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- Author or Editor: Alain Colin de Verdière x

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## Abstract

This note draws attention to the natural instabilities of a mean zonal flow that arise with the planetary geostrophic equations increasingly used in theories of large-scale oceanic circulation. Baroclinic instability is not excised by the absence of the relative acceleration terms in the momentum equations. The growth rate is shown to increase linearly with wavenumber, yielding an ill-posed mathematical problem. A small amount of lateral friction cures the problem, however, as shown in a three-layer model, which possesses the minimal vertical structure to exhibit the instability.

## Abstract

This note draws attention to the natural instabilities of a mean zonal flow that arise with the planetary geostrophic equations increasingly used in theories of large-scale oceanic circulation. Baroclinic instability is not excised by the absence of the relative acceleration terms in the momentum equations. The growth rate is shown to increase linearly with wavenumber, yielding an ill-posed mathematical problem. A small amount of lateral friction cures the problem, however, as shown in a three-layer model, which possesses the minimal vertical structure to exhibit the instability.

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## Abstract

Although the instability of the thermohaline circulation has been widely observed in numerical ocean models, theoretical advances have been hindered by the nonlinearity of heat and salt transports, a circulation governed by lateral temperature, and salinity gradients. Because the instability occurs initially in polar waters through the formation of haloclines and the halt of convection, any explanatory model must have at least a surface and a deep layer. The model proposed here (two surface boxes above a deep one) reduces to a 2 degrees-of-freedom dynamical system when convection is active and 3 degrees when it is interrupted. The instability that is induced by a negative freshwater perturbation in polar waters has three stages. The first stage is a rapid 5-yr adjustment to a transient thermal attractor that results from an approximate balance between heat advection and air–sea heat fluxes. The second stage is a slow evolution that self-organizes near this attractor, which preconditions the instability, as it can be shown that the circulation becomes more sensitive to changes in salinity gradients than in temperature gradients. The slow *O*(100 yr) growth of salinity in the subtropics is the critical precursor of the instability while at the same time the subpolar salinity rises against the initial perturbation to stabilize the system by increasing the overturning and restoring convection. When the overturning becomes smaller than the value at the unstable fixed point, the third stage occurs, which is when the subpolar salinity decreases at last on a fast *O*(10 yr) time scale, precipitating the fall of the overturning. During the last two stages of the instability, the horizontal thermal gradient increases, but its stabilizing effect is just barely unable to prevent the outcome. The return to stability occurs frequently through a regime of multidecadal oscillations with intermittent convection. The hypothesis of mixed boundary conditions has been relaxed by coupling the ocean box model to an atmospheric energy balance model to show that the coupling increases the stability of the oceanic circulation; however, the precursors of the instability are unchanged.

## Abstract

Although the instability of the thermohaline circulation has been widely observed in numerical ocean models, theoretical advances have been hindered by the nonlinearity of heat and salt transports, a circulation governed by lateral temperature, and salinity gradients. Because the instability occurs initially in polar waters through the formation of haloclines and the halt of convection, any explanatory model must have at least a surface and a deep layer. The model proposed here (two surface boxes above a deep one) reduces to a 2 degrees-of-freedom dynamical system when convection is active and 3 degrees when it is interrupted. The instability that is induced by a negative freshwater perturbation in polar waters has three stages. The first stage is a rapid 5-yr adjustment to a transient thermal attractor that results from an approximate balance between heat advection and air–sea heat fluxes. The second stage is a slow evolution that self-organizes near this attractor, which preconditions the instability, as it can be shown that the circulation becomes more sensitive to changes in salinity gradients than in temperature gradients. The slow *O*(100 yr) growth of salinity in the subtropics is the critical precursor of the instability while at the same time the subpolar salinity rises against the initial perturbation to stabilize the system by increasing the overturning and restoring convection. When the overturning becomes smaller than the value at the unstable fixed point, the third stage occurs, which is when the subpolar salinity decreases at last on a fast *O*(10 yr) time scale, precipitating the fall of the overturning. During the last two stages of the instability, the horizontal thermal gradient increases, but its stabilizing effect is just barely unable to prevent the outcome. The return to stability occurs frequently through a regime of multidecadal oscillations with intermittent convection. The hypothesis of mixed boundary conditions has been relaxed by coupling the ocean box model to an atmospheric energy balance model to show that the coupling increases the stability of the oceanic circulation; however, the precursors of the instability are unchanged.

