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Near-Inertial Wave Propagation in the Western Arctic

ROBERT PINKEL Marine Physical Laboratory, Scripps Institution of Oceanography, La Jolla, California

(Manuscript received 11 April 2003, in final form 8 October 2004)

ABSTRACT

From October 1997 through October 1998, the Surface Heat Budget of the Arctic (SHEBA) ice camp drifted across the western Arctic Ocean, from the central Canada Basin over the Northwind Ridge and across the Chukchi Cap. During much of this period, the velocity and shear fields in the upper ocean were monitored by Doppler sonar. Near-inertial internal waves are found to be the dominant contributors to the superinertial motion field. Typical rms velocities are 1–2 cm sϪ1. In this work, the velocity and shear variances associated with upward- and downward-propagating wave groups are quantified. Patterns are detected in these variances that correlate with underlying seafloor depth. These are explored with the objective of assessing the role that these extremely low-energy near-inertial waves play in the larger-scale evolution of the Canada Basin. The specific focus is the energy flux delivered to the slopes and shelves of the basin, available for driving mixing at the ocean boundaries. The energy and shear variances associated with downward-propagating waves are relatively uniform over the entire SHEBA drift, independent of the season and depth of the underlying topography. Variances associated with upward-propagating waves follow a (depth)Ϫ1/2 dependence. Over the deep slopes, vertical wavenumber spectra of upward-propagating waves are blue-shifted relative to their downward counterparts, perhaps a result of reflection from a sloping seafloor. To aid in interpretation of the observations, a simple, linear model is used to compare the effects of viscous (volume) versus underice (surface) dissipation for near-inertial waves. The latter is found to be the dominant mechanism. A parallel examination of the topography of the western Arctic shows that much of the continental slope is close to critical for near-inertial wave reflection. The picture that emerges is consistent with “one bounce” rather than trans-Arctic propagation. The dominant surface-generated waves are substantially absorbed in the underice boundary layer following a single roundtrip to the seafloor. However, surface-generated waves can interact strongly with nearby (Ͻ300 km) slopes, potentially contributing to dissipation rates of order 10Ϫ6–10Ϫ7 WmϪ3 in a zone several hundred meters above the bottom. The waves that survive the reflection process (and are not back-reflected) display a measurable blue shift over the slopes and contribute to the observed dependence of energy on seafloor depth that is seen in these upper-ocean observations.

1. Introduction Merrifield and Pinkel 1996). Assuming propagation without significant energy loss, this corresponds to a Near-inertial waves represent a resonant response of shoreward flux of 10–100 W mϪ1, propagating across the ocean to impulsive forcing. The resonant frequency the 104 km circumference of the shelf break. If this flux f ϭ ⍀ ⌽ ⍀ is 2 sin( ), where is the earth’s rate of rotation dissipates in an annular region of 50-km width, 200-m and ⌽ is the latitude. On a spherical earth, the waves, Ϫ Ϫ average depth, a mean dissipation of 10 6 to 10 5 W once generated, are constrained to refract toward lower Ϫ m 3 results. These mixing rates are comparable to en- latitudes (Munk and Phillips 1968). In the Arctic, near- ergetic open-ocean and coastal sites. If real, they would inertial waves will invariably refract southward toward have a great impact on property distributions in the the ocean boundaries, potentially contributing to dia- Arctic. pycnal mixing processes on the slopes and shelves. This naive, “straw man” scenario posits that near- To estimate the magnitude of this process, we imagine sea surface (wind) forcing over the 1013 m2 area of inertial waves propagate through the Arctic like surface the Arctic Ocean resulting in an average downward swell propagates across the open sea. Despite the ran- near-inertial energy flux of order 0.01–0.1 mW mϪ2 dom (in space, time, and direction) midocean forcing of (D’Asaro and Morehead 1991; Halle and Pinkel 2003; the swell, one always sees a landward energy flux at the borders of the ocean. The arriving flux represents the cumulative effect of past generation events distributed Corresponding author address: Dr. Robert Pinkel, MPL/SIO, across a large area of the ocean surface, modestly di- 9500 Gilman Dr., La Jolla, CA 92093-0213. minished by dissipation and nonlinear transfers. E-mail: [email protected] If dissipative effects are severe and trans-Arctic

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JPO2715 646 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 propagation is unrealistic, can local near-inertial gen- loss.” Surface attenuation, volume attenuation, back re- eration contribute significantly to deep mixing on flection, and dissipation on deep slopes are the princi- nearby continental slopes? Consider a second straw pal unknowns. Assessing the relative importance of man, where the energy input at a single square meter of these various processes is the focus of this paper. the sea surface is mapped to a reflection area on the The yearlong Surface Heat Budget of the Arctic deep slope. For simplicity, a 2D situation is taken, with (SHEBA) experiment provided an opportunity to in- all of the energy radiated at a single near-inertial fre- vestigate both the temporal and geographic variability quency. The total downward flux, Fdn, is delivered to a of the Arctic wave field. The experiment was staged seafloor area sin(␪)/sin(␪ ϩ ␥), where ␪ and ␥ are the from the Canadian Coast Guard icebreaker des angles of energy propagation and seafloor slope rela- Groseilliers, which was frozen into the southern Canada ϭ tive to the horizontal. If the full incident flux ( 2Fdn at Basin (75°N, 144°W) in October 1997 (Fig. 1). The ship the critical frequency, ␪ ϭ ␥) is available for mixing an and a surrounding ice camp subsequently drifted west- “internal surf zone” (Thorpe 2001) of depth D, the dis- ward across the basin (November 1997–February 1998) ⑀ ϭ ␪ ϩ ␥ ␪ ϭ sipation is (Fdn/D) sin( )/sin( ). Taking Fdn over the Northwind Ridge (March–April 1998) and 0.1 mW mϪ2, D ϭ 200 m, and ␪ ϭ ␥, the resulting then northward along the Chukchi slope and over the dissipation is ⑀ ϭ 10Ϫ6 WmϪ3. Chukchi Cap (June 1998). The camp was recovered in Timmermans et al. (2003) find that a dissipation rate October 1998 (80.5°N, 165°W). of 10Ϫ7 WmϪ3 along the deep (Ͼ2500 m) boundaries of Downward-looking Doppler sonars documented the the Canada Basin is sufficient to balance the geother- internal wave field and lower-frequency current struc- mal heat flux in the interior. Acknowledging that sur- tures through much of the drift. The sonars profiled to face-generated energy does not always propagate 250 m, with a depth resolution of 3 m. Operation was shoreward, is not completely dissipated on the slope, interrupted by the periodic breakup of the camp over and so on, this straw man flux estimate can be degraded the winter and by problems associated with the partial by a factor of 10 and still contribute significantly to the melting of the ice cover in midsummer. Nevertheless, geothermal flux balance. Much of the incident flux excellent views of the wave field were obtained in the might be left to reflect upslope and shoal. central Canada Basin (December 1997), over the Chuk- Given the near-absence of direct dissipation mea- chi Cap (spring–summer 1998), and in the deep ocean surements over arctic slopes and shelves, there is some to the northwest of the cap (August–September 1998). motivation for exploring the details of these straw man The Arctic Ocean is a unique wave propagation en- scenarios. A key issue is that of “propagation without vironment. Typical wave field energy levels are a factor

FIG. 1. The topography of the Western Canada Basin and Chukchi Slope. The SHEBA drift track is indicated.

