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

DECEMBER 2003 SMALL 2657

JUSTIN SMALL International Paci®c Research Center, School of Ocean and Earth Science and Technology, University of Hawaii at Manoa, Honolulu, Hawaii

(Manuscript received 4 December 2001, in ®nal 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 1995. The model is ®rst 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 contained online at http://iprc.soest.hawaii.edu/ϳmiyata/ IWavesPublicationList.htm and also see http://atlas.cms. Recently the study of nonlinear and solitary internal udel.edu), and the detailed structure of the solitary waves waves has received much attention, often motivated by (e.g., Choi and Camassa 1999; Stanton and Ostrovsky the ability of the waves to disrupt acoustic propagation 1998; Small et al. 1999a). (Headrick et al. 2000) and sea-drilling operations (Bole et The aim of this paper is to investigate the importance al. 1994), to displace and sometimes transport nutrients of refraction and shoaling effects on nonlinear internal (Lamb 1997; Sharples et al. 2001), to contribute to cross- shelf exchange (Inall et al. 2001), and to cause ocean waves of the type observed in deep water off the Malin turbulence (Inall et al. 2000; Pinkell 2000). Recent studies slope off the United Kingdom. Oceanographic data are have focused on the generation of the internal tide and utilized from the Shelf Edge Study (SES) and Shelf associated nonlinear waves at topography (e.g., Farmer Edge Study Acoustic Measurement Experiment (SES- and Armi 1999; Gerkema 1996; Lamb 1994), the evolution AME) experiments, which took place at the Malin slope of solitary internal waves from long internal waves (e.g., and shelf in 1995 and 1996 (see left panel of Fig. 1 for Holloway et al. 1997, 1999; New and Pingree 2000; Lamb location and positions of instrumentation). The current and Yan 1996), the detection of internal waves by synthetic paper investigates nonlinear internal waves that cross aperture radar (SAR; Brandt et al. 1996; Zheng et al. 2001; the slope and are associated with an internal tide. Nu- Small et al. 1999a; and many othersÐsee the references merical 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 * International Paci®c Research Center Contribution Number 212 study signi®cantly extends that of Small et al. (1999a,b) and School of Ocean and Earth Science and Technology Contribution to include numerical simulations over a sloping bottom, Number 6231. including the effects of second-order nonlinearity and refraction. It is believed to be the ®rst time these effects Corresponding author address: Dr. Justin Small, International Pa- have been simulated in unison in a realistic application, ci®c Research Center, School of Ocean and Earth Science and Tech- and it follows on from the idealized studies of Small nology, University of Hawaii at Manoa, 2525 Correa Rd., Honolulu, HI 96822. (2001a,b). E-mail: [email protected] This paper focuses on particular packets of high-fre-

᭧ 2003 American Meteorological Society

Unauthenticated | Downloaded 10/04/21 06:19 AM UTC 2658 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 SAR image of the Malin shelf edge, 1136 UTC 20 Aug 1995. SES bathymetry ERS-1 (right) contours of 180,internal-wave 500, feature and 1000 markedcontours. m A Farther are onshelf is there markedwhich present are in may sets on black have of the resulted internal (dashed),edge from waves, slope fromof generation the the between right ®rst during swath. two the to themoorings Also marked previous are marked left. 500- B1 tidal are marked and An cycle. and internal-wave as B2, Darkto interaction diamonds, 1000-m area right. zones, denoting is as S700, the I. S400, Positions S300, of SES S200, and S140 from left . 1. (left) Location of the main experiment area discussed in this paper. SES moorings IG F are marked with0000±0200 diamonds UTC 19 and Aug 1995measurements labeled is of shown internal S700±S140. as waves the are The thick markedare dashed with labeled thermistor line. asterisks. T700 The The and chain times leading-wave T400 measurements of andof survey the are thermistor de®ned the in track the following text.internal-wave The waves times fronts of from thermistor in measurements the the SARlines image for packet of the 20 are right-handjoining Aug marked panel asterisks are (1136 2 for marked UTC), 21 as (i.e.,in Aug thick including the second solid 1995 text. the (1136 wave), Bathymetry following UTC).on 3, contour The waves, the levels feature and and right (m) A 4.longitude are as is and axes, at The the a respectively; getting every main right thick deeper 100-m wave andoutside solid top discussed intervals, to the show line starting the the plot at corresponding refer left. range 200 (km). to m Left (Tick range marks and and bottom inside axes the show plot refer the to latitude longitude and or latitude.)

