Abstract

Recent finescale observations of shear and stratification in temperate shelf sea thermoclines show that they are of marginal stability, suggesting that episodes of enhanced shear could potentially lead to shear instability and diapcynal mixing. The bulk shear between the upper and lower boundary layers in seasonally stratified shelf seas shows remarkable variability on tidal, inertial, and synoptic time scales that has yet to be explained. In this paper observations from the seasonally stratified northern North Sea are presented for a time when the water column has a distinct two-layer structure. Bulk shear estimates, based on ADCP measurements, show a bulk shear vector that rotates in a clockwise direction at the local inertial period, with episodes of bulk shear spikes that have an approximately twice daily period, and occur in bursts that last for several days. To explain this observation, a simple two-layer model based on layer averaging of the one-dimensional momentum equation is developed, forced at the surface by wind stress and damped by (tidally dominated) sea bed friction. The two layers are then linked through an interfacial stress term. The model reproduces the observations, showing that the bulk shear spikes are a result of the alignment of the wind stress, tidal bed stress, and (clockwise rotating) bulk shear vectors. Velocity microstructure measurements are then used to confirm enhanced levels of mixing during a period of bulk shear spikes. A numerical study demonstrates the sensitivity of the spike generation mechanism to the local tidal conditions and the phasing and duration of wind events.

1. Introduction

A central part of the current agenda in the physical oceanography of continental shelf seas is the identification and parameterization of processes that drive vertical mixing across the seasonal thermocline. The seasonal thermocline acts as an important physical barrier separating the sunlit surface layers from the dark nutrient-rich deep water. The mixing of material across the seasonal thermocline therefore represents a key biogeochemical pathway (Sharples et al. 2001).

Coincident high-resolution measurements of shear and stratification (e.g., van Haren et al. 1999; MacKinnon and Gregg 2005; Rippeth et al. 2005; Rippeth 2005) for a range of temperate seasonally stratified shelf seas show that the thermocline is in a state of marginal stability (gradient Richardson number, Ri ≈ 1), implying that the addition of extra shear could potentially reduce Ri sufficiently to trigger shear instability, thus transferring energy to thermocline turbulence and resulting in vertical mixing. While flow in continental shelf seas tends to be dominated by the tides (e.g., Rippeth et al. 2005), currents also arise because of wind-driven slab motion of the surface layer. These near-inertial oscillations are the response to abrupt changes in wind forcing. They have long been recognized as energetic features throughout the ocean (e.g., Pollard 1980; Itsweire et al. 1989) and continental shelf seas (Chant 2001; Chen et al. 1996). Indeed, they are observed to account for a significant proportion of the observed current variance in some tidally energetic coastal seas (e.g., Shearman 2005). Layers of strong shear tend to coincide with layers of strong stratification and cover extended areas with strong vertical veering but little horizontal variability in shear direction (Itsweire et al. 1989). Itsweire et al. (1989) found for Monterey Bay that the mean shear direction within the shear layer rotated at the local inertial frequency.

In coastal seas the presence of a coastline produces a barotropic response resulting in a 180° phase shift across the thermocline (Krauß 1979; Craig 1989; Rippeth et al. 2002; Simpson et al. 2002; Shearman 2005), resulting in enhanced shear and a potentially significant source of mixing in shelf seas. Estimates of the rate at which near-inertial oscillations are damped (Sherwin 1987) are suggestively consistent with TKE dissipation rates observed in the seasonal thermocline (Rippeth 2005).

In this paper we develop a simple analytical model to investigate a mechanism for the generation of periods of enhanced shear across the seasonal thermocline, which mainly result from interaction between shear itself and surface wind stress. In the first section of the paper we present observations of wind and bulk shear from the seasonally stratified northern North Sea, for a period when the water column exhibited a two-layer structure (section 2). We then derive a theory for bulk shear generation in two-layer flows (section 3), which is then applied to the observations from the northern North Sea (section 4a). In section 4b of the paper the correlation between bulk shear, wind stress, and observed mixing estimates, based on microstructure measurements, is investigated. Finally, a numerical sensitivity study is undertaken (section 5) and conclusions are drawn (section 6).

Note that for simplicity the sense of rotation is always related to the Northern Hemisphere. Thus, clockwise (counterclockwise) means anticyclonic (cyclonic) sense of rotation.