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FIG. 2. Two-dimensional shoaling internal waves scenarios in the geometric optics limit. For sufficiently high frequency waves, (a) forward reflection leads to a buildup of wave energy as wave groups shoal. Scenarios for (b) back reflection, (c) spatially uniform surface forcing, and (d) surface attenuation are described in the text. The symbols used in (c) indicate separate surface-generation events. of 10–100 less than in the open ocean (Levine et al. the energy density, E (J mϪ3), of both upward- and 1985; D’Asaro and Morison 1992). Shear variance is downward-propagating waves should grow, varying in- also reduced, although to a lesser extent than energy.1 versely with bottom depth H in the absence of dissipa- Nonlinear interactions, which proceed at a rate depen- tion, generation, or lateral ray divergence (Fig. 2a). If, dent on wave field energy, must be very weak in the in contrast, the waves are back-reflected into the deep Arctic. sea (Fig. 2b), there will be a reduction in upward arriv- In the deep Arctic Ocean, the waves are generated ing wave energy over the continental slopes (and a cor- primarily at the sea surface by ice motion resulting from responding increase where the packet “resurfaces” off- passing storms (Morison 1986; McPhee and Kantha shore). If the waves have a lifetime that is long in com- 1989; Halle and Pinkel 2003). Unlike the open sea, parison with a single round-trip surface to seafloor where the horizontal scale of the surface stress is set by transit (the straw man scenario; Fig. 2c), the energy the scale of the storm, in the Arctic, the stress scale is density of both upward- and downward-propagating also influenced by the morphology and relative motion waves will increase as the shelf is approached. Both will of the ice. Ice keels and other small-scale features im- experience the HϪ1 “shoaling” effect. If waves decay pose a vertical motion in the mixed layer, leading to the after a single round-trip cycle, say because of underice generation of relatively small-scale (Ͻ1 km), high- dissipation (Fig. 2d), down-going energy will depend frequency waves (McPhee and Kantha 1989). Larger- only on local generation conditions and will be inde- scale, low-frequency waves result from lateral stresses pendent of the depth of the seafloor below. In this case, imposed at the scale of the floe. the spatial concentration (shoaling) of upward- D’Asaro (1989) emphasizes that a relatively broad propagating wave packets over the slopes still occurs, bandwidth of generated motion is needed if energy is to although the magnitude of the effect is reduced relative escape from the mixed layer into the thermocline be- to the nondissipative case. An examination of the spa- low. There is now evidence (Halle and Pinkel 2003) tial variability of upward- and downward-propagating that the efficiency with which an arctic wind event gen- wave energy, interpreted through simple model studies erates propagating near-inertial motions is inversely re- of propagation, dissipation, and reflection, can advance lated to the horizontal scale of the floes, in accord with understanding of this process.2 D’Asaro’s argument. While both high and low modes The SHEBA observations are presented in section 2. are generated, the relative proportion of high modes (lz These include waves that are found in the upper ocean Ͻ 100 m) is greater in the Arctic than in the open (50–300 m), either immediately following local surface ocean, a consequence of the ϳ1–10-km scale of typical generation/reflection (downward energy propagation) floes. or more distant seafloor generation/reflection (upward To assist in interpreting the SHEBA observations, it energy propagation). is useful to consider several simple propagation scenarios. As propagating near-inertial waves shoal (over the course of multiple surface and seafloor reflections), 2 For simplicity, the potentially significant effects of nonlinear- ity, refraction by large-scale currents, and seafloor scattering are not considered. D’Asaro (1991) includes these processes in his 1 Nevertheless, the Gregg (1989) parameterization relating tur- discussion of open-ocean propagation. The failure of the linear bulent dissipation to background finescale shear predicts an eddy model presented here to account for the occurrence of small-scale diffusivity that is smaller than molecular viscosity at subther- upward-propagating waves in the deep Arctic Ocean attests to the mocline depths. significance of these neglected processes.

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A spatial propagation filter (section 3) is used to A replacement 161-kHz sonar was hastily con- separate upward- and downward-propagating waves structed. This device used conventional 6-in. disk trans- and to track wave energy and shear as a function of the ducers angled 60° downward from horizontal. An 18-bit depth of the underlying seafloor. The observations dis- code of 12.5-kHz bandwidth was transmitted, resulting play a complicated dependence of E on H. Temporal in depth resolution of 3.69 m. The data record resumed variability appears as a significant “noise,” complicat- in late March and continued through the end of the ing the effort to identify geographic patterns. In gen- experiment in early October 1998, with a few gaps over eral, both energy and shear variance associated with the summer. downward-propagating waves appear to be nearly in- During the course of the drift, a daily record of sea- dependent of seafloor depth. For upward-propagating floor depth was maintained using the ship’s echo waves, a significant dependence is seen, with E ϳ 1/H1/2. sounder. These data are used in the studies of wave The issue of the reflection of near-inertial waves shoaling presented here. from a sloping seafloor is explored in section 4. Using a The thermocline can be characterized by the buoy- z]1/2, where g representsץ/␳ץ(full ocean depth buoyancy profile and a digital bathy- ancy frequency N ϭ [(g/␳ metric map, it is found that large regions of the western gravitational acceleration, ␳ is the potential density of Arctic slopes are back-reflective or near-critically re- the fluid, and z is positive downward. Profiles of N are flective to near-inertial waves. Deep local mixing along presented in Fig. 3 for three representative observa- the Arctic slopes is likely, fueled by near-critically re- tional periods.3 In December in the Central Beaufort flected waves. The signature of the bottom reflection Sea, high stratification occurs at the base of the mixed process in the upper-ocean SHEBA observations is in- layer (ϳ40 m) and at the deeper boundary between the vestigated in section 5. halocline and Atlantic waters, centered at 200 m. In A linear wave propagation model is applied in the addition, an anomalous fresh (salinity ϭ 28 psu) lens of appendix to address whether volume, seafloor water is found just beneath the ice (McPhee et al. 1998), (D’Asaro 1982), or underice attenuation (Morison et al. providing significant stratification to the region. In 1985) materially affect arctic wave propagation. The June, over the Chuckchi Cap, the distinct underice lens conclusion is that underice attenuation is a critical fac- is no longer present. The seasonal pycnocline is shal- tor. Only near-inertial waves (␴ ϳ f ), which propagate lower (20 m), with greatly reduced stability. By Sep- great distances between successive surface reflections, tember, a new, seasonal pycnocline has formed with and high-frequency waves (␴ ӷ f ), which rapidly high stratification within 15 m of the surface. A deep traverse the underice turbulent boundary layer, can secondary maximum in buoyancy frequency is not seen. propagate over significant distances in the Arctic. 2. Observations 3. Wave field description The Doppler sonar initially used in SHEBA was a 140-kHz four-beam instrument constructed specifically a. Depth–time variability for Arctic operations. The transducers, each of dimen- For initial analysis, the 1-min sonar velocity records sion 12 in. vertical by 8 in. horizontal, had narrower ϳ ϫϳ have been hourly averaged and rotated into earth co- beam widths ( 2° 3°) than typical commercially ordinates. The shear field is approximated as a 1.65-m available systems. These minimized sidelobe contami- vertical difference prior to February 1998, 1.85 m there- nation from reflections associated with ice keels and after. Representative plots of the zonal component of other underice structures. Sonar beams were oriented shear are presented in Fig. 3. The record exhibits the 45° down from horizontal. A 4.5-ms repeat sequence natural tendency of shear to be concentrated in regions code (Pinkel and Smith 1992) was transmitted with of higher buoyancy frequency, a consequence of linear 4.16-kHz bandwidth, yielding a depth resolution of refraction. 3.27 m. The December record shows a low energy wave field The SHEBA sonar was designed for deployment under a surface layer whose thickness and stratification from a well on the base ship. However, when the vessel vary in time. A storm on 6–8 December generates ultimately selected for SHEBA proved to not have such strong shears under the ice. The storm coincides with a well, the sonar was deployed from a heated tent over the passage of the camp over a coherent baroclinic eddy a hole in the ice. The initial installation was performed that extends throughout the low stratified region from in November 1997. One-minute-average profiles of 50–250 m. Following the storm and passage of the eddy, echo covariance (velocity) and acoustic scattering in- the downward propagation (upward-propagating tensity were recorded on optical disk. The winter rec- crests) of a near-inertial wave packet is seen. The ap- ord was interrupted in January by storms that broke up the SHEBA floe, temporarily disrupting electrical power. It terminated on 9 February during a period of floe rafting, when the sonar transducers were de- 3 CTD data were graciously provided by J. Morison and R. stroyed. Anderson of the University of Washington.

