Looking for new problems to solve? Consider the climate

Marston, Brad

doi:10.1103/physics.4.20

Cited by 17 publications

(19 citation statements)

References 40 publications

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“…Imposing rotation to the sphere is not sufficient to recover these transfers and allow full condensation of the energy; however, it is expected that interactions with a bottom topography would do so. Note that the flow topologyfour vortices -resulting from partial energy condensation coincides with stationary states observed in numerical simulations on a spherical domain [29,61]. It is sometimes argued that the presence of a quadrupole instead of a dipole is due to angular momentum conservation.…”

Section: Discussionsupporting

confidence: 69%

Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a Spherical Domain

Herbert

2013

J Stat Phys

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The role of the domain geometry for the statistical mechanics of 2D Euler flows is investigated. It is shown that for a spherical domain, there exists invariant subspaces in phase space which yield additional angular momentum, energy and enstrophy invariants. The microcanonical measure taking into account these invariants is built and a mean-field, Robert-Sommeria-Miller theory is developed in the simple case of the energy-enstrophy measure. The variational problem is solved analytically and a partial energy condensation is obtained. The thermodynamic properties of the system are also discussed.Keywords Incompressible Euler fluid flow on S 2 · Turbulence · Robert-Sommeria-Miller theory · Dynamical Invariants IntroductionThe tools of equilibrium statistical mechanics have been applied to the study of turbulent flows in a variety of ways. From the pioneering theory of point vortices initiated by Onsager ([74], see also [39]) and developed by many others [67,75,59,43,6,17,38,30,52,53] to the mean-field theory of Robert, Sommeria and Miller (RSM) [80,78,65] (see also [66,84,20,21,60,14]) through the spectral approach by Kraichnan [54,55], several theories are available, with their strengths and weaknesses. The models for turbulent flows investigated range from the Euler equations to quasi-geostrophic [82,41,63,34,13,92,49] or shallow-water equations [27,24]. In all these cases, the major feature that statistical mechanics enabled us to better understand is the large-scale organization of the flow. While Onsager's theory gave birth to the early prediction of the existence of negative temperature, and Kraichnan's work yielded the notion of energy condensation for fluid flows, it is arguably the RSM theory which provides the most convenient framework to effectively compute the statistical equilibria of the system. Very often, several statistical equilibria can coexist for a given value of the external parameters, which leads to interesting phase transitions. The long-range nature of the interactions at work in turbulent flows is responsible for new types of phase transitions [10]. Like with many other systems with long-range interactions [33,16], the energy is not additive in turbulent flows, which has the crucial consequence that the different statistical ensembles may not give equivalent results, even in the thermodynamic limit [36]. Ensemble inequivalence has many manifestations at the thermodynamic level, like the existence of negative specific heats, non concave entropies,... [92] For all the above mentioned properties, the domain geometry plays an important role. In the first place, it bears connections with the dynamical invariants of the system, which are the cornerstones

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Section: Discussionsupporting

confidence: 69%

Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a Spherical Domain

Herbert

2013

J Stat Phys

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“…Conceptually, our analysis supports the view that non-equilibrium approaches can provide analytical insights into the dynamics of planetary flows (Delplace et al 2017) and atmospheres (Marston 2011(Marston , 2012. Moreover, in view of the recent successful application of phenomenological GNS models to active fluids (Dunkel et al 2013;S lomka & Dunkel 2017b), the results of this study can also help advance the understanding of active matter propagation on curved surfaces (Sanchez et al 2012;Sknepnek & Henkes 2015;Zhang et al 2016;Henkes et al 2018;Nitschke et al 2019) and in rotating frames (Löwen 2019).…”

