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ARNEBORG AND LILJEBLADH
The Internal Seiches in Gullmar Fjord. Part II: Contribution to Basin Water Mixing
LARS ARNEBORG
AND
BENGT LILJEBLADH
Department of Oceanography, University of Göteborg, Göteborg, Sweden
(Manuscript received 26 July 1999, in final 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 reflection 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 efficiency, defined 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
The diapycnal mixing below the upper mixed layer
is important in the oceans because it drives the thermohaline circulation, in lakes because it brings oxygen
down and nutrients up and in fjords mainly because it
decreases the density of the basin water (the water below
sill level), so that new oxygen-rich water can come in
from outside and replace the ‘‘old’’ water. Stigebrandt
(1976) proposed that the basin water mixing in fjords
was mainly caused by breaking of internal tides near
sloping bottoms. This hypothesis was strengthened in
Stigebrandt and Aure (1989), where they showed that
the basin water mixing in a large number of Norwegian
fjords was related to dissipation of internal tides generated at the sills. In fjords with low tidal energy, other
processes may, however, become important. A reasonable hypothesis is that internal seiche motions become
relevant for the mixing in the deep waters, similar to
the case in stratified lakes (Wiegand and Chamberlain
1987; Münnich et al. 1992) and this is investigated in
the present paper.
The forcing and damping of the internal seiches in
Gullmar Fjord is investigated in a companion paper (Arneborg and Liljebladh 2001, hereafter AL), based on
data from two mooring datasets from 1997. Gullmar
Fjord, located on the Swedish West Coast, is a fjord
Corresponding author address: Lars Arneborg, Department of
Oceanography, Earth Sciences Centre, University of Göteborg, Box
460, 405 30 Göteborg, Sweden.
E-mail: [email protected]
q 2001 American Meteorological Society
with low tidal forcing. Energy is put into the internal
seiches by both the direct action of wind on the fjord
and coastal forcing due to oscillations in the coastal
stratification, the two energy sources being of about the
same magnitude. Analyses of horizontal velocities and
vertical displacements inside the sill indicate that the
motions observed in the basin water are progressive
wave motions rather than standing wave motions, with
down- and inward group velocity emanating from the
sill region. An alongfjord section of the topography with
characteristics corresponding to the actual stratification
and seiche frequency (Fig. 1, same as Fig. 11 in AL)
shows that most of the slopes in the fjord basin are near
critical to reflection of the internal seiche. It is therefore
suggested that the energy radiating downward is lost to
turbulence in the basin water after near-critical reflection
from the bottom. Estimates of the energy flux radiating
downward from the sill region show that it is only about
2% of the total energy loss in the internal seiches, but
that this contribution may be important for the basin
water mixing relative to the contribution from internal
tides. The main purpose with the present paper is to
investigate if there is any evidence that some of the
energy radiating downward is used to mix the basin
water and how important this contribution is compared
with other energy sources.
One measure for the relation between mixing and
energy sources is the mixing efficiency g , defined as
g5
O
W
,
F
(1)
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FIG. 1. Long-section of Gullmar Fjord, showing deepest topography
and selected characteristics corresponding to stratification at day 245
and frequency 0.023 cph. Also shown is a group velocity vector and
the suggested area of dissipation.
where W is the total work against buoyancy forces in
the basin water caused by mixing and the denominator
is the sum of all mechanical energy inputs F to the basin
water. Based on lab experiments and microstructure
measurements, Imberger and Ivey (1991) concluded that
the maximum fraction of turbulent energy that can be
used for mixing is 0.2 and that this efficiency is generally obtained in turbulence far from boundaries, while
it can be much smaller for benthic boundary-layer turbulence. Stigebrandt and Aure (1989) computed the energy input to internal tides and compared it with the
work against buoyancy forces in the basin waters of
fjords, assuming that all of the internal tidal energy was
lost to turbulence in the basin water. They found an
efficiency factor, g 5 0.06, which indicates that the
mixing mainly takes place in benthic boundary layers.
That benthic boundary layers dominate vertical diffusion has been confirmed by tracer experiments in fjords,
lakes and, oceans (Stigebrandt 1979; Goudsmit et al.