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FIG. 3. Depth–time maps of (left) zonal shear and (right) the corresponding monthly averaged buoyancy frequency for three representative observational periods. The dominant shear is associated with near-inertial internal waves, propagating through a background of intermittent mesoscale activity. For these waves, the upward propagation of crests with time corresponds to the downward propagation of energy. Note the refractive enhancement of the shear field in regions of high stability and the baroclinic eddy that passes under the camp on 8 Dec. Shear levels in the interior of the eddy are very low. The horizontal line at 150 m in the Dec record is an artifact. The broad scar occasionally seen in the Jun data is the echo return from the seafloor associated with a previously transmitted acoustic pulse. Here and in Fig. 4, the “green” color represents offscale values less than –0.005 sϪ1. Offscale values greater than ϩ0.005 are pegged “red.” parent vertical group speed is 30–50 m weekϪ1.4 Nu- and is confined to a smaller region in depth, 25–150 m. merous upward-propagating packets are also seen. Further subinertial activity is seen around 27 June. By The June shear field, over the Chukchi Cap, is much open-ocean standards, energy levels over the Chukchi more energetic, with a distinct upward-propagating Cap are low. Shear levels, however, are comparable to wave group dominating the first 5 days of the record. energetic open-ocean or coastal sites. The small vertical There is some hint of a corresponding downward re- scale of the motions, relative to lower latitude wave flected wave packet on 9–12 June. The picture is ob- fields, leads to the large shears. The depth–time inter- scured by a subinertial feature passing under the camp mittency of the wave groups appears to be greater than during 10–12 June, between 75–150 m. The pattern of a in the December deep-sea observations. shear maximum with a contrasting maximum negative The August–September shears are relatively uniform shear below indicates a jetlike velocity field. The fea- in variance over the upper 150 m. A few distinct down- ture might be one edge of a baroclinic vortex or an ward-propagating wave packets stand out above the independent frontal structure. The camp passes directly background. Subinertial activity is seen on 31 August at over a small vortex on 15–17 June. This vortex has 25-m depth and 14–15 September from 20 to 100 m. substantially greater shear than a Beaufort Sea eddy While a continuous spectrum of motions is present, analyses to be presented in a subsequent work indicate that the field is primarily near-inertial. The apparently broad range of frequencies is a result of simple Doppler 4 During the period of observation, 1–25 December, the camp drifted 150 km southwestward. The apparent vertical expansion of shifting. For near-inertial waves, the vertical direction the wave packet into the upper ocean is in fact a mix of vertical of propagation is given by the rotation of the wave- propagation and lateral variability current pattern with depth (Leaman and Sanford 1975).

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In the Northern Hemisphere, downward-propagating specifies the sense of phase propagation in time, the wave energy is associated with a clockwise rotation of assumption that the shear is primarily near-inertial is horizontal currents with increasing depth. confirmed. The rotary polarization of the waves (with depth) can Examination of rotary filtered hourly profiles reveals be used to separate the overall wave field into upward- coherent patterns of vertical phase propagation, with and downward-propagating constituents. Unlike tradi- the proper relationship between rotation in depth and tional wavenumber–frequency spectral analysis (e.g., propagation in time (Fig. 4). Thus, despite the appar- Pinkel 1984), continuous time series are not required. ently continuous spectrum of temporal frequencies ob- Intermittent data gaps associated with camp power fail- served, these arctic waves are highly polarized and must ures, for example, do not degrade the quality of the have intrinsic frequencies near inertial. Data from separation. However, higher-frequency internal waves throughout the drift have been rotary filtered, provid- that are not rotationally polarized will contribute vari- ing a look at wave variability with both season and ance to both positive and negative wavenumbers in a location. rotary spectral separation. The daily averaged velocity (kinetic energy) and To perform the separation, vertical profiles of com- shear variances of the wave field are presented as a ϭ ϩ ␷ plex current, U(z) uEast i North, are high-pass fil- function of time in Fig. 5. Individual points represent an tered in time to include motions of period 15 h and integration over 20–160-m vertical scale of the corre- shorter and are Väisäla [Wentzel–Kramers–Brillouin sponding vertical wavenumber spectra of velocity and (WKB)] stretched (Pinkel 1984) in both amplitude and shear. The time history of upward-propagating kinetic vertical phase. The stretching is normalized such that energy density (Fig. 5a) shows a variation of about a the total kinetic energy and the vertical extent of the factor of 5 over the year, with peak values occurring observation window are preserved. The stretched pro- over the Chukchi Cap in midsummer. Smallest values files are then first differenced in depth and Fourier are seen in deep water, in the center of the basin (De- transformed between depths of 39 and 295 m, below the cember) and north of the cap (October 1998). Variabil- region of most rapid variation of the buoyancy fre- ity in downward energy density (Fig. 5b) is slightly less. quency. Fourier coefficients are then “recolored.” Ve- Values center on ϳ0.7 cm2 sϪ2, except during May– locity spectral variance is accumulated in specific ver- June, when enhanced energy is seen. A similar pattern tical wavenumber bands, for both positive and negative is seen in the daily averaged shear (Figs. 5d,e). Again, wavenumbers. Shear products are obtained by weight- slightly greater variability in shear is associated with ing the velocity Fourier coefficients by ikz and process- upward propagation. The seafloor depth, obtained ing in parallel with velocity. Daily averaged vertical from an echo sounder on the des Groseilliers is given in wavenumber spectra are then archived for the geo- Fig. 5c, with slope angle magnitude given in Fig. 5f. graphic variability studies presented below.5 Reconstructed “depth profiles” of strictly upward b. Wave field variability with seafloor depth and strictly downward wave motion can be obtained by The dependence of the upper-ocean velocity and inverse transforming “reconstituted” sequences of Fou- shear variance on underlying seafloor depth is pre- rier coefficients. To produce profiles of purely upward sented in Fig. 6. Individual entries represent an integra- (downward) propagation, the negative (positive) wave- tion over 20–160-m vertical scale of the corresponding number coefficients are replaced by the complex con- daily averaged vertical wavenumber spectra6 of velocity jugate of the corresponding positive (negative) wave- and shear. The vertical extent of each entry indicates number coefficients. Normalization of the profiles by Ϫ1/2 the 80% confidence interval, calculated under the as- 2 preserves the variance of the overall field. sumption that the process is spatially homogenous. The quality of this rotary separation can be investi- At depths shallower than 500 m, velocity and shear gated by comparing the many independent hourly ve- variances are at maximum levels and are similar for locity profiles. Downward energy propagation should both upward and downward propagation. This cluster be associated with the upward propagation of wave of energetic values is associated with the June–July phase with time. If the sense of rotation in depth indeed Chukchi Cap passage, when subinertial activity is high (Fig. 3), the potential for seafloor generation is great- est, and the travel time between surface and seafloor is 5 For the shear variance estimates of Figs. 5 and 6, and the spectral presentations of Figs. 7, 8, 9 and 12, individual hourly the smallest. For the moment, let us exclude these ob- ⌬ 4 spectral estimates are divided by the factor sin(kz zres) to account for the smoothing resulting from the finite duration of the ⌬ 6 transmitted pulse (vertical extent zres) and subsequent averaging To assess the statistical stability of the various spectral quan- in range. A modeled noise corresponding to 5 mm sϪ1 rms hourly tities presented, representative wavenumber frequency spectral averaged velocity uncertainty is then removed. The combined ef- estimates of shear are first formed. These are subsequently in- fect of these various corrections is minimal at scales greater than verse-transformed in frequency and normalized to form estimates 10–15 m. To avoid presenting results that are overly sensitive to of coherence as a function of time lag and vertical wavenumber. the “corrections,” spectral displays are here truncated at 10-m Statistical stability is then estimated using the method of D’Asaro vertical scale. and Perkins (1984).

