A study of the bottom boundary layer over the Eastward Scarp of the Bermuda Rise

Velocity and temperature measurements from the bottom boundary layer (BBL) over the Eastward Scarp of the Bermuda Rise in water depth of 4620 m show little variability over an 8-month period. The free-stream flow 62 m above the bottom was south-southeasterly following the isobaths in the region with an average speed of 22 cm/s. The current vector in the BBL rotated an average of 5° in a counterclockwise sense between 62 and 0.8 m above the bottom. The thickness of the BBL was ∼40 m and the average magnitude of the bottom stress was ∼0.7 dynes/cm2. Mean speed profiles, height of the BBL, and the magnitude of the bottom stress predicted by a model compare favorably with the observations, but the model predicts a rotation of the current vector between 62 and 0.8 m more than twice that measured. The time-dependent nature of the flow field is also reproduced by the model. The Bermuda Rise data and speed profile measurements at the base of the Scotian Rise show that the M2 clockwise polarized tide is damped more than the mean current as the bottom is approached in the BBL. This phenomenon is reproduced by the model and can be explained by differing effective Ekman layer thicknesses associated with tidal and steady components of the flow.


INTRODUCTION
This study was undertaken for two purposes.The first was to examine an 8-month near-bottom velocity and temperature record from the Eastward Scarp of the Bermuda Rise at 32ø52.5'N, 57ø29.0'W in water depth of 4620 m, and to compare these observations with a numerical bottom boundary layer (BBL) model described by Weatherly et al. [1980] and Weatherly and Martin [1978].Simulations of the BBL with the parameters at this site are used to explore the effects of a sloping bottom, density stratification, tidal oscillations, and unsteady free-stream flow on the structure and dynamics of the boundary layer.The second purpose was to extend the study begun in Weatherly et al. [1980] of the combined effects of tidal and mean currents on the structure of the BBL.It has been observed in flows composed of a mean flow and a small oscillatory flow that these components are damped unequally as the bottom is approached in the B BL [Weatherly and Van Leer, 1977;Weatherly and Wimbush, 1980].The model reproduces this feature, and a mechanism is presented.

