Processes influencing the non-linear interaction between inertial oscillations, near inertial internal waves and internal tides



[1] A numerical model of the Hebrides shelf edge (represented by a cross section) is used to examine the non-linear interactions producing energy at the fM2 frequency. Calculations show that in the near coastal ocean this is primarily due to coupling between wind induced inertial oscillations (although near inertial internal waves are present) and the M2 internal tide. The major non-linear interaction, and hence largest fM2 currents, occurs in the region of the thermocline and is associated with shear in the inertial oscillation and the vertical velocity due to the internal tide. This non-linear process represents an important contribution to the energy cascade from the wind and tide into higher frequency waves and eventually mixing. The fM2 current is shown to be a maximum in the shelf edge region and hence measurements in this area will be particularly valuable in determining the extent to which models can represent non-linear processes.

1. Introduction

[2] Recent ideas [Munk and Wunsch, 1998], suggest that a major part of the mixing in the ocean is due to boundary layer mixing at the shelf edge. Two major sources of energy for mixing are the tides and wind. Oceanic tidal mixing arises through the production of internal tides at the shelf edge [Xing and Davies, 1998] which can then propagate into the deep ocean. Away from the shelf edge internal tides can be generated over topographic features [e.g. the Hawaiian ridge, Merifield et al., 2001], 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 mixing associated with it [Xing and Davies, 1998]. Wind generated inertial oscillations and near-inertial internal waves are also a major source of mixing since they are at the resonant frequency, and hence appreciable currents can be generated by light winds. In this paper we show that in near coastal and shelf edge regions they are accompanied by a 180° phase shift in currents across the thermocline, due to the presence of the coast [Rippeth et al., 2002]. This phase shift gives rise to maximum shear across the thermocline. As we will show the non-linear interaction between this shear, at the local inertial frequency f, (period 14.32 hrs) and the vertical velocity associated with the semi-diurnal M2 internal tide (period 12.42 hrs) is a major source of energy at the sum of their frequencies, termed the fM2 frequency (period 6.65 hrs). By examining the non-linear processes giving rise to energy at the fM2 frequency and ultimately by comparison with measurements it should be possible to determine how accurately models can represent the non-linear interaction between tide and wind. This interaction is important as it is the first stage in the cascade of energy to shorter wavelengths and eventually to mixing. This mixing is in addition to the vertical shear production of turbulence. Although it could be substantially smaller than that due to other processes e.g. solitons and associated breaking, which require a non-hydrostatic model, it can be studied with a hydrostatic model and conclusions drawn as to the nature of non-linear interactions giving energy at the fM2 frequency; the focus of this paper.

[3] Recently measurements have shown that there is appreciable energy in both shallow sea [Van Haren et al., 1999] and oceanic currents [Mihaly et al., 1998] at the fM2 frequency to indicate significant coupling between near-inertial currents and those due to the M2 internal tide. In this paper a three dimensional model applied previously to examine the M2 internal tide [Xing and Davies, 1998], is used in cross sectional form to determine the various interaction processes giving rise to energy at the fM2 frequency.

2. Numerical Model

[4] Since the three dimensional model and the associated parameterization of vertical exchange using a Mellor-Yamada level 2.5 turbulence scheme has been described in detail by Xing and Davies [1998], only the essential details will be considered here. The model with sixty sigma levels in the vertical is applied in cross sectional form with idealized topography representing the shelf edge at 57°N off the west coast of Scotland. This area is chosen as the three dimensional model successfully reproduced the appreciable 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. The model domain extends from the coastline up to 240 km offshore (although only a limited region, namely the shelf edge is shown in the figures). A grid resolution of 0.6 km is applied in the horizontal.

3. Calculations

[5] The model starts from a state of rest with horizontal temperature surfaces, and motion is induced by barotropic tidal forcing at the M2 tidal period using a radiation condition at the oceanic boundary. After 5 days the internal tide is established and wind forcing in the form of a horizontally uniform clockwise rotating pulse in time with a maximum magnitude of 0.5 Pa and duration of less than 0.5 days is applied everywhere. This wind impulse produces inertial oscillations in the surface layer with an associated on-offshore flow.

[6] Before considering the interaction of the internal tide with the wind we initially consider the wind forced current only. Results from these calculations (not presented) show that in the coastal ocean at the land the no-flow condition is satisfied by an associated off-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 off-onshore flow due to the inertial oscillations gives rise to the generation of near inertial internal waves at frequencies of order 1.05 f. Calculations (not presented) show that these internal waves propagate very slowly away from their generation point. In essence in the coastal ocean for the stratification, topography and time scale considered here (of order 10 days after the imposition of the wind) the response at the near inertial period is dominated by inertial oscillations in the surface layer with similar flows at depth, phase shift by 180° across the thermocline due to the presence of the coast. A weak current (of order 0.5 cm s−1, see later) associated with a near inertial internal wave generated at the shelf edge, is also present although its contribution through wave-wave interaction to energy at the fM2 frequency is small (see later).