Unauthenticated | Downloaded 10/04/21 06:19 AM UTC DECEMBER 2003 SMALL 2659 quency internal waves that were observed on each tidal 1999) to describe the wave evolution and a ray method cycle between 19 and 21 August 1995 during neap tides to describe the refraction. The present model modi®es [see, e.g., Inall et al. (2001), their Fig. 3a]. Inall et al. the EKdV ray model of Small (2001a,b) to handle re- (2000) showed that in August 1995 the energy of high- alistic ocean strati®cation and bathymetry. A normal- frequency waves was greatest during the neap tide pe- mode approach is taken as described in the appendix. riod, and this may be due to the in¯uence of distantly Only the dominant mode is chosen for analysis, and generated internal tides that would not necessarily arrive here the ®rst mode is selected in accordance with the at the Malin shelf break at the time of greatest local observations described in Inall et al. (2000) and Small tidal forcing. et al. (1999a). The EKdV ray model is given by In fact, the particular internal waves being studied 2 here do not appear to be products of a locally forced ␩t ϩ c0(x)␩xxϩ ␣(x)␩␩ ϩ ␣1(x)␩␩xxxxϩ ␥(x)␩ internal tide. This is suggested by the depth of water where the waves were ®rst observed (500±900 m, much ϩ ␯(x)␩ ϩ S(x) ϭ 0, (1) deeper than the shelf edge around 200-m depth), com- where ␩(x, t) is a waveform, (approximating the max- bined with the onshelf direction of propagation (see imum displacement found in the water column), t is Small et al. 1999a). The signature of the internal waves time, x is range along the ray of interest, and subscripts may be seen on the SAR image of 1136 UTC 20 August denote differentiation. The variable coef®cients of (1) 1995 [right panel of Fig. 1; a portion of this image was [c (x), ␣(x), ␣ (x), ␥(x), see appendix] are here termed shown in Small et al. (1999a)]. The feature labeled A 0 1 the environmental parameters, and ␯(x) and S(x) are the on the SAR image corresponds to the internal-wave shoaling and spreading terms (see Small 2001a,b). The packet that passed the moorings late on the afternoon of 20 August, while those labeled B1 and B2 are spec- numerical scheme of the model was described in Small ulated to have passed the moorings early on 20 August (2001a,b), and model testing was reported in Small (and have moved farther onshelf by the time of the (2000), which veri®ed that in idealized situations the model predictions were accurate in comparison with an- image), an M2 tidal cycle earlier. Further, the tidal ¯ow U, which at the time of ob- alytical EKdV solutions. servation was at its weakest around 0.1±0.2 m sϪ1, never The effect of rotation was also brie¯y investigated by approached the phase speed c of the internal waves considering the rotation-modi®ed EKdV equation (Hol- (measured at between 0.6 and 0.9 m sϪ1), indicating that loway et al. 1999), which includes rotation to ®rst order. the Froude number Fr ϭ U/c never approached a critical In fact it turned out that in this situation the effect of value of unity for hydraulic jump formation (see e.g., rotation on the internal-wave amplitudes was a very Brandt et al. 1996). These preliminary results suggest small reduction in amplitude. Small (2000) gives some that the features observed during the time of interest to discussion of this point and notes that the observed this paper, 19±21 August 1995, at the Malin slope during phase speeds suggest that rotation is not signi®cantly neap tides, were generated elsewhere. Small (2000) sug- affecting the internal waves because of the relatively gested that likely generation sites were Rockall Bank, high frequency of the waves. It is acknowledged that a a known generator of internal tides (DeWitt et al. 1986), complete description of the development of the internal and/or Anton Dohrn seamount, one of the seamounts tide from its long linear form to the high-frequency that Xing and Davies (1998) demonstrated numerically disintegration may involve the use of a governing equa- to have an important in¯uence on the internal tides in tion with full rotation, such as developed by New and the region. A more comprehensive discussion of the Pingree (2000). possibility of local and distant generation of internal To compute exact environmental parameters at each tides at the Malin shelf break is given in Small (2000). water depth, knowledge of the strati®cation and back- The main topic of this present paper is how the waves ground current shear is required. Data for these quan- behave as they reach the continental slope and undergo tities were taken from the Land Ocean Interaction Study refraction and shoaling. (LOIS)±SES CD-ROM, available from the British The paper is organized as follows. Section 2 introduces Oceanographic Data Centre at the address given in the the numerical model. Section 3 describes numerical sim- acknowledgements. The background density strati®ca- ulations of the transformation process, using both ide- tion of the Malin shelf/slope environment was domi- alized and realistic initial conditions, together with adi- nated by a strong seasonal thermocline that extended abatic predictions of solitary-wave behavior. Section 4 from near the surface to 50 m (Fig. 2a). The background then compares the simulations with the measurements buoyancy frequency N(z) given by from SES and SESAME and discusses the limitations of the model. Last, concluding remarks are presented. *␳ץ 1 N 2(z) ϭϪg , (2) zץ ␳ 2. The numerical model of nonlinear internal wave 0 ΂΃ refraction where g is gravitational acceleration and ␳ 0 and ␳*(z) The model employs second-order nonlinear extended are a reference and a pro®le of background potential Korteweg±de Vries (EKdV) theory (Holloway et al. density, respectively, is a measure of the strati®cation.

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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 ®rst

FIG. 2. Density strati®cation, 19±21 Aug 1995: (a) potential density indication that the refraction will be limited over the pro®le to 1000-m depth and (b) buoyancy frequency to 150-m depth. 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, The peak buoyancy frequency was around 10 cph cen- where f is the Coriolis frequency, consistent with the tered around 30-m depth (Fig. 2b). statement made above that we are assuming the waves In this paper there will be no discussion of how do not fall under the in¯uence of rotation. Because the changes in strati®cation with range affect internal-wave internal-wave period along the thermocline is around paths. This is because reliable density pro®les over a 0.5±1 h, f has a period of 14.4 h, and the value of N

FIG. 3. Variation of (a) linear phase speed c0, (b) dispersive coef®cient ␥, (c) nonlinear coef®cient

␣, and (d) second-order nonlinear coef®cient ␣1 as a function of water depth. The coef®cients ␣ Ϫ2 Ϫ1 Ϫ4 Ϫ1 Ϫ1 and ␣1 are in units of 10 s and 10 m s respectively.