2. Observations

During the European Union (EU) funded Processes of Vertical Stratification in Shelf Seas (PROVESS) project, intense water column measurements were carried out in the northern North Sea (NNS) at 59°20′N, 1°E in water depth 110 m. At the central station A (see Fig. 1 for the location) over the period 8 September–2 November 1998, current velocities were observed using a bottom-mounted 150-kHz broadband RD Instruments acoustic Doppler current profiler (ADCP), returning 10-min averages between 11 and 87 m below the sea surface, with a bin size of 4 m (for details, see Knight et al. 2002). These observations were accompanied by conductivity–temperature–depth (CTD) casts and thermistor chain observations at adjacent locations for measurements of the stratification, as well as velocity microstructure profiles (Prandke et al. 2000), from which the rate of turbulent dissipation (ɛ) and, thus, the vertical mixing rate are estimated (Burchard et al. 2002). Meteorological parameters have been obtained from the Frigg oil rig at 59°54′N, 2°6′E (about 48 n mi northeast of the central station A, see Fig. 1). These data have been used to calculate surface stresses using bulk formulae (Kondo 1975).

Fig. 1.

Location of the PROVESS station A in the northern North Sea at 59°20′N, 1°E (marked by a star) in water depth of 110 m. The location of the Frigg oil rig, where meteorological parameters have been observed, is marked by a circle (59°54′N, 2°6′E). Contour lines and shading show the water depth; contour interval is 50 m.

Fig. 1.

Location of the PROVESS station A in the northern North Sea at 59°20′N, 1°E (marked by a star) in water depth of 110 m. The location of the Frigg oil rig, where meteorological parameters have been observed, is marked by a circle (59°54′N, 2°6′E). Contour lines and shading show the water depth; contour interval is 50 m.

Tides are predominantly semidiurnal, rotating in a clockwise sense, with M2 and S2 amplitudes of 0.20 and 0.07 m s−1, respectively (Knight et al. 2002), and the ellipticity of the M2 tide is about 1:3 (Davies et al. 1997) with the major axis oriented in the meridional direction.

A characteristic feature that Knight et al. (2002) analyzed from the PROVESS-NNS current velocity data is that near-inertial oscillations that clearly dominate the signal in the surface and the bottom mixed layer are absent in the vertically averaged current velocities, due to a 180° phase shift between the two layers. Knight et al. could partially explain this feature by applying the theory of Craig (1989) as due to the presence of an adjacent coast, which set up a barotropic pressure gradient accelerating the water column in the opposite direction of the wind-driven Ekman transport.

The observations of temperature and salinity at the PROVESS-NNS site showed a distinct three-layer structure of the water column at the beginning of the measurements in early September, with a bottom boundary layer and a surface boundary layer being separated by a 30–40-m-thick diffuse thermocline. However, the erosion of this intermediate layer by surface cooling resulted in the development of a clear two-layer structure with surface and bottom boundary layers separated by a stratified region of less than 20-m thickness that persisted until the end of the observations.

A 10-day period of the observations, between 16 October (day 288) and 26 October 1998 (day 298), is selected to be the focus of this study because the water column has a clear two-layer structure at this time (Fig. 2).

Fig. 2.

Observed profiles of potential temperature, salinity, potential density, and buoyancy frequency squared at the beginning (0703 UTC 16 Oct 1998) and one day before the end (1704 UTC 24 Oct 1998) of the investigated 10-day period.

Fig. 2.

Observed profiles of potential temperature, salinity, potential density, and buoyancy frequency squared at the beginning (0703 UTC 16 Oct 1998) and one day before the end (1704 UTC 24 Oct 1998) of the investigated 10-day period.

Figure 3 shows the observed 10-m wind, the bulk shear squared, and the shear direction. Here the bulk shear is calculated by first identifying the depth of strongest vertical stratification (typically about middepth), then averaging the observed current velocities over the resulting surface and bottom layers, and finally dividing the difference of the two velocities by half of the water depth. The shear direction is the direction of the resulting shear vector with respect to north (0°) in a clockwise-rotating sense. The period of interest is characterized by typically strong but changeable winds of up to 20 m s−1 velocity. In contrast to that, the observed bulk shear shows distinct peaks twice daily. Periods of an enhanced background bulk shear are observed during days 289 and 290 and again between days 294 and 297. In contrast, the bulk shear is small between days 291 and 293, despite the occasionally strong winds at this time.

Fig. 3.

Observations of wind and bulk shear vector in the NNS: (a) time series of wind speed and direction (sticks) and wind speed (bold line) corrected to 10-m height, (b) bulk shear squared, and (c) direction of bulk shear (dots) compared to rotation at local inertial frequency (lines). Bulk shear squared and bulk shear direction have been filtered with a box filter of 2-h length.

Fig. 3.

Observations of wind and bulk shear vector in the NNS: (a) time series of wind speed and direction (sticks) and wind speed (bold line) corrected to 10-m height, (b) bulk shear squared, and (c) direction of bulk shear (dots) compared to rotation at local inertial frequency (lines). Bulk shear squared and bulk shear direction have been filtered with a box filter of 2-h length.

