Double-gyre Model Research Articles

Ocean Plastic Assimilator v0.2: assimilation of plastic concentration data into Lagrangian dispersion models

Abstract. A numerical scheme to perform data assimilation of concentration measurements in Lagrangian models is presented, along with its first implementation called Ocean Plastic Assimilator, which aims to improve predictions of the distributions of plastics over the oceans. This scheme uses an ensemble method over a set of particle dispersion simulations. At each step, concentration observations are assimilated across the ensemble members by switching back and forth between Eulerian and Lagrangian representations. We design two experiments to assess the scheme efficacy and efficiency when assimilating simulated data in a simple double-gyre model. Analysis convergence is observed with higher accuracy when lowering observation variance or using a circulation model closer to the real circulation. Results show that the distribution of the mass of plastics in an area can effectively be improved with this simple assimilation scheme. Direct application to a real ocean dispersion model of the Great Pacific Garbage Patch is presented with simulated observations, which gives similarly encouraging results. Thus, this method is considered a suitable candidate for creating a tool to assimilate plastic concentration observations in real-world applications to estimate and forecast plastic distributions in the oceans. Finally, several improvements that could further enhance the method efficiency are identified.

Data-driven discovery of Koopman eigenfunctions for control

Data-driven transformations that reformulate nonlinear systems in a linear framework have the potential to enable the prediction, estimation, and control of strongly nonlinear dynamics using linear systems theory. The Koopman operator has emerged as a principled linear embedding of nonlinear dynamics, and its eigenfunctions establish intrinsic coordinates along which the dynamics behave linearly. Previous studies have used finite-dimensional approximations of the Koopman operator for model-predictive control approaches. In this work, we illustrate a fundamental closure issue of this approach and argue that it is beneficial to first validate eigenfunctions and then construct reduced-order models in these validated eigenfunctions. These coordinates form a Koopman-invariant subspace by design and, thus, have improved predictive power. We show then how the control can be formulated directly in these intrinsic coordinates and discuss potential benefits and caveats of this perspective. The resulting control architecture is termed Koopman Reduced Order Nonlinear Identification and Control (KRONIC). It is further demonstrated that these eigenfunctions can be approximated with data-driven regression and power series expansions, based on the partial differential equation governing the infinitesimal generator of the Koopman operator. Validating discovered eigenfunctions is crucial and we show that lightly damped eigenfunctions may be faithfully extracted from EDMD or an implicit formulation. These lightly damped eigenfunctions are particularly relevant for control, as they correspond to nearly conserved quantities that are associated with persistent dynamics, such as the Hamiltonian. KRONIC is then demonstrated on a number of relevant examples, including (a) a nonlinear system with a known linear embedding, (b) a variety of Hamiltonian systems, and (c) a high-dimensional double-gyre model for ocean mixing.

Comparing Finite-Time Lyapunov Exponents and Lagrangian Descriptors for identifying phase space structures in a simple two-dimensional, time-periodic double-gyre model

This paper compares the advantages, limitations, and computational considerations of using Finite-Time Lyapunov Exponents (FTLEs) and Descriptors (LDs) as tools for identifying barriers and mechanisms of fluid transport in two-dimensional time-periodic flows. These barriers and mechanisms of transport are often referred to as Lagrangian Coherent Structures, though this term often changes meaning depending on the author or context. This paper will specifically focus on using FTLEs and LDs to identify stable and unstable manifolds of hyperbolic stagnation points, and the Kolmogorov-Arnold-Moser (KAM) tori associated with elliptic stagnation points. The background and theory behind both methods and their associated phase space structures will be presented, and then examples of FTLEs and LDs will be shown based on a simple, periodic, time-dependent double-gyre toy model with varying parameters.

Observability of laminar bidimensional fluid flows seen as autonomous chaotic systems.

Lagrangian transport in the dynamical systems approach has so far been investigated disregarding the connection between the whole state space and the concept of observability. Key issues such as the definitions of Lagrangian and chaotic mixing are revisited under this light, establishing the importance of rewriting nonautonomous flow systems derived from a stream function in autonomous form, and of not restricting the characterization of their dynamics in subspaces. The observability of Lagrangian chaos from a reduced set of measurements is illustrated with two canonical examples: the Lorenz system derived as a low-dimensional truncation of the Rayleigh-Bénard convection equations and the driven double-gyre system introduced as a kinematic model of configurations observed in the ocean. A symmetrized version of the driven double-gyre model is proposed.

