Ana M. Mancho

Computation of stable and unstable manifolds of hyperbolic trajectories in two-dimensional, aperiodically time-dependent vector fields

In this paper, we develop two accurate and fast algorithms for the computation of the stable and unstable manifolds of hyperbolic trajectories of two-dimensional, aperiodically time-dependent vector fields. First we develop a benchmark method in which all the trajectories composing the manifold are integrated from the neighborhood of the hyperbolic trajectory. This choice, although very accurate, is not fast and has limited usage. A faster and more powerful algorithm requires the insertion of new points in the manifold as it evolves in time. Its numerical implementation requires a criterion for determining when to insert those points in the manifold, and an interpolation method for determining where to insert them. We compare four different point insertion criteria and four different interpolation methods. We discuss the computational requirements of all of these methods. We find two of the four point insertion criteria to be accurate and robust. One is a variant of a criterion originally proposed by Hobson. The other is a slight variant of a method due to Dritschel and Ambaum arising from their studies of contour dynamics. The preferred interpolation method is also due to Dritschel. These methods are then applied to the computation of the stable and unstable manifolds of the hyperbolic trajectories of several aperiodically time-dependent variants of the Duffing equation.

Hidden Geometry of Ocean Flows

We introduce a new global Lagrangian descriptor that is applied to flows with general time dependence (altimetric data sets). It succeeds in detecting simultaneously, with great accuracy, invariant manifolds, hyperbolic and nonhyperbolic flow regions.

Distinguished trajectories in time dependent vector fields

We introduce a new definition of distinguished trajectory that generalizes the concepts of fixed point and periodic orbit to aperiodic dynamical systems. This new definition is valid for identifying distinguished trajectories with hyperbolic and nonhyperbolic types of stability. The definition is implemented numerically and the procedure consists of determining a path of limit coordinates. It has been successfully applied to known examples of distinguished trajectories. In the context of highly aperiodic realistic flows our definition characterizes distinguished trajectories in finite time intervals, and states that outside these intervals trajectories are no longer distinguished.

Lagrangian descriptors: A method for revealing phase space structures of general time dependent dynamical systems

In this paper we develop new techniques for revealing geometrical structures in phase space that are valid for aperiodically time dependent dynamical systems, which we refer to as Lagrangian descriptors. These quantities are based on the integration, for a finite time, along trajectories of an intrinsic bounded, positive geometrical and/or physical property of the trajectory itself. We discuss a general methodology for constructing Lagrangian descriptors, and we discuss a “heuristic argument” that explains why this method is successful for revealing geometrical structures in the phase space of a dynamical system. We support this argument by explicit calculations on a benchmark problem having a hyperbolic fixed point with stable and unstable manifolds that are known analytically. Several other benchmark examples are considered that allow us the assess the performance of Lagrangian descriptors in revealing invariant tori and regions of shear. Throughout the paper “side-by-side” comparisons of the performance of Lagrangian descriptors with both finite time Lyapunov exponents (FTLEs) and finite time averages of certain components of the vector field (“time averages”) are carried out and discussed. In all cases Lagrangian descriptors are shown to be both more accurate and computationally efficient than these methods. We also perform computations for an explicitly three dimensional, aperiodically time-dependent vector field and an aperiodically time dependent vector field defined as a data set. Comparisons with FTLEs and time averages for these examples are also carried out, with similar conclusions as for the benchmark examples.

Lagrangian transport through an ocean front in the North-Western Mediterranean Sea

Abstract With the tools of lobe dynamics, the authors analyze the structures present in the velocity field obtained from a numerical simulation of the surface circulation in the northwestern Mediterranean Sea. In particular, focus is placed on the North Balearic Front, the westernmost part of the transition zone between saltier and fresher waters in the western Mediterranean, which is here interpreted in terms of the presence of a semipermanent “Lagrangian barrier,” across which little transport occurs. Identified are relevant hyperbolic trajectories and their manifolds, and it is shown that the transport mechanism known as the turnstile, previously identified in abstract dynamical systems and simplified model flows, is also at work in this complex and realistic ocean flow. In addition, nonlinear dynamics techniques are shown to be powerful enough to identify the key geometric structures in this part of the Mediterranean. The construction also reveals the spatiotemporal routes along which this transport h...

