Stability of the Slow Manifold in the Primitive Equations (original) (raw)
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Journal of Fluid Mechanics, 2003
The problem of nonlinear adjustment of localized front-like perturbations to a state of geostrophic equilibrium (balanced state) is studied in the framework of rotating shallow-water equations with no dependence on the along-front coordinate. We work in Lagrangian coordinates, which turns out to be conceptually and technically advantageous. First, a perturbation approach in the cross-front Rossby number is developed and splitting of the motion into slow and fast components is demonstrated for non-negative potential vorticities. We then give a non-perturbative proof of existence and uniqueness of the adjusted state, again for configurations with nonnegative initial potential vorticities. We prove that wave trapping is impossible within this adjusted state and, hence, adjustment is always complete for small enough departures from balance. However, we show that retarded adjustment occurs if the adjusted state admits quasi-stationary states decaying via tunnelling across a potential barrier. A description of finite-amplitude periodic nonlinear waves known to exist in configurations with constant potential vorticity in this model is given in terms of Lagrangian variables. Finally, shock formation is analysed and semi-quantitative criteria based on the values of initial gradients and the relative vorticity of initial states are established for wave breaking showing, again, essential differences between the regions of positive and negative vorticity. † As usual in geophysical applications, the centrifugal force will be neglected and molecular dissipation will be absent in what follows.
The route to dissipation in strongly stratified and rotating flows
Journal of Fluid Mechanics, 2013
We investigate the route to dissipation in strongly stratified and rotating systems through high-resolution numerical simulations of the Boussinesq equations (BQs) and the primitive equations (PEs) in a triply periodic domain forced at large scales. By applying geostrophic scaling to the BQs and using the same horizontal length scale in defining the Rossby and the Froude numbers, mathitRo\mathit{Ro}mathitRo and mathitFr\mathit{Fr}mathitFr, we show that the PEs can be obtained from the BQs by taking the limit mathitFr2/mathitRo2rightarrow0{\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} \rightarrow 0mathitFr2/mathitRo2rightarrow0. When mathitFr2/mathitRo2{\mathit{Fr}}^{2} / {\mathit{Ro}}^{2} mathitFr2/mathitRo2 is small the difference between the results from the BQ and the PE simulations is shown to be small. For large rotation rates, quasi-geostrophic dynamics are recovered with a forward enstrophy cascade and an inverse energy cascade. As the rotation rate is reduced, a fraction of the energy starts to cascade towards smaller scales, leading to a shallowing of the horizontal spectra from kh−3{ k}_{h}^{- 3} kh−3 to ${...
2022
We study the effect of the fast rotation and vertical viscosity on the lifespan of solutions to the three-dimensional primitive equations (also known as the hydrostatic Navier-Stokes equations) with impermeable and stress-free boundary conditions. Firstly, for a short time interval, independent of the rate of rotation ∣Omega∣|\Omega|∣Omega∣, we establish the local well-posedness of solutions with initial data that is analytic in the horizontal variables and only L2L^2L2 in the vertical variable. Moreover, it is shown that the solutions immediately become analytic in all the variables with increasing-in-time (at least linearly) radius of analyticity in the vertical variable for as long as the solutions exist. On the other hand, the radius of analyticity in the horizontal variables might decrease with time, but as long as it remains positive the solution exists. Secondly, with fast rotation, i.e., large ∣Omega∣|\Omega|∣Omega∣, we show that the existence time of the solution can be prolonged, with "well-prepared" initial data. Finally, in the case of two spatial dimensions with Omega=0\Omega=0Omega=0, we establish the global well-posedness provided that the initial data is small enough. The smallness condition on the initial data depends on the vertical viscosity and the initial radius of analyticity in the horizontal variables.
The primitive equations on the large scale ocean under the small depth hypothesis
Discrete and Continuous Dynamical Systems, 2002
In this article we study the global existence of strong solutions of the Primitive Equations (PEs) for the large scale ocean under the small depth hypothesis. The small depth hypothesis implies that the domain Mε occupied by the ocean is a thin domain, its thickness parameter ε is the aspect ratio between its vertical and horizontal scales. Using and generalizing the methods developed in [23],[24], we establish the global existence of strong solutions for initial data and volume and boundary 'forces', which belong to large sets in their respective phase spaces, provided ε is sufficiently small. Our proof of the existence results for the PEs is based on precise estimates of the dependence of a number of classical constants on the thickness ε of the domain. The extension of the results to the atmosphere or the coupled ocean and atmosphere or to other relevant boundary conditions will appear elsewhere.