonlinear predictions is that the nonlinear waves move faster, as expected. The phase speed of the lead wave along the ray highlighted in Fig. 5a is shown in Fig. 6a along with the corresponding bathymetry in Fig. 6b. The phase speed varies from 0.85 m s 1 in m depth to 0.61 m s 1 in 140-m depth (as compared with 0.61 m s 1 and 0.48 m s 1, respectively, in the linear case, Fig. 3a) As a result the waves travel some 3 5 km farther in the nonlinear case over the 12.4 hours of computation, relative to the linear case. The ratio of deep-water phase speed to shallow-water phase speed is quite small (1.4) in this nonlinear case: this is similar to the ratio for the linear case (1.3) and implies that here nonlinearity has not significantly affected the extent of refraction. This can be seen from the change in propagation direction expected from Snell s law: ref ref sin c, (3) in in sin c where in and ref refer to the incident and refracted angles, respectively, between the wave direction and the gradient of bathymetry and c in and c ref are the corresponding phase speeds of the incident and refracted wave. Taking a typical initial angle to the bathymetry gradient as 20 and using the linear (nonlinear) phase speeds in deep water and shallow water quoted above as the incident and refracted speeds gives a final angle of 16 (14 ), a rather small extent of refraction of 4 (6 ). Also plotted in Fig. 6a is the expected theoretical EKdV phase speed for a solitary wave of the amplitude shown by the asterisks. There is reasonable agreement between the expected speed and the speed from the model results, suggesting that the internal waves produced by the model become pure solitary waves. A summary of the results of all the nonlinear runs is shown in Fig. 7 (dotted lines), for different initial wave amplitudes from 15 to 40 m and for the ray highlighted on Fig. 5a. In the deep water, the curves show similar trends with a rapid initial increase in absolute amplitude between 770 and 750 m as the high-frequency waves start to form (cf. Fig. 7a with Fig. 6a). The smallest-

8 2664 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 FIG. 7. Summary of all numerical model results and comparison with observations: (a) amplitude variation with depth, (b) variation of phase speed with depth. The simulations with idealized initial conditions (section 3a) are shown as dotted lines and annotated at left with initial wave amplitude D. The simulations with realistic initial conditions (section 3c) are shown as a dashed line (with spreading) and as a dot dash line (no spreading) see legend in (b). The observations are shown as symbols connected by solid lines and are annotated in (a), and the same symbols are used in (b). For clarity, in (b) error bars for the observations are not shown see Tables 2 4. amplitude cases (15 and 20 m) continue to grow slightly in amplitude until the end at 140 m. It will be shown in section 3b that this is because the wave is sufficiently small to be governed by KdV dynamics. However, as the initial absolute wave amplitude increases, the initial growth is reversed after around 500-m depth, and in the shallow water of 140 m the wave amplitude rapidly decreases toward a value around 40 m. In section 3b it will be shown that this is because these larger waves are being governed by EKdV dynamics. The phase-speed variation is shown in Fig. 7b (dotted lines): because this is a derived quantity from the model output, it tends to be less smooth than the amplitude variation, and the kinks in the curve may be due to an insufficient number of samples available (in the water depths of interest) to make an estimate. However, a general trend is discernable of a decrease in speed with decreasing depth after the initial period of wave development (in m), and the phase speed increases with wave amplitude in the depth range m. On the continental shelf, the tendency of the amplitudes to be limited leads to a capping in phase speed at around 0.6ms 1, in agreement with the EKdV solitary wave value seen in Fig. 6a (solid line). b. Adiabatic predictions of solitary internal waves In the previous section an example was shown of how the refraction model predicted solitary-wave evolution out of an initial drop of the thermocline. The behavior

