## Abstract

A baroclinically forced deep-water renewal event in a small Swedish sill fjord is investigated using a one-dimensional numerical model and a dataset that resolves temporal variations in salinity and oxygen. The observations indicate an almost complete renewal of the basin water within a period of 2–3 weeks. The details of the renewal process are emphasized by modeling the sill flow as well as the resulting dense bottom plume, with various rates of entrainment. It is found that sill mixing is relatively unimportant, but that entrainment increases the deep-water inflow by a factor of 2–4. Different formulas for calculating plume entrainment are compared and the model's sensitivity to variations in sill flow and bottom friction is investigated. It is shown that even weak entrainment, occurring at shallow depths where the density difference between the plume and the resident water is large, has a significant impact. Entrainment prolongs the time it takes for a complete renewal (i.e., to flush out all resident water) and, on the moderate timescales considered here, yields lower post-renewal salinity and oxygen concentrations. This implies that entrainment (and mixing) during renewal may be as important as basin water diffusion in setting the timescale for forthcoming renewal events.

## 1. Introduction

Deep water renewal in sill fjords is the process by which dense *juvenile* water from outside enters the fjord and sinks to the bottom, replacing the less dense *resident* water. In this study we use observations and a numerical 1D model to investigate a renewal event in the Swedish sill fjord Byfjorden. This renewal took place during the spring of 1974 as a result of a strong upwelling event along the coast. Juvenile water entered the fjord as a primarily baroclinic inflow during a period of two to three weeks, thereby almost completely exchanging the deep water volume. We investigate the influence of sill flow and bottom plume mixing on the post-renewal state of the basin waters. In addition, the relative performance of different entrainment formulas is compared.

Comprehensive early summaries of fjord processes including bottom-water renewal are given by Gade and Edwards (1980) and Farmer and Freeland (1983). However, though much work has been done in attempting to determine what causes renewals (Gade 1973; Edwards and Edelsten 1977; Stigebrandt and Aure 1989; Cannon et al. 1990; Gillibrand et al. 1995; Allen and Simpson 1998; Liungman 2000), only a few of these authors address the details of such events. An obvious reason is their episodic nature. Large-scale renewals are rare and often difficult to predict.

Renewals may occur on a variety of timescales (Farmer and Freeland 1983), from the spring–neap cycle (Geyer and Cannon 1982; Allen and Simpson 1998) to decades or more (Gade 1973). The majority of fjords, however, appear to experience seasonal deep water inflows, often activated by a corresponding seasonal variation in the outer hydrography (Farmer and Freeland 1983; Stigebrandt and Aure 1989). Between renewals the basin water stagnates. Interior diffusion, driven by, for example, interaction between the flow and the topography at the sill and/or local winds (Stigebrandt and Aure 1989), will slowly homogenize the water column and decrease the deep water density.

Renewals may be partial or complete. In the former case not all resident basin water is flushed out of the fjord during the renewal. Either the juvenile water is not dense enough to reach the bottom, such that it interleaves at an intermediate depth, or the volume of inflowing water is insufficient to replace the entire water mass below the sill level. However, higher post-renewal densities at all depths need not necessarily indicate complete renewal, though this is a common assumption (Allen and Simpson 1998). An important factor is to what extent mixing takes place between juvenile and resident water during the renewal. Mixing will recirculate a fraction of the resident water within the fjord as new basin water is formed. As a result the volume required for complete renewal will be larger than the total basin volume below sill level.

It is often implicitly assumed that the interior diffusion is the dominant factor determining when the next renewal event will occur (Gade and Edwards 1980). If, on the other hand, the renewal is not complete due to substantial mixing between juvenile and resident water, the resulting post-renewal basin density will be lower than the original density of the juvenile water. Thus, further mixing by interior diffusion is not necessary for another renewal to occur.

A renewal involves two key processes: exchange across the sill and the descent of inflowing dense water as an entraining bottom plume. Sill exchange, in relation to fjords, has been treated extensively by Stigebrandt (1977, 1980, 1990), whereas Farmer and Armi (1986) presented a detailed analysis of the dynamics of combined baroclinic and barotropic two-layer flow over a sill. Dense bottom plumes have been the subject of a great deal of work, with reference to marginal sea overflows (Price and O'Neil Baringer 1994; Price and Yang 1998; Borenäs and Wåhlin 2000). The particular problem of entrainment in plumes has been addressed by, for example, Ellison and Turner (1959), Bo Pedersen (1980), and Turner (1986). Even so, there are few studies of smaller-scale bottom plumes in the sea or in lakes, and almost none in relation to fjords. One exception is the paper of Edwards and Edelsten (1977) in which a particular renewal event in the Scottish fjord Loch Etive is investigated.

In the present paper we address the issue of partial versus complete renewal. By use of a fjord model, including submodels of sill flow and an entraining bottom plume, we attempt to improve our understanding of deep water exchange in fjords. A detailed study of different entrainment formulas, including comparisons to observations, is hopefully of some importance not only when considering fjords but in the context of oceanic overflows as well. We begin with a description of Byfjorden, followed in section 3 by a presentation and analysis of available data. The paper continues in section 4 with a theoretical discussion about different entrainment parameterizations. In section 5 the model is described. The results are presented in section 6. This is followed by a discussion in section 7, and the paper ends with a summary and some general conclusions.

## 2. Byfjorden

Byfjorden, shown in Fig. 1, is a small sill fjord on the Swedish west coast. It was the subject of a large-scale investigation, including an extensive field campaign, between 1970 and 1974 (Göransson and Svensson 1975; Svensson 1980b). The maximum depth of Byfjorden is 50 m, the surface area 6 × 10^{6} m^{2}, and the total volume 138 × 10^{6} m^{3}. The volume below sill depth is approximately 80 × 10^{6} m^{3}. The sill, which separates Byfjorden from the larger Havstensfjorden, consists of a strait 500 m long and 300 m wide. The strait is shallow (depth less than 1–2 m) except for a central channel, which has been dredged on several occasions. At the time of the renewal event in question the width of the channel was approximately 145 m at the surface and 80 m at the maximum depth, which was 11.5 m. Inside the sill there is a roughly v-shaped canyon running from the strait's inner end to the deep basin, a distance of approximately 700 m. The width of this canyon is between 50 and 100 m with an average bottom angle of about 170° (see Fig. 11). The bottom slope, defined as the change in depth per meter horizontal distance, increases from 1%–2% (1°) near the sill to 18% (10°) near the end of the canyon, with a mean of 5.5% (3.1°).

