On the interaction between internal tides and wind-induced near-inertial currents at the shelf edge

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. C3, 3099, doi: /2002jc001375, 2003 On the interaction between internal tides and wind-induced near-inertial currents at the shelf edge Alan M. Davies and Jiuxing Xing Proudman Oceanographic Laboratory, Bidston Observatory, Birkenhead, UK Received 5 March 2002; revised 18 October 2002; accepted 12 November 2002; published 28 March [1] Numerical calculations are performed at the shelf edge to examine the role of nonlinear processes in transferring energy from the M 2 tide- and wind-induced currents close to the inertial frequency f into a wave at the sum of their frequencies, termed the fm 2 frequency. A numerical model of the Hebrides shelf edge (represented by a cross section), initially with idealized topography and subsequently with realistic topography is used in these calculations. Results show that in the near-coastal ocean, currents at the fm 2 frequency are primarily due to coupling between wind-induced inertial oscillations and the M 2 internal tide. A major source is associated with vertical shear in the inertial oscillations and the vertical velocity due to the internal tide. A secondary source is due to the nonlinear momentum advection term. In the case in which eddy viscosity is computed from a turbulence energy model, shear across the thermocline is larger than when a constant viscosity is used. The reduction in shear with a constant viscosity reduces the role of the nonlinear term involving vertical shear, and hence the magnitude of the fm 2 current in the region of the thermocline. Increasing the wind stress leads to a deeper thermocline, and hence the location of maximum fm 2 current in the water column. Changing the vertical stratification influences the intensity of the inertial oscillations in the surface layer and the distribution of the internal tide, and hence changes the pattern of fm 2 currents. Results with realistic topography confirm the major conclusions. INDEX TERMS: 4219 Oceanography: General: Continental shelf processes; 4255 Oceanography: General: Numerical modeling; 4544 Oceanography: Physical: Internal and inertial waves; KEYWORDS: internal tides, near-inertial currents, shelf edge, nonlinear interaction, turbulence energy model Citation: Davies, A. M., and J. Xing, On the interaction between internal tides and wind-induced near-inertial currents at the shelf edge, J. Geophys. Res., 108(C3), 3099, doi: /2002jc001375, Introduction [2] In recent years, there have been extensive measurements of the internal tide and associated mixing in shelf edge regions [Sherwin, 1988; New and Pingree, 1990; Inall et al., 2000, 2001; Holloway, 2001] aimed at determining the generation and propagation of the internal tide and associated mixing. In particular, New and Pingree [1990] related enhanced shelf edge cooling to internal tide mixing in the shelf edge region. As the internal tide propagates away from the shelf edge, it can also increase mixing in the ocean. [3] Away from the shelf edge, internal tides can be generated over topographic features (e.g., the Hawaiian ridge, Merifield et al. [2001] and sea mounts, Xing and Davies [1998]), and these also contribute to oceanic mixing. Recent calculations using a three-dimensional prognostic model forced with the barotropic tide and incorporating a turbulence energy closure scheme have shown that it can reproduce the internal tide and enhanced shelf edge cooling [New and Pingree, 1990] associated with it [Xing and Copyright 2003 by the American Geophysical Union /03/2002JC Davies, 1996a]. Besides the tides, wind forcing is a major source of mixing, particularly in shallow seas [Van Haren et al., 1999; Van Haren, 2000] and shelf edge regions, where the continental slope influences the internal wave spectrum [Van Haren et al., 2002; Gemmrich and Van Haren, 2002]. Recent ideas [Munk and Wunsch, 1998] suggest that a major part of the mixing in the ocean is due to boundary layer mixing of tidal and wind origin at the shelf edge, which then diffuses into the ocean. Wind generated inertial oscillations and near-inertial internal waves are a major source of mixing since they are at the resonant frequency, and hence appreciable currents can be generated by light winds. As will be shown, in near-coastal and shelf edge regions, they are accompanied by a 180 phase shift in currents across the thermocline. This phase shift gives rise to maximum shear across the thermocline, and hence is a major source in the shear production of turbulence and shelf edge mixing. [4] Although the decay of inertial oscillations in the oceanic surface layer can be reproduced by a single-point model by including an empirical linear friction term [Mellor, 2001], this cannot represent the transfer of energy to nearinertial internal waves. To account for the important nonlinear processes which allow energy transfer between waves (wave-wave interaction) [Muller et al., 1986], and hence the 44-1