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## Abstract

The interaction of internal waves with geostrophic flows is found to be strongly dependent upon the background stratification. Under the traditional approximation of neglecting the horizontal component of the earth’s rotation vector, the well-known inertial and symmetric instabilities highlight the asymmetry between positive and negative vertical components of relative vorticity (horizontal shear) of the mean flow, the former being stable. This is a strong stratification limit but, if it becomes too low, the traditional approximation cannot be made and the Coriolis terms caused by the earth’s rotation vector must be kept in full. A new asymmetry then appears between positive and negative horizontal components of relative vorticity (vertical shear) of the mean flow, the latter becoming more unstable. Particularly conspicuous at low latitudes, this new asymmetry does not require vanishing stratification to occur as it operates readily for rotation/stratification ratios 2Ω/*N* as small as 0.25 (the stratification still dominates over rotation) for realistic vertical shears. Given that such ratios are easily found in ocean–atmosphere boundary layers or in the deep ocean, such ageostrophic instabilities may be important for the routes to dissipation of the energy of the large-scale motions. The energetics show that, depending on the orientation of the internal wave crests with respect to the mean isopycnal surfaces, the unstable motions can draw their energy either from the kinetic energy or from the available potential energy of the mean flow. The kinetic energy source is usually the leading contribution when the growth rates reach their maxima.

## Abstract

The interaction of internal waves with geostrophic flows is found to be strongly dependent upon the background stratification. Under the traditional approximation of neglecting the horizontal component of the earth’s rotation vector, the well-known inertial and symmetric instabilities highlight the asymmetry between positive and negative vertical components of relative vorticity (horizontal shear) of the mean flow, the former being stable. This is a strong stratification limit but, if it becomes too low, the traditional approximation cannot be made and the Coriolis terms caused by the earth’s rotation vector must be kept in full. A new asymmetry then appears between positive and negative horizontal components of relative vorticity (vertical shear) of the mean flow, the latter becoming more unstable. Particularly conspicuous at low latitudes, this new asymmetry does not require vanishing stratification to occur as it operates readily for rotation/stratification ratios 2Ω/*N* as small as 0.25 (the stratification still dominates over rotation) for realistic vertical shears. Given that such ratios are easily found in ocean–atmosphere boundary layers or in the deep ocean, such ageostrophic instabilities may be important for the routes to dissipation of the energy of the large-scale motions. The energetics show that, depending on the orientation of the internal wave crests with respect to the mean isopycnal surfaces, the unstable motions can draw their energy either from the kinetic energy or from the available potential energy of the mean flow. The kinetic energy source is usually the leading contribution when the growth rates reach their maxima.

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## Abstract

A new mechanism is proposed to account for the southward motion of salt lenses in the Canary Basin. It relies on the active mixing that is observed at the periphery of such eddies between the warm salty Mediterranean waters of the core and the surrounding fresher and cooler Atlantic waters. Intrusions driven by this thermohaline contrast feed on the eddy potential energy and drain off the eddy volume anomaly. Submitted to the continuous entrainment of its own fluid by the frontal turbulence, the salt lens suffers a velocity divergence at its periphery, induced by a geostrophic adjustment process. This causes in turn a squashing of the density surfaces and the salt lens must then move southward to conserve its potential vorticity, a possibility offered by the spherical nature of the earth. It appears that mixing may load to the coherent latitudinal displacement of an ensemble of fluid parcels on a β plant, while at the same time dissipating the energy of the structure itself.