9 DECEMBER 2003 SMALL 2665 FIG. 8. (a) Adiabatic predictions of the shoaling of internal solitary waves initially in a water depth of 400 m: different initial amplitudes of 10, 25, and 50 m. Dashed lines represent KdV theory and solid lines are EKdV predictions. (b) The limiting amplitude c predicted by (4) from the EKdV coefficients of Fig. 3. of the solitary waves can be interpreted using theoretical predictions of the evolution and shoaling of EKdV internal solitary waves. Under the adiabatic assumption that the horizontal scale over which the depth varies is long in comparison with the internal-wave length, so that the internal wave always has time to adjust to the local solitary-wave solution, the conservation of energy can be used to derive how the solitary wave changes with depth. In Small (2001a, section 2c) this technique was used to study the evolution of KdV and EKdV waves. Here this method is used to compute the adiabatic evolution of internal solitary waves based on the KdV and EKdV coefficients shown in Fig. 3. The evolution of solitary waves of different amplitudes initially in 400-m water depth has been investigated. This initial depth of 400 m was chosen because the observations suggested that the internal waves had reached a solitary state by that location (Fig. 10), appropriate for application of the theory. The adiabatic predictions (Fig. 8a) show that the difference between the KdV and EKdV predictions increases with increasing initial amplitude. For a small 10-m initial wave, KdV predicts a gradual shoaling to a final amplitude of 15 m in 140-m water depth, and the EKdV prediction is similar with a final amplitude of 14 m. For higher initial amplitudes the KdV prediction remains that the final amplitude will be 1.5 times the initial amplitude, so that a 50-m initial wave grows to 75-m amplitude in 140-m water depth, an unrealistic amplitude that would imply the seasonal thermocline would intersect the seabed. (In reality it is expected that turbulent effects would dominate before this amplitude is reached.) In contrast, as the initial amplitude is increased, the EKdV predicts that the ratio of final amplitude to initial amplitude actually decreases as the initial amplitude increases. The effect is notable for an initial 25-m wave and striking for an initial 50-m wave, where the final amplitude is less than the initial. An important parameter of relevance here is the maximum wave amplitude c according to EKdV dynamics (Stanton and Ostrovsky 1998): c. (4) Using the environmental parameters displayed in Fig. 3, the absolute value of c was calculated and has been shown in Fig. 8b and Table 1. The values of c generally decrease with decreasing water depth, reaching 64 m in 200-m depth and 46 m in 140-m depth. Note that where and 1 are negative, then c will also be negative, implying that maximum amplitude wave is one of depression. For KdV waves there is no limiting wave amplitude. It can be seen from Fig. 8 that while shoaling the small-amplitude initial wave nowhere approaches the limiting value c but, as the initial amplitude is increased, the wave progressively becomes closer to the limiting values on its path. For the large-amplitude initial wave (50 m), the result is that the wave amplitude is severely capped and decays to the value of c 46 m in the final depth of 140 m. These predictions help to explain the EKdV refraction model results presented in section 3a. There, an idealized waveform of a drop in thermocline level D rapidly developed into a set of solitary waves. In Fig. 7a (model results), the cases with the smallest values of initial displacement D show a slow and steady increase in amplitude as the shallow water is approached. In contrast, for the larger values of D, the solitary waves stop growing after reaching m water depth and subsequently decay subject to the effect of the limiting amplitude of c, eventually falling to a value close to that of c when in 140-m water depth. c. Realistic case In the next experiment the initial waveform was taken from towed thermistor chain data, which gave the finest resolution of the internal waves (the thermistor chain track is shown in Fig. 1a: its sampling interval was 1 s). Initial conditions for the model were derived from the measurements at T700 by removing the Doppler shift of an internal wave moving relative to the ship, as 1

10 2666 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 described in Small et al. (1999a). A single representative waveform was derived from the towed thermistor data as follows. First the mean depth of each isotherm was calculated from the moored timeseries at S700, over a 24-h period from 0000 UTC 19 August to 0000 UTC 20 August. The displacement of the isotherms at T700 from the mean values was then constructed for seven isotherms (from 10.5 to 13.5 C at intervals of 0.5 C), and these displacement series were simply averaged. The waveforms were then inserted into a domain for numerical simulation of evolution. For the simulations it was required that the waveform was equal to zero at the boundaries, and so a Gaussian tail was added to the waveforms to make it decay it to zero well before the boundaries as shown in Fig. 9b (bottom plot). This new waveform was then used as initial conditions for the model. The Gaussian tail had an e-folding length of 1 km: further experiments showed that the final results were not sensitive to this length scale. The experiment assumed an initial wavefront of the arc form of Fig. 5a, but located 4800 m farther onshelf so that the