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FIG. 4. The zonal component of shear for Aug–Oct following rotary filtering in depth. The sense of rotation in depth is substantially consistent with the direction of vertical phase propagation, as given by linear internal wave theory. servations from the discussion and concentrate on a deep sea. Spectral levels increase significantly with de- comparison between variance levels over the slopes and creasing depth over slopes shallower than 2000 m. over the deep seafloor. Spectra associated with downward propagation (Fig. For upward-propagating waves, the depth variability 7b) are essentially independent of the depth of the un- of upper-ocean velocity variance (Fig. 6a) displays a derlying seafloor. Notable exceptions occur for H Յ ϳ Ϫ1/2 Ͼ somewhat irregular trend, with Eup H . The pat- 500 m, over the Chukchi Cap (not plotted), and for H tern associated with downward-propagating waves 3000 m. These deep data are primarily from the central shows no perceptible trend (Fig. 6b). The suggestion is Beaufort Gyre in December 1997. that, in the upper ocean, downward-propagating wave The up-down ratio (Fig. 7c) indicates a general sur- energy density is independent of underlying seafloor plus of downward-propagating energy in the deep sea, depth, at all depths greater than 500 m. Shear variance with the excess most pronounced in the 20–40-m verti- exhibits a slightly more convincing dependence on cal wavelength band. Over the slopes (H Ͻ 2000 m) the ocean depth (Fig. 6c) for upward-propagating waves, situation reverses, and upward propagation dominates, with little dependence seen for downward waves (Fig. again most strongly in the 20–40-m band. 6d), provided the Chuckchi Cap data are excluded. Shear spectra (Figs. 7d, e) are related to the velocity 2 Of particular interest is the disparity between the spectra by the deterministic factor kz. As expected, they predicted 80% confidence limits and the observed vari- too show a growth in the variance of upward-propa- ability of the signals. For the downward-propagating gating waves with decreasing seafloor depth. The waves, variability in velocity variance (energy density) downward shear spectra are more nearly uniform in presumably results from episodic forcing events. For depth, excepting the ranges H Ͻ 500 m and H Ͼ 3500 m. the upward-propagating waves, the variability over the The ratio of upward- to downward-propagating shear slopes is much greater than would be predicted for a (Fig. 7f) is essentially identical to that of velocity, as spatially homogenous process. expected. In the deep sea the variance associated with One can minimize the influence of temporal variabil- downward propagation exceeds that of upward. In wa- ity by averaging the daily velocity and shear spectra ter depths less than ϳ2000 m, the situation reverses, from the entire experiment, binned at like values of with the change most pronounced for 20–40-m scale underlying seafloor depth. The horizontal velocity (ki- motions. netic energy) spectra associated with upward- The depth-binned spectra of Fig. 7 are replotted in propagating motions (Fig. 7a) are smallest when in the Fig. 8, in log–log format. Wavenumber spectra associ-

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FIG. 5. A chronological summary of the SHEBA wave field. Daily averages of (a), (b) velocity and (d), (e) shear variance are presented as a function of time for both upward- and downward-propagating waves. The rate of change in (c) depth and (f) slope magnitude of the underlying seafloor reflects both topographic variation and the varying rate of drift of the camp.

ated with downward-propagating energy and shear The daily spectra, E(kz; t, H), are again sorted into (Figs. 8b,d) are substantially independent of seafloor categories based on the underlying seafloor depth. depth. The spectra of upward-propagating waves (Figs. Within each category the variability of the daily spectra

8a,c) have less variance over the deep sea, but grow to relative to the cruise-long mean, E(kz; H), is calculated have greater variance at high wavenumber as the water as a function of wavenumber. Since, formally, all bands shoals. of the spectrum are estimated at the same statistical If wave reflection from a sloping seafloor is the cause precision, the underlying uncertainty in the estimates of these changes, some component of the increased en- should not be a function of wavenumber.7 Plots of the ͗ Ϫ ergy density might result from the spatial compression ratio of the variance in spectral level, [E(kz; t, H) 2͘ 2 of the individual reflected wave packets. It follows that E(kz; H)] /E(kz; H) between upward and downward there might be an observable increase in the space–time waves (positive versus negative wavenumber bands in variability of the vertical wavenumber spectrum as the the same spectral estimate) are presented in Fig. 9 as a ice camp drifts over the varied topography of the western Arctic. From Fig. 6, it is apparent that spatial inhomogene- 7 In fact, there is a possibility that the effective number of de- ity, rather than statistical imprecision, sets the variabil- grees of freedom does vary as a function of wavenumber. If the ity of the spectral estimates. This variability can be depth–time extent of the packets in one wavenumber band is longer than in another, there will be fewer independent observa- quantified by examining the temporal fluctuation in the tions of these waves per unit depth–time. The effective number of daily vertical wavenumber spectra used in the vertical degrees of freedom will be less in this band, with increased spec- separation. tral variability as a consequence.

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FIG. 6. Daily averages of velocity and shear variance are presented as a function of underlying seafloor depth for both upward- and downward-propagating waves. Vertical lines give the 80% confidence interval for each estimate. For downward-propagating waves, there is little relationship between variance and depth, except for depths less than 500 m. For upward-propagating waves, there is a clear increase in variance with decreasing depth, although the pattern is contaminated by strong secular variability. The curves give the least squares linear fit of log(depth) to log(variance), for depths below 500 m. function of underlying seafloor depth. As with the level ␥ ϭ tanϪ1(|ٌH|), relative to the vertical inclination of the spectrum, an increase in the intermittency of the angle of wave energy propagation, ⌰ϭtanϪ1(⌬/N)(␴ spectrum is seen for upward-propagating waves over Ӷ N), and ⌬ϭ(␴2 Ϫ f 2)1/2. Formally, pure inertial topography (depths less than 2000 m). The increase is waves (⌬ϭ0) are back-reflected by arbitrarily small again most pronounced for the shorter waves (10–40 topographic slopes (Sandstrom 1969; Phillips 1966; m). Variability in the spectra at low wavenumbers, both Eriksen 1982, 1985; Garrett and Gilbert 1988). positive and negative, is similar. Consider an idealized situation (the Garrett 2001 scenario) where the near-inertial waves arriving at the continental slope are generated as purely inertial waves at 4. Reflection from continental boundaries: ␴ ϭ ⍀ ⌽ ⌽ A geographic summary of the western Arctic higher latitudes. Thus, 2 sin( 0), where 0 is the latitude of origin of the wave group, Surface-generated wave groups that are “within ⌬ ϭ ⍀͑ 2⌽ Ϫ 2⌽͒1ր2 ͑ ͒ range” will eventually encounter the continental slope. 2 sin 0 sin , 1 They can then forward reflect into shallow water, back- and reflect into the deep sea, or dissipate in a near-bottom ⌰ ϭ −1͓͑ ⍀ր ͒͑ 2⌽ Ϫ 2⌽͒1ր2͔ ͑ ͒ boundary layer. Forward reflection leads to increasing tan 2 N sin 0 sin . 2 energy densities as the waves shoal. To assess the like- ⌰ lihood of these three possibilities, the topography of the Forward reflection occurs when is greater than the |␥ | western Arctic can be examined from the perspective of encountered topographic slope magnitude, (x) , wave reflection. −1 2 2 1ր2 ⌰ր|␥| ϭ tan ͓͑2⍀րN ͒͑sin ⌽ Ϫ sin ⌽͒ ͔ր|␥͑x͒| The issue of forward versus back reflection is deter- bottom 0 mined by the angle of the seafloor at the reflection site, Ͼ 1. ͑3͒

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FIG. 7. Daily-averaged vertical wavenumber spectra of (left) upper-ocean kinetic energy and (right) shear, further averaged according to underlying seafloor depth. For upward-propagating waves, there is a systematic increase in the (a) energy spectrum with decreasing ocean depth. This contrasts with the corresponding spectrum for (b) downward-propagating motions, which displays little variation. (c) The up/down spectral ratio demonstrates a net downward energy flux when the camp is over the deep sea, evolving to a net upward flux over waters shallower than about 2 km. (d), (e) The corresponding shear spectra are band limited in vertical wavenumber. The shear spectral level associated with (d) upward-propagating motions increases as the water shoals. (e) There is a weaker change in spectral level for the downward-propagating waves.

