Mathematics Colloquia and Seminars

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In turbulence, climate simulation and in all other multiscale fluids problems, a major challenge is ‘scale-up.’ This is the challenge of deriving models for the averaged dynamics that correctly capture the mean, or large scale flow—including the influence on it of the rapid or small scale dynamics. In his 99 Physica D paper, Holm formulates equations for the slow time dynamics of fluid motion that self consistently account for the effects of the variability upon the mean. By transporting the covariance of small-scale fluctuations, the pressure gradient is dispersively regularized in an adaptive fashion depending on the velocity shear through the evolution of a dynamical Helmholtz operator. In this talk, I discuss how this non-linear dispersive mechanism can be captured in simulation of Lagrange averaged shallow water (shallow water alpha) using particle-mesh methods. At each time step, initially mesh-aligned particles move with the mean transport velocity - advecting the covariance of fast-scale fluid fluctuations. Subsequent (mass conservative) remapping of the particles to the mesh and solution of a Galerkin finite element approximation of the dynamical Helmholtz equation gives an Eulerian representation of the non-linear dispersively regularized pressure gradient. I present numerical simulations of 1D shallow water alpha flow in a bounded domain with piece-wise continuous bottom topography. I will close the talk by showing the application of particle-mesh methods to mesoscale weather simulation.