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- Author or Editor: Lars Petter Røed x

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## Abstract

A coupled ice-ocean model for thermodynamic growth of sea ice suitable for coupling with a similar dynamic model is considered. The model is balanced locally in that no horizontal (or vertical) advection or diffusion of properties are considered. Furthermore, the emphasis is on short time scales and focusing on applicability to the marginal ice zone. A main assumption is the inclusion of lateral growth and decay only, in that imbalances in the vertical heat fluxes give rise to a change in compactness rather than thickness of ice: The vertical heat fluxes are simply parameterized to be proportional to the temperature difference across thin boundary layers adjacent to the ice-atmosphere and ice-ocean interfaces. The ocean is treated as a two-layer model in which the lower deep layer acts as a heat source. The temperature of the upper (mixed) layer is predicted. The model exhibits time scale for freezing (∼2 days), and melting (∼6.5 days), which are of the same order of magnitude as for similar dynamic models. Thus, interesting interaction between dynamics and thermodynamics may occur if the two models are coupled.

## Abstract

A coupled ice-ocean model for thermodynamic growth of sea ice suitable for coupling with a similar dynamic model is considered. The model is balanced locally in that no horizontal (or vertical) advection or diffusion of properties are considered. Furthermore, the emphasis is on short time scales and focusing on applicability to the marginal ice zone. A main assumption is the inclusion of lateral growth and decay only, in that imbalances in the vertical heat fluxes give rise to a change in compactness rather than thickness of ice: The vertical heat fluxes are simply parameterized to be proportional to the temperature difference across thin boundary layers adjacent to the ice-atmosphere and ice-ocean interfaces. The ocean is treated as a two-layer model in which the lower deep layer acts as a heat source. The temperature of the upper (mixed) layer is predicted. The model exhibits time scale for freezing (∼2 days), and melting (∼6.5 days), which are of the same order of magnitude as for similar dynamic models. Thus, interesting interaction between dynamics and thermodynamics may occur if the two models are coupled.

## Abstract

Considered is a pointwise energy diagnostic scheme for a multilayer, primitive equation, nonisopycnic ocean model. Both conservative as well as nonconservative energy exchange terms are considered. Moreover, the scheme is worked out for both the finite depth as well as the reduced gravity versions of the model. The work is motivated by the need to discern the various instability processes responsible for the observed and modeled mesoscale flow structures commonly found in oceanic frontal regions, for example, upwelling areas, and regions separating coastal and adjacent deep ocean currents. As is common the mathematical form of the conservative energy exchange terms are ambiguous. A careful analysis is therefore effectuated to interpret them in terms of known physical processes. The analysis reveals that four basic instability processes are supported. One is the barotropic or horizontal shear instability, which extracts its energy from the horizontal shear in the mean current. The remaining three are the vertical shear instability, the frontal instability, and the conventional baroclinic instability and are, thus, different forms of baroclinic instability. The first, the vertical shear instability, obtains its energy from the velocity difference between adjacent layers (the model’s rendition of a vertical shear). The second, the frontal instability, elicits the potential energy stored in the lateral layer density gradients, while the third, the conventional baroclinic instability, gets its energy from the lateral gradient in the layer thicknesses (the model’s rendition of a vertical density gradient). It is also further shown that the bottom topography contributes to the conservative energy exchange by releasing potential energy when the integrated mass transport in a water column is directed downslope. Moreover, the analysis reveals that the traditional reduced gravity models, that is, models employing uniform layer densities, only support horizontal and vertical shear instabilities. Finally, it is shown that the entrainment process always leads to a loss of kinetic energy and that some of this lost energy may, under certain circumstances, be retrieved as potential energy.

## Abstract

Considered is a pointwise energy diagnostic scheme for a multilayer, primitive equation, nonisopycnic ocean model. Both conservative as well as nonconservative energy exchange terms are considered. Moreover, the scheme is worked out for both the finite depth as well as the reduced gravity versions of the model. The work is motivated by the need to discern the various instability processes responsible for the observed and modeled mesoscale flow structures commonly found in oceanic frontal regions, for example, upwelling areas, and regions separating coastal and adjacent deep ocean currents. As is common the mathematical form of the conservative energy exchange terms are ambiguous. A careful analysis is therefore effectuated to interpret them in terms of known physical processes. The analysis reveals that four basic instability processes are supported. One is the barotropic or horizontal shear instability, which extracts its energy from the horizontal shear in the mean current. The remaining three are the vertical shear instability, the frontal instability, and the conventional baroclinic instability and are, thus, different forms of baroclinic instability. The first, the vertical shear instability, obtains its energy from the velocity difference between adjacent layers (the model’s rendition of a vertical shear). The second, the frontal instability, elicits the potential energy stored in the lateral layer density gradients, while the third, the conventional baroclinic instability, gets its energy from the lateral gradient in the layer thicknesses (the model’s rendition of a vertical density gradient). It is also further shown that the bottom topography contributes to the conservative energy exchange by releasing potential energy when the integrated mass transport in a water column is directed downslope. Moreover, the analysis reveals that the traditional reduced gravity models, that is, models employing uniform layer densities, only support horizontal and vertical shear instabilities. Finally, it is shown that the entrainment process always leads to a loss of kinetic energy and that some of this lost energy may, under certain circumstances, be retrieved as potential energy.

