Spatial Variability of Diapycnal Mixing and Turbulent Dissipation ...

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Spatial Variability of Diapycnal Mixing and Turbulent Dissipation Rates in a Stagnant Fjord Basin LARS ARNEBORG Department of Oceanography, Go¨teborg University, Go¨teborg, Sweden

CAROL JANZEN School of Ocean Sciences, University of Wales Bangor, Bangor, Gwynedd, United Kingdom

BENGT LILJEBLADH Department of Oceanography, Go¨teborg University, Go¨teborg, Sweden

TOM P. RIPPETH

AND

JOHN H. SIMPSON

School of Ocean Sciences, University of Wales Bangor, Bangor, Gwynedd, United Kingdom

ANDERS STIGEBRANDT Department of Oceanography, Go¨teborg University, Go¨teborg, Sweden (Manuscript received 18 February 2003, in final form 9 February 2004) ABSTRACT Two microstructure profilers, two ships, and four moorings with acoustic Doppler current profilers and conductivity–temperature loggers were used in an intensive effort to map the spatial and temporal variations of vertical mixing in the stagnant deep basin of Gullmar Fjord, Sweden. During three days in the beginning of August 2001 a continuous time series of turbulent kinetic energy dissipation profiles was obtained with one microstructure profiler at a fixed position near the deepest part of the fjord. During the same period the other microstructure profiler was used to obtain six sections of dissipation through the length of the basin. Two moorings were deployed in the fjord basin for one month from the end of July to the end of August. The mapping of dissipation rates reveals that the dissipation in the deep basin is confined to areas just inside the sill. More than 77% of the dissipation in the fjord basin happens above the sloping bottoms closest to the sill.

1. Introduction a. Background One of the important missing links in the understanding of ocean circulation is the diapycnal mixing below the wind-mixed layer. It is still not known how the downward mixing of warm water that drives the upwelling at midlatitudes will respond to a changing climate and changing properties of the deep water generated essentially at high latitudes. A large step forward was achieved during the last decade, when it was realized that tides loose much of their energy in the deep ocean, and thereby contribute a large part of the work needed to mix the water below 1000 m (e.g., Egbert Corresponding author address: Lars Arneborg, Go¨teborg University, Box 460, 405 30 Go¨teborg, Sweden. E-mail: [email protected]

q 2004 American Meteorological Society

and Ray 2000; Munk and Wunsch 1998). One of the pioneering works leading to this understanding (Sjo¨berg and Stigebrandt 1992) was based on experience about basin water mixing in fjords. The term ‘‘basin water’’ is here used for the body of water below sill level that is cut off from direct communication with the water outside the fjord due to the presence of the sill and due to the stable stratification. The advantages of studying mixing processes in fjords rather than in the deep ocean are that they can be investigated with small and relatively inexpensive expeditions and that the mixing can be calculated directly from changes in stratification within the fjord basin. At the same time, fjords house similar processes as the oceans, namely, tides and some wind-generated movements. This means that processes, such as generation of internal waves and turbulence that transfer energy out of barotropic tides and large-scale wind-driven

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motions to dissipation and mixing, can be studied much more conveniently than in the deep ocean. Until recently the fjord research on diapycnal mixing has been suffering from the lack of direct turbulence measurements. Inall and Rippeth (2002) introduced turbulence measurements into fjord studies by sampling two stations within the Clyde Sea, a large Scottish fjord. Their main result was that the basinwide diapcynal mixing, which reduces the deep-water density, is significantly larger than the mixing deduced from dissipation rate measurements both near the sill and in the interior basin. The most likely reason for this discrepancy is that the two stations were not representative of the basin as a whole. In order to obtain a more complete map of the dissipation rates in a fjord basin, a field experiment was designed for the simpler (shorter and much narrower) Gullmar Fjord on the Swedish West Coast. In this experiment two microstructure shear probes were used simultaneously onboard two ships to sample five stations along the fjord axis during a period of three days. It is the results of this experiment that are presented in the present paper. b. Gullmar Fjord Gullmar Fjord (Fig. 1) is a 28-km-long and 1–2-kmwide silled fjord, with a sill depth of 43 m and a maximum depth of about 120 m. Gullmar Fjord is characterized by weak tides (,0.2-m amplitude) and weak freshwater runoff (;25 m 3 s 21 ). The residence times above sill level are short (1–4 weeks) because of vivid baroclinic exchanges forced by upwelling and downwelling outside the fjord (Bjo¨rk and Nordberg 2003; Arneborg 2004). The basin water below 50–60 m is stagnant most years from May through December and renewed one or several times during the winter months. Arneborg and Liljebladh (2001b) investigated the basinaverage diapycnal mixing and related it to the dynamics in the fjord based on time series from moored acoustic Doppler current profilers (ADCPs) and conductivity– temperature (CT) loggers. They concluded that the magnitude and time variation of the work against buoyancy forces by diapycnal mixing in the deep basin can be explained by three equally important energy sources: (i) internal waves of tidal frequency generated by interaction of the barotropic tides with the sill, (ii) internal waves generated by an external seiche (see also Parsmar and Stigebrandt 1997), and (iii) internal seiches in the fjord. However, they did not have coincidental turbulence measurements in the fjord. They could therefore only speculate about the spatial distribution of the mixing and the detailed processes leading from the barotropic tides, the barotropic seiche, and internal seiches down to turbulence and mixing.