2 44-2 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS energy cascade to higher frequencies and mixing, it is necessary to use a three-dimensional model incorporating the nonlinear momentum terms. [5] In shallow seas, these nonlinear terms together with quadratic bottom friction are responsible for transferring energy from the principal semidiurnal tide (the M 2 tide) to its higher harmonics (e.g., M 4, M 6,...). Besides a transfer of energy to higher harmonics, a tidal residual can be produced by various nonlinear processes, and at the shelf edge the residual flow due to the internal tide can be significant [Xing and Davies, 2001a; Inall et al., 2001]. Also, in shallow water, tidal and wind forced currents can interact not only through the nonlinear momentum terms, but also due to time variations in the coefficient of vertical eddy viscosity and wind-induced shear to modify the profile of the M 4 tidal current [Davies and Lawrence, 1994; Davies and Xing, 2001]. [6] These nonlinear effects and the resulting cascade of energy from longer waves to shorter waves through wavewave interaction [Muller et al., 1986] and wave breaking, and eventually to turbulence are important mixing processes. In shelf edge regions, local enhancements in mixing [Samelson, 1998; Spall, 2001] can have a significant influence upon the oceanic circulation. Consequently, a study of the shelf edge nonlinear interaction between the internal tide- and wind-induced currents, particularly, those at the inertial frequency (the resonant frequency at which there is significant energy in the internal wave spectrum [Garrett, 2001]) is of importance as a means of enhancing mixing through internal waves of tidal and wind origin [Garrett, 2001]. Recently, Ledwell et al. [2000] have shown the importance of enhanced mixing over topography upon the large-scale oceanic circulation. [7] On and offshelf measurements have shown that there is significant energy in both shallow sea [Van Haren et al., 1999] and oceanic currents [Mihaly et al., 1998] at the fm 2 frequency to indicate significant coupling between nearinertial currents and those due to the M 2 internal tide. In this paper we concentrate upon the shelf edge region and a spatially uniform wind stress. However, in the deep ocean away from coastal influence, temporal changes and divergences in the wind stress generate inertial oscillations in the mixed layer, which subsequently decay as horizontally and vertically propagating near-inertial internal gravity waves carry energy away from the generation region. The vertical propagation of these waves is responsible for the generation of inertial energy at depth [Gill, 1984]. Even though this process of propagating inertial energy to depth is excluded here by the use of a uniform wind field, as will be shown, there is another mechanism in the nearshore region. In the coastal ocean, changes in sea surface elevation due to no flow at the coast drive inertial oscillations at depths which are phase shifted by 180 from the surface oscillation [Rippeth et al., 2002]. In this case, the vertical propagation of internal waves is not required to produce inertial energy at depth. [8] The focus of this paper is the use of a three-dimensional model applied previously to examine nonlinear effects producing the M 4 tide in shallow water [Xing and Davies, 1996b]; the M 2 internal tide and its residual at the shelf edge [Xing and Davies, 1996a, 1998, 2001a], to determine the various processes giving rise to energy at the fm 2 frequency at the shelf edge. Although previous calculations [Xing and Davies, 1997] have shown that a steady wind can modify the internal tide by a change in the density field, to date no work has been done on the nonlinear coupling of the internal tide and near-inertial internal waves. This topic is examined here. 2. Numerical Model [9] Only the details of the essential model will be considered here, since the three-dimensional model has been described elsewhere [Xing and Davies, 1998]. The continuity equation, momentum equations, and transport equations for temperature in transport form using s coordinates, where s =(z z)/h with s = 0 sea surface and s = 1 the seabed, are given by 0 1 H~V ds A ¼ þr Hu~V þ þ BPF x þ 1@ H þr þ fhu þ BPF y þ 1@ H A V with the pressure P at any depth s ¼ rgh; ; ; and w computed diagnostically from the continuity equation. The time evolution of temperature T is þr ¼ H ð4þ : ð5þ [10] A simple equation of state was used to convert temperature into density [Xing and Davies, 2001b]. In these equations, ~V =(u, v), u, v, and w are the velocity components corresponding to the x, y, and s coordinates, respectively, r is density, T is the temperature, H = h + z is the water depth, z is the elevation of the sea surface above the undisturbed level h, z is the water depth increasing vertically upward with z = z the free surface and z = h the seabed, f is the Coriolis parameter, g is the gravitational acceleration, t is time, and BPF x, BPF y are the Baroclinic pressure force terms. The vertical eddy viscosity and diffusivity are denoted by A V and K V, respectively. At the land boundary, the normal component of flow was set to zero, while offshore, a radiation condition was applied [Xing and Davies, 1998, 2001b]. The surface stress was set equal to the wind stress. At the seabed, a linear friction law was applied. For temperature, the heat flux was zero at sea surface and seabed. The eddy viscosity and diffusivity used in the hydrodynamic model were computed using a turbu-