The tides are dominated by the semidiurnal components with a spring tidal range of about 0.35 m (Björk et al. 2000). At the head riverine input from the small river Bäveån averages approximately 4 m^{3} s^{−1}. During winter and spring the runoff may reach 10 m^{3} s^{−1} or more, whereas summer values are often below 1 m^{3} s^{−1}.

Different from many other fjords, the hydrography of the system of fjords to which Byfjorden belongs is dominated by the coastal stratification, which in turn depends on the low saline Baltic water outflow. Local freshwater supply is normally of minor importance (Björk et al. 2000). The shallow sill at the entrance to Byfjorden prevents exchange of the water mass below the sill in Byfjorden, whereas barotropic and baroclinic exchange with Havstensfjorden yields a surface layer with properties very similar to those of the surface water in Havstensfjorden. The main halocline is almost consistently located in the depth range from 10 to 18 m, with the strongest gradients usually found at or just below sill level (Svensson 1980b). Surface layer salinities above the halocline vary in the range 22 to 28 psu and temperatures between −1°C (ice cover) and 20°C. During periods of high freshwater supply, mean near-surface salinities may be as low as 15 (Svensson 1980b). The temperature variations are primarily seasonal. Below sill level the salinities and temperatures show minor variations. Typical values are 31 and 4.5°C, respectively.

In addition to the 1970–74 investigations, there is a nearly 40-yr long record of hydrographic observations available for Byfjorden. Before 1990 observations were made up to four times per year, thereafter once every month. Figure 2 shows the salinity for depths between 40 and 48 m. It is clear that the deep water of Byfjorden is stagnant for long periods, with renewals occurring every 2–5 years. During stagnation the salinity in the deep water decreases by about 0.2 yr^{−1} (Svensson 1980a). This slow diffusion is probably forced by energy flux to the internal tide, which in turn arises from the interaction of the barotropic tide with the sill (Svensson 1980b; Liungman 2000).

## 3. Data

### a. Field measurements

In the spring of 1974 an intensive field campaign was carried out in Byfjorden, in order to investigate the effect of simultaneous dredging operations. When the renewal event became evident, the campaign was intensified. Vertical profiles of salinity and temperature were measured at stations both within and outside Byfjorden during the renewal. The sampling interval varied from a few days to two weeks. Some additional profiles of oxygen and hydrogen sulphide were also taken.

The data on which we will base our analysis are shown in Figs. 3–8. Figure 3 (left panel) shows six salinity profiles from the deepest part of the fjord, obtained during the period from 3 April to 10 May. In Fig. 4 the isohaline movements in Havstensfjorden, outside the sill, are plotted from 1 April to 31 May. Profiles of oxygen and hydrogen sulphide (expressed as negative oxygen; see section 5) between 17 and 26 April are shown in Fig. 5 (left panel).

Measurements were also made in the sill area. Salinity was recorded continuously at three different depths (1, 5, and 9 m) from 24 April to 4 May, though with limited absolute accuracy. The salinity time series are plotted in Fig. 6. Some additional salinity profiles from the sill are shown in Fig. 7. Current measurements were also made on and inside the sill using gelatine pendulum current meters, and are shown in Fig. 8 [see Cederlöf et al. (1996) for a description of these instruments].

Meteorological and sea level data used to force the model are described in Liungman (2000).

### b. Analysis

#### 1) Dredging

The dredging operations, which began in late January and ended in May, caused a pronounced decrease in the deep water salinities prior to the renewal event. This cannot be seen in Fig. 2 due to the low sampling frequency of these measurements. During the dredging, approximately 1.4 m^{3} s^{−1} of slurry was ejected as a negatively buoyant turbidity jet in the central parts of Byfjorden. As a result, from late January to early April the bottom water salinity decreased from 30.7 to 28.2 psu, or almost 0.2 psu week^{−1}. Likewise, deep-water hydrogen sulphide concentrations decreased from more than 500 *μ*mol l^{−1} on 23 January to about 100 *μ*mol l^{−1} on 20 March. Since this dredging continued during the renewal event, it will to some degree disrupt the results of the forthcoming analysis. However, it should be emphasized that the rate of change of deep water salinity induced by the dredging was one order of magnitude smaller than that caused by the renewal (see below). This issue will be further discussed in section 7.

#### 2) The renewal period

The occurrence of a renewal event is evident from the salinity profiles shown in the left panel of Fig. 3. Between 17 April and 3 May the basin water salinity increased from about 28 to near 32 psu. The renewal began just before 17 April and ended soon after 3 May (see also Fig. 4). Figure 3 also shows a gradual increase of the deep water salinities at all depths, indicating that the inflow has been more or less continuous and is well distributed within the water column. This, on the other hand, does not preclude short-term variations in the inflow due to tides and varying winds.

#### 3) Some characteristics of the renewal

Figure 4 indicates that the salinity of the inflowing water increases from 28–29 psu on 15 April to maximum values of 33–34 psu between 23 April and 3 May. Salinity data from the sill region (Figs. 6 and 7) support these figures. The sill measurements also show that the sill flow is layered (Fig. 7), such that the entering high salinity water is usually found at depths below 8 m. There is also a considerable tidal variability in the sill salinity (Fig. 6). Thus, we expect a corresponding short-term variability in volume fluxes across the sill. It is also obvious that the salinities of the juvenile water are higher than those of the basin water during the entire inflow period. This difference immediately brings us to one of the main issues of this paper, namely to what extent the entering juvenile water is mixed with resident basin water and to what extent former resident water remains within the basin after the renewal has ended. A quick estimate may be performed on the basis of the oxygen profiles shown in Fig. 5 (left panel), which include hydrogen sulphide as negative oxygen. Between 17 and 23 April the inflowing water on the sill had an oxygen concentration in the range 5–6 ml l^{−1}, whereas on 23 April the oxygen concentration in the bottom water of Byfjorden was approximately 3 ml l^{−1}. As the basin water concentrations were close to zero before the start of the renewal, this implies that by 23 April the bottom water is made up of approximately equal amounts of resident and juvenile water.

#### 4) Flux estimates

The pendulum measurements shown in Fig. 8 provide an order of magnitude estimate of the volume flux on the sill during the renewal. Mean velocities of 20 cm s^{−1} over a depth of 4 m and an approximate channel width of 100 m yield an estimated flux of 80 m^{3} s^{−1}, which is of the same magnitude as the tidal flux at spring.

