33 research outputs found
The Lagrangian description of aperiodic flows: a case study of the Kuroshio Current
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 2D
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. 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.Comment: 40 pages, 24 figure
Symmetry and plate-like convection in fluids with temperature-dependent viscosity
We explore the instabilities developed in a fluid in which viscosity depends
on temperature. In particular, we consider a dependency that models a very
viscous (and thus rather rigid) lithosphere over a convecting mantle. To this
end, we study a 2D convection problem in which viscosity depends on temperature
by abruptly changing its value by a factor of 400 within a narrow temperature
gap. We conduct a study which combines bifurcation analysis and time-dependent
simulations. Solutions such as limit cycles are found that are fundamentally
related to the presence of symmetry. Spontaneous plate-like behaviors that
rapidly evolve towards a stagnant lid regime emerge sporadically through abrupt
bursts during these cycles. The plate-like evolution alternates motions towards
either the right or the left, thereby introducing temporary asymmetries on the
convecting styles. Further time-dependent regimes with stagnant and plate-like
lids are found and described.Comment: 19 pages, 8 figures. arXiv admin note: text overlap with
arXiv:1302.073
Bifurcations and dynamics in convection with temperature-dependent viscosity in the presence of the O(2) symmetry
We focus the study of a convection problem in a 2D setup in the presence of
the O(2) symmetry. The viscosity in the fluid depends on the temperature as it
changes its value abruptly in an interval around a temperature of transition.
The influence of the viscosity law on the morphology of the plumes is examined
for several parameter settings, and a variety of shapes ranging from spout to
mushroom shaped is found. We explore the impact of the symmetry on the time
evolution of this type of fluid, and find solutions which are greatly
influenced by its presence: at a large aspect ratio and high Rayleigh numbers,
traveling waves, heteroclinic connections and chaotic regimes are found. These
solutions, which are due to the symmetry presence, have not been previously
described in the context of temperature dependent viscosities. However,
similarities are found with solutions described in other contexts such as flame
propagation problems or convection problems with constant viscosity also under
the presence of the O(2) symmetry, thus confirming the determining role of the
symmetry in the dynamics.Comment: 21 pages, 10 figure
The hidden geometry of ocean flows
We introduce a new global Lagrangian descriptor that is applied to flows with
general time dependence (altimetric datasets). It succeeds in detecting
simultaneously, with great accuracy, invariant manifolds, hyperbolic and
non-hyperbolic flow regions.Comment: 4 pages, 4 figure
The turnstile mechanism across the Kuroshio current: analysis of dynamics in altimeter velocity fields
In this article we explore the ability of dynamical systems tools to describe
transport in oceanic flows characterized by data sets measured from satellite.
In particular we have studied the geometrical skeleton describing transport in
the Kuroshio region. For this purpose we have computed special hyperbolic
trajectories, recognized as distinguished hyperbolic trajectories, that act as
organizing centres of the flow. We have computed their stable and unstable
manifolds, and they reveal that the turnstile mechanism is at work during
several spring months in the year 2003 across the Kuroshio current. We have
found that near the hyperbolic trajectories takes place a filamentous transport
front-cross the current that mixes waters at both sides.Comment: Nonlinear Processes in Geophysics 17, 2010 (in press
Lagrangian Descriptors for Stochastic Differential Equations: A Tool for Revealing the Phase Portrait of Stochastic Dynamical Systems
In this paper we introduce a new technique for depicting the phase portrait
of stochastic differential equations. Following previous work for deterministic
systems, we represent the phase space by means of a generalization of the
method of Lagrangian descriptors to stochastic differential equations.
Analogously to the deterministic differential equations setting, the Lagrangian
descriptors graphically provide the distinguished trajectories and hyperbolic
structures arising within the stochastic dynamics, such as random fixed points
and their stable and unstable manifolds. We analyze the sense in which
structures form barriers to transport in stochastic systems. We apply the
method to several benchmark examples where the deterministic phase space
structures are well-understood. In particular, we apply our method to the noisy
saddle, the stochastically forced Duffing equation, and the stochastic double
gyre model that is a benchmark for analyzing fluid transport
A Theoretical Framework for Lagrangian Descriptors
This paper provides a theoretical background for Lagrangian Descriptors
(LDs). The goal of achieving rigourous proofs that justify the ability of LDs
to detect invariant manifolds is simplified by introducing an alternative
definition for LDs. The definition is stated for -dimensional systems with
general time dependence, however we rigorously prove that this method reveals
the stable and unstable manifolds of hyperbolic points in four particular 2D
cases: a hyperbolic saddle point for linear autonomous systems, a hyperbolic
saddle point for nonlinear autonomous systems, a hyperbolic saddle point for
linear nonautonomous systems and a hyperbolic saddle point for nonlinear
nonautonomous systems. We also discuss further rigorous results which show the
ability of LDs to highlight additional invariants sets, such as -tori. These
results are just a simple extension of the ergodic partition theory which we
illustrate by applying this methodology to well-known examples, such as the
planar field of the harmonic oscillator and the 3D ABC flow. Finally, we
provide a thorough discussion on the requirement of the objectivity
(frame-invariance) property for tools designed to reveal phase space structures
and their implications for Lagrangian descriptors