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TABLE 1. Variation of EKdV coef®cients with water depth. Also TABLE 3. Speed across ground calculated and direction from included is the depth at which the ®rst mode has its maximum value, mooring triangle 2 (S700, S300, S200). referred to as the scale depth, and the maximum internal-wave am- Speed across ground Propagation direction plitude ␩c predicted by EKdV theory. The large value of ␩c at 800- (cm sϪ1) (ЊTrue) m depth is probably due to a weakness of EKdV when applied at Time of wave this large dpeth: this value does not affect the present simulations packet Mean Lower Upper Mean Lower Upper where the wave amplitude is O(10 m). The coef®cients are de®ned in the appendix. 19 Aug, A.M. 77 69 77 115 100 130 19 Aug, P.M. 92 80 102 122 105 138 Phase ␣1 Scale 20 Aug, A.M. 75 65 85 134 120 145 Ϫ4 Ϫ1 Water speed c0 ␣ ␥ (10 m depth H 20 Aug, P.M. 74 66 83 127 113 140 Ϫ1 Ϫ2 Ϫ1 3 Ϫ1 Ϫ1 depth (m) (m s ) (10 s ) (m s ) s )|␩c| (m) (m) 21 Aug, A.M. 72 64 81 130 116 141 21 Aug, P.M. 73 65 85 140 126 150 140 0.48 Ϫ1.3 377 Ϫ2.8 46 45 200 0.52 Ϫ1.5 703 Ϫ2.4 64 54 300 0.55 Ϫ1.7 1351 Ϫ2.0 81 65 400 0.57 Ϫ1.7 2110 Ϫ1.7 100 77 500 0.59 Ϫ1.7 3049 Ϫ1.3 122 88 The environmental parameters were calculated for 600 0.60 Ϫ1.7 4229 Ϫ1.0 165 102 140, 200 m, and then every 100 m of water depth to 700 0.61 Ϫ1.6 5623 Ϫ0.5 288 123 1000 m: for points in between these depths the coef®- 800 0.62 Ϫ1.5 7696 0.2 787 170 cients were found by linear interpolation. The spreading 900 0.63 Ϫ1.4 10 543 1.1 124 234 1000 0.65 Ϫ1.2 15 064 2.1 55 317 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 in the thermocline is around 10 cph (see Fig. 2b), the bilinear interpolation of the SES bathymetry (data from inequality seems to be justi®ed. When the frequency of the LOIS±SES CD). the internal waves is much greater than f, the phase It should be noted that Doppler shifting of internal speed is only weakly dependent on water depth, as seen waves by the barotropic tidal current will not be con- in Fig. 3a. sidered, for both observations and model. As discussed The dispersive coef®cient increases rapidly with ␥ in Small et al. (1999a), the barotropic tidal and slope water depth (Fig. 3b), while the nonlinear coef®cient ␣ currents were weak during the period of interest and changes little and is always negative (Fig. 3c). This is were typically smaller than 0.1 m sϪ1. This value is less typical of situations in which the strati®cation is con- than the typical uncertainty in measured phase speed centrated in the upper part of the water column, and it (Tables 2±4), and so the barotropic currents are ne- may be compared with the two-layer case with upper- glected. and lower-layer thicknesses h1 and h 2, respectively, where ␣ ϰ (h1 Ϫ h 2)/h1h 2 (Small 2001a). Consequently in the two-layer case with h1 Ͻ h 2, ␣ is also negative. 3. Simulation of the extent of refraction of the [When ␣ Ͻ 0, KdV theory predicts that solitary internal internal waves waves must depress the interface (Ostrovsky and Ste- In this section a simulation of the extent of refraction panyants 1989), as observed in this experiment.] The of the internal waves is made. To this end the initial second-order coef®cient ␣ (Fig. 3d) smoothly varies 1 condition is chosen to be the same waveform on each between 2.8 10Ϫ4 and 1 10Ϫ4 mϪ1 sϪ1, showing Ϫ ϫ ϫ ray, with an initial wavefront approximating the position that ␣ can take both signs in real density strati®cation 1 of the internal wave A in Fig. 1. Throughout this paper as opposed to being always negative in two layers [Pe- a convention is made that the amplitude of an internal linovskii et al. (2000): however, they also found that ␣ 1 wave refers to the absolute value of the peak-to-trough can be positive in a three-layer ocean]. displacement. In all cases the internal waves are of depression, and in most cases the peak-to-trough ampli- 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 TABLE 4. Additional estimates of speed across ground and direction. times due to the instrument sampling interval. The calculations marked * were derived from arrival times from the towed thermistor chain measurements and the S700 mooring. Those Speed across ground Propagation direction marked ϩ and # were derived from two positions on the wavefront in (cm sϪ1) (ЊTrue) Time of wave the SAR image and the S700 mooring for 20 Aug and 21 Aug. packet Mean Lower Upper Mean Lower Upper Speed across ground Propagation direction (cm sϪ1) (ЊTrue) 19 Aug, A.M. 71 63 79 109 99 118 Time of wave 19 Aug, P.M. 77 65 90 115 104 124 packet Mean Lower Upper Mean Lower Upper 20 Aug, A.M. 64 57 73 120 111 127 20 Aug, P.M. 69 60 79 120 111 128 19 Aug, A.M.* 79 75 82 113 110 116 21 Aug, A.M. 65 57 74 120 111 128 20 Aug, P.M. ϩ 79 76 83 131 131 131 21 Aug, P.M. 66 57 77 126 118 133 21 Aug, P.M. # 85 82 88 130 130 130

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FIG. 4. Initial conditions for the internal-wave simulations applied to each ray. Here the drop in the thermocline of D ϭ 30 m occurs over 200±300 m either side of the 6-km-wide ``bucket'': (a) complete domain, and (b) zoom on right-hand edge of feature. 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 ϩ30 m. The water depth where the tude corresponds to the depression from the mean back- lead solitary wave is located is annotated at right (m). Initial wave- ground level (unless speci®ed otherwise). form is bottom plot. Range axis is x Ϫ ct, where c is the leading- wave speed. a. Idealized simulation In this case the initial internal wave is an idealized in the thermocline from the resting level. Various initial representation of the observations shown in Small et al. amplitudes of the drop (hereinafter called D) between (1999a) that are summarized later in Fig. 10. The therm- 15 and 40 m have been tried to see if they evolve into istor chain data of the internal-wave packet at the po- similar features. The thermocline was assumed to be sition T700 (see left-hand panel of Fig. 1 for location) level behind the jump for a distance of 6 km (chosen show that the thermocline drops 30±40 m over a period to be similar to the typical observed wave packet of about 5 min, equivalently 200 m. The idealized initial length). At the end of this drop a restoration to the zero waveform has hence been chosen to be a smooth drop 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 every1hofpropagation: 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).

