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## Abstract

A model for the exchange of salt and water between the Baltic and the Sea (the Skagerrak) is presented. Because of strong inter-basin interactions in the Baltic entrance area, the model must include the Kattegat and the Belt Sea. These are modeled by horizontally homogeneous two-layer sub-models. The most prominent dynamical properties of the sub-models are wind-driven entrainment flows and rotational-baroclinic, hydraulic controls. The model is driven by a meteorologically forced barotropic transport *Q _{h}
* [calculated from the freshwater supply to the Baltic (

*Q*) and the sea level fluctuations in the Kattegat], and turbulent entrainment flows coupled to the wind speed

_{f}*W*and, in the Belt Sea, also to the barotropic transport. The most important bathymetric features of the basins are included.

The model equations are integrated numerically for a test period of 1½ years. The stratification in the Kattegat, as well as in the Belt Sea, is quite well predicted. It is found that approximately one-half of the salt transport into the Baltic is carried out by the dispersive mode associated with the barotropic fluctuations.

The effects of (short-term) changes in the external parameters *Q _{h}
*,

*Q*,

_{f}*W*

*S*

_{2K}(the salinity of the Skagerrak water) upon the stratification in the Belt Sea and the Kattegat are also investigated. Finally, the effects of long-term changes in the external parameters upon the surface salinity of the Baltic are investigated.

## Abstract

A model for the exchange of salt and water between the Baltic and the Sea (the Skagerrak) is presented. Because of strong inter-basin interactions in the Baltic entrance area, the model must include the Kattegat and the Belt Sea. These are modeled by horizontally homogeneous two-layer sub-models. The most prominent dynamical properties of the sub-models are wind-driven entrainment flows and rotational-baroclinic, hydraulic controls. The model is driven by a meteorologically forced barotropic transport *Q _{h}
* [calculated from the freshwater supply to the Baltic (

*Q*) and the sea level fluctuations in the Kattegat], and turbulent entrainment flows coupled to the wind speed

_{f}*W*and, in the Belt Sea, also to the barotropic transport. The most important bathymetric features of the basins are included.

The model equations are integrated numerically for a test period of 1½ years. The stratification in the Kattegat, as well as in the Belt Sea, is quite well predicted. It is found that approximately one-half of the salt transport into the Baltic is carried out by the dispersive mode associated with the barotropic fluctuations.

The effects of (short-term) changes in the external parameters *Q _{h}
*,

*Q*,

_{f}*W*

*S*

_{2K}(the salinity of the Skagerrak water) upon the stratification in the Belt Sea and the Kattegat are also investigated. Finally, the effects of long-term changes in the external parameters upon the surface salinity of the Baltic are investigated.

## Abstract

A one-dimensional seasonal pycnocline model, primarily intended for use in long-term circulation models, is developed. The model is of the two-layer (integral) type with a turbulent, well mixed surface layer and a nonturbulent, stratified lower layer. The parameterization of the entrainment velocity at the pycnocline accounts for the effects of buoyancy fluxes through the sea surface upon the entrainment flow. Also the “retreat” of the pycnocline, caused by large positive buoyancy supplies through the sea surface, is allowed for in the model. Effects of the rotation of the system are included. Under stable or neutral conditions the rotation may limit the penetration of mechanically generated turbulence. Hence, for weak positive buoyancy fluxes through the sea surface the rotation will cause the “retreat” of a deep pycnocline.

The model is utilized to simulate the climatological mean annual cycles of temperature and salinity in the upper parts of the Baltic (above the perennial main halocline at about 60 m depth). The test of seasonal pycnocline models introduced by Gill and Turner is applied. Two empirical constants are determined, the well known *m*
_{0}, which appears in formulas for the entrainment velocity, and 𝒱, which determines the thickness of the Ekman layer. For the best-fit case, which is obtained for *m*
_{0}=0.6 and 𝒱=0.20, the computed annual cycles of temperature and salinity appear quite realistic.

The estuarine circulation of the Baltic is accounted for. Brackish surface water is exported to the ocean and a dense bottom current carries the import of saltier “sea” water from the mouth. The seasonal pycnocline model correctly predicts the depth of the main halocline.