Assuming a quasi-steady two-layer flow over the sill, the sill inflow can also be estimated from conservation of salt according to

Here, *Q*_{in}*S*_{in} is the flux of salt into Byfjorden over the sill (lower layer) and *Q*_{out}*S*_{out} is the flux of salt out of the fjord (upper layer). Continuity of volume implies *Q*_{out} = *Q*_{in} + *Q*_{f}, where *Q*_{f} is the freshwater input to the fjord. The integral can be determined from the hypsography and observed salinity profiles (Fig. 3), whereas *S*_{in} and *S*_{out} are estimated from Fig. 7. The freshwater input was small during this period, that is, *Q*_{f} ≈ 1 m^{3} s^{−1}. The results, shown in Table 1, indicate an average sill inflow of about 100 m^{3} s^{−1}.

A method for obtaining quantitative estimates of volume fluxes to the deep water is to use the approach of Walin (1977). The volume distribution as a function of salinity *υ*(*S*), defined as the volume of water with a salinity in the interval between *S* − Δ*S* and *S*, was calculated for the four dates 17, 23, and 26 April and 3 May. Taking the difference and dividing by the time interval between these instances yields the average rate of change *dυ*(*S*)/*dt*. The results, using Δ*S* = 0.5, are shown in Table 2. From 17 to 23 April the volume of low salinity water *υ*(*S* = 28 psu) + *υ*(*S* = 28.5 psu) decreased by 127 m^{3} s^{−1} (corresponding to 66 × 10^{6} m^{3} for the 6-day period), whereas the volume of water with higher salinities (*S* > 28.5 psu) increased by the same amount. For the second period (23–26 April) there is a volume increase for salinities above 30 psu of 169 m^{3} s^{−1}, and during the third period (26 April–3 May) the increase of the volume of water with salinities above 31 psu is 125 m^{3} s^{−1}. Assuming that these increases of the volume of high salinity water *M*_{b} is due to a volume flux of juvenile water *M*_{j} with mean salinity *S*_{j} and entrainment of basin water *M*_{e} with mean salinity *S*_{e}, conservation of salt and volume yield

Here, *S*_{b} is the resulting mean salinity of the new deep water. This purely advective approach assumes a stationary sill flow and neglects diapycnal mixing caused by the dredging (see discussion). The mean salinities were estimated from Figs. 3 (*S*_{e} and *S*_{b}) and 4 (*S*_{j}). These are shown together with the calculated values of *M*_{j} and *M*_{e} in Table 3. Though the results are rather sensitive to the salinity estimates, they indicate an average inflow of juvenile water *M*_{j} of approximately 55 m^{3} s^{−1}. Hence, about half the sill flow is dense enough to take part in the flushing of the deep basin. The average of the entrainment is 84 m^{3} s^{−1}, indicating that about 60% of the volume flux interleaving in the deep basin consists of entrained basin water from intermediary depths.

In summary, the observations show a renewal event caused by an increase in the surface salinity outside the fjord. This renewal coincides with inflow events reported from the Scottish fjord Loch Etive (Edwards and Edelsten 1977) and Frierfjord in the south of Norway (Molvær 1980). A 17-day-long period with exterior salinities up to 6 psu higher than those of the resident water above the sill level give rise to a baroclinic sill flow with varying salinity and volume fluxes. The sill flow is modulated by a barotropic tidal flow of the same order of magnitude. The inflowing juvenile water is mixed with resident water to yield the basin product water, most likely as a result of entrainment into a dense bottom plume. In the following sections we will discuss different parameterizations of this entrainment process and show how the renewal event can be modeled.

## 4. Bottom plume dynamics and entrainment

To model the renewal, we will use a one-dimensional stream tube approach where the bottom plume (carrying dense juvenile water down into the fjord basin) is considered homogeneous and well-mixed in all properties (see, e.g., Price and O'Neil Baringer 1994). The plume is accelerated by gravity down the slope and retarded by bottom and interfacial friction, as well as by entrainment of ambient water. The density and volume transport of the plume can only be changed via entrainment. The ambient water is assumed to be stratified and at rest, thus nonturbulent. The plume is assumed to descend until it reaches a level of equal density, alternatively the bottom, where it interleaves. For further details see section 5a. In addition to determining the friction coefficient and initial conditions, the model requires a formula for the entrainment. Below we will present some different parameterizations, which will later be tested in the numerical model.

It is common to express the normalized entrainment rate *E* as a function of a suitable Richardson number, where *E* is is the entrainment velocity *υ*_{e} divided by the flow speed *U*. For a homogeneous bottom boundary layer separated from the ambient water by an interface, the bulk Richardson number is defined as

where *g*′ = *g*Δ*ρ*/*ρ* is the reduced gravity, *h* the layer thickness, and Δ*U* the velocity difference across the interface. Since, in our case, the ambient water is at rest, Δ*U* = *U*. The acceleration of gravity is *g,* Δg is the density difference across the interface, and *ρ* is the layer density.

Laboratory results for interfacial mixing over a wide span of Ri values were summarized by Fernando (1991). Different mixing regimes from low to high Ri values are discussed in a recent paper by Sullivan and List (1994), who also present a model for the dependence of *E* on Ri, though only for Ri > 1. However, there still does not appear to exist any clear consensus on how entrainment in stratified fluids should be parameterized.

Based on the early laboratory measurements of Ellison and Turner (1959), Turner (1986) suggested that the entrainment rate be calculated according to

where *ϕ* is the angle of the bottom to the horizontal. Since in our case the slope does not exceed 10°, we will henceforth neglect the factor cos *ϕ*. Note that the results of Ellison and Turner (1959) were limited to slopes of 12° or more, which are rare in nature.

Bo Pedersen (1980) suggested a formula for *E* that matches the data of both Ellison and Turner (1959) (Ri < 1) and Löfquist (1960) (Ri > 1), as well as some other empirical estimates of entrainment. In Bo Pedersen's formula, *E* is a function of the bottom slope *s* = sin*ϕ*, the overall friction coefficient *C*_{D}, a parameter describing the degree of turbulence and a bulk flux Richardson number, *R*^{T}_{f}, which includes the gain of turbulent kinetic energy via the entrainment process. This new Richardson flux number is considered to be nearly constant, but with different values depending on whether the flow is sub- or supercritical with respect to a long interfacial wave. Inserting the parameter values suggested by Bo Pedersen, the formula reads

where Ri_{cr} ∼ 1 is the critical Richardson number. Unfortunately, some of the required parameters are difficult to determine, for example *C*_{D}. Bo Pedersen estimates the turbulence parameter *b*′ to lie between 5 for low values of *C*_{D} and 3 for high values. In the subcritical case this is of little importance, as values of *C*_{D} between 10^{−3} and 10^{−2} yield very similar results. For a supercritical plume the best agreement with data appears to be for *C*_{D} ∼ 10^{−2}. However, Bo Pedersen also presents the following simple entrainment function, which fits most of the data very well:

As can be seen in Fig. 9, the major deviations between this expression and Eq. (2) are found for supercritical flow on low slopes, that is, in the range 10^{−2} < *s* < 10^{−1}, which is what we find in Byfjorden. It should be noted that the measurements in this range referred to by Bo Pedersen show substantial scatter, and hence it is not obvious that Eq. (2) is more correct than Eq. (3).