FIG. 1. Map of Gullmar Fjord showing the locations of moorings (G1, G2, G3, G4, and G5) and dissipation measurement stations (G1, D1, G2, G3, and D2).

c. Structure of paper We will set out by describing the field experiment and go on with a description of the data processing methods. The results section includes a general presentation of data, an estimate of diapycnal mixing rates in the fjord basin calculated from the rate of density decrease in the basin (budget method), and a presentation of the dissipation rates estimated using microstructure profilers. The results section continues with an effort to calculate the volume integrated dissipation rates in the basin, which are then compared with the diapcynal mixing rates, and it ends with an estimate of the internal wave fluxes in the deep basin. The paper ends with a discussion followed by our conclusions. 2. The experiment a. Moorings Five moorings with acoustic Doppler current profilers (ADCPs) and conductivity–temperature (CT) loggers were placed in and outside the Gullmar Fjord from 26 July to 21 August 2001 (Fig. 1). Moorings G1 and G3

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ARNEBORG ET AL. TABLE 1. Mooring instrumentation. Date (2001)

Mooring

Depth (m)

G1

90

G2 G3

110 79

G4 G5

65 45

Instruments 1 RDI Workhorse ADCP, 8 Seabird Microcat CT(D) loggers, and 8 Richard Brancker T loggers. 2 RDI Workhorse ADCPs 1 RDI Workhorse ADCP, 7 Seabird Microcat CT(D) loggers, 7 Richard Brancker T loggers, and one Aanderaa CT chain with 5 sensors. 1 RDI Workhorse ADCP 8 Seabird Microcat CT(D) loggers.

were placed in each end of the deep basin, and were each equipped with one upward-looking, bottommounted ADCP, and a number of CT loggers distributed from the bottom to the surface (see Table 1). Mooring G2 was placed near the deepest part of the fjord and was equipped with two upward-looking ADCPs, one at the bottom and one at 37 m to cover the surface water. Mooring G4, placed in the narrowest part of the main entrance to the fjord, was equipped with a bottommounted, upward-looking ADCP. Mooring G5 was situated outside the fjord and was equipped with eight CT loggers distributed from the bottom to the surface. All ADCPs, except the upper instrument at mooring G2, were 300-kHz models configured with 2-m bins. The upper ADCP at mooring G2 was a 600-kHz model configured with 1-m bins. All ADCPs were configured with 50 pings per ensemble giving a velocity error standard deviation of 1 cm s 21 . b. Microstructure measurements During three days from 31 July to 3 August 2001, two microstructure profilers onboard two research vessels (Skagerrak and Alice) were used in an intensive effort to map the spatial and temporal variations of vertical mixing in the deep stagnant basin of Gullmar Fjord. One profiler, the MSS profiler (see Prandke et al. 2000), was used to obtain a continuous time series of the rate of turbulent kinetic energy (TKE) dissipation at station G2 near the deepest part of the fjord (Fig. 1). The measurements were performed as bursts of four profiles. The interval between individual profiles was about 8 min, and the interval between bursts was 1–2 h. The FLY profiler (see Dewey et al. 1987) simultaneously measured six longitudinal sections of the rate of dissipation of TKE along the axis of the fjord. Measurements were made at stations D1, G1, G2, G3, and D2 (Fig. 1). One extra station visit was made at G2 as time allowed. Bursts of six profiles were performed at each station visit. The profiles were taken while steaming at approximately 1 kt (;0.5 m s 21), and one burst took approximately 20 min, which means that one burst was distributed over a distance of about 600 m.

Start

End

26 Jul

21 Aug

Sampling interval (min) 5

27 Jul 26 Jul

21 Aug 21 Aug

5 5 (2 for Aanderaa)

30 Jun 30 Jun

21 Aug 21 Aug

10 10

3. Data processing a. Estimating diapycnal mixing from mooring data: The budget method The water below sill level in Gullmar Fjord is stagnant during most of the year, being renewed only in winter and early spring when denser water outside the fjord reaches above sill depth and spills over. The stagnant conditions enable us to use a budget method to determine the horizontally averaged diapycnal mixing below sill level. The budget method has been used widely to estimate diapycnal mixing in laterally bounded basins during periods with no deep-water renewal (e.g., Axell 1998), and it is discussed in Gargett (1984) and Stigebrandt and Aure (1989). Under stagnant conditions, the total decrease in mass below a given level z 0 must be accompanied by a corresponding upward diapycnal mass flux, F d , through level z 0 . After integration of the mass conservation equation, this can be expressed as Fd 5 2