Unauthenticated | Downloaded 10/04/21 06:19 AM UTC DECEMBER 2003 SMALL 2663 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 200±300 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, neg- ligible in shallow water) and of 0.05 m sϪ1 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 ϭ 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 approxi- mation 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 ϭ 30 m FIG. 6. (a) Evolution of wave amplitude and phase speed along the with the wave front traces from Fig. 1 plotted for com- ray marked in black and white in Fig. 5a. Amplitudes are shown as parison. The extent of refraction is small for reasons asterisks. The analytical solitary-wave phase speed for these ampli- discussed below. tudes and water depths is shown as diamonds. The ray-model phase speed is shown as a solid line. The ®gure is for nonlinear EKdV For the ray highlighted in Fig. 5a, the evolution of model with initial amplitude Ϫ30 m. (b) Bathymetry along ray. the waveform is shown in Fig. 5b. An undular bore of three±four oscillations rapidly develops out of the initial form. Quickly the ®rst three waves disperse and separate fected the extent of refraction. This can be seen from into solitary waveforms (by water depths of 500±400 the change in propagation direction expected from m). When the lead wave has reached 443-m depth it Snell's law: has an amplitude of 50 m and is separated from the sin␪ refc ref second wave by 1300 m. By the time the lead wave ϭ , (3) sin inc in reaches 213-m depth the ®rst and second waves have ␪ amplitudes of 50 and 40 m, respectively, and have not where ␪ in and ␪ ref refer to the incident and refracted dispersed further. As the waves move onto the shelf, the angles, respectively, between the wave direction and the lead waves remain separated by 1000±1200 m and the gradient of bathymetry and cin and cref are the corre- ®rst two waves have similar amplitudes of 40 m. The sponding phase speeds of the incident and refracted third wave is considerably weaker and lags the second wave. Taking a typical initial angle to the bathymetry wave by 2 km by the end of the run. Figure 6a sum- gradient as 20Њ and using the linear (nonlinear) phase marizes how the amplitude of the lead wave varies with speeds in deep water and shallow water quoted above range and water depth. as the incident and refracted speeds gives a ®nal angle A signi®cant difference between linear and nonlinear of 16Њ (14Њ), a rather small extent of refraction of 4Њ predictions is that the nonlinear waves move faster, as (6Њ). expected. The phase speed of the lead wave along the Also plotted in Fig. 6a is the expected theoretical ray highlighted in Fig. 5a is shown in Fig. 6a along EKdV phase speed for a solitary wave of the amplitude with the corresponding bathymetry in Fig. 6b. The phase shown by the asterisks. There is reasonable agreement speed varies from 0.85 m sϪ1 in 700±800-m depth to between the expected speed and the speed from the 0.61 m sϪ1 in 140-m depth (as compared with 0.61 m model results, suggesting that the internal waves pro- sϪ1 and 0.48 m sϪ1, respectively, in the linear case, Fig. duced by the model become pure solitary waves. 3a) As a result the waves travel some 3±5 km farther A summary of the results of all the nonlinear runs is in the nonlinear case over the 12.4 hours of computation, shown in Fig. 7 (dotted lines), for different initial wave relative to the linear case. amplitudes from 15 to 40 m and for the ray highlighted The ratio of deep-water phase speed to shallow-water on Fig. 5a. In the deep water, the curves show similar phase speed is quite small (1.4) in this nonlinear case: trends with a rapid initial increase in absolute amplitude this is similar to the ratio for the linear case (1.3) and between 770 and 750 m as the high-frequency waves implies that here nonlinearity has not signi®cantly af- start to form (cf. Fig. 7a with Fig. 6a). The smallest-

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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)Ðsee 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 shownÐsee Tables 2±4. amplitude cases (15 and 20 m) continue to grow slightly depths of interest) to make an estimate. However, a gen- in amplitude until the end at 140 m. It will be shown eral trend is discernable of a decrease in speed with in section 3b that this is because the wave is suf®ciently decreasing depth after the initial period of wave devel- small to be governed by KdV dynamics. However, as opment (in 770±700 m), and the phase speed increases the initial absolute wave amplitude increases, the initial with wave amplitude in the depth range 700±200 m. On growth is reversed after around 500-m depth, and in the the continental shelf, the tendency of the amplitudes to shallow water of 140 m the wave amplitude rapidly be limited leads to a capping in phase speed at around decreases toward a value around 40 m. In section 3b it 0.6msϪ1, in agreement with the EKdV solitary wave will be shown that this is because these larger waves value seen in Fig. 6a (solid line). are being governed by EKdV dynamics. The phase-speed variation is shown in Fig. 7b (dotted b. Adiabatic predictions of solitary internal waves lines): because this is a derived quantity from the model output, it tends to be less smooth than the amplitude In the previous section an example was shown of how variation, and the kinks in the curve may be due to an the refraction model predicted solitary-wave evolution insuf®cient number of samples available (in the water out of an initial drop of the thermocline. The behavior

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of 14 m. For higher initial amplitudes the KdV prediction remains that the ®nal 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 ®nal 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 ®nal amplitude is less than the initial. An important parameter of relevance here is the max-

imum wave amplitude ␩c according to EKdV dynamics (Stanton and Ostrovsky 1998): ␣ ␩c ϭϪ . (4) ␣1 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 am- FIG. 8. (a) Adiabatic predictions of the shoaling of internal solitary plitude. waves initially in a water depth of 400 m: different initial amplitudes It can be seen from Fig. 8 that while shoaling the of 10, 25, and 50 m. Dashed lines represent KdV theory and solid small-amplitude initial wave nowhere approaches the lines are EKdV predictions. (b) The limiting amplitude ␩c predicted by (4) from the EKdV coef®cients of Fig. 3. 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 ini- of the solitary waves can be interpreted using theoretical tial wave (50 m), the result is that the wave amplitude predictions of the evolution and shoaling of EKdV in- is severely capped and decays to the value of ␩c ϭ 46 ternal solitary waves. Under the adiabatic assumption m in the ®nal depth of 140 m. that the horizontal scale over which the depth varies is These predictions help to explain the EKdV refraction long in comparison with the internal-wave length, so model results presented in section 3a. There, an ideal- that the internal wave always has time to adjust to the ized waveform of a drop in thermocline level D rapidly local solitary-wave solution, the conservation of energy developed into a set of solitary waves. In Fig. 7a (model can be used to derive how the solitary wave changes results), the cases with the smallest values of initial with depth. In Small (2001a, section 2c) this technique displacement D show a slow and steady increase in was used to study the evolution of KdV and EKdV amplitude as the shallow water is approached. In con- waves. Here this method is used to compute the adia- trast, for the larger values of D, the solitary waves stop batic evolution of internal solitary waves based on the growing after reaching 500±400-m water depth and sub- KdV and EKdV coef®cients shown in Fig. 3. The evo- sequently decay subject to the effect of the limiting lution of solitary waves of different amplitudes initially amplitude of ␩c, eventually falling to a value close to in 400-m water depth has been investigated. This initial that of ␩c when in 140-m water depth. 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 appli- c. Realistic case cation of the theory. In the next experiment the initial waveform was taken The adiabatic predictions (Fig. 8a) show that the dif- from towed thermistor chain data, which gave the ®nest ference between the KdV and EKdV predictions in- resolution of the internal waves (the thermistor chain creases with increasing initial amplitude. For a small track is shown in Fig. 1a: its sampling interval was 1 10-m initial wave, KdV predicts a gradual shoaling to s). Initial conditions for the model were derived from a ®nal amplitude of 15 m in 140-m water depth, and the measurements at T700 by removing the Doppler the EKdV prediction is similar with a ®nal amplitude shift of an internal wave moving relative to the ship, as

