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- Author or Editor: Jonas Nycander x

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## Abstract

A local neutral plane is defined so that a water parcel that is displaced adiabatically a small distance along the plane continues to have the same density as the surrounding water. Since such a displacement does not change the density field or the gravitational potential energy, it is generally assumed that it does not produce a restoring buoyancy force. However, it is here shown that because of the nonlinear character of the equation of state (in particular the thermobaric effect) such a neutral displacement is accompanied by a conversion between internal energy *E* and gravitational potential energy *U*, and an equal conversion between *U* and kinetic energy *K*. While there is thus no net change of *U*, *K* does change. This implies that a force is in fact required for the displacement.

It is further shown that displacements that are orthogonal to a vector **P** do not induce conversion between *U* and *K*, and therefore do not require a force. Analogously to neutral surfaces, which are defined to be approximately orthogonal to the dianeutral vector **N**, one may define “**P** surfaces” to be approximately orthogonal to **P**. These **P** surfaces are intermediate between neutral surfaces and surfaces of constant *σ*
_{0} (potential density reference to the surface).

If the equation of state is linear, there exists a well-known expression for the mixing energy in terms of the diapycnal flow. This expression is here generalized for a general nonlinear equation of state. The generalized expression involves the velocity component along **P**. Since **P** is not orthogonal to neutral surfaces, this means that stationary flow along neutral surfaces in general requires mixing energy.

## Abstract

A local neutral plane is defined so that a water parcel that is displaced adiabatically a small distance along the plane continues to have the same density as the surrounding water. Since such a displacement does not change the density field or the gravitational potential energy, it is generally assumed that it does not produce a restoring buoyancy force. However, it is here shown that because of the nonlinear character of the equation of state (in particular the thermobaric effect) such a neutral displacement is accompanied by a conversion between internal energy *E* and gravitational potential energy *U*, and an equal conversion between *U* and kinetic energy *K*. While there is thus no net change of *U*, *K* does change. This implies that a force is in fact required for the displacement.

It is further shown that displacements that are orthogonal to a vector **P** do not induce conversion between *U* and *K*, and therefore do not require a force. Analogously to neutral surfaces, which are defined to be approximately orthogonal to the dianeutral vector **N**, one may define “**P** surfaces” to be approximately orthogonal to **P**. These **P** surfaces are intermediate between neutral surfaces and surfaces of constant *σ*
_{0} (potential density reference to the surface).

If the equation of state is linear, there exists a well-known expression for the mixing energy in terms of the diapycnal flow. This expression is here generalized for a general nonlinear equation of state. The generalized expression involves the velocity component along **P**. Since **P** is not orthogonal to neutral surfaces, this means that stationary flow along neutral surfaces in general requires mixing energy.

## ABSTRACT

The overturning circulations in the atmosphere and ocean transport energy from the tropics to higher latitudes and thereby modulate Earth’s climate. The interannual variability in the overturning over the last 40 years is found to be dominated by two coupled atmosphere–ocean modes. The first is related to the meridional motion of the intertropical convergence zone and the second to El Niño. Both modes have a strong influence on the sea level variability in the tropical Indo-Pacific Ocean. The interannual variability of the cross-equatorial energy transport is dominated by the first mode, and the variability is larger in the Indo-Pacific Ocean than in the Atlantic Ocean or the atmosphere. Our results suggest an important role of oceanic energy transport in setting precipitation patterns in the tropics and a key role of the Indo-Pacific Ocean as a climate modulator.

## ABSTRACT

The overturning circulations in the atmosphere and ocean transport energy from the tropics to higher latitudes and thereby modulate Earth’s climate. The interannual variability in the overturning over the last 40 years is found to be dominated by two coupled atmosphere–ocean modes. The first is related to the meridional motion of the intertropical convergence zone and the second to El Niño. Both modes have a strong influence on the sea level variability in the tropical Indo-Pacific Ocean. The interannual variability of the cross-equatorial energy transport is dominated by the first mode, and the variability is larger in the Indo-Pacific Ocean than in the Atlantic Ocean or the atmosphere. Our results suggest an important role of oceanic energy transport in setting precipitation patterns in the tropics and a key role of the Indo-Pacific Ocean as a climate modulator.