It is thus concluded that the magnitude of the individual shear spikes is not directly correlated to the magnitude or direction of the wind. An analysis of the shear direction shows that it rotates in a clockwise sense at the local inertial period of 13.91 h, with occasional deviations from the inertial period (see the bottom panel in Fig. 3). This observation is in agreement with the findings of Itsweire et al. (1989).

The focus of the present study is to explain the development of the observed shear spikes as a function of wind and tide. This is achieved using a simple two-layer theory, which will be developed in the next section of the paper.

3. Theory

The theory for bulk shear dynamics is based on the one-dimensional momentum equations on the rotating earth:

 
formula

with the eastward and northward velocity components u and υ, respectively; the shear stress components normalized by density,τx and τy; the Coriolis parameter f; the gravitational acceleration g; and the surface elevation slopes, ∂xη and ∂yη, the latter two terms providing a prescribed barotropic pressure gradient forcing. Horizontal density gradients, lateral advection, and mixing and variations in water depth h are neglected as well shear generated by internal waves, assuming a distance from the nearest coast or frontal feature of at least a few internal Rossby radii.

Definition of an arbitrary intermediate depth zi with −h < zi < 0 and upper- and lower-layer thicknesses hs = −zi and hb = zi + h, respectively, leads to the definition of upper and lower layer velocity components

 
formula

and depth-mean velocity components

 
formula

such that

 
formula

Integrating the momentum equations (1) over the upper and the lower layer, respectively, and defining the surface stress as τsx = τx(0), τsy = τy(0), the interfacial stress as τix = τx(zi), τiy = τy(zi), and the bottom stress as τbx = τx(−h), τby = τy(−h), the dynamical equations for the upper and lower layer velocity components are obtained:

 
formula

and

 
formula

After defining the bulk shear vector components as

 
formula

equations for the bulk shear may be derived from (5) and (6):

 
formula

After parameterizing the interfacial shear stress by means of a quadratic friction law as

 
formula

with the interfacial drag coefficient ci, and the bulk shear squared,

 
formula

then multiplication of the first equation in (8) by 2Su and the second equation by 2Sυ and adding the two, a dynamical equation for the bulk shear squared S2 is obtained:

 
formula

with the bulk shear vector, S = (Su, Sυ) the surface shear stress vector, τs = (τsx, τsy); and the bottom shear stress vector, τb = (τbx, τby). Equation (11) shows that bulk shear is generated or reduced by the scalar product of bulk shear and the weighted sum of the surface and bottom stress (first term rhs), P(S2) = Ps(S2) + Pb(S2), and dissipated by interfacial friction (second term rhs), Di(S2). The maximum bulk shear production rate for given absolute values of the surface and bottom shear stress is obtained for a perfect alignment of the three vectors: surface shear stress, bottom shear stress, and bulk shear. Although the barotropic pressure gradient does not explicitly appear in Eq. (11), it still has an indirect effect by changing the bottom-layer velocity and thus the bottom friction. With this, a potential 180° phase shift, as suggested by the Craig (1989) condition, is implicitly included in the present theory.

To highlight the role of tidal forcing, a parameterization for the bottom stress is used:

 
formula

where

 
formula

has been used as derived from (4). In (12), cd is the bottom drag coefficient with respect to the bottom-layer thickness hb.

With (12), the dynamical Eq. (11) for the bulk shear squared can be reformulated:

 
formula

with the depth-mean velocity vector, Vm = (um, υm), and the lower-layer velocity vector, Vb = (ub, υb) In (14) the production of bulk shear is now formulated as the scalar product of bulk shear with the weighted sum of the surface shear stress and the depth-mean velocity, Ps(S2) and Pm(S2), respectively. It should be noted that both Ps(S2) and Pm(S2) may increase or decrease S2. The bed stress appears here as an additional sink of bulk shear, Db(S2), such that Pm(S2) −Db(S2) = Pb(S2).

Although the direct effect of the earth’s rotation on bulk shear squared is eliminated in Eqs. (11) and (14), the bulk shear vector S is subject to earth rotation—see Eq. (8). In the Northern Hemisphere, S therefore has the tendency to rotate in a clockwise direction with period 2π/f (the local inertial period).

The formulation of the bottom stress by means of the barotropic velocity, (um, υm), see Eq. (12), gives another interesting interpretation of the bulk shear equations (8):

 
formula

With this, the bulk shear can be interpreted as a pendulum in a rotating reference frame, forced by wind and tide (first two terms rhs) and damped by interfacial and bottom friction (last term rhs).