3 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS 44-3 lence energy submodel. Further details of the model are given by Xing and Davies [2001b]. [11] The model with 60 s levels in the vertical is applied in cross-sectional form with initially idealized topography representing the shelf edge at 57 N off the west coast of Scotland, and subsequently, using realistic topography. The idealized topography takes the form of a shelf of constant depth (125 m) extending to 85 km offshore, with a linear shelf slope (a 1% gradient) extending down to a constant depth ocean of 500 m. The model domain extends from the coastline up to 240 km offshore (although only the shelf edge region is given in the figures). A grid resolution of 0.6 km is applied in the horizontal. Idealized topography is chosen initially to see the main features of the nonlinear processes independent of the significant spatial variability produced by realistic bottom topography. This is used subsequently to check that the findings with idealized topography are realistic. Calculations were also performed with a moderate (0.5 Pa) and strong (1.0 Pa) wind stress to examine the influence of wind strength upon mixed layer depth and the corresponding changes in amplitude of inertial oscillations and energy at the fm 2 frequency. Although changes in stratification between moderate and strong wind forcing can influence the nature of the internal tide, changing it from subcritical to supercritical depending upon the ratio of vertical density gradient to across shelf depth gradient [Craig, 1987], the main influence here is upon the magnitude and vertical distribution of the inertial oscillations. Similarly, the main effect of changing the temperature profile is to influence the intensity of surface inertial oscillations for a given wind stress, although this does influence the internal tide. [12] The area off the west coast of Scotland is chosen as the three-dimensional model successfully reproduced the significant internal tide in this region. Also, the area is subject to strong wind forcing from the Atlantic, and hence is a region of significant inertial oscillations as measured during JASIN [Weller, 1982]. Momentum advection in the model is represented using central differencing, with a total variation derivative method used for density advection over steep topography. A turbulence energy submodel is used to compute vertical eddy viscosity and diffusivity. Consequently, eddy viscosity and diffusivity evolve with the flow field giving rise to an additional source of nonlinearity besides the momentum advection terms. 3. Calculations 3.1. Wind Only Forcing [13] In this case (calculation 1, Table 1) aimed at determining the wind forced response, the model starts from rest with horizontal temperature surfaces, and wind forcing in the form of a clockwise rotating pulse of maximum magnitude 0.5 Pa and duration 0.5 days applied everywhere. Before considering the near-coastal response, it is useful to discuss the open ocean situation. In an oceanic situation with no divergence in the wind field and no coastline (infinite unbounded ocean), calculations [Davies, 1985a, 1985b] show that the wind s momentum diffuses down in the vertical, to the level of the thermocline where near-zero eddy viscosity prevents further downward diffusion. In essence, the wind s momentum is distributed over the mixed Table 1. Summary of Parameters Used in the Calculations Calculation Maximum Wind, Pa Tide Mixing Topography Stratification t.k.e. idealized weak yes t.k.e. idealized weak yes t.k.e. idealized weak yes t.k.e. idealized strong yes A V idealized strong yes t.k.e. realistic weak layer, giving rise to inertial oscillations (at the inertial frequency f, which for the latitude considered here is f = s 1, namely a period of hours) in this region with no flow below. In the coastal ocean at the land, the no-flow condition is satisfied by an associated offshore/ onshore flow below the mixed layer at 180 phase difference from the surface layer. In situations in which the stratification intersects the topography, the upwelling/ downwelling associated with the offshore/onshore flow due to the inertial oscillations gives rise to the generation of near-inertial internal waves (in addition to the inertial oscillations) at frequencies above the inertial period on the order up to 1.05f. As these are at the superinertial frequency, they can propagate in the horizontal. Since the frequency of these internal waves is close to the inertial period, they propagate very slowly (compared with the M 2 internal tide with a frequency of s 1 and period of hours) away from their generation point. In the case of a flat bottom region where near-inertial internal gravity waves are generated at the coast [Tintoré etal., 1995], then it takes some time (significantly longer than 10 days in our case) for them to propagate to the shelf edge. However, at the shelf edge the downwelling/upwelling associated with offshore/ onshore flows at the near-inertial period can produce oscillations of the thermocline, and hence internal waves in the region of significant internal tide. In essence, in the coastal ocean on the timescale considered here (namely less than 10 days after the imposition of the wind) the response at the near-inertial period is characterized by inertial oscillations in the surface layer with similar flows at depth, and a 180 phase shift across the thermocline. [14] In this calculation, the vertical stratification is characterized by a surface mixed layer depth on the order of 40 m, and a surface temperature of 10 C, separated by a sharp thermocline from a bottom layer of 6 C. Associated with this, the buoyancy frequency N varies through the vertical. Details of this temperature profile (referred to as Profile A) and associated buoyancy frequency are given by Xing and Davies [1998]. The wind forcing produces inertial oscillations in the surface layer with an associated onshore/offshore flow. Contours of the amplitude of the clockwise component at the inertial frequency ( f ) at the sea surface show an increase in amplitude from zero at the coast (due to the coastal inhibition of flow) to a value above 20 cm s 1 at the shelf edge (Figure 1a). Their amplitude increases to a surface maximum of about 27 cm s 1 in the ocean, rapidly decreasing in this region to near zero below the thermocline. On the shelf, the oscillatory flow in the mixed layer due to the surface inertial current drives a flow at 180 phase difference (as discussed earlier) below the thermocline due to the noflow coastal boundary. The amplitude of the anticlockwise (cyclonic) component of the current at the inertial period

4 44-4 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS Figure 1. (a) Contours in the shelf edge region (a subdomain of the model) of the amplitude of the (top) clockwise and (bottom) anticlockwise components of current at the inertial frequency ( f ), due to an impulsive wind stress of 0.5 Pa computed with temperature profile A (calculation 1). Note different contour intervals. (b) Time series of (top) temperature anomaly (contour interval 0.1 C), and (bottom) u- component of velocity (contour interval 5 cm s 1 ) at a location 77.5 km from the coast due to an impulsive wind stress of 0.5 Pa (calculation 1). (Figure 1a) is negligible (on the order of 0.25 cm s 1 ) and is confined to the shelf edge region, suggesting that upwelling/ downwelling at the shelf edge is responsible for a small nearinertia gravity wave in this region that has propagated upward from its shelf edge generation point. Time series of contours of the u component of velocity, (Figure 1b) at position 1 (77.5 km from the coast (Figure 1a), as we will show the location of near-maximum fm 2 current amplitude) are characterized by oscillatory flow at the inertial period in the surface layer. Time series of the temperature anomaly (Figure 1b) suggests that with the topography and stratification used here and on the timescale considered here, no significant near-inertial internal wave has propagated from the coastline or the shelf edge (where stratification intersects topography) into the region Tide and Wind Forcing [15] In this calculation (calculation 2, Table 1), we examine the interaction between tidal and wind forced motion. The initial conditions are as before, and tidal motion is induced by barotropic tidal forcing at the M 2 period at the oceanic boundary. The forcing is applied using a radiation condition, although experiments with different boundary conditions and a larger model domain yielded similar results in the shelf edge region. [16] An initial spin-up period of 5 days was found [Xing and Davies, 1998] to be sufficient to obtain a near-periodic internal tide at the shelf edge after which an identical wind forcing to that described previously (calculation 1) was applied. As mentioned previously, this wind impulse produces inertial oscillations in the surface layer. The following 5-day period was harmonically analyzed to determine the rotary components of the currents at the f, M 2, and higher frequencies produced by nonlinear interaction between the forcing frequencies giving rise to M 4 (frequency = s 1, period = 6.21 hours due to M 2 interacting with itself) and fm 2 (frequency = s 1, period = 6.78 hours, due to f and M 2 interaction) frequencies. By postprocessing, the calculations in terms of a harmonic analysis in rotary space, it is possible to determine the contributions at the various frequencies from the external forcing. Contours of the clockwise (anticyclonic) compo-

5 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS 44-5 Figure 2. Contours in the shelf edge region of the amplitude of the clockwise and anticlockwise components at (a) M 2,(b)M 4, and (c) fm 2 frequencies; due to tidal and wind stress (0.5 Pa) forcing (calculation 2).