Unauthenticated | Downloaded 10/04/21 06:19 AM UTC 2666 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 described in Small et al. (1999a). A single representative come closer as they propagate across the shelf). By the waveform was derived from the towed thermistor data end of the model evolution (after 8.5 h of propagation), as follows. First the mean depth of each isotherm was all wave amplitudes are 50 m or less. calculated from the moored timeseries at S700, over a The phase speed in the model initially rises as the 24-h period from 0000 UTC 19 August to 0000 UTC wave transformation takes place, from 0.6 to 0.7 m sϪ1 20 August. The displacement of the isotherms at T700 (between depths of 700 and 600 m, not shown) before from the mean values was then constructed for seven decreasing to 0.55 m sϪ1 on the shelf (summarized in isotherms (from 10.5Њ to 13.5ЊC at intervals of 0.5ЊC), Fig. 7b, dashed line). These values are less than the and these displacement series were simply averaged. corresponding EKdV speeds for waves of those ampli- The waveforms were then inserted into a domain for tudes (by 0.05±0.1 m sϪ1), although there is evidence numerical simulation of evolution. For the simulations that the model and the EKdV solitary wave speeds are it was required that the waveform was equal to zero at converging as the waves propagate farther onshelf. This the boundaries, and so a Gaussian tail was added to the is discussed further in the following section. waveforms to make it decay it to zero well before the Last, the impact of ray spreading on wave amplitude boundaries as shown in Fig. 9b (bottom plot). This new was assessed by running an extra simulation with the waveform was then used as initial conditions for the spreading term S(x) in (1) switched off. The simulated model. The Gaussian tail had an e-folding length of 1 wave amplitudes and phase speeds in this case are km: further experiments showed that the ®nal results shown in Fig. 7 (dot±dash line). It can be seen that the were not sensitive to this length scale. amplitudes are some 5±10 m greater than for the case The experiment assumed an initial wavefront of the with spreading, while the phase speed is slightly larger arc form of Fig. 5a, but located 4800 m farther onshelf (by 0.01 m sϪ1) with no spreading. Hence, it may be so that the wavefront passes through T700 (see Fig. 9a). seen that the wave front spreading in this case is im- A group of rays were constructed, each perpendicular portant to modeling the shoaling of the waves. to the wavefront. For each ray, the initial waveform just described was placed in the evolution model domain 4. Comparison of simulations with observations such that the leading edge of the feature in the waveforms was located at the position of the wave front. The numerical simulations presented above may be It should be mentioned here that there is no a priori compared with observations from the SES and SESA- reason for assuming the waveform should be the same ME experiments. Previous analysis of nonlinear inter- along each ray: in fact, this is unlikely because each ray nal-wave conditions in the experiments has discussed has passed over different bathymetry tracks and so the the SES (Inall et al. 2000, 2001) and SESAME (Small wave evolution is likely to be different. However, this et al. 1999a,b) data separately. Small (2000) then de- section will focus on the evolution along the ray that tailed the inferences from the combined datasets, from passes through the observation point, and the other rays which the examples shown here are taken. are only used to simulate any approximate spreading effects on that ray. a. Shoaling Figure 9a shows the refraction diagram for the ®rst 8.5 h of evolution. The extent of spreading is similar The most complete picture of the internal tide trans- to that shown in Fig. 5a. The ray marked in black in formation from in situ measurements was obtained on Fig. 9a starts close to T700, and the evolution plot for the morning of 19 August when a towed thermistor this ray is displayed in Fig. 9b. The initial waveform chain survey took place. The towed and moored data quickly develops into a set of two primary waves and were converted to an approximate ``snapshot'' by re- following small oscillations by 530-m depth. By the moving the Doppler shift caused by the waves moving time the lead wave reaches the depth of 423 m, the relative to the ship or the mooring, using the method peak±trough amplitude is 48 m (see close-up in Fig. 9c) described in Small et al. (1999a). and the second wave has an amplitude of 35±40 m. At The transformation of thermocline depth (Fig. 10a) 293-m depth, the lead wave has an amplitude of 55 m and upper layer current (Fig. 10b) on 19 August is then and that of the following wave is 40 m. The separation shown as a function of range. The ``range series'' for of the leading waves in the model is 1 km. By the time each measurement are each incremented by 10 km for it reaches a depth of 202 m (shelf break) the lead wave display purposes. The plot is arranged so that the right has a peak±trough amplitude of around 62 m (Fig. 9c). of each record is the eastward (shelf) side, and the left After this the waves slowly reduce in size as they move is the westward (ocean) side. over the shelf, and by the time the lead wave is in 146 The data on the morning of 19 August show a con- m of water (Fig. 9c), after 5.5 h of propagation, it has sistent pattern from the thermistor records and the cur- a 55-m peak±trough amplitude, while the second wave rent records (Figs. 10a,b). At T700 there is a drop of is actually larger (60 m) and has approached the lead nearly 40 m in thermocline depth followed by a partial wave, being 600 m separated (the waves never merge, restoration to the original level: (the drop, or bore or because their amplitudes and hence phase speeds be- jump, is from a shallow level on the right-hand, shelf

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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 every1hofpropagation: 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 ϩ60 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 Ϫ Vt, where V is a constant velocity: 0.6 m sϪ1. (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 then a small wave of 5-m amplitude. At S300 the ther- record at S700 describes a bore followed by an oscil- mocline displacement only resolves two waves (owing lation, slightly undersampled: the total current change to the coarse 10-min sampling) of similar amplitude to across the bore is 0.55 m sϪ1. The T400 record shows those at T400, while the current record indicates that a leading ``solitary'' wave of amplitude 50 m followed the lead wave has a current pulse of 0.6 m sϪ1 (the 5- by another slightly smaller wave splitting in two, and min sampling here gives slightly better resolution). The

