The quasigeostrophic model is a simplified geophysical fluid model at asymptotically high rotation rate or at small Rossby number. We consider the quasigeostrophic equation with dissipation under random forcing in bounded domains. We show that global unique solutions exist for appropriate initial data. Unlike the deterministic quasigeostrophic equation whose well-posedness is well-known, there seems no rigorous result on global existence and uniqueness of the randomly forced quasigeostrophic equation. Our work provides such a rigorous result on global existence and uniqueness, under very mild conditions.
The quasigeostrophic equation is a prototypical geophysical fluid model. In this paper, we consider time-periodic motions of this model under dissipation and time-dependent wind forcing. We show that when the wind forcing is time-periodic and the spatial square-integral of the wind forcing is bounded in time, the full nonlinear quasigeostrophic model has time-periodic motions, under some conditions on $β$ parameter, Ekman number, viscousity and the domain size.
We consider time-periodic patterns of the dissipative three dimensional baroclinic quasigeostrophic model in spherical coordinates, under time-dependent forcing. We show that when the forcing is time-periodic and the spatial square-integral of the forcing is bounded in time, the model has time-periodic solutions.
he quasigeostrophic model describes large scale and relatively slow fluid motion in geophysical flows. We investigate the quasigeostrophic model under random forcing and random boundary conditions. We first transform the model into a partial differential equation with random coefficients. Then we show that, under suitable conditions on the random forcing, random boundary conditions, viscosity, Ekman constant and Coriolis parameter, all quasigeostrophic motion approach a unique stationary state exponentially fast. This stationary state corresponds to a unique invariant Dirac measure
The three-dimensional baroclinic quasigeostrophic flow model has been widely used to study basic mechanisms in oceanic flows and climate dynamics. In this paper, we consider this flow model under random wind forcing and time-periodic fluctuations on fluid boundary (the interface between the oceans and the atmosphere). The time-periodic fluctuations are due to periodic rotation of the earth and thus periodic exposure of the earth to the solar radiation. After establishing the well-posedness of the baroclinic quasigeostrophic flow model in the state space, we demonstrate the existence of the ran…
We investigate the ergodicity of 2D large scale quasigeostrophic flows under random wind forcing. We show that the quasigeostrophic flows are ergodic under suitable conditions on the random forcing and on the fluid domain, and under no restrictions on viscosity, Ekman constant or Coriolis parameter. When these conditions are satisfied, then for any observable of the quasigeostrophic flows, its time average approximates the statistical ensemble average, as long as the time interval is sufficiently long.
The two-layer quasigeostrophic flow model is an intermidiate system between the single-layer 2D barotropic flow model and the continuously stratified, 3D baroclinic flow model. This model is widely used to investigate basic mechanisms in geophysical flows, such as baroclinic effects, the Gulf Stream and subtropical gyres. The wind forcing acts only on the top layer. We consider the two-layer quasigeostrophic model under stochastic wind forcing. We first transformed this system into a coupled system of random partial differential equations and then show that the asymptotic probabilistic dynamic…
We describe a topological method to study the dynamics of dissipative PDEs on a torus with rapidly oscillating forcing terms. We show that a dissipative PDE, which is invariant with respect to Galilean transformations, with a large average initial velocity can be reduced to a problem with rapidly oscillating forcing terms. We apply the technique to the Burgers equation, and the incompressible 2D Navier-Stokes equations with a time-dependent forcing. We prove that for a large initial average speed the equation admits a bounded eternal solution, which attracts all other solutions forward in time…
We consider a class of ordinary differential equations describing one-dimensional systems with a quasi-periodic forcing term and in the presence of large damping. We discuss the conditions to be assumed on the mechanical force and the forcing term for the existence of quasi-periodic solutions which have the same frequency vector as the forcing.