6 44-6 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS nent at the inertial period (not given) were only slightly (namely a few centimeters per second reduction in nearsurface amplitude) different from those presented with wind forcing only. Separation of the internal tide from the barotropic tide requires the removal of the depth mean tidal current before performing the harmonic analysis. In the near-bed region, frictional effects reduce the amplitude of the barotropic tide, and consequently, when this tide is subtracted, a spurious amplitude in the internal tide in the near-bed region is produced (Figure 2a). The nature of the specified tidal forcing which took the form of an onshore/ offshore current at the open boundary, was such as to give a near-circular tidal current ellipse, consequently, u and v tidal current amplitudes (not shown) were nearly equal, and the clockwise component (Figure 2a) dominated the flow. The anticlockwise component (Figure 2a) was small on the order of 0.5 cm s 1 (about 10% of the clockwise component) and was confined to the shelf edge region, with a maximum at the level of the thermocline. Figure 2a shows that there are near-surface regions of intensified amplitude of the clockwise component of the internal tide close to the shelf edge both on the shelf and in the ocean. This together with the phase distribution (not shown) suggests that the internal tide propagates both on and offshelf from its generation point at the shelf edge [Craig, 1987]. On the shelf, there is a 180 phase shift across the thermocline due to the dominance of the first mode internal tide. [17] The significant contribution (values above 0.5 cm s 1 ) to the clockwise component of the M 4 tide at the surface occur close to the shelf edge (Figure 2b), where the M 2 internal tide is largest. It is located approximately between the two surface maxima in the clockwise component of the M 2 tide (Figure 2a), where the nonlinear term u@u/@x which generates the M 4 tide has a significant effect. Below the surface layer on the shelf, there is a region of local intensification with a maximum close to the top of the bottom boundary layer, which appears to be associated with vertical shear. The anticlockwise component (Figure 2b) is significantly smaller (on the order of 20%) with maximum occurring at the bottom of the mixed layer and above the bottom boundary layer, where shear is significant. This is the position of a nonzero anticlockwise contribution at the M 2 and f period. [18] The clockwise component of current at the fm 2 frequency has a significantly different vertical structure to that for the M 4 tide (compare Figures 2b and 2c). It exhibits a maximum in the region of the thermocline, where the shear in the amplitude of the inertial current is greatest (Figure 1a). However, both frequencies show a maximum above the bottom boundary layer, where frictional effects give rise to significant shear in the current. The close correlation between changes in the amplitude of the inertial period current in the vertical and the fm 2 current can be Figure 3. (opposite) Time series of (a) u-component of velocity (contour interval 5 cm s 1 ), (b) w@u/@z (unit 10 5 ms 2 ), (c) u@u/@x (unit 10 5 ms 2 ) at position 1, 77.5 km from the coast, and (d) w@u/@z (unit 10 5 ms 2 ) and (e) u@u/@x (unit 10 5 ms 2 ) at position 2, 100 km from the coast. (Solid line is positive, dashed and dotted line is the zero contour, and dashed line is negative.)

7 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS 44-7 Figure 4. Profiles of amplitude of (solid line) and line) at position 1, computed with tidal and wind forcing, for (a) wind stress of 0.5 Pa (calculation 2) and (b) wind stress of 1.0 Pa (calculation 3). clearly seen from Figures 1a and 2c. For the M 4 clockwise component of the tide, there is no intensification in the highshear region (about 40 m below the surface) to match that at the fm 2 frequency. The anticlockwise components of current at the M 4 and fm 2 frequencies are significantly smaller (on the order of 20%) than the clockwise component (Figures 2b and 2c). They exhibit local maxima at the shelf edge in the region of the thermocline, and at the top of the bottom boundary layer. [19] To examine the importance of the various nonlinear processes giving rise to the M 4 and fm 2 frequencies, we examine time series of the nonlinear terms u@u/@x and w@u/@z with x and z as horizontal and vertical axis, and u and w as corresponding cross shelf and vertical velocities, respectively. It is only necessary to consider these derivatives in the present cross-sectional model since the alongshelf derivative is zero. Also, for circular motion, the u@u/@x and w@u/@z terms (with v as alongshelf flow) are identical to the corresponding u terms except for a phase shift. As shown by Davies [1990] and Davies and Lawrence [1994], the nonlinear term involving the vertical eddy viscosity can also contribute to the interaction between various frequencies. However, this can only occur in regions of high vertical shear and when A (with A V as vertical eddy viscosity) is significant, such as wind forcing at the sea surface and a significant value of A V due to enhanced turbulence due to wind wave breaking. In the present calculation, is large in the thermocline, this is the region of small A V due to the suppression of turbulence by buoyancy. The suppression of turbulence by buoyancy is a feature of the turbulence energy model used here. In practice, internal wave breaking within the thermocline [Inall et al., 2001] will lead to an additional source of turbulence, which the present hydrostatic model cannot account for. Consequently, to examine the cascade to turbulence a nonhydrostatic model would be required. However, as this paper is only concerned with the first stage of this cascade, namely the first higher harmonic of M 2 and f, then a hydrostatic model is applicable. Another consequence of near-circular motion, as described previously in connection with the inertial oscillations, is that vertical eddy viscosity A V computed with the turbulence energy model does not show significant time variability. [20] Time series of the u component of velocity (Figure 3a) shows the modulation of the inertial currents by the tidal current giving rise to a longer period variation on the order of 3 days is associated with the tidal current and inertial current going in and out of phase with each other. This is not due to interaction, but is just a linear combination of two time series, at the f and M 2 period. Also, there is a surface layer of