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FIG. 10. Transformation of the wave packet on the morning of 19 Aug: (a) depth of the thermocline from thermistor records for the 14ЊC isotherm at T700, T400, and S140, and the 12ЊC isotherm at S300. (b) Current velocity in direction of propagation (120ЊT) 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 of 19±21 August 1995 (presented as symbols joined by well-de®ned waves, the lead wave having the largest lines) are a combination of real measurements and in- current of 0.50±0.55 m sϪ1, and the leading wave sep- ference from current pro®les. The amplitude of ther- aration is 1000±1500 m. (The thermocline displacement mocline displacement was only precisely measured on for S200 is not shown because the full depth range of the morning of 19 August when the towed thermistor the thermistors was limited to between 2- and 42-m chain was in operation. From the adjacent thermistor depth: however, it did indicate the presence of the three and current-meter measurements for that morning, the signi®cant waves without capturing the full amplitude). ratio of current velocity in the upper ocean in the di- At S140 three large waves followed by smaller os- rection of wave propagation to the thermocline dis- cillations are clearly seen in the thermistor and current placement was calculated. Then this ratio was applied records: the lead wave has amplitude of at least 30 m on the subsequent tidal cycles (when accurate displace- (the ``¯at bottom'' of the wave suggests that the 14ЊC ment data were not available) to infer the amplitude of isotherm is displaced somewhat below the deepest mea- displacement from the current. (Note that this manner surement at 55-m depth) and an associated current pulse of conversion is exact for a linear wave but only ap- of 0.55±0.60 m sϪ1. The thermistor string also showed proximate for a nonlinear wave, and hence it is used that the displacement of the 15ЊC isotherm was at least only as a guideline.) 35 m.1 The waves appear still to be rank-ordered at The observations of lead-wave amplitude show sim- S140, and the ®rst wave is separated by 1500±2000 m ilar trends to the simulations with the idealized initial from the following wave. conditions (shown as dotted lines in Fig. 7a) with an Figure 7 summarizes the simulations of section 3 and initial growth in amplitude followed by a reduction be- compares the results with the observations. Here the tween depths of 300 and 140 m. For the simulations, observed amplitudes (Fig. 7a) from the six tidal cycles 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 dynam- 1 For comparison, the maximum measured amplitude at S140 (dur- ics, and possibly damping through mixing (see below). ing a period when a deeper thermistor string was in place, 15±16 The idealized simulations also showed wave separations August 1995, data not published) was 50 m. The signature of that of 1000±1200 m at the front of the packet: this may be wave at 55-m depth was similar to that observed on 19 August, as was the background density strati®cation, suggesting that the waves compared with the observed 1000±1500 m over the had a similar amplitude. Hence, a reasonable estimate of the actual slope and 1500±2000 m at S140 on the shelf (see Fig. wave amplitude on 19 August (A.M.) is between 35 and 50 m. 10).

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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-to- trough 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 ®nal 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 mecha- nism) and that the observations were not all gathered along one ray path and hence show along-wavefront variations that are dif®cult to compare with the model.

On some tidal cycles the observed ®nal amplitude FIG. 11. (a) Richardson numbers in modeled internal-wave packets, was just 10±20 m. In one case (the afternoon of 20 from the model run illustrated in Fig. 5b, lead wave in a water depth August: Fig. 7a, plus symbol) the amplitude drops dra- of 443 m. Solid line contour is of Ri ϭ 0.25; dashed line is Ri ϭ 1. matically from 55 m in 300-m depth to 10 m in 140-m (b) Comparison of the phase speed (solid line) and maximum particle velocity (asterisk) corresponding to the simulation shown in Fig. 5b. 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- direction of propagation, density perturbation ␳Ј(x, z, t), AME. Inall et al. (2000) measured turbulent dissipation wave amplitude ␩(x, t), and mode ␾(z): during the passage of the internal waves past S140 dur- M 2 ing the afternoon of 21 August 1995 and found an av- Ri ϭ : z)2ץ/uץ) ,erage diapycnal eddy diffusivity of 5 cm2 sϪ1. Further Small (2000) found evidence that the necessary con- [(␳*(z) ϩ ␳Ј(x, z, t]ץ g ditions for gravitational (Orlanski and Bryan 1969) and M 2 ϭϪ , zץ shear instability (Miles 1961) were satis®ed in the large ␳ 0 internal solitary waves observed on the shelf. Turbu- (␾(zץ lence and breaking ensuing from these instabilities are u(x, z, t) ϭ c0␩(x, t) , and zץ likely to explain the small amplitudes of some observed (␳*(zץ waves at S140 after the waves have crossed the slope (Fig. 7a). ␳Ј(x, z, t) ϭϪ␩(x, t) ␾(z). (5) zץ A contour plot of the Richardson number Ri within the idealized model simulation in 443-m water depth Modi®cations to this formulation to include higher-order (Fig. 11a) shows patches where Ri Ͻ 0.25 near the expansions of u and ␳Ј still resulted in Ri Ͻ 0.25 in the thermocline in the internal-wave troughs, indicating that troughs of the waves.] Values of Ri Ͻ 0.25 continued the waves are susceptible to shear instability (Miles to be found as the waves propagated onshelf. Further,

1961). [For these computations Ri has been calculated a comparison of maximum onshelf current Umax with the using the instantaneous buoyancy frequency M 2, and phase speed c of the leading internal wave (Fig. 11b) the linear relationships between current u(x, z, t) in the shows that Umax Ͼ c in depths of 500 m or less, sug-