E

z0

A(z)

zb

]r dz, ]t

(1)

where A(z) is the hypsograph, z b is the level of the deepest point in the fjord, r is the density, and the overbar denotes averaging in time over the horizontal area at each depth. The derivation of this expression is described in more detail in Arneborg and Liljebladh (2001b). The diapcynal mixing can also be expressed in terms of a horizontally averaged diapycnal diffusion coefficient, K z , defined as the horizontally averaged diapycnal mass flux divided by the horizontally averaged density gradient; that is, K z (z) [

2Fd (z) . A(z)]r/]z

(2)

The rate of work against buoyancy due to diapycnal mixing is the upward mass flux multiplied with the gravitational acceleration. The horizontally integrated value, W d , can therefore be written as

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Wd 5 gFd 5 r 0 AK z N 2 ,

(3)

where N is the buoyancy frequency. The density decrease rates were determined with the continuous time series of density, obtained from the CT loggers at moorings G1 and G3. The decrease rates were determined as an average over the whole 26-day deployment period to remove influences of short-term fluctuations caused by for example; internal seiche motions. The vertical density gradients were obtained from a mean density profile based on all MSS casts at mooring G2 during the 3-day cruise. b. Estimating dissipation rates and diapycnal diffusion from microstructure shear measurements As an alternative to the budget method, where one deduces the turbulence and mixing from the effects it causes, one can measure the turbulence, or more correctly the dissipation rate of turbulent kinetic energy, directly. The FLY and MSS profilers were used to do this. The principle behind the FLY and MSS probes are identical in that they both are free-falling probes with airfoil sensors that measure velocity fluctuations perpendicular to the fall direction. However, the designs of the two probes are widely different with regard to size, airfoil design, electronics, operation, and so on. For details on the various specifications of each of the microstructure profilers, the reader is referred to Dewey et al. (1987) (FLY profiler), and Prandke et al. (2000) (MSS profiler). The MSS and FLY profilers fall with speeds of ;0.68 (MSS) and ;0.83 m s 21 (FLY), sampling from 5–10-m depth (i.e., after exclusion of ship-generated turbulence) and down to within 15 cm of the seabed. They provide measurements of velocity fluctuations on a scale down to ;1 cm. The time series of horizontal velocity fluctuations are differentiated and transformed to vertical shear using Taylor’s frozen field hypothesis. The rate of dissipation of turbulent kinetic energy « is estimated from the vertical shear, using « 5 7.5n

1 2 ]u ]z

2

(W kg21 ),

(4)

where n is kinematic viscosity of seawater. The assumption behind (4) is that the turbulence is isotropic on scales within the dissipation range. The mean square shear is calculated by deriving the power spectrum for 0.34 m (MSS) and 1.5 m (FLY) sections of the record and relies on the assumption that the turbulence is homogeneous and stationary within each depth interval. High- and low-frequency noise caused by for example, probe vibrations is eliminated, and the loss of spectral coverage is corrected using the Nasmyth spectrum (Simpson et al. 1996; Prandke et al. 2000). The FLY measures .70% of the dissipation spectrum in the range from 10 29 to 10 25 W kg 21 (Rippeth et al. 2003), with

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an error in « estimates of approximately 50% (Simpson et al. 1996). The MSS sections are shifted so that the last depth bin ends exactly at the last data record above the bottom. Confidence limits on mean dissipation values are calculated with a bootstrapping method described in Efron and Gong (1983) and Emery and Thomson (1997). The bootstrapping method provides a means of resampling the original data and generating multiple new artificial realizations of the data without having to repeat the observations. These artificial realizations are then used to estimate the reliability of the statistic of interest, in our case the binned mean dissipation profiles. The diapycnal diffusion can be estimated from the dissipation rate, using

kz 5

Rf « 1 2 Rf N 2

(5)

(Osborn 1980), where N is the buoyancy frequency and R f is the flux Richardson number. The local diapycnal mass flux per unit horizontal area, f d , is through the definition of k z given as f d 5 2k z

]r r N2 5 kz 0 . ]z g

(6)

By use of (5) this can also be written in terms of the dissipation as fd 5

R f r 0« . 1 2 Rf g

(7)

Assuming that the flux Richardson number is constant, the diapycnal mass flux is therefore proportional to the dissipation rate. 4. Results a. Mooring data Figure 2 shows the stratification on day 212.0 obtained with the CT sensors on the MSS probe, and the corresponding values obtained with the loggers mounted on moorings G1 and G3. There is a strong pycnocline at about 10 m and a weaker one at 50 m, just below sill level. This is typical for Gullmar Fjord, where one often sees a relatively fresh upper layer of varying thickness (5–20 m) dominated by Kattegat water flowing northward along the Swedish west coast, and an intermediary more saline layer dominated by Skagerrak water. The basin water is weakly stratified below 60 m, with a buoyancy frequency squared on the order of 10 25 s 22 . It is cold and salty due to its origin as surface water in Skagerrak during the previous winter. The density structure at stations G1 and G3 is very similar to that at station G2, but the water column in the interior basin is slightly colder and less saline than the water near the sill. Figure 3 shows horizontal, along-fjord velocities at moorings G1 to G4 and winds at Kristineberg (Fig. 1)