1997; Ledwell and Bratkovich 1995), and by microstructure measurements in oceans and lakes (Polzin et
al. 1997; MacIntyre et al. 1999). The most interesting
aspect of the results from Stigebrandt and Aure (1989)
is that the efficiency is relatively constant for a large
number of fjords with different geometry, which indicates that the efficiency of mixing within benthic boundary layers may be controlled by some, yet unknown,
mechanism. One important property of the internal
seiches is that their amplitudes are highly varying in
response to the forcing. Thus, if the internal seiches are
important for the basin water mixing, the mixing will
change correspondingly.
In Gullmar Fjord there is a third energy source, besides the semidiurnal tides and the internal seiches,
which needs to be taken into account. This is the external
(barotropic) seiche (period ; 2 h) investigated by Parsmar and Stigebrandt (1997). They showed that the
damping of the external seiche could be explained solely
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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 internal (IS) seiche frequencies and the M 2 tidal frequency; 95% confidence intervals are based on 20–320 degrees of freedom in the
spectral estimates.
by generation of internal waves at the sill. These internal
waves may be assumed to contribute to the basin water
mixing in a manner similar to the internal tides, and
more importantly the amplitude of these are highly varying. Variations in mixing can therefore be caused by
both internal and external seiches, which means that we
need to quantify both.
Based on the summer–fall dataset from 1997 presented in AL and briefly recapitulated in section 2, we
calculate the basin water mixing for three different periods, using the budget method, and estimate the energy
sources for the same three periods. The budget method
and the results from it are described in section 3, while
the energy sources are estimated in section 4, and compared with the mixing in section 5. The paper is discussed and concluded in section 6.
2. The dataset
Gullmar Fjord is 28 km long and 1–2 km wide, with
maximum depth 116 m and sill depth 43 m. During the
period August to mid-October 1997 a mooring equipped
with two upward-looking acoustic Doppler current profilers (ADCPs) and 29 temperature sensors, some of
them also measuring conductivity and pressure, was
placed at depth 108 m in the central part of the fjord.
The dataset from this mooring (M1s) is described in
more detail in AL. In the present paper we only use
data from the bottom-mounted ADCP and nine temperature sensors located below sill level. The ADCP
(RDI 600 kHz SC-ADCP) was configured with 4-m
bins, 60 pings/ensemble giving an ensemble standard
deviation ;1 cm s 21 . The temperature sensors were
distributed vertically with 5–10 m separations. All instruments were set to record 10-min averaged data. The
sensors were calibrated against three CTD profiles taken
close to the mooring during the period.
Figure 2 shows a spectrum of the velocities projected
in the fjord direction at depth 59 m. The spectrum has
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ARNEBORG AND LILJEBLADH
2569
FIG. 3. Temperature time series and least squares fitted 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 (;0.54 cph), at
semidiurnal tidal frequencies (;0.08 cph), and at the
internal seiche frequencies at around 0.023 and 0.048
cph, discussed more thoroughly in AL. Figure 3 shows
temperature time series from the basin water. These
show (i) large oscillations with period 1–3 days related
to internal seiche motions and (ii) general warming
trends related with vertical mixing. The oscillations are
largest in the first 28 days, followed by a relatively calm
14-day period, which is again followed by a 14-day
period with larger oscillations. Wind data from the
mouth of the fjord (not shown) are consistent with this,
with strong wind events during the periods with largeamplitude internal seiche motions and no wind during
the period with small-amplitude internal seiche motions.
In the following we estimate the mixing during these
three periods to see if there is any connection with the
energy supply from the internal seiches, the external
seiche, and the tides.
3. Mixing
a. The budget method
The water below the sill-level pycnocline in Gullmar
Fjord is stagnant during most of the year, being renewed
only in winter and early spring. The stagnant conditions
enable us to use a budget method to determine the diapycnal mixing below the sill-level pycnocline. The budget method has been used widely to estimate diapycnal
mixing in laterally bounded basins during periods with
no deep-water renewal (see, e.g., Stigebrandt and Aure
1989; Axell 1998).