8 ⌬ ⌽ The tendency for forward reflection is enhanced in re- the western Arctic, maps of /f and 0 are presented gions of low Nbottom. Inertial waves originating far pole- in Figs. 10a and 10b. The southwestern boundary of the ward of the topography are also more likely to forward- Canada Basin is seen to be back-reflective to locally reflect. generated waves over a significant fraction of its slope Upon reflection, wave frequency is maintained, but (Fig. 10b). Waves generated in the high Arctic are more both the cross-slope horizontal wavenumber and the likely to forward reflect if they can reach this section of vertical wavenumber change such that both the normal coastline. flow and normal energy flux vanish at the boundary. There are a number of possible characterizations of The ratio of reflected to incident vertical wavenumbers, the western Arctic that can be distilled from the geo- Rk, a useful metric of this change, is given by graphic map. In Fig. 11a, the average amplification of forward-reflected waves, calculated at five values of ␴/f R ϭ ͑a2 ϩ 2Ca ϩ 1͒ր͑a2 Ϫ 1͒, ͑4͒ at each of the geographic points in Fig. 10, is presented k as a function of seafloor depth. For simplicity, reflection at a normal incidence angle, C ϭ 1, is considered at ϭ ␥ ⌰ ϭ ␾ ␾ where a tan( )/tan( ), C cos( ), and is the all locations. The “mean” amplification is significant for azimuth of the incident wave relative to the upslope normal to the isobaths (Eriksen 1982). For forward up- Ͼ slope reflection, Rk 1. The energy density of the re- 8 2 The profile was obtained on the 1997 trans-Arctic cruise of flected wave packet is increased by the factor Rk. the RV Northwind. Data were graciously provided by Dr. J. Using a single buoyancy profile as characteristic of Swift.

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FIG. 8. Vertical wavenumber spectra of (a), (b) velocity and (c), (d) shear as presented in Fig. 7, obtained in depth intervals of 500–1000 m (green) through 3000–3500 m (blue). While the up–down difference in the mean spectral levels (thick lines) is small, a deterministic trend is seen in the spectra associated with upward (but not downward) propagation. (c) Upward shear spectra increase in level and shift to the right as the water depth shoals. A parallel shift is seen in the velocity spectra in (a), although the net variance increase is significantly reduced relative to the shear. The vertical bar gives the 95% confidence interval for the 500–1000-m “depth interval,” which has the fewest independent data. Vari- ability in the “upward” spectra generally exceeds this limit, a consequence of the spatial inhomogeneity of the wave field. waves of frequency Ͻ1.1f. However, it is strongly de- with greater than fivefold amplification to the total area pendent on those few sites where near critical reflection is presented. For waves ␴ ϭ 1.01f, 20% of the deep occurs and the amplification ratio approaches infinity. arctic slope is associated with extreme amplification Several other statistics are more useful. (The ratio of and potential deep mixing. This ratio rises to 40% at backscattering to forward scattering topographic area is depths less than 1 km. Higher-frequency (␴ Ն 1.03f) given in Fig. 11b). Backscattering is significant at waves display a similar trend, at reduced levels. depths above 2000 m for waves of frequency 1.03f or less. Higher-frequency motions are generally able to progress up-slope in the western Arctic. 5. Upper-ocean estimates of deep seafloor slope A significant concern is the ratio of topographic area where the waves are forward scattered but not signifi- For an individual wave packet, upslope reflection 2 cantly amplified/spatially compacted to the total area leads to an increase in energy density of Rk (Phillips (Fig. 11c). These regions are associated with the trans- 1966; Eriksen 1982). One factor of Rk results from the mission of wave energy to shallower waters, where vertical compaction of the wave group and one from breaking might subsequently occur. At depths greater the horizontal. The signature of this process in a than 2000 m, most of the slope reflects with little am- moored measurement can be either an increase or a plification (for ␴ Ͼ 1.03f ). At shallower depths, the decrease in energy, depending on whether the sensor is likelihood of significant amplification on reflection in- receiving the reflected “glint” off a topographic “facet” creases. or not. For horizontal or vertical profiling measure- Extreme amplification might lead to local mixing just ments (such as these SHEBA acoustic profiles) there above the deep slopes (Eriksen 1982; Garrett and Gil- should be no net energy change observed, as the in- bert 1988). In Fig. 11d, the ratio of topographic area creased energy density of the packet is offset by the

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product H͗S2͘ should change only as a consequence of the shift in spectral scale to higher wavenumber by the

factor Rk. Thus, upper-ocean observations of vertically propagating shear contain information about the char-

acteristic Rk at the seafloor. Can this signal be detected? To attempt an estimate of Rk, the linear propagation/ dissipation model (see the appendix) is applied. With the observed kinetic energy spectrum associated with downward propagation taken as the initial state, the model [Eq. (A1)] is applied band by band to propagate the spectrum to the seafloor. At this point, the wavenumber scale of the reflected spectrum is shifted by the

factor Rk. The spectral density is increased by Rk,as well, assuming reflection without loss. This corresponds 2 to an increase in packet energy density by Rk spread over the broader spectral bandwidth. The “upward spectrum” is next propagated to the surface, again using Eq. (A1). Overall velocity variance is then reduced 2 by the factor Rk, accounting for the reduced likelihood of observing the spatially compacted wave groups that comprise the spectrum. The modeled shear spectrum is derived from the velocity spectrum by multiplication 2 by kz. Plots of the modeled “reflected” spectrum, as it would appear in the upper ocean, are compared with the observed spectrum of upward propagation in Fig. 12. The resemblance is not unreasonable, given that the upward waves have been perhaps generated weeks to months earlier and many kilometers from the site where they are eventually observed. Perhaps not sur- prisingly, the best agreement is seen for modest as- ϭ sumed values of Rk ( 1.25). More extreme “blue shifting” is probably associated with local mixing near the seafloor. In the upper ocean, one gets to observe the “survivors” of the reflection process. 2 The dashed spectral level N /kz is often taken as a saturation reference for open ocean (low latitude) and atmospheric gravity wave spectra (e.g., Muller et al. FIG. 9. The variance of daily spectral estimates normalized by 1991). The observed upward shear spectra approach the square of the cruise mean spectrum for each of six 500-m this threshold and all modeled spectra, for R Ͼ 1.25, depth zones. Spectral variability associated with (a) upward k propagation increases at depths less than 2000 m, in contrast to (b) exceed it. While turbulent mixing has never been ob- the variability of spectra associated with downward propagation. served in the central Canada Basin, Fig. 12 strongly (c) The log of the up/down variability ratio demonstrates that the suggests that dissipative processes are active over the time variability of the short, upward-propagating motions also deep slopes. exceeds that of the downward waves. This is consistent with the The deep-water flat-bottom case (Fig. 12d) is in- spatial compaction of wave groups that can result from bottom reflection. cluded for contrast. Clearly, simple viscosity is insuffi- cient to explain the change in spectral levels between upward- and downward-propagating waves in these reduced probability of encountering it. A shift of the December 1997 observations. Previous observations in packet to smaller vertical scale, with an associated in- the deep Canada Basin (Merrifield and Pinkel 1996; crease in shear variance, is expected, however. Halle and Pinkel 2003) do not exhibit this degree of Barring back-reflection and bottom boundary layer vertical anisotropy. dissipation, the spatial density of wave packets will increase with shoaling, leading to an expected factor of HϪ1 increase in the climatological velocity and shear 6. Summary, discussion, and speculation variances. If upslope reflection is indeed responsible for the in- With typical velocities of only 0.01–0.02 m sϪ1, the crease in observed upward-propagating shear, the measurement of arctic internal waves represents a sig-

