Temporal variability of the standing internal tide in the Browse Basin, Western Australia

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 117,, doi: /2011jc007523, 2012 Temporal variability of the standing internal tide in the Browse Basin, Western Australia M. D. Rayson, 1 N. L. Jones, 1 and G. N. Ivey 1 Received 17 August 2011; revised 23 April 2012; accepted 2 May 2012; published 15 June [1] We have investigated the seasonal evolution of the internal tide using a yearlong mooring record in the Browse Basin on the Australian North-West Shelf. Using a modal harmonic-decomposition technique, we studied the modulation of the semi-diurnal kinetic energy, potential energy, energy flux, group velocity and sea surface height anomaly for different internal wave modes. A very coherent and energetic, locally generated internal tide dominated the mooring record during spring-tide periods. Relationships amongst the wave variables suggest that the interaction of multiple waves formed a standing wave pattern. To assist in explaining the seasonal variability of this standing wave pattern, we used an analytical model of internal wave generation and propagation in a double-shelf system, representative of the area, to highlight the effects of changes in the stratification and barotropic forcing. A comparison of the low-mode wave energy indicated that the analytical solution overestimated the observed wave energy by a factor of two and failed to detect an apparent seasonal shift in the conversion rate. We discuss potential mechanisms for this discrepancy. Citation: Rayson, M. D., N. L. Jones, and G. N. Ivey (2012), Temporal variability of the standing internal tide in the Browse Basin, Western Australia, J. Geophys. Res., 117,, doi: /2011jc Introduction [2] Understanding the variability in the temporal and spatial scales of velocity and buoyancy perturbations associated with internal waves is pivotal to understanding the ocean dynamics of a region. The interaction of internal waves is common in regions with multiple generation and/or reflection zones, such as continental shelves and small ocean basins [e.g., Nash et al., 2004; Martini et al., 2007; Rainville et al., 2010; Rayson et al., 2011]. One form of interaction is a standing wave an idealized description of this phenomenon is the interaction of two mode-one internal waves of equal amplitude propagating in opposite directions (see the derivation of the equations in Nash et al. [2004]). In practice, wave energy propagates along beam trajectories composed of a sum of vertical modes, making it difficult to isolate the signature of individual waves from typically sparse point measurements. [3] In situ and satellite observations of internal tides have noted the existence of standing wave interference patterns. Nash et al. [2006] observed, with repeated vertical profiles, spatially variable patterns in both the ratio of horizontal 1 School of Environmental Systems Engineering and Oceans Institute, University of Western Australia, Crawley, Western, Australia. Corresponding author: M. D. Rayson, School of Environmental Systems Engineering, University of Western Australia, M015, 35 Stirling Hwy., Crawley, WA 6009, Australia. (rayson@sese.uwa.edu.au) American Geophysical Union. All Rights Reserved. kinetic energy (HKE) to available potential energy (APE) and along-shelf energy flux at Kaena Ridge where two beams cross. Martini et al. [2007] noted a similar response at Mamala Bay where beams cross from two different generation regions. Zhao and Alford [2009] used satellite altimetry to identify a wave interference pattern that caused spatially variable phase patterns of the sea surface height anomaly due to the interaction of beams generated at the Hawaiian and Aleutian ridges. Measurements of HKE to APE ratios and energy flux magnitude from six moorings located along this satellite altimetry track confirmed this interaction. Zhao et al. [2010] used 80 moorings distributed globally and identified standing wave behavior at 41 N in the Pacific [Alford and Zhao, 2007a] and North Atlantic [Alford and Zhao, 2007b]. Of note is that none of the 80 moorings analyzed in these papers were located in the Eastern Indian Ocean- the location of the present study. [4] The Browse Basin on the Australian North-West Shelf is a region with a prominent semi-diurnal internal tide generated by a large barotropic tide interacting with complex topography [Rayson et al., 2011]. The region is characterized by two shelf breaks, separated by 90 km in the offshore direction, that are significant internal wave generation regions (Figure 1, longitude: E and E). During spring tide conditions in the summer, the interaction of waves propagating onshore from the outer-shelf break and offshore from the inner-shelf break form a partly standing internal wave, resulting in a spatially variable pattern of energy density and flux over the shelf [Rayson et al., 2011]. The focus of this paper is to investigate the temporal variability of this standing internal tide due to seasonal changes in forcing and stratification in the Browse Basin. 1of17

2 Figure 1. (a) The Browse Basin bathymetry contour plot. The B2 mooring is indicated by the gray triangle. (b) Cross-section of the bathymetry at dashed line in Figure 1a showing instrument locations on B2 mooring. [5] In a rotating stratified fluid, under the hydrostatic approximation, linear internal gravity waves adhere to the dispersion relation [Gill, 1982] w 2 f 2 þ N 2 k 2 x k 2 z ; ð1þ where w is the wave frequency, f is the inertial frequency, N is the buoyancy frequency and k x and k z are the horizontal and vertical wave numbers, respectively. A purely progressive wave in this regime will propagate at a constant speed given by the horizontal group speed c g Nðw 2 f 2 Þ 1=2 ¼ : ð2þ wk z In the absence of planetary rotation, the horizontal group velocity of a perfect standing wave is zero, while with rotation the group velocity will vary horizontally according to [cf. Alford and Zhao, 2007b] c s g ¼ 2wF sin ð 2k xxþ w 2 f 2 cosð2k x xþ c g: ð3þ In the Browse Basin, representative depth-averaged buoyancy frequencies for summer and winter are N 0 = s 1 and N 0 = s 1, respectively. This small change in N 0 can result in significant changes in the standing internal wave characteristics, as illustrated by the example of a perfect mode-one standing internal tide on a uniform-depth shelf (Figure 2). The model standing wave is characterized by the oscillation of HKE and APE at twice the horizontal wave number with a 90 phase lag. The location of the peaks in energy are shifted by as much as 10 km from summer to winter (Figure 2a). The energy flux is zero in the direction of propagation (cross-shelf), the along-shelf energy flux alternated in direction, and the pattern is displaced horizontally from summer to winter (Figure 2b). Group velocity peaked at 0.8 m s 1, approximately half the group velocity of a progressive wave (Figure 2c) and decreased from summer to winter due to the shortening of the horizontal wavelength (as given by (1)). The idealized standing wave model demonstrates how even small variations in stratification can cause large variations in the observed internal APE, HKE and energy flux at a given location. However, the model cannot be used as a predictive tool as it only describes a single vertical mode and does not account for wave generation by topography; therefore, the wave amplitude must be known a priori. [6] Ray tracing is an alternative approach to studying internal wave interaction. The raypath is found by tracing the internal wave angle of propagation a ¼ w 2 f N 2 ðz; tþ w 2 ð4þ through space from the critical slope location. The origin of these rays are assumed to be where the slope criticality parameter g is equal to one; where g = rh /a and H is the water depth. Ray tracing reveals subtle shifts in the ray trajectories over the Browse Basin double shelf break system between summer and winter stratification conditions (Figure 3). Although ray tracing provides a kinematic description of the wavefield it is of limited use due to the lack of information about the energetics. [7] Analytical solutions to the two-dimensional linear internal wave equations [e.g., Echeverri and Peacock, 2010; Llewellyn-Smith and Young, 2003; Petrelis et al., 2006] provide additional insight to ray tracing and, unlike numerical models, dissipation and grid resolution do not affect the solution. These models predict barotropic to baroclinic conversion and therefore a priori information about the wave amplitude is not necessary. The simplicity of these solutions allows us to evaluate the effect of the model parameters on the wavefield and, as distinct from ray tracing, they predict the energy converted into the internal tide. 2of17

3 Here N 0 is a representative buoyancy frequency and, u 0 and H are the barotropic velocity amplitude and depth at the offshore boundary, respectively. Numerical studies of internal tide generation at a super-critical, finite length shelfbreak have validated these analytical predictions of wave conversion [e.g., Gerkema et al., 2004], however, few comparisons have been made with oceanic observations. Part of the reason for this is that there are few locations where incoherent, remotely generated waves do not contaminate the observations; making it difficult to isolate the locally generated response [e.g., Kelly and Nash, 2010; Martini et al., 2011]. In contrast, the Browse Basin region has a large-amplitude, coherent internal tide signal [Rayson et al., 2011] that can even be detected with temporally coarse sampled XBT profiles along a repeated line [Katsumata and Wijffels, 2006]. The region is therefore a good candidate for comparison between observations and an analytical model for wave generation. [9] The objective of this paper is to use both field observations and the analytical model of Echeverri and Peacock [2010] to understand the internal tide seasonal variability in the Browse Basin due to varying stratification and forcing conditions. The structure of this paper is as follows: in sections 2 and 3 we present our field data analysis method- Figure 2. Spatial distribution of (a) energy density; (b) cross-shelf, u p, and along-shelf, v p, energy flux; and (c) perceived group velocity of a theoretical mode-one standing internal tide. Solid lines are values for a representative summer stratification while dashed lines are for a representative winter stratification. [8] The energy flux, F [W m 1 ], for an internal tide that propagates away from a supercritical topographic feature with small topographic excursion parameter, is [e.g., St. Laurent et al., 2003; Petrelis et al., 2006; Garrett and Kunze, 2007] F ¼ F 0 f ðtopographyþ; ð5þ where the (dimensional) forcing parameter F 0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi F 0 ¼ 1 2p r N0 2 w2 ð w2 f 2 Þ 0 u 2 0 w H 2 : ð6þ Figure 3. The theoretical M 2 ray characteristic paths for (a) summer and (b) winter stratification conditions over the two-dimensional topography in Figure 1b. The black arrows indicate the direction of energy propagation. (c) The slope criticality parameter (g) for summer (red) and winter (blue) stratification. 3of17

4 Table 1. The Vertical Location of the B2 Mooring Instruments Instrument Type Variables Depths (m above seabed) InterOcean CM04 u, v 2.6, 130, a 250, 330, 390, 450, 490, 530 Starmon mini thermistor T 2.6, 50, 90, 130, 170, 210, 250, 270, 290, 330, 350, 370, 390, 410, 430, 450, 470, 490, 510, 530 a This instrument failed in March 2007 until re-deployed in April. ology and analysis of the mooring results; in section 4 we describe the results of the analytical model; in section 5 we discuss the energy conversion for both the model and the observations, and in section 6 we present the conclusions. 2. Data Analysis [10] In this study, we focus on mooring data collected at the outer-shelf break in 550 m of water in the Browse Basin, at site B2, over a 12-month period (Figure 1). The mooring was deployed from October 2006 September 2007 with a brief servicing period in April We studied the temporal variability of the internal wave response via the cross-shelf and along-shelf baroclinic velocity, (u, v ) and buoyancy perturbation, b, where b ¼ gr =r; here r is the time-mean density taken as the 14-d low-pass filtered value. The crossshelf component was defined as 32 degrees clockwise from east. The mooring consisted of 20 thermistors (Starmon mini) and eight current meters (InterOcean CM04) which sampled at 60 s. The instruments were staggered vertically with higher resolution in the upper 250 m (Table 1) Stratification [11] The ocean stratification, along with the mean flow, plays a pivotal role in modulating the internal wave dynamics in a given region. To determine whether the observation period was representative of the regional climatology, we investigated the stratification variability using two independent but complementary data sources: a fixed mooring that measured temperature only, and the temporally and spatially sparse Argo profiling floats that provide a longer and deeper record. The B2 mooring provided a fixed-point record of the ocean thermal structure with 20 m resolution in the upper 200 m and approximately 40 m resolution below