Multi-layer Quasi-geostrophic Model Research Articles

Modon solutions in an N-layer quasi-geostrophic model

Modons, or dipolar vortices, are common and long-lived features of the upper ocean, consisting of a pair of counter-rotating monopolar vortices moving through self-advection. Such structures remain stable over long times and may be important for fluid transport over large distances. Here, we present a semi-analytical method for finding fully nonlinear modon solutions in a multi-layer quasi-geostrophic model with arbitrarily many layers. Our approach is to reduce the problem to a multi-parameter linear eigenvalue problem which can be solved using numerical techniques from linear algebra. The method is shown to replicate previous results for one- and two-layer models and is applied to a three-layer model to find a solution describing a mid-depth propagating, topographic vortex.

Back-and-forth nudging for the quasi-geostrophic ocean dynamics with altimetry: Theoretical convergence study and numerical experiments with the future SWOT observations

<p style='text-indent:20px;'>In data assimilation for geophysical problems, the increasing amount of satellite data to analyze makes it more and more challenging to guarantee near real time forecasting. Thus, low time and memory consuming data assimilation methods become very attractive. The back-and-forth nudging (BFN) method is a non-classical data assimilation method that can be seen as a deterministic and smoothing version of the Kalman filter. From a practical point of view, the BFN method is very valuable for its simplicity of implementation (no optimization, no differentiation, ...) and its rapidity of convergence. Under observability conditions, we prove the mathematical convergence of BFN at deep layers for a multi-layer quasi-geostrophic (MQG) ocean circulation model using an infinite dimensional variant of LaSalle's invariance principle. We also extend the BFN to the problem of joint state-parameter identification. The numerical experiments, performed on 120km large swath sea surface height (SSH) simulated data of the Surface Water Ocean Topography (SWOT) satellite, show the high robustness of the algorithm to uncertainties and the few iterations needed to reach convergence, whereas some problems remain due to non-reversibility properties in time. We also give a strategy to improve geophysical model accuracy, considering the large number of uncertain parameters inherent to models and their impacts on state estimation performance. We propose here a joint state-parameter estimation, tested on the baroclinic wavenumber as an unobserved parameter.</p>

Multi-layer quasi-geostrophic ocean dynamics in Eddy-resolving regimes

The multi-layer quasi-geostrophic model of the wind-driven ocean gyres is numerically investigated using a combination of long-time runs (200 years) needed for accurate statistics, spatial resolutions (grid interval of less than one kilometer) needed for accurate representation of mesoscale eddies, and large Reynolds number (Re > 104) needed for more realistic flow regimes. We gradually increased the Reynolds number by lowering the eddy viscosity and analysed the corresponding changes of the large-scale circulation, energetics and eddy fluxes, with the goal to understand how the nonlinear eddy dynamics affects the large-scale ocean circulation, as more and more degrees of freedom become dynamically available. Three- and six-layer configurations of the model are considered in order to understand effects of higher baroclinic modes. A parameter sensitivity study is also carried out to show that the explored flow regime is robust.As Re increases, most properties of the flow show no signs of approaching an asymptote, and the following tendencies are found. The time-mean flow properties tend to an asymptote in the three-layer model but not in the six-layer one, suggesting that higher baroclinic modes are dynamically more active at larger Re. The eddy kinetic and potential energies grow faster in the six-layer case. The intensity of the eddy forcing (eddy flux divergence) increases with Re. The inter-gyre eddy potential vorticity flux is predominantly northward and up-gradient for all Re studied. A comparison of the three- and six-layer model solutions revealed an inhibitory influence of high baroclinic modes on the penetration length of the eastward jet extension of the western boundary currents and on the strength of the adjacent recirculation zones. In large-Re regimes, the population of eddies is mostly sustained by the eddy generation at the eastern end of the eastward jet rather than in its central section. Finally, by studying the numerical convergence of the solutions, we found the empirical dependency between the eddy viscosity and the required grid resolution: halving the viscosity requires halving the grid spacing.