wavefront passes through T700 (see Fig. 9a). A group of rays were constructed, each perpendicular to the wavefront. For each ray, the initial waveform just described was placed in the evolution model domain such that the leading edge of the feature in the waveforms was located at the position of the wave front. It should be mentioned here that there is no a priori reason for assuming the waveform should be the same along each ray: in fact, this is unlikely because each ray has passed over different bathymetry tracks and so the wave evolution is likely to be different. However, this section will focus on the evolution along the ray that passes through the observation point, and the other rays are only used to simulate any approximate spreading effects on that ray. Figure 9a shows the refraction diagram for the first 8.5 h of evolution. The extent of spreading is similar to that shown in Fig. 5a. The ray marked in black in Fig. 9a starts close to T700, and the evolution plot for this ray is displayed in Fig. 9b. The initial waveform quickly develops into a set of two primary waves and following small oscillations by 530-m depth. By the time the lead wave reaches the depth of 423 m, the peak trough amplitude is 48 m (see close-up in Fig. 9c) and the second wave has an amplitude of m. At 293-m depth, the lead wave has an amplitude of 55 m and that of the following wave is 40 m. The separation of the leading waves in the model is 1 km. By the time it reaches a depth of 202 m (shelf break) the lead wave has a peak trough amplitude of around 62 m (Fig. 9c). After this the waves slowly reduce in size as they move over the shelf, and by the time the lead wave is in 146 m of water (Fig. 9c), after 5.5 h of propagation, it has a 55-m peak trough amplitude, while the second wave is actually larger (60 m) and has approached the lead wave, being 600 m separated (the waves never merge, because their amplitudes and hence phase speeds become closer as they propagate across the shelf). By the end of the model evolution (after 8.5 h of propagation), all wave amplitudes are 50 m or less. The phase speed in the model initially rises as the wave transformation takes place, from 0.6 to 0.7 m s 1 (between depths of 700 and 600 m, not shown) before decreasing to 0.55 m s 1 on the shelf (summarized in Fig. 7b, dashed line). These values are less than the corresponding EKdV speeds for waves of those amplitudes (by m s 1 ), although there is evidence that the model and the EKdV solitary wave speeds are converging as the waves propagate farther onshelf. This is discussed further in the following section. Last, the impact of ray spreading on wave amplitude was assessed by running an extra simulation with the spreading term S(x) in (1) switched off. The simulated wave amplitudes and phase speeds in this case are shown in Fig. 7 (dot dash line). It can be seen that the amplitudes are some 5 10 m greater than for the case with spreading, while the phase speed is slightly larger (by 0.01 m s 1 ) with no spreading. Hence, it may be seen that the wave front spreading in this case is important to modeling the shoaling of the waves. 4. Comparison of simulations with observations The numerical simulations presented above may be compared with observations from the SES and SESA- ME experiments. Previous analysis of nonlinear internal-wave conditions in the experiments has discussed the SES (Inall et al. 2000, 2001) and SESAME (Small et al. 1999a,b) data separately. Small (2000) then detailed the inferences from the combined datasets, from which the examples shown here are taken. a. Shoaling The most complete picture of the internal tide transformation from in situ measurements was obtained on the morning of 19 August when a towed thermistor chain survey took place. The towed and moored data were converted to an approximate snapshot by removing the Doppler shift caused by the waves moving relative to the ship or the mooring, using the method described in Small et al. (1999a). The transformation of thermocline depth (Fig. 10a) and upper layer current (Fig. 10b) on 19 August is then shown as a function of range. The range series for each measurement are each incremented by 10 km for display purposes. The plot is arranged so that the right of each record is the eastward (shelf) side, and the left is the westward (ocean) side. The data on the morning of 19 August show a consistent pattern from the thermistor records and the current records (Figs. 10a,b). At T700 there is a drop of nearly 40 m in thermocline depth followed by a partial restoration to the original level: (the drop, or bore or jump, is from a shallow level on the right-hand, shelf