this depth. We calculated the time-variable stratification from the mooring by low-pass filtering the measured temperature with a 14-d cut-off, and used a constant salinity of 34.6 to calculate density. To ascertain if the observed thermal structure was representative of a typical year, we used temperature and salinity data from the Argo profiling float network (www. argo.net) to develop the seasonal climatology for the upper 2000 m. All 300 Argo profiles in the domain 17 to 12 S and 118 to 124 E from the period 1999 to 2010 were retrieved and interpolated onto a common-depth grid. We then harmonically fit the Argo data to both the annual and semi-annual frequencies at each depth layer to build a climatological record of the seasonal temperature cycle over the last decade Wave Number-Frequency Decomposition [12] We decomposed the raw internal tide signal from the mooring into its vertical wave number-frequency (modalharmonic) form: Y(z, t) =Y 0 exp(i(w m t k z z)) where Y(z, t) is any wave quantity. We used a methodology similar to Buijsman et al. [2010] to decompose the wavefield into a modal-harmonic form Yðz; tþ ¼ X X Y mn expð iw m tþw mn ðþ: z ð7þ m n [13] Here Y mn is the modal-harmonic amplitude (complex) for a given harmonic m and mode number n. W mn (z) are the eigenfunctions for the vertical velocity obtained by numerically solving the non-hydrostatic normal mode equation for given profiles of N(z) [cf. Gerkema and Zimmerman, 2008] d 2 W mn ðþ z dz 2 þ kmn 2 N 2 ðþ w z 2 m w 2 m f 2 W mn ðþ¼0: z This solution assumes a flat bottom that was considered to be reasonable at the mooring location given g < 1 (Figure 2c, distance =50 km).following Buijsman et al. [2010], the solutions to W mn (z) were normalized by multiplying with ð8þ q mn ¼ 1 Z 0 N 2 ðz Þ w 2 0:5 m H w 2 m f 2 Wmn 2 ðz Þdz : ð9þ H The vertical eigenfunctions for velocity in the direction of wave propagation, U mn ðþ z, perpendicular to wave propagation, V mn ðþ, z and buoyancy perturbation, B mn ðþ, z were then found using the internal wave polarization relationships [Gerkema and Zimmerman, 2008] U mn ðþ¼ z V mn ðþ¼ z i dw mn ðþ z ; ð10þ k mn dz f dw mn ðþ z ; ð11þ w m k mn dz B mn ðþ¼ z in 2 ðþ z W mn ðþ: z w m ð12þ Splitting the velocity into these two components assumes that the wavefield is two-dimensional and that waves are propagating in one horizontal plane, an approximation that will not be accurate for sites with a three-dimensional wavefield. It is important, however, to include both velocity components for energy considerations. We derived the barotropic velocity during this step by setting the barotropic mode (n = 0) to unity. [14] To perform the wave decomposition, first we used harmonic analysis to determine the amplitude of the semidiurnal signal via a least-squares fit to Yðx; z; tþ ¼ X m A m ðx; z Þcosðw m tþþb m ðx; zþsinðw m tþ: ð13þ Unlike the barotropic tide, the internal tide was nonstationary and therefore traditional harmonic analysis 4of17

5 techniques with many constituents were not useful for describing the variance in the signal. To capture the nonstationarity of the signal, and hence the temporal modulation, we fit the M 2 frequency (12.42 h) wave to 3-d segments of the data shifting forward 12 h at a time so there was an 84% overlap in the fits. Instead of summing many harmonics as in (13), we assumed the signal could be approximately modeled with a time varying amplitude and phase at the M 2 semi-diurnal frequency, i.e., Yðx; z; t Þ ¼ A m ðx; z; tþcosðw m tþþb m ðx; z; tþsinðw m tþ: ð14þ [15] During spring tides, the percentage of variance explained by the harmonic fit of velocity and buoyancy perturbation to 3-d segments of data was, on average, 60% and greater than 80%, respectively. We attribute this high coherency to the dominance of large-amplitude locally generated waves, assuming that wave generation u 0 2 according to (6). During neap tides, however, the semidiurnal fit accounted for less than 20% of the total signal variance of both quantities. This suggests the locally generated internal tide signal was significantly weaker, and the signal was dominated by remotely generated waves as found in other shelf regions [e.g., Martini et al., 2011]. We are primarily concerned with the locally generated wavefield in this paper and do not further consider the influence of these remotely generated waves other than noting their existence. Doppler shifting of the fundamental wave frequencies by background circulation may also influence the harmonic fit. The maximum background flow of 20 cm s 1 shifted the M 2 frequency mode-one waves by up to 10 % according to Dw ¼ku, where k is the wave vector and u is the mean depth-averaged flow. Higher-modes were more likely to be influenced by Doppler shifting as k increases (wavelength decreases). [16] Finally, we used the normal mode eigenfunctions to decompose the harmonic cosine and sine amplitudes, A m (z) and B m (z), into modal amplitudes via a least-squares fit A m ðx; zþ ¼ X n B m ðx; zþ ¼ X n A mn ðþw x mn ðþ z B mn ðþw x mn ðþ: z ð15þ [17] Note that the velocity and buoyancy eigenfunctions (U mn ðþ, z V mn ðþ z and B mn ðþ) z were substituted into (15) instead of W mn to fit the respective quantities. The modalharmonic amplitudes and phases for velocity (u mn, f u mn ) and buoyancy perturbation p (b mn, f b mn ) were calculated from (15) via u mn ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 mn þ B2 mn and f = tan 1 (B mn /A mn ). To minimize the uncertainty in the modal fits, we determined the optimal number of modes to be 7 for buoyancy and 5 for velocity. See Appendix A for a discussion of the uncertainty analysis procedure and results Energy Density and Energy Flux [18] The time-averaged and depth-integrated equations for available potential energy (APE = r 0 b 2 /2N 2 ), horizontal kinetic energy (HKE ¼ 1=2r 0 u u ) and energy flux (F ¼ u p ) in modal-harmonic (wave number-frequency) form are r APE mn ¼ 0 b 2 Z 0 mn B 2 mn ðþ z 4 H N 2 ðþ z dz; ð16þ HKE mn r ¼ 0 u 2 Z 0 mn U 2 mn 4 ðþdz; z H r F mn ¼ 0 2 u mnb mn cos f b mn Z 0 fu mn H U mn ðþ ^z Z 0 z B mn ðz Þdz ð17þ d^z: ð18þ [19] Here we have used angle brackets to denote a timeaverage and an over-bar to denote a depth-integral Sea Surface Height Anomaly [20] The surface manifestation of internal waves can be derived from the vertical pressure profile by using the constraint that the depth-integral of the baroclinic pressure, p, must be zero [see, e.g., Zhao et al., 2010]. The surface pressure induced by the baroclinic pressure is therefore p surf ¼ r 0 Z 0 H The sea surface height anomaly is SSHA ¼ psurf r 0 g : bz ðþdz : In modal-harmonic form the SSHA is SSHA mn ¼ b Z 0 mn B mn ðþdz : z g H ð19þ ð20þ ð21þ 2.5. Group Velocity [21] The group velocity was determined in two ways. First by taking the eigenvalues, k mn, from the solution to (8) and inserting them into c g w 2 ¼ m f 2 : ð22þ w m k mn Secondly, the group velocity was determined from the energy relationship c g ¼ FE 1. Following Alford and Zhao [2007b], we refer to these independent measures of group velocity as the theoretical and perceived group velocity, respectively. 