## Abstract

The interaction of the barotropic tide with bottom topography when the tidal frequency *ω* is smaller than the Coriolis frequency *f* is examined. The resulting waves are called bottom-trapped internal tides. The energy density associated with these waves is computed using linear wave theory and vertical normal-mode decomposition in an ocean of finite depth. The global calculation of the modal energy density is performed for the semidiurnal M_{2} tidal constituent and the two major diurnal tidal constituents K_{1} and O_{1}. An observationally based decay time scale of 3 days is then used to transform the energy density to energy flux in units of watts per square meter. The globally integrated energy fluxes are found to be 1.99 and 1.43 GW for the K_{1} and O_{1} tidal constituents, respectively. For the M_{2} tidal constituent, it is found to be 1.15 GW. The Pacific Ocean is found to be the most energetic basin for the bottom-trapped diurnal tides. Two regional estimates of the bottom-trapped energy flux are given for the Kuril Islands and the Arctic Ocean, in which the bottom-trapped waves play a role for the tidally induced vertical mixing. The results of this study can be incorporated into ocean general circulation models and coupled climate models to improve the parameterization of the vertical mixing induced by breaking of the internal tides.

## Abstract

The interaction of the barotropic tide with bottom topography when the tidal frequency *ω* is smaller than the Coriolis frequency *f* is examined. The resulting waves are called bottom-trapped internal tides. The energy density associated with these waves is computed using linear wave theory and vertical normal-mode decomposition in an ocean of finite depth. The global calculation of the modal energy density is performed for the semidiurnal M_{2} tidal constituent and the two major diurnal tidal constituents K_{1} and O_{1}. An observationally based decay time scale of 3 days is then used to transform the energy density to energy flux in units of watts per square meter. The globally integrated energy fluxes are found to be 1.99 and 1.43 GW for the K_{1} and O_{1} tidal constituents, respectively. For the M_{2} tidal constituent, it is found to be 1.15 GW. The Pacific Ocean is found to be the most energetic basin for the bottom-trapped diurnal tides. Two regional estimates of the bottom-trapped energy flux are given for the Kuril Islands and the Arctic Ocean, in which the bottom-trapped waves play a role for the tidally induced vertical mixing. The results of this study can be incorporated into ocean general circulation models and coupled climate models to improve the parameterization of the vertical mixing induced by breaking of the internal tides.

## Abstract

Breaking internal tides contribute substantially to small-scale turbulent mixing in the ocean interior and hence to maintaining the large-scale overturning circulation. How much internal tide energy is available for ocean mixing can be estimated by using semianalytical methods based on linear theory. Until recently, a method resolving the horizontal direction of the internal waves generated by conversion of the barotropic tide was lacking. We here present the first global application of such a method to the first vertical mode of the principal lunar semidiurnal internal tide. We also show that the effect of supercritical slopes on the modally decomposed internal tides is different than previously suggested. To deal with this the continental shelf and the shelf slope are masked in the global computation. The global energy conversion obtained agrees roughly with the previous results by Falahat et al. if the mask is applied to their result, which decreases their energy conversion by half. Thus, around half of the energy conversion obtained by their linear calculations occurs at continental slopes and shelves, where linear theory tends to break down. The barotropic-to-baroclinic energy flux at subcritical slopes away from the continental margins is shown to vary substantially with direction depending on the shape and orientation of topographic obstacles and the direction of the local tidal currents. Taking this additional information into account in tidal mixing parameterizations could have important ramifications for vertical mixing and water mass properties in global numerical simulations.

## Abstract

Breaking internal tides contribute substantially to small-scale turbulent mixing in the ocean interior and hence to maintaining the large-scale overturning circulation. How much internal tide energy is available for ocean mixing can be estimated by using semianalytical methods based on linear theory. Until recently, a method resolving the horizontal direction of the internal waves generated by conversion of the barotropic tide was lacking. We here present the first global application of such a method to the first vertical mode of the principal lunar semidiurnal internal tide. We also show that the effect of supercritical slopes on the modally decomposed internal tides is different than previously suggested. To deal with this the continental shelf and the shelf slope are masked in the global computation. The global energy conversion obtained agrees roughly with the previous results by Falahat et al. if the mask is applied to their result, which decreases their energy conversion by half. Thus, around half of the energy conversion obtained by their linear calculations occurs at continental slopes and shelves, where linear theory tends to break down. The barotropic-to-baroclinic energy flux at subcritical slopes away from the continental margins is shown to vary substantially with direction depending on the shape and orientation of topographic obstacles and the direction of the local tidal currents. Taking this additional information into account in tidal mixing parameterizations could have important ramifications for vertical mixing and water mass properties in global numerical simulations.