## Abstract

The development of a pointwise (in the horizontal) energy diagnostic scheme applicable to a 1½-layer, nonisopycnic, primitive equation model is presented. The scheme utilizes the concept of available gravitational energy to replace the conventional potential energy. This gives a total energy (kinetic plus potential) that is zero and a minimum with respect to a given reference state (a positive definite quantity) locally. Mean and eddy components of the kinetic and available gravitational energy forms are defined by introducing a thickness-weighted mean for velocity and density. Finally, mathematical formulations for the conversion terms, that is, those terms responsible for a reversible exchange of energy between the four energy compartments, are derived.

## Abstract

The development of a pointwise (in the horizontal) energy diagnostic scheme applicable to a 1½-layer, nonisopycnic, primitive equation model is presented. The scheme utilizes the concept of available gravitational energy to replace the conventional potential energy. This gives a total energy (kinetic plus potential) that is zero and a minimum with respect to a given reference state (a positive definite quantity) locally. Mean and eddy components of the kinetic and available gravitational energy forms are defined by introducing a thickness-weighted mean for velocity and density. Finally, mathematical formulations for the conversion terms, that is, those terms responsible for a reversible exchange of energy between the four energy compartments, are derived.

## Abstract

A linear stability analysis combined with an energy analysis is performed to discriminate between the various instabilities that may develop at upwelling fronts. In the present study, a two-active-layer model of finite depth is considered. Thus, the model includes a variable across-front bottom topography, a sloping interface, a surface elevation, and variable densities in the two layers. In addition, the energy analysis departs from earlier studies in that it makes use of the available gravitational energy to replace the conventional potential energy. The concept of available gravitational energy is akin to available potential energy, but avoids the constraint of considering a closed basin. Interestingly, the earlier findings of two preferred bands of unstable waves are retained in the present model. The first band (wavelengths of 10–30 km) is associated with the so-called frontal instability (frontal mode), and the second band (wavelengths of 60–70 km) is associated with a mixed barotropic–baroclinic instability (mixed mode). The growth rate of the frontal mode is typically in the range of one to two days, while the mixed mode is typically three to five days. Although the frontal mode dominates in most cases, an exception occurs when the horizontal shear (in terms of the jet speed divided by the frontal width) becomes large. Indeed, the frontal mode ceases to exist when the frontal width becomes small enough, depending on the horizontal viscosity. Another exception occurs when the frontal jet is caused by the sloping interface only (no upper-layer density front). In this case the frontal mode is cut off, lending further support to the theory that the smaller-scale waves found in the coastal transition zones of the world oceans indeed owe their presence to the existence of the upwelling front. When the vertical shear is increased, the present analysis reveals that the growth rates of all the unstable waves, in particular the waves associated with the frontal mode, are increased. Moreover, the mixed mode ceases to exist as a preferred band of unstable waves. A final case shows that the frontal mode is unaffected by a sloping bottom topography. This is in support of the suggestion that the frontal mode is trapped to the upper layer. Experiments with a numerical multilayer, primitive-equation ocean model support the findings of the linear stability analysis, both qualitatively and quantitatively. They also reveal a complicated nonlinear wave–wave interaction causing a transition from the well-organized linear instability wave pattern toward a new organized pattern of much longer scale, filament-type, structures.

## Abstract

A linear stability analysis combined with an energy analysis is performed to discriminate between the various instabilities that may develop at upwelling fronts. In the present study, a two-active-layer model of finite depth is considered. Thus, the model includes a variable across-front bottom topography, a sloping interface, a surface elevation, and variable densities in the two layers. In addition, the energy analysis departs from earlier studies in that it makes use of the available gravitational energy to replace the conventional potential energy. The concept of available gravitational energy is akin to available potential energy, but avoids the constraint of considering a closed basin. Interestingly, the earlier findings of two preferred bands of unstable waves are retained in the present model. The first band (wavelengths of 10–30 km) is associated with the so-called frontal instability (frontal mode), and the second band (wavelengths of 60–70 km) is associated with a mixed barotropic–baroclinic instability (mixed mode). The growth rate of the frontal mode is typically in the range of one to two days, while the mixed mode is typically three to five days. Although the frontal mode dominates in most cases, an exception occurs when the horizontal shear (in terms of the jet speed divided by the frontal width) becomes large. Indeed, the frontal mode ceases to exist when the frontal width becomes small enough, depending on the horizontal viscosity. Another exception occurs when the frontal jet is caused by the sloping interface only (no upper-layer density front). In this case the frontal mode is cut off, lending further support to the theory that the smaller-scale waves found in the coastal transition zones of the world oceans indeed owe their presence to the existence of the upwelling front. When the vertical shear is increased, the present analysis reveals that the growth rates of all the unstable waves, in particular the waves associated with the frontal mode, are increased. Moreover, the mixed mode ceases to exist as a preferred band of unstable waves. A final case shows that the frontal mode is unaffected by a sloping bottom topography. This is in support of the suggestion that the frontal mode is trapped to the upper layer. Experiments with a numerical multilayer, primitive-equation ocean model support the findings of the linear stability analysis, both qualitatively and quantitatively. They also reveal a complicated nonlinear wave–wave interaction causing a transition from the well-organized linear instability wave pattern toward a new organized pattern of much longer scale, filament-type, structures.