REVIEW OF THEORY AND LITERATURE
The equations used in this study of the B BL are Here U and V are the x and y components of velocity relative to the geostrophic velocity components, U• and V•, in the interior above the boundary layer, and z is the vertical coordinate, positive upward from the bottom; a' is the bottom slope and a is the isotherm slope relative to the bottom, both assumed to be uniform and small and tilted in the x direction only; f is the Coriolis parameter and X =/3 g is a buoyancy parameter, where /3 is the coefficient of thermal expansion for seawater; 0 is the potential tempera- The friction velocity defined by ( 6) is an important scaling parameter for turbulent boundary layers.For the BBL it is a function of i7• I, oe, z0, and No. Weatherly and Martin [1978] argue that for values of No typical of the ocean, the dependence of u, on No is very weak.Thus, using similarity expressions given in Csanady [1967], Blackadar and Tennekes [1968], and values of surface Rossby number typical of the deep ocean, u, can be estimated as 3-5% of the geostrophic speed.Further discussion of the dependence of u, on stratification may be found in Bird [1981].
The total veering angle, a0, is also a function of surface Rossby number [Csanady, 1967] and No [Bird, 1981], with values of 10 ø to 20 ø expected for surface Rossby numbers and values of No typical of the deep ocean.In the lowest part of the BBL, the logarithmic layer, the current vector is not expected to turn appreciably with depth because Coriolis effects are negligible there.Kundu [ 1976] reported an Ekman veering angle of 6 ø between 5 and 20 m above the ocean bottom on the Oregon shelf, well above the logarithmic layer (--•3 m) thick.The current veered counterclockwise (CCW) as the bottom was approached, as expected in the northern hemisphere.Weatherly [1972], in a study of the BBL beneath the Florida Current, observed a mean veering of The velocity components and speeds are in units of cm/s, direction is degrees from true north, and the temperature is the in situ temperature in øC.
10øCCW in the lowest 3 m and no mean veering above the logarithmic layer over a period of 6 days.Veering above the logarithmic layer was observed only when the current was very strong (---30 cm/s).It was suggested that the variability in direction of the current due to the presence of a strong diurnal tide prevented the Ekman layer from forming above the logarithmic layer.In a study of the BBL in the eastern tropical North Pacific, a mean veering of 11 ø CCW was observed between 50 and 8 m above the bottom, but between 8 and 4 m, the current vector turned 6 ø in a clockwise sense [Hayes, 1980].Hayes attributes this clockwise rotation to small-scale topography near the mooring; the discrepancy is too large to be explained by an instrumental effect.Between the two current meters at 6.9 and 0.8 m, a mean veering of 1.5 ø was observed over the entire record, but it was in a clockwise sense.The rms accuracy of the direction for a VACM is estimated to be 2.0 ø [Bryden, 1976], so that this small veering may have been an instrumental effect.
Furthermore, the vane of the current meter nearest to the ocean floor was between the legs of the tripod, and wakes shed by the legs may have offset its orientation.Near the end, the veering became slightly positive (CCW).Within the accuracy of the instrument, however, there was no significant veering in the lowest 6.9 m of the BBL, consistent with theory.
A mixed layer depth typically greater than 12 m but less than 62 m may be inferred from the temperature data.Inspection of a portion of the temperature record (Figure 3) show s such good coherence between the lowest three sensors that they must have been in the bottom mixed layer.
The differences in temperature between the lowest three sensors were nearly constant in time, while the difference between at 62 and 0.8 m was highly variable (Figure 4).Two 5-day sections of the data were chosen for comparison with model simulations.Five days is long enough to make direct comparisons between model and data but not so long as to be prohibitively expensive in computer time.The two 5-day intervals chosen are indicated on a stick plot of the entire time series in Figure 6.The first section will be analyzed and discussed here; analysis of the second section led to similar conclusions [Bird, 1981]  nature of the time series, the velocity measured at 62 m above the bottom was used as input for the model freestream velocity.In all runs, the free-stream velocity was linearly increased to its starting value over 1 day to lessen the inertial oscillations associated with initializing the model.Table 3a summarizes speed and direction relative to Vz at the four heights closest to the current meter positions for the three steady model simulations and for the 5-day averaged data.In the a = 0, the bottom parallel isopycnal case, the speed at every level in the BBL is higher than the observed speed.In the a • 0 simulations, the speeds at all levels in the B BL are reduced in comparison to the a = 0 case.The profile for the observed speed averaged over the 5oday period (Figure 10) was drawn assuming a BBL thickness of 40 m as indicated by calculations using equations ( 5) and (7).
Hodographs using the mean observed and the simulated velocity components for the parallel and sloping isopycnal cases are depicted in Figure 11.A coordinate system was chosen such that the interior velocity has no u component for comparison to the model.The angle that the hodograph makes in the lowest part of the BBL represents the total veering angle, or total rotation of the current vector with depth.The veering predicted by the model is greater than that observed.The error in direction for the VACM is 2.0 ø [Bryden, 1976], so that the total error in the veering angle between any two current meters is at most 4.0 ø .Thus, the discrepancy between the data and the model results is significant.Table 3b summarizes        In the model the water column is initially linearly stratified.In the a = 0 situation, as time progresses, the bottom layers become well mixed with a temperature equal to the average of the initial temperature at the top of the mixed layer and that at the bottom.Within 2-3 days a steady state is reached in the model with respect to all of the variables.However, a nonzero slope in the heat equation prohibits a steady temperature field from being attained.The model is formulated such that when the geostrophic current flows parallel to the isobaths with higher terrain to its right, the bottom slope is positive.As the bottom is approached in the BBL, the current vector rotates counterclockwise, giving the flow near the bottom a downslope component.If the isopycnal surfaces are level, the flow transports warmer water down the slope and the BBL continues to warm with time.After 1« days after a slope of 0.01 is introduced, the bottom mixed layer is 13 m thicker and 0.005øC warmer than in the a = 0 case (Table 4).The free-stream velocity was increased linearly from rest to 18.92 cm/s from day 0 to day 1.A slope of 0.01 was increased linearly from t+28%• 0 between day 2 and day 2.5.In the model h is resolved to •,-22%;.To reproduce some of the time-dependent features of the data, the strongest tidal constituent was included in the model.A steady geostrophic current was increased linearly from rest to the mean value as before.Then, a rotary tidal current was added to the geostrophic velocity field.From rotary spectral analysis, the dominant tidal constituent at this location was the M2 clockwise polarized component, with an amplitude of 1.35 cm/s and a frequency of 1.4 x 10 -4 s -• (Table 2): Vt' = Vt -amp sin(at),