## Abstract

A one-dimensional seasonal pycnocline model, primarily intended for use in long-term circulation models, is developed. The model is of the two-layer (integral) type with a turbulent, well mixed surface layer and a nonturbulent, stratified lower layer. The parameterization of the entrainment velocity at the pycnocline accounts for the effects of buoyancy fluxes through the sea surface upon the entrainment flow. Also the “retreat” of the pycnocline, caused by large positive buoyancy supplies through the sea surface, is allowed for in the model. Effects of the rotation of the system are included. Under stable or neutral conditions the rotation may limit the penetration of mechanically generated turbulence. Hence, for weak positive buoyancy fluxes through the sea surface the rotation will cause the “retreat” of a deep pycnocline.

The model is utilized to simulate the climatological mean annual cycles of temperature and salinity in the upper parts of the Baltic (above the perennial main halocline at about 60 m depth). The test of seasonal pycnocline models introduced by Gill and Turner is applied. Two empirical constants are determined, the well known *m*
_{0}, which appears in formulas for the entrainment velocity, and 𝒱, which determines the thickness of the Ekman layer. For the best-fit case, which is obtained for *m*
_{0}=0.6 and 𝒱=0.20, the computed annual cycles of temperature and salinity appear quite realistic.

The estuarine circulation of the Baltic is accounted for. Brackish surface water is exported to the ocean and a dense bottom current carries the import of saltier “sea” water from the mouth. The seasonal pycnocline model correctly predicts the depth of the main halocline.

## Abstract

This paper presents a dynamical model for the salinity and thickness of the upper layer in the Arctic. The parameters are the river runoff to the Arctic, the buoyancy supply through the Bering Strait, the export of ice from the Arctic and a parameter characterizing the vertical mixing. An ice model is formulated, having the following two important properties: 1) the horizontal surface area of the exported ice is essentially determined by external parameters (the wind field over the Arctic); and 2) there is a relationship between the ice thickness and the fraction of open water in the Arctic. The model for the upper layer and the ice model are used together with a heat budget for the Arctic, also including the effect of different albedo for ice and open water. A relationship between the freshwater supply and the ice thickness is derived. Also investigated are the effects on the ice thickness of a changed export of ice area and changed properties of the flow through the Bering Strait. It is found that a decrease of the fresh-water supply by as much as 50% would have only a small effect upon the ice thickness and the fraction of open water in the present Arctic Ocean. However, if such a decrease of the freshwater supply is combined with a moderate decrease of the flow through the Bering Strait and with a likewise moderate increase of the area of the exported ice, the pack ice might disappear.

## Abstract

This paper presents a dynamical model for the salinity and thickness of the upper layer in the Arctic. The parameters are the river runoff to the Arctic, the buoyancy supply through the Bering Strait, the export of ice from the Arctic and a parameter characterizing the vertical mixing. An ice model is formulated, having the following two important properties: 1) the horizontal surface area of the exported ice is essentially determined by external parameters (the wind field over the Arctic); and 2) there is a relationship between the ice thickness and the fraction of open water in the Arctic. The model for the upper layer and the ice model are used together with a heat budget for the Arctic, also including the effect of different albedo for ice and open water. A relationship between the freshwater supply and the ice thickness is derived. Also investigated are the effects on the ice thickness of a changed export of ice area and changed properties of the flow through the Bering Strait. It is found that a decrease of the fresh-water supply by as much as 50% would have only a small effect upon the ice thickness and the fraction of open water in the present Arctic Ocean. However, if such a decrease of the freshwater supply is combined with a moderate decrease of the flow through the Bering Strait and with a likewise moderate increase of the area of the exported ice, the pack ice might disappear.

## Abstract

The time-dependent vertical circulation of the Baltic Proper is modeled using a horizontally integrated model of high vertical resolution. A seasonal pycnocline model computes the properties of the mixed layer. Below this an advection-vertical diffusion model computes the evolution of the salinity and temperature fields. A simple model for an entraining dense bottom current—which carries the intruding seawater and drives the vertical advection in the basin—is developed and used. In the derivation of entrainment velocity *w _{e}
* it is shown that

*E*(=

*w*/

_{e}*u*, where

*u*is the speed of the bottom current) may be expressed in the well-known empirical constants

*m*

_{0}and

*C*.

_{d}The hypsographic features of the Baltic are accounted for in the model. The model is forced using realistic meteorological and hydrological time series. The inflow of dense seawater to the Baltic, with large fluctuations in flow rate and salinity, is realistically described and constitutes the upstream boundary condition for the bottom current.