For subcritical flow on gentle slopes (*s* < *C*_{D}) Stigebrandt (1987) proposed the expression

for the entrainment velocity. Here, *m*_{0} = 0.6 is an empirical constant adopted from a seasonal pycnocline model and *u*∗ = *C*_{D}*U*^{2} is the friction velocity. Using

where *q* is the specific volume transport and *s* is assumed small (sin*s* ≈ *s*), Stigebrandt then determines

Inserting the value for *C*_{D} used by Stigebrandt (3.5 × 10^{−3}) yields *E* = 0.071*s*, which is almost identical to Eq. (3).

Christodoulou (1986) suggested a combination of different expressions for *E*, to describe the different types of mixing at high and low Ri values. Based on a compilation of laboratory results [see Fig. 10 and Fernando (1991)], Christodoulou proposed the following formula:

The limits 0.08163 and 12.25 are the intersection points of the respective formulas. Equation (4) has been applied to deep-sea bottom plumes by Alendal et al. (1994).

In Fig. 10 the formulas (1), (2), and (4) are compared. The formula of Christodoulou appears to show the best overall agreement with data. The formula of Bo Pedersen, Eq. (2), is discontinuous and nonunique in the approximate range 0.2 < Ri < 1.5, and the resulting entrainment rate depends on the state of the flow with respect to criticality. Note how Eq. (1) decreases rapidly toward zero as Ri → 0.8.

In our case it is possible to arrive at a crude estimate of the observed entrainment by using the current profiles from the sill and some 400 m inside the sill taken on 5 May (Fig. 8) and 23 April (not shown). It is assumed that the available data points at the inner site define the vertical extension of the bottom plume and that the cross-sectional area is given by the topography. The resulting increase in volume flux of the dense plume Δ*Q*_{d}, *υ*_{e} calculated using an estimate of the plume's interfacial area and *E* based on a mean plume velocity of 0.25 m s^{−1}, are shown in Table 4.

The mean slope *s* is approximately 0.015 along this part of the canyon inside the sill. Equation (3) then yields *E* = 1.1 × 10^{−3}, which is approximately of the same order of magnitude as the results in Table 4. It is not unlikely that the estimated values of *E* suffer from large errors. For example, the profiles at the sill and at the inner site were obtained at different times and are from different parts of the tidal cycle. This highlights the inherent difficulties of estimating entrainment from velocity profiles (cf. Edwards and Edelsten (1977)).

## 5. The model

The process-oriented, one-dimensional model used in this work is described in detail by Liungman (2000). The model includes the following processes:

baroclinic and barotropic exchange across the sill;

turbulent vertical mixing, forced by vertical velocity shears and internal wave activity;

development of the vertical velocity profiles forced by wind stress and barotropic pressure gradients; and

entrainment and interleaving of a dense bottom plume.

The major model assumption is horizontal homogeneity in all variables. The only exception is the inclusion of horizontal pressure gradients in the horizontal momentum equations, caused by tilting of the sea surface due to wind-driven advection. Note that the model includes actual hypsography. For completeness the fundamental model equations have been included in the appendix. In the following two subsections we will present detailed descriptions of the submodels describing the two processes particularly relevant in this study: the dense bottom plume and the sill exchange.

### a. The dense bottom plume

Water entering the fjord basin with a density less than or equal to that at sill level inside the fjord is interleaved directly at a level of neutral buoyancy. Inflowing layers with densities exceeding that at sill level are assumed to form a well-mixed dense bottom plume, which entrains ambient fjord water and finally also interleaves at a level of equal density. To model this flow, we consider a quasi-stationary and nonrotating plume with quadratic bottom friction. Quasi stationarity assumes that the plume adjusts to changes in the boundary conditions on a timescale shorter than the model time step. The plume is also assumed to be thin; that is, pressure gradients due to alongstream variations in plume height and density are small compared to the buoyancy acceleration caused by the density difference between the plume and the ambient water (Borenäs and Wåhlin 2000). The equations of continuity for momentum, salinity, temperature, and volume are

Here, *U*_{d}(*x*′) is the plume velocity in the downslope direction averaged over the cross-sectional area *A*_{d}(*x*′) and *g*′ = *g*(*ρ*_{d} − *ρ*)/*ρ*_{0}, where *ρ*_{d} is the plume density and *ρ* the density of the ambient water inside the fjord; *W*_{d}(*x*′) is the plume width at the interface; *S*_{d}(*x*′) and *T*_{d}(*x*′) are the salinity and temperature of the bottom plume; and *x*′ is a coordinate in the downslope direction, that is

The sum of the bottom and interfacial stress is given by

and it is assumed that the length of the bottom perimeter is approximately equal to the width at the interface. The value of *C*_{D} will be estimated from the results of Mofjeld (1988), except when using the entrainment formula given in Eq. (2) in which case the values proposed by Bo Pedersen (1980) will be adopted (see section 4). Based on the roughly triangular shape of the submarine canyon inside the sill (see section 2 and Fig. 11) we can write

where *h*_{d}(*x*′) is the maximum height of the plume and *β* = 85° (half the bottom angle). This implies that *A*_{d} = *h*^{2}_{d} tan *β*, and all equations can be rewritten in terms of only *h*_{d} instead of *A*_{d} and *W*_{d}.

The initial conditions on *U*_{d} and *h*_{d} are determined from the total transport across the sill sufficiently dense to form a bottom plume below sill level, *Q*_{d}(0), combined with the assumption that the flow rapidly becomes critical as it begins to flow down the slope at the inner end of the entrance. Hence,

where

Using the expression for *A*_{d} in terms of *h*_{d}, Eqs. (9) and (10) yield the initial condition for *h*_{d},

It is not a serious problem if the criticality assumption is not always fulfilled. The Runge–Kutta routine used to solve the set of equations (5)–(8) employs an adaptive stepsize technique and hence converges to the appropriate local solution in a very short distance.