## Abstract

A new mechanism is proposed to account for the southward motion of salt lenses in the Canary Basin. It relies on the active mixing that is observed at the periphery of such eddies between the warm salty Mediterranean waters of the core and the surrounding fresher and cooler Atlantic waters. Intrusions driven by this thermohaline contrast feed on the eddy potential energy and drain off the eddy volume anomaly. Submitted to the continuous entrainment of its own fluid by the frontal turbulence, the salt lens suffers a velocity divergence at its periphery, induced by a geostrophic adjustment process. This causes in turn a squashing of the density surfaces and the salt lens must then move southward to conserve its potential vorticity, a possibility offered by the spherical nature of the earth. It appears that mixing may load to the coherent latitudinal displacement of an ensemble of fluid parcels on a β plant, while at the same time dissipating the energy of the structure itself.

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## Abstract

The behavior of the intense anticyclonic eddy observed during the Tourbillon Experiment (September-November 1979) is studied within the framework of quasi-geostrophic dynamics. The nondivergent part of the pressure held is estimated through objective analysis at the times of four quasi-synoptic CTD arrays, making use as well of velocity data from the current-meter mooring array and subsurface acoustic floats. Maps of relative vorticity, vortex stretching and potential vorticity allow identification of the signature of the eddy. Nearby companion anomalies caused by an intrusion of Mediterranean Water (MW) can be singled out and are comparable to those of the main eddy around 1200 m. Eulerian and Lagrangian tests of the conservation of potential vorticity are presented and the close similarities of salinity and potential vorticity as tracers of mesoscale motions are vividly demonstrated. Computations of advection of vortex stretching and relative vorticity show that the anticyclonic path of the main eddy above 1000 m is controlled by a genuine interaction with the MW intrusion whose ultimate cited is to sheer the vertical axis of the eddy. It is argued that such interactions must be rather common in the eastern North Atlantic for warm salty patches of Mediterranean origin are often associated with low density anomalies.

## Abstract

The behavior of the intense anticyclonic eddy observed during the Tourbillon Experiment (September-November 1979) is studied within the framework of quasi-geostrophic dynamics. The nondivergent part of the pressure held is estimated through objective analysis at the times of four quasi-synoptic CTD arrays, making use as well of velocity data from the current-meter mooring array and subsurface acoustic floats. Maps of relative vorticity, vortex stretching and potential vorticity allow identification of the signature of the eddy. Nearby companion anomalies caused by an intrusion of Mediterranean Water (MW) can be singled out and are comparable to those of the main eddy around 1200 m. Eulerian and Lagrangian tests of the conservation of potential vorticity are presented and the close similarities of salinity and potential vorticity as tracers of mesoscale motions are vividly demonstrated. Computations of advection of vortex stretching and relative vorticity show that the anticyclonic path of the main eddy above 1000 m is controlled by a genuine interaction with the MW intrusion whose ultimate cited is to sheer the vertical axis of the eddy. It is argued that such interactions must be rather common in the eastern North Atlantic for warm salty patches of Mediterranean origin are often associated with low density anomalies.

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## Abstract

The Tourbillon experiment carried out in 1979–80 over the Porcupine Abyssal Plain in the eastern North Atlantic is the fist experiment providing data that allow an efficient description of the spatial structure of the mesoscale low-frequency flows in an eastern midlatitude basin. The eddies appear to scale in the horizontal as the internal Rossby radius of deformation and, although of an energy level comparable to those observed during MODE, they are intrinsically more nonlinear because of reduced horizontal scale. In much the same way the time scales are less than what is observed in western basin experiments MODE, Polymode III A, B suggesting some kind of turbulent dispersion relation with frequency proportional to wavenumber in the eddy energy-containing range. We observe in Tourbillon an equipartition of eddy kinetic and potential energy. The frequency kinetic-energy spectra have slopes of order −2 in a log–log form, hence are more white than in western basins (−2.5, −3). The distribution of eddy kinetic is highly intensified above the main pycnocline (more so at high rather than low frequencies). An empirical-orthogonal-function decomposition has been carried out in the vertical direction and indicates that the lowest frequencies (450–64 days) are vertically coherent, 72% of the total energy being described by only one EOF with baroclinic structure. The vertical coherence decreases with period. Finally the relevance of variable atmospheric driving to induce eddy motions is discussed.

