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Atmospheric dynamics span a range of time-scales. The projection of measured data to a slow manifold, ${cal M}$, removes fast gravity waves from the initial state for numerical simulations of the atmosphere. We explore further the slow manifold for a simple atmospheric model introduced by Lorenz and anticipate that our results will relevant to the vastly more detailed dynamics of atmospheres and oceans. Within the dynamics of the Lorenz model, we make clear the relation between a slow manifold $cal M$ and the ``slowest invariant manifold ({SIM}), which was constructed by Lorenz in order to avoid the divergence of approximation schemes for $cal M$. These manifolds are shown to be identical to within exponentially small terms, and so the {SIM} in fact shares the asymptotic nature of $cal M$. We also investigate the issue of balancing initial data in order to remove gravity waves. This is a question of how to compute an ``initialized point on $cal M$ whose subsequent evolution matches that from the measured initial data that in general lie off $cal M$. We propose a choice based on the intuitive idea that the initialization procedure should not significantly alter the forecast. Numerical results demonstrate the utility of our initialization scheme. The normal form for Lorenz atmospheric model shows clearly how to separate the dynamics of the different atmospheric waves. However, its construction demonstrates that {em any} initialization procedure must eventually alter the forecast--the time-scale of the divergence between the initialized and the uninitialized solutions is inevitable and is inversely proportional to the square of initial level of gravity-wave activity.
The behaviour of turbulent flows within the single-layer quasi-geostrophic (Charney--Hasegawa--Mima) model is shown to be strongly dependent on the Rossby deformation wavenumber $lambda$ (or free-surface elasticity). Herein, we derive a bound on the
We present a simple technique for the computation of coarse-scale steady states of dynamical systems with time scale separation in the form of a wrapper around a fine-scale simulator. We discuss how this approach alleviates certain problems encounter
Consider the surface quasi-geostrophic equation with random diffusion, white in time. We show global existence and uniqueness in high probability for the associated Cauchy problem satisfying a Gevrey type bound. This article is inspired by recent work of Glatt-Holtz and Vicol.
This work involves theoretical and numerical analysis of the Thermal Quasi-Geostrophic (TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, the TQG model involves thermal geostrophic balance, in which the Rossby number, the Froud
We establish a regularity criterion for weak solutions of the dissipative quasi-geostrophic equations in mixed time-space Besov spaces.