Ut' = amp cos (•ot)
where Vt is the 5-day average speed at 62 m (18.9 cm/s, Table 3a) and where amp is the tidal amplitude (1.35 cm/s Table 2) and •o is the tidal frequency (1.4 x 10 -4 s-l).In this simulation, isopycnal slope relative to the bottom slope was set to zero in order to examine only the effects of the tide.
The oscillatory component of the flow was small relative to the mean flow at all levels, and the effects of the tide on the flow were small (Figure 12).The magnitude of the bottom stress averaged over a tidal cycle was slightly greater for the combined flow than for the steady flow because of increased dissipation by the tidal flow.The bottom stress due to the M2 clockwise polarized tide alone was 0.0038 dynes/cm 2 in magnitude, that due to the mean flow was 0.4689 dynes/cm 2, and that of the combined flow, averaged over a tidal period, was 0.4708 dynes/cm 2. In this case, the model shows that the tidal amplitude is too small to alter the flow parameters significantly.
To test fully the agreement between model and the data, measured values of the interior velocity can be used as input for the model in place of a steady geostrophic velocity or a combined steady tidal current.Although the VACM recorded an average velocity every 7.5 min, the study of Weatherly and Wimbush [1980] suggest that an appropriate Reynolds averaging interval for the BBL should be several hours.Thus, the 2-hour averaged velocity time series at 62 m was used as the interior velocity for the model, linearly interpolated into the 20-min time intervals required by the model.In the a 9 0 simulations, the basic state was taken to be the model output at the beginning of day 3 (Table 4); after day 3 the B BL thickness was too great.
The time series predicted by the model for speed and direction in the a = 0 case are plotted in Figures 7b and 8b, and many observed features were reproduced by the simulation.For example, the observed speed at 12 m is higher than at 62 m at times 2.25-2.50days and at 5.5 days, and this feature is reproduced in the simulation.Note that the predicted and observed records at 62 m should and do agree since the model B BL thickness is less than 62 m.The fluctuations of the temperature time series were not reproduced by the model.Between 2.5 and 3.5 days the temperature decreased at all levels by 0.02 ø C (Figure 9a).In the a = 0 simulations, the temperature remained constant in time.In the a = 0.01 case, there was a decrease in temperature in the middle portion of the 5-day period, but not of the same magnitude as that observed.The cooling occurred during a period of relatively slow current speed and small veering angle, indicating that the mechanism that brings warmer water downslope was inhibited at that time due to a slowing of the free-stream flow.
The model temperature field responds to cross-stream advection but downstream advection is not modeled.The interior temperature fluctuations could be reproduced by using the 62-m temperature as the interior temperature.The BBL temperature field would fluctuate with the interior temperature.The observations indicate that the physics involved are more complex; for example, in Figure 9a, between 5.5 and 6.0 days, a drop of 0.2øC occurred in the temperature records at 0.8, 6.9, and 12 m, while a warming was felt at 62 m.Events of this nature cannot be reproduced by a one-dimensional model of the BBL.