Below sill level the rate of change in horizontally
averaged temperature can be written as (neglecting molecular diffusion, and heat transport through the side
boundaries)
]^T &
1 ]
52
(A^wT &),
]t
A ]z
(2)
where angle brackets denote horizontal averaging over the
whole basin area A(z). The advective transports on the
right-hand side can be separated into reversible and irreversible contributions. The reversible transports by internal
waves and turbulence can be removed by averaging over
timescales longer than the longest internal wave periods.
Irreversible contributions can be caused by dense intrusions and by turbulent mixing. During stagnant conditions
the contribution from dense intrusions is zero, leaving turbulent mixing as the only way to cause irreversible vertical
transports. Although molecular diffusion can be neglected
as a direct tranport mechanism in (2), it is molecular diffusion on small scales that causes the irreversible net contribution from turbulent advective motions, as discussed
in Winters et al. (1995).
The horizontally and time averaged vertical transports
caused by vertical mixing can be parameterized with a
turbulent diffusion coefficient, k t , in the following manner:
^ wT & 5 2k t ]^ T &/]z,
(3)
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TABLE 1. Values of b, defined in (10), and work against buoyance
forces in the basin water, W.
Day
b
W (W)
240–260
263–273
276–268
1.2
1020
1.5
178
1.1
254
where overbar denotes time averaging over a timescale
long enough to remove reversible contributions. Combining (2) and (3) and integrating vertically the diffusion
coefficient can be found from the temperature field as
E
z0
A]^ T &/]t dz
2H
k t (z 0 ) 5
(A]^ T &/]z)| z 0
,
(4)
where heat transports through the bottom have been
neglected. If both salt and temperature are stably stratified, the salt and temperatures are mixed by the same
advective turbulent motions. The salt transports can
therefore be parameterized with the same diffusion coefficient,
^ wS & 5 2k t ]^ S &/]z.
(5)
Assuming a linear equation of state, r 5 aT 1 bS, the
vertical mass transports can also be parameterized as
^ wr & 5 2k t ]^ r &/]z.
(6)
The work against gravity caused by moving dense fluid
upward and light fluid downward below level z 0 is
W5
E
z0
g^ rw &A dz.
(7)
2H
Using (6) this can also be written as
W5
E
z0
k t ^ r &N 2 A dz,
(8)
2H
where N is the buoyancy frequency based on time and
horizontally averaged densities. Now W, which is a measure for the increase in potential energy, can be estimated from the changes in temperature by calculating
the diffusion coefficients from (4) and inserting in (8).
b. Results
In Gullmar Fjord the velocity spectra below sill level
decrease rapidly for frequencies lower than the lowestfrequency internal seiche (see Fig. 2). The contributions
from reversible internal wave motions are therefore removed from (2) by using an averaging timescale longer
than the inverse of this frequency. We assume that the
active mixing regions and the rest of the basin are in
balance on such a timescale, so the density profile observed at one point is representative for the horizontally
averaged density profile. The density profile is stable
FIG. 4. Turbulent diffusion coefficients, k t , vs buoyancy frequency
squared N 2 for the periods, day 240–260, day 263–273, and day 276–
288.
with respect to both temperature and salt, so there should
be no potential for double-diffusive processes, and the
assumption of one diffusion coefficient for salt and temperature must therefore be valid (see also section 4c).
The heat exchange through the bottom and sides of the
basin are assumed negligible.
The basin water mixing is estimated for three different
periods: day 240–260, day 263–273, and day 276–288.
These are chosen to cover the two relatively energetic
periods and the calm period in the middle. The temperature derivatives are found from least squares fitting
of linear lines to the temperature time series (Fig. 3).
The spiky events around day 237 and 275 are left out
because they tend to have too large an influence on the
least squares fitting. Vertical gradients and fluxes are
found at the midpoint between two sensors using central
differences. We integrate (8) up to the level z 0 5 258
m, which is the midpoint between sensors at 253 and
262 m, since this is always located below the sill-level
pycnocline. Below this level T–S plots give no indication of advective inflow of ‘‘new’’ water from outside
the fjord, so we assume that the required condition of
stagnancy is fulfilled.