## Abstract

The Tourbillon experiment carried out in 1979–80 over the Porcupine Abyssal Plain in the eastern North Atlantic is the fist experiment providing data that allow an efficient description of the spatial structure of the mesoscale low-frequency flows in an eastern midlatitude basin. The eddies appear to scale in the horizontal as the internal Rossby radius of deformation and, although of an energy level comparable to those observed during MODE, they are intrinsically more nonlinear because of reduced horizontal scale. In much the same way the time scales are less than what is observed in western basin experiments MODE, Polymode III A, B suggesting some kind of turbulent dispersion relation with frequency proportional to wavenumber in the eddy energy-containing range. We observe in Tourbillon an equipartition of eddy kinetic and potential energy. The frequency kinetic-energy spectra have slopes of order −2 in a log–log form, hence are more white than in western basins (−2.5, −3). The distribution of eddy kinetic is highly intensified above the main pycnocline (more so at high rather than low frequencies). An empirical-orthogonal-function decomposition has been carried out in the vertical direction and indicates that the lowest frequencies (450–64 days) are vertically coherent, 72% of the total energy being described by only one EOF with baroclinic structure. The vertical coherence decreases with period. Finally the relevance of variable atmospheric driving to induce eddy motions is discussed.

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## Abstract

The turbulent diapycnal mixing in the ocean is currently obtained from microstructure and finestructure measurements, dye experiments, and inverse models. This study presents a new method that infers the diapycnal mixing from low-resolution numerical calculations of the World Ocean whose temperatures and salinities are restored to the climatology. At the difference of robust general circulation ocean models, diapycnal diffusion is not prescribed but inferred. At steady state the buoyancy equation shows an equilibrium between the large-scale diapycnal advection and the restoring terms that take the place of the divergence of eddy buoyancy fluxes. The geography of the diapycnal flow reveals a strong regional variability of water mass transformations. Positive values of the diapycnal flow indicate an erosion of a deep-water mass and negative values indicate a creation. When the diapycnal flow is upward, a diffusion law can be fitted in the vertical and the diapycnal eddy diffusivity is obtained throughout the water column. The basin averages of diapycnal diffusivities are small in the first 1500 m [*O*(10^{−5}) m^{2} s^{−1}] and increase downward with bottom values of about 2.5 × 10^{−4} m^{2} s^{−1} in all ocean basins, with the exception of the Southern Ocean (50°–30°S), where they reach 12 × 10^{−4} m^{2} s^{−1}. This study confirms the small diffusivity in the thermocline and the robustness of the higher canonical Munk’s value in the abyssal ocean. It indicates that the upward dianeutral transport in the Atlantic mostly takes place in the abyss and the upper ocean, supporting the quasi-adiabatic character of the middepth overturning.

## Abstract

The turbulent diapycnal mixing in the ocean is currently obtained from microstructure and finestructure measurements, dye experiments, and inverse models. This study presents a new method that infers the diapycnal mixing from low-resolution numerical calculations of the World Ocean whose temperatures and salinities are restored to the climatology. At the difference of robust general circulation ocean models, diapycnal diffusion is not prescribed but inferred. At steady state the buoyancy equation shows an equilibrium between the large-scale diapycnal advection and the restoring terms that take the place of the divergence of eddy buoyancy fluxes. The geography of the diapycnal flow reveals a strong regional variability of water mass transformations. Positive values of the diapycnal flow indicate an erosion of a deep-water mass and negative values indicate a creation. When the diapycnal flow is upward, a diffusion law can be fitted in the vertical and the diapycnal eddy diffusivity is obtained throughout the water column. The basin averages of diapycnal diffusivities are small in the first 1500 m [*O*(10^{−5}) m^{2} s^{−1}] and increase downward with bottom values of about 2.5 × 10^{−4} m^{2} s^{−1} in all ocean basins, with the exception of the Southern Ocean (50°–30°S), where they reach 12 × 10^{−4} m^{2} s^{−1}. This study confirms the small diffusivity in the thermocline and the robustness of the higher canonical Munk’s value in the abyssal ocean. It indicates that the upward dianeutral transport in the Atlantic mostly takes place in the abyss and the upper ocean, supporting the quasi-adiabatic character of the middepth overturning.