3. Temporal Variability of the Internal Tide [22] In this section, we examine the temporal variability of the internal wave induced velocity, buoyancy, HKE, APE, energy flux, SSHA and group velocity at the B2 mooring. We conclude the section with a summary of evidence supporting the standing wave hypothesis Stratification and Barotropic Forcing [23] The vertical thermal structure derived from both the mooring and the harmonic fit to the Argo data exhibited similar annual cycles of surface heating and cooling (Figures 4a and 4b). The filtered mooring data was approximately 1 C cooler than the Argo data, most likely due to inter-annual variability. The warmest surface waters occurred around April 2007, toward the end of the austral summer, with surface temperatures in excess of 30 C. From May 2007 to September 2007, the surface waters began cooling and a 5of17

6 Figure 4. (a) Low-pass filtered depth-time temperature plot at the B2 mooring. Gray circles show location of thermistors. (b) Annual and semi-annual harmonic fit of temperature from 10 years of Argo profiles. Black crosses show the Argo record depths and black triangles indicate characteristic stratification conditions used for both summer and winter. surface mixing layer developed. The depth of this surface layer peaked at 100 m at the start of August, coinciding with the coolest period of the year (25 C). We categorized the stratification into two distinct seasons: summer, with larger average N in the upper 250 m and weaker stratification below 250 m; and winter, with a surface mixing layer in the upper m, the base of the thermocline at 250 m and weaker stratification below 250 m (Figure 5). These changes in stratification affect the shapes of the vertical eigenfunctions for velocity and buoyancy by increasing the depth of the maxima of the lowest three modes from summer to winter (Figures 5b and 5c). The stratification altered the eigenvalues and hence the group velocities, c g, at the mooring by approximately 10% for each mode: m s 1 for mode-one and m s 1 for mode-two. [24] The semi-diurnal cross-shelf barotropic forcing at the site underwent fortnightly modulations (Figure 6a). This is the result of the beating of the M 2 and S 2 constituents, which are similar in magnitude, causing a distinct spring-neap cycle. Furthermore, the spring tide intensity underwent a semi-annual modulation with the greatest amplitude in March and September (Equinox) and the smallest amplitude in December and June (Solstice). The amplitude of the crossshelf barotropic velocity during spring tides varied between 0.08 m s 1 and 0.15 m s 1. This is the result of the solar semi-diurnal tide amplitude being larger when the tilt of the earth is in plane with the sun [Pugh, 1987] Velocity and Buoyancy Perturbations [25] Both the semi-diurnal baroclinic velocity and buoyancy perturbation underwent a fortnightly modulation over the spring-neap barotropic tidal cycle (Figures 6b 6d). The periods with the largest amplitude baroclinic velocity and buoyancy perturbation generally corresponded with the spring tides. However, as we will show, other processes influenced the observed intensity of the internal tide during spring tide conditions, namely the interaction of multiple waves. [26] During the period November 2006 to February 2007, the baroclinic velocity exhibited a single peak of 0.15 m s 1 in the upper 100 m (Figure 6b); loosely corresponding to the predicted velocity profile shape for a mode-one wave during summer stratification conditions (Figure 5b). From the period of April 2007 to September 2007 there were three peaks in the velocity profile: one peak at the surface, a second peak at 150 m, and a third peak at 300 m depth. The deepening of the upper peak corresponded with the seasonal change in the eigenfunction vertical structure (Figure 5b); however, the peak at 300 m did not correspond to a peak in the lowest three vertical modes, indicating the possible 6of17

7 Figure 5. (a) Vertical profile of buoyancy frequency for summer and winter stratifications (b) velocity and (c) buoyancy perturbation eigenfunctions for the first three baroclinic modes. The summer and winter profiles correspond to the periods indicated by the black triangles in Figure 4b. The vertical position of the current meters are indicated by the triangles plotted on Figure 5b and the thermistors by the circles on Figure 5c. presence of a beam, composed of higher modes, intersecting the mooring at this depth. [27] Modal decomposition of the velocity field in the direction of wave propagation emphasized that the first mode was usually largest (u 21 = 0.08 m s 1 ) by up to a factor of two (Figure 7). There were periods when the second and third baroclinic modes were equal in amplitude to mode-one, e.g., Nov 06 and Apr 07, and therefore these modes cannot be neglected in the analysis. We will explore the modulation of the first two modes in more detail in the next section using wave energetics. [28] The semi-diurnal buoyancy perturbation amplitude also exhibited a distinct spring-neap modulation throughout the year (Figure 6d). Peak amplitudes of up to ms 2 were observed in the upper m of the water column. The peak perturbation deepened by 50 m during the winter months, consistent with the deepening of the peaks in the low mode eigenfunctions (Figure 5c). The buoyancy perturbation amplitude was very small below 250 m depth where the stratification was weaker (0.005 N 0.01 s 1 ). The peak vertical displacement amplitude, z(=b/n 2 ), was 40 m (80 m peak to trough) and occurred in the lower half of the water column (not shown). The greatest buoyancy perturbation amplitude generally coincided with periods of strongest barotropic tide, e.g., Mar 07, Apr 07 and Aug 07; however, this was not always the case, e.g., start of Nov 07 and Jan 07. The periods of largest perturbation coincided with the largest amplitude of the first baroclinic mode (Figure 7c). We observed no significant increase in the higher buoyancy perturbation modes during these periods. The increased buoyancy perturbation amplitude during larger spring tides indicated that the strength of the