Initial values for the plume salinity and temperature, and hence for *g*′, are calculated by summing all fluxes to be included in the plume and dividing by the total initial transport *Q*_{d}(0). For the oxygen concentration in the bottom plume we use the same continuity equation as for salinity and temperature, that is Eq. (6) with salinity replaced by oxygen concentration.

### b. The sill exchange

The flow across the sill is determined by calculating the combined baroclinic and barotropic pressure difference across the sill and then assuming a Bernoulli-type flow at all depths (Stigebrandt 1990). The velocity on the sill *u*_{s} is then given by

where

Here, Δ*p*_{c} is the baroclinic pressure difference and Δ*p*_{t} the barotropic. The parameter *δ* = (Δ*p*_{c} + Δ*p*_{t})/|Δ*p*_{c} + Δ*p*_{t}| ensures that *u*_{s} is real and of the correct sign, whereas *α* is the fraction of the total pressure difference used to accelerate the flow into the entrance (Stigebrandt 1990). The density outside the sill, *ρ*_{s}, is given by the salinity and temperature profiles in Havstensfjorden, and Δ*η* is the instantaneous sea level difference between the exterior and interior. The system is closed by invoking volume conservation in the fjord,

where *η* is the observed sea level inside the fjord, *F*_{w}(*t*) the net freshwater input per surface area, including both river discharge and precipitation−evaporation, and *W*_{s}(*z*) the width distribution of the sill. The set of equations above is solved through an iterative process that determines the unknown sea level difference Δ*η* such that Eq. (12) is fulfilled.

This model assumes that the fjord entrance is a short and narrow channel, such that frictional effects and the current velocities upstream the channel may be neglected. It also neglects mixing on the sill (see discussion). For the parameter *α *Stigebrandt (1990) recommends a value of about 0.5, based on comparisons with specific two-layer cases. [Liungman (2000) used 0.8 in Byfjorden, but *α* was then related only to the baroclinic part of the flow, since the equations were somewhat simplified.] However, when considering the overmixed case of Stommel and Farmer (1953), Stigebrandt assumed that the maximum baroclinic two-layer exchange will occur when the interface between the two layers on the sill is at mid depth, that is, the two layers are of equal thickness *H*_{s}/2. According to Farmer and Armi (1986), this is only true for a contraction, not for flow over a sill. Instead, the interface for the maximal exchange solution will be found at a depth of 0.625 *H*_{s}. The current speed on the sill is then given by (cf. Stigebrandt 1990)

where we have used Eq. (11). Integrating over the thickness of the upper layer yields the upper-layer transport per unit width

Equating this with the result for the maximal exchange rate given by Farmer and Armi, 0.208 *g*′*H*^{3/2}_{s}, we find *α* ≈ 0.2. If frictional effects are included, an even lower value for *α* may be expected. However, it should be emphasized that it is not clear whether the results for the two-layer case may be extended to a multilayered or continuously stratified flow. We will consider *α* a tuning parameter and investigate its effect on the model results.

## 6. Model results

The model was run from 15 April to 5 May in two configurations: a standard configuration, in which the bottom plume model described in section 5a was used, and a nonentraining configuration in which all inflows were interleaved immediately at a level of equal density without formation of an entraining bottom plume. The value of *α* was adjusted until the model—in the standard configuration—produced the best agreement between simulated and observed salinity profiles. The standard configuration was also used to investigate how the choice of entrainment formula, *α*, and *C*_{D} affects the behavior of the bottom plume. In the following comparison between model results and observations, focus will be on the deep water properties.

### a. Comparison to data

#### 1) Standard configuration

Figure 3 (center panel) shows the modeled salinity profiles for the standard configuration using *α* = 0.1 and *E* and *C*_{D} according to the formula of Bo Pedersen [see Eq. (2) and Fig. 9]. Modeled and observed profiles (left panel) are in very good agreement below the halocline. The model appears to produce marginally lower bottom salinities and weaker near-bottom stratification, compared to observations. However, there is an obvious discrepancy in the surface salinities, where the model predicts lower values. In our case the discrepancy is probably due to insufficient near-surface mixing in the model, as indicated by a better agreement if the wind speed was increased (not shown). Alternatively, there could be deficiencies in the description of the outflow of low-salinity surface water over the sill.

There is also a good agreement between observed and modeled oxygen concentrations (Fig. 5). The oxygen minima on 23 and 26 April are found at almost exactly the same depths as in the observations, indicating that the final plume transport is correctly estimated (see below). In the model some H_{2}S is still present on these dates, whereas the observations indicate oxygen concentrations above zero at all depths. The main difference is in the surface concentrations, where the boundary condition on O_{2} (surface saturation) does not allow for the observed decrease caused by the lifting of resident, oxygen-poor water from the deep basin toward the surface.

The temporal development of the sill inflow calculated by the model is shown in Fig. 12. It averages 71 m^{3} s^{−1}, compared to about 100 m^{3} s^{−1} estimated from measurements [see section 3b(4)]. Figure 12 also shows the modeled sill inflow in the case of only baroclinic forcing. It can be clearly seen how the daily average sill inflow is larger than that calculated without tidal forcing [as pointed out by Stigebrandt (1977)], particularly in the beginning and at the end of the simulated period when the tidal flow is large (spring) compared to the purely baroclinic flow. Between 21 and 30 April, when the tidal fluctuations are smaller (neap), the baroclinic inflow is close to its maximum value and the two curves coincide.

Figure 13 shows the temporal development of the initial and final plume properties. The entrainment flux is several times larger than the initial transport (top panel). The final plume salinity, however, is relatively insensitive to the tidal variations in the initial salinity (middle panel). Weak inflows of juvenile water correlate with high salinities, as only the densest water enters when the barotropic flow is directed outward. In these cases the final plume salinity is lower than otherwise since entrainment will have stronger impact on a small, dense plume. The bottom panel of Fig. 13, showing the interleaving depth, clearly indicates how the plume penetrates all the way to the bottom during a major part of the renewal period, though interleaving at intermediary depths is not uncommon.

The mean values of the initial plume transport and total entrainment between 17 April and 3 May are 61 and 145 m^{3} s^{−1}, respectively, yielding a total modeled mean inflow to the basin water of 206 m^{3} s^{−1}. This is clearly higher than the estimates from observations (*M*_{b} = 140 m^{3} s^{−1}), primarily due to higher entrainment. However, the basin product water was defined in section 3b(4) as water with salinities higher than 28.5, 30, and 31 psu for the three periods 17–23 April, 23–26 April, and 26 April–3 May, respectively. As indicated by the varying interleaving depths in Fig. 13, the modeled plume does not always interleave below these isohalines. Calculating the plume transports for the three periods, but including only the instances when the final plume salinity was higher than the salinities above, yields 42 m^{3} s^{−1} for the mean initial plume transport and 111 m^{3} s^{−1} for the mean total entrainment. This implies a mean deep water inflow of 153 m^{3} s^{−1}, which is close to *M*_{b}. To summarize, the model yields a slightly lower inflow of juvenile water but approximately 30% higher entrainment compared to estimates from observations (see Table 3).