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FIG. 10. The topography of the Northwind Ridge and Chukchi Cap. Colors indicate (a) the critical frequency for waves reflecting on the steeper slopes and (b) the latitudes of origin of inertial waves that are critical at each location. In near-critical regions, enhanced mixing in close proximity to the slopes is likely. nificant technical challenge. The attempt to monitor supply energy to these scales, reducing the anisotropy. these motions over the yearlong SHEBA drift pro- (The effectiveness of the vertical propagation filter is duced a number of surprises. Internal wave shears are also questionable if the vertical waveform of these so strongly concentrated at near-inertial intrinsic fre- small-scale motions is significantly nonsinusoidal.) quency that the sense of shear rotation in depth is an The initial “straw man” experiment posed in the In- effective index of the (vertical) direction of propagation troduction suggested a growth in near-inertial energy in time. Velocity and shear variances associated with density from the center to the edges of the deep Arctic downward-propagating waves fluctuate irregularly over Ocean and an additional 1/H increase in wave energy the course of the drift, except during June and July, density from the deep sea up over the slope. In fact, no when the camp is over the Chukchi Cap. Here, levels correlation between the downward energy flux and lo- are elevated. Variances associated with upward propa- cation or seafloor depth is seen (excluding the very gation are clearly related to the depth of the underlying energetic downward waves over the Chukchi Cap). seafloor, with E ϳ HϪ0.5. Over sloping terrain, upward This is consistent with signature of “one-bounce propa- energy frequently exceeds downward. This requires ex- gation” (Fig. 2d; appendix) and suggests that underice planation, given the belief that these motions are gen- dissipation is effective in inhibiting the long-range erated at the sea surface. The space–time variability of propagation of Arctic near-inertial waves. The model the upward-propagating waves is also much greater of Morison et al. (1985) predicts the limiting effect of than the variability of downward propagation groups, underice dissipation (although the SHEBA observa- and is consistent with significant spatial nonhomogene- tions are less than a definitive test of their model). If the ity. Vertical wavenumber spectra of upward-propa- stability of the underice boundary layer is radically in- gating shear are blue-shifted relative to their down- creased, by greater river runoff or summer melt, the ward-propagating counterparts. The shift is an upper- dissipative dynamics of the wave field might change ocean signature of the seafloor slope, thousands of significantly. In past or future epochs of an ice-free meters below. Arctic, the near-inertial energy delivered to the basin Vertical anisotropy is reduced at the smallest vertical boundaries and available for boundary mixing might be scales resolvable by the sonars (ϳ5–10 m). A linear significantly larger than at present. propagation model (see the appendix) suggests that The one-bounce propagation physics of the Arctic even molecular viscosity would be sufficient to inhibit suggests that only energy from atmospheric forcing round-trip propagation at these scales, increasing the within 300–500 km of the shelves will contribute to apparent anisotropy. Presumably, nonlinear transfers boundary mixing. However, roughly 60% of the Arctic

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FIG. 11. Scattering properties for this quadrant of the Arctic are summarized in terms of regions of like water depth by considering waves of frequency ␴ ϭ (1.01, 1.03, 1.10, 1.30)f. Here f is evaluated locally at each geographic point in Fig. 10, and normally incident scattering is assumed. (a) The mean amplification of forward-scattered waves is significant on both the lower and upper slopes. This average is dominated by the few critical locations where amplification approaches infinity. (b) For low-frequency waves (␴ Ͻ 1.01f ), a significant fraction of the upper slope becomes back reflective. (c) With the exception of ␴ Ͻ 1.03f waves, most of the western Arctic slope is forward reflective, with moderate amplification. (d) Low-frequency waves can experience extreme amplification and presumably contribute to deep local mixing.

Ocean is within 300 km of a boundary. A significant a similar behavior for continental slopes at semidiurnal concern is with the direction of initial propagation. frequency. It is likely that the back-reflection and near- Generated waves are not constrained to propagate to slope dissipation of the incoming wave energy flux re- the nearest coast. Equatorward refraction will be a lim- duces the observed energy enhancement relative to the ited factor over the one-bounce path. Waves propagat- reference dependence E ϭ HϪ1. The relative incidence ing parallel to the coast or offshore will not survive the of back reflection and/or extreme amplification is sen- transbasin journey. It is thus probably more realistic to sitive to small departures, ⌬, of the wave intrinsic fre- posit a 1–10WmϪ1 energy flux impinging on the Arctic quency from f. The SHEBA data are not adequate to slopes, arbitrarily “downsized” by a factor of 10 from resolve intrinsic frequency to the required precision. the straw man value presented in the introduction. Given that a typical wind generation/ice motion event Large areas of the continental slopes of the western has a lifetime of 3–6f Ϫ1, one can imagine spectral band- Arctic are near-critical for waves in the near-inertial widths of order 0.16–0.3f being energized. Following frequency band. Recently, Cacchione (2002) has found dispersive propagation to the seafloor, approximately

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FIG. 12. Averaged power spectra associated with observed downward (thick solid line) and upward (thick line with stars) propagation, presented for four ranges of seafloor depth. Modeled “upward” spectra (thin lines) are calculated under the assumptions that the downward waves reflect off the seafloor ϭ with vertical wavenumber increased by the factor Rk ( 1, 1.25, 1.5, 1.75) and that total energy is preserved. Viscosity acts along the propagation path as specified by the linear model of section 4. Its ϭ cumulative effect is seen in the lowest model curve (Rk 1), corresponding to reflection from a flat 2 ϭ bottom. The dashed line indicates the spectral reference level N /kz, for N 3.6 cph. Listed days indicate the total data available in each depth interval.

5%–20% of the incident flux will be at frequencies suf- ment of the reflected group, the less likely that it will be ficiently low to interact strongly with topography (Fig. passing through the measurement volume at any given A1). time. This principle applies independent of the azimuth Even in the Arctic, the ␤ effect is significant in the of the incident wave relative to the topographic slope. propagation of equatorward traveling near-inertial (␴ Spectral changes are primarily associated with changes Ͻ 1.1f ) waves. Both vertical group speed and propaga- in the scale and bandwidth of the reflected waves, as tion slope increase along the propagation path. This well as the overall increase in proximity of independent improves the chance for forward reflection on conti- wave groups as the water shoals. nental slopes and reduces the contact time with the Although the total energy of a wave group is un- dissipative underice boundary layer. However, the dis- changed on reflection from a slope, the increase in en- tance traveled before reencountering the surface is sig- ergy density within the group combines with the change 4 nificantly shortened. Zonally propagating near-inertial in scale to alter the shear variance by the factor Rk. waves experience these effects only following ␤-in- Gregg (1989) has found empirical evidence that mean duced equatorward refraction. turbulent dissipation levels are proportional to the 8 The process of interpreting the complex 3D scatter- square of shear variance, or Rk. Thus turbulent mixing ing problem (suggested by Figs. 2 and 10) in terms of in the near field of upslope reflection regions might be simplified 2D models is, of course, suspect. The ap- greatly enhanced, particularly in more energetic tem- proach has proven worthwhile, in part because the en- perate oceans. ergy enhancement associated with the upslope reflec- Given an established vertical mixing rate, Young et tion process is proportional to the spatial compaction of al. (1982) have suggested that lateral diffusion in the the reflected wave groups. The greater the enhance- ocean is significantly enhanced in the presence of inter-

