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## Abstract

The part of the meridional overturning circulation driven by time-varying winds is usually assumed to be an Ekman flux within a mixed layer, and a depth- and laterally independent return flow beneath. For a simple linear frictional ocean model, the return flow is studied for a range of frequencies from several days to decades. It is shown that while the east–west integral of the return flow is usually, but not always, almost independent of depth, the spatial distribution of the return flow varies strongly with both horizontal and vertical position. This can have important consequences for calculations of the northward heat flux, which traditionally assumes a spatially uniform return flow.

## Abstract

The part of the meridional overturning circulation driven by time-varying winds is usually assumed to be an Ekman flux within a mixed layer, and a depth- and laterally independent return flow beneath. For a simple linear frictional ocean model, the return flow is studied for a range of frequencies from several days to decades. It is shown that while the east–west integral of the return flow is usually, but not always, almost independent of depth, the spatial distribution of the return flow varies strongly with both horizontal and vertical position. This can have important consequences for calculations of the northward heat flux, which traditionally assumes a spatially uniform return flow.

## Abstract

This paper examines the representation of eddy fluxes by bolus velocities. In particular, it asks the following: 1) Can an arbitrary eddy flux divergence of density be represented accurately by a nondivergent bolus flux that satisfies the condition of no normal flow at boundaries? 2) If not, how close can such a representation come? 3) If such a representation can exist in some circumstances, what is the size of the smallest bolus velocity that fits the data?

The author finds, in agreement with earlier authors, that the answer to the first question is no, although under certain conditions, which include a modification to the eddy flux divergence, a bolus representation becomes possible. One such condition is when the eddy flux divergence is required to balance the time-mean flux divergence. The smallest bolus flow is easily found by solving a thickness-weighted Poisson equation on each density level. This problem is solved for the North Pacific using time-mean data from an eddy-permitting model. The minimum bolus flow is found to be very small at depth but larger than is usually assumed near the surface. The magnitude of this minimum flow is of order one-tenth of the mean flow. Similar but larger results are found for a coarse-resolution model.

## Abstract

This paper examines the representation of eddy fluxes by bolus velocities. In particular, it asks the following: 1) Can an arbitrary eddy flux divergence of density be represented accurately by a nondivergent bolus flux that satisfies the condition of no normal flow at boundaries? 2) If not, how close can such a representation come? 3) If such a representation can exist in some circumstances, what is the size of the smallest bolus velocity that fits the data?

The author finds, in agreement with earlier authors, that the answer to the first question is no, although under certain conditions, which include a modification to the eddy flux divergence, a bolus representation becomes possible. One such condition is when the eddy flux divergence is required to balance the time-mean flux divergence. The smallest bolus flow is easily found by solving a thickness-weighted Poisson equation on each density level. This problem is solved for the North Pacific using time-mean data from an eddy-permitting model. The minimum bolus flow is found to be very small at depth but larger than is usually assumed near the surface. The magnitude of this minimum flow is of order one-tenth of the mean flow. Similar but larger results are found for a coarse-resolution model.

## Abstract

The trajectories of inertial flows on a rotating earth are calculated, in an attempt to reconcile the differing heuristic suggestions in the literature on the subject. It is shown that westward propagating “nearly closed” orbits are possible away from the equator. For orbits crossing the equator, we find a stationary, “figure-eight- like” orbit, together with eastward and westward propagating modes. Near the pole, the convergence of longitudes causes the trajectories to be deflected cyclonically in contrast to the deflection of the Coriolis force, giving rise to a westward propagating mode that meanders about a central latitude.

## Abstract

The trajectories of inertial flows on a rotating earth are calculated, in an attempt to reconcile the differing heuristic suggestions in the literature on the subject. It is shown that westward propagating “nearly closed” orbits are possible away from the equator. For orbits crossing the equator, we find a stationary, “figure-eight- like” orbit, together with eastward and westward propagating modes. Near the pole, the convergence of longitudes causes the trajectories to be deflected cyclonically in contrast to the deflection of the Coriolis force, giving rise to a westward propagating mode that meanders about a central latitude.