8 44-8 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS corresponding shear. The time series of vertical velocity (not shown) has a maximum w at about 40 m below the surface with a smaller maximum at about 90 m. The time variation of these terms produces maxima in the w@u/@z term (Figure 3b) at about 40 and 90 m below the surface. This suggests that it has a significant contribution to the fm 2 intensification shown in Figure 2c at 40 and 90 m below the surface. In the surface layer, this is due to the vertical shear in the current at the f frequency across the thermocline (at 40 m below the surface), and regions of enhanced upwelling and downwelling in the M 2 internal tide. In the near-bed region, the vertical shear is associated with frictional effects in the bottom boundary layer. For the M 4 tide, the surface shear is absent in the M 2 tide, and hence there is no local enhancement at 40 m. However, at 90 m, bottom frictional effects give rise to shear at both the f and M 2 frequencies, and hence there is a local maximum at both M 4 and fm 2 frequencies. Figure 5. Contours in the shelf edge region of the amplitude of the clockwise rotating component of current at (a) f, (b)m 4, and (c) fm 2 frequencies due to tidal and increased wind stress (1.0 Pa) forcing (calculation 3). thickness on the order of 40 m corresponding to the region of surface inertial oscillations, separated from a lower layer of inertial oscillations, phase shifted by 180 with a region of maximum between the two. Similarly, bottom friction produces a bottom boundary layer extending from the seabed to approximately 90 m below the surface, with a Figure 6. Time series of (a) u-component of velocity (contour interval 5 cm s 1 ), (b) w@u/@z, and (c) u@u/@x (units 10 5 ms 2 ) at position 1, 77.5 km from the coast, from calculation 3.

9 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS 44-9 Figure 7. Contours in the shelf edge region of the amplitude of the clockwise rotating component of current at (a) f, (b)m 2,(c)M 4, and (d) fm 2 frequencies computed with a strong thermocline (temperature profile B) and tidal and wind (0.5 Pa) forcing (calculation 4). [21] The time variation of u@u/@x shows a maximum in the surface layer at times of strong surface current, near the start of the time series (Figure 3c). The position of the maximum varies with time reflecting variations in the current time series (Figure 3a). A small intensification is present in the lower part of the water column with a maximum about 90 m below the surface. In the surface layer (top 30 m), this term is clearly larger than the w@u/@z term, which is near zero in this region, suggesting that the u@u/@x term is responsible for the surface intensification of the M 4 current shown in Figure 2b. Profiles of u@u/@x and w@u/@z at position 1 at the M 4 and fm 2 frequencies are given in Figure 4a. At the M 4 frequency, a maximum of u@u/@x is evident in the surface layer (Figure 4a) with a local maximum about 90 m below the surface. The w@u/@z term is negligible in the upper part of the water column, with a maximum at about 90 m below the surface. At the fm 2 frequency, u@u/@x initially increases with distance below the surface up to 30 m, then decreases. The w@u/@z term shows a maximum at the level of the thermocline, and a local maximum at 90 m below the surface. This suggests that u@u/@x is the main source of generation of M 4 and fm 2 in the near-surface layer, with w@u/@z producing the fm 2 maximum in the region of the thermocline. Bottom friction effects through the w@u/@z term in the near-bed region intensify both of these terms. In deep water (position 2, 100 km from the coast) time series of u@u/@x and w@u/@z (Figures 3d and 3e) have no significant contribution at depth, suggesting that bottom friction on the shelf is the source of the near-bed intensification of M 4 and fm Influence of Wind Magnitude [22] To investigate the role of the depth of the thermocline and wind intensity upon the nonlinear interaction between the M 2 and f frequencies, the previous calculation was repeated with the wind stress maximum increased to 1.0 Pa (calculation 3). [23] Although increasing the wind stress increases the magnitude of the inertial oscillation in the surface layer and the depth of the thermocline, the spatial distribution of the clockwise component (Figure 5a) is comparable with that found previously (Figure 1a) (as previously shown, the clockwise component dominates the solution, and in the following discussion we will only consider this component). The spatial distribution and amplitude of the M 2 internal tide (not shown) is only slightly affected by the increased wind stress, in particular, the region of surface enhance internal tide increases and their position changes slightly.