## Abstract

A numerical model based on the shallow-water equations is developed to represent the flow of Antarctic Bottom Water (AABW) in the Brazil Basin (southwest Atlantic Ocean). The aim is twofold. First, an attempt is made to identify in a model that includes both simplified dynamics and realistic bathymetry (at 1/6° resolution) the impacts of the elevated diapycnal mixing rates near the Mid-Atlantic Ridge (MAR) documented by dissipation data of the Deep Basin Experiment (DBE). To this end, different assumptions regarding the distribution of the velocity across the AABW layer interface (*w*) are considered. Second, the extent to which the shallow-water model can replicate observations relative to AABW circulation in the basin, in particular the trajectory and velocity of neutrally buoyant floats released in the AABW during the DBE, is examined. The model flows are characterized by small Rossby numbers, except in the northward-flowing western boundary current where kinetic energy is largely concentrated. To interpret the flows, model streamlines are compared with isopleths of linear potential vorticity *f*/*h*
_{0} of the shallow-water theory (*f* is the planetary vorticity and *h*
_{0} is the layer thickness in the absence of motion). The *f*/*h*
_{0} contours are oriented northwest–southeast in the western part of the basin and southwest–northeast in the eastern part, reflecting the bowl-shaped topography of the Southern Hemisphere basin. With a spatially uniform (positive) *w,* the ubiquitous vortex stretching produces a flow to the southeast, consistent with the Stommel–Arons theory. This flow occurs in most of the basin interior, even in the east where *f*/*h*
_{0} contours converge to the northeastern end of the basin. With strongly positive *w* near the ridge and zero or slightly negative *w* elsewhere, the flow follows more closely *f*/*h*
_{0} contours in the western interior and intersects them near the ridge. The confinement of the diapycnal mass flux near the MAR drastically reduces the southward flow in the interior or even reverses its direction, leading to a circulation quite distinct from that of the Stommel– Arons theory. The model results compare favorably to some (but not all) hydrographic estimates of AABW circulation patterns and rates. On the other hand, the model streamlines and velocities show important differences with, respectively, the trajectory and the velocity of the floats launched in the AABW layer. The prescription of vanishing *w* in the interior does not systematically improve the fit of the model streamlines to the float trajectories, and the model velocities simulated with spatially uniform *w* or spatially variable *w* are on average smaller by one order of magnitude than the float velocities. A variety of mechanisms, which are not included in the numerical experiments, may explain the differences between the model results and the float data.

## Abstract

A numerical model based on the shallow-water equations is developed to represent the flow of Antarctic Bottom Water (AABW) in the Brazil Basin (southwest Atlantic Ocean). The aim is twofold. First, an attempt is made to identify in a model that includes both simplified dynamics and realistic bathymetry (at 1/6° resolution) the impacts of the elevated diapycnal mixing rates near the Mid-Atlantic Ridge (MAR) documented by dissipation data of the Deep Basin Experiment (DBE). To this end, different assumptions regarding the distribution of the velocity across the AABW layer interface (*w*) are considered. Second, the extent to which the shallow-water model can replicate observations relative to AABW circulation in the basin, in particular the trajectory and velocity of neutrally buoyant floats released in the AABW during the DBE, is examined. The model flows are characterized by small Rossby numbers, except in the northward-flowing western boundary current where kinetic energy is largely concentrated. To interpret the flows, model streamlines are compared with isopleths of linear potential vorticity *f*/*h*
_{0} of the shallow-water theory (*f* is the planetary vorticity and *h*
_{0} is the layer thickness in the absence of motion). The *f*/*h*
_{0} contours are oriented northwest–southeast in the western part of the basin and southwest–northeast in the eastern part, reflecting the bowl-shaped topography of the Southern Hemisphere basin. With a spatially uniform (positive) *w,* the ubiquitous vortex stretching produces a flow to the southeast, consistent with the Stommel–Arons theory. This flow occurs in most of the basin interior, even in the east where *f*/*h*
_{0} contours converge to the northeastern end of the basin. With strongly positive *w* near the ridge and zero or slightly negative *w* elsewhere, the flow follows more closely *f*/*h*
_{0} contours in the western interior and intersects them near the ridge. The confinement of the diapycnal mass flux near the MAR drastically reduces the southward flow in the interior or even reverses its direction, leading to a circulation quite distinct from that of the Stommel– Arons theory. The model results compare favorably to some (but not all) hydrographic estimates of AABW circulation patterns and rates. On the other hand, the model streamlines and velocities show important differences with, respectively, the trajectory and the velocity of the floats launched in the AABW layer. The prescription of vanishing *w* in the interior does not systematically improve the fit of the model streamlines to the float trajectories, and the model velocities simulated with spatially uniform *w* or spatially variable *w* are on average smaller by one order of magnitude than the float velocities. A variety of mechanisms, which are not included in the numerical experiments, may explain the differences between the model results and the float data.