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FIG. 2. (a) Potential density anomaly, (b) temperature, (c) salinity, and (d) buoyancy frequency squared vs depth at day 212.0. The continuous line is the average of one burst of four profiles, taken from the MSS conductivity and temperature sensors. The circles and stars are data from the mooring G1 and G3 conductivity and temperature sensors, respectively.

from 25 June to 21 August 2001. Positive velocities are directed into the fjord. The white vertical lines show the times when each station was visited with the FLY profiler, and the time span of the MSS measurements at G2 is indicated with the gray box in the background in Fig. 3d. Figure 4 shows the potential density anomaly s u at moorings G1, G3, and G5 and winds at Kristineberg. The wind record shows relatively calm conditions up until day 209 (two days before the first dissipation measurements). Strong westerly winds started blowing directly into the fjord on day 210, one day before the first dissipation series commenced, and the westerly wind reached its maximum of about 15 m s 21 . Thereafter, it gradually decreased, and the last day it shifted to an easterly wind of about 8 m s 21 . The rest of the mooring deployment period was characterized by variable winds, generally below 10 m s 21 . The jumpwise changes in isopycnal depths above 20m depth at stations G1 and G3 (Fig. 4) are artificial. They are caused by the sharp pycnocline, as seen in Fig. 2, that migrates vertically combined with errors from linear interpolation between sensors that are situated too far from each other to adequately capture the pycnocline movements. The jumps occur when the pycnocline migrates past the depth of a sensor. Figures 3 and 4 show strong baroclinic currents above sill level and large vertical migrations of the upper pycnocline. These kinds of motions are described in Djurfeldt (1987) and Arneborg (2004), thus will only be briefly explained here since the focus of this paper is on the deeper water. Basically, these low-frequency (periods longer than ;3 days), large-amplitude motions are adjustments of the fjord stratification to upwelling and downwelling motions of the stratification outside the fjord. Below sill level the velocities are weaker since the

low-frequency motions are blocked by the sill (Arneborg and Liljebladh 2001a). Here the internal seiches and the tidal motions become more dominant. Note the large vertical oscillations with amplitudes on the order of 5 m and periods around 1–2 days in Fig. 4. These are about 1808 out of phase at moorings G1 and G3 at each end of the deep basin and represent the internal seiche motions. Figure 5 shows the results of spectral analysis on the velocities at 70 m at G1, G2, and G3. The peaks labeled M 2 and S f are the M 2 tidal frequency and the barotropic seiche spectral peaks, respectively. The internal seiches are represented by the wide and energy containing frequency band from 0.01–0.06 cph (17 to 100 h). There are no distinct peaks for the individual internal seiches. The reason for this is that the periods change rather dramatically with the change in first- and second-mode internal phase speed, caused by the changing location of the upper pycnocline and by the low-frequency baroclinic currents. The reader is referred to Arneborg and Liljebladh (2001a) for more details about the internal seiches. One feature in Fig. 5 that is very distinct is that the spectrum at G3 contains much larger variance than the spectra at G2 and G1 at all frequencies. This means that kinetic energy at 70 m is enhanced close to the sill relative to the central and inner part of the basin. This is worth keeping in mind when the horizontal distribution of turbulent dissipation is presented. b. Basin integrated diapycnal mixing obtained with the budget method As described in section 3 the diapycnal mass flux at a given level in the fjord basin can be determined from the rate of decrease in total mass below that level [see Eq. (1)]. The density decrease rates were determined from CT data at moorings G1 and G3. Figure 6 shows

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FIG. 3. (a) Westerly (thick line) and southerly (thin line) wind speeds (m s 21 ) from Kristineberg. (b)–(e) The alongfjord velocities (m s 21 ) at station G4, G3, G2, and G1, respectively, as a function of time (UTC) and depth. Positive velocities are directed into the fjord. The white vertical lines show when each station was visited with FLY measurements. The gray box in the background of (d) shows the time span of the MSS measurements at G2.

the potential density anomaly time series from the CT loggers at 90, 80, 70, and 55 m on mooring G1, and at 49, 47, and 45 m on mooring G3. The large density fluctuations with periods less than 3 days are caused by vertical oscillations related to, for example, the internal seiches (section 2). Note how the density fluctuations at 55 m at G1 are 1808 out of phase with the density fluctuations at 49 m at G3, as mentioned earlier. Initially, we assume that the long-term density decreases seen in all series are related to dilution with less

dense water in the upper water column, caused by diapycnal mixing. The potential density decrease rates are shown as a function of depth in Fig. 7a. These were found by least square fitting a straight line to the potential density time series. The density decrease rates were interpolated (linearly in the logarithmic scale) to 1-m bins to give the continuous curve shown in Fig. 7a and multiplied by the horizontal area at each depth. The sum of these values from the bottom and up to a given depth gives the total decrease in mass below that depth,

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FIG. 4. (a) As in Fig. 3 except (b)–(d) contours of potential density anomalies (kg m 23 ) at station G5 (outside the fjord), G3, and G1, respectively, as function of time (UTC) and depth. The contours have been calculated based on linear interpolation between the sensor depths. No contours have been drawn between 55- and 35-m depth at mooring G1 because of the large distance between these sensors.