We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping. As a physical application one can think of a resistor-inductor-varactor circuit with a periodic (or quasi-periodic) forcing function, even if the range of applicability of the theory is much wider. In the limit of large damping we look for quasi-periodic solutions which have the same frequency vector of the forcing term, and we study their analyticity properties in the inverse of the damping coefficient. We find that …
We consider several models of nonlinear wave equations subject to very strong damping and quasi-periodic external forcing. This is a singular perturbation, since the damping is not the highest order term. We study the existence of response solutions (i.e., quasi-periodic solutions with the same frequency as the forcing). Under very general non-resonance conditions on the frequency, we show the existence of asymptotic expansions of the response solution; moreover, we prove that the response solution indeed exists and depends analytically on $\varepsilon$ (where $\varepsilon$ is the inverse of t…
We consider a class of singular ordinary differential equations describing analytic systems of arbitrary finite dimension, subject to a quasi-periodic forcing term and in the presence of dissipation. We study the existence of response solutions, i.e. quasi-periodic solutions with the same frequency vector as the forcing term, in the case of large dissipation. We assume the system to be conservative in the absence dissipation, so that the forcing term is --- up to the sign --- the gradient of a potential energy, and both the mass and damping matrices to be symmetric and positive definite. Furth…
We prove that when subjected to periodic forcing of the form $p_{μ, \rh, \om} (t) = μ(\rh h(x,y) + \sin (\om t))$, certain second order systems of differential equations with dissipative homoclinic loops admit strange attractors with SRB measures for a set of forcing parameters $(μ, \rh, \om)$ of positive measure. Our proof applies the recent theory of rank one maps, developed by Wang and Young based on the analysis of strongly dissipative Hénon maps by Benedicks and Carleson.
We consider a class of ordinary differential equations describing one-dimensional analytic systems with a quasi-periodic forcing term and in the presence of damping. In the limit of large damping, under some generic non-degeneracy condition on the force, there are quasi-periodic solutions which have the same frequency vector as the forcing term. We prove that such solutions are Borel summable at the origin when the frequency vector is either any one-dimensional number or a two-dimensional vector such that the ratio of its components is an irrational number of constant type. In the first case t…
We present a computer assisted method for proving the existence of globally attracting fixed points of dissipative PDEs. An application to the viscous Burgers equation with periodic boundary conditions and a forcing function constant in time is presented as a case study. We establish the existence of a locally attracting fixed point by using rigorous numerics techniques. To prove that the fixed point is, in fact, globally attracting we introduce a technique relying on a construction of an absorbing set, capturing any sufficiently regular initial condition after a finite time. Then the absorbin…
We consider a class of ordinary differential equations describing one-dimensional quasiperiodically forced systems in the presence of large damping. We give a fully constructive proof of the existence of response solutions, that is quasi-periodic solutions which have the same frequency vector as the forcing. This requires dealing with a degenerate implicit function equation: we prove that the latter has a unique solution, which can be explicitly determined. As a by-product we obtain an explicit estimate of the minimal size of the damping coefficient.
This paper develops further and systematically the asymptotic expansion theory that was initiated by Foias and Saut in [11]. We study the long-time dynamics of a large class of dissipative systems of nonlinear ordinary differential equations with time-decaying forcing functions. The nonlinear term can be, but not restricted to, any smooth vector field which, together with its first derivative, vanishes at the origin. The forcing function can be approximated, as time tends to infinity, by a series of functions which are coherent combinations of exponential, power and iterated logarithmic functi…
We consider a class of differential equations, $\ddot x + γ\dot x + g(x) = f(ωt)$, with $ω\in {\bf R}^{d}$, describing one-dimensional dissipative systems subject to a periodic or quasi-periodic (Diophantine) forcing. We study existence and properties of the limit cycle described by the trajectory with the same quasi-periodicity as the forcing. For $g(x)=x^{2p+1}$, $p\in {\bf N}$, we show that, when the dissipation coefficient is large enough, there is only one limit cycle and that it is a global attractor. In the case of other forces, including $g(x)=x^{2p}$ (with $p=1$ describing the varacto…