[7] We now consider the interaction between the tide and wind forced motion. After an initial spin up, the following 5 day period was harmonically analysed to determine the rotary components of the currents at the f, M2, M4 (the first higher harmonic of M2) and fM2 frequency. Contours of the clockwise component at the inertial period, 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 26 cm s−1 in the ocean, rapidly decreasing in this region to near zero below the thermocline. On the shelf the on-offshore oscillatory flow in the mixed layer due to the surface inertial current drives an off-shore flow at 180° phase difference below the thermocline. The amplitude of the anticlockwise component of the current at the inertial period (not shown) is negligible (of order 0.5 cm s−1) and is confined to the shelf edge region. The dominance of the current associated with the clockwise inertial oscillation in the area compared to the anticlockwise component of current suggests that upwelling/downwelling at the shelf edge only produces a small near inertial internal wave that propagates from its shelf edge generation point.

Figure 1.

Contours (Note differences in contour intervals (ci) and maximum values (max)) of the amplitude of the clockwise component of the baroclinic current (cm s−1) in the shelf edge region for various frequencies, namely (a) f (ci = 5, max = 25.17), (b) M2 (ci = 2.5, max = 12.35), (c) M4 (ci = 0.5, max = 1.44), (d) fM2 (ci = 0.5, max = 3.51)

[8] In order to separate the internal tide from the barotropic tide it is necessary to remove 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 M2 component of the internal tide is produced (Figure 1b). The nature of the specified barotropic tidal forcing was such as to give a near circular tidal current ellipse, consequently u and v tidal current amplitudes (Figure 2) were nearly equal. Results from the harmonic analysis of the internal tide showed that the clockwise component (Figure 1b) dominated the flow. Neglecting the spurious near bed maxima (due to frictional effects), 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. The phase distribution (not shown) suggests that the M2 internal tide propagates both on and off shelf from its generation point at the shelf edge. On the shelf there is an 180° phase shift across the thermocline due to the dominance of the first mode internal tide. The anti-clockwise component (not shown) is small of the order of 0.5 cm s−1 and is confined to the shelf edge region, with a maximum at the level of the thermocline.

Figure 2.

Profiles of the amplitude of the u (solid) and v (dashed) components of the total velocity at a location 77.5 km from the coast at (a) M2, (b) f, (c) M4 and (d) fM2 frequency.

[9] The clockwise component of the M4 internal tidal current has a surface maximum on the shelf side of the shelf edge (Figure 1c) approximately between the two surface maxima in the clockwise component of the M2 tide. This is the region where the horizontal gradient in the u velocity (across shelf current), namely the term ∂u/∂x is a maximum and hence the non-linear term u∂u/∂x which generates the M4 tide has a significant effect. This is considered later in connection with the contribution of the various non-linear terms. In general the significant contribution (values above 0.5 cm s−1) to the clockwise component of the M4 tide in the surface layer occurs close to the shelf edge where the M2 internal tide is largest. 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. Although this is physically realistic, it must be borne in mind that in this figure the depth mean M4 tide has been removed as in the M2 case. However the total M4 tidal current is shown in Figure 2, where the maximum is clearly evident. The anticlockwise component (not shown) is significantly smaller with a maximum occurring at the bottom of the mixed layer and above the bottom boundary layer where shear is significant.

[10] The clockwise component at the fM2 frequency in the surface layer has a significantly different vertical structure to the M4 tide (compare Figures 1c and 1d) with a maximum in the region of the thermocline, where the shear in the amplitude of the inertial current is greatest (Figure 2). A second maximum is also evident above the bottom boundary layer where frictional effects give rise to significant shear in the current at the inertial period and a reduction in the total M2 tidal current (Figure 2). The close correlation between changes in the amplitude of the inertial period current in the vertical and the fM2 current can be clearly seen (Figure 2) at a point located 77.5 km from the coastline (the location of maximum fM2 current). For the M4 tide at this point there is no surface intensification to match that in the fM2 current, but an increase in the near bed layer corresponding to a local increase in the internal M2 tide (Figure 1b) at a similar height to that found at the fM2 frequency does occur (Figure 2).