## Abstract

Considered are the capabilities of a recently developed pulse-to-pulse coherent sonar called the High Resolution Current Profiler (HRCP). Special emphasis is placed on methods whereby reliable and accurate vertical profiles of turbulence parameters, such as turbulent kinetic energy and Reynolds stresses, may be extracted from such sonars. The prototype HRCP has been developed in order to obtain precise and reliable measurements of both the mean and the fluctuating components of the velocity vector profile in the bottom boundary layer (lowermost 10 m).

The prototype HRCP developed through the project BSEX is a 307 KHz pulse-to-pulse coherent sonar with four beams mounted 30° off the vertical in 90° azimuthal increments. It has 50 range cells covering a vertical profiling distance of 10 meters. The data collected are radial speeds along the four acoustic beams. Thus, care has to be exercised in interpreting the measurements in terms of turbulence parameters. It is shown, based upon reasonable assumptions about oceanic turbulence and turbulence characteristics, that it is possible to construct methods whereby reliable estimates of Reynolds stresses and turbulent kinetic energy profiles may be obtained utilizing the HRCP technology.

The HRCP was deployed during the summer of 1986 for 52 hours at a site in the northern North Sea. Reliable data with the HRCP were collected with 4 Hz sampling rate at 46 vertical levels 20 cm apart in the range 88–948 cm above the bottom. First, the observed velocities are decomposed into mean and fluctuating components based on spectral analysis. The mean velocities are then fitted to a logarithmic profile to provide estimates of bottom parameters, such as shear velocities and roughness length. Finally, the associated bottom boundary-layer turbulence is discussed in terms of Reynolds stress components and turbulent kinetic energy. The results are in agreement with similar studies of both bottom parameters and bottom turbulence. The estimates are also shown to be precise enough to estimate vertical mixing coefficients commonly applied in planetary boundary layer models, i.e., the parameterization of the Reynolds stresses.

The success in estimating reliable turbulence parameters by use of the pulse-to-pulse coherent Doppler sonars is encouraging for the future applications and use of this technology.

## Abstract

Considered are the capabilities of a recently developed pulse-to-pulse coherent sonar called the High Resolution Current Profiler (HRCP). Special emphasis is placed on methods whereby reliable and accurate vertical profiles of turbulence parameters, such as turbulent kinetic energy and Reynolds stresses, may be extracted from such sonars. The prototype HRCP has been developed in order to obtain precise and reliable measurements of both the mean and the fluctuating components of the velocity vector profile in the bottom boundary layer (lowermost 10 m).

The prototype HRCP developed through the project BSEX is a 307 KHz pulse-to-pulse coherent sonar with four beams mounted 30° off the vertical in 90° azimuthal increments. It has 50 range cells covering a vertical profiling distance of 10 meters. The data collected are radial speeds along the four acoustic beams. Thus, care has to be exercised in interpreting the measurements in terms of turbulence parameters. It is shown, based upon reasonable assumptions about oceanic turbulence and turbulence characteristics, that it is possible to construct methods whereby reliable estimates of Reynolds stresses and turbulent kinetic energy profiles may be obtained utilizing the HRCP technology.

The HRCP was deployed during the summer of 1986 for 52 hours at a site in the northern North Sea. Reliable data with the HRCP were collected with 4 Hz sampling rate at 46 vertical levels 20 cm apart in the range 88–948 cm above the bottom. First, the observed velocities are decomposed into mean and fluctuating components based on spectral analysis. The mean velocities are then fitted to a logarithmic profile to provide estimates of bottom parameters, such as shear velocities and roughness length. Finally, the associated bottom boundary-layer turbulence is discussed in terms of Reynolds stress components and turbulent kinetic energy. The results are in agreement with similar studies of both bottom parameters and bottom turbulence. The estimates are also shown to be precise enough to estimate vertical mixing coefficients commonly applied in planetary boundary layer models, i.e., the parameterization of the Reynolds stresses.

The success in estimating reliable turbulence parameters by use of the pulse-to-pulse coherent Doppler sonars is encouraging for the future applications and use of this technology.