10 44-10 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS is significant. The effect of the increased depth of the thermocline, and increase in magnitude of the inertial oscillations is clearly evident from a comparison of the w@u/@z term between calculations 2 and 3 (Figures 4a and 4b). This suggests that as the energy in the wind at the inertial period increases, the fm 2 current amplitude will increase, with its maximum value occurring at a greater depth as the mixed layer becomes thicker. Figure 8. Time series of (a) w@u/@z and (b) u@u/@x (units 10 5 ms 1 ) at position 1, 77.5 km from the coast, from calculation 4. Some differences in the near-bed region are found on the shelf. [24] Associated with these changes in the M 2 internal tide are corresponding modifications of the M 4 tide (compare Figures 5b and 2b). The most significant change, however, occurs at the fm 2 frequency (compare Figures 5c and 2c). The increase in magnitude can be readily appreciated in terms of the enhancement in the inertial oscillations (Figure 5a). Although the horizontal spatial distribution of current amplitude at the fm 2 frequency has remained about the same, being determined by the decay of inertial oscillations as the coast is approached and the magnitude of the internal tide, it is evident that the maximum value occurs at a greater depth. This is due to the increased depth of the mixed layer. The height above the seabed of the maximum at depth has also increased (Figures 2c and 5c), suggesting a thickening of the bottom boundary layer associated with increased friction and viscosity at the bed. [25] Although the time series of u, w@u/@z, and u@u/@x (Figure 6) show comparable time variations to those found previously (Figure 3), it is evident that the depth below the free surface of maximum shear in the u velocity and the maximum value of w@u/@z has increased as has the magnitude of this term compared with u@u/@x. The corresponding increase in magnitude of fm 2 and location of its maximum in the water column, confirms that w@u/@z the term with w determined from the tide from the inertial oscillations is the major source of fm 2. Although the term can also contribute, this does not have a maximum in the surface layer, but can influence the near-bed maximum. For the M 4 tide, the u@u/@x term has the greatest contribution near the surface, but at depth the w@u/@z term 3.4. Influence of Stratification [26] In this calculation (calculation 4, Table 1) aimed at examining the influence of stratification upon the intensity of the fm 2 current through its effect upon the energy at the inertial frequency, the tidal and wind stress forcing were identical to those used in calculation 2, but the thermocline was much stronger. Details of the temperature profile (Profile B) and buoyancy frequency associated with this stronger thermocline are given by Xing and Davies [1998]. This has the effect of increasing the magnitude of the inertial oscillations in the surface layer and increasing the shear across it (compare Figures 7a and 1a). Although the primary effect of changing the stratification is upon the inertial oscillations, it does influence the internal tide [Craig, 1987]. On the shelf, the magnitude of the near-surface internal tide at about 70 km from the coast is increased, with a substantial increase in its surface value in the ocean (Figure 7b, 120 km offshore). This offshore increase in the M 2 internal tide gives rise to an enhanced M 4 tide in the surface layer away from the shelf edge (compare Figures 7c and 2b). A similar offshelf surface intensification of the current at the fm 2 frequency occurs compared with that found previously (see Figures 7d and 2c). [27] A detailed analysis of the contribution from the nonlinear terms (Figure 8) showed that, as before, the term involving vertical shear dominated the generation of currents at the fm 2 frequency giving rise to the near-surface maximum shown in Figure 7d. In essence, changes in the thermocline which reduce the extent to which the wind s momentum diffuses to depth increased the near-surface fm 2 current, the horizontal spatial distribution of which was determined by the distribution of the vertical velocity associated with the M 2 tide. Consequently, any change in stratification which enhances the offshelf magnitude of the internal tide would enhance the fm 2 current in similar regions. This is because the magnitude of the inertial oscillations became constant in the ocean where their coastal reduction was absent Influence of the Parameterization of Vertical Eddy Viscosity [28] In the previous calculations, eddy viscosity was computed from the turbulence closure model, and showed a rapid decrease in the region of the thermocline accompanied by a rapid reduction in current. In the present calculation (calculation 5, Table 1), eddy viscosity and diffusivity were fixed at m 2 s 1, and the same temperature profile (Profile B, a strong thermocline) as previously (calculation 4) was used. With this value of eddy viscosity, the surface magnitude of the inertial currents is comparable with that found previously (compare Figures 9a and 7a), enabling a consistent comparison with earlier results to be made.

11 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS Figure 9. Contours in the shelf edge region of the amplitude of the clockwise rotating component of current at (a) f, (b)m 2,(c)M 4, and (d) fm 2 frequencies computed with a strong thermocline (temperature profile B) and tidal and wind (0.5 Pa) forcing, with a fixed eddy viscosity (calculation 5). [29] The effect of using a fixed eddy viscosity is to produce a smoother decrease in inertial currents below the thermocline (compare Figures 9a and 7a). Also, the spatial distribution of the M 2 internal tide away from the bottom boundary layer is comparable with that found previously. However, the thickness of the bottom boundary layer is significantly reduced (compare Figures 9b and 7b). [30] The M 4 tide in the surface layer shows a similar variability to that found previously, although the intensification at the top of the bottom boundary layer found earlier (Figure 7c) is no longer present (Figure 9c). For the fm 2 component, the limited vertical intensification found previously in the region of the thermocline (Figure 7d) is no longer present (Figure 9d), but has been replaced in deep water (about 110 km offshore) by a surface intensification decreasing with distance below the surface. Although on the shelf at about 75 km from the coast, there is still a local maximum between the surface and the thermocline. Similarly, in the bottom boundary layer, a local intensification has been replaced by a region extending from the seabed up to the thermocline (Figure 9d). [31] Time series of the w@u/@z term at position 1 (Figure 10) no longer shows a narrow band of local intensification at the level of the thermocline (Figure 8), but a much broader region and a significant reduction in magnitude is evident. The region of local intensification above the bottom boundary layer is no longer present. The u@u/@x time series is comparable with that found previously (compare Figures 10 and 8). A harmonic analysis of the various nonlinear terms, showed that the term involving vertical shear played a smaller role in determining the fm 2 current than before. In the surface region, the u@u/@x term was increased compared with that found earlier. This explains why the vertical profile of fm 2 current in the surface layer is comparable with that found for M 4. This will be examined further in the case of real topography. This calculation suggests that because a primary mechanism producing the fm 2 current involves vertical shear, the vertical profile of the fm 2 current may be particularly sensitive to changes in vertical eddy viscosity Realistic Topography [32] In a subsequent calculation (calculation 6, Table 1), realistic topography taken from a west-east cross-section of the shelf edge off the west coast of Scotland at 57 N was used. This calculation was performed to examine the extent to which conclusions concerning the fm 2 generation are modified by using realistic shelf edge topography. A major feature of this change in topography was that the offshore water depth was increased to over 1000 m. Also,