Vortex stability in a multi-layer quasi-geostrophic model: application to Mediterranean Water eddies

The stability of circular vortices to normal mode perturbations is studied in a multi-layer quasi-geostrophic model. The stratification is fitted on the Gulf of Cadiz where many Mediterranean Water (MW) eddies are generated. Observations of MW eddies are used to determine the parameters of the reference experiment; sensitivity tests are conducted around this basic case. The objective of the study is two-fold: (a) determine the growth rates and nonlinear evolutions of unstable perturbations for different three-dimensional (3D) velocity structures of the vortices, (b) check if the different structure of our idealized vortices, mimicking MW cyclones and anticyclones, can induce different stability properties in a model that conserves parity symmetry, and apply these results to observed MW eddies. The linear stability analysis reveals that, among many 3D distributions of velocity, the observed eddies are close to maximal stability, with instability time scales longer than 100 days (these time scales would be less than 10 days for vertically more sheared eddies). The elliptical deformation is most unstable for realistic eddies (the antisymmetric one dominates for small eddies and the triangular one for large eddies); the antisymmetric mode is stronger for cyclones than for anticyclones. Nonlinear evolutions of eddies with radii of about 30 km, and elliptically perturbed, lead to their re-organization into 3D tripoles; smaller eddies are stable and larger eddies break into 3D dipoles. Horizontally more sheared eddies are more unstable and sustain more asymmetric instabilities. In summary, few differences were found between cyclone and anticyclone stability, except for strong horizontal velocity shears.

Energy and potential enstrophy flux constraints in quasi-geostrophic models

We investigate an inequality constraining the energy and potential enstrophy flux spectra in two-layer and multi-layer quasi-geostrophic models. Its physical significance is that it can diagnose whether any given multi-layer model that allows co-existing downscale cascades of energy and potential enstrophy can allow the downscale energy flux to become large enough to yield a mixed energy spectrum where the dominant k−3 scaling is overtaken by a subdominant k−5/3 contribution beyond a transition wavenumber kt situated in the inertial range. The validity of the flux inequality implies that this scaling transition cannot occur within the inertial range, whereas a violation of the flux inequality beyond some wavenumber kt implies the existence of a scaling transition near that wavenumber. This flux inequality holds unconditionally in two-dimensional Navier–Stokes turbulence, however, it is far from obvious that it continues to hold in multi-layer quasi-geostrophic models, because the dissipation rate spectra for energy and potential enstrophy no longer relate in a trivial way, as in two-dimensional Navier–Stokes. We derive the general form of the energy and potential enstrophy dissipation rate spectra for a generalized symmetrically coupled multi-layer model. From this result, we prove that in a symmetrically coupled multi-layer quasi-geostrophic model, where the dissipation terms for each layer consist of the same Fourier-diagonal linear operator applied on the streamfunction field of only the same layer, the flux inequality continues to hold. It follows that a necessary condition to violate the flux inequality is the use of asymmetric dissipation where different operators are used on different layers. We explore dissipation asymmetry further in the context of a two-layer quasi-geostrophic model and derive upper bounds on the asymmetry that will allow the flux inequality to continue to hold. Asymmetry is introduced both via an extrapolated Ekman term, based on a 1980 model by Salmon, and via differential small-scale dissipation. The results given are mathematically rigorous and require no phenomenological assumptions about the inertial range. Sufficient conditions for violating the flux inequality, on the other hand, require phenomenological hypotheses, and will be explored in future work.

Pullback attractors for the multi-layer quasi-geostrophic equations of the ocean

This article studies the pullback asymptotic behavior of solutions for a non-autonomous multi-layer quasi-geostrophic model in a two-dimensional domain. We prove the existence of pullback attractors AV in V (the velocity has the H1-regularity) and AH in H (the velocity has the L2-regularity). Then we verify the regularity of the pullback attractors by proving that AV=AH, which implies the pullback asymptotic smoothing effect of the model in the sense that the solutions eventually become more regular than the initial data. The method used in this article is similar to the one used in Zhao and Zhou (2007) in the case of the non-autonomous incompressible non-Newtonian fluid in a two-dimensional domain. Let us mention that the non-homogeneous boundary conditions (and the non-local constraint) present in the multi-layer quasi-geostrophic model makes the estimates more complicated, see Bernier (1994). These difficulties are overcome using the new formulation presented in Tachim Medjo (2009).