For a circular tide with frequency ω and zero interfacial friction (ci = 0) and zero surface stress (τs = 0) the bulk shear equations (15) can be simplified to

 
formula

with

 
formula

where At denotes the tidal velocity amplitude. For circular tides, |Vb| can be assumed to be a constant with |Vb| ≈ At. For ω > 0, tides are rotating in a counterclockwise sense, and for ω < 0 rotation is in a clockwise sense.

An analytical solution to (16) is found by defining the complex shear Sc = Su + iSυ with i = −1, and transforming (16) into a dynamical equation for Sc by multiplying the second equation in (16) with i and adding the two equations:

 
formula

With the solution of the resulting equation,

 
formula

the shear amplitude is analytically obtained as the absolute value of A:

 
formula

Thus, for clockwise tides, shear is strongly enhanced while it is suppressed by counterclockwise tides. This result is consistent with theoretical analysis (Prandle 1982), which predicts that tides rotating in a clockwise sense create considerably more shear than tides rotating in a counterclockwise sense.

With this, shear rotating at tidal frequency and amplitude is the background state for zero wind stress. Wind events will generate shear rotating in a clockwise sense at inertial frequency, overlaying this background rotation.

4. Analysis of field data

a. Impact of wind stress on shear spikes

A comparison of wind stress with bulk shear does not reveal any clear relationship between these two quantities: see the discussion in section 2. For example, peaks in the wind stress at d = 293.0 and d = 296.8 do not coincide with maxima in bulk shear and bulk shear maxima at d = 290.0 and d = 295.7 and are not associated with distinct wind stress maxima, see Fig. 4.

Fig. 4.

Bulk properties at the station NNS during days 288–298 (16–26 Oct) in 1998: (a) absolute value of surface wind stress, |τs|; (b) bulk shear squared, S2; and (c) time derivative of the bulk shear squared, ∂tS2, and production of bulk shear squared, P (S2), according to Eq. (11). Both ∂tS2 and P (S2) were first computed from instantaneous ADCP data and filtered afterward. (d) Direction of surface shear stress and bulk shear. Circles denote time of alignment between bulk shear and wind direction for S2 > 2 × 10−5s−2. Both S2 and P (S2) have been filtered with a box filter of 2-h length; ∂tS2 has been filtered with a box filter of 4-h length.

Fig. 4.

Bulk properties at the station NNS during days 288–298 (16–26 Oct) in 1998: (a) absolute value of surface wind stress, |τs|; (b) bulk shear squared, S2; and (c) time derivative of the bulk shear squared, ∂tS2, and production of bulk shear squared, P (S2), according to Eq. (11). Both ∂tS2 and P (S2) were first computed from instantaneous ADCP data and filtered afterward. (d) Direction of surface shear stress and bulk shear. Circles denote time of alignment between bulk shear and wind direction for S2 > 2 × 10−5s−2. Both S2 and P (S2) have been filtered with a box filter of 2-h length; ∂tS2 has been filtered with a box filter of 4-h length.

To reproduce the bulk shear production term in Eq. (11), P(S2), the bed stress τb is estimated from the current measurements. Using a bed roughness length of zb0 = 0.001 m for this site (Bolding et al. 2002), a bed friction coefficient, cd = 0.0015, is derived from the law of the wall, based on the average bottom-layer thickness of 55 m. This leads to a bed stress estimate significantly smaller than the wind stress for the period of interest, such that the wind stress predominantly drives the bulk shear (Fig. 5a).

Fig. 5.

The role of bed stress for the generation of bulk shear squared: (a) bed stress τb calculated from ub and ub; (b) bed stress direction calculated from ub and υb (line), overlaid with bulk shear direction (dots); and (c) surface stress and bed stress contributions to the total generation of bulk shear squared: Ps, Pb, respectively. All time series have been filtered with a box filter of 2-h length.

Fig. 5.

The role of bed stress for the generation of bulk shear squared: (a) bed stress τb calculated from ub and ub; (b) bed stress direction calculated from ub and υb (line), overlaid with bulk shear direction (dots); and (c) surface stress and bed stress contributions to the total generation of bulk shear squared: Ps, Pb, respectively. All time series have been filtered with a box filter of 2-h length.

With this bottom drag coefficient, the curves for the time derivative of the observed bulk shear squared, ∂t S2, and the production of bulk shear squared, P(S2), agree well (Fig. 4c), thus providing clear validation of Eq. (11) as a description of the dynamics of this two-layer flow.