## Abstract

Attempts to estimate the state of the ocean usually involve one of two approaches: either an assimilation of data (typically altimetric surface height) is performed or an inversion is carried out according to some minimization scheme. The former case normally retains some version of the time-dependent equations of motion; the latter is usually steady. Data sources are frequently not ideal for either approach, usually being spatially and temporally confined (e.g., from an oceanographic cruise). This raises particular difficulties for inversions, whose physics seldom includes much beyond the geostrophic balance. In this paper the authors examine an approach midway between the two, examining several questions. (i) What is the impact of data assimilated continuously to a steady state on regions outside the data sources? (ii) Can remote data improve the long-term mean of a model whose natural response is not close to climatology? (iii) Can an eddy-free model assimilate data containing eddies?

The authors employ an inversion using a simple North Atlantic model, which permits no eddies, but contains better dynamics than geostrophy (the frictional planetary geostrophic equations), and an assimilative scheme rather simpler than those normally employed, almost equivalent to direct data insertion, run to a steady state. The data used are real subsurface data, which do contain eddies, from World Ocean Circulation Experiment cruises in the northern North Atlantic. The presence of noise in these data is found to cause no numerical difficulties, and the authors show that the impact of even one vertical profile can strongly modify the water mass properties of the solution far from the data region through a combination of wave propagation, advection, and diffusion. Because the model can be run for very long times, the region of impact is thus somewhat wider than would occur for assimilations over short intervals, such as a year.

## Abstract

Attempts to estimate the state of the ocean usually involve one of two approaches: either an assimilation of data (typically altimetric surface height) is performed or an inversion is carried out according to some minimization scheme. The former case normally retains some version of the time-dependent equations of motion; the latter is usually steady. Data sources are frequently not ideal for either approach, usually being spatially and temporally confined (e.g., from an oceanographic cruise). This raises particular difficulties for inversions, whose physics seldom includes much beyond the geostrophic balance. In this paper the authors examine an approach midway between the two, examining several questions. (i) What is the impact of data assimilated continuously to a steady state on regions outside the data sources? (ii) Can remote data improve the long-term mean of a model whose natural response is not close to climatology? (iii) Can an eddy-free model assimilate data containing eddies?

The authors employ an inversion using a simple North Atlantic model, which permits no eddies, but contains better dynamics than geostrophy (the frictional planetary geostrophic equations), and an assimilative scheme rather simpler than those normally employed, almost equivalent to direct data insertion, run to a steady state. The data used are real subsurface data, which do contain eddies, from World Ocean Circulation Experiment cruises in the northern North Atlantic. The presence of noise in these data is found to cause no numerical difficulties, and the authors show that the impact of even one vertical profile can strongly modify the water mass properties of the solution far from the data region through a combination of wave propagation, advection, and diffusion. Because the model can be run for very long times, the region of impact is thus somewhat wider than would occur for assimilations over short intervals, such as a year.

## Abstract

Although data assimilation is now an established oceanographic technique, little work has been done on the interaction of the assimilation scheme and the physics of the underlying model. The way in which even a simple assimilation scheme (here nudging) can significantly alter the response of the model to which it is applied is illustrated here.

Using analytic and semianalytic models, the assimilation of sea surface height, density, and velocity is studied. It is shown that the assimilation can act to alter the high inertia–gravity wave frequency to be the order of the Coriolis parameter, a result that is of relevance to the problems of initialization. The theory also predicts an optimum strength of nudging, normally dependent on wavelength, wave speed, and latitude, which can give convergence of the assimilation on a timescale as short as a day. The results are verified by identical twin experiments using a full primitive equation model, the Free Surface Cox Code, both in barotropic spinup (results presented here) and in a more realistic baroclinic situation (results presented in Part II).