The work against buoyancy forces, W, for the three
periods is given in Table 1. It is seen that it changes
considerably over the three periods. In the first period
the work is almost a factor of 6 larger than in the middle
period, and in the last period the work is, again, a little
larger than in the middle period. In the next sections
we look at the sources of energy, to see if they can
explain these large variations in ‘‘observed’’ mixing.
The diffusion coefficients, k t , are shown as function
of the buoyancy frequency N, in Fig. 4. The value of
k t varies with about one decade from the bottom to depth
58 m, and at each depth the values vary with up to a
factor of 5, consistent with the variations in W and the
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ARNEBORG AND LILJEBLADH
linear dependency between k t and W found from (8).
The linear least squares fits shown in the log–log plot
each correspond to a relation
k t } N 2b ,
(9)
where b is a constant. The values of b are given for
each period in Table 1. Gargett (1984) reviewed budget
estimates of basinwide diffusion coefficients in lakes
and fjords, and found the value of b to vary between
0.8 and 1.2. Stigebrandt and Aure (1989) found the
value of b to be in the range 1.47 6 0.35 in Norwegian
fjords. Imberger (1998) reviewed the values of b to the
range from 1–2 in lakes, but also noted that the vertical
variation of basin-average diffusion is dependent on the
type of forcing and basin geometry. No satisfactory explanation has as yet been given for this dependency or
for the observed values of b. The values in Table 1 are
seen to lay within the ranges found in the literature.
There is a slight indication of larger N dependency in
the middle period than in the other periods.
It is worth noticing the importance of having either
fine resolution in time or a large integration time, when
estimating long-term time trends. Consider that the mixing is found from two single CTD profiles, as is often
done. If one profile is situated near the trough of an
internal wave, while the second is situated near a neighboring peak, the result will be a measure for the changes
in potential energy of the internal wave field, rather than
for the mixing. From Fig. 3 it can be seen that the
irreversible changes caused by the mixing are first exceeded by the reversible internal wave changes on timescales on the order of a month. Mixing estimates based
on two CTD profiles with less than one month separation
therefore does not make sense. We avoid this problem
by having high time resolution.
4. Energy sources
a. Energy flux related to internal seiches
The downward-radiating wave observed at M1s (see
introduction) is assumed to be the main source of mechanical energy to the basin water from the internal
seiches. The energy flux, FIS , in this wave is estimated
from the observed velocities at M1s by integrating the
horizontal energy flux at M1s. As shown in AL this can
be written as
FIS 5
E
z2
z1
1 2N
ru B dz,
2 0m
TABLE 2. Work against buoyancy forces in the basin water, W and
energy inputs to basin water, FIS , FSD , FES , related with internal
seiches, semidiurnal tides, and external seiche, and efficiencies of
mixing with various combinations of energy sources.
(10)
where N is the buoyancy frequency, u 0 is the horizontal
velocity amplitude, m is the vertical wavenumber of the
internal seiche motion, and B is the z-dependent width
of the fjord. As in AL we integrate from the bottom (z1
5 2H) to 60-m depth (z 2 5 260 m).
In order to relate the energy fluxes to the mixing, we
are interested in determining the mean energy fluxes
during each of the same three periods used in section
Day
Averages
240–
260
263–
273
276–
288
Ensemble
Time
W (kW)
FIS (kW)
FSD (kW)
FES (kW)
1.02
3.9
2.7
2.3
0.178
0.36
3.1
0.51
0.254
1.5
2.1
1.6
0.48
1.9
2.6
1.5
W/(FES 1 FSD)
W/(FES 1 FSD 1 FIS)
0.20
0.11
0.049
0.045
0.069
0.049
0.11
0.069
0.60
2.4
2.6
1.7
3. To do this it is necessary to determine the horizontal
velocity amplitudes for the seiche frequencies during
these periods. Each of the periods is too short relative
to the seiche periods to get any confidence in a spectral
estimate of the amplitude. An alternative method is to
bandpass filter the whole data series and determine the
velocity amplitudes from the variance of the filtered
series. This method is used here. The time series are
bandpass filtered with a seventh-order Butterworth filter
with cutoff frequencies 0.015 and 0.06 cph, keeping
99% of the variance at 0.02 and 0.043 cph. This way
we keep most of the motions connected with internal
seiches and cut away low-frequency and tidal motions.