11 DECEMBER 2003 SMALL 2667 FIG. 9. Results of nonlinear refraction model, real initial conditions. (a) Ray diagram. Thick white lines show rays: thin white lines show wave fronts plotted every 1hofpropagation: grayscale contours are depth (m). Thick black line is the ray analyzed in (b) and (c). Mooring positions are shown as asterisks and thermistor chain measurements of the internal wave packet as diamonds. (b) Evolution and transformation of the waveform along the ray marked in black on (a). The amplitudes are incremented for each plot by 60 m. The water depth where the lead solitary wave is located is annotated at right (m). Initial waveform is bottom plot. Range axis is x Vt, where V is a constant velocity: 0.6 m s 1. (c) Close-up of numerical results for realistic simulation; 8-km-long segments of the main wave packet from four of the timesteps are shown. Each section is incremented by 10-km range. The water depth at which the lead solitary wave was located is annotated beneath each section. side to a deeper level on the left, ocean side). The current record at S700 describes a bore followed by an oscillation, slightly undersampled: the total current change across the bore is 0.55 m s 1. The T400 record shows a leading solitary wave of amplitude 50 m followed by another slightly smaller wave splitting in two, and then a small wave of 5-m amplitude. At S300 the thermocline displacement only resolves two waves (owing to the coarse 10-min sampling) of similar amplitude to those at T400, while the current record indicates that the lead wave has a current pulse of 0.6 m s 1 (the 5- min sampling here gives slightly better resolution). The

12 2668 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 FIG. 10. Transformation of the wave packet on the morning of 19 Aug: (a) depth of the thermocline from thermistor records for the 14 C isotherm at T700, T400, and S140, and the 12 C isotherm at S300. (b) Current velocity in direction of propagation (120 T) at 30-m depth. The approximate range series are annotated with the mooring identity and the isotherm in (a). The origin of each section is incremented by 10 km. waves are separated by 1500 m. At S200 there are three well-defined waves, the lead wave having the largest current of m s 1, and the leading wave separation is m. (The thermocline displacement for S200 is not shown because the full depth range of the thermistors was limited to between 2- and 42-m depth: however, it did indicate the presence of the three significant waves without capturing the full amplitude). At S140 three large waves followed by smaller oscillations are clearly seen in the thermistor and current records: the lead wave has amplitude of at least 30 m (the flat bottom of the wave suggests that the 14 C isotherm is displaced somewhat below the deepest measurement at 55-m depth) and an associated current pulse of m s 1. The thermistor string also showed that the displacement of the 15 C isotherm was at least 35 m. 1 The waves appear still to be rank-ordered at S140, and the first wave is separated by m from the following wave. Figure 7 summarizes the simulations of section 3 and compares the results with the observations. Here the observed amplitudes (Fig. 7a) from the six tidal cycles 1 For comparison, the maximum measured amplitude at S140 (during a period when a deeper thermistor string was in place, August 1995, data not published) was 50 m. The signature of that wave at 55-m depth was similar to that observed on 19 August, as was the background density stratification, suggesting that the waves had a similar amplitude. Hence, a reasonable estimate of the actual wave amplitude on 19 August (A.M.) is between 35 and 50 m. of August 1995 (presented as symbols joined by lines) are a combination of real measurements and inference from current profiles. The amplitude of thermocline displacement was only precisely measured on the morning of 19 August when the towed thermistor chain was in operation. From the adjacent thermistor and current-meter measurements for that morning, the ratio of current velocity in the upper ocean in the direction of wave propagation to the thermocline displacement was calculated. Then this ratio was applied on the subsequent tidal cycles (when accurate displacement data were not available) to infer the amplitude of displacement from the current. (Note that this manner of conversion is exact for a linear wave but only approximate for a nonlinear wave, and hence it is used only as a guideline.) The observations of lead-wave amplitude show similar trends to the simulations with the idealized initial conditions (shown as dotted lines in Fig. 7a) with an initial growth in amplitude followed by a reduction between depths of 300 and 140 m. For the simulations, this was due to the limited growth of EKdV solitary waves (section 3b), while for the observations, this is due to a combination of laminar internal-wave dynamics, and possibly damping through mixing (see below). The idealized simulations also showed wave separations of m at the front of the packet: this may be compared with the observed m over the slope and m at S140 on the shelf (see Fig. 10).