Averaging of a multi-layer quasi-geostrophic equations with oscillating external forces

In this article, we consider a non-autonomous multi-layer quasi-geostrophic equations of the ocean with a singularly oscillating external force $g^{\epsilon}= g_0(t) + \epsilon^{-\rho} g_1(t/\epsilon) $ depending on a small parameter $ \epsilon > 0 $ and $ \rho \in [0, 1) $ together with the averaged system with the external force $g_0(t),$ formally corresponding to the case $\epsilon = 0. $ Under suitable assumptions on the external force, we prove as in [10] the boundness of the uniform global attractor $\mathcal{A}^{\epsilon} $ as well as the upper semi-continuity of the attractors $\mathcal{A}^{\epsilon} $ of the singular systems to the attractor $\mathcal{A}^0 $ of the averaged system as $ \epsilon \rightarrow 0^+. $ When the external force is small enough and the viscosity is large enough, the convergence rate is controlled by $K \epsilon^{(1 -\rho)}. $ Let us mention that the non-homogenous boundary conditions (and the non-local constraint) present in the multi-layer quasi-geostrophic model makes the estimates more complicated, [3]. These difficulties are overcome using the new formulation presented in [25].

Annular Mode–Like Variation in a Multilayer Quasigeostrophic Model

Abstract Eddy–zonal flow interactions in the annular modes are investigated in this study using a modified beta-plane multilayer quasigeostrophic (QG) channel model. This study shows the different response of high- and low-phase-speed (frequency) eddies to the zonal wind anomalies and suggests a baroclinic mechanism through which the two eddies work symbiotically maintaining the positive eddy feedback in the annular modes. Analysis also indicates that the different roles played by these two eddies in the annular modes are related to the differences in their critical line distributions. Eddies with higher phase speeds experience a low-level critical layer at the center of the jet. They drive the zonal wind anomalies associated with the annular mode but weaken the baroclinicity of the jet in the process. Lower-phase-speed eddies encounter low-level critical lines on the jet flanks. While their momentum fluxes are not as important for the jet shift, they play an important role by restoring the lower-level baroclinicity at the jet center, creating a positive feedback loop with the fast eddies that extends the persistence of the jet shift. The importance of the lower-level baroclinicity restoration by the low-phase-speed eddies in the annular modes is further demonstrated in sensitivity runs, in which surface friction on eddies is increased to selectively damp the low-phase-speed eddies. For simulations in which the low-phase-speed eddies become inactive, the leading mode of the zonal wind variability shifts from the position fluctuation to a pulsing of the jet intensity. Further studies indicate that the response of the lower-level baroclinicity to the zonal wind anomalies caused by the low-phase-speed eddies can be crucial in maintaining the annular mode–like variations.

Uniform attractors of finite-difference schemes for the multilayer quasigeostrophic model of ocean dynamics

Uniform attractors of finite-difference schemes for the multilayer quasigeostrophic model of ocean dynamics

Baroclinic Eddy Equilibration under Specified Seasonal Forcing

Abstract Baroclinic eddy equilibration under a Northern Hemisphere–like seasonal forcing is studied using a modified multilayer quasigeostrophic channel model to investigate the widely used “quick baroclinic eddy equilibration” assumption and to understand to what extent baroclinic adjustment can be applied to interpret the midlatitude climate. Under a slowly varying seasonal forcing, the eddy and mean flow seasonal behavior is characterized by four clearly divided time intervals: an eddy inactive time interval in summer, a mainly dynamically determined eddy spinup time interval starting in midfall and lasting less than one month, and a quasi-equilibrium time interval for the zonal mean flow available potential energy from late fall to late spring, with a mainly external forcing determined spindown time interval for eddy activity from late winter to late spring. The baroclinic adjustment can be clearly observed from late fall to late spring. The sensitivity study of the eddy equilibration to the time scale of the external forcing indicates that the time scale separation between the baroclinic adjustment and the external forcing in midlatitudes is only visible for external forcing cycles one year and longer. In spite of the strong seasonality of the eddy activity, similar to the observations, a robust potential vorticity (PV) structure is still observed through all the seasons. However, it is found that baroclinic eddy is not the only candidate mechanism to maintain the robust PV structure. The role of the boundary layer thermal forcing and the moist convection in maintaining the lower-level PV structure is discussed. The adjustment and the vertical variation of the lower-level stratification play an important role in all of these mechanisms.

Convergence of solutions to the problem of data assimilation for a multilayer quasigeostrophic model of ocean dynamics

An altimetry data assimilation problem is considered for a multilayer quasigeostrophic model of ocean dynamics. An implicit finite difference scheme is used for the construction of approximate solutions to the problem. The convergence of numerical solutions to the exact solution to this data assimilation problem is proved.

