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The Internal Seiches in Gullmar Fjord. Part II: Contribution to Basin Water Mixing

LARS ARNEBORG AND BENGT LILJEBLADH Department of Oceanography, University of GoÈteborg, GoÈteborg, Sweden

(Manuscript received 26 July 1999, in ®nal form 20 December 2000)

ABSTRACT The mixing in the basin water (the water below sill level) of Gullmar Fjord has been investigated with the main focus on the contribution from internal seiches. A companion paper reports evidence for dissipation of internal seiche energy in the basin water after near-critical re¯ection from the bottom. In the present paper the magnitude and variation of basin water mixing is investigated, using the budget method. The results are related to variations of three energy sources, namely (i) the internal seiches, (ii) the internal tides, and (iii) the internal waves generated by the external seiche. The mixing ef®ciency, de®ned as the irreversible work against buoyancy forces due to mixing divided by the total mechanical energy loss from the sources mentioned above, is about 7%, similar to results obtained for other fjords. Large variations in mixing are shown to be related to large variations of the energy sources. The internal seiches are found to be important for the mixing, with a contribution that is 144% of the internal tide contribution during the most energetic period and 92% on average over the investigated periods. Including contributions from the external seiche, the wind forcing is responsible for 61% of the basin water mixing, while tidal forcing is responsible for 39%.

1. Introduction with low tidal forcing. Energy is put into the internal The diapycnal mixing below the upper mixed layer seiches by both the direct action of wind on the fjord is important in the oceans because it drives the ther- and coastal forcing due to oscillations in the coastal mohaline circulation, in lakes because it brings oxygen strati®cation, the two energy sources being of about the down and nutrients up and in fjords mainly because it same magnitude. Analyses of horizontal velocities and decreases the density of the basin water (the water below vertical displacements inside the sill indicate that the sill level), so that new oxygen-rich water can come in motions observed in the basin water are progressive from outside and replace the ``old'' water. Stigebrandt wave motions rather than standing wave motions, with (1976) proposed that the basin water mixing in fjords down- and inward group velocity emanating from the was mainly caused by breaking of internal tides near sill region. An alongfjord section of the topography with sloping bottoms. This hypothesis was strengthened in characteristics corresponding to the actual strati®cation Stigebrandt and Aure (1989), where they showed that and seiche frequency (Fig. 1, same as Fig. 11 in AL) the basin water mixing in a large number of Norwegian shows that most of the slopes in the fjord basin are near fjords was related to dissipation of internal tides gen- critical to re¯ection of the internal seiche. It is therefore erated at the sills. In fjords with low tidal energy, other suggested that the energy radiating downward is lost to processes may, however, become important. A reason- turbulence in the basin water after near-critical re¯ection able hypothesis is that internal seiche motions become from the bottom. Estimates of the energy ¯ux radiating relevant for the mixing in the deep waters, similar to downward from the sill region show that it is only about the case in strati®ed lakes (Wiegand and Chamberlain 2% of the total energy loss in the internal seiches, but 1987; MuÈnnich et al. 1992) and this is investigated in that this contribution may be important for the basin the present paper. water mixing relative to the contribution from internal The forcing and damping of the internal seiches in tides. The main purpose with the present paper is to Gullmar Fjord is investigated in a companion paper (Ar- investigate if there is any evidence that some of the neborg and Liljebladh 2001, hereafter AL), based on energy radiating downward is used to mix the basin data from two mooring datasets from 1997. Gullmar water and how important this contribution is compared Fjord, located on the Swedish West Coast, is a fjord with other energy sources. One measure for the relation between mixing and energy sources is the mixing ef®ciency ␥, de®ned as Corresponding author address: Lars Arneborg, Department of Oceanography, Earth Sciences Centre, University of GoÈteborg, Box W 460, 405 30 GoÈteborg, Sweden. ␥ ϭ , (1) E-mail: [email protected] ͸ F

᭧ 2001 American Meteorological Society

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FIG. 2. Spectrum of alongfjord velocities at 59-m depth, based on the 56-day time series. Vertical lines indicate external (ES) and in- FIG. 1. Long-section of Gullmar Fjord, showing deepest topography ternal (IS) seiche frequencies and the M2 tidal frequency; 95% con- and selected characteristics corresponding to strati®cation at day 245 ®dence intervals are based on 20±320 degrees of freedom in the and frequency 0.023 cph. Also shown is a group velocity vector and spectral estimates. the suggested area of dissipation.

