Dissipative Quasi-geostrophic Dynamics under Random Forcing

Brannan, James R.; Duan, Jinqiao; Wanner, Thomas

doi:10.1006/jmaa.1998.6128

Cited by 24 publications

(28 citation statements)

References 10 publications

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“…In this section we discuss preliminary routine properties of solutions for eq. (0.2); most of them are well known (e.g., see [BDW98] and cf. [KS12]).…”

Section: Effective Equation and Main Resultsmentioning

confidence: 99%

The limit of small Rossby numbers for randomly forced quasi-geostrophic equation onβ-plane

Kuksin

Maiocchi

2015

Nonlinearity

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We consider the 2d quasigeostrophic equation on the β-plane for the stream function ψ, with dissipation and a random force:For typical values of the horizontal period L we prove that the law of the action-vector of a solution for ( * ) (formed by the halves of the squared norms of its complex Fourier coefficients) converges, as β → ∞, to the law of an action-vector for solution of an auxiliary effective equation, and the stationary distribution of the action-vector for solutions of ( * ) converges to that of the effective equation. Moreover, this convergence is uniform in κ ∈ (0, 1]. The effective equation is an infinite system of stochastic equations which splits into invariant subsystems of complex dimension ≤ 3; each of these subsystems is an integrable hamiltonian system, coupled with a Langevin thermostat. Under the iterated limits lim L=ρ→∞ lim β→∞ and lim κ→0 lim β→∞ we get similar systems. In particular, none of the three limiting systems exhibits the energy cascade to high frequencies.

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“…In this section we discuss preliminary routine properties of solutions for eq. (0.2); most of them are well known (e.g., see [BDW98] and cf. [KS12]).…”

Section: Effective Equation and Main Resultsmentioning

confidence: 99%

The limit of small Rossby numbers for randomly forced quasi-geostrophic equation onβ-plane

Kuksin

Maiocchi

2015

Nonlinearity

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“…The above QG equation has structural similarities to equations of Navier-Stokes type. To be able to adapt well known results of such equations, we need to replace these boundary conditions by zero boundary conditions, which is possible with particular types of stationary transformations; see Crauel and Flandoli [5] or Brannan, Duan and Wanner [4], or in a more general context Keller and Schmalfuss [11] or Imkeller and Schmalfuss [10]. In particular, we transform (12) into…”

Section: Transformation Of the Quasigeostrophic Equationmentioning

confidence: 99%

Exponential stability of the quasigeostrophic equation under random perturbations

Duan

Kloeden

Schmalfuß

2001

Stochastic Climate Models

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Progress in Probability 49(2001), 241-256. The 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. 1991 Mathematics Subject Classification. Primary 60H25, 47H10; Secondary 34D35. Key words and phrases. Quasigeostrophic equation, random dynamical systems, random fixed point theorem, stable stationary solutions.

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“…This random jet may also be viewed as satisfying, approximately in the spirit of del-castillo-Negrete and Morrison ( [6]), the randomly wind forced quasigeostrophic model. Several authors have considered the randomly wind forced quasigeostrophic model in order to incorporate the impact of uncertain geophysical forces ( [30], [13], [15], [23], [7], [4]). They studied statistical issues such as estimating correlation coefficients for the linearized quasigeostrophic equation with random forcing.…”

Section: A Quasigeostrophic Jet Modelmentioning

confidence: 99%

Escape probability, mean residence time and geophysical fluid particle dynamics

Brannan

Duan

Ervin

1999

Physica D: Nonlinear Phenomena

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Stochastic dynamical systems arise as models for fluid particle motion in geophysical flows with random velocity fields. Escape probability (from a fluid domain) and mean residence time (in a fluid domain) quantify fluid transport between flow regimes of different characteristic motion.We consider a quasigeostrophic meandering jet model with random perturbations. This jet is parameterized by the parameter β = 2Ω r cos(θ), where Ω is the rotation rate of the earth, r the earth's radius and θ the latitude. Note that Ω and r are fixed, so β is a monotonic decreasing function of the latitude. The unperturbed jet (for 0 < β < 2 3 ) consists of a basic flow with attached eddies. With random perturbations, there is fluid exchange between regimes of different characteristic motion. We quantify the exchange by escape probability and mean residence time.For an eddy, the average escape probability for fluid particles (initially inside the eddy) escape into the exterior retrograde region is smaller than escape into the jet core for 0 < β < 0.3333, while for 0.3333 < β < 2 3 , the opposite holds. For a unit jet core near the jet troughs, the average escape probability for fluid particles (initially inside the jet core) escape into the northern recirculating region is greater than escape into the southern recirculating region for 0 < β < 0.115, while for 0.385 < β < 2 3 , the opposite holds. Moreover, for 0.115 < β < 0.385, fluid particles are about equally likely to escape into either recirculating regions. Furthermore, for a unit jet core near the jet crests, the situation is the opposite as for near the jet troughs.The maximal mean residence time of fluid particles initially in an eddy increases as β increases from 0 to 0.432 (or as latitude decreases accordingly), then decreases as β increases from 0.432 to 2 3 (or as latitude decreases accordingly). However, the maximal mean residence time of fluid particles initially in a unit jet core always increases as β increases (or as latitude decreases).

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Dissipative Quasi-geostrophic Dynamics under Random Forcing

Cited by 24 publications

References 10 publications

The limit of small Rossby numbers for randomly forced quasi-geostrophic equation onβ-plane

The limit of small Rossby numbers for randomly forced quasi-geostrophic equation onβ-plane

Exponential stability of the quasigeostrophic equation under random perturbations

Escape probability, mean residence time and geophysical fluid particle dynamics

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