Exponential stability of the multi-layer quasi-geostrophic ocean model with delays

In this paper, the multi-layer quasi-geostrophic ocean model with delays is studied. First, a few criteria are derived for asymptotic stability of the stationary solutions. Then, to quantify the impact of delay on the overall system evolution, an estimate is derived for comparing the quasi-geostrophic ocean states with or without delays.

Generalization of the dual variational data assimilation algorithm to a nonlinear layered quasi-geostrophic ocean model

In this paper, we present a generalization to nonlinear models of the four-dimensional variational dual method, the 4D-PSAS algorithm. The idea of 4D-PSAS (physical space analysis system) is to perform the minimization in the space of the observations, rather than in the model space as in the primal 4D-VAR scheme. Despite the formal equivalence between 4D-VAR and 4D-PSAS in a linear situation (both for model equations and observation operators), the dual method has several important advantages: in oceanographic cases, the observation space is smaller than the model space, which should improve the minimization process; for no additional cost, it provides an estimation of the model error; and finally, it does not have any singularities when the covariance error matrices tend to zero. The idea of this paper is to extend this algorithm to a fully nonlinear situation, as has been done in previous years with other classical data assimilation schemes: the 4D-VAR and the Kalman filter. For this purpose, we consider a nonlinear multi-layer quasi-geostrophic ocean model, which mimics quite well the mid-latitude circulation. We recall the standard primal 4D-VAR scheme applied to this model, and then introduce an extended 4D-PSAS algorithm in the particular case of this nonlinear QG model. We report then the results of extensive numerical experiments that have been carried out to compare this extended algorithm to the classical variational formulation, and to study its sensitivity to many parameters such as the nonlinearities, the number of available observations, the presence of an unknown term in the assimilation model and to study the detection of the model error. As a matter of fact, it is found that this extended algorithm has kept the same advantages as in the linear case (model error detection, smaller sensitivity to various perturbations, more efficient minimization process). All these experiments suggest that it is an efficient assimilation scheme for oceanographic problems.

A Variance-Minimizing Filter for Large-Scale Applications

Abstract A truly variance-minimizing filter is introduced and its performance is demonstrated with the Korteweg–DeVries (KdV) equation and with a multilayer quasigeostrophic model of the ocean area around South Africa. It is recalled that Kalman-like filters are not variance minimizing for nonlinear model dynamics and that four-dimensional variational data assimilation (4DVAR)-like methods relying on perfect model dynamics have difficulty with providing error estimates. The new method does not have these drawbacks. In fact, it combines advantages from both methods in that it does provide error estimates while automatically having balanced states after analysis, without extra computations. It is based on ensemble or Monte Carlo integrations to simulate the probability density of the model evolution. When observations are available, the so-called importance resampling algorithm is applied. From Bayes's theorem it follows that each ensemble member receives a new weight dependent on its “distance” to the obse...

The Net Advective Effect of a Vertically Sheared Current on a Coherent Vortex

Abstract The nonlinear interaction of a localized vortex with a vertically sheared mean flow has been studied using numerical and asymptotic methods in a multilayer quasigeostrophic model on the beta plane. Numerical solutions indicate that baroclinic large-scale flows have a weak influence on the translation of coherent vortices, even when the advective effect of the mean flow is important. This is opposed to what is observed in general for the dynamics of linear Rossby waves, which are sensitive to the presence of a baroclinic current. Thus, the nonlinear nature of vortices has to be taken into account to explain this reduction of the net advective effect of a vertically sheared current on a coherent vortex. The asymptotic method of Sutyrin and Morel is generalized to describe analytically the development of the beta gyres and corresponding vortex motion in the presence of a vertically sheared current. The initial vortex structure is prescribed as a piecewise constant potential vorticity anomaly in one ...

The finiteness of determining degrees of freedom for the quasi-geostrophic multi-layer ocean model

In this paper, we consider a multi-layer quasi-geostrophic model of the ocean dynamics and we prove that the long-time behaviour of its solutions can be described by a finite number of determining parameters. Under some additional condition we also show that the dynamics of the bottom layer is completely determined by parameters connected with upper layers only. It means that the information about the bottom layer is not essential for a description of the long-time behaviour of the system under consideration.