## Abstract

The nonlinear equation of state of seawater introduces a sink or source of buoyancy when water parcels of unequal salinities and temperatures are mixed. This article contains quantitative estimates of these nonlinear effects on the buoyancy budget of the global ocean. It is shown that the interior buoyancy sink can be determined from surface buoyancy fluxes. These surface buoyancy fluxes are calculated using two surface heat flux climatologies, one based on in situ measurements and the other on a reanalysis, in both cases using a nonlinear equation of state. It is also found that the buoyancy budget in the ocean general circulation model Nucleus for European Modeling of the Ocean (NEMO) is in good agreement with the buoyancy budgets based on the heat flux climatologies. Moreover, an examination of the vertically resolved buoyancy budget in NEMO shows that in large parts of the ocean the nonlinear buoyancy sink gives the largest contribution to this budget.

## Abstract

The nonlinear equation of state of seawater introduces a sink or source of buoyancy when water parcels of unequal salinities and temperatures are mixed. This article contains quantitative estimates of these nonlinear effects on the buoyancy budget of the global ocean. It is shown that the interior buoyancy sink can be determined from surface buoyancy fluxes. These surface buoyancy fluxes are calculated using two surface heat flux climatologies, one based on in situ measurements and the other on a reanalysis, in both cases using a nonlinear equation of state. It is also found that the buoyancy budget in the ocean general circulation model Nucleus for European Modeling of the Ocean (NEMO) is in good agreement with the buoyancy budgets based on the heat flux climatologies. Moreover, an examination of the vertically resolved buoyancy budget in NEMO shows that in large parts of the ocean the nonlinear buoyancy sink gives the largest contribution to this budget.

## Abstract

The problem of finding the state of minimum potential energy through the rearrangement of water parcels with a nonlinear equation of state is discussed in the context of a combinatorial optimization problem. It is found that the problem is identical to a classical optimization problem called the linear assignment problem. This problem belongs to a problem class known as P, a class of problems that have known efficient solutions. This is very fortunate since this study’s problem has been suggested to be an asymmetric traveling salesman problem. A problem that belongs to a class called NP-hard, for which no efficient solutions are known. The difference between the linear assignment problem and the traveling salesman problem is discussed and made clear by looking at the different constraints used for the two problems. It is also shown how the rearrangement of water parcels that minimizes the potential energy can be found in polynomial time using the so-called Hungarian algorithm. The Hungarian algorithm is then applied to a simplified ocean stratification, and the result is compared to a few different approximate solutions to the minimization problem. It is found that the improved accuracy over the approximate methods comes at a high computational cost. Last, the algorithm is applied to a realistic ocean stratification using a technique that splits the minimization problem into smaller bits.

## Abstract

The problem of finding the state of minimum potential energy through the rearrangement of water parcels with a nonlinear equation of state is discussed in the context of a combinatorial optimization problem. It is found that the problem is identical to a classical optimization problem called the linear assignment problem. This problem belongs to a problem class known as P, a class of problems that have known efficient solutions. This is very fortunate since this study’s problem has been suggested to be an asymmetric traveling salesman problem. A problem that belongs to a class called NP-hard, for which no efficient solutions are known. The difference between the linear assignment problem and the traveling salesman problem is discussed and made clear by looking at the different constraints used for the two problems. It is also shown how the rearrangement of water parcels that minimizes the potential energy can be found in polynomial time using the so-called Hungarian algorithm. The Hungarian algorithm is then applied to a simplified ocean stratification, and the result is compared to a few different approximate solutions to the minimization problem. It is found that the improved accuracy over the approximate methods comes at a high computational cost. Last, the algorithm is applied to a realistic ocean stratification using a technique that splits the minimization problem into smaller bits.

## Abstract

This article presents a new framework for studying water mass transformations in salinity–temperature space that can, with equal ease, be applied to study water mass transformation in spaces defined by any two conservative tracers. It is shown how the flow across isothermal and isohaline surfaces in the ocean can be quantified from knowledge of the nonadvective fluxes of heat and salt. It is also shown how these cross-isothermal and cross-isohaline flows can be used to form a continuity equation in salinity–temperature space. These flows are then quantified in a state-of-the-art ocean model. Two major transformation cells are found: a tropical cell driven primarily by surface fluxes and dianeutral diffusion and a conveyor belt cell where isoneutral diffusion is also important. Both cells are similar to cells found in earlier work on the thermohaline streamfunction. A key benefit with this framework over a streamfunction approach is that transformation due to different diabatic processes can be studied individually. The distributions of volume and surface area in *S*–*T* space are found to be useful for determining how transformations due to these different processes affect the water masses in the model. The surface area distribution shows that the water mass transformations due to surface fluxes tend to be directed away from *S*–*T* regions that occupy large areas at the sea surface.