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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 Ϫ1. (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 parison of the EKdV predictions with those from a fully is to be expected that mixing processes should be im- nonlinear computational ¯uid dynamics (CFD) model portant in this idealized simulation, and also in the re- (Hornby and Small 2002). The CFD model has the ca- alistic simulation in which the amplitudes are larger. pacity to model turbulent kinetic energy and its dissi- The lack of turbulent dissipation in the model is one pation rate and, hence, should address the importance reason why the simulated amplitudes on the shelf in of mixing and strong nonlinearity in the evolution of 140-m water depth (60 m) in the realistic experiment internal solitary waves. are larger than observed. In terms of the modeling results, the wave amplitudes b. Refraction may easily be reduced to a value for which the instability criteria are not satis®ed by imposing a simple Laplacian Details on the propagation speed and direction of the form of horizontal diffusion with suf®ciently large value internal waves were derived from the arrival times at of the constant eddy viscosity coef®cient. This method the moorings and the SAR images, where available, by may also bring the simulated wave amplitudes closer to applying simple geometry. From the internal-wave sig- those observed. This approach is not taken here because natures on Fig. 1 it can be seen that the internal wave- it is not likely to realistically model the behavior of fronts can be assumed to be quasi-planar on O(km) turbulent internal solitary waves where dissipation is length scales. The longest distance between successive likely to be sporadic and localized (see, e.g., Inall et al. moorings along the south line is 6±7 km. So within any 2000; Pinkel 2000). Nor is the model of Bogucki and contiguous trio of moorings a planar wavefront has been Garrett (1993) used here, because their method is most assumed in order to compute the wave speed and di- appropriate for KdV interfacial waves: for a realistic rection from the arrival times. This was performed in strati®cation their method is dependent on a parameter Small (2000) and is summarized in Tables 2±4. (the ratio of downward to upward mixing thickness, Two illustrative examples of the nature of the ob- their ␤), the value of which is not fully understood. served phase velocity of the internal waves are con- Last, it may be said that the inclusion of dissipation is tained in Fig. 12, for the tidal cycles on the morning of not appropriate in the framework of EKdV theory, 19 August when extra towed thermistor chain data were which is weakly nonlinear and excludes the strongly available and the afternoon of 20 August when SAR nonlinear effects that lead to dissipation. imagery supported the analysis. Idealized planar wave Instead, current and future work is focusing on a com- fronts are shown aligned at right angles to the calculated

Unauthenticated | Downloaded 10/04/21 06:19 AM UTC DECEMBER 2003 SMALL 2671 direction of propagation and passing inside the trio of • Nonlinear internal waves develop out of a distantly moorings used to derive the values. The phase speed is generated internal tide. Between depths of 700 and annotated alongside each wave front. 400 m there is a growth in wave amplitude and the Figure 12 and Tables 2±4 show how the lead wave number of waves. This was seen in the observations does tend to slow as it approaches shallow water. Typical and all simulations. speeds in the triangle (S700, S300, S200) are 0.7±0.8 • Between depths of 400 and 140 m the observed msϪ1, while in the shallower triangle (S300, S200, growth slows initially and then there is a decay in S140) speeds lie generally between 0.6 and 0.7 m sϪ1. amplitude as the wave propagates across the shelf into The two examples in Fig. 12 show a small bending of 140-m depth. This property is also seen in simulations wavefronts more parallel to the isobaths as the wave of idealised EKdV solitary waves but is not seen in crosses the slope: by 4Њ on 19 August and 11Њ on 20 the simulated evolution of a realistic initial condition. August (using the mean directions listed in Tables 2± • The observations showed a limited extent of refraction 4). because the phase speeds in shallow water (0.65±0.75 The simulated and observed phase speeds are sum- msϪ1) are not signi®cantly different from those in the marized in Fig. 7b. The simulations bracket most of the deeper water (0.7±0.85 m sϪ1). observations in water deeper than 200 m. However, the • The simulations tended to underpredict the phase idealized simulations tend to show higher speeds than speeds relative to the observed values, particularly in observed at around 500-m depth (with one observed shallow water (by 0.1±0.2 m sϪ1). However, because exception), while the realistic predictions give slightly the ratio of shallow-water phase speed to deep-water lower speeds at that depth. On the continental shelf in phase speed was similar in the simulations to the ob- 140 m, all the simulations appear to underpredict the servations, the model gave a similar extent of refrac- observed speeds. This either is a limitation of EKdV dynamics (which severely limits the amplitude and tion to the observations. This was con®rmed by com- speed in 140 m) or is due to uncertain knowledge of parison of the orientation of leading wave fronts on the observed phase speeds (note the ``error bars'' in SAR imagery with those from the simulations. Table 2: in fact the model results are quite close to the • Numerical predictions using a realistic initial condi- lower end of the observed phase speed uncertainties). tion showed larger wave amplitudes on the shelf than Both idealized (Fig. 5a) and realistic (Fig. 9a) sim- observed (by around 10±20 m). Part of the reason for ulations predicted ®nal wavefronts that had orientations this discrepancy may be that the observations were consistent with the observed packets B1 and B2 in the not conducted along a single ray path, and hence can- right panel of Fig. 1, indicating that the model is cor- not strictly be compared with model predictions along rectly predicting the extent of refraction. After the 12.4 a single ray. Further, discrepancies may arise from the h of idealized simulation, the ®nal position of the lead use of a weakly nonlinear model with no turbulent wavefront (Fig. 5a) is in the vicinity of the lead wave- dissipation to simulate internal waves with strongly front of the observed packet B2, which was speculated nonlinear aspects (the ratio of upper-layer current to be from the previous tidal cycle to feature A. So, the speed to phase speed often approaches unity). Present nonlinear model predicts a range of propagation over work is addressing these aspects. one complete tidal cycle similar to the observations. The phase speeds arising from the realistic initial conditions are smaller than those from the idealized simulations (Fig. 7b). The reason again relates to the pres- Acknowledgments. This paper is in memory of John ence of the background long wave in the realistic sim- Apel and Peter Holloway whose pioneering work in the ulation. Small et al. (1999b) compared the evolution of ®eld inspired many. The suggestions of Steve Thorpe two initial waveforms, one identical to the other except are gratefully acknowledged. The U.K. Ministry of De- that the height of the background level was different. fence funded the Ph.D. studentship of the author. Thanks In one the wave was all a depression relative to the go out to Dr. John Scott and colleagues at the Defence background and in the other the wave oscillated above Evaluation Research Agency, United Kingdom, for or- and below the background level: the latter case gave ganizing the experiment. Discussions and exchange of rise to signi®cantly lower phase speeds than the former. data with Mark Inall and Toby Sherwin of the SES group This result is partly a consequence of the fact that waves were essential to the development of the study. Exten- of depression are more nonlinear than elevation waves sive discussions with E®m Pelinovsky and Tatyana Tal- in cases of negative nonlinear coef®cient ␣ (see, e.g., ipova were of great assistance. The comments of three Osborne and Burch 1980) and consequently move faster. anonymous reviewers were welcomed. The SES data are contained in the LOIS±SES CD, available from the 5. Conclusions British Oceanographic Data Centre at BODC, Bidston This study has elucidated some of the properties of Observatory, Bidston Hill, Prenton, Merseyside, CH43 nonlinear internal wave shoaling and refraction at the 7RA, United Kingdom. The IPRC is partly supported Malin slope. The main conclusions are the following. by the Frontier Research System for Global Change.