Thermal convection in a cylindrical annulus heated laterally

In this paper we study thermoconvective instabilities appearing in a fluid within a cylindrical annulus heated laterally. As soon as a horizontal temperature gradient is applied a convective state appears. As the temperature gradient reaches a critical value a stationary or oscillatory bifurcation takes place. The problem is modelled with a novel method which extends the one described in Herrero and Mancho (Herrero H and Mancho A M 2001 Int. J. Numer. Methods Fluids Preprint math.AP/0109200). The Navier–Stokes equation is solved in the primitive variable formulation, with appropriate boundary conditions for pressure. This is a low-order formulation which in cylindrical coordinates introduces lower order singularities. The problem is discretized with a Chebyshev collocation method easily implemented and its convergence has been checked. The results obtained are not only in very good agreement with those obtained in experiments, but also provide a deeper insight into important physical parameters developing the instability, which has not been reported before.

Review Article: "The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current"

Abstract. This article reviews several recently developed Lagrangian tools and shows how their combined use succeeds in obtaining a detailed description of purely advective transport events in general aperiodic flows. In particular, because of the climate impact of ocean transport processes, we illustrate a 2-D application on altimeter data sets over the area of the Kuroshio Current, although the proposed techniques are general and applicable to arbitrary time dependent aperiodic flows. The first challenge for describing transport in aperiodical time dependent flows is obtaining a representation of the phase portrait where the most relevant dynamical features may be identified. areas that are related to confinement regions. This representation is accomplished by using global Lagrangian descriptors that when applied for instance to the altimeter data sets retrieve over the ocean surface a phase portrait where the geometry of interconnected dynamical systems is visible. The phase portrait picture is essential because it evinces which transport routes are acting on the whole flow. Once these routes are roughly recognised, it is possible to complete a detailed description by the direct computation of the finite time stable and unstable manifolds of special hyperbolic trajectories that act as organising centres of the flow.

Routes of Transport across the Antarctic Polar Vortex in the Southern Spring

AbstractTransport in the lower stratosphere over Antarctica has been studied in the past by means of several approaches, such as contour dynamics or Lyapunov exponents. This paper examines the problem by means of a new Lagrangian descriptor, which is referred to as the function M. The focus is on the southern spring of 2005, which allows for a comparison with previous analyses based on Lyapunov exponents. With the methodology based on the function M, a much sharper depiction of key Lagrangian features is achieved and routes of large-scale horizontal transport across the vortex edge are captured. These results highlight the importance of lobe dynamics as a transport mechanism across the Antarctic polar vortex.

Instabilities in buoyant flows under localized heating

We study, from the numerical point of view, instabilities developed in a fluid layer with a free surface in a cylindrical container which is nonhomogeneously heated from below. In particular, we consider the case in which the applied heat is localized around the origin. An axisymmetric basic state appears as soon as a nonzero horizontal temperature gradient is imposed. The basic state may bifurcate to different solutions depending on vertical and lateral temperature gradients and on the shape of the heating function. We find different kinds of instabilities: extended patterns growing on the whole domain, which include those known as targets, and spiral waves. Spirals are present even for infinite Prandtl number. Localized structures both at the origin and at the outer part of the cylinder may appear either as Hopf or stationary bifurcations. An overview of the developed instabilities as functions of the dimensionless parameters is presented in this article.

Bénard–Marangoni convection in a differentially heated cylindrical cavity

The work described in this paper concerns the study of a Benard–Marangoni convection problem in a differentially heated cylindrical cavity. The study had two main aims; first to justify from a numerical point of view the transitions that have been reported in several experiments as the aspect ratio is varied and, second, to study both theoretically and experimentally the role of vertical and horizontal temperature differences in lateral heating convection. Initially, we analyzed the role of the aspect ratio in layers where a dynamic flow is imposed through a nonzero temperature gradient at the bottom. The basic solutions are linear or return flows depending on different parameters. Depending on the vertical temperature difference and other heat-related parameters, the problem bifurcates either to stationary or oscillatory structures. Competing solutions at codimension two bifurcation points were found: stationary radial rolls with different wavenumbers and radial rolls together with hydrothermal waves. Fo...