## Abstract

Although data assimilation is now an established oceanographic technique, little work has been done on the interaction of the assimilation scheme and the physics of the underlying model. The way in which even a simple assimilation scheme (here nudging) can significantly alter the response of the model to which it is applied is illustrated here.

Using analytic and semianalytic models, the assimilation of sea surface height, density, and velocity is studied. It is shown that the assimilation can act to alter the high inertia–gravity wave frequency to be the order of the Coriolis parameter, a result that is of relevance to the problems of initialization. The theory also predicts an optimum strength of nudging, normally dependent on wavelength, wave speed, and latitude, which can give convergence of the assimilation on a timescale as short as a day. The results are verified by identical twin experiments using a full primitive equation model, the Free Surface Cox Code, both in barotropic spinup (results presented here) and in a more realistic baroclinic situation (results presented in Part II).

## Abstract

A preoperational scheme has been implemented to calculate sea surface height fields at 7-day intervals over the North Atlantic. Input data from Argo floats is downloaded and processed in near–real time. The solution method is by Bernoulli inverse. Early results are encouraging. Features of the results are compared with both model and satellite data and show good agreement.

## Abstract

A preoperational scheme has been implemented to calculate sea surface height fields at 7-day intervals over the North Atlantic. Input data from Argo floats is downloaded and processed in near–real time. The solution method is by Bernoulli inverse. Early results are encouraging. Features of the results are compared with both model and satellite data and show good agreement.

## Abstract

A shallow-water beta-channel model was used to carry out numerical experiments with cyclonic and anticyclonic disturbances of various strengths. The model is inviscid, so fluid elements conserve potential vorticity *q* when unforced. Regions of closed *q* contours correspond to Lagrangian (material) eddies. (All fluid within a Lagrangian eddy travels with the eddy—in contrast to regions of closed height contours.)

Motion is wavelike for very weak disturbances (maximum particle speed *Û*; ≪ long planetary wave speed *ĉ*). The height field disperses like a group of linear Rossby waves, and tracers have small, oscillatory (mainly north-south) displacements, with very little scatter.

When *Û*≈*ĉ*, the planetary *q* field is sufficiently distorted for small Lagrangian eddies to appear. Very small eddies are simply bodily advected by the linear wave field. Small eddies are to some extent “self propelling”: they move westward and north (cyclone) or south (anticyclone), moving fluid elements towards their “rest” latitudes. Tracers within such eddies are moved away from neighboring tracers initially outside the eddy (which have largely wavelike motion). The eddy and the height extremum, initially together, gradually separate. (The position of a height extremum is not a good indicator of tracer movement.)

When *Ü*≫*ĉ*, the *q* field is grossly distorted, and the motion is dominated by a nonlinear eddy which is strong enough to advect ambient *q* (and fluid elements) around itself. This wrapping effect leads to relatively strong mixing (by wave breaking?) around the fringes of the eddy, which slowly decays by this mechanism. Movement of the eddy is predominantly westward, at almost the same speed as the center-of-mass anomaly (for a buoyancy-generated disturbance).

Analytic center-of-mass calculations predict that the center-of-mass of an anticyclone travels westward faster than the linear long-wave speed *ĉ*, whereas a cyclone travels slower than *ĉ*. The predictions are confirmed by the numerical experiments.

Some estimates of mixing based on tracer separation are given.

## Abstract

A shallow-water beta-channel model was used to carry out numerical experiments with cyclonic and anticyclonic disturbances of various strengths. The model is inviscid, so fluid elements conserve potential vorticity *q* when unforced. Regions of closed *q* contours correspond to Lagrangian (material) eddies. (All fluid within a Lagrangian eddy travels with the eddy—in contrast to regions of closed height contours.)

Motion is wavelike for very weak disturbances (maximum particle speed *Û*; ≪ long planetary wave speed *ĉ*). The height field disperses like a group of linear Rossby waves, and tracers have small, oscillatory (mainly north-south) displacements, with very little scatter.