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APPENDIX TABLE A1. Coef®cients of the KdV equation at 400-m depth computed for the case with no background shear current (U ϭ 0) and for the case with a shear current U(z). Computation of the Environmental Parameters Water depth (m) C (m sϪ1) ␣ ϫ 10Ϫ2 (sϪ1) ␥ (m3 sϪ1) The environmental parameters of the EKdV equation 0 are dependent on the normal-mode solutions to the lin- U ϭ 0 0.57 Ϫ1.70 2110 ear internal-wave equation and related quantities. Lamb U ϭ U(z) 0.53 Ϫ1.64 2439 and Yan (1996) have succinctly summarized the pro- cedure and further discussion is contained in Holloway in the direction of the wavenumber vector. In the ab- et al. (1999). This approach separates the ocean ®elds sence of current shear the equation is simpli®ed and into vertical normal modes ␾(z) and horizontal/temporal linear normal modes are found from solutions to wave functions ␩(x, t); for example, the streamfunction (22N (zץ ␺(x, t, z) is written as (␾ ) ϩ ␾ ϭ 0. (A3) 22ii zc0ץ ϱ ␺(x, t, z) ϭ c ␾ (z)␩ (x, t) ϩ O(␦, ␧), (A1) ͸ 0 ii In both these expressions [(A2), (A3)] it has been as- iϭ1 sumed that the initial waveform has a frequency ␻ that where ␦ and ␧ are parameters that describe the extent lies in the band of f K ␻ K N, where f is the Coriolis to which the waves are nonhydrostatic and nonlinear. frequency. In the presence of a vertical shear of horizontal current, For the simulations only the ®rst mode has been used, the normal modes are given by solutions to the Taylor± following observations that suggested this was the dom- Goldstein equation (Thorpe 1969): inant mode (Inall et al. 2000; Small et al. 1999a,b). Consequently the results for this mode are used for cal- 22␾ N (z) dU 22/dzץ i ϩϪ␾ ϭ 0, (A2a) culating the environmental coef®cients. The ®rst-order 22 i .z Ά·[U(z) Ϫ c00][U(z) Ϫ c ] environmental parameters are given by Holloway et alץ where (1999): 0 ␾ (z) ϭ 0onz ϭ 0, and z ϭϪH. (A2b) i [c Ϫ U(z)]23␳␾ dz ͵ 0 z Here U is the component of the horizontal velocity 3 ϪH ␣ ϭ 2 0 [c Ϫ U(z)]␳␾ 2 d ͵ 0 z ϪH

0 [c Ϫ U(z)]22␳␾ dz ͵ 0 1 ϪH ␥ ϭ . (A4) 2 0 [c Ϫ U(z)]␳␾ 2 d ͵ 0 z ϪH As discussed in section 2, the dominant in¯uence on the nonlinear dynamics in the SES environment is the strati®cation. The buoyancy frequency in the top 140 m was calculated from a 24-h time series of CTDs conducted by RRS Challenger near S140, while the N(z) pro®le below that was computed from a set of deeper pro®les at much coarser temporal and spatial resolution that spanned the continental slope along a line con- necting the SES south moorings (down to 1500 m; the shallower moorings are shown in Fig. 1). For each water depth of interest, the N(z) pro®le was taken from this mean pro®le, cut off at the appropriate water depth. It should be noted that true mean pro®les could not be determined at every location individually because the CTD sampling was too coarse (except at S140). The resulting potential density and buoyancy frequency pro- FIG. A1. (a) Vertical pro®le of summer mean slope current, as well ®les were shown in Fig. 2. as the component of slope current in the internal-wave propagation direction (U ´ k), and a zero-current case for comparison (see legend). The background slope current U(z) observed during (b) The ®rst linear normal mode computed with and without currents SES was discussed by Sousa et al. (2001). The effect (see legend). on the environmental coef®cients of the slope currents

Unauthenticated | Downloaded 10/04/21 06:19 AM UTC DECEMBER 2003 SMALL 2673 observed in SES during summer has been assessed using by Gaussian elimination. The computation of a partic- the mean U(z) pro®le presented in Fig. 5a of Sousa et ular solution follows Holloway et al. (1999, Annex). al. [(2001); derived from measurements for 12 August± 6 September 1995, relevant to the period of interest]. This pro®le shows a core alongslope velocity between REFERENCES Ϫ1 0.14 and 0.16 m s centered at 200-m depth and located Bogucki, D., and C. Garrett, 1993: A simple model for the shear- over the 400-m isobath and deeper. The velocity decays induced decay of an internal solitary wave. J. Phys. Oceanogr., to 0.06 m sϪ1 at the surface and 0.1 m sϪ1 at 400-m 23, 1767±1776. depth. The corresponding across-slope velocity is weak Bole, J. B., C. C. Ebbesmeyer, and R. D. Romea, 1994: Soliton Ϫ1 currents in the South China Sea: Measurements and theoretical and reaches a maximum of 0.02 m s . A simpli®ed modelling. Proc. Offshore Technology Conference, Houston, version of the ␷ pro®le is shown in Fig. A1a (dotted TX, 367±377. line). Brandt, P., W. Alpers, and J. O. Backhaus, 1996: Study of the generation The Taylor±Goldstein equation [(A2)] requires input and propagation of internal waves in the Strait of Gibraltar using of the background current in the direction of internal- a numerical model and radar images from the European ERS-1 satellite. J. Geophys. Res., 101 (C6), 14 237±14 252. wave propagation, that is, U ´ k, where k is the internal- Choi, W., and R. Camassa, 1999: Fully nonlinear internal waves in wave wavenumber vector. As discussed in section 4, a a two-¯uid system. J. Fluid Mech., 396, 1±36. typical propagation direction was 125ЊTrue. A plot of DeWitt, L. M., M. D. Levine, C. A. Paulson, and W. V. Burt, 1986: the corresponding U ´ k pro®le is shown in Fig. A1a Semidiurnal tide in JASIN: Observations and modelling. J. Geo- phys. Res., 91, 2581±2592. 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