To maximize P(S2), the product of the surface stress and the bulk shear must be large with the two vectors well aligned. In Fig. 4 all alignments of the bulk shear and wind direction preceding a shear spike with S2 > 2 × 10−5 s−2 are indicated with a circle. Nine such shear spikes are identified for the 10-day period under consideration. The instants of alignment coincide with distinct maxima of P(S2) and, thus, with the maxima in ∂tS2. Therefore, these alignments also mark the steepest ascent to maxima of S2, typically occurring a quarter of an inertial period before the bulk shear maxima, which coincide with zero values of P(S2). The two largest shear spikes occur at days 290.0 and 295.7 due to the coincidence of high shear stresses >0.4 N m−2 and shear squared values >2 × 10−5 s−2 with the alignment of wind and shear directions. In contrast the wind stress maximum on day 293.0 coincided with a small shear squared value, resulting in only a moderate subsequent bulk shear peak.

The role of the bed stress in the dynamics of bulk shear is shown in Fig. 5. From the second panel in the figure it is obvious that shear and bed stress are generally out of phase by ∼180°, which is pronounced during days 295 and 296. The explanation for this is given by the 180° phase shift between the bottom and surface layers. With relatively weak tides this results in a positive east component of bulk shear coinciding with negative near-bed velocity and thus negative bed stress in the eastern direction, and vice versa. Consequently, the bed stress contribution to the bulk shear is predominantly negative here, specifically during days 295 and 296 (Fig. 5c). For example, on day 293.0 the bed stress peaks at a phase shift of 180° to the bulk shear, thus reducing the impact of the strongest wind stress peak on the generation of bulk shear.

To discriminate between the barotropic (mainly tidal) and bed stress contributions to the bulk shear, the mean flow production, Pm(S2), and the bed-friction-related dissipation, Db(S2), are calculated according to Eq. (14), Fig. 6a. Clearly the tidal contribution may be positive during periods of up to 2 days (e.g., days 288 and 289). This is because the tide is rotating in a clockwise sense and is in phase with the shear over longer periods (Fig. 6b). Part of the positive tidal contribution is always counterbalanced by the bed friction.

Fig. 6.

Barotropic contribution to bulk shear: (a) contributions of barotropic flow Pm and bed dissipation Db to bulk shear loss due to bed stress and (b) depth-mean flow direction (line) overlaid with bulk shear direction (dots). All time series filtered with a box filter of 2-h length.

Fig. 6.

Barotropic contribution to bulk shear: (a) contributions of barotropic flow Pm and bed dissipation Db to bulk shear loss due to bed stress and (b) depth-mean flow direction (line) overlaid with bulk shear direction (dots). All time series filtered with a box filter of 2-h length.

b. Impact of shear spikes on isopycnal mixing

A major consequence of increased bulk shear may be shear instability (as the gradient Richardson number Ri is reduced) resulting in increased diapycnal mixing. The reduction of Ri is however directly related to local shear across the thermocline, and not to the bulk shear.

To connect increased bulk shear with enhanced diapycnal mixing across the thermocline, bulk shear is compared to shear across the thermocline (the position of which is derived from thermistor chain data), Fig. 7. Naturally, the local shear, Si, calculated as the shear across the two 4-m ADCP bins located at the maximum stratification, is much larger than the bulk shear, but high local shears seem to be largely induced by high bulk shears.

Fig. 7.

Time series of bulk shear squared S2 compared to the interfacial shear squared Si2 both filtered with a box filter of 2-h length. Note the factor of 50 difference in the scales.

Fig. 7.

Time series of bulk shear squared S2 compared to the interfacial shear squared Si2 both filtered with a box filter of 2-h length. Note the factor of 50 difference in the scales.

With the aid of the local shear and assumptions about the interfacial eddy viscosity, Km, the interfacial drag coefficient may be roughly estimated:

 
formula

Taking a typical value of Km = 10−5 m2 s−1, Table 1, the interfacial drag coefficient resulted in ci ≈ 4 × 10−6. A comparison between the bulk shear loss due to interfacial friction, Di(S2), calculated with this value of ci, with the bed friction loss, Db(S2), shows that interfacial friction is dynamically negligible here.

Table 1.

Average values of bulk shear cubed, S3; surface wind stress to the power of 3/2, (τs)3/2; dissipation rate, ɛ; and eddy diffusivity, Kρ, for two periods of comparable wind strength and different bulk shear. The latter two quantities have been derived from MST microstructure profiler data only. The 95% confidence intervals, calculated using the bootstrap method, are given in parentheses.

Average values of bulk shear cubed, S3; surface wind stress to the power of 3/2, (τs)3/2; dissipation rate, ɛ; and eddy diffusivity, Kρ, for two periods of comparable wind strength and different bulk shear. The latter two quantities have been derived from MST microstructure profiler data only. The 95% confidence intervals, calculated using the bootstrap method, are given in parentheses.
Average values of bulk shear cubed, S3; surface wind stress to the power of 3/2, (τs)3/2; dissipation rate, ɛ; and eddy diffusivity, Kρ, for two periods of comparable wind strength and different bulk shear. The latter two quantities have been derived from MST microstructure profiler data only. The 95% confidence intervals, calculated using the bootstrap method, are given in parentheses.