Section: Introductionsupporting

confidence: 79%

Linearly forced fluid flow on a rotating sphere

et al. 2020

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Motivated in part by the complex flow patterns observed in planetary atmospheres, we investigate generalized Navier-Stokes (GNS) equations that couple nonlinear advection with a generic linear instability. This analytically tractable minimal model for fluid flows driven by internal active stresses has recently been shown to permit exact solutions on a stationary 2D sphere. Here, we extend the analysis to linearly driven flows on rotating spheres, as relevant to quasi-2D atmospheres. We derive exact solutions of the GNS equations corresponding to time-independent zonal jets and superposed westwardpropagating Rossby waves. Direct numerical simulations with large rotation rates obtain statistically stationary states close to these exact solutions. The measured phase speeds of waves in the GNS simulations agree with analytical predictions for Rossby waves.

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“…Farrell and Ioannou have also used this model in a deterministic context [22], where the correlation C m is then added phenomenologically in order to model the effect of small scale turbulence (the effect of the neglected non linear terms in the quasi-linear dynamics). This model, or an analogous one in a deterministic context, is also referred to as CE2 (Cumulant expansion of order 2) [52,51,50].…”

Section: The Quasi-linear Dynamicsmentioning

confidence: 99%

“…In order to prove the result, we rely on two main ingredients. First we use the fact that the stationary solution of the Lyapunov equation can be computed from solutions of the deterministic linearized dynamics, as expressed by formulas (50) and (51). Second we prove that these formulas have limits for large times based on results on the asymptotic behavior of the linearized 2D Euler equations, discussed in the work [7].…”

Section: Stationary Solutions To the Lyapunov Equation In The Inertiamentioning

confidence: 99%

“…It is also related to a quasi-linear approximation plus an hypothesis where zonal average and ensemble average are assumed to be the same, as discussed in details in section 2.4. More recently, an interpretation in terms of a second order closure (CE2) has also been given [51,50,74]. All these different forms of quasi-linear approximations have thoroughly been studied numerically, sometimes with stochastic forces and sometimes with deterministic ones [22].…”

Section: Introductionmentioning

confidence: 99%

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Kinetic Theory of Jet Dynamics in the Stochastic Barotropic and 2D Navier-Stokes Equations

2013

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We discuss the dynamics of zonal (or unidirectional) jets for barotropic flows forced by Gaussian stochastic fields with white in time correlation functions. This problem contains the stochastic dynamics of 2D Navier-Stokes equation as a special case. We consider the limit of weak forces and dissipation, when there is a time scale separation between the inertial time scale (fast) and the spin-up or spin-down time (large) needed to reach an average energy balance. In this limit, we show that an adiabatic reduction (or stochastic averaging) of the dynamics can be performed. We then obtain a kinetic equation that describes the slow evolution of zonal jets over a very long time scale, where the effect of non-zonal turbulence has been integrated out. The main theoretical difficulty, achieved in this work, is to analyze the stationary distribution of a Lyapunov equation that describes quasi-Gaussian fluctuations around each zonal jet, in the inertial limit. This is necessary to prove that there is no ultraviolet divergence at leading order, in such a way that the asymptotic expansion is self-consistent. We obtain at leading order a Fokker-Planck equation, associated to a stochastic kinetic equation, that describes the slow jet dynamics. Its deterministic part is related to well known phenomenological theories (for instance Stochastic Structural Stability Theory) and to quasi-linear approximations, whereas the stochastic part allows to go beyond the computation of the most probable zonal jet. We argue that the effect of the stochastic part may be of huge importance when, as for instance in the proximity of phase transitions, more than one attractor of the dynamics is present.

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Looking for new problems to solve? Consider the climate

Abstract: Even though global warming remains a heated political topic, physicists should not ignore the intellectual challenge of trying to model climate change.

Cited by 17 publications

References 40 publications

Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a Spherical Domain

Additional Invariants and Statistical Equilibria for the 2D Euler Equations on a Spherical Domain

Linearly forced fluid flow on a rotating sphere

Kinetic Theory of Jet Dynamics in the Stochastic Barotropic and 2D Navier-Stokes Equations

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