by generation of internal waves at the sill. These internal where W is the total work against buoyancy forces in waves may be assumed to contribute to the basin water the basin water caused by mixing and the denominator mixing in a manner similar to the internal tides, and is the sum of all mechanical energy inputs F to the basin more importantly the amplitude of these are highly vary- water. Based on lab experiments and microstructure ing. Variations in mixing can therefore be caused by measurements, Imberger and Ivey (1991) concluded that both internal and external seiches, which means that we the maximum fraction of turbulent energy that can be need to quantify both. used for mixing is 0.2 and that this ef®ciency is gen- Based on the summer±fall dataset from 1997 pre- erally obtained in turbulence far from boundaries, while sented in AL and brie¯y recapitulated in section 2, we it can be much smaller for benthic boundary-layer tur- calculate the basin water mixing for three different pe- bulence. Stigebrandt and Aure (1989) computed the en- riods, using the budget method, and estimate the energy ergy input to internal tides and compared it with the sources for the same three periods. The budget method work against buoyancy forces in the basin waters of and the results from it are described in section 3, while fjords, assuming that all of the internal tidal energy was the energy sources are estimated in section 4, and com- lost to turbulence in the basin water. They found an pared with the mixing in section 5. The paper is dis- ef®ciency factor, ␥ ϭ 0.06, which indicates that the cussed and concluded in section 6. mixing mainly takes place in benthic boundary layers. That benthic boundary layers dominate vertical diffu- sion has been con®rmed by tracer experiments in fjords, 2. The dataset lakes and, oceans (Stigebrandt 1979; Goudsmit et al. Gullmar Fjord is 28 km long and 1±2 km wide, with 1997; Ledwell and Bratkovich 1995), and by micro- maximum depth 116 m and sill depth 43 m. During the structure measurements in oceans and lakes (Polzin et period August to mid-October 1997 a mooring equipped al. 1997; MacIntyre et al. 1999). The most interesting with two upward-looking acoustic Doppler current pro- aspect of the results from Stigebrandt and Aure (1989) ®lers (ADCPs) and 29 temperature sensors, some of is that the ef®ciency is relatively constant for a large them also measuring conductivity and pressure, was number of fjords with different geometry, which indi- placed at depth 108 m in the central part of the fjord. cates that the ef®ciency of mixing within benthic bound- The dataset from this mooring (M1s) is described in ary layers may be controlled by some, yet unknown, more detail in AL. In the present paper we only use mechanism. One important property of the internal data from the bottom-mounted ADCP and nine tem- seiches is that their amplitudes are highly varying in perature sensors located below sill level. The ADCP response to the forcing. Thus, if the internal seiches are (RDI 600 kHz SC-ADCP) was con®gured with 4-m important for the basin water mixing, the mixing will bins, 60 pings/ensemble giving an ensemble standard change correspondingly. deviation ϳ1cmsϪ1. The temperature sensors were In Gullmar Fjord there is a third energy source, be- distributed vertically with 5±10 m separations. All in- sides the semidiurnal tides and the internal seiches, struments were set to record 10-min averaged data. The which needs to be taken into account. This is the external sensors were calibrated against three CTD pro®les taken (barotropic) seiche (period ϳ 2 h) investigated by Pars- close to the mooring during the period. mar and Stigebrandt (1997). They showed that the Figure 2 shows a spectrum of the velocities projected damping of the external seiche could be explained solely in the fjord direction at depth 59 m. The spectrum has