#### 2) Nonentraining configuration

The picture is quite different when all inflows over the sill are interleaved without mixing (Fig. 3, right panel). The bottom salinities exceed the observed values by about 2 psu, and a much stronger stratification appears below sill level, compared to both observations and the standard configuration. The same is true for the oxygen concentrations (Fig. 5). Indeed, this figure shows the importance of mixing between juvenile and resident water, since the nonentraining configuration produces about 2 ml l^{−1} higher deep water concentrations.

The mean sill inflow for the nonentraining configuration is 73 m^{3} s^{−1}. This is similar to that of the standard configuration, mainly because the sill flow is only weakly dependent on changes in the basin stratification. Decreasing *α* by a factor of 10, which reduces the mean sill flow by 34%, still yields too high bottom salinities compared to observations. In summary, it is obvious that a nonentraining renewal can by no means reproduce the observed stratification.

### b. Sensitivity analysis

To illustrate the differences in behavior between the entrainment formulas presented in section 4, median vertical profiles of the plume salinity 〈*S*_{d}〉 and volume transport 〈*Q*_{d}〉 are plotted in Fig. 14. The median is calculated over time for each depth, but only for those instances when the plume penetrated all the way to the bottom. Hence, 〈*S*_{d}〉 and 〈*Q*_{d}〉 are not the same as the estimates of the mean plume properties referred to in the previous section.

The results show that the formulas of Bo Pedersen (1980) and Christodoulou (1986) both produce lower final plume salinities than the formula of Turner (1986), despite that the latter yields a substantially larger total entrainment. This is because the Bo Pedersen and Christodoulou formulas yield more entrainment in the upper part of the plume, where the salinity difference between the plume and the resident water is largest, producing a strong decrease in the plume salinity. Turner's formula produces very little entrainment above 20 m (Ri ≥ 0.8). The formula of Bo Pedersen, being primarily a function of the slope, yields more or less constant entrainment at all depths, whereas the Ri-dependent formulas produce the greatest entrainment near the bottom (cf. Figs. 9 and 10). Despite their differences, both the Christodoulou and Bo Pedersen formulas produce realistic salinity profiles. The formula of Turner, however, yields too high salinities in the deep basin.

The simulations discussed in this section were performed using *α* = 0.1 and *C*_{D} = 8 × 10^{−3}. The choice of *C*_{D} was based on the results of Mofjeld (1988), assuming a plume height of 3–9 m and a bottom roughness of between 0.01 and 0.05 m. Using this constant value of *C*_{D}, instead of different values for the “subcritical” and “supercritical” conditions in the formula of Bo Pedersen, produced negligible differences. To investigate the sensitivity of the model to changes in *α* and *C*_{D}, some numerical experiments were performed. The results for the entrainment formula of Bo Pedersen are presented in Figs. 15 and 16.

The parameter *α* affects the sill flow such that larger values increase the relative importance of the baroclinic flows, and vice versa for smaller values. Hence, we would expect a stronger and more dense plume for higher values of *α*, as shown in Fig. 15. Increasing *C*_{D} yields less entrainment and vice versa, but the plume salinity is only weakly affected (Fig. 16). Note the asymmetric response of the final plume transport and salinity to changes in *α* compared to changes in *C*_{D}. The interleaving volume flux varies similarly in the two cases. The salinity of the interleaving plume, on the other hand, is strongly dependent on *α* but responds only very weakly to variations of *C*_{D}.

For the entrainment formulas of Turner and Christodoulou [Eqs. (1) and (4)] the response to changes in *α* is similar, but both are much more sensitive to the value of *C*_{D} (not shown). These results apply to both the plume transport and the salinity and are due to the direct dependence on Ri. The sensitivity of the post-renewal bottom salinity to the variations in *α* and *C*_{D} was about ±0.5 psu.

To summarize, all entrainment formulas produce better agreement with the observed stratification than if the nonentraining configuration is used. However, the poorest agreement in this case is for the formula of Turner (1986).

## 7. Discussion

### a. General importance of entrainment

The results presented in the previous section clearly reveal the importance of entrainment during basin water renewal in fjords. Though the appearance of dense water on the sill is in general determined by the baroclinic response to winds and tides of the water mass outside a fjord (Cannon et al. 1990; Stigebrandt 1990), or weak runoff into the fjord (Edwards and Edelsten 1977; Gillibrand et al. 1995), no renewal will occur unless the density in the fjord basin is below that of the exterior juvenile water. Hence, either the basin density must decrease substantially between renewals or mixing during the renewals yields a post-renewal product water of lower densities than that of the juvenile water. Since the energy available for mixing below the pycnocline is small in most sill fjords (Stigebrandt and Aure 1989), it is clear that the entrainment of resident water into inflowing juvenile water during a renewal (either as plume entrainment or sill mixing; see below) will significantly affect the frequency and size of subsequent renewals. Basically, less entrainment will increase the time between renewals, unless they are complete.

If entrainment occurs, much larger inflows are required for a renewal to be complete. To illustrate this, the proportion of resident water below sill level is shown in Fig. 17 as a function of time for both the standard and the nonentraining model configurations. The proportion of resident water was determined by setting the initial concentration of a passive tracer to 1 everywhere in the fjord, giving the juvenile water a concentration of 0, and then calculating the average tracer concentration in the fjord basin. The plot shows that there is almost 20% resident water left in the basin by the end of the renewal for the entraining case, whereas, if there is no entrainment, almost all the resident water has been flushed out by 3 May. The importance of mixing between juvenile and resident water during a renewal is also evident in the measured and simulated oxygen concentrations, as discussed in section 6a(2).

Although the early work by Edwards and Edelsten (1977) and Edwards et al. (1980) identify entrainment during renewal as an important process, it seems as if its importance for consecutive renewals has not been commented on, neither in their work nor elsewhere. At least in fjords with weak interior diffusion, such as Byfjorden, short periods of intensive entrainment may very well dominate the total vertical mixing in the fjord. Furthermore, entrainment will also, as clearly seen from a comparison between the entraining and nonentraining case, reduce the vertical density gradient, thereby indirectly contributing to a more efficient vertical diffusion.

