Low Rossby limiting dynamics for stably stratified flow with finite Froude number

Wingate, Beth; Embid, Pedro; Holmes-Cerfon, Miranda; Taylor, Mark A.

doi:10.1017/jfm.2011.69

Cited by 27 publications

(53 citation statements)

References 34 publications

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“…A basic observation behind efficiently constructing a slow solution is that the solution u (t) to (1.1) has the asymptotic approximation u (t) = exp (−t/ǫL) u (t) + O (ǫ) (cf. [29], [30], [38]), where the slowly varying function u (t) satisfies a reduced equation of the form…”

Section: Introductionmentioning

confidence: 99%

An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs

Haut¹,

Wingate²

2014

SIAM J. Sci. Comput.

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Abstract. We present a new time-stepping algorithm for nonlinear PDEs that exhibit scale separation in time of a highly oscillatory nature. The algorithm combines the parareal method-a parallel-in-time scheme introduced in [24]-with techniques from the Heterogeneous Multiscale Method (HMM) (cf. [13,2]), which make use of the slow asymptotic structure of the equations [32], [27], [28], [36]. By numerically computing a locally asymptotic solution we are able to "factor out" the fast oscillatory part of the solution, and solve for the remaining slow part of the solution using time steps that are orders of magnitude larger than standard time-stepping methods allow. The scheme has two elements. First, we use HMM to numerically advance the asymptotic form of the equations with a large time step ∆T . Second, we refine the solution in parallel on the subintervals [n∆T, (n + 1) ∆T ] using small time steps ∆t and the iterative scheme scheme in [24]; the intermediate solutions on the sub-intervals [n∆T, (n + 1) ∆T ] are ephemeral, and are used only to converge the overall solution at the large time steps n∆T . Using the asymptotic structure allows the computed solutions to be close enough to the actual solution that parallel-in-time methods converge to high accuracy and with significant parallel speedup.We present error bounds, based on the analysis in [17], that demonstrate convergence of the method. A complexity analysis also demonstrates that the parallel speedup increases arbitrarily with greater scale separation. Finally, we demonstrate the accuracy and efficiency of the method on the (onedimensional) rotating shallow water equations, which is a standard test problem for new algorithms in geophysical fluid problems. Compared to exponential integrators such as ETDRK4 and Strang splitting-which solve the stiff oscillatory part exactly-we find that we can use coarse time steps ∆T that are orders of magnitude larger (for a comparable accuracy), yielding an estimated parallel speedup of approximately 100 for physically realistic parameter values. For the (one-dimensional) shallow water equations, we also show that the estimated parallel speedup of this "asymptotic parareal method" is more than a factor of 10 greater than the speedup obtained from the standard parareal method.

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Section: Introductionmentioning

confidence: 99%

An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs

Haut¹,

Wingate²

2014

SIAM J. Sci. Comput.

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“…As a result of this variety of regimes and of the limitations in computer power, in recent years progress has been made in understanding rotating stratified flows thanks to a combination of tools, including, e.g., wave turbulence approaches [6][7][8][9], reduced equations based on asymptotic expansions in a small parameter [10][11][12][13], experiments [14][15][16][17][18], observations in the atmosphere and the ocean [19,20], and direct numerical simulations [5,[21][22][23][24] (see also the reviews in [25,26]). However, some fundamental issues have not been clarified yet.…”

Section: Introductionmentioning

confidence: 99%

Large-scale anisotropy in stably stratified rotating flows

et al. 2014

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We present results from direct numerical simulations of the Boussinesq equations in the presence of rotation and/or stratification, both in the vertical direction. The runs are forced isotropically and randomly at small scales and have spatial resolutions of up to 1024 3 grid points and Reynolds numbers of ≈ 1000. We first show that solutions with negative energy flux and inverse cascades develop in rotating turbulence, whether or not stratification is present. However, the purely stratified case is characterized instead by an early-time, highly anisotropic transfer to large scales with almost zero net isotropic energy flux. This is consistent with previous studies that observed the development of vertically sheared horizontal winds, although only at substantially later times. However, and unlike previous works, when sufficient scale separation is allowed between the forcing scale and the domain size, the total energy displays a perpendicular (horizontal) spectrum with power law behavior compatible with ∼ k −5/3 ⊥ , including in the absence of rotation. In this latter purely stratified case, such a spectrum is the result of a direct cascade of the energy contained in the large-scale horizontal wind, as is evidenced by a strong positive flux of energy in the parallel direction at all scales including the largest resolved scales.

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“…For this limiting case, a suite of approximations, simpler and easier to integrate than the primitive equations, has been developed using asymptotic expansions (see, e.g., Refs. [1,[7][8][9][10][11][12][13]), two-point closures of turbulence [14][15][16], or weak turbulence statistical approaches [17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Helicity dynamics in stratified turbulence in the absence of forcing

et al. 2013

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A numerical study of decaying stably stratified flows is performed. Relatively high stratification (Froude number ≈ 10 −2 -10 −1 ) and moderate Reynolds (Re) numbers (Re ≈ 3-6 × 10 3 ) are considered and a particular emphasis is placed on the role of helicity (velocity-vorticity correlations), which is not an invariant of the nondissipative equations. The problem is tackled by integrating the Boussinesq equations in a periodic cubical domain using different initial conditions: a nonhelical Taylor-Green (TG) flow, a fully helical Beltrami [ArnoldBeltrami-Childress (ABC)] flow, and random flows with a tunable helicity. We show that for stratified ABC flows helicity undergoes a substantially slower decay than for unstratified ABC flows. This fact is likely associated to the combined effect of stratification and large-scale coherent structures. Indeed, when the latter are missing, as in random flows, helicity is rapidly destroyed by the onset of gravitational waves. A type of large-scale dissipative "cyclostrophic" balance can be invoked to explain this behavior. No production of helicity is observed, contrary to the case of rotating and stratified flows. When helicity survives in the system, it strongly affects the temporal energy decay and the energy distribution among Fourier modes. We discover in fact that the decay rate of energy for stratified helical flows is much slower than for stratified nonhelical flows and can be considered with a phenomenological model in a way similar to what is done for unstratified rotating flows. We also show that helicity, when strong, has a measurable effect on the Fourier spectra, in particular at scales larger than the buoyancy scale, for which it displays a rather flat scaling associated with vertical shear, as observed in the planetary boundary layer.

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Low Rossby limiting dynamics for stably stratified flow with finite Froude number

Cited by 27 publications

References 34 publications

An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs

An Asymptotic Parallel-in-Time Method for Highly Oscillatory PDEs

Large-scale anisotropy in stably stratified rotating flows

Helicity dynamics in stratified turbulence in the absence of forcing

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