To investigate the relationship between the generation of turbulence and consequent diapycnal mixing, and the surface wind stress and bulk shear, time series of bulk shear and wind stress are shown together with estimates for the dissipation rate and eddy diffusivity from microstructure data, Fig. 8. Each microstructure data point represents an average of all data within the thermocline with N2 > 5 × 10−4 s−2 (where N is the buoyancy frequency) averaged over one burst (which corresponds to approximately five profiles taken over a period of about 30 min). The dissipation rate and stratification data are then combined to obtain estimates for the eddy diffusivity Kρ, using the local equilibrium assumption for the turbulent kinetic energy equation, that is, the assumption of a balance between shear production, buoyancy production, and dissipation, resulting in

 
formula

with the mixing efficiency Γ = −B/ɛ, where B is the buoyancy production (negative for stable stratification). Osborn (1980) estimated Γ = 0.2 to be an upper limit for the mixing efficiency in stably stratified flow, a value used here for estimating Kρ using (22).

Fig. 8.

Comparison between (a),(b) bulk shear and (a),(c) dissipation rate and bulk shear and (b),(d) eddy diffusivity averaged over the thermocline. Each data point represents averages over one burst sample. The observations for dissipation rates and eddy diffusivity are from the MST (open circles) and the FLY (bullets) velocity microstructure profilers. See Prandke et al. (2000) for the MST details and Rippeth et al. (2003) for the FLY details. The horizontal bold lines indicate the averages of dissipation rate and eddy diffusivity calculated from the MST profiles for the two 2.5-day evaluation periods compared in Table 1.

Fig. 8.

Comparison between (a),(b) bulk shear and (a),(c) dissipation rate and bulk shear and (b),(d) eddy diffusivity averaged over the thermocline. Each data point represents averages over one burst sample. The observations for dissipation rates and eddy diffusivity are from the MST (open circles) and the FLY (bullets) velocity microstructure profilers. See Prandke et al. (2000) for the MST details and Rippeth et al. (2003) for the FLY details. The horizontal bold lines indicate the averages of dissipation rate and eddy diffusivity calculated from the MST profiles for the two 2.5-day evaluation periods compared in Table 1.

Although there is considerable scatter (Fig. 8), the average values of dissipation rate and eddy diffusivity are clearly higher during the high shear phase, day 294.0, than during the low shear phase after day 290.0.

A more rigorous comparison is achieved by comparing two 2.5-day periods of data: days 290.0–292.5 and days 294.0–296.5. The wind strength is similar during both periods; however, during the first period there is relatively small bulk shear and during the second period there is relatively large bulk shear. Since near-surface dissipation rates scale with the cube of the surface friction velocity (according to the law of the wall) and the dissipation of bulk shear squared is proportional to the cube of the bulk shear, averages of S3 and (τs)3/2 are calculated for comparison (Table 1). For both periods, the average dissipation rate and eddy diffusivity are calculated. The results show that the eddy diffusivity correlates more strongly with the bulk shear than with the wind stress, with a threefold increase in S3 during the second period when compared to the first, coinciding with a fivefold increase in the value of Kρ, despite the fact that the average (τs)3/2 is smaller during the second period (Fig. 8). The average dissipation rate also increases by about a factor of 2.5 between the first and second periods. The bootstrap method has been used to show the significance of these results (Table 1). The comparison shows clearly that bulk shear has a stronger impact on diapycnal mixing than surface wind stress.

5. Sensitivity studies

In section 4a we have shown that the bulk shear is highly sensitive to the duration of the wind events and the phasing of the wind and bulk shear directions.

To better understand the impact of the wind on the shear spike generation, a systematic sensitivity analysis, using the simple two-layer model, is carried out. The numerical model is a discretization of Eqs. (5) and (6), with the surface slopes calculated in such a way that a prescribed (tidal) depth-mean flow results. This is obtained numerically by adding (for each time step) a constant value to the calculated velocity vectors in the upper and lower layer in such a way that the prescribed depth-mean velocity vector is obtained (see Burchard 1999). The interfacial drag coefficient has been set to ci = 10−5 and the bottom drag coefficient to cd = 1.5 × 10−3. The time step is chosen such that each tidal period is resolved with 1000 time steps, short enough to exclude significant discretization errors.