The velocity amplitudes are found as u o 5 21/2 urms ,
where urms is the root-mean-square of the bandpass filtered velocity.
The energy flux is found from (10), using N 5 0.006
s 21 , which is a representative value for the buoyancy
frequency between the bottom and 260 m, and using
m 5 0.07 m 21 for the vertical wavenumber, as found
in AL from FEOF analysis of the horizontal velocities.
The results are shown in Table 2.
The energy flux into the basin water from the internal
seiches is seen to vary with more than a factor of 10
and to be large in the same period as the mixing is large
and small in the same period as the mixing is small.
This indicates that the internal seiches can be responsible for some of the mixing, but before concluding
anything we need to look at the other energy sources
b. Energy fluxes related to tides and external seiche
The energy input to internal waves caused by the
interaction of oscillatory barotropic motions with steep
topography and stratification can be estimated with Stacey’s (1984) expression, which is an extension of the
Stigebrandt (1976) model, to take into account continuous stratification. Using Stacey’s expression the energy
flux in the progressive internal waves can be written as
`
˜ n2 (2d)
1 r 0 Bm cn Um2 W
F5
,
(11)
0
n51 2
2
W̃ nz dz
O
E
2h
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where W̃ n (z) is the nth vertical velocity mode, c n is the
nth mode phase velocity, d is the depth of the sill, B m
is the width of the mouth, h is a representative depth
inside the sill, and U m is the amplitude of the external
velocity over the sill. The amplitude of the barotropic
velocity at the sill can be found from the amplitude of
the surface elevation a at a position inside the sill using
the relation
Um 5 ca
vA s
,
Am
(12)
where v is the angular frequency, A s the surface area
of the fjord, A m the cross-sectional area of the mouth,
and c a factor taking into account the alongfjord variation of the surface elevation. The surface elevation of
the semidiurnal tides is close to constant over the length
of the fjord since the wavelength is much longer than
the fjord length; therefore c can be set to 1. For the
external seiche, which has a node at the sill, the coefficient may be different from one as discussed below.
The variation of F/U 2m caused by changes in the stratification is found from (11) by determining the normal
modes from CTD data from different periods and inserting d 5 43 m and B m 5 500 m. For the first period
we use a CTD profile from day 145 taken at M1s. For
the second period we use a profile taken closer to the
mouth at day 171 and for the last period a profile from
day 187 at M1s. The representative depth inside the sill
is chosen to be h 5 70 m, which means that the normal
modes are calculated for a water column between z 5
270 m and the surface.
The amplitude of the barotropic velocity in the mouth,
U m , is found from the surface elevation a at M1s using
(12), and a is found from spectral analysis of a pressure
time series at the bottom of M1s.
VOLUME 31
amplitude changes with distance from the sill, but also
with width and cross-sectional area of the fjord. We
assume that the surface elevation at M1s is representative for the average surface elevation and use c 5
1.0. The surface amplitude is found from spectral analysis of a pressure time series from the bottom of M1s.
The spectral analysis is performed on 43-h-long time
segments, and in each segment the seiche amplitude is
found by averaging bins in the frequency range from
0.45 to 0.6 cph. This gives 12 degrees of freedom in
the spectral estimate, which is satisfactory in terms of
reducing the confidence interval. The energy fluxes into
internal waves are calculated for each segment using
the method outlined above. The average energy input
within each of the three periods is then calculated as
the mean of the segments located within each period.
The results are shown in Table 2.
The energy input by the external seiche varies with
a factor of 4.5, and the calmest period is coincident with
the period of least mixing and least internal seiche input
(day 263–273). The external seiche is forced either by
local wind or by oscillations in the surface elevations
outside the fjord, which are also caused by wind. It is
therefore not surprising that the external seiche varies
with the strength of the wind.