The Dependence of Rossby Wave Breaking on the Vertical Structure of the Polar Vortex

The three-dimensional structure of wave propagation and breaking on the edge of polar vortices is examined using a multilayer quasigeostrophic model, with piecewise constant potential vorticity (PV) in each layer. The linear propagation of waves up the edge of a vortex is found to be sensitive to vertical variations in the vortex structure, with reduced propagation if the PV or area of the vortex increases with height; this reduction is dramatic for a cylindrical vortex with increasing PV. The characteristics of the nonlinear evolution and wave breaking is examined using high-resolution contour dynamics simulations and is also found to be sensitive to the vertical structure of the vortex. The amplitude of the forcing required for wave breaking to occur is larger for baroclinic vortices (with PV or area increasing with height) than for barotropic vortices. For cylindrical vortices with PV increasing with height the variation of wave breaking with forcing amplitude is qualitatively different from that of a barotropic vortex. Wave breaking occurs in the upper layers for only a limited, intermediate range of forcing amplitudes: there is no wave breaking in upper layers for weak forcing and for large forcing there is only wave breaking at the bottom of the vortex (i.e., the wave breaking is more vertically confined than for a barotropic vortex). For vortices with both PV and area increasing with height there is again a regime with wave breaking in the upper layers for weak amplitude forcing. However, the characteristics of the filaments produced by the wave breaking in upper layers is different from that in the barotropic case, with the filaments rolling up into a series of small vortices.

Nonlinear stability of zonally symmetric quasi-Geostrophic flow

By using the conservation laws and the method of variational principle, an improved Arnol'd's second nonlinear stability theorem for the two-dimensional multilayer quasi-geostrophic model in periodic channel is obtained.

Topography and barotropic transport control by bottom friction

The problem of the stratified general circulation in the presence of topography is revisited. The novel effect examined here is that of localized, but large-scale, topographic anomalies on the wind-driven circulation, a problem whose relevance is found in the occurrence of many such features in the open ocean. Using the classical methods of homogenization theory, it is argued that the barotropic transport near topography can come under the direct control of bottom friction. This result differs substantively from either the well-known Sverdrup constraint (which applies to a flat-bottomed ocean, or to one with a resting deep layer) or its recent extensions that allow for planar bottom topographic profiles. Bottom friction emerges as a controlling parameter roughly in the event that the topography forms closed f/(H - h b ) contours, where H - hb is the total fluid depth, although the theoretical minimum requirements are somewhat looser than this. Our analytical predictions are supported by numerical experimentation with a multi-layer quasi-geostrophic model, and we examine some mean flow observations from the North and South Atlantic in light of the theory. In particular, the theory can rationalize the 100 Sv transport observed recently around the Zapiola Drift.

Variational assimilation of altimeter data into a non-linear ocean model: temporal strategies

In this paper, we explore the use of the adjoint method in the oceanographic context for assimilating satellite altimeter data. Experiments with simulated altimeter data are performed in a multi-layer quasigeostrophic ocean model. The interest is mainly with controlling the mesoscale eddy active ocean circulation observed in the mid-latitudes. Due to the surface nature of the observations, one key aspect in the success of is its ability to transfer the surface data information downwards to the deep flows. Test experiments are performed first, in a coarse resolution regional basin within which several eddies are interacting on the f-plane and second, in a high resolution basin size domain on the s-plane which mimics the Gulfstream-like behavior of mid-latitude jets and western boundary currents and the associated eddy system. As a matter of fact, it is found that the length of the cycle is crucial to the success of this assimilation. Short cycles may be efficient in the control of surface flows but rather ineffective with respect to the downward penetration of information. Conversely, long cycles lead to rather coherent results in terms of efficiency on the vertical but this global efficiency is poor, especially identifying the initial control state. It is suggested that an efficient strategy can be constructed by dividing the global time sequence in several time sub-periods the individual duration of which must be less than the typical predictability time scale of the flow. However, the whole cycle must be long enough and larger than the vertical penetration time scale. A sliced assimilation strategy which satisfies these conditions is studied in which the period is divided into equal-length time-intervals. With this approach, an acceptable accuracy can be reached for the recovery of the final flow state. This meets the requirements of the filtering objectives of data and therefore the forecast purposes. A so-called progressive assimilation strategy is also studied in which successive iterations of increasing durations are performed in order to progressively improve the control of the initial state control variable. This more expensive strategy is more adequate for smoothing objectives.