## Abstract

This article presents a new framework for studying water mass transformations in salinity–temperature space that can, with equal ease, be applied to study water mass transformation in spaces defined by any two conservative tracers. It is shown how the flow across isothermal and isohaline surfaces in the ocean can be quantified from knowledge of the nonadvective fluxes of heat and salt. It is also shown how these cross-isothermal and cross-isohaline flows can be used to form a continuity equation in salinity–temperature space. These flows are then quantified in a state-of-the-art ocean model. Two major transformation cells are found: a tropical cell driven primarily by surface fluxes and dianeutral diffusion and a conveyor belt cell where isoneutral diffusion is also important. Both cells are similar to cells found in earlier work on the thermohaline streamfunction. A key benefit with this framework over a streamfunction approach is that transformation due to different diabatic processes can be studied individually. The distributions of volume and surface area in *S*–*T* space are found to be useful for determining how transformations due to these different processes affect the water masses in the model. The surface area distribution shows that the water mass transformations due to surface fluxes tend to be directed away from *S*–*T* regions that occupy large areas at the sea surface.

## Abstract

Convective flow in baroclinic vortices is studied analytically, taking viscosity ν, and thermal diffusivity κ into account. The meridional circulation depends strongly on the Prandtl number Pr = ν/κ. If Pr > 1, there is upwelling in the interior of the vortex and the vertical heal diffusion is therefore inhibited by advection. The radial flow is inward in most of the vortex, which is compensated by outward flow in a viscous boundary layer just below the surface. The authors focus on the strongly nonlinear regime, when the background stratification is much weaker than that of the vortex. It is found that the nonlinear equation governing the flow in the limit Pr ≫ 1 has a class of exact time-dependent solutions. In this class the evolution of the vertical temperature profile is determined by Burger's equation, whereas the horizontal profile is determined by the Liouville equation. Both these equations can be solved analytically.

## Abstract

Convective flow in baroclinic vortices is studied analytically, taking viscosity ν, and thermal diffusivity κ into account. The meridional circulation depends strongly on the Prandtl number Pr = ν/κ. If Pr > 1, there is upwelling in the interior of the vortex and the vertical heal diffusion is therefore inhibited by advection. The radial flow is inward in most of the vortex, which is compensated by outward flow in a viscous boundary layer just below the surface. The authors focus on the strongly nonlinear regime, when the background stratification is much weaker than that of the vortex. It is found that the nonlinear equation governing the flow in the limit Pr ≫ 1 has a class of exact time-dependent solutions. In this class the evolution of the vertical temperature profile is determined by Burger's equation, whereas the horizontal profile is determined by the Liouville equation. Both these equations can be solved analytically.

## Abstract

The conversion of barotropic to baroclinic tidal energy in the global abyssal ocean is calculated using three different formulations. The calculations are done both “offline,” that is, using externally given tidal currents to estimate the energy conversion, and “online,” that is, by using the formulations to parameterize linear wave drag in a prognostic tidal model. All three schemes produce globally integrated offline dissipation rates beneath 500-m depth of ~0.6–0.8 TW for the M_{2} constituent, but the spatial structures vary significantly between the parameterizations. Detailed investigations of the energy transfer in local areas confirm the global results: there are large differences between the schemes, although the horizontally integrated conversion rates are similar. The online simulations are evaluated by comparing the sea surface elevation with data from the TOPEX/Poseidon database, and the error is then significantly lower when using the parameterization provided by Nycander than with the other two parameterizations examined.

## Abstract

The conversion of barotropic to baroclinic tidal energy in the global abyssal ocean is calculated using three different formulations. The calculations are done both “offline,” that is, using externally given tidal currents to estimate the energy conversion, and “online,” that is, by using the formulations to parameterize linear wave drag in a prognostic tidal model. All three schemes produce globally integrated offline dissipation rates beneath 500-m depth of ~0.6–0.8 TW for the M_{2} constituent, but the spatial structures vary significantly between the parameterizations. Detailed investigations of the energy transfer in local areas confirm the global results: there are large differences between the schemes, although the horizontally integrated conversion rates are similar. The online simulations are evaluated by comparing the sea surface elevation with data from the TOPEX/Poseidon database, and the error is then significantly lower when using the parameterization provided by Nycander than with the other two parameterizations examined.