13 DECEMBER 2003 SMALL 2669 As discussed above, the cases with realistic initial conditions (Fig. 7a, dashed line) gave rise to larger amplitudes on the shelf than the idealized simulation (Fig. 7a, dotted lines). The amplitudes are larger than those predicted by (4) for EKdV solitary waves in this depth (46 m: see Table 1). This is because the internal waves in this case are riding on a longer wave that is elevating the thermocline (see Fig. 9c) and are no longer pure EKdV solitary waves. This situation arises because of the complexity of the initial condition, which both raised and lowered the thermocline: compare it with the situation in section 3a where a smooth depression of the interface gave rise to theoretical EKdV internal solitary waves. In fact, it appears that the amount by which the internal waves in Fig. 9c depress the interface below the mean level is similar to the EKdV maximum amplitude, but in this case it is not equal to the peak-totrough amplitude because of the additional effect of the long background wave raising the thermocline. The simulated amplitudes for the realistic case at the shelf depth of 140 m (55 60 m) are slightly larger than those observed (on 19 August the final wave amplitude was between 35 and 50 m; see footnote 1). However, the model amplitude decayed to 50 m as the waves continued propagating across the shelf. Possible reasons for the difference between model and observations include the limiting assumptions inherent in the model (such as weak nonlinearity and no dissipation mechanism) and that the observations were not all gathered along one ray path and hence show along-wavefront variations that are difficult to compare with the model. On some tidal cycles the observed final amplitude was just m. In one case (the afternoon of 20 August: Fig. 7a, plus symbol) the amplitude drops dramatically from 55 m in 300-m depth to 10 m in 140-m depth. One possible reason for this decay is mixing. Turbulent effects were evident in some of the observations of internal solitary waves during SES and SES- AME. Inall et al. (2000) measured turbulent dissipation during the passage of the internal waves past S140 during the afternoon of 21 August 1995 and found an average diapycnal eddy diffusivity of 5 cm 2 s 1. Further, Small (2000) found evidence that the necessary conditions for gravitational (Orlanski and Bryan 1969) and shear instability (Miles 1961) were satisfied in the large internal solitary waves observed on the shelf. Turbulence and breaking ensuing from these instabilities are likely to explain the small amplitudes of some observed waves at S140 after the waves have crossed the slope (Fig. 7a). A contour plot of the Richardson number Ri within the idealized model simulation in 443-m water depth (Fig. 11a) shows patches where Ri 0.25 near the thermocline in the internal-wave troughs, indicating that the waves are susceptible to shear instability (Miles 1961). [For these computations Ri has been calculated using the instantaneous buoyancy frequency M 2, and the linear relationships between current u(x, z, t) in the FIG. 11. (a) Richardson numbers in modeled internal-wave packets, from the model run illustrated in Fig. 5b, lead wave in a water depth of 443 m. Solid line contour is of Ri 0.25; dashed line is Ri 1. (b) Comparison of the phase speed (solid line) and maximum particle velocity (asterisk) corresponding to the simulation shown in Fig. 5b. direction of propagation, density perturbation (x, z, t), wave amplitude (x, t), and mode (z): M Ri 2 : ( u/ z) 2 g [ *(z) (x, z, t)] 2 M, z 0 (z) u(x, z, t) c0 (x, t), and z *(z) (x, z, t) (x, t) (z). (5) z Modifications to this formulation to include higher-order expansions of u and still resulted in Ri 0.25 in the troughs of the waves.] Values of Ri 0.25 continued to be found as the waves propagated onshelf. Further, a comparison of maximum onshelf current U max with the phase speed c of the leading internal wave (Fig. 11b) shows that U max c in depths of 500 m or less, sug-