The energy supply from the external seiche is at most
85% of the energy supply from the semidiurnal tides
and 65% on average over the investigated periods. Considering the length of the experiment, which may not
be representative for a whole year, these numbers are
in good accordance with earlier results (Parsmar and
Stigebrandt 1997), which suggest a yearly average contribution from the external seiche of 48% relative to the
tides.
c. Turbulent dissipation
1) TIDES
For the semidiurnal tides the parameter values in (12)
are A s 5 51 3 10 6 m 2 , A m 5 23 3 10 3 m 2 , and c 5
1. The amplitude of the semidiurnal tides does not vary
much during the measuring period, so we find the rms
value from spectral analysis of the whole pressure time
series as an average in the band 0.078–0.086 cph. The
surface amplitude is a SD 5 0.12 m. The final results for
the tidal energy input are shown in Table 2. The numbers
are reasonably constant and there is no correlation with
the variations in mixing.
2) EXTERNAL
SEICHE
The external seiche is a little more complicated since
the amplitude varies in time and in space along the fjord
(the seiche has a node at the sill). The variation in space
comes in through the parameter c, which is the relation
between the fjord-averaged surface elevation and the
elevation at M1s. This is quite complicated since the
Before looking at the mixing efficiencies we want to
make sure that the estimated energy inputs are strong
enough to generate turbulence in the stratified basin water. To do this we introduce the small-scale Froude number (Imberger and Ivey 1991) defined as
Frg 5
1 2
«
nN 2
1/2
,
(13)
where « is the dissipation of turbulent kinetic energy,
n is the viscosity, and N is the buoyancy frequency.
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,
the volume-average dissipation can be found from
^«& 5
F(1 2 g)
,
rV
(14)
where g is the fraction used for mixing and V is the
volume of the basin water. In the least energetic period
the total energy input is F 5 4 kW (Table 2). Inserting
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ARNEBORG AND LILJEBLADH
this value in (14) with g K 1 and V ø 0.5 km 3 , the
average dissipation is ^«& 5 8 3 10 29 W kg 21 . Inserting
into (13) with N 5 0.006 s 21 and n 5 1 3 10 26 m 2 s 21
we get Fr g 5 15. The dissipation within turbulent patches is easily one to two decades larger than the volume
average, so there is no doubt that the energy is lost to
turbulence rather than directly to viscous diffusion.
5. Mixing efficiency
In order to estimate the mixing efficiencies we assume
that the energy inputs, estimated in the previous section,
are lost to turbulence in the basin water of the fjord.
Some fraction of this energy is finally lost to heat by
viscous dissipation, while the rest is used to increase
the basin-water potential energy by mixing. This last
fraction is the mixing efficiency g , defined in (1).
The results from the previous sections are shown in
Table 2. As mentioned earlier, the mixing is much larger
in the first than in the second period, and then again
somewhat larger in the third period. The tides cannot
explain these variations since the tidal contribution is
almost constant. The external seiche contribution can
explain some variations since it is smallest in the middle
period. Best resemblance with the mixing variation is,
however, obtained by the internal seiche contribution.
The agreement is not perfect since the ratio between the
first and second period is larger than the corresponding
ratio for the mixing, while the ratio between the first
and third period is smaller than the corresponding ratio
for the mixing.
Table 2 also shows the mixing efficiencies with and
without the internal seiche contributions included. As a
consequence of the properties mentioned above the mixing efficiency varies less if the internal seiche contribution is taken into account than if it is not. The ratio
of the largest to the smallest mixing efficiency is 2.4
with the internal seiche included and 4.1 without. The
largest change by including the internal seiche is obtained in the first period, where the efficiency is reduced
from 0.2 to 0.11. When looking at average mixing efficiencies we get the numbers 0.07 and 0.11 with and
without the internal seiche included. The first value corresponds well with the results from other fjords (g ø
0.06: Stigebrandt and Aure 1989).
Although the mixing efficiency varies less with the
internal seiches included than without, the results above
are not excessively convincing, since the estimated mixing efficiency is large during a period with large mixing
and small during periods with little mixing. The reasons
for these variations may be
R an underestimate of the internal seiche contribution.
A larger contribution from the internal seiche would
mainly decrease the efficiency in the first period.
R an overestimate of the contribution from the tides. By
reducing the almost constant contribution from the
2573
tides the efficiency will become larger for all periods,
but relatively more for the periods with least mixing.
R an overestimate of the mixing during the first period
or an underestimate for the last period.