DAMPING OF FLOW IN THE BBL
The Bermuda Rise measurements and output from the numerical model indicate that as the bottom is approached the semidiurnal tidal current is more damped in the BBL than the mean flow (Table 5).Unequal damping of tidal and steady flows has been observed in other BBL data sets [Weatherly and Van Leer, 1977;Weatherly and Wimbush, 1980].This phenomenon is particularly evident in speed profile measurements made at the base of the ScotJan Rise (40ø06'N,62ø29'W) in water depth of 4900 m.Savonius rotors located at heights of 0.7, 2.0, 3.9, 8.6, and 28.8 m above the bottom recorded speeds for 9 days in September 1979.Fiveminute averages are plotted in Figure 13.Note that the flow exhibits turbulent behavior near the bottom becoming less turbulent higher in the BBL, implying a turbulent Ekman layer and justifying the use of a turbulence closure model.The discrepancy in veering angle between the data and the model is significant and cannot be attributed solely to instrumental error.One possible explanation for a reduced veering in the data lies in the topography.Furrows were found upstream of the mooring although none were found in the immediate vicinity [McCave et al., 1982].Brown [1970] solved the equation of motion to obtain helical rolls as finite perturbations on a mean large-scale flow in a boundary layer.In these solutions, the velocity profiles were altered so that the veering angle was reduced; velocity profiles made in the atmospheric boundary layer beneath parallel cloud lines give similarly reduced veering angles.Also, helical rolls result in reversed veering in the lower part of the boundary layer [Brown, 1980] and reverse (but not significant) veering is seen between 6.9 and 0.8 m in Figure 6.Furrows can set

Fig. 1 .
Fig. 1.Topographic map of the Eastward Scarp of the Bermuda Rise [from Lonsdale, 1978].Site of observations is indicated by cross.Depths are in meters.

Fig. 2 .
Fig. 2. Progressive vector diagrams of the Bermuda Rise data.Each point represents a 1-day average; cross denotes the beginning of a monthly interval.The dashed line represents the ørientation of the isobaths at the mooring site.

Fig. 4 .
Fig. 4. Temperature time series at 6.9 m, 12 m, and 62 m relative to the temperatures measured at 0.8 m above the bottom.

Fig. 5 .
Fig. 5. (a) Starting from lower left and proceeding clockwise: autospectrum of temperature 0.8 m, autospectrum of temperature at 12 m, coherency squared of cross spectra, and phase of cross spectra.For frequencies <0.01 cph smoothing over three band widths and for frequencies >0.01 cph smoothing over 20 band widths.For positive phase higher record leads lower record.Phase is drawn only if coherency squared •0.5.(b) As in Figure 5a except temperature time series at 12 m and 62 m.

Fig. 6 .
Fig. 6.Stick plot of daily averaged Bermuda Rise data.Horizontal bar over "September" shows time interval considered here.Horizontal bar over "April" shows time interval considered in Bird (1981) but not here.Note the counterclockwise rotation with decreasing height above bottom between 62 m and 6.9 m.
. The first 5-day sample from the observations starts 1 day after the record begins; this 5-day section was chosen because it was closest in time to a deep-tow survey, from which the stratification was derived.Time series of observed speeds, direction, veering and temperature are plotted in Figures 7a, 8a, and 9a.To gain insight into the processes determining the structure and dynamics of the BBL, the simulations proceeded in stages.First, the mean features of the 5-day period were simulated with a steady free-stream velocity which was obtained by averaging the velocity at 62 m over that period.As a next step in reproducing the characteristics of the time series, the effect of the major tidal component was reproduced by using a free-stream velocity consisting of an oscillating flow superposed on the mean velocity field.ß Finally, to reproduce more accurately the time-dependent the magnitude and direction of the bottom stress, friction velocity, total veering angle, and the thickness of the BBL.Inclusion of a slope in the model between the isopycnals and the bottom reduces the magnitude of the bottom stress and the friction velocity in comparison to the a = 0 simulation.The bottom stress was computed from the 5-day averaged velocity observations as described in section 3.In the model, all of the terms in equation (8) are computed; that is, the time-dependent and slope terms are not neglected.The time-dependent terms are small (---10% of I•'• ).The slope term calculated by the model is larger (---17% of for a slope of 0.01 after 1« days, 39% after 2« days) and has the effect of decreasing the x component of the stress.Thus, as discussed earlier, if this term could be calculated from the data, the bottom stress might