14 2670 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 FIG. 12. Propagation directions and speed across ground deduced from wave packet arrival times at moorings. The dashed lines show the calculated wave front orientation, (with an arbitrary length of 10 km, comparable to that seen in SAR) and the numbers beside the wave front are the speed across ground in cm s 1. (a) Morning of 19 Aug. (b) Afternoon of 20 Aug. The SES mooring positions are shown as diamonds. In (a) the positions of the wave packet crossing by the towed thermistor chain are marked with asterisks. In (b) the position of the lead wave front on the SAR image is shown as a thick line marked SAR. More details are contained in Table 2, which describes the uncertainty of the measurements. Bathymetric contours at 180 and 200 m and then at every 100 m are shown. gesting that gravitational collapse may occur. Hence, it is to be expected that mixing processes should be important in this idealized simulation, and also in the realistic simulation in which the amplitudes are larger. The lack of turbulent dissipation in the model is one reason why the simulated amplitudes on the shelf in 140-m water depth (60 m) in the realistic experiment are larger than observed. In terms of the modeling results, the wave amplitudes may easily be reduced to a value for which the instability criteria are not satisfied by imposing a simple Laplacian form of horizontal diffusion with sufficiently large value of the constant eddy viscosity coefficient. This method may also bring the simulated wave amplitudes closer to those observed. This approach is not taken here because it is not likely to realistically model the behavior of turbulent internal solitary waves where dissipation is likely to be sporadic and localized (see, e.g., Inall et al. 2000; Pinkel 2000). Nor is the model of Bogucki and Garrett (1993) used here, because their method is most appropriate for KdV interfacial waves: for a realistic stratification their method is dependent on a parameter (the ratio of downward to upward mixing thickness, their ), the value of which is not fully understood. Last, it may be said that the inclusion of dissipation is not appropriate in the framework of EKdV theory, which is weakly nonlinear and excludes the strongly nonlinear effects that lead to dissipation. Instead, current and future work is focusing on a comparison of the EKdV predictions with those from a fully nonlinear computational fluid dynamics (CFD) model (Hornby and Small 2002). The CFD model has the capacity to model turbulent kinetic energy and its dissipation rate and, hence, should address the importance of mixing and strong nonlinearity in the evolution of internal solitary waves. b. Refraction Details on the propagation speed and direction of the internal waves were derived from the arrival times at the moorings and the SAR images, where available, by applying simple geometry. From the internal-wave signatures on Fig. 1 it can be seen that the internal wavefronts can be assumed to be quasi-planar on O(km) length scales. The longest distance between successive moorings along the south line is 6 7 km. So within any contiguous trio of moorings a planar wavefront has been assumed in order to compute the wave speed and direction from the arrival times. This was performed in Small (2000) and is summarized in Tables 2 4. Two illustrative examples of the nature of the observed phase velocity of the internal waves are contained in Fig. 12, for the tidal cycles on the morning of 19 August when extra towed thermistor chain data were available and the afternoon of 20 August when SAR imagery supported the analysis. Idealized planar wave fronts are shown aligned at right angles to the calculated