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FIG. 3. Temperature time series and least squares ®tted linear trends for the periods, day 240± 260, day 263±273, and day 276±288. Depths are 107, 102, 93, 88, 83, 78, 73, 67, and 62 m. peaks at the external seiche frequency (ϳ0.54 cph), at no deep-water renewal (see, e.g., Stigebrandt and Aure semidiurnal tidal frequencies (ϳ0.08 cph), and at the 1989; Axell 1998). internal seiche frequencies at around 0.023 and 0.048 Below sill level the rate of change in horizontally cph, discussed more thoroughly in AL. Figure 3 shows averaged temperature can be written as (neglecting mo- temperature time series from the basin water. These lecular diffusion, and heat transport through the side show (i) large oscillations with period 1±3 days related boundaries) to internal seiche motions and (ii) general warming ץ T͘ 1͗ץ trends related with vertical mixing. The oscillations are ϭϪ (A͗wT͘), (2) zץtAץ largest in the ®rst 28 days, followed by a relatively calm 14-day period, which is again followed by a 14-day where angle brackets denote horizontal averaging over the period with larger oscillations. Wind data from the whole basin area A(z). The advective transports on the mouth of the fjord (not shown) are consistent with this, right-hand side can be separated into reversible and irre- with strong wind events during the periods with large- versible contributions. The reversible transports by internal amplitude internal seiche motions and no wind during waves and turbulence can be removed by averaging over the period with small-amplitude internal seiche motions. timescales longer than the longest internal wave periods. In the following we estimate the mixing during these Irreversible contributions can be caused by dense intru- three periods to see if there is any connection with the sions and by turbulent mixing. During stagnant conditions energy supply from the internal seiches, the external the contribution from dense intrusions is zero, leaving tur- seiche, and the tides. bulent mixing as the only way to cause irreversible vertical transports. Although molecular diffusion can be neglected 3. Mixing as a direct tranport mechanism in (2), it is molecular dif- fusion on small scales that causes the irreversible net con- a. The budget method tribution from turbulent advective motions, as discussed The water below the sill-level pycnocline in Gullmar in Winters et al. (1995). Fjord is stagnant during most of the year, being renewed The horizontally and time averaged vertical transports only in winter and early spring. The stagnant conditions caused by vertical mixing can be parameterized with a enable us to use a budget method to determine the dia- turbulent diffusion coef®cient, ␬t, in the following man- pycnal mixing below the sill-level pycnocline. The bud- ner: get method has been used widely to estimate diapycnal (z, (3ץ/T ͗͘ץmixing in laterally bounded basins during periods with ͗wT ͘ϭϪ␬t

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TABLE 1. Values of ␤, de®ned in (10), and work against buoyance forces in the basin water, W. Day 240±260 263±273 276±268 ␤ 1.2 1.5 1.1 W (W) 1020 178 254 where overbar denotes time averaging over a timescale long enough to remove reversible contributions. Com- bining (2) and (3) and integrating vertically the diffusion coef®cient can be found from the temperature ®eld as

z0 tdzץ/T ͗͘ץA ͵ ϪH ␬t(z0) ϭ , (4) z)|z0ץ/T ͗͘ץA) where heat transports through the bottom have been FIG. 4. Turbulent diffusion coef®cients, ␬t, vs buoyancy frequency neglected. If both salt and temperature are stably strat- squared N 2 for the periods, day 240±260, day 263±273, and day 276± i®ed, the salt and temperatures are mixed by the same 288. advective turbulent motions. The salt transports can therefore be parameterized with the same diffusion co- with respect to both temperature and salt, so there should ef®cient, be no potential for double-diffusive processes, and the

-z. (5) assumption of one diffusion coef®cient for salt and temץ/S ͗͘ץwS ͘ϭϪ␬t͗ perature must therefore be valid (see also section 4c). Assuming a linear equation of state, ␳ ϭ ␣T ϩ ␤S, the The heat exchange through the bottom and sides of the vertical mass transports can also be parameterized as basin are assumed negligible. z. (6) The basin water mixing is estimated for three differentץ/␳ ͗͘ץw␳ ͘ϭϪ␬t͗ periods: day 240±260, day 263±273, and day 276±288. The work against gravity caused by moving dense ¯uid These are chosen to cover the two relatively energetic upward and light ¯uid downward below level z is 0 periods and the calm period in the middle. The tem- z0 perature derivatives are found from least squares ®tting W ϭ ͵ g͗␳w ͘Adz. (7) of linear lines to the temperature time series (Fig. 3). ϪH The spiky events around day 237 and 275 are left out Using (6) this can also be written as because they tend to have too large an in¯uence on the least squares ®tting. Vertical gradients and ¯uxes are z0 2 found at the midpoint between two sensors using central W ϭ ␬t͗␳ ͘N A dz, (8) ͵ differences. We integrate (8) up to the level z 0 ϭϪ58 ϪH m, which is the midpoint between sensors at Ϫ53 and where N is the buoyancy frequency based on time and Ϫ62 m, since this is always located below the sill-level horizontally averaged densities. Now W, which is a mea- pycnocline. Below this level T±S plots give no indi- sure for the increase in potential energy, can be esti- cation of advective in¯ow of ``new'' water from outside mated from the changes in temperature by calculating the fjord, so we assume that the required condition of the diffusion coef®cients from (4) and inserting in (8). stagnancy is ful®lled. The work against buoyancy forces, W, for the three periods is given in Table 1. It is seen that it changes b. Results considerably over the three periods. In the ®rst period In Gullmar Fjord the velocity spectra below sill level the work is almost a factor of 6 larger than in the middle decrease rapidly for frequencies lower than the lowest- period, and in the last period the work is, again, a little frequency internal seiche (see Fig. 2). The contributions larger than in the middle period. In the next sections from reversible internal wave motions are therefore re- we look at the sources of energy, to see if they can moved from (2) by using an averaging timescale longer explain these large variations in ``observed'' mixing. than the inverse of this frequency. We assume that the The diffusion coef®cients, ␬t, are shown as function active mixing regions and the rest of the basin are in of the buoyancy frequency N, in Fig. 4. The value of balance on such a timescale, so the density pro®le ob- ␬t varies with about one decade from the bottom to depth served at one point is representative for the horizontally 58 m, and at each depth the values vary with up to a averaged density pro®le. The density pro®le is stable factor of 5, consistent with the variations in W and the