For comparison with the field data, the water depth (110 m) and latitude (59°20′N) of station A are used, with the interface between the two layers set at mid depth. A tide with a tidal amplitude At = 0.3 m s−1 for a rectilinear tide and of At = 0.3/2 m s−1 for circular tides has been prescribed, giving the same average tidal kinetic energy for both cases. All model simulations have been integrated during 50 periods of the M2 tide (period T = 44 714 s), of which the last 10 periods are then analyzed. Periodic wind events from the west have been prescribed by means of a Gaussian wind evolution with a maximum wind speed of Wmax = 20 m s−1:

 
formula

with the duration of the wind event, d, and the wind peak instants ti/T = 5.0, 15.0, 25.0, 35.0, and 45.0.

From these given wind speeds, surface stresses have been calculated using a quadratic drag law with a drag coefficient of 10−3.

A total of 24 simulations have been carried out, with variations in

  • tidal ellipticity (clockwise, circular; rectilinear, east–west; counterclockwise, circular),

  • initial phase of tide relative to wind (0°, 90°, 180°, and 270°), and

  • duration of wind (one tidal period, d = T; three tidal periods, d = 3T).

To investigate the impact of the phase of the wind in detail, the model is first run with a clockwise circular tidal forcing with a wind event duration of d = T and initial tidal phase shifts of 270° and 90° (Figs. 9 and 10). For the 270° initial phase shift, the bulk shear squared peaks at a value of 1.6 × 10−5 s−2, while for the 90° initial phase shift, bulk shear is close to a minimum value during the wind peak with values < 5 × 10−6 s−2. The strong shear peak in the former case is because the wind direction (constantly 90°) and shear direction are almost aligned during the maximum wind stress, leading to a large peak in shear production Ps at t/T = 5. In contrast for the second period, the angle between the wind and shear direction remains close to 90° during the wind peak, resulting in a small values of Ps. An interesting feature is that the shear remains in a southerly direction throughout the wind event as a result of the westerly wind blocking the southerly, clockwise-rotating shear from further rotating toward the west. For both initial phase shifts, shear production due to depth-mean velocity, Pm, is small, slightly positive when the angle between shear and the depth-mean velocity is <90°, and slightly negative when this angle is >90°. In both cases, dissipation due to bed friction, Db, is dominated by, and largely proportional to, S2, with the interfacial friction, Di, negligible.

Fig. 9.

Two-layer model sensitivity study with (top) a clockwise tide and a Gaussian eastward wind burst centered at t/T = 5 with 20 m s−1 wind velocity and duration of one tidal period. Shown are (middle left) an initial tidal phase of 270° for the bulk shear squared, S2; (middle right) the production of bulk shear squared due to wind stress, Ps, and due to depth-mean current, Pm; (bottom left) the phase angle of the shear and the tide; and (bottom right) the dissipation of bulk shear squared due to interfacial and bottom friction, −Di, and −Db, respectively. Results are shown for a periodical steady state.

Fig. 9.

Two-layer model sensitivity study with (top) a clockwise tide and a Gaussian eastward wind burst centered at t/T = 5 with 20 m s−1 wind velocity and duration of one tidal period. Shown are (middle left) an initial tidal phase of 270° for the bulk shear squared, S2; (middle right) the production of bulk shear squared due to wind stress, Ps, and due to depth-mean current, Pm; (bottom left) the phase angle of the shear and the tide; and (bottom right) the dissipation of bulk shear squared due to interfacial and bottom friction, −Di, and −Db, respectively. Results are shown for a periodical steady state.

Fig. 10.

As in Fig. 9, but for an initial tidal phase of 90°.

Fig. 10.

As in Fig. 9, but for an initial tidal phase of 90°.

Figures 11 and 12 show time series of bulk shear squared, S2 (thin lines), and its shear production, Ps (bold lines), for all 24 simulations outlined above. A number of general observations can be made here:

  • For short and intense wind events the phasing with the bulk shear direction is critical in determining their impact on the intensity of the shear. Short wind peaks may either strongly enhance, or significantly reduce, the bulk shear. The latter may be compared to the relatively small impact of the wind peaks during the observational period days 293.0 and 297.6, Fig. 4.

  • For wind events of a duration significantly longer than the inertial period, the relative phase to the shear direction does not play such an important role. Longer wind events result in a number of subsequent shear peaks, each occurring approximately one inertial period after the preceding peak. This may be compared to the series of shear peaks occurring during the extended wind event between days 288.5 and 290.3 (Fig. 4).