A Conservative Formulation and a Numerical Algorithm for the Double-Gyre Nonlinear Shallow-Water Model

AbstractWe present a conservative formulation and a numerical algorithm for the reduced-gravity shallow-water equations on a beta plane, subjected to a constant wind forcing that leads to the formation of double-gyre circulation in a closed ocean basin. The novelty of the paper is that we reformulate the governing equations into a nonlinear hyperbolic conservation law plus source terms. A second-order fractional-step algorithm is used to solve the reformulated equations. In the first step of the fractional-step algorithm, we solve the homogeneous hyperbolic shallow-water equations by the wave-propagation finite volume method. The resulting intermediate solution is then used as the initial condition for the initial-boundary value problem in the second step. As a result, the proposed method is not sensitive to the choice of viscosity and gives high-resolution results for coarse grids, as long as the Rossby deformation radius is resolved. We discuss the boundary conditions in each step, when no-slip boundary conditions are imposed to the problem. We validate the algorithm by a periodic flow on an f-plane with exact solutions. The order-of-accuracy for the proposed algorithm is tested numerically. We illustrate a quasi-steady-state solution of the double-gyre model via the height anomaly and the contour of stream function for the formation of double-gyre circulation in a closed basin. Our calculations are highly consistent with the results reported in the literature. Finally, we present an application, in which the double-gyre model is coupled with the advection equation for modeling transport of a pollutant in a closed ocean basin.

Dynamically consistent parameterization of mesoscale eddies. Part I: Simple model

This work aims at developing a framework for dynamically consistent parameterization of mesoscale eddy effects for use in non-eddy-resolving ocean circulation models. The proposed eddy parameterization framework is successfully tested on the classical, wind-driven double-gyre model, which is solved both with explicitly resolved vigorous eddy field and in the non-eddy-resolving configuration with the eddy parameterization replacing the eddy effects. The parameterization locally approximates transient eddy flux divergence by spatially localized and temporally periodic forcing, referred to as the plunger, and focuses on the linear-dynamics flow solution induced by it. The nonlinear self-interaction of this solution, referred to as the footprint, characterizes and quantifies the induced cumulative eddy forcing exerted on the large-scale flow. We find that spatial pattern and amplitude of the footprint strongly depend on the underlying large-scale and the corresponding relationships provide the basis for the eddy parameterization and its closure on the large-scale flow properties. Dependencies of the footprints on other important parameters of the problem are also systematically analyzed. The parameterization utilizes the local large-scale flow information, constructs and scales the corresponding footprints, and then sums them up over the gyres to produce the resulting eddy forcing field, which is interactively added to the model as an extra forcing. The parameterization framework is implemented in the simplest way, but it provides a systematic strategy for improving the implementation algorithm.

Path-integrated Lagrangian measures from the velocity gradient tensor

Abstract. Spatial maps of the finite-time Lyapunov exponent (FTLE) have been used extensively to study LCS in two-dimensional dynamical systems, in particular with application to transport in unsteady fluid flows. We use the time-periodic double-gyre model to compare spatial fields of FTLE and the path-integrated Eulerian Okubo–Weiss parameter (OW). Both fields correlate strongly, and by solving the dynamics of the deformation gradient tensor, a theoretical relationship between both magnitudes has been obtained. While for long integration times more and more FTLE ridges appear that do not seem to coincide with the stable manifold, ridges in the field of path-integrated OW represent fewer additional structures.

When complexity leads to simplicity: Ocean surface mixing simplified by vertical convection

The effect of weak vertical motion on the dynamics of materials that are limited to move on the ocean surface is an unresolved problem with important environmental and ecological implications (e.g., oil spills and larvae dispersion). We investigate this effect by introducing into the classical horizontal time-periodic double-gyre model vertical motion associated with diurnal convection. The classical model produces chaotic advection on the surface. In contrast, the weak vertical motion simplifies this chaotic surface mixing pattern for a wide range of parameters. Melnikov analysis is employed to demonstrate that these conclusions are general and may be applicable to realistic cases. This counter intuitive result that the very weak nocturnal convection simplifies ocean surface mixing has significant outcomes.

Coherence Resonance in a Double-Gyre Model of the Kuroshio Extension

Abstract The effect of stochastic winds on the intrinsic low-frequency variability of the Kuroshio Extension (KE) is analyzed through a double-gyre (DG) model forced by a steady climatological wind plus an idealized Ornstein–Uhlenbeck wind noise. A DG model of the KE bimodality, whose results compare well to altimeter data, is first shown to be an excitable system. In fact, the relaxation oscillation (forced by steady winds) with decadal time scale that describes the bimodality is recognized to be an internal mode of the system, which can be excited also in a dissipative parameter range (PR) in which it does not arise spontaneously, provided appropriate initial conditions are chosen. It is then shown that, if the additive wind noise is included in the forcing, the actual excitation of the relaxation oscillation in PR occurs if the noise is red with a decorrelation time greater than a minimum time scale ranging from 1 month to 1 year, depending on the dissipation. This behavior, known as “coherence resonance,” is likely to be paradigmatic of the low-frequency variability of western boundary current extensions of intrinsic origin, when it is in the form of relaxation oscillations resulting from a homoclinic bifurcation. General considerations concerning the interpretation of model results obtained within different parameter ranges are applied to this study.