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TABLE 2. Work against buoyancy forces in the basin water, W and linear dependency between ␬t and W found from (8). energy inputs to basin water, F , F , F , related with internal The linear least squares ®ts shown in the log±log plot IS SD ES seiches, semidiurnal tides, and external seiche, and ef®ciencies of each correspond to a relation mixing with various combinations of energy sources. Ϫ␤ ␬t ϰ N , (9) Day Averages where ␤ is a constant. The values of ␤ are given for 240± 263± 276± Ensem- each period in Table 1. Gargett (1984) reviewed budget 260 273 288 ble Time estimates of basinwide diffusion coef®cients in lakes W (kW) 1.02 0.178 0.254 0.48 0.60 and fjords, and found the value of ␤ to vary between FIS (kW) 3.9 0.36 1.5 1.9 2.4 F (kW) 2.7 3.1 2.1 2.6 2.6 0.8 and 1.2. Stigebrandt and Aure (1989) found the SD F (kW) 2.3 0.51 1.6 1.5 1.7 value of ␤ to be in the range 1.47 Ϯ 0.35 in Norwegian ES W/(F ϩ F ) 0.20 0.049 0.069 0.11 fjords. Imberger (1998) reviewed the values of ␤ to the ES SD W/(F ϩ F ϩ F ) 0.11 0.045 0.049 0.069 range from 1±2 in lakes, but also noted that the vertical ES SD IS variation of basin-average diffusion is dependent on the type of forcing and basin geometry. No satisfactory ex- 3. To do this it is necessary to determine the horizontal planation has as yet been given for this dependency or velocity amplitudes for the seiche frequencies during for the observed values of ␤. The values in Table 1 are these periods. Each of the periods is too short relative seen to lay within the ranges found in the literature. to the seiche periods to get any con®dence in a spectral There is a slight indication of larger N dependency in estimate of the amplitude. An alternative method is to the middle period than in the other periods. bandpass ®lter the whole data series and determine the It is worth noticing the importance of having either velocity amplitudes from the variance of the ®ltered ®ne resolution in time or a large integration time, when series. This method is used here. The time series are estimating long-term time trends. Consider that the mix- bandpass ®ltered with a seventh-order Butterworth ®lter ing is found from two single CTD pro®les, as is often with cutoff frequencies 0.015 and 0.06 cph, keeping done. If one pro®le is situated near the trough of an 99% of the variance at 0.02 and 0.043 cph. This way internal wave, while the second is situated near a neigh- we keep most of the motions connected with internal boring peak, the result will be a measure for the changes seiches and cut away low-frequency and tidal motions. in potential energy of the internal wave ®eld, rather than The velocity amplitudes are found as u ϭ 21/2 u , for the mixing. From Fig. 3 it can be seen that the o rms where u is the root-mean-square of the bandpass ®l- irreversible changes caused by the mixing are ®rst ex- rms tered velocity. ceeded by the reversible internal wave changes on time- The energy ¯ux is found from (10), using N 0.006 scales on the order of a month. Mixing estimates based ϭ sϪ1, which is a representative value for the buoyancy on two CTD pro®les with less than one month separation frequency between the bottom and 60 m, and using therefore does not make sense. We avoid this problem Ϫ m 0.07 mϪ1 for the vertical wavenumber, as found by having high time resolution. ϭ in AL from FEOF analysis of the horizontal velocities. The results are shown in Table 2. 4. Energy sources The energy ¯ux into the basin water from the internal seiches is seen to vary with more than a factor of 10 a. Energy ¯ux related to internal seiches and to be large in the same period as the mixing is large The downward-radiating wave observed at M1s (see and small in the same period as the mixing is small. introduction) is assumed to be the main source of me- This indicates that the internal seiches can be respon- chanical energy to the basin water from the internal sible for some of the mixing, but before concluding seiches. The energy ¯ux, FIS, in this wave is estimated anything we need to look at the other energy sources from the observed velocities at M1s by integrating the horizontal energy ¯ux at M1s. As shown in AL this can b. Energy ¯uxes related to tides and external seiche be written as The energy input to internal waves caused by the z2 1 N F ϭ ␳u2 B dz, (10) interaction of oscillatory barotropic motions with steep IS ͵ 2 0 m topography and strati®cation can be estimated with Sta- z1 cey's (1984) expression, which is an extension of the where N is the buoyancy frequency, u 0 is the horizontal velocity amplitude, m is the vertical wavenumber of the Stigebrandt (1976) model, to take into account contin- internal seiche motion, and B is the z-dependent width uous strati®cation. Using Stacey's expression the energy ¯ux in the progressive internal waves can be written as of the fjord. As in AL we integrate from the bottom (z1 ϱ 22 ϭϪH) to 60-m depth (z2 ϭϪ60 m). 1 ␳ BcUWÄ (Ϫd) F ϭ 0 mn m n , (11) In order to relate the energy ¯uxes to the mixing, we ͸ 0 nϭ1 2 are interested in determining the mean energy ¯uxes WdzÄ 2 ͵ nz during each of the same three periods used in section Ϫh