Unauthenticated | Downloaded 10/03/21 08:00 PM UTC 660 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 nal wave shears. They express this enhancement [their • For simplicity, strictly meridional propagation is con- ⌬ Eq. (45)] in terms of an eddy diffusivity, KH, where sidered. • The cross-scale transfer of energy by nonlinear pro- ⌬ ϭ ր ͗ 2͘ ր 2 ͑ ͒ KH 1 4K␷ S 1 f . 5 cesses is neglected. It is presumed that nonlinear ͗ 2͘ transfer rates are low in the Arctic, because of the Here K␷ is the vertical eddy diffusivity and S 1 is the wave field shear variance, considering motions down to low overall energy level of the wave field. a vertical scale of order 1 m. For an rms (1 m) shear of • Refraction caused by variable background currents, a Ϫ1 ϳ potentially significant process in the Arctic, is not order 0.005 s , KH 300K␷ in the polar ocean. If the effective vertical diffusivity is associated with considered. wave breaking, the combination of Gregg’s observation • The lateral divergence of wave energy with decreas- with the Young et al. (1982) model suggests that lateral ing latitude, associated with the sphericity of the diffusivities vary as the sixth power of rms wave field earth, is neglected as well. The model can be easily shear, further emphasizing the role of wave field vari- extended to include this effect. For purely meridional ⌽ ability. Armi (1979) and more recently Samelson (1998) propagation, E cos( ) is the appropriate conserved emphasize the significant effect of a spatially variable quantity. Over the latitude range of the SHEBA diffusivity on the large-scale circulation. Observations drift, 75°–82°, the signature of relative divergence is in the near field of likely slope-mixing sites are called small in comparison with the observed scatter in the for to establish the reality of these conjectures. data.

For propagation from depths z0 to z1, the modeled Acknowledgments. The sonar systems were designed, variation in wave energy density takes the form constructed, and deployed by Eric Slater, Lloyd Green, Ϫ ͒ ͑ ͒ ϭ ͑ ͒ ͑ ͒ր ͑ ͒ q͑z0,z1 ͑ ͒ Mike Goldin, and Chris Neely at the Marine Physical E z1 E z0 N z1 N z0 e . A1 Laboratory of Scripps Institution of Oceanography. Here The author thanks Jay Ardai, John Bitters, Andreas Heiberg, Dean Stewart, Roger Anderson, and the crew z1 ͑ ͒ ϭ ͑ ␯ 3␴ր⌬5͒ͯ͵ 3 ͯ ͑ ͒ of the CCG des Groseilliers for their support during q z0, z1 2 kh N dz A2 SHEBA. Comments of the two reviewers were greatly z0 appreciated. This work was supported by the National describes the cumulative effects of viscous decay, with ␯ Science Foundation. being a coefficient of viscosity, kh the horizontal wavenumber, and ␴ the wave frequency, and ⌬ϭ(␴2 Ϫ APPENDIX f 2)1/2 (see following section). To estimate surface and bottom reflection losses, the Modeling Near-Inertial Propagation and turbulent boundary layer model of Morison et al. Attenuation in the Western Arctic (1985) is applied (again, see following section). The central feature of the model is the “no-slip” condition In Figs. 5–9 we find no trend in the magnitude of the that is applied at the ice–water interface. Internal wave downward-propagating energy/shear over the course of horizontal velocities are constrained to decay as the the drift. A weak (inverse) dependence of variance on interface is approached. Momentum diffuses down the seafloor depth is found for upward-propagating waves. resulting velocity gradients at a rate established by the This is consistent with the propagation scenario illus- background near-surface flow, which is assumed to trated in Fig. 2d, where underice absorption limits the have established a turbulent underice boundary lay- propagation of surface-generated waves to a single er.A1 Morison et al. assert that the rate of wave energy round trip. In this appendix, a simple linear propaga- loss during reflection is tion/dissipation model is introduced to assess the cumu- ␧ ϭ ␳ 2 ϭ ͑ Ϫ2͒ lative effect of both volume and surface (underice) dis- BL CdUmeanuwave 2CdUmeanEwave Wm , sipation. The model is described in detail in the second ͑A3͒ section of this appendix. In the first section, the model ϭ is briefly introduced and the scientific findings are sum- where Cd is a characteristic drag coefficient (Cd marized. 0.0034 for the underice surface), Umean is the mean, low-frequency flow that establishes the underice a. Application of a linear near-inertial propagation boundary layer, and uwave is the perturbation velocity model to the Arctic associated with the waves. Key assumptions of the model are the following. A1 • The underice boundary layer is the generation region In the open sea, “preexisting” wind driven turbulence is typi- cally much greater than the turbulence found under arctic sea ice. for downward-propagating near-inertial waves. However, there is no analog of the “no slip” condition in the open • Near-inertial wave dissipation at the flat seafloor of sea. The predictions of extreme energy loss obtained from this the Canada Basin is modest. model are thus not relevant to open-ocean conditions.