barotropic tide had an influence on the observed magnitude of the internal tide, although other processes such as the interaction of multiple waves also influenced the response. We examine the relative contribution of the interaction of multiple waves using an analytical internal tide model in section Energy Density and Energy Flux [29] We calculated the temporal modulation of the modeone and mode-two energy density and flux for the entire record (Figures 8 and 9). The magnitude of the mode one APE, HKE and energy flux all exhibited a spring-neap modulation. The amplitude of the APE generally exceeded the HKE by a factor of two or more, with peak APE and HKE in the range of 1 5 kj m 2 and kj m 2, respectively (Figure 8a). The magnitude and direction of the mode-one energy flux also varied widely, with magnitudes in the range of kwm 1. The energy density and flux were more variable in the summer months (December April) than in the winter months (May August) when the peak magnitude of energy flux was consistently between kw m 1 and the direction was primarily positive and along-shelf (north-east). In the winter months, the APE was consistently around 2 kj m 2, and the ratio of HKE to APE was around 0.5, significantly different from the progressive wave value ((w 2 + f 2 )/(w 2 f 2 ) = 1.13) and supporting the idea that the mooring was located in a standing wave region. This observation was first noted for this region by Rayson et al. [2011] for spring tide conditions during February 2007 and this suggests standing waves are persistent throughout the year. [30] The mode-two energy density and flux at the B2 mooring was highly variable throughout the year and even the spring-neap modulation was not readily apparent (Figure 9). The peak APE and HKE varied between kj m 2. In contrast to mode-one, the mode-2 HKE was often much larger than the APE and the HKE to APE ratio ranged between 1 to greater than 5. The magnitude of the mode-two energy flux ranged from kw m 1 and the direction was variable, although the predominant direction was off-shelf. 7of17

8 Figure 6. (a) Modulation of the semi-diurnal barotropic major ellipse amplitude (m s 1 ) at the B2 mooring. Time-depth plot of the amplitude modulation of the: (b) semi-diurnal velocity (m s 1 ) in the direction of wave propagation, (c) semi-diurnal velocity (m s 1 )perpendicular to the direction of propagation and (d) buoyancy perturbation (m s 2 ) at the B2 mooring. The vertical positions of the current meters are indicated by the triangles plotted on Figures 6b and 6c and the thermistors by the circles on Figure 6d Group Velocity [31] Following Alford and Zhao [2007b] we plot the observed energy flux versus energy density. All data are less than the theoretical group velocity for a progressive wave (Figures 10a and 10c). To identify temporal variability in the group velocity, we fit a line to monthly segments of jfj versus E, where the slope of the line is the perceived group speed. The perceived group speed (c g ¼ jfje 1 ) of the observed mode-one and mode-two waves varied significantly throughout the year (Figures 10b and 10d). The perceived mode-one group velocity, for example, was more variable in the summer period, reflecting the greater variability in energy density and flux during the summer, but did follow an annual cycle with a peak at the end of winter. In all cases, the perceived group velocity was much smaller than the theoretical value indicating that the wave response at the mooring was more likely due to multiple waves interacting rather than a single progressive wave Sea Surface Height Anomaly (SSHA) [32] The internal tide induced SSHA has been detected in the Browse Basin region with satellite altimetry [Katsumata and Wijffels, 2006]. Katsumata and Wijffels [2006] did, however, acknowledge that only the spatially and temporally coherent portion of the internal tide signal could be resolved. The key assumptions for using satellite altimetry to detect the surface expression of internal tides are that the signal is phase-locked with the barotropic tide and the amplitudes are stationary over multiyear periods. Time varying stratification and meso-scale flows both affect the propagation and generation of internal waves and hence temporal variability of the SSHA signal at any given point [e.g., Mitchum and Chiswell, 2000; Ray and Mitchum, 1996]. Our in situ observations of the mode-one and mode-two SSHA temporal modulation indicate that these assumptions are likely violated in the Browse Basin (Figures 8c and 9c, respectively). The peak amplitudes of the mode-one SSHA varied between 3 and 10 cm, with an average of 6 cm. While the peaks in the mode-one signal generally coincided with peaks in the local spring-neap tidal forcing (Figure 6a), the mode-one SSHA was not well correlated with u 0 (r 2 = 0.63). The low correlation can likely be attributed to the spatial shift of the standing internal tide as the stratification changes with season (Figure 2). In the Browse Basin longer altimetry records would have to be used in order to separate and analyze data 8of17

9 Figure 7. Modulation of the amplitude of the first three baroclinic modes for (a) velocity in the direction of wave propagation, (b) velocity perpendicular to the direction of propagation and (c) buoyancy perturbation at the B2 mooring. by season to account for the seasonal modulation of the mode-one internal wavelength by stratification [e.g., Ray and Zaron, 2011]. In section 4 we will use an analytical model to show how modulation of a standing wave by stratification can alter the potential energy amplitude and hence SSHA at a discrete location. [33] The amplitude of the mode-two SSHA varied between 0.1 and 0.5 cm and the correlation with the surface tide was poor (r 2 = 0.34). The small amplitude and lack of correlation with the surface tide, means that satellite altimetry will have difficulty detecting the surface signature of the mode-two internal tide at this site due to instrument precision and temporal sampling [e.g., Ray and Cartwright, 2001] Summary of Observations [34] Several observations indicate that a standing internal tide pattern dominated the dynamics at the B2 mooring: the HKE to APE ratio deviated from a progressive wave value of 1.13; the direction of the energy flux was highly variable; and the perceived group velocity was much smaller than the progressive wave value of 1.6 to 1.8 ms 1. The spring-neap tidal forcing strongly modulated the temporal variability of the mode-one internal tide amplitude, but the amplitude of HKE, APE and energy flux exhibited considerable variation amongst spring tides, suggesting tidal forcing was not the only influence on the wave response. Baroclinic modes two and higher exhibited less coherence with the surface tide, possibly due to their slower group velocity and longer travel times away from their generation region, making their influence less predictable. It was difficult to infer from the mooring data alone that the evolving stratification was a primary cause of this variation; therefore, in the next section we use an analytical model to highlight how stratification alters the spatial wavefield, and hence, the temporal modulation at a single point. 