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Ä where Wn(z)isthenth vertical velocity mode, cn is the amplitude changes with distance from the sill, but also nth mode phase velocity, d is the depth of the sill, Bm with width and cross-sectional area of the fjord. We is the width of the mouth, h is a representative depth assume that the surface elevation at M1s is represen- inside the sill, and Um is the amplitude of the external tative for the average surface elevation and use ␺ ϭ velocity over the sill. The amplitude of the barotropic 1.0. The surface amplitude is found from spectral anal- velocity at the sill can be found from the amplitude of ysis of a pressure time series from the bottom of M1s. the surface elevation a at a position inside the sill using The spectral analysis is performed on 43-h-long time the relation segments, and in each segment the seiche amplitude is found by averaging bins in the frequency range from ␻As 0.45 to 0.6 cph. This gives 12 degrees of freedom in Um ϭ ␺a , (12) Am the spectral estimate, which is satisfactory in terms of where ␻ is the angular frequency, A the surface area reducing the con®dence interval. The energy ¯uxes into s internal waves are calculated for each segment using of the fjord, Am the cross-sectional area of the mouth, and ␺ a factor taking into account the alongfjord vari- the method outlined above. The average energy input ation of the surface elevation. The surface elevation of within each of the three periods is then calculated as the semidiurnal tides is close to constant over the length the mean of the segments located within each period. of the fjord since the wavelength is much longer than The results are shown in Table 2. the fjord length; therefore ␺ can be set to 1. For the The energy input by the external seiche varies with external seiche, which has a node at the sill, the coef- a factor of 4.5, and the calmest period is coincident with ®cient may be different from one as discussed below. the period of least mixing and least internal seiche input 2 (day 263±273). The external seiche is forced either by The variation of F/U m caused by changes in the strat- i®cation is found from (11) by determining the normal local wind or by oscillations in the surface elevations modes from CTD data from different periods and in- outside the fjord, which are also caused by wind. It is therefore not surprising that the external seiche varies serting d ϭ 43 m and Bm ϭ 500 m. For the ®rst period we use a CTD pro®le from day 145 taken at M1s. For with the strength of the wind. the second period we use a pro®le taken closer to the The energy supply from the external seiche is at most mouth at day 171 and for the last period a pro®le from 85% of the energy supply from the semidiurnal tides day 187 at M1s. The representative depth inside the sill and 65% on average over the investigated periods. Con- is chosen to be h ϭ 70 m, which means that the normal sidering the length of the experiment, which may not modes are calculated for a water column between z ϭ be representative for a whole year, these numbers are Ϫ70 m and the surface. in good accordance with earlier results (Parsmar and The amplitude of the barotropic velocity in the mouth, Stigebrandt 1997), which suggest a yearly average con- tribution from the external seiche of 48% relative to the Um, is found from the surface elevation a at M1s using (12), and a is found from spectral analysis of a pressure tides. time series at the bottom of M1s. c. Turbulent dissipation