Tht vertical diffusiivty of salt and temperature κ, applicable in the interior of the basin, i.e. outside the dense bottom current and below the mixed layer, is taken to be κ = α/*N*, where *N* is the Brunt-Väisälä frequency. By comapring model results with field data it is concluded that α = 2×10^{−7}∓35% (m^{2} s^{−2}). The accompanying κ-values are generally lower than values for the Baltic reported by other investigators. The reason for this is that the present study manages to separate the mixing performed by the dense bottom current from the mixing performed in the interior of the basin.

The model produces a vertical stratification quite similr to that observed in the real Baltic, with a parennial hallocline at about 60 m depth and below this a strongly stratified deepwater. The observed intermittent exchange of the deepest deepwater is well described by the model. The model has been run also with constant volume flow and salinity of the intruding seawater. In this case the stratification below the perrennial halocline becomes only weakly stratified. This experiment clearly demonstrates that the characteristics of the inflow of seawater have a profound influence upon the resulting stratification.

## Abstract

The time-dependent vertical circulation of the Baltic Proper is modeled using a horizontally integrated model of high vertical resolution. A seasonal pycnocline model computes the properties of the mixed layer. Below this an advection-vertical diffusion model computes the evolution of the salinity and temperature fields. A simple model for an entraining dense bottom current—which carries the intruding seawater and drives the vertical advection in the basin—is developed and used. In the derivation of entrainment velocity *w _{e}
* it is shown that

*E*(=

*w*/

_{e}*u*, where

*u*is the speed of the bottom current) may be expressed in the well-known empirical constants

*m*

_{0}and

*C*.

_{d}The hypsographic features of the Baltic are accounted for in the model. The model is forced using realistic meteorological and hydrological time series. The inflow of dense seawater to the Baltic, with large fluctuations in flow rate and salinity, is realistically described and constitutes the upstream boundary condition for the bottom current.

Tht vertical diffusiivty of salt and temperature κ, applicable in the interior of the basin, i.e. outside the dense bottom current and below the mixed layer, is taken to be κ = α/*N*, where *N* is the Brunt-Väisälä frequency. By comapring model results with field data it is concluded that α = 2×10^{−7}∓35% (m^{2} s^{−2}). The accompanying κ-values are generally lower than values for the Baltic reported by other investigators. The reason for this is that the present study manages to separate the mixing performed by the dense bottom current from the mixing performed in the interior of the basin.

The model produces a vertical stratification quite similr to that observed in the real Baltic, with a parennial hallocline at about 60 m depth and below this a strongly stratified deepwater. The observed intermittent exchange of the deepest deepwater is well described by the model. The model has been run also with constant volume flow and salinity of the intruding seawater. In this case the stratification below the perrennial halocline becomes only weakly stratified. This experiment clearly demonstrates that the characteristics of the inflow of seawater have a profound influence upon the resulting stratification.

## Abstract

Energy transfer from barotropic tides to baroclinic motions may take place at the ends of straits connecting stratified basins, implying generation of internal waves propagating into the basins. Different aspects of this have been described in the literature, including quantification of the energy transfer, studies of the resulting internal tides, and the relationship to diapycnal mixing in the basins. However, the accompanying resistance to oscillatory barotropic strait flows, by so-called baroclinic wave drag, has not been implemented earlier in models for strait flow. In the present paper a formulation for the instantaneous baroclinic wave drag force is suggested. This leads to a linear relationship between the strait flow and the synoptic sea level difference between the adjacent basins. The model is used to study tidal response in land-locked basins. The response is computed for three fjords where the resistance to the barotropic flow comes essentially from baroclinic wave drag. It is concluded that the strait-flow model handles baroclinic wave drag in a satisfactory way since the observed responses are well predicted by the model. Baroclinic wave drag should provide the major resistance to surface tides in the deep ocean, which indicates its importance in a wider context.