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## Abstract

The mean ocean circulation near 1000-m depth is estimated with 100-km resolution from the Argo float displacements collected before 1 January 2010. After a thorough validation, the 400 000 or so displacements found in the 950–1150 dbar layer and with parking times between 4 and 17 days allow the currents to be mapped at intermediate depths with unprecedented details. The Antarctic Circumpolar Current (ACC) is the most prominent feature, but western boundary currents (and their recirculations) and alternating zonal jets in the tropical Atlantic and Pacific are also well defined. Eddy kinetic energy (EKE) gives the mesoscale variability (on the order of 10 cm^{2} s^{−2} in the interior), which is compared to the surface geostrophic altimetric EKE showing *e*-folding depths greater than 700 m in the ACC and northern subpolar regions. Assuming planetary geostrophy, the geopotential height of the 1000-dbar isobar is estimated to obtain an absolute and deep reference level worldwide. This is done by solving numerically the Poisson equation that results from taking the divergence of the geostrophic equations on the sphere, assuming Neumann boundary conditions.

## Abstract

The mean ocean circulation near 1000-m depth is estimated with 100-km resolution from the Argo float displacements collected before 1 January 2010. After a thorough validation, the 400 000 or so displacements found in the 950–1150 dbar layer and with parking times between 4 and 17 days allow the currents to be mapped at intermediate depths with unprecedented details. The Antarctic Circumpolar Current (ACC) is the most prominent feature, but western boundary currents (and their recirculations) and alternating zonal jets in the tropical Atlantic and Pacific are also well defined. Eddy kinetic energy (EKE) gives the mesoscale variability (on the order of 10 cm^{2} s^{−2} in the interior), which is compared to the surface geostrophic altimetric EKE showing *e*-folding depths greater than 700 m in the ACC and northern subpolar regions. Assuming planetary geostrophy, the geopotential height of the 1000-dbar isobar is estimated to obtain an absolute and deep reference level worldwide. This is done by solving numerically the Poisson equation that results from taking the divergence of the geostrophic equations on the sphere, assuming Neumann boundary conditions.

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## Abstract

The time-mean Argo float displacements and the *World Ocean Atlas 2009* temperature–salinity climatology are used to obtain the total, top to bottom, mass transports. Outside of an equatorial band, the total transports are the sum of the vertical integrals of geostrophic- and wind-driven Ekman currents. However, these transports are generally divergent, and to obtain a mass conserving circulation, a Poisson equation is solved for the streamfunction with Dirichlet boundary conditions at solid boundaries. The value of the streamfunction on islands is also part of the unknowns. This study presents and discusses an energetic circulation in three basins: the North Atlantic, the North Pacific, and the Southern Ocean. This global method leads to new estimations of the time-mean western Eulerian boundary current transports maxima of 97 Sverdrups (Sv; 1 Sv ≡ 10^{6} m^{3} s^{−1}) at 60°W for the Gulf Stream, 84 Sv at 157°E for the Kuroshio, 80 Sv for the Agulhas Current between 32° and 36°S, and finally 175 Sv for the Antarctic Circumpolar Current at Drake Passage. Although the large-scale structure and boundary of the interior gyres is well predicted by the Sverdrup relation, the transports derived from the wind stress curl are lower than the observed transports in the interior by roughly a factor of 2, suggesting an important contribution of the bottom torques. With additional Argo displacement data, the errors caused by the presence of remaining transient terms at the 1000-db reference level will continue to decrease, allowing this method to produce increasingly accurate results in the future.