When *Û*≈*ĉ*, the planetary *q* field is sufficiently distorted for small Lagrangian eddies to appear. Very small eddies are simply bodily advected by the linear wave field. Small eddies are to some extent “self propelling”: they move westward and north (cyclone) or south (anticyclone), moving fluid elements towards their “rest” latitudes. Tracers within such eddies are moved away from neighboring tracers initially outside the eddy (which have largely wavelike motion). The eddy and the height extremum, initially together, gradually separate. (The position of a height extremum is not a good indicator of tracer movement.)

When *Ü*≫*ĉ*, the *q* field is grossly distorted, and the motion is dominated by a nonlinear eddy which is strong enough to advect ambient *q* (and fluid elements) around itself. This wrapping effect leads to relatively strong mixing (by wave breaking?) around the fringes of the eddy, which slowly decays by this mechanism. Movement of the eddy is predominantly westward, at almost the same speed as the center-of-mass anomaly (for a buoyancy-generated disturbance).

Analytic center-of-mass calculations predict that the center-of-mass of an anticyclone travels westward faster than the linear long-wave speed *ĉ*, whereas a cyclone travels slower than *ĉ*. The predictions are confirmed by the numerical experiments.

Some estimates of mixing based on tracer separation are given.

## Abstract

Three inverse methods (the Bernoulli, beta-spiral, and box inverse methods) are used on mean data from an eddy-resolving oceanic general circulation model, in an attempt to reconstruct the observed mean flow field. Inversions are performed in the Gulf Stream extension, a quiet region which is relatively eddy-free, the center of the region of homogenized potential vorticity, and a near-equatorial area, together with an inversion of the flow across a transoceanic sector. Resolutions for the inversions of ⅓°, 1° and 2° are used. Numerical estimates of geostrophy using the wider resolutions can give top-to-bottom thermal wind shears in error by up to 1 cm s^{−1} in a flow change of around 8 cm s^{−1}. Two “scores” for the methods are created, one which tests pointwise accuracy (the “global” score) and one which tests fluxes of mass through a section (the “flux” score). The Bernoulli method yields accurate global scores except in the homogenized region; the box inverse method yields fairly accurate global scores everywhere; and the beta-spiral only gives accurate global scores near the equator. No method gives reliable flux scores, although the box inverse was the least inaccurate, as might be expected from the nature of this method. The hypothesis of no flow at the bottom gives a predicted velocity field which is more accurate than any of the inversions most of the time. The Bernoulli and beta-spiral methods contain an internal measure which is well correlated with their accuracy, so that it is possible to estimate the accuracy of an inversion on real data.

## Abstract

Three inverse methods (the Bernoulli, beta-spiral, and box inverse methods) are used on mean data from an eddy-resolving oceanic general circulation model, in an attempt to reconstruct the observed mean flow field. Inversions are performed in the Gulf Stream extension, a quiet region which is relatively eddy-free, the center of the region of homogenized potential vorticity, and a near-equatorial area, together with an inversion of the flow across a transoceanic sector. Resolutions for the inversions of ⅓°, 1° and 2° are used. Numerical estimates of geostrophy using the wider resolutions can give top-to-bottom thermal wind shears in error by up to 1 cm s^{−1} in a flow change of around 8 cm s^{−1}. Two “scores” for the methods are created, one which tests pointwise accuracy (the “global” score) and one which tests fluxes of mass through a section (the “flux” score). The Bernoulli method yields accurate global scores except in the homogenized region; the box inverse method yields fairly accurate global scores everywhere; and the beta-spiral only gives accurate global scores near the equator. No method gives reliable flux scores, although the box inverse was the least inaccurate, as might be expected from the nature of this method. The hypothesis of no flow at the bottom gives a predicted velocity field which is more accurate than any of the inversions most of the time. The Bernoulli and beta-spiral methods contain an internal measure which is well correlated with their accuracy, so that it is possible to estimate the accuracy of an inversion on real data.