FIG. 5. Spectra of horizontal velocities at 70-m depth at G1, G2, and G3. The spectra are calculated as the average over three segments, using a Hanning window and no overlapping. This gives 6 degrees of freedom in the spectral estimate.

which according to (1) is equal to the upward, diapycnal mass flux F d . The horizontally averaged upward diapycnal mass flux is shown in Fig. 7b. The buoyancy frequency squared, obtained from an average of all MSS density profiles, is given in Fig. 7c, and the horizontally averaged diffusion coefficient defined in (2) is shown in Fig. 7d. The diapycnal diffusion coefficient ranges from below 10 25 m 2 s 21 in the most stratified water to above 10 24 m 2 s 21 in the deepest parts of the fjord (Fig. 7d). These results agree well with those from the most energetic period of an experiment in Gullmar Fjord in the summer of 1997 (Arneborg and Liljebladh 2001b). The total rate of work against buoyancy in the fjord basin, obtained by using (3) and integrating from the bottom and up to 60 m, is 0.8 kW. This is only slightly smaller than the 1.0 kW reported for the most energetic period and the same depth interval in the 1997 study (Arneborg and Liljebladh 2001b). The diapycnal fluxes were calculated under the assumption of stagnant conditions in the basin water. The main source of error is the potential advective fluxes in

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riod as an average from the bottom up to 49 m. The density gradient profile used for the diffusion coefficient calculation in our analysis is based on the mean of all MSS profiles. These were distributed around day 6 of the 26-day period. Assuming that the density gradients at each depth increases linearly by 40% during the 26day period, the density gradient profiles at day 6 may be expected to be ;11% smaller than the mean profile occurring at day 13. This means that the diffusion coefficients are overestimated by about 12% due to this effect. This effect is smaller than the interpolation error, and is therefore not shown in Fig. 7. c. Dissipation levels and diffusion coefficients at station G2 from the MSS probe

FIG. 6. Time series of potential density anomaly (kg m 23 ) at various depths as calculated from measured CT series at moorings G1 and G3. The insert is a zoom of the deepest three time series. The straight lines are trend lines calculated with least squares fitting.

and out of the fjord below sill level. Oxygen profiles, taken as part of regular monitoring programs before and after the present experiment, show clear decreasing trends below 50 m, indicating that new oxygen-rich water has not entered the fjord below 50 m. The isopycnal lines at mooring G3 (Fig. 4c) show vertical fluctuations of about 5-m height at sill depth (43 m). These may be enhanced closer to the sill, which means that it is not impossible that water at 50 m can be flushed out of the fjord. Such a loss of water would lead to an overestimate in the diapycnal diffusion calculations. In order to avoid any effects of advection, we therefore only use the results below 55 m in the following comparison with dissipation measurements. Another source of error in the diapycnal flux estimate is the interpolation of density decrease rates. In order to estimate the magnitude of this error, the diapycnal mass flux and diffusion coefficients were recalculated based on the alternative density decrease rate interpolations shown as dashed lines in Fig. 7a. The results are shown as dashed lines in Figs. 7b and 7d. It is seen that these extreme interpolations, which must be said to be less realistic than the linear interpolation, can change the result by a factor of 2. A source of error for the diffusion coefficient estimate, but not for the diapycnal flux, is the choice of buoyancy frequency profile. According to the moored observations the density at 49 m decreases by about 0.48 kg m 23 during the 26-day period, while the bottom density decreases much less. The mean density difference from the bottom and up to 49 m is 1.2 kg m 23 . This means that the vertical density gradient increases by 40% during the 26-day pe-

Figure 8 shows the dissipation rates of TKE at station G2 as function of time and depth. The contours are based on burst averages of four profiles, taken at 1–2-h intervals. The values are calculated in 1-m bins as described in section 3. Above sill depth, the dissipation rate levels are seen to be variable, ranging between 10 29 and 10 26 W kg 21 . The largest values are centered on the upper pycnocline, which rises from about 15 m at day 211 to 7 m at day 214. These enhanced values are probably caused by the large shear in the pycnocline seen in Fig. 3d. Below sill level there are some intermittently enhanced dissipation rate levels between 40 and 50 m related to the shear over the pycnocline between stagnant basin water and intermediary Skagerrak water (section 2). There are also enhanced dissipation rate levels in the deep water around day 213.0. In addition, there is a less than 5-m-thick bottom boundary layer showing enhanced dissipation rates. Otherwise the water column below 60 m is very calm with dissipation levels on the order of 2 3 10 29 W kg 21 . Mean values of the rate of dissipation of TKE, diffusion coefficients calculated with (5), and buoyancy frequency squared, calculated in isopycnal bands corresponding to approximately 5-m bins, are shown in Fig. 9. Confidence limits (95%) calculated with the bootstrapping method are shown for the dissipation rate and buoyancy frequency squared curves (Figs. 9a and 9b). The error limits shown for the diffusion coefficient (Fig. 9c) take into account the range of possible flux Richardson values (R f 5 0.06–0.17), mentioned below (in section 4f), in addition to the 95% confidence limit on the dissipation estimates. The diffusion coefficient ranges from 6 3 10 27 m 2 s 21 in the deeper pycnocline just below sill level to 3 3 10 25 m 2 s 21 in the deep basin. This is a factor of 5– 10 less than the basin averages calculated with the budget method. Neither the very pessimistic error estimates for the basin average diffusion coefficient nor the large span of the flux Richardson number values can bring the two estimates to be compatible with each other. It is therefore clear that the dissipation rate levels at G2