4. Analytical Internal Tide Generation Model [35] In this section, we examine how stratification affects the spatial variability of velocity and buoyancy perturbations in an idealised analytical model of a double shelf break. The analytical model for internal tide generation and propagation of Echeverri and Peacock [2010], (referred to as EP10 herein) allows us to identify the generation locations and study the wavefield structure and energetics over the shelf where the onshore and offshore propagating waves interact Model Assumptions and Setup [36] The EP10 model solves the two-dimensional, linear equations of motion using a Green s function approach, which reduces the problem to a single equation subject to boundary conditions) see Echeverri and Peacock [2010], Balmforth and Peacock [2009] and Llewellyn-Smith and 9of17

10 Figure 8. For the first baroclinic mode, time series of (a) semi-diurnal, depth-integrated HKE and APE and (b) semi-diurnal, depth-integrated wave energy flux magnitude and direction and (c) internal wave induced sea surface height anomaly. CS (+/ ) represents the positive/negative cross-shelf directions that are 32 clockwise from east and 148 counter-clockwise from east, respectively; AS (+/ ) represents the positive/negative along-shelf directions. Young [2003] for a derivation and details of the solution technique). The advantage of this approach over other generation models [e.g., Llewellyn-Smith and Young, 2003] is that it solves the 2D linear equations for arbitrary topography and non-uniform stratification. To achieve this, the solution invokes the Wentzel-Kramers-Brillouin (WKB) approximation that assumes the stratification is a slowly varying function of depth (i.e., no sharp density gradients) and the topography is smooth. The inviscid and linear assumptions limit the interpretation of the analytical solution. If the local Richardson number, Ri < 0.25, then the wave-induced flow field is unstable along the beam path. The linear qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi assumption requires that the nonlinearity parameter pu 0 1 ðf =wþ 2 = ðnhþ 1 [Llewellyn-Smith and Young, 2003] and that the ratio of tidal excursion to topographic length scale is far less than one. The nonlinearity parameter is essentially a Froude number for a rotating gravity wave in an ocean with constant N. At the inner-shelf break, where the barotropic flow exceeds 0.8 m s 1, the nonlinearity parameter will exceed one, particularly for higher vertical wave numbers. Higher modes are thought to be dissipated locally at generation due to hydraulic effects when the barotropic flow exceeds the internal wave group velocity [e.g., Klymak et al., 2010]. This is supported by laboratory experiments which found that a high-mode beam will not form under energetic nonlinear flow conditions [Lim et al., 2010] and instead low-mode internal waves will propagate away from the generation region [Echeverri et al., 2009]. The analytical model provides an upper-bound for the energetics on the shelf, but is useful for identifying the high mode beam trajectories and the effects of variation in the stratification and tidal forcing. [37] We used a smoothed version of the two-dimensional slice in Figure 1 to represent the topography in the model. We chose this slice as it was both aligned with the major tidal ellipse axis and the dominant direction of internal wave energy propagation [see Rayson et al., 2011]. The model requires the water depth to be equal at both ends of the domain and, to satisfy this condition, the topography was gently sloped downward from the continental shelf on the right hand side (not shown). We ran the model for the range of stratification (Figure 4) and barotropic forcing conditions (Figure 6a) observed at the mooring. We first use the model to demonstrate the effect of changing stratification on the wave structure and the wave energy over the shelf,in particular the alteration of the HKE-APE ratio. In Section 5, we directly compare the observations of wave energy at the mooring of summer to the analytical solution and discuss the implications Summer and Winter Solutions [38] We applied the EP10 model for two representative density profiles for summer (April) and winter (July) 10 of 17

11 Figure 9. As in Figure 8 but for mode two. conditions (black arrows in Figure 5b). The M 2 barotropic forcing amplitude, u 0, was set to 0.03 m s 1, representative of maximum spring tide conditions at the offshore boundary. We begin our discussion with a kinematic description of the internal tide propagation under these stratification conditions before moving onto a more quantitative analysis of the wave energetics. We focus our attention solely on the region of the domain between the two shelf breaks where the mooring and the standing wave were located. [39] The double shelf topography generated two beams, the first at the inner-shelf break (x = 110 km) and the second at the outer-shelf break (x = 35 km), each propagating away from the respective generation sites in both upward and downward directions (Figure 11). For the summer solution, the onshore (rightward) beam from the outer-break reflected from the surface and bottom before intersecting the inner shelf (Figure 11a). This beam then back-reflected from the supercritical inner-shelf (x = 110 km), forming the beam that propagated upwards and offshore from this site (and eventually intersecting the mooring location near the surface). A second beam was generated at the inner-shelf that propagated downward and reflected off both the bottom and the surface before intersecting the mooring location essentially overlying the first beam coming in from offshore. Ray tracing using (4) confirmed that there were two overlying beams along the same characteristic path (Figure 3). As can be seen, the onshore and offshore propagating beams intersected the B2 mooring near z = 400 m, while the backreflected beam intersected it near the surface. For the winter solution, the two generation regions at the inner- and outershelf breaks were again