## Abstract

Energy transfer from barotropic tides to baroclinic motions may take place at the ends of straits connecting stratified basins, implying generation of internal waves propagating into the basins. Different aspects of this have been described in the literature, including quantification of the energy transfer, studies of the resulting internal tides, and the relationship to diapycnal mixing in the basins. However, the accompanying resistance to oscillatory barotropic strait flows, by so-called baroclinic wave drag, has not been implemented earlier in models for strait flow. In the present paper a formulation for the instantaneous baroclinic wave drag force is suggested. This leads to a linear relationship between the strait flow and the synoptic sea level difference between the adjacent basins. The model is used to study tidal response in land-locked basins. The response is computed for three fjords where the resistance to the barotropic flow comes essentially from baroclinic wave drag. It is concluded that the strait-flow model handles baroclinic wave drag in a satisfactory way since the observed responses are well predicted by the model. Baroclinic wave drag should provide the major resistance to surface tides in the deep ocean, which indicates its importance in a wider context.

## Abstract

The atmospheric net flow of water from the Atlantic to the Pacific Ocean is supposed to maintain the salinity difference between the two oceans. Assuming the existence of a subsurface level of no horizontal pressure gradient in the ocean, the mean sea level in the northern Pacific must be higher than in the Arctic Ocean. This mean sea level difference is supposed to drive the observed mean flow through the Bering Strait.

The estimated flow of freshwater through the Bering Strait is approximately equal to the estimated atmospheric net flow of water from the North Atlantic to the North Pacific. This justifies the formulation of a simple estuary model for the North Pacific in which the “brackish” water exits through the Bering Strait. The salinity difference between the two oceans is shown to be controlled by the topography of the Bering Strait. The estuary model gives residence times for water in the upper layer (∼1000 m thick) of approximately 1000 years and in the lower layer of approximately 4000 years.

## Abstract

The atmospheric net flow of water from the Atlantic to the Pacific Ocean is supposed to maintain the salinity difference between the two oceans. Assuming the existence of a subsurface level of no horizontal pressure gradient in the ocean, the mean sea level in the northern Pacific must be higher than in the Arctic Ocean. This mean sea level difference is supposed to drive the observed mean flow through the Bering Strait.

The estimated flow of freshwater through the Bering Strait is approximately equal to the estimated atmospheric net flow of water from the North Atlantic to the North Pacific. This justifies the formulation of a simple estuary model for the North Pacific in which the “brackish” water exits through the Bering Strait. The salinity difference between the two oceans is shown to be controlled by the topography of the Bering Strait. The estuary model gives residence times for water in the upper layer (∼1000 m thick) of approximately 1000 years and in the lower layer of approximately 4000 years.

## Abstract

In an earlier paper (Stigebrandt, 1976), a model was proposed for internal wave-induced vertical mixing in a sill fjord. The observational evidence for the theory at that time was rather sparse, but the void is materially filled by observations presented in this note. Current measurements show, in accordance with the theory, that progressive internal waves radiate out from the Dröbak sill in the Oslofjord. An experiment with tracer dye below the sill depth in the same fjord showed that the coefficient of vertical diffusion of the tracer was an order of magnitude less than the overall coefficient for vertical diffusion of density during the same period. This finding, indicating large horizontal inhomogeneities in the mixing field, gives strong support to the mixing model proposed in the earlier paper referred to above.

## Abstract

In an earlier paper (Stigebrandt, 1976), a model was proposed for internal wave-induced vertical mixing in a sill fjord. The observational evidence for the theory at that time was rather sparse, but the void is materially filled by observations presented in this note. Current measurements show, in accordance with the theory, that progressive internal waves radiate out from the Dröbak sill in the Oslofjord. An experiment with tracer dye below the sill depth in the same fjord showed that the coefficient of vertical diffusion of the tracer was an order of magnitude less than the overall coefficient for vertical diffusion of density during the same period. This finding, indicating large horizontal inhomogeneities in the mixing field, gives strong support to the mixing model proposed in the earlier paper referred to above.

## Abstract

A new mechanism, the breaking of internal waves, is proposed to explain vertical mixing within the lower layers of sill fjords. The generation of the waves at the sill of a fjord is modelled assuming constant depth except at the narrow sill whose height is the thickness of the lower layer. A barotropic tide oscillating across the still creates internal waves which propagate both seaward and landward from the still. These waves break against the bottom, creating boundary turbulence which mixes water of different density in the lower layer. This is demonstrated experimentally. The mixture flows away from the boundaries into the interior of the fjord, causing an effective vertical mixing. The energy input into the internal waves and the damping of barotropic seiches are computed using linear theory. Possible instability, except at the bottom, is discounted by considering representative Richardson and Froude numbers. The theory is then qualitatively applied to the Oslofjord with particular attention given to the effects caused by changing the sill geometry. An estimate of the Richardson flux number from the Oslofjord data gives a value of 0.05.