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FIG. 7. (a) Density decrease rates obtained from the time series shown in Fig. 6, (b) horizontally averaged diapycnal mass flux, (c) buoyancy frequency squared obtained from an average of all MSS density profiles, and (d) horizontally averaged diapycnal diffusion calculated using (2). Maximum and minimum estimates are shown as dashed lines.

in the deepest parts of the fjord are too low to explain the basin average mixing in the fjord. The discrepancy between the budget and dissipation method estimates of vertical mixing rates are similar to those reported for the Clyde Sea (Inall and Rippeth 2002). d. Sections of dissipation levels from the FLY probe Figure 10 shows the FLY profiler estimates of the mean rate of dissipation of TKE and their respective 95% confidence intervals at stations G1, D1, G2, G3, and D2. The mean values are calculated in 5-m bins and are based on all FLY profiles from all visits to each station. The dissipation rates at G2 obtained with the FLY profiler correspond well with those obtained from the MSS profiler at the same station. Mean FLY dissipation rates are on the order of 2–3 (310 29 W kg 21 ) in the deep basin with some enhancement close to the bottom and are on the order of 10 28 W kg 21 above the lower pycnocline, rising to 10 27 W kg 21 in the upper pycnocline. The dissipation rates above sill level (43 m) are relatively constant through the fjord, as shown in the FLY dissipation results (Fig. 10). There is some increased dissipation at station D1 between 20 and 40 m, probably related to interaction with the sill at the entrance to Saltka¨lle Fjord (Fig. 1). For the remainder of this paper, we concentrate on the water below sill level. Below sill level, in particular at the three inner stations (D1, G1, and G2), we observe enhanced dissipation rates in the three bins closest to the bottom. The enhanced dissipation spread over three bars is artificial. This artifact could be removed using the height above the bed to average the data. With this averaging procedure the enhanced dissipation rates are seen (Fig. 10,

lower panel) to be concentrated within the bottom 5-m bin, and the enhancement is probably caused by bottom boundary layer turbulence. Below sill level, but above the bottom boundary layer, there is a clear enhancement of the dissipation rates at the two stations closest to the Gullmar sill (D2 and G3) in comparison to the three inner fjord stations (G2, G1, and D1). Below 50–60 m the dissipation rates range between 2 and 5 (310 29 W kg 21 ) at stations D1, G1, and G2, with the highest levels at D1 and the lowest at G2. At station G3 the levels below 50–60 m increase to 2–6 (310 28 W kg 21 ) and at D2, closest to the sill, they are 5–10 (310 28 W kg 21 ). This means that the levels at G3 and D2 are a factor of 10 and 20, respectively, larger than the levels in the central and inner fjord basin. e. Basin integrated dissipation In order to estimate the horizontally integrated dissipation rates, we assume (i) that the mean dissipation rates are independent of depth below 55 m and above a bottom boundary layer of thickness, d, and (ii) that the mean interior and bottom boundary layer dissipation rates at each station represent the dissipation rates in the surrounding parts of the fjord. We next describe how we define those partial volumes. The separation between the partial basin volumes represented by each station are shown with dashed lines in Fig. 10. These are chosen based on the distance between the stations and the topography of the fjord. The separation between the boxes of station G2 and G3 is chosen to be closest to station G3 since G3 is assumed to represent the relatively steep slopes, while G2 is assumed to represent the deep and gently sloping areas. Because of a lack of detailed data on the bottom and

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FIG. 8. Contour plot of dissipation levels at G2 as a function of time (UTC) and depth. The dissipation values are burst averages over four profiles calculated in 1-m bins. The interval between bursts is 1–2 h. The times when FLY profiles were performed at G2 are indicated with vertical lines.

lateral topography, we apply a width function that gives the width, W, as function of depth and distance up the fjord. The width function applied here is given as W(z, x) 5 W0

[

]

z 1 h(x) z 0 1 h(x)

1/4

,

(8)

