31 research outputs found
Fast and slow resonant triads in the two layer rotating shallow water equations
In this paper we examine triad resonances in a rotating shallow water system
when there are two free interfaces. This allows for an examination in a
relatively simple model of the interplay between baroclinic and barotropic
dynamics in a context where there is also a geostrophic mode. In contrast to
the much-studied one-layer rotating shallow water system, we find that as well
as the usual slow geostrophic mode, there are now two fast waves, a barotropic
mode and a baroclinic mode. This feature permits triad resonances to occur
between three fast waves, with a mixture of barotropic and baroclinic modes, an
aspect which cannot occur in the one-layer system. There are now also two
branches of the slow geostrophic mode with a repeated branch of the dispersion
relation. The consequences are explored in a derivation of the full set of
triad interaction equations, using a multi-scale asymptotic expansion based on
a small amplitude parameter. The derived nonlinear interaction coefficients are
confirmed using energy and enstrophy conservation. These triad interaction
equations are explored with an emphasis on the parameter regime with small
Rossby and Froude numbers
On the Two-point Correlation of Potential Vorticity in Rotating and Stratified Turbulence
A framework is developed to describe the two-point statistics of potential
vorticity in rotating and stratified turbulence as described by the Boussinesq
equations. The Karman-Howarth equation for the dynamics of the two-point
correlation function of potential vorticity reveals the possibility of
inertial-range dynamics in certain regimes in the Rossby, Froude, Prandtl and
Reynolds number parameters. For the case of large Rossby and Froude numbers,
and for the case of quasi-geostrophic dynamics, a linear scaling law with 2/3
prefactor is derived for the third-order mixed correlation between potential
vorticity and velocity, a result that is analogous to the Kolmogorov 4/5-law
for the third-order velocity structure function in turbulence theory.Comment: 10 pages, to appear in Journal of Fluid Mechanics (2006
A Decentralized Parallelization-in-Time Approach with Parareal
With steadily increasing parallelism for high-performance architectures,
simulations requiring a good strong scalability are prone to be limited in
scalability with standard spatial-decomposition strategies at a certain amount
of parallel processors. This can be a show-stopper if the simulation results
have to be computed with wallclock time restrictions (e.g.\,for weather
forecasts) or as fast as possible (e.g. for urgent computing). Here, the
time-dimension is the only one left for parallelization and we focus on
Parareal as one particular parallelization-in-time method.
We discuss a software approach for making Parareal parallelization
transparent for application developers, hence allowing fast prototyping for
Parareal. Further, we introduce a decentralized Parareal which results in
autonomous simulation instances which only require communicating with the
previous and next simulation instances, hence with strong locality for
communication. This concept is evaluated by a prototypical solver for the
rotational shallow-water equations which we use as a representative black-box
solver
An asymptotic parallel-in-time method for highly oscillatory PDEs
© 2014, Society for Industrial and Applied Mathematics. Available online at http://epubs.siam.org/doi/abs/10.1137/130914577We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time. Our scheme combines asymptotic techniques (which are inexpensive but can have insufficient accuracy) with parallel-in-time methods (which, alone, can be inefficient for equations that exhibit rapid temporal oscillations). In particular, we use an asymptotic numerical method for computing, in serial, a solution with low accuracy, and a more expensive fine solver for iteratively refining the solutions in parallel. We present examples on the rotating shallow water equations that demonstrate that significant parallel speedup and high accuracy are achievable
Anisotropic small-scale constraints on energy in rotating stratified turbulence
Author's version issued as working paper on Arxiv.orgRapidly rotating, stably stratified three-dimensional inviscid flows conserve both energy and potential enstrophy. We show that in such flows, the forward cascade of potential enstrophy imposes anisotropic constraints on the wavenumber distribution of kinetic and potential energy. The horizontal kinetic energy is suppressed in the large, nearly horizontal wave modes, and should decay with the horizontal wavenumber as . The potential energy is suppressed in the large, nearly vertical wave modes, and should decay with the vertical wavenumber as . These results augment the only other exact prediction for the scaling of energy spectra due to constraints by potential enstrophy obtained by Charney (J. Atmos. Sci. 28, 1087 (1971)), who showed that in the quasi-geostrophic approximation for rotating stratified flows, the energy spectra must scale isotropically with total wavenumber as . We test our predicted scaling estimates using resolved numerical simulations of the Boussinesq equations in the relevant parameter regimes, and find reasonable agreement
Multilevel Parareal Algorithm with Averaging for Oscillatory Problems
The present study is an extension of the work done by Peddle, Haut, and Wingate [SIAM J. Sci. Comput., 41 (2019), pp. A3476–A3497] and Haut and Wingate [SIAM J. Sci. Comput., 36 (2014), pp. A693–A713], where a two-level Parareal method with mapping and averaging is examined. The method proposed in this paper is a multilevel Parareal method with arbitrarily many levels, which is not restricted to the two-level case. We give an asymptotic error estimate which reduces to the two-level estimate for the case when only two levels are considered. Introducing more than two levels has important consequences for the averaging procedure, as we choose separate averaging windows for each of the different levels, which is an additional new feature of the present study. The different averaging windows make the proposed method especially appropriate for nonlinear multiscale problems, because we can introduce a level for each intrinsic scale of the problem and adapt the averaging procedure such that we reproduce the behavior of the model on the particular scale resolved by the level. The method is applied to nonlinear differential equations. The nonlinearities can generate a range of frequencies in the problem. The computational cost of the new method is investigated and studied on several examples