1) TIDES Before looking at the mixing ef®ciencies we want to make sure that the estimated energy inputs are strong For the semidiurnal tides the parameter values in (12) enough to generate turbulence in the strati®ed basin wa- 6 2 3 2 are As ϭ 51 ϫ 10 m , Am ϭ 23 ϫ 10 m , and ␺ ϭ ter. To do this we introduce the small-scale Froude num- 1. The amplitude of the semidiurnal tides does not vary ber (Imberger and Ivey 1991) de®ned as much during the measuring period, so we ®nd the rms value from spectral analysis of the whole pressure time ␧ 1/2 series as an average in the band 0.078±0.086 cph. The Fr␥ ϭ , (13) ΂΃␯N2 surface amplitude is aSD ϭ 0.12 m. The ®nal results for the tidal energy input are shown in Table 2. The numbers where ␧ is the dissipation of turbulent kinetic energy, are reasonably constant and there is no correlation with ␯ is the viscosity, and N is the buoyancy frequency. the variations in mixing. According to Imberger and Ivey (1991) this number has to exceed 3.9 in order to have turbulence. Given the rate of mechanical energy, F, lost in the basin water, 2) EXTERNAL SEICHE the volume-average dissipation can be found from The external seiche is a little more complicated since F(1 Ϫ ␥) the amplitude varies in time and in space along the fjord ͗␧͘ ϭ , (14) ␳V (the seiche has a node at the sill). The variation in space comes in through the parameter ␺, which is the relation where ␥ is the fraction used for mixing and V is the between the fjord-averaged surface elevation and the volume of the basin water. In the least energetic period elevation at M1s. This is quite complicated since the the total energy input is F ϭ 4 kW (Table 2). Inserting