present; the key difference with the summer solution was the clear offset between the characteristic paths of the onshore and offshore propagating beams (Figure 11b). The onshore and offshore propagating beams intersected the B2 mooring at z = 400 and z = 550 m, respectively. The change in stratification shortened the distance between the beam reflection sites, thus offsetting the two beams. For the winter solution the onshore propagating beam propagated shoreward into depths of less than 200 m rather than back-reflecting at the inner-shelf slope. [40] There were several properties and features of the wavefield in the analytical model that likely differ from reality in this macro-tidal shelf region. First, the sharp velocity gradients associated with the beam would likely dissipate due to local shear instabilities and overturning and hence would rapidly dissipate the higher vertical modes [e.g., Lien and Gregg, 2001]. Second, near-critical reflection of the beam at the inner-shelf during summer, would result in a large local loss of energy due to turbulent motion, and thus a considerable reduction or perhaps even elimination of the energy in the back-reflected wave [e.g., Ivey and Nokes, 1989]. Third, energy losses are also likely during reflection at near horizontal sections of the shelf due to high levels of turbulence in the bottom boundary layer associated with the intense barotropic tidal motion [Bluteau et al., 2011; Rayson et al., 2011]. The other features that may not exist in reality were the beams that emanated from the concave section of the inner-shelf at x = km. Although these beams are predicted by the linear theory of Gilbert and Garrett [1989], the non-linear advective terms in the equations of motion would lead to greater dissipation near this particular type of 11 of 17

12 Figure 10. (a) Scatterplot of energy flux magnitude versus energy density for the mode one internal tide at the B2 mooring. The gradient of the dashed line represents the theoretical group velocity of a purely progressive wave during February. (b) Time series of the monthly least-squares fit to the perceived group velocity (black line) and the theoretical group velocity (dashed). The error bars indicate the standard error of the fit. (c and d) As in Figures 10a and 10b but for mode-two. topography [e.g., Legg and Adcroft, 2003]. Despite these limitations, it is still illuminating to investigate the variability of the low-mode wavefield and associated energy over the shelf since turbulent dissipation is likely to have less influence on these waves Low-Mode Energy Density [41] We applied the modal decomposition procedure described in Section 2 to the velocity and buoyancy fields at each point in the analytical model to assess the spatial variability of the energy in each mode. We only applied the modal decomposition to the relatively flat subcritical (g <1) region between the two shelf breaks where the decomposition to orthogonal vertical modes over a flat bottom is a reasonable assumption. The spatial variability of the modeone APE and HKE behaved similarly to the expected result for a standing wave- oscillating at twice the rate of the horizontal wavelength and 90 out of phase (Figure 11c). The effect of the change in stratification from the summer to the winter solution was a horizontal shift in the nodesand antinodes by approximately 10 km. This shift in the nodes and anti-nodes modulated the HKE to APE ratio from 0.55 during summer to 0.9 during winter; deviating from the progressive wave ratio of 1.13 at latitude 14 S. The modetwo HKE and APE variation behaved similarly to the mode-one amplitudes, although there were more nodes and anti-nodes in the standing wave pattern due to the shorter horizontal wavelength (Figure 11d). The HKE to APE ratio at the B2 mooring location in these two example scenarios was in qualitative agreement with the observations: in particular, the mode-one ratio was <1 and was temporally modulated by the stratification. As expected, the total energy in the model wavefield was considerably greater than seen in the observations, and we now present a quantitative comparison over a wider parameter space. 5. Energy Conversion Comparison [42] Here we present a comparison between the analytical model and the observed internal tide at the B2 mooring under the full range of observed forcing and stratification conditions. We have plotted the observed mode-one, semidiurnal energy density, E ¼ HKE 21 þ APE21, versus the theoretical energy density E 0 [J m 2 ], derived from equation (6) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi E 0 ¼ F 0 ¼ 1 N0 2 w2 ð w2 f 2 Þ c g;n 2pc n r 0 u 2 0 g w H 2 : ð23þ 12 of 17

13 Figure 11. (a and b) Internal tide velocity amplitude (m s 1 ) from the summer and winter analytical solutions, respectively. The yellow dashed lines indicate the M 2 ray characteristic paths. Spatial variation of the APE and HKE for summer and winter stratification for (c) mode one and (d) mode two waves. The location of the mooring and instruments are marked with the gray triangles. [43] Here c g,n is the mode-n group velocity and u 0, H and N 0 are representative values of the barotropic velocity, depth and buoyancy frequency, respectively. The ideal choice of these parameters are the respective values away from the generation site; however, given there were two generation sites, we used the value of N 0 and c g, n at the mooring depth (550 m). The value of u 0 2 H 2 in (23) is the square of the tidal transport, Q [m 2 s 1 ]. The ranges of these parameters relevant to the region and used in the model were Q =9 81 m 2 s 1, N = s 1, c g,1 = m s 1 and E 0 = J m 2. Our decision to use energy, and not energy flux, as the metric for comparison was to avoid issues with wave interference reducing or canceling the energy flux (e.g., Figure 2). We compare only the mode-one energy due to the uncertainty in estimating the higher mode energy content from the mooring (see Appendix A). [44] The observed internal tide energy, E, exhibited no significant linear relationship with E 0 other than a general positive trend (Figure 12). There was, however, a clear seasonal shift in the region of the graph inhabited by the summer and winter clusters of points. The gradient of the linear best fit for the summer (December to March inclusive) cluster of points was 0.75, while for the winter data (May to August inclusive) the gradient was 1.51 (not shown). The maximum seasonal shift in energy for a perfect horizontally standing wave is 1 (w 2 f 2 )/(w 2 + f 2 ) = 13% at 14 S, which does not account for the >50% shift observed here (see Figure 2b). We suspect that