Quantifying long time statistical properties of a few fluid models

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WangDMS-1008852 The principal investigator and colleagues study the issue ofquantifying the long-time statistical properties of a fewprototype fluid systems via long-time statistical properties ofsuitable discrete dynamical systems related to temporal and/orspatial approximations. The physical problems considered are theRayleigh-Benard convection at large Prandtl number and/or smallEkman number regime, and a few related simplified models. Inparticular, the methodology developed is applied to numericallyquantify an important physical long-time statistical quantity,the averaged heat transport, in a few convection models. The keyissue here is the design, analysis and implementation of schemesthat are efficient and convergent (in the sense that thestationary statistical properties of the discrete system convergeto those of the underlying system). Approximating long-timebehavior of large complex systems is a well-known challengebecause small errors could accumulate and amplify. Additionaldifficulties related to multiple scales (induced by large Prandtlnumber, small Ekman number, large Rayleigh number), andgeneralised dynamical system (such as the 3D Boussinesq system)are also addressed. Suitable random perturbations of the fluidsystems are considered in order to ensure convergence to thephysically relevant long-time behaviour. Quantifying long-time statistical properties is of greatimportance in applications. Besides well-known applications inclassical turbulence theory, it is also extremely important inclimate studies because the predicted climate is the long-timestatistical behaviour of the underlying climate model. Themodels to be investigated, although far from practical climatemodels, share several important mechanisms that are crucial torealistic climate models, such as energy-preserving nonlinearadvection, rotation, convection, dissipation/damping and forcing. A clearer understanding of long-time statistical behavior in thissetting helps us better understand many geophysical fluidphenomena, and provides guidelines for accurate numerical studyof climate changes. The project also provides abundantopportunities for graduate students, including student fromunderrepresented group, to participate in the modeling, analysis,and computation of many physically motivated problems.