12 44-12 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS Figure 10. Time series of (a) and (b) (units 10 5 ms 1 ) at position 1, 77.5 km from the coast from calculation 5. the shelf slope was not constant, but varied from shelf to ocean. As shown by Xing and Davies [1999], small-scale variations in the shelf slope can lead to more spatial variability in the internal tide. For this reason, it is important to confirm the earlier results with a model containing realistic topography. Also, a realistic crossshelf M 2 tidal forcing [Xing and Davies, 1998] was applied. A consequence of using a cross-sectional model was that uniform alongshelf topography was assumed. However, as our primary aim here is to determine the extent to which the conclusions as to the processes influencing fm 2 current generation over idealized topography are modified by using realistic topography, the assumption of uniform alongshelf topography is consistent with the earlier calculations. [33] In this calculation, the wind stress forcing, stratification, and subgrid scale mixing were identical to those used in calculation 2. As mentioned previously, the clockwise component of current is dominant and this is the component we will examine in detail. The clockwise component of the near-surface inertial oscillation has a similar distribution (Figure 11a) to that found previously (Figure 1a), although the region of coastal inhibition (reduced inertial current amplitude) extends over the shelf edge. An area of increased amplitude (maximum 1.7 cm s 1 ) of the anticlockwise current component, centered on the shelf edge (not shown) was found. This is associated with near-inertial internal wave energy being generated at the shelf edge and propagating upward. The region does not extend very far onto the shelf, suggesting local generation. The significantly larger component of anticlockwise near-inertial wave at the shelf edge in this case compared with the previous one appears to be associated with the steeper shelf edge topography. [34] Differences in the amplitude and distribution of the clockwise component of the M 2 internal tide (Figures 11b and 2a) can be attributed to differences in tidal forcing and bottom topography. At the shelf edge, there is a significant contribution to the anticlockwise component that was not found with idealized topography. Both the clockwise (Figure 11c) and anticlockwise (not shown) components of the M 4 tide are appreciably larger than that found previously (Figure 2b), although they have a similar spatial pattern to those found with idealized topography. The distribution of the clockwise component at the fm 2 frequency (Figure 11d) is close to that found previously (Figure 2c) with regions of increased magnitude just above the thermocline and above the bottom boundary layer. For the anticlockwise component (not shown), a region of rapidly changing magnitude was evident in the shelf edge region, although outside this area its magnitude was negligible. [35] Profiles of u@u/@x and w@u/@z at position 1 (77.5 km from the coast) (Figure 12), at the M 4 and fm 2 frequencies, show similar variations to those found with idealized topography (Figure 4a). For M 4 current, as before, the contribution from u@u/@x is a maximum in the near-surface layer where the M 2 tide is a maximum. The increase in magnitude above that found previously, reflects the increased M 2 tidal current in the surface layer. The contribution to the M 4 current from w@u/@z, as before, is a maximum just above the bottom boundary layer and is negligible in the surface layer. For the current at the fm 2 frequency, the u@u/@x term is significant in the surface layer, and exhibits a reduction in the region of the thermocline that was not found previously (Figure 4a). The reason for this is not clear, but this term is small and exhibits significant vertical variations. As mentioned previously, the contribution from w@u/@z is a maximum at the level of the thermocline. A similar local maximum is found above the bottom boundary layer. These maxima are larger than before (compare Figures 12 and 4a), due to an increase in magnitude of the M 2 internal tide. [36] This calculation and others using realistic topography with fixed diffusion coefficients and strong stratification (not presented) confirm the results found with the idealized topography, with a calculation involving realistic topography and strong stratification (not presented) emphasizing that the horizontal distribution of the fm 2 current is closely related to the spatial variability of the M 2 internal tide. As shown by a number of authors [e.g., Craig, 1987], the distribution of the M 2 internal tide is critically dependent on the profile of vertical stratification and shelf slope. Consequently, changes in these will significantly influence the fm 2 distribution in the shelf edge region. 4. Conclusions [37] Calculations using a three-dimensional nonlinear prognostic model in cross-sectional form forced by both the M 2 tide and a wind impulse have shown that coupling between inertial oscillations and the internal tide can produce significant fm 2 currents. For the case considered

13 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS Figure 11. Contours in the shelf edge region, of the clockwise rotating component of current computed with realistic topography at (a) f, (b)m 2,(c)M 4, and (d) fm 2 frequencies, using viscosity computed with the turbulence model (calculation 6). here of a horizontally uniform wind field in a coastal ocean environment, the predominant source of energy at the f frequency takes the form of inertial oscillations in the surface layer. Due to the proximity of the coastal boundary, corresponding inertial oscillations with a 180 phase shift occur below the thermocline. Although near-inertial internal waves can be generated where stratification intersects topography, namely the shelf edge and coastline for the cases examined here, currents associated with these internal waves are small. Also, their frequency is only Figure 12. Profiles of amplitude of u@u/@x (solid line) and w@u/@z (dashed line) at position 1, computed with realistic topography (calculation 6).