changes in the relative amount of internal wave energy reaching the mooring from the inner- and outer-shelf generation regions are the biggest influence on the seasonal shift in the slope of E versus E 0. [45] A mechanism that may cause this factor of two decrease in the slope of E versus E 0 from summer to winter, is destructive interference of the pressure field at the generation sites [Kelly and Nash, 2010]. The tidally averaged barotropic-to-baroclinic conversion rate, C [W m 2 ], is hci ¼ 1 2 u ð 0 rhþp jz¼ H cos f u f p ; ð24þ where u 0 is the barotropic velocity amplitude, p jz¼ H is the harmonic amplitude of the baroclinic pressure at the seabed and, f u and f p are the respective phases. The cos(f u f p ) term highlights how a phase difference between the forcing and the pressure field will modulate the conversion term. Zilberman et al. [2011] measured a temporal modulation of energy conversion at a generation site on the Hawaiian Ridge and attributed the difference to relative changes in the phasing of u 0 and p by the seasonal stratification. The EP10 model indicates that there may some seasonally dependent wave interference at the inner-shelf caused by the onshore propagating beam intersecting the slope during summer (Figure 11a). The linear solution does not account for the effect of this process on wave generation and numerical modeling or field measurements would therefore be required to validate this idea that stratification can alter the energy conversion rate at the inner-shelf. 13 of 17

14 are the topographic height and length scales, respectively. Formally, the analytical solution applied here is not valid at the inner-shelf and a fully non-linear, non-hydrostatic solution is necessary to calculate the conversion [e.g., Legg and Huijts, 2006]. However, the analytical solution provides an upper bound for the potential mode-one internal tide energy and therefore provides an important foundation for comparison with observations. Figure 12. Scatterplot of the dimensional energy conversion term, E 0 (23), versus the semi-diurnal, mode-one internal tide energy, E, for the range of stratification and forcing conditions observed at the B2 mooring location. Colored dots represent the observed values (colored by time) while the black dashed line indicates the analytical model solution. The gray patch indicates the range of variability in the analytical solution due to two wave interference at 14 S. [46] The analytical model slope of E versus E 0 was 2.45 and provided an upper bound for the observed data points. There were some observed energy densities at low E 0 (<1000 J m 2 ) that exceeded the model but generally the model over-predicted the observations. This result is unsurprising given that the model is frictionless and therefore conversion efficiency, that is, the ratio of topographic conversion to energy flux that radiates away, is 100%. 3-D numerical modeling of the Browse Basin internal tides using ROMS found that the conversion efficiencies at the innerand outer-shelf breaks during spring tides and summer stratification were 52% and 74%, respectively [Rayson et al., 2011]. A 40 50% relative difference between the modeled and observed slope of E versus E 0 is therefore reasonable. Furthermore, observations by Klymak et al. [2006] of openwater turbulence away from a generation region indicate that dissipation scales with E (10.5), providing further explanation for the differences between the linear solution and observations. [47] We expect that dissipation and non-linear advection effects at the inner-shelf generation site to be strong as the topography at this site falls into region 4 of the Garrett and Kunze [2007] parameter regime diagram (finite topographic excursion (u 0 /wl s 1), inverse topographic Froude number (N 0 h s /u 0 ) > 1 and slope criticality (g) > 1); where h s and l s 6. Summary and Conclusions [48] We have investigated the temporal modulation of the coherent semi-diurnal internal tide between a double-shelf break. The low-mode, coherent portion of the internal tide dominated the signal especially during spring tide periods. We estimate that our analysis of the coherent portion of the low-mode internal tide accounted for 50% of the total internal tide variability at this site; the rest of the entire frequency-vertical wave number continuum accounted for the other half. We attribute this single constituent dominance to locally generated waves given that conversion scales with u 2 0. This is in contrast to other shelf regions where the wavefield is generally less coherent and made up of a combination of locally and remotely generated waves (e.g., Oregon Shelf) [Martini et al., 2011]. [49] The coherent portion of the internal tide exhibited properties characteristic of a standing internal tide: variable HKE-APE ratio; variable energy flux direction; and a lower group velocity than expected from linear theory. A onedimensional model demonstrated how stratification altered the horizontal position of the standing wave s nodes and anti-nodes over a flat bottom (Figure 2). The low-mode wavefield from the EP10 analytical solution exhibited a similar response to the one-dimensional model, in particular the spatial variability of HKE and APE (Figures 11c and 11d). These models demonstrate the influence of seasonal changes in stratification on data from a single point measurement. [50] Quantifying the proportion of internal wave energy that propagates away from a generation region in low-modes is important for understanding the global distribution of internal wave energy and mixing [Alford, 2003]. This requires the prediction of barotropic-baroclinic conversion and the direction and magnitude of the energy flux away from generation regions. In places where wave interference is strong, the interpretation of energy flux from individual observations needs to be treated with care as energy flux is not aligned with the direction of wave propagation; as observed at the B2 mooring. Furthermore, analytical solutions, as demonstrated here, provide an upper-bound prediction of wave energy in response to local forcing and stratification. A higher order prediction of the low-mode wavefield would require estimates of both dissipation and energy transfer to the rest of the wave continuum [e.g., Klymak et al., 2006]. The temporally variable effect of wave interference at generation sites is another factor to consider. Appendix A: Modal Uncertainty Analysis [51] Our interpretation of the linear internal wave quantities at the mooring is dependent on our level of confidence in the modal fits. To gauge the errors to the modal fits, we 14 of 17