R that the mixing efficiency may not be constant, but
change with type and/or strength of energy input. During a period where all energy sources are energetic,
nonlinear interactions between the different types of
internal waves could lead to increased mixing away
from the boundaries, and thereby an increased efficiency, as discussed in the introduction.
The first point seems to be the most reasonable cause
since the method for determining the internal seiche
contribution is very approximate. For example, we only
include the energy flux passing M1s, while energy may
already be lost between the sill and M1s. There may
also be downward energy fluxes from the inner part of
the fjord, which are not observed at M1s.
The last point may be supported by the findings in
section 3 that the diffusion coefficient parameterization
(9) has a larger value of b in the least energetic period
than in the other periods, suggesting a different mixing
mechanism. It is, however, not possible to draw such
conclusions based on this dataset. There are only three
data points and each of these are related with quite large
uncertainties (maybe up to 650%), so the amount of
significant statistic information is rather limited.
What we can conclude is that (i) the observed work
against buoyancy by mixing is highly varying, and that
the energy input by the internal seiches varies in a similar manner; (ii) the efficiency of mixing for the combined internal wave field from tides, external seiche,
and internal seiches is 7% on average, similar to what
has been found in other fjords; and iii) without the internal seiches included the mixing efficiency varies
more, the average is 11%, which is too large compared
with other fjords. It therefore seems reasonable to conclude that the internal seiches contribute to the basin
water mixing, and that the proposed energy path gives
the correct order of magnitude for the contribution, although it may be an underestimate.
6. Concluding remarks
Basin water mixing in tidally energetic fjords is mainly caused by dissipation of internal tides, but other processes may be important in fjords with weak tidal forcing, similar to the case in lakes, where internal seiches
are important for the deep-water mixing. In the present
paper we investigate the basin water mixing in Gullmar
Fjord with the main focus on the contribution from internal seiches. In a companion paper (AL) it was proposed that the internal seiches lose some of their energy
in the basin water after critical reflection and that a
fraction of this should be available for mixing. In order
to investigate this we estimate the work against buoyancy forces for three different periods and compare the
2574
JOURNAL OF PHYSICAL OCEANOGRAPHY
results with estimates of the energy input from the internal seiches to the basin water. The work against buoyancy forces due to mixing varies with almost a factor
of 6 between a period with large internal seiche motions
and a period without internal seiche motions. There are,
however, other energy sources that need to be taken into
account, namely the internal tides and internal waves
generated by the external seiche at the sill. The tidal
contribution is almost constant, but the external seiche
contribution is highly varying, with large contributions
during periods of large mixing and small contributions
during periods of little mixing. Taking all contributions
into account the average mixing efficiency is 7%, similar
to what has been found in tidally dominated fjords.
On average over the investigated periods 36% of the
mixing is caused by the internal seiche and 25% by the
external seiche, leaving 39% for the tides. This means
that the internal seiches are almost as important for the
basin water mixing as are the tides. In the end both the
external and internal seiches are forced by local or remote winds, which means that 61% of the mixing is
caused by wind forcing. Although the actual numbers
are dependent on how representative the investigated
period is, it is sure that the basin water mixing will vary
between windy and calm years. During a windy year
there will be more mixing than during a calm year. This
has a double effect on the oxygen conditions in the basin
water. Besides increasing the turbulent transport of oxygen, the mixing decreases the basin water density,
which increases the possibility that an oxygen-rich
dense inflow can penetrate all the way to the bottom.
There are, however, some missing links before we
can estimate the mixing variations caused by wind variations. These are (i) the relation between the downward-propagating component of the internal seiche and
the total seiche damping, (ii) the relation between wind
and coastal density variations, and (iii) the forcing of
the external seiche. Regarding (i), it was shown in AL
that the downward-radiating component only carried 2%
of the total energy loss from the seiches. This is, however, dependent on the actual stratification, and this dependency has not as yet been sorted out.
VOLUME 31
Acknowledgments. During this work L. Arneborg was
supported by a grant from the Danish Research Council,
while B. Liljebladh and the field measurements were
supported by a grant to Anders Stigebrandt from the
Swedish Natural Science Research Council (NFR).
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