On the Crucial Role of Basin Geometry in Double-Gyre Models of the Kuroshio Extension

Abstract The decadal chaotic relaxation oscillation obtained in a recent double-gyre model study of the Kuroshio Extension intrinsic low-frequency variability was found to compare surprisingly well with the real variability of the jet as revealed by altimeter data, despite the high degree of idealization of the model. In this note it is shown that elements of realism in the basin geometry, present in that study and absent in previous double-gyre models applied to the Kuroshio Extension, play a crucial role in shaping the low-frequency variability of the jet, and can explain the good performance of the model. A series of numerical experiments with different basin geometries of increasing degrees of simplicity are analyzed. If the schematic western boundary representing the coastline south of Japan is removed, the strong decadal variability completely disappears and only a very weak periodic oscillation about an elongated state of the jet is found. If the large zonal width of the basin (representing correctly the extension of the North Pacific Ocean) is reduced by a half, then the total meridional Sverdrup transport is reduced by the same factor, and so is the intensity of the Kuroshio and Oyashio western boundary currents: as a result, the modeled Kuroshio Extension is totally unrealistic in shape and is steady. If both simplifications are introduced the resulting jet is, again, totally unrealistic, yielding a weak periodic bimodal cycle. On the basis of these results, two main conclusions are drawn: (i) the introduction of appropriate geometrical elements of realism in double-gyre model studies of the Kuroshio Extension is essential, and (ii) the Kuroshio Extension intrinsic low-frequency variability would be dramatically different if the southwestern coastline of Japan were more meridionally oriented.

Regime Transitions in a Stochastically Forced Double-Gyre Model

A reduced-gravity double-gyre ocean model is used to study the influence of an additive stochastic wind stress component on the regime behavior of the wind-driven circulation. The variance of the stochastic component (spatially coherent white noise) representing the effect of atmospheric transient eddies is chosen to be spatially inhomogeneous. This is done to account for the observed concentration of eddy activity along the North Atlantic and North Pacific storm tracks. As a result the double-gyre model with a spatially inhomogeneous stochastic forcing shows a bimodal behavior. One regime shows a quasi-antisymmetric; the second regime a nonsymmetric flow pattern. It is suggested that the nonsymmetric regime corresponds to one member of a known nonsymmetric pair of stationary solutions. Actually no stationary solutions are explicitly calculated in this study. The bimodality does not appear without a spatially inhomogeneous stochastic forcing nor with spatially homogeneous stochastic forcing. Therefore, the regime transitions are induced by the inhomogeneity of the white noise variance. The study suggests that the stochastic forcing enables the system to reach the neighborhood of an unstable fixed point that is not reached without the spatially inhomogeneous stochastic wind field. The unstable fixed point then acts to steer the model in a temporarily persistent regime.

Phenomenology of the Low-Frequency Variability in a Reduced-Gravity, Quasigeostrophic Double-Gyre Model

The low-frequency variability of the oceanic wind-driven circulation is investigated by use of a reduced-gravity, quasigeostrophic model with slight variations on the classic double-gyre wind forcing. Approximately 30 eddy-resolving simulations of 100–1000 years duration are analyzed to determine the types of low-frequency variability and to estimate statistical uncertainties in the results. For parameters close to those leading to a stable antisymmetric solution, the system appears to have several preferred phenomenological regimes, each with distinct total energy levels. These states include a high-energy quasi-stable state; a low-energy, weakly penetrating state; and a state of intermediate energy and modest eddy/ring generation. The low-frequency variability of the model is strongly linked to the irregular transitions between these dynamical regimes. For a central set of reference parameters, the behavior of the system is investigated for each period in which the total energy remains in certain ranges. The structure of the time-averaged streamfunction and eddy energy fields are observed to have remarkable repeatability from event to event for each state. A parameter study documents the ways in which the probability distribution function of the total energy depends on the strength and asymmetry of the wind forcing field. As the parameters shift away from those leading to a steady antisymmetric solution, we find that increasing the asymmetry of the wind field or reducing the viscosity decreases the occurrences of the high-energy, quasi-stable state. The low-energy, weakly penetrating state is more robust and exists whenever there is both instability and a certain minimal asymmetry in the forcing. As the wind asymmetry is increased, the distributions shift smoothly (but rapidly) away from the higher-energy states, until only the low-energy state remains.

The Statistics and Sensitivity of a Double-Gyre Model: The Reduced-Gravity, Quasigeostrophic Case

Abstract The sensitivity of the time-averaged circulation of an oceanic double-gyre model to variations in the model's parameters and forcing is studied. Unresolved low-frequency variability in the solution leads to statistical uncertainty in the estimates of the time-averaged quantities. The authors utilize bootstrap analyses of a number of multicentury integrations of a reduced-gravity, quasigeostrophic, eddy-resolving ocean model to estimate these statistical uncertainties. An analysis is then presented of the sensitivity of the system to variations in the strength and asymmetry of the wind forcing and to variations in several other physical and numerical parameters of the system. The bootstrap results enable us to estimate error bounds on these sensitivities The physical measures of the system investigated include the means and variances of the total energy, the peak transport, the location of the separation point, and the penetration scale of the free jet.