FIG. 9. (a) Dissipation levels, (b) buoyancy frequency squared, and (c) diffusion coefficient calculated with (5), based on all 3 days of MSS data at G2. The data are averaged in isopycnal bands chosen to be approximately 5 m thick. 95% confidence estimates are calculated for the dissipation and buoyancy frequency estimates. The flux Richardson number used for the diffusion coefficient calculation is 0.11. The error band is calculated using R f 5 0.2 and 0.06 and the upper and lower 95% confidence values for the mean dissipation rate. In (c) the basin average diffusion coefficient from Fig. 7 (fully drawn line without dots) and the corresponding error limits (dashed lines) are also shown.

where z 0 is a reference level, W 0 is the width at that level, and h is the maximum depth at distance x from the entrance. We use z 0 5 250 m as reference level because the width is relatively constant at that depth and because it will give some variation of the width along the fjord at z 5 255 m, which is our upper integration level. The width at 50 m is taken as W 0 5 1000 m, which corresponds well with the available maps. For each station, the volume represented by this station was found by integrating (8) between the separation lines shown in Fig. 10 and between the bottom and 55 m. In order to estimate the influence of bottom boundary mixing, the horizontal areas for a d 5 5 m thick bottom boundary layer and the interior were calculated separately for the three inner stations (G2, G1, and D1). The width above the boundary layer, W i , was calculated by using h b 5 h 2 d instead of h in (8). The width of the boundary layer was then calculated as W 2 W i . The exponent in (8) is chosen so that the sum of the five area profiles fits well with the basin hypsograph. The basin integrated dissipation rates were computed by multiplying the interior and bottom boundary layer averages of dissipation rates at each station by their representative areas and summing the values. In terms of volume averages, stations D2 and G3 represent 77% of the vertical mass flux, while the interior and bottom boundary layer of stations D1, G1, and G2 together represent only 13% and 10% of the vertical mass flux,

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FIG. 10. (upper left) Dissipation levels at station (from the left to the right) D2, G3, G2, G1, and D1 shown as function of depth and distance from the entrance. Mean values in 5-m bins, based on distance from surface, are shown in a logarithmic scale with the vertical axis located at 10 29 W kg 21 . The dark gray part of the bars indicate the 95% confidence interval. Also shown is the fjord depth as function of distance from the entrance. (upper right) The buoyancy frequency squared at G2, based on the MSS profiles. The lower-left panel is similar to the upper-left panel, except that the dissipation rates are averaged in 5-m bins based on the distance from the bottom. The dashed lines in the upper panel show the separation between the boxes used for horizontal averaging of the dissipation rates.

respectively (Fig. 11). This means that 77% of the vertical mass transport due to mixing takes place within only 19% of the basin volume. The total dissipation rate in the fjord basin below 55 m is 7.3 kW. This is in the same order of magnitude as

the 8.9 kW (semidiurnal tides: 2.7 kW, internal seiches: 3.9 kW, barotropic seiche: 2.3 kW), estimated for the most energetic period of 1997 (Arneborg and Liljebladh 2001b). f. Mixing efficiency

FIG. 11. Volume integrated dissipation rates for the sum of boxes D2 and G3; the sum of the interior of boxes D1, G1, and G2; and the sum of the boundary layer contribution of boxes D1, G1, and G2.

The volume-integrated rate of work against buoyancy below 55 m is 1.2 kW, as found with the budget method. With a volume-integrated dissipation rate of 7.3 kW this gives a flux Richardson number of R f 5 0.14 for the whole basin below 55 m [calculated from Eq. (7)]. This value is within the range of those proposed in the literature, R f 5 0.05–0.07 (Stigebrandt 1979; Stigebrandt and Aure 1989; De Young and Pond 1989; Simpson and Rippeth 1993; Arneborg and Liljebladh 2001b), R f 5 0.11 (Arneborg 2002), and R f 5 0.17 (Osborn 1980). This is reassuring for our dissipation rate estimates. However, the dataset is not good enough to discriminate between any of these values. Our estimate may easily be a factor of 2 or more in error because of undersampling in space and time of the dissipation rate of turbulent kinetic energy. In addition, it is probably biased high rather than low because of the extremely nonuniform distribution of the dissipation rates in space and time.

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TABLE 2. Depth integrated kinetic and potential energies below 55 m. Station

E k (J m22 )

E p (J m22 )

G1 G2 G3

17.1 19.5 45.0

16.2 — 29.5

g. Internal wave energy fluxes Rough estimates of the energy fluxes in the fjord can be obtained from the kinetic and potential energies at each station and the group velocities of the relevant vertical modes. With such an approach, it is necessary to take into consideration the existence of standing rather than progressive modes. It is more correct to use normal-mode decomposition (e.g., Webb and Pond 1986), but this is not possible in our case because of the bad representation of the upper-layer halocline movements (Fig. 4). The depth-integrated kinetic and potential energies below 55 m, calculated as 1 Ek 5 r 0 2 Ep 5

1 g2 2 r0

E E

5. Discussion and conclusions

z0

u dz and 2

(9)

2H z0

2H

FIG. 12. Lagged correlation coefficients between vertical displacements at 55 m and depth-averaged baroclinic velocities below 55 m at stations G1 (dashed line) and G3 (fully drawn line). A positive lag means that vertical displacements lag behind the velocities.