## Abstract

A new mechanism, the breaking of internal waves, is proposed to explain vertical mixing within the lower layers of sill fjords. The generation of the waves at the sill of a fjord is modelled assuming constant depth except at the narrow sill whose height is the thickness of the lower layer. A barotropic tide oscillating across the still creates internal waves which propagate both seaward and landward from the still. These waves break against the bottom, creating boundary turbulence which mixes water of different density in the lower layer. This is demonstrated experimentally. The mixture flows away from the boundaries into the interior of the fjord, causing an effective vertical mixing. The energy input into the internal waves and the damping of barotropic seiches are computed using linear theory. Possible instability, except at the bottom, is discounted by considering representative Richardson and Froude numbers. The theory is then qualitatively applied to the Oslofjord with particular attention given to the effects caused by changing the sill geometry. An estimate of the Richardson flux number from the Oslofjord data gives a value of 0.05.

## Abstract

A simple theory shows that the two-layer transport capacity of a constriction may be increased considerably by barotropic current fluctuations. This is confirmed by laboratory experiments. The effect may be of great importance for the deep-water renewal process in some sill fjords and for the hydrographic conditions in some overmixed estuaries.

## Abstract

A simple theory shows that the two-layer transport capacity of a constriction may be increased considerably by barotropic current fluctuations. This is confirmed by laboratory experiments. The effect may be of great importance for the deep-water renewal process in some sill fjords and for the hydrographic conditions in some overmixed estuaries.

## Abstract

The rate of work against the buoyancy forces due to vertical mixing (*W*) has been determined from repeated measurements of vertical density profiles in a large number of fjordic sill basins (basins dammed by sills). It is found that there is a weak “background” rate of work *W*
_{0}, probably driven by the local wind. Superposed upon this is work driven by the tide. Thus *W* = *W*
_{0} + *R _{f}E*, where

*E*is the mean energy flux from the surface tide to turbulence in the sill basin and

*R*is an efficiency factor. We distinguish between “wave basins” and “jet basins.” In the former category progressive internal tides are generated in the mouths, while in the latter there are tidal jets at the mouths. For wave basins, about 5.6% of the energy flux

_{f}*E*from the surface tide is used for work against the buoyancy forces in the basin water (i.e.,

*R*≈ 0.056). The corresponding figure for jet basins appears to be less than 1%.

_{f}We have also studied the dependence of the vertical diffusivity κ upon the vertical stratification *N*. For well-behaved vertical distributions of *N*, it is found that κ ∼ *N*
^{−1.5}. A formula for κ, which appears to be applicable to many wave sill basins in fjords, is derived. From this, κ may be predicted if the vertical stratification *N*N(*z*), the characteristics of the topography and the sea level statistics are known.

## Abstract

The rate of work against the buoyancy forces due to vertical mixing (*W*) has been determined from repeated measurements of vertical density profiles in a large number of fjordic sill basins (basins dammed by sills). It is found that there is a weak “background” rate of work *W*
_{0}, probably driven by the local wind. Superposed upon this is work driven by the tide. Thus *W* = *W*
_{0} + *R _{f}E*, where

*E*is the mean energy flux from the surface tide to turbulence in the sill basin and

*R*is an efficiency factor. We distinguish between “wave basins” and “jet basins.” In the former category progressive internal tides are generated in the mouths, while in the latter there are tidal jets at the mouths. For wave basins, about 5.6% of the energy flux

_{f}*E*from the surface tide is used for work against the buoyancy forces in the basin water (i.e.,

*R*≈ 0.056). The corresponding figure for jet basins appears to be less than 1%.

_{f}We have also studied the dependence of the vertical diffusivity κ upon the vertical stratification *N*. For well-behaved vertical distributions of *N*, it is found that κ ∼ *N*
^{−1.5}. A formula for κ, which appears to be applicable to many wave sill basins in fjords, is derived. From this, κ may be predicted if the vertical stratification *N*N(*z*), the characteristics of the topography and the sea level statistics are known.