[11] To understand the relative importance of the various non-linear processes giving rise to these variations at the M4 and fM2 frequency it is necessary to examine time series of the non linear terms u∂u/∂x, w∂u/∂z and equation image with x and z horizontal and vertical axis, and u, w corresponding velocities, and AV the vertical eddy viscosity. In a cross sectional model the along shelf derivative is zero, and for near circular motion the u∂v/∂x, w∂v/∂z and equation image (with v alongshelf flow) is similar to the corresponding u terms except for a phase shift. Another consequence of circular motion is that AV computed with the turbulence energy model is constant. However in the case of both tidal and inertial periods being present the beat frequency with period (T = 3.82 days) leads to the modulation of the flow giving rise to a slight variation in AV (Figure 3). However this time variation of AV was not sufficient to produce a significant contribution from the non-linear term involving AV (not shown).

Figure 3.

Time series (Note differences in contour intervals (ci) and units) of (a) temperature anomaly (ci = 0.1°C), (b) u-component of velocity (ci = 5 cm s−1) (Note, positive values denoted by solid line), (c) viscosity (ci = 1, units log10 m2 s−1), (d) u∂u/∂x (ci = 0.25, units 10−5 m s−2) and (e) w∂u/∂z (ci = 0.25, units 10−5 m s−2), at a location 77.5 km from the coast.

[12] The time series of u, AV and temperature anomaly (Figure 3), show the presence of a surface layer of inertial currents of thickness of order 40 m separated from a lower layer of inertial currents phase shifted by 180° from those above, with a region of maximum shear ∂u/∂z between the two. Similarly bottom friction produces a bottom boundary layer extending from the sea bed to approximately 90 m below the surface, with a corresponding shear. The time series of the temperature anomaly has a maximum, and hence maximum w at about 40 m below the surface with a smaller maximum at about 90 m. The time variation of these terms closely follows the current time series and produces maxima in the w∂u/∂z term (Figure 3) at about 40 m and 90 m below the surface. This suggests that it has a significant contribution to the fM2 intensification shown in Figures 1d and 2 at 40 m below the surface, due to the vertical shear in f and regions of enhanced upwelling and downwelling in the M2 internal tide. For the M4 tide the surface shear is absent in the M2 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 M2 frequency and hence there is a local maximum in both M4 and fM2.

[13] 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 3). 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 40 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 M4 current shown in Figure 1c. The fact that the time series (Figure 3) shows a depth variation in the term goes someway to explain the more complex depth variation for the M4 tide shown in Figure 1c than that found at the fM2 frequency (Figure 1b). The presence of a near bed intensification of u∂u/∂x, although over a broader depth range than w∂u/∂z explains why the intensification of the M4 profile shown in Figure 2 is over a broader depth range than for fM2. This suggests that u∂u/∂x is the main source of generation of M4 in the surface layer, with w∂u/∂z producing the fM2 maximum in the region of the thermocline, with bottom friction producing the near bed intensification of both. In deep water (at a second location, 100 km from the coast) time series of u∂u/∂x and w∂u/∂z (not shown) have no significant contribution at depth, suggesting that bottom friction on the shelf is the source of the near bed intensification of M4 and fM2.

[14] A detailed analysis of a calculation (not presented) with increased wind stress and a deeper thermocline showed an increase in magnitude of fM2 and the depth of its maximum below the surface. This confirmed that the w∂u/∂z term with w determined from the tide and ∂u/∂z from the inertial oscillations was the major source of fM2.

4. Conclusions

[15] Calculations have shown that coupling between inertial oscillations and the internal tide can produce significant fM2 currents. For the case considered here, energy at the f frequency primarily takes the form of inertial oscillations, with currents phase shifted by 180° across the thermocline. Consequently the main source of fM2 production is through shear in the inertial oscillations at the level of the thermocline, and the vertical velocity associated with the internal tide. The major generation of the M4 tide in the surface layer is the non-linear horizontal momentum term. At depth, shear above the bottom boundary layer gives rise to enhanced fM2 and M4 currents.

[16] In the near coastal ocean, non-linear interaction between the M2 internal tide and wind produced inertial oscillations is an important mechanism for transferring energy to higher frequencies and eventually to mixing. Despite this mixing process being just one of many at the shelf edge, the fact that the fM2 signal appears strong in this region suggests that it is valuable to study its generation in the area. A detailed model/data comparison would be a good test of the model's ability to correctly reproduce the non-linear interaction between the M2 tide and the wind induced inertial currents. This interaction is important as it is the first stage of the cascade of energy to shorter wavelengths through wave-wave interaction and hence to shelf edge mixing with associated important implications to the ocean circulation. Further work on this topic using a range of detailed models to quantify this mixing is warranted.