## Abstract

The time-mean Argo float displacements and the *World Ocean Atlas 2009* temperature–salinity climatology are used to obtain the total, top to bottom, mass transports. Outside of an equatorial band, the total transports are the sum of the vertical integrals of geostrophic- and wind-driven Ekman currents. However, these transports are generally divergent, and to obtain a mass conserving circulation, a Poisson equation is solved for the streamfunction with Dirichlet boundary conditions at solid boundaries. The value of the streamfunction on islands is also part of the unknowns. This study presents and discusses an energetic circulation in three basins: the North Atlantic, the North Pacific, and the Southern Ocean. This global method leads to new estimations of the time-mean western Eulerian boundary current transports maxima of 97 Sverdrups (Sv; 1 Sv ≡ 10^{6} m^{3} s^{−1}) at 60°W for the Gulf Stream, 84 Sv at 157°E for the Kuroshio, 80 Sv for the Agulhas Current between 32° and 36°S, and finally 175 Sv for the Antarctic Circumpolar Current at Drake Passage. Although the large-scale structure and boundary of the interior gyres is well predicted by the Sverdrup relation, the transports derived from the wind stress curl are lower than the observed transports in the interior by roughly a factor of 2, suggesting an important contribution of the bottom torques. With additional Argo displacement data, the errors caused by the presence of remaining transient terms at the 1000-db reference level will continue to decrease, allowing this method to produce increasingly accurate results in the future.

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## Abstract

Numerical simulations of coarse-resolution, idealized ocean basins under constant surface heat flux are analyzed to show that the interdecadal oscillations that emerge naturally in such configurations are driven by baroclinic instability of the mean state and damped by horizontal diffusion. When the surface heat fluxes are diagnosed from a spinup in which surface temperatures are strongly restored to apparent atmospheric temperatures, the most unstable regions diagnosed by large downgradient eddy heat fluxes are located in the basin northwest corner where the surface heat losses are largest. The long-wave limit of the baroclinic instability of idealized mean flows in a three-layer model with vertical shears as observed in the GCMs demonstrates that growth rates of order one cycle per year can be produced locally, large enough to amplify thermal anomalies in the face of lateral diffusion. The proposed instability mechanism that favors surface-intensified perturbations also explains the lack of oscillations if the restoring to a surface climatology is too strong. To assess whether this instability process of oceanic origin is robust enough to cause interdecadal variability of coupled ocean–atmosphere models, a four-box ocean–atmosphere model is constructed. Given the large heat capacity of the ocean as compared to the atmosphere, the dynamical system that governs the model evolution is reduced to only two degrees of freedom, the oceanic overturning thermohaline circulation and the interior north–south temperature gradient. The authors show that, when the baroclinic instability growth rate exceeds the overall dissipation caused by turbulent eddy diffusion in the atmosphere and ocean and infrared back radiation, the dynamical system undergoes a Hopf bifurcation, and interdecadal oscillations emerge through a limit cycle.

## Abstract

Numerical simulations of coarse-resolution, idealized ocean basins under constant surface heat flux are analyzed to show that the interdecadal oscillations that emerge naturally in such configurations are driven by baroclinic instability of the mean state and damped by horizontal diffusion. When the surface heat fluxes are diagnosed from a spinup in which surface temperatures are strongly restored to apparent atmospheric temperatures, the most unstable regions diagnosed by large downgradient eddy heat fluxes are located in the basin northwest corner where the surface heat losses are largest. The long-wave limit of the baroclinic instability of idealized mean flows in a three-layer model with vertical shears as observed in the GCMs demonstrates that growth rates of order one cycle per year can be produced locally, large enough to amplify thermal anomalies in the face of lateral diffusion. The proposed instability mechanism that favors surface-intensified perturbations also explains the lack of oscillations if the restoring to a surface climatology is too strong. To assess whether this instability process of oceanic origin is robust enough to cause interdecadal variability of coupled ocean–atmosphere models, a four-box ocean–atmosphere model is constructed. Given the large heat capacity of the ocean as compared to the atmosphere, the dynamical system that governs the model evolution is reduced to only two degrees of freedom, the oceanic overturning thermohaline circulation and the interior north–south temperature gradient. The authors show that, when the baroclinic instability growth rate exceeds the overall dissipation caused by turbulent eddy diffusion in the atmosphere and ocean and infrared back radiation, the dynamical system undergoes a Hopf bifurcation, and interdecadal oscillations emerge through a limit cycle.