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At the seafloor, both Umean and uwave are small. band and short, higher-frequency waves will not survive Morison et al. (1985) and D’Asaro (1982) conclude that a single encounter. Only the low-mode high-frequency seafloor attenuation is negligible. At the underice sur- waves [reported to have significant vertical displace- face, we write ment and coherence by Levine (1990)] can survive the reflection process. ␧ ϭ c ͑E Ϫ E ͒ ϭ 2C U ͑E ϩ E ͒. BL gz up dn d mean up dn Representative maps of “propagation distance” are ͑A4͒ given as a function of frequency and vertical wavenum- ϭ ␣ ␣ ber in Figs. A1d–f, for assumed mean ice speeds of 0.03 For a single reflection, Edn iEup, where i is the (Fig. A1d), 0.10 (Fig. A1e), and 0.30 (Fig. A1f) m sϪ1 underice reflection coefficient, which, from (A4), is and viscosities of 1.8 ϫ 10Ϫ6 (Fig. A1d), 5 ϫ 10Ϫ6 (Fig. ␣ ϭ ͑ Ϫ ͒ր͑ ϩ ͒ ͑ ͒ A1e), and 10Ϫ4 (Fig. A1f) m2 sϪ1. Propagation distance i cgz 2CdUmean cgz 2CdUmean . A5 is defined as the path traveled before the wave packet An attractive aspect of the Morison et al. model is attenuates to eϪ1 of its original energy. The highest- that it is linear in wave energy. Thus, the expression for wavenumber, low-frequency waves (area 1 in Fig. A1f) ␣ i derived for a single reflection also applies to a su- are attenuated before completing a single round trip. perposition of wave groups (of identical group velocity) The influence of the varying levels of viscosity is seen in arriving at the reflection site from a variety of distant the region of horizontal contour lines at high kz. sources. A weakness in the model is that the rate at Low-wavenumber near-inertial motions can travel in which a wave packet looses energy in the boundary ␧ excess of 500 km, by virtue of their shallow propagation layer, BL, is not constrained by the rate of energy sup- angle. These waves do not survive a single surface en- ply to the boundary, c E . This formally enables ␣ to gz up i counter even at low ice speeds. Hence their propaga- assume nonphysical negative values. However, over tion distance is unaffected by changes in ice speed. most of the wavenumber–frequency domain, and for Ϫ They occupy the region of ⌬–k space characterized by typical ice speeds of U Ͻ 0.3 ms 1, ␣ is positive. z mean i vertical contours (area 2). To examine the interplay of the factors that affect To the right of the curved boundary are the waves propagation, we evaluate (A1)–(A5) assuming a range Ϫ6 2 Ϫ1 that can survive one (area 3) or more (area 4) encoun- of values for viscosity (1.8 ϫ 10 m s , molecular, representative of the deep Arctic Ocean, 5 ϫ 10Ϫ6, typi- ters with the surface. In this region, the expected propa- cal of the midlatitude open ocean, and 10Ϫ4, represent- gation distance is highly sensitive to ice speed. Periods ϭ of rapid ice motion might sweep the established high- ing a canonical region of great mixing), for Umean (0.03, 0.1, and 0.3 m sϪ1) and for a variety of vertical frequency wave field from the upper halocline, perhaps wavenumbers and frequencies. A double exponential replacing it with freshly generated waves. The “saw buoyancy profile (Fig. A2a) is used in this calculation. tooth” pattern seen at intermediate vertical wavenum- Initially examining a single round-trip cycle through bers reflects the degree of attenuation associated with the exponential waveguide, Figure A1a presents the each incremental surface bounce. The slope of the in- ratio of viscous to surface reflection losses. In the individual “teeth” reflects the fact that the lowest fre- termediate–wavenumber, low-frequency section of the quency waves, for a given number of surface reflec- plot, absorption in the underice boundary layer is total. tions, will propagate farthest, because of their shallow Throughout the regime, underice absorption equals or propagation angle. The lateral separation between the exceeds volume dissipation by up to 5 orders of mag- teeth indicates the increment in frequency necessary for nitude. Viscosity only becomes a significant influence at a wave of fixed kz to survive an additional reflection. scales so small and frequencies so low that the waves These simple results require modification if the are unable to complete a single round trip. earth’s curvature is considered. Specifically, equator- Attention is thus focused on the discretization of the ward-propagating near-inertial waves return to the sur- propagation into distinct surface encounters. In Fig. face (Figs. A2a, b) with a vertical component of group A1b, the horizontal propagation distance between suc- velocity (ϳ⌬2) that can be 2–3 times that at the gen- cessive surface reflections is presented as a function of eration site (Fig. A2c). Underice attenuation will be frequency for ocean depths of 500 and 4000 m. The reduced, given the correspondingly faster transit times low-frequency limit to the calculation is ␴ ϭ 1.005f,at through the boundary layer. Zonally propagating waves which point propagation distances of ϳ1000 km sepa- do not share this benefit. Curvature effects depend on rate successive surface reflections in the deep basins. the latitude and ⌬ at initial generation. They are gen- ␣ ␴ Ͼ Given the fractional energy loss per reflection, i, erally inconsequential in the Arctic for 1.1f at gen- Յ Ϫ1 how many reflections occur before E/E0 e , that is, eration. ␣p ϭ Ϫ1 i e ? Figure 10c presents this value as a function of The central finding of the propagation study is that, vertical wavenumber and frequency. Apparently, a for typical ice speeds, underice dissipation is a domi- large vertical component of group velocity is needed to nant process. Finescale near-inertial wave propagation survive attenuation in the underice boundary layer. Us- in the Arctic is essentially a “one bounce” phenom- ing this e-folding criterion, waves in the near-inertial enon. Surface energy input greater than ϳ500 km from

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FIG. A1. (a) The ratio of viscous energy loss, over a round-trip path from the sea surface to seafloor, to loss in the underice boundary layer, as a function of vertical wavenumber and frequency. (b) The propagation distance per round-trip surface to seafloor cycle, as a function of wave frequency and ocean depth. For this calculation, the inertial frequency is held constant along the propagation path. (c) The number of surface encounters required before wave packet energy drops to eϪ1 of its initial value. An ice speed of 0.1 m sϪ1 and viscosity of 5 ϫ 10Ϫ6 are assumed for (a) and (c). (d)–(f) The horizontal propagation distance attainable before wave energy drops to eϪ1 of its initial value, for mean ice speeds of 0.03, 0.10, and 0.30 msϪ1.

topography should not contribute significantly to refractive geometry of the waves on a spherical earth. boundary mixing. Here, for initial simplicity, various propagation properties are derived under the assumption that ␤ ϭ df/dy is b. A linear model of near-inertial propagation negligibly small. Comparison with a more accurate model that includes a changing inertial frequency over In the steady state, the two-dimensional energy con- the propagation path is presented in Fig. A2. servation equation for wave groups takes the form The aspect ratio of the surface-generated waves is specified by N/⌬ where ͒ ͑ иٌ ϩ ٌи ϭ Ϫ cg E E cg sources sinks. A6 ⌬ ϭ ͑␴2 Ϫ f 2͒1ր2 ͑A7͒ Here, E is the local energy per unit mass in (m sϪ1)2, ␴ cg is the vector wave group velocity, and “sources and and is the wave frequency. For near-inertial waves, sinks” account for wave generation and dissipation. ⌬ ϭ ր ␴ Ӷ ͑ ͒ The application of (A6) is a trivial simplification of khN kz, N, A8 Garrett (2001), who has recently discussed the long-

range propagation of near-inertial waves at mid- and where kh and kz are horizontal and vertical wavenum- low latitudes. Garrett was primarily concerned with the bers, respectively.

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Fig A1 live 4/C MAY 2005 P I N K E L 663

FIG. A2. Near-inertial propagation on a spherical earth is considered for (a) modeled buoyancy profiles both with and without an idealized halocline. For purely equatorward propagation, (b) the ratio of vertical group speed at the first surface bounce to that at generation is presented as a function of generation latitude for wave frequencies (at generation) of (1.01, 1.03, 1.1, 1.3, 2)f. For initial wave frequencies greater than 1.1f, curvature effects are relatively unimportant. The round-trip propagation distance for meridionally propagating near-inertial packets on an f plane (upper curves) and a ␤ plane (lower curves) both (c) including and (d) neglecting the modeled halocline. Proper modeling of the earth’s sphericity leads to significantly shorter propagation paths, for ␴ Ͻ 1.1f. Inclusion of the halocline lengthens path predictions for ␴ Ͻ 1.1f. Neither effect is significant for ␴ Ͼ 1.1f.

z1 To evaluate the consequences of the conservation Ϫ ր ͑ ͒ ϭ ͵ ր ϭ ͑␴ ր⌬3͓͒ ͑ Ϫ z0 b0͒ model, a canonical (Garrett and Munk 1972, 1975) ex- T z0, z1 1 cgz dz kh b0N0 1 e ponential buoyancy frequency profile, modified to in- z0 Ϫ ր clude an arctic halocline, can be applied: ϩ ͑ Ϫ z0 b1͔͒ ͑ ͒ b1N1 1 e . A11 Ϫ ր Ϫ ր ͑ ͒ ϭ ͑ ͒ ϩ ͑ ͒ ϭ z b0 ϩ z b1 N z N z basin N z halo N0e N1e ͑ ͒ Note that the waves spend much of their lives in the A9 upper ocean. With a traditional Garrett–Munk (GM) ϭ ϭ ϭ ϭ ϭ with N0 3.6 cph, b0 1000 m, N1 6 cph, and b1 buoyancy profile (N1 0), the travel time from the 100 m (Fig. A2a). For ␴ Ӷ N, the vertical component of surface to 700 m is one-half of the time required for wave group velocity is propagation to infinite depth. Adding the model halocline, the “residence depth” shrinks to 520 m. The in- c ϭϪ⌬2ր͑␴k ͒ ϭϪ⌬3ր͑␴k N͒ ͑ ͒ gz z h . A10 teraction of the waves with baroclinic eddies and other

The propagation time from depth z0 to depth z1 is mesoscale denizens of the halocline is thus a significant

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⌬Ϫ3 ␣ aspect of Arctic propagation. The factor of in the surface reflection i and the characteristic number of travel time emphasizes the extreme sensitivity of the reflections before attenuation. process to the horizontal scale of the waves. The horizontal component of group velocity is REFERENCES

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