### b. Entrainment formulas

It is interesting to note the quite different behavior of the three entrainment formulas considered here (Fig. 14). While the Bo Pedersen formula produces a similar entrainment at all depths, the other two yield an entrainment that increases with depth, as the density difference relative the ambient water decreases. The formula of Turner yields hardly any entrainment at all above 20 m depth.

Turner's formula [Eq. (1)] has been used extensively in streamtube models of marginal sea overflows [see, e.g., Price and O'Neil Baringer (1994) and Borenäs and Wåhlin (2000)]. This is somewhat surprising considering the abrupt cutoff of entrainment at Ri = 0.8. However, if the slope is steep early in the path of the plume and rotation is important, it is possible that all mixing takes place at an early stage, after which the plume slows down and enters a regime where the buoyancy force is approximately balanced by the Coriolis force. This appears to be the case for several marginal sea overflows where entrainment is confined to a region on the shelf break (Price and O'Neil Baringer 1994; Price and Yang 1998). Hence, the end result may still be realistic. If, on the other hand, the plume is subcritical for most of its path (Ri > 1), it is unlikely that the formula of Turner will produce a correct net entrainment. Turner's formula is also more sensitive to the value of the friction coefficient *C*_{D} (not shown). Setting *C*_{D} = 1.6 × 10^{−2} yields a final median plume transport 〈*Q*_{d}〉_{final} of only 255 m^{3} s^{−1} and a final median plume salinity 〈*S*_{d}〉_{final} of 30.8 psu. For low friction (*C*_{D} = 4 × 10^{−3}) the corresponding results are 〈*Q*_{d}〉_{final} ≈ 632 m^{3} s^{−1} and 〈*S*_{d}〉_{final} ≈ 30.0 psu. A similar sensitivity is found for the formula of Christodoulou. The formula of Bo Pedersen on the other hand, is fairly insensitive to the value of *C*_{D}, as *E* is primarily a function of the bottom slope. However, this obviously raises doubts concerning the use of this formula in rotating plumes, which may flow parallel to the isobaths. The sensitivity of the three entrainment formulas to the parameter *α* is similar.

It should be emphasized that none of the formulas tested here are capable of fully describing the 3D turbulent nature of the flow. For example, one may expect vigorous mixing in regions where the plume is thin relative to the bottom roughness length, such as at the edges of the plume where the interface intersects the bottom. Also, the plume is most likely layered, such that less dense water in the upper parts of the plume interleave at shallower depths. To our knowledge, attempts to model geophysical flows in three dimensions are scarce. In their 2D model of the Denmark Strait overflow, Jungclaus and Backhaus (1994) determine the entrainment velocity from a turbulent vertical exchange coefficient, which in turn is parameterized by an algebraic function of the turbulent Schmidt number (calculated as a function of Ri), the plume velocities, and the density difference.

### c. Sill and plume flow

In the model used here, the initial conditions of the bottom plume are determined by the inflow across the sill. In this context the parameter *α*, introduced by Stigebrandt (1990), is important. Since the net flow over the sill is fixed by continuity of volume [see Eq. (12)] and the baroclinic pressure difference Δ*p*_{c} is given by the stratification on either side of the sill, changes in *α* can only affect the calculated sea level difference Δ*η*. Decreasing *α* yields a larger sea level difference and hence stronger barotropic flow, and vice versa.

The results indicate that the model underestimates the volume flux over the sill. This implies that the two-layered baroclinic flow component is weaker in the model than in the observations, although a somewhat larger fraction of the modeled sill inflow is dense enough to form a plume. If *α* is increased, however, the model produces too high deep water salinities. Furthermore, the simulated initial plume transport is slightly lower than the estimates of the deep water inflow over the sill *M*_{j}, whereas the entrainment in the model is approximately 30% higher than the entrainment *M*_{e} estimated from observations. The most likely explanation for these discrepancies is that the model produces a higher initial plume salinity compared to the estimate based on observations (*S*_{j}). That this is actually the case can be seen from Fig. 13. The mean initial plume salinity is 31.1 psu for the period 17 to 23 April, 32.8 psu for 23 to 26 April and 33.2 psu for 26 April to 3 May. These values are 0.5 psu higher than the corresponding values for *S*_{j} (Table 3). The reason is that in the model the initial plume properties are calculated from the modeled salt fluxes over the sill and because of the baroclinic forcing these fluxes will be biased toward the highest salinities. The estimates of *S*_{j}, on the other hand, are arithmetic mean values of all the observed exterior salinities higher than the mean deep water salinity inside the fjord during the period in question. Using the mean initial plume salinities calculated by the model as estimates of *S*_{j} yields averages for *M*_{j} and *M*_{e} of 44 and 96 m^{3} s^{−1}, respectively, which agrees well with model results. Thus, the entrainment estimated from observations is slightly more than twice the initial inflow.

In the model we have neglected mixing on the sill, though this process is implicitly included in our estimates of *M*_{j} and *M*_{e}. However, this mixing will be limited by the short length of the channel and the sharp density gradient. A rough estimate, assuming a two-layered flow of equal layer thicknesses in a rectangular channel of width 100 m, depth 10 m, and length 500 m, yields a decrease in the salinity of the inflowing water of approximately 0.4 psu. [This is based on average current velocities and salinities in the two layers of 0.2 and −0.2 m s^{−1}, and 32 and 28 psu, respectively. Thus, Ri is approximately 1.0 for both layers, and we may expect *E* ∼ 10^{−3} (Bo Pedersen 1980; Fernando 1991).] This shows that in our case mixing on the sill is much less significant than plume entrainment. Neglecting sill mixing will cause the model to underestimate the inflow over the sill and overestimate the entrainment. Including mixing on the sill would result in a slightly higher value for *α*. Gillibrand et al. (1995) use a formula by Cokelet and Stewart (1985) to determine how the salinity of the inflowing water was reduced due to sill mixing. They found that in Loch Sunart up to 70% of the outflowing surface water was mixed with inflowing juvenile water to form new deep water during weak stratification, but only 20% during strong stratification. Undoubtedly, the influence of sill mixing may vary a great deal from one fjord to another, particularly due to different tidal fluxes, and the relative importance of sill mixing versus entrainment is an issue for future studies.