AFig. 10 .
Fig. 10.Observed and model speed profilesß The model profiles were plotted from day 3 of the simulation.The data are average speeds over the 5-day period.
With nine meters, a 2 ø change in direction in any meter results in as much as a 9% change in magnitude of the stress and an 11% change in the direction.With four current meters, a change of 0.35 cm/s in speed, the error in speed measured by a VACM[Bryden, 1976], results in as much as a 3% change in the magnitude of the stress and as much as a 13% change in direction.With nine current meters, a 0.35 cm/s change in speed of any one current meter results in as much as a 2% change in the magnitude and a 10% change in the direction of the stress.
nature the BBL over a sloping bottom does not continue to warm indefinitely at a fixed point, the warming trend should be removed in the model and a steady state attained.First, a computer run is made with a nonzero slope and a mean velocity field whose free-stream velocity components are (0, Vt).After several days of integration, the values of all the variables are stored for subsequent use as a 'basic state' and are denoted by the subscript 'BS.'The choice of the length of time of integration of the model determines only the actual temperature and thickness of the BBL: the longer the time, the warmer and thicker the mixed layer.Next, these basic state values are used to initialize the model in a new 'basic state' simulation.In the heat equation, the basic Fig. 12.Time series of speeds at 0.8, 7.0, 12, and 62 m predicted in a tidal simulation with a = 0.

Figure 14 BBL
Figure 14 depicts the same data, now averaged over 1-hour intervals.The semidiurnal tide is evident; rotary spectral analysis of currents in this region shows that this tidal component was primarily clockwise polarized.The ampli-tude of the tidal oscillations, which was about 8% of the mean speed at 28.8 m, decreased 60% to 0.7 m, while the mean current decreased only 45% between the same heights.To determine whether this phenomenon is consistent with comparing mean quantities, the versatility of the model allows accurate reproduction of the time-dependent nature of the flow.A first approximation was made by using a free-stream velocity field consisting of the major tidal component, the M2 clockwise polarized tide, superposed on the steady flow.Still better agreement is reached by using the velocity time series as measured at 62.0 m above the bottom to drive the model.The time series generated at the heights of the current meters correspond well to the ob, served time series.The large temperature fluctuations in the time series are not adequately reproduced by the model and are therefore not caused by cross-stream advection.Differential damping with depth in the BBL of different constituents of the flow has been observed in the Bermuda Rise data and in speed measurements from the ScotJan Rise BBL.In the free stream the amplitude of the M2 clockwise polarized tide was 6% of the mean speed on the Bermuda Rise and 8% of the mean speed on the ScotJan Rise.Model simulations with a steady flow plus a relatively small clockwise M2 tidal current also show differential damping.The thickness of the Ekman layer due to a constituent of the flow depends uniquely on the effective eddy diffusivity and on the effective rotation rate.The former is determined primarily by the steady flow since the tidal amplitude is small relative to the mean speed, and the latter is determined by the latitude and by the frequency and polarization of the tide.The components of the flow that have thicker boundary layers associated with them feel the effects of bottom friction higher in the water column than do those associated with thinner boundary layers.Thus, different components of the flow are damped unequally in the BBL as the bottom is approached.