14 44-14 DAVIES AND XING: INTERNAL TIDES AND NEAR-INERTIAL CURRENTS slightly (on the order of 1.05f) above the inertial waves. Therefore on the timescale considered here, they cannot propagate far from their generation points. Consequently, in the calculations with eddy viscosity computed using a turbulence model, the main source of fm 2 production is through shear in the inertial oscillations at the level of the thermocline, and the vertical velocity associated with the internal tide. In the case of a fixed eddy viscosity (slightly larger than that found in the turbulence model), shear across the thermocline is reduced, and hence its contribution to the fm 2 current is decreased. However, the near-surface contribution from the nonlinear horizontal momentum term is increased. For the M 4 tide in the surface layer, the nonlinear horizontal momentum term is the major generation term. At depth, shear above the bottom boundary layer gives rise to enhanced fm 2 and M 4 currents. The nonlinear term involving the time variation of eddy viscosity and vertical shear does not contribute to fm 2 or M 4 generation. [38] Increasing the wind stress forcing in the case of eddy viscosity and diffusivity derived from the turbulence model leads to a deeper thermocline, and hence the location of maximum fm 2 currents occurs farther below the surface. Calculations also show that changing the vertical stratification in the model with eddy viscosity derived from the turbulence model influences the magnitude of the inertial oscillations in the surface layer. The distribution of the M 2 internal tidal current is also sensitive to changes in stratification, and hence both M 4 and fm 2 currents are influenced by the stratification. [39] The main results of the calculations performed with idealized topography are confirmed by those using realistic topography and density profile. In the case of realistic water depths, Xing and Davies [1999] showed that small variations in the local shelf edge slope, and localized changes in stratification could yield multiple generation points for the internal tide along the shelf slope. In this case, the spatial variability of the M 2 internal tide, and hence currents at the M 4 and fm 2 frequencies could be larger than shown here. Also, in the case of more realistic and longer duration wind forcing, as shown by Xing and Davies [1997], this could influence the stratification in the nearbed region by upwelling or downwelling processes, and hence the distribution of the internal tide. [40] Here we have considered a uniform wind field, and a short timescale, however, with realistic wind forcing, nearinertial internal waves will be generated in the surface layer and on the longer timescale have time to propagate from their generation point. In this case, which is more typical of the open ocean, the horizontal momentum terms will contribute more to the fm 2 production. However, in the nearcoastal ocean, where mixing in the oceanic boundary layer is most likely to occur, nonlinear interaction between the M 2 internal tide and wind-produced inertial oscillations are an important mechanism for transferring energy from the tide and wind to shorter waves. [41] Acknowledgments. The authors are indebted to L. Parry for typing the paper, and R. A. Smith for his help in preparing diagrams. References Craig, P. D., Solutions for internal tide generation over coastal topography, J. Mar. Res., 45, , Davies, A. M., A three-dimensional modal model of wind induced flow in a sea region, Prog. Oceanogr., 15, , 1985a. Davies, A. M., Application of a sigma cooordinate sea model to the calculation of wind-induced currents, Cont. Shelf Res., 4, , 1985b. Davies, A. M., On the importance of time varying eddy viscosity in generating higher tidal harmonics, J. Geophys. Res., 95, 20,287 20,312, Davies, A. M., and J. Lawrence, Examining the influence of wind and wind wave turbulence on tidal currents, using a three-dimensional hydrodynamic model including wave-current interaction, J. Phys. Oceanogr., 24, , Davies, A. M., and J. Xing, The influence of eddy viscosity parameterization and turbulence energy closure scheme upon the coupling of tidal and wind induced currents, Estuarine Coastal Shelf Sci., 53, , Garrett, C., What is the near-inertial band and why is it different from the rest of the internal wave spectrum, J. Phys. Oceanogr., 31, , Gemmrich, J. R., and H. Van Haren, Internal wave band eddy fluxes above a continental slope, J. Mar. Res., 60, , Gill, A. E., On the behaviour of internal waves in the wake of storms, J. Phys. Oceanogr., 14, , Holloway, P. E., A regional model of the semi-diurnal internal tide on the Australian North West shelf, J. Geophys. Res., 106, 19,625 19,639, Inall, M. E., T. P. Rippeth, and T. J. Sherwin, Impact of nonlinear waves on the dissipation of internal tidal energy at a shelf break, J. Geophys. Res., 105, , Inall, M. E., G. I. Shapiro, and T. J. Sherwin, Mass transport by non-linear internal waves on the Malin shelf, Cont. Shelf Res., 21, , Ledwell, J. R., E. T. Montgomery, K. L. Polzin, L. C. St. Laurent, R. W. Schmitt, and J. M. Toole, Evidence for enhanced mixing over rough topography in the abyssal ocean, Nature, 403, , Mellor, G. L., One-dimensional, ocean surface layer modeling: A problem and a solution, J. Phys. Oceanogr., 31, , Merifield, M. A., P. E. Holloway, and T. M. Johnston, The generation of internal tides at the Hawaiian Ridge, Geophys. Res. Lett., 28, , Mihaly, S. F., R. E. Thomson, and A. B. Rabinovich, Evidence for nonlinear interaction between internal waves of inertial and semidiurnal frequency, Geophys. Res. Lett., 25, , Muller, P., G. Holloway, F. Henyey, and N. Pomphrey, Non-linear interactions among internal gravity waves, Rev. Geophys., 24, , Munk, W., and C. Wunsch, Abyssal recipes, part II, Energetics of tidal and wind mixing, Deep Sea Res., Part I, 45, , New, A. L., and R. D. Pingree, Evidence for internal tidal mixing near the shelf break in the Bay of Biscay, Deep Sea Res., 37, , Rippeth, T. P., J. H. Simpson, R. J. Player, and M. Garcia, Current oscillations in the diurnal-inertial band on the Catalonian shelf in spring, Cont. Shelf Res., 22, , Samelson, R. M., Large scale circulation with locally enhanced vertical mixing, J. Phys. Oceanogr., 28, , Sherwin, T. J., Analysis of an internal tide observed on the Malin shelf, north of Ireland, J. Phys. Oceanogr., 18, , Spall, M. A., Large-scale circulations forced by localized mixing over a sloping bottom, J. Phys. Oceanogr., 31, , Tintoré, J., D.-P. Wang, E. Garcia, and A. Viudez, Near-inertial motions in the coastal ocean, J. Mar. Syst., 6, , Van Haren, H., Properties of vertical current shear across stratification in the North Sea, J. Mar. Res., 58, , Van Haren, H., L. Maas, J. T. F. Zimmermann, H. Ridderinkhof, and H. Malschaert, Strong inertial currents and marginal internal wave stability in the central North Sea, Geophys. Res. Lett., 26, , Van Haren, H., L. Maas, and H. Van Aken, On the nature of internal wave spectra near a continental slope, Geophys. Res. Lett., 29, 1615, doi: /2001gl014341, Weller, R. A., relation of near-inertial motions observed in the mixed layer during the JASIN [1978] experiment to the local wind stress and to the quasi-geostrophic flow field, J. Phys. Oceanogr., 12, , Xing, J., and A. M. Davies, Processes influencing the internal tide, its higher harmonics and tidally induced mixing on the Malin-Hebrides shelf, Prog. Oceanogr., 38, , 1996a. Xing, J., and A. M. Davies, Application of a range of turbulence energy models to the determination of M 4 tidal current profiles, Cont. Shelf Res., 16, , 1996b. Xing, J., and A. M. Davies, The influence of wind effects upon internal tides in shelf edge regions, J. Phys. Oceanogr., 27, , 1997.

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