  • For counterclockwise tides, the effect of wind events on the bulk shear is significantly weaker than for clockwise tides. Inserting the parameters used for the present study into the analytical solution (20) for the shear amplitude (neglecting wind stress and interfacial friction), |A|2 = 2.1 × 10−6 s−2 results for the clockwise tide and |A|2 = 7.0 × 10−9 s−2 results for the counterclockwise tide. These values are almost identical to those shear square values that result for the clockwise and counterclockwise numerical experiments with the extended wind event, when the effect of the wind has ceased (see Fig. 12).

  • The angle between wind and shear directions thus plays a far less important role for counterclockwise-rotating tides than it does for clockwise-rotating tides.

Fig. 11.

Two-layer model sensitivity study with (top) a Gaussian eastward wind burst centered at t/T = 5 with 20 m s−1 wind velocity and duration of one tidal period. Shown are the shear squared, S2 (thin lines), and the wind forcing, Ps (bold lines). Simulations were carried out periodically at a period of 10 tidal periods with the wind burst repeating each 10th tide. Results are shown for aperiodical steady state. The character of the tide with a tidal velocity amplitude of 0.3 m s−1 was varied between (left) clockwise, (middle) rectilinear, and (right) counterclockwise. The initial tidal phase was varied by steps of 90° (rows).

Fig. 11.

Two-layer model sensitivity study with (top) a Gaussian eastward wind burst centered at t/T = 5 with 20 m s−1 wind velocity and duration of one tidal period. Shown are the shear squared, S2 (thin lines), and the wind forcing, Ps (bold lines). Simulations were carried out periodically at a period of 10 tidal periods with the wind burst repeating each 10th tide. Results are shown for aperiodical steady state. The character of the tide with a tidal velocity amplitude of 0.3 m s−1 was varied between (left) clockwise, (middle) rectilinear, and (right) counterclockwise. The initial tidal phase was varied by steps of 90° (rows).

Fig. 12.

As in Fig. 11, but with a wind burst duration of three tidal periods.

Fig. 12.

As in Fig. 11, but with a wind burst duration of three tidal periods.

6. Discussion and conclusions

Recent finescale observations of shear and stratification in temperate shelf sea thermoclines show that they are of marginal stability, suggesting that episodes of enhanced shear could potentially lead to shear instability and diapcynal mixing (van Haren et al. 1999; MacKinnon and Gregg 2005; Rippeth et al. 2005; Rippeth 2005). Here we present observations from the northern North Sea, which show that the bulk shear vector rotates in a clockwise direction at the local inertial frequency with periods of enhanced bulk shear taking the form of shear spikes that are separated by approximately one inertial period and occur in bursts lasting several days. Similar observations of the bulk shear are reported for the seasonally stratified western Irish Sea (Rippeth et al. 2008) and Celtic Sea (Palmer et al. 2008).

A two-layer analytical model has been derived to explain the observed characteristics of the bulk shear. The model results show that the shear “spikes” arise because of the alignment of the surface wind stress, bulk shear, and bed stress vectors, thus indicating the sensitivity of the system to both phase and direction of the wind. The results also highlight the important role of the sense of rotation of the tide in determining the level of diapcynal shear and mixing, thus confirming the theoretical result of Prandle (1982) and the observations of Simpson and Tinker (2009) in the bottom boundary layer.

The results will present particular problems for numerical models due to the requirement of the alignment of the “local” wind vector with the bulk shear direction, the phase of which will have been set by some previous event. These problems are compounded by the further result that short wind events have the potential to generate stronger shear spikes than longer wind events, which typically generate a sequence of shear spikes.

Application of the model to other areas requires velocity and density profiles together with local wind conditions. Because the interfacial stresses are negligible compared to typical bottom and surface stresses, the theory can easily be extended to three layers, in order to accommodate a diffuse thermocline. The model does not, however, include shear associated with long internal waves, which may form an important source of shear in some areas.

Acknowledgments

The work of Hans Burchard has been supported by a Kirby Laing Fellowship at the School of Ocean Sciences of the University of Wales in Bangor and the EU-funded European Coastal-Shelf Sea Operational Monitoring and Forecasting System project (ECOOP, Contract 36355). Additional work was carried out under NERC Grant NE/F002432. All field data have been obtained during the EU-funded Processes of Vertical Stratification in Shelf Seas (PROVESS) project. The authors are grateful to Adolf Stips (Ispra, Italy) for providing the MST microstructure profiler data. Comments on the manuscript by Hans Ulrich Lass (Warnemünde, Germany) and two anonymous referees were highly appreciated.

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Footnotes

* Visiting scientist at School of Ocean Sciences, College of Natural Sciences, Bangor University, Anglesey, United Kingdom

Corresponding author address: Hans Burchard, Leibniz Institute for Baltic Sea Research Warnemünde, Seestraße 15, D-18119 Rostock, Germany. Email: hans.burchard@io-warnemuende.de