Unauthenticated | Downloaded 09/27/21 04:25 AM UTC SEPTEMBER 2001 ARNEBORG AND LILJEBLADH 2573 this value in (14) with ␥ K 1 and V ഠ 0.5 km3, the tides the ef®ciency will become larger for all periods, average dissipation is ͗␧͘ ϭ 8 ϫ 10Ϫ9 WkgϪ1. Inserting but relatively more for the periods with least mixing. into (13) with N ϭ 0.006 sϪ1 and ␯ ϭ 1 ϫ 10Ϫ6 m 2 sϪ1 R an overestimate of the mixing during the ®rst period we get Fr␥ ϭ 15. The dissipation within turbulent patch- or an underestimate for the last period. es is easily one to two decades larger than the volume R that the mixing ef®ciency may not be constant, but average, so there is no doubt that the energy is lost to change with type and/or strength of energy input. Dur- turbulence rather than directly to viscous diffusion. ing a period where all energy sources are energetic, nonlinear interactions between the different types of internal waves could lead to increased mixing away 5. Mixing ef®ciency from the boundaries, and thereby an increased ef®- ciency, as discussed in the introduction. In order to estimate the mixing ef®ciencies we assume that the energy inputs, estimated in the previous section, The ®rst point seems to be the most reasonable cause are lost to turbulence in the basin water of the fjord. since the method for determining the internal seiche Some fraction of this energy is ®nally lost to heat by contribution is very approximate. For example, we only viscous dissipation, while the rest is used to increase include the energy ¯ux passing M1s, while energy may the basin-water potential energy by mixing. This last already be lost between the sill and M1s. There may fraction is the mixing ef®ciency ␥, de®ned in (1). also be downward energy ¯uxes from the inner part of The results from the previous sections are shown in the fjord, which are not observed at M1s. Table 2. As mentioned earlier, the mixing is much larger The last point may be supported by the ®ndings in in the ®rst than in the second period, and then again section 3 that the diffusion coef®cient parameterization somewhat larger in the third period. The tides cannot (9) has a larger value of ␤ in the least energetic period explain these variations since the tidal contribution is than in the other periods, suggesting a different mixing almost constant. The external seiche contribution can mechanism. It is, however, not possible to draw such explain some variations since it is smallest in the middle conclusions based on this dataset. There are only three period. Best resemblance with the mixing variation is, data points and each of these are related with quite large however, obtained by the internal seiche contribution. uncertainties (maybe up to Ϯ50%), so the amount of The agreement is not perfect since the ratio between the signi®cant statistic information is rather limited. ®rst and second period is larger than the corresponding What we can conclude is that (i) the observed work ratio for the mixing, while the ratio between the ®rst against buoyancy by mixing is highly varying, and that and third period is smaller than the corresponding ratio the energy input by the internal seiches varies in a sim- for the mixing. ilar manner; (ii) the ef®ciency of mixing for the com- Table 2 also shows the mixing ef®ciencies with and bined internal wave ®eld from tides, external seiche, without the internal seiche contributions included. As a and internal seiches is 7% on average, similar to what consequence of the properties mentioned above the mix- has been found in other fjords; and iii) without the in- ing ef®ciency varies less if the internal seiche contri- ternal seiches included the mixing ef®ciency varies bution is taken into account than if it is not. The ratio more, the average is 11%, which is too large compared of the largest to the smallest mixing ef®ciency is 2.4 with other fjords. It therefore seems reasonable to con- with the internal seiche included and 4.1 without. The clude that the internal seiches contribute to the basin largest change by including the internal seiche is ob- water mixing, and that the proposed energy path gives tained in the ®rst period, where the ef®ciency is reduced the correct order of magnitude for the contribution, al- from 0.2 to 0.11. When looking at average mixing ef- though it may be an underestimate. ®ciencies we get the numbers 0.07 and 0.11 with and without the internal seiche included. The ®rst value cor- 6. Concluding remarks responds well with the results from other fjords (␥ ഠ 0.06: Stigebrandt and Aure 1989). Basin water mixing in tidally energetic fjords is main- Although the mixing ef®ciency varies less with the ly caused by dissipation of internal tides, but other pro- internal seiches included than without, the results above cesses may be important in fjords with weak tidal forc- are not excessively convincing, since the estimated mix- ing, similar to the case in lakes, where internal seiches ing ef®ciency is large during a period with large mixing are important for the deep-water mixing. In the present and small during periods with little mixing. The reasons paper we investigate the basin water mixing in Gullmar for these variations may be Fjord with the main focus on the contribution from in- ternal seiches. In a companion paper (AL) it was pro- R an underestimate of the internal seiche contribution. posed that the internal seiches lose some of their energy A larger contribution from the internal seiche would in the basin water after critical re¯ection and that a mainly decrease the ef®ciency in the ®rst period. fraction of this should be available for mixing. In order R an overestimate of the contribution from the tides. By to investigate this we estimate the work against buoy- reducing the almost constant contribution from the ancy forces for three different periods and compare the

Unauthenticated | Downloaded 09/27/21 04:25 AM UTC 2574 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 31 results with estimates of the energy input from the in- Acknowledgments. During this work L. Arneborg was ternal seiches to the basin water. The work against buoy- supported by a grant from the Danish Research Council, ancy forces due to mixing varies with almost a factor while B. Liljebladh and the ®eld measurements were of 6 between a period with large internal seiche motions supported by a grant to Anders Stigebrandt from the and a period without internal seiche motions. There are, Swedish Natural Science Research Council (NFR). however, other energy sources that need to be taken into account, namely the internal tides and internal waves REFERENCES generated by the external seiche at the sill. The tidal Arneborg, L., and B. Liljebladh, 2001: The internal seiches in Gullmar contribution is almost constant, but the external seiche Fjord. Part I: Dynamics. J. Phys. 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During a windy year the second vertical mode of the internal seiche in an alpine lake. there will be more mixing than during a calm year. This Limnol. Oceanogr., 37, 1705±1719. Parsmar, R., and A. Stigebrandt, 1997: Observed damping of baro- has a double effect on the oxygen conditions in the basin tropic seiches through baroclinic wave drag in the Gullmar Fjord. water. Besides increasing the turbulent transport of ox- J. Phys. Oceanogr., 27, 849±857. ygen, the mixing decreases the basin water density, Polzin, K. L., J. M. Toole, J. R. Ledwell, and R. W. Schmitt, 1997: Spatial variability of turbulent mixing in the abyssal ocean. Sci- which increases the possibility that an oxygen-rich ence, 276, 93±96. dense in¯ow can penetrate all the way to the bottom. Stacey, M. W., 1984: The interaction of tides with the sill of a tidally There are, however, some missing links before we energetic inlet. J. Phys. 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