On the breakdown into turbulence of propagating internal waves

Lombard, Peter N.; Riley, James J.

doi:10.1016/0377-0265(95)00431-9

Cited by 52 publications

(38 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The latter result showing that w given by (44) represents the exact non-linear solution to the inviscid equations of motion (4)-(6) follows from the fact that it solves the linear part of (2)-(4) while the non-linear convective-derivative terms in (4)-(6) are identically zero (for the case f ¼ 0 and a single plane wave solution, this observation has also been discussed previously in Lombard and Riley [23]). …”

Section: Generalization Of the Invariant Solutionmentioning

confidence: 62%

“…(4)- (6) have been studied extensively in many previous works. For example, in the particular case when the Earth rotation is ignored, f ¼ 0, these equations have been studied by Tabaei [30,31] (colliding and reflecting internal wave beams), Lombard and Riley [23] (linearized stability of interacting exact solutions) and Dauxois and Young [7] (reflection of internal waves off a near-critical slope). The case of nonzero earth rotation, f -0, has been considered in our previous studies [13] to model weakly non-linear wave interactions.…”

Section: Model Equationsmentioning

confidence: 98%

“…Nonlinear interactions of two-dimensional monochromatic not uni-directional beams ignoring the effects of the earth rotation have been studied in Kistovich and Chasheschkin [19]. The stability problem of two interacting plane waves have been studied by Lombrad and Riley [23], using the Floquet theory. The problems on beam impinging onto a curved topography in a fluid have not been studied analytically to present day because of complicated scattering effects at the bottom boundary (waves of different frequencies propagate in different directions).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

Ibragimov¹,

Ibragimov²

2010

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

Section: Generalization Of the Invariant Solutionmentioning

confidence: 62%

Section: Model Equationsmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

Ibragimov¹,

Ibragimov²

2010

Communications in Nonlinear Science and Numerical Simulation

View full text Add to dashboard Cite

“…One of the advantages of Lie group analysis consists in the fact that it can be used to integrate nonlinear equations that are not of Hamiltonian type. As a particular example, it is demonstrated in this paper, that the invariant η = kx + mz of the translation symmetry provides the class of exact solutions that were deduced in the previous studies from the anisotropic nature of internal wave propagation (see e.g., [45], [44], [5] and [34]). However, the more general forms of invariant solutions, which can not be guessed from the anisotropic property and correspondingly were not reported in previous studies, are presented by the infinite-dimensional Lie algebra spanned by the infinitesimal symmetries (4.29) -(4.33).…”

Section: Concluding Discussionmentioning

confidence: 63%

Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

Ibragimov

Ibragimov²

2012

Math. Model. Nat. Phenom.

View full text Add to dashboard Cite

Abstract. Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie's group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

show abstract

“…From the evidence presented here, we conjecture that such instabilities in these and related flow fields at large Richardson numbers may contribute significantly to the step-like microstructures often observed in buoyancy measurements in the ocean (9,10). Furthermore, these mechanisms with spontaneous generation of instability are fundamentally different from the traditional ones involving larger amplitude gravity wave breaking (11).…”

Section: Discussionmentioning

confidence: 67%

The instability of stratified flows at large Richardson numbers

Madja

Shefter

1998

Proc. Natl. Acad. Sci. U.S.A.

View full text Add to dashboard Cite

In contrast to conventional expectations based on the stability of steady shear flows, elementary time-periodic stratified flows that are unstable at arbitrarily large Richardson numbers are presented here. The fundamental instability is a parametric one with twice the period of the basic state. This instability spontaneously generates local shears on buoyancy time scales near a specific angle of inclination that saturates into a localized regime of strong mixing with density overturning. We speculate that such instabilities may contribute significantly to the step-like microstructure often observed in buoyancy measurements in the ocean.One of the basic analytical results for stably stratified fluid flows is the celebrated Miles-Howard theorem (1, 2). This theorem states that steady shear flows V = v z 0 0 in an inviscid stably stratified fluid are linearly stable for all Richardson numbers, i, satisfyingwith N 2 = −g ∂ρ ∂z /ρ b , the square of the buoyancy or BruntVaisala frequency. This criterion for stability is often interpreted and applied literally for time-dependent flow fields in both theoretical and numerical modeling for the atmosphere or ocean. For example, a popular turbulent eddy diffusivity used in numerical simulations in the atmosphere/ocean community is the Lilly-Smagorinsky eddy diffusivity (3, 4), where the turbulent eddy diffusivity is completely switched off and set to zero for i > i > 1 4 with i of order unity. Here, we present elementary examples, with firm mathematical underpinnings, which demonstrate that such reasoning can be violated in dramatic fashion for time-dependent strongly stratified flows. With appropriate nondimensional units explained below, we consider solutions of the inviscid two-dimensional Boussinessq equations in vorticity-stream form,whereρ is fluid density,is the vorticity, ψ is the stream function, and the field velocity v is given byNext, we build elementary time-periodic mean states for the equations in 2, and then we demonstrate their linearized and nonlinear instability for large Richardson numbers. Time-Periodic Mean States and the Nonlinear PendulumWe construct elementary time-periodic exact solutions of the Boussinessq equations with constant spatial gradients withThe publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. $1734 solely to indicate this fact.© 1998 by The National Academy of Sciences 0027-8424/98/957850-4$2.00/0 PNAS is available online at http://www.pnas.org.the formρDirect calculation yields the fact that special solutions of 2 have the structure in 4 provided that the phase function, θ t , satisfies the nonlinear pendulum equationwith the initial dataImplicit in both 2 and 6 is a nondimensionalization where we use the ambient density, ρ b , the stably stratified initial vertical density gradient, and gravity, g, to set the relevant scales. Thus, the unit of time in 2 is determined by the constant buoyancy frequency, N, w...

show abstract

On the breakdown into turbulence of propagating internal waves

Cited by 52 publications

References 9 publications

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

Internal gravity wave beams as invariant solutions of Boussinesq equations in geophysical fluid dynamics

Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences

The instability of stratified flows at large Richardson numbers

Contact Info

Product

Resources

About