## Abstract

The response of an ocean with a single active dynamical layer (notionally with an infinitely thick upper layer above it, of slightly less density) to localized buoyancy forcing on a beta-plane is considered. It is shown that three regimes exist. When the forcing is very weak, the response is linear, and consists of a quasi-steady “tube” of fluid stretching westwards from the forcing region, with a front advancing at the long Rossby wave speed, and some transient structure in the vicinity of the forcing. When the amplitude of the forcing is increased, potential vorticity contours are sufficiently deformed to permit instability both in the forced region and to its west. The response becomes a series of shed eddies each of which propagates westwards. The time scale to generate an eddy is proportional to the time taken for a long Rossby wave to propagate across the forced region. Further increase in forcing amplitude yields a completely unsteady response.

## Abstract

The response of an ocean with a single active dynamical layer (notionally with an infinitely thick upper layer above it, of slightly less density) to localized buoyancy forcing on a beta-plane is considered. It is shown that three regimes exist. When the forcing is very weak, the response is linear, and consists of a quasi-steady “tube” of fluid stretching westwards from the forcing region, with a front advancing at the long Rossby wave speed, and some transient structure in the vicinity of the forcing. When the amplitude of the forcing is increased, potential vorticity contours are sufficiently deformed to permit instability both in the forced region and to its west. The response becomes a series of shed eddies each of which propagates westwards. The time scale to generate an eddy is proportional to the time taken for a long Rossby wave to propagate across the forced region. Further increase in forcing amplitude yields a completely unsteady response.

## Abstract

The Rossby adjustment of an initially circular column of water, the so-called collapse of a cylinder, continues to be a widely used method for forming lenslike eddies in the laboratory. Here, we consider the structure of an eddy so formed as well as some ramifications of that formation. We demonstrate that the calculation of the eddy structure can be reduced to the extraction of the roots of two nonlinear, coupled algebraic equations. Analytical solutions in the limit of the collapse of a needle are given and roots are obtained numerically otherwise. It is concluded that in the collapse of a cylinder initially spanning the entire column of water, the eddy always maintains contact with both surfaces. (This is not the case in the seemingly equivalent two-dimensional case with no variation in one Cartesian direction.) In the event the initial cold column is separated only slightly from the surface, the above solution acts as the lowest order solution in a regular perturbation sequence.

Next, these “collapse eddy” solutions, which possess motions in both layers and finite energies, are used to examine lens merger. Two collapse eddies of equal volume jointly possess less energy than one collapse eddy of twice the volume. However, we argue that two collapse eddies of equal volume can have more energy than the circularly symmetric end-state eddy formed from them if the two initial eddies “mix.” We also offer evidence that the energy budgets may be balanced exactly if the end-state eddy is slightly asymmetric. Comparisons with some previous laboratory experiments are made.

## Abstract

The Rossby adjustment of an initially circular column of water, the so-called collapse of a cylinder, continues to be a widely used method for forming lenslike eddies in the laboratory. Here, we consider the structure of an eddy so formed as well as some ramifications of that formation. We demonstrate that the calculation of the eddy structure can be reduced to the extraction of the roots of two nonlinear, coupled algebraic equations. Analytical solutions in the limit of the collapse of a needle are given and roots are obtained numerically otherwise. It is concluded that in the collapse of a cylinder initially spanning the entire column of water, the eddy always maintains contact with both surfaces. (This is not the case in the seemingly equivalent two-dimensional case with no variation in one Cartesian direction.) In the event the initial cold column is separated only slightly from the surface, the above solution acts as the lowest order solution in a regular perturbation sequence.

Next, these “collapse eddy” solutions, which possess motions in both layers and finite energies, are used to examine lens merger. Two collapse eddies of equal volume jointly possess less energy than one collapse eddy of twice the volume. However, we argue that two collapse eddies of equal volume can have more energy than the circularly symmetric end-state eddy formed from them if the two initial eddies “mix.” We also offer evidence that the energy budgets may be balanced exactly if the end-state eddy is slightly asymmetric. Comparisons with some previous laboratory experiments are made.