r 92 dz N2

(10)

(Gill 1982), where r9 is the density perturbation, H is the depth, u is the baroclinic (total minus depth averaged) horizontal velocity, and z 0 5 255 m, are given in Table 2. The second baroclinic mode, which is relevant in this case because it is mainly a wave on the sill-level pycnocline, has a group velocity of c g2 ø 0.5 m s 21 . If the energies in Table 2 were related to progressive waves, the energy fluxes would be in the order of 17 kW [5W(E k 1 E p )c g2 where W is the width] at the inner station (G1) and 37 kW at the outer station (G3). However, lagged correlation coefficients between the depth integrated baroclinic velocities below 55 m and vertical oscillations at 55 m at stations G1 and G3, show that these time series are almost in quadrature rather than in phase (Fig. 12). This indicates that the baroclinic motions are standing waves rather than progressive waves. This applies both for the tidal and subtidal frequencies. The fact that the interface elevation lags behind the velocity at the inner station and is in front of the velocity at the outer station confirms this interpretation. This means that the net energy fluxes are much smaller than the 17 and 37 kW estimated above. This is reassuring for our dissipation estimates since the net energy flux should at least be less than the total rate of energy loss 8.5 kW (dissipation 1 work against buoyancy) in the deep basin. The further interpretation of these results in terms of turbulence generating mechanisms is discussed in the next section.

Our effort to map the dissipation rates in a fjord basin have shown that more than 77% of the dissipation below the sill level pycnocline takes place in 19% of the basin volume closest to the sill. If we assume a constant flux Richardson number, this means that also 77% of the diapycnal mass flux caused by mixing takes place within the same volume. The observed dissipation rates at the sill are distributed nearly uniformly over the depth and are not confined to a well-mixed bottom boundary layer. In the central and inner parts of the fjord, about onehalf of the remaining dissipation happens in the stratified interior, while one-half is concentrated to a less-than-5m-thick bottom boundary layer. A comparison of the dissipation rates with the work against buoyancy implied from the observed density decrease rates in the fjord basin gives a flux Richardson number of about 0.14. This is within the range of theoretical, numerical, and experimental values reported in the literature, which makes us confident that we have covered a large part of the dissipation that takes place in the fjord. The internal wave velocities and vertical oscillations in the deep basin are nearly in quadrature, which indicates that the internal motions primarily consist of standing waves. This is consistent with the low observed dissipation rates in the inner part of the basin, but leaves us with a number of possible explanations for the sources of enhanced dissipation at the sill. We know that the main area of internal tide generation is the sill, but this is also the main interaction area between the quarterwave internal seiche on the upper pycnocline and the half-wave internal seiche on the lower pycnocline (Arneborg and Liljebladh 2001a). Therefore the dissipation at the sill can be caused (i) by the interaction between

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the barotropic tide and the sill, (ii) by the interaction between the internal quarter-wave seiche and the sill, or (iii) by breaking of internal waves that have been generated by processes i and ii, after having been completely reflected from the head of the fjord. The energy fluxes in the internal wave field are large enough that iii is possible. The time resolution of the dissipation data at the sill is not good enough to discriminate between these mechanisms, and probably the answer is a combination of them. Although we have not managed to explain the coupling between large-scale forcing and microscale turbulence and dissipation, the results of the present work is a large step forward toward that goal. Now, we know where to look for the dissipation and mixing, and in a future experiment we plan to focus the observational effort to the sill region. The present results are also of relevance for similar efforts in the ocean. In a recent experiment at Hawaii (Rudnick et al. 2003) there was problems closing the energy budget. About 20 GW is lost from the barotropic tides at the Hawaiian ridge, and about one-half of this radiates away as low-mode internal waves. A reasonable conclusion would be that the other half is lost to dissipation and mixing in the vicinity of the ridge, but the dissipation measurements performed during the experiment did not support this conclusion. However, dissipation measurements were not performed over the steepest flanks of the ridge, which with the present results in mind would be the place to find the dissipation. Acknowledgments. We thank Carina P. Erlandsson, Arne Sjo¨quist, and the crew of R/V Skagerrak for their help during the cruise. The MSS system was maintained by Dr. Harmut Prandke at ISW Wassermesstechnik, who also did the dissipation rate calculations for the MSS data. The work was performed as part of the EU project OAERRE: EVK3-CT1999-00002. Most of the moored instruments were funded by the Swedish Research Council (VR), and the new research vessel, Alice, is a donation from K&A Wallenberg foundation. REFERENCES Arneborg, L., 2002: Mixing efficiencies in patchy turbulence. J. Phys. Oceanogr., 32, 1496–1506. ——, 2004: Turnover times for the water above sill level in Gullmar Fjord. Cont. Shelf Res., 24, 443–460. ——, and B. Liljebladh, 2001a: The internal seiches in Gullmar Fjord. Part I: Dynamics. J. Phys. Oceanogr., 31, 2549–2566. ——, and ——, 2001b: The internal seiches in Gullmar Fjord. Part

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