Different from Loch Sunart, a fjord with strong tides and supercritical flow on the sill (Gillibrand et al. 1995), the sill flow in Byfjorden is generally subcritical. Following Farmer and Armi (1986), a two-layer flow is said to be critical where the composite Froude number

The densimetric Froude number *F*_{i} for layers *i* = 1 (upper) and 2 (lower) is given by

where *u*_{i} is the layer speed, *h*_{i} the layer thickness, and *g*′ = *g*(*ρ*_{2} − *ρ*_{1})/*ρ*_{2}.

Measurements on the sill of Byfjorden during the renewal indicate velocities of about 0.1–0.2 m s^{−1} and layer thicknesses of approximately 7 and 4 m for the upper and lower layer, respectively (Fig. 8). The average salinities of the layers differed by about 4–6 psu (Table 1), which implies that *g*′ ≈ 0.03–0.05 m s^{−2}. This yields *G*^{2} ∼ 0.1; thus the flow was subcritical on the sill and there was no hydraulic control.

Possibly the controlling section was divided, such that the upper layer was critical at the outer end of the entrance channel and the lower, inflowing layer was critical at the inner end. The alongchannel tilt of the interface would be caused by interfacial friction, producing a thinner upper (lower) layer at the outer (inner) end. If we estimate the lower layer transport per unit width *q*_{2} = *u*_{2}*h*_{2} using the numbers above, we find *q*_{2} ≈ 0.8 m^{2} s^{−1}. If we assume continuity, that is, *q*_{1} ≈ *q*_{2}, and neglect entrainment, we can calculate the critical layer depth for which *F*^{2}_{i} = 1. This yields *h*_{i} ≈ 2.5 m, which seems plausible enough.

### d. Dredging

As mentioned above, the original reason for the field campaign was to monitor the effects of intensive dredging operations in the fjord. Although the dredging continued during the course of the renewal, we have neglected its effects on deep water mixing. It can be seen that after the inflow ended, the basin salinities decreased by about 0.2 psu during the last week (Fig. 3), which agrees with the mean change before the renewal (see section 3). Hence, without dredging the post-renewal basin salinities would have been at most 0.5 psu higher, that is, 32.5 instead of 32 psu. Furthermore, because of the dredging less saline surface water (mixed with resident water) could have been transported all the way down to the bottom as a negatively buoyant turbidity plume. If the suspended sediment then settled out on to the bottom, instabilities in the stratification would result, causing increased mixing within the deep basin.

## 8. Summary and conclusions

A main goal of this paper was to show the importance of mixing during renewal of the basin water in a fjord. Our results from Byfjorden indicate that the deep water inflow increases by a factor of 2–3, mainly due to entrainment of resident water. Mixing of the inflowing water results in longer timescales for a complete renewal, as well as weaker post-renewal stratification of the basin water. In many fjords, mixing by plume entrainment (and/or sill mixing) may be more important than mixing by vertical diffusion in determining the long-term deep water density and thereby the time between renewals. Finally, the results of the numerical model were found to agree well with observations, and the small differences could be explained by different estimates of the initial properties of the juvenile water.

## Acknowledgments

The authors would like to thank Anders Stigebrandt for inspiring discussions, and Agneta Malm for preparing the map.

## REFERENCES

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

_{2}S in Framvaren Fjord.

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

**,**

### APPENDIX

#### The Fundamental Model Equations

The equations to be solved are the horizontally averaged equations of continuity for salinity and temperature (cf. Stigebrandt 1987):

Here *S*(*z*, *t*) and *T*(*z*, *t*) are the fjord basin salinity and temperature, *A*(*z*) is the horizontal area, and *z* the vertical coordinate with origin at the undisturbed surface. The bottom is at *z* = −*H*; *D*_{S}(*z*, *t*) and *D*_{T}(*z*, *t*) are the vertical diffusivities for salt and heat, *w*(*z*, *t*) is the vertical velocity, *ρ*_{0} a reference density, and *c*_{p}(*S*, *T*) the heat capacity of water. Finally, *q*_{in}(*z*, *t*) represents the net inflow per unit depth with salinity *S*_{in}(*z*, *t*) and temperature *T*_{in}(*z*, *t*), and *Q*_{T}(*z*, *t*) is the net penetrative heat flux through the surface absorbed per unit depth. Note that *S*_{in} and *T*_{in} are set equal to *S* and *T*, respectively, if *q*_{in} < 0 at that level.

To solve for *S* and *T* we require equations for the diffusivities in the diffusion terms, the vertical velocity in the advection terms, the inflows—and for the temperature also the cross-surface heat flux—in the source terms, as well as appropriate boundary conditions. The diffusivities *D*_{S} and *D*_{T} are determined using the standard *k*–*ɛ* model, as described by, for example, Burchard et al. (1998). The model has been extended to include a source term for internal wave energy to correctly model the weak interior diffusion mentioned in the introduction (Liungman 2000).

The boundary conditions for *S* and *T* are zero flux through the surface and the bottom. Heat exchange through the surface, inflow over the sill, and freshwater input are handled via the source terms. To begin, the net inflow at each depth is given by

where *q*_{s}(*z*, *t*) is the net inflow per unit depth over the sill that interleaves above the sill depth *H*_{s} and *Q*_{d}(*z*, *t*) is the volume flow in a possible dense bottom plume. This is further discussed in sections 5a and 5b. The vertical advection is determined from continuity of volume, which yields

with the boundary condition *w*(−*H*) = 0.

A simplified equation of state is used:

The maximum difference between this formula and the UNESCO equation of state for typical profiles of *S* and *T* in Byfjorden is about 0.1 kg m^{−3} and occurs for low salinities.

Finally, oxygen is modeled as a tracer in the same fashion as salinity, but with a sink due to oxidation of hydrogen sulphide. This oxidation is a complicated chemical process that, among other factors, depends on the levels of Fe, Mn, available oxygen, and bacterial processes (Millero 1991). We consider the fastest reactions only and assume that they are immediate. Hydrogen sulphide is consequently included as negative oxygen concentrations, based on the reactions described by the stoichiometric formulae

(Millero 1991). Hence, for every mol H_{2}S that is oxidized one mol of O_{2} is consumed. Thus the oxidation of SO^{2−}_{3} to sulphate as well as the oxidation of S_{2}O^{2−}_{3} to sulphate and sulphur are neglected. Our approach implies that during the renewal event advection, mixing, and oxidation of hydrogen sulphide will dominate over other biological and chemical processes consuming oxygen. At the surface we simplify by setting the oxygen concentration in the uppermost computational cell to the value of the saturation concentration.

## Footnotes

*Corresponding author address:* Olof Liungman, SMHI, Nya Varvet 31, Västra Frölunda S-426 71 Sweden. Email: olof.liungman@smhi.se