A simple one‐dimensional model for the three‐dimensional vorticity equation

Constantin, Peter; Lax, Peter D.; Majda, Andrew J.

doi:10.1002/cpa.3160380605

Cited by 265 publications

(318 citation statements)

References 8 publications

(16 reference statements)

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“…Equation (32) with F = ψ, and γ 0 τ = t is equivalent to the asymptotic equation (4). After accounting for the coefficients, we find that (32) is identical to the asymptotic equation derived in Section 4 for the Burgers-Hilbert equation…”

Section: Surface Waves On a Vorticity Discontinuitymentioning

confidence: 58%

“…In this section, we use the method of multiple scales to show that weakly nonlinear solutions of (1) satisfy the asymptotic equation (4). Before doing so, we briefly compare (1) with some other nonlinear, nonlocal wave equations that have been studied previously.…”

Section: The Burgers-hilbert Equationmentioning

confidence: 99%

“…introduced in [4] as a one-dimensional model for vortex stretching, includes a Hilbert transform in the nonlinearity, as does the equation…”

Section: The Burgers-hilbert Equationmentioning

confidence: 99%

“…In this section, we describe the formal Hamiltonian structure of the asymptotic equation (4), and derive equivalent spectral and real forms.…”

Section: The Asymptotic Equationmentioning

confidence: 99%

“…To derive a semi-classical approximation for (4) it is convenient to proceed informally. The same results can be obtained by the explicit introduction of a small parameter.…”

Section: Semi-classical Solutionsmentioning

confidence: 99%

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Nonlinear Hamiltonian waves with constant frequency and surface waves on vorticity discontinuities

Biello

Hunter

2009

Comm Pure Appl Math

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Waves with constant, nonzero linearized frequency form an interesting class of nondispersive waves whose properties differ from those of nondispersive hyperbolic waves. We propose an inviscid Burgers-Hilbert equation as a model equation for such waves, and give a dimensional argument to show that it models Hamiltonian surface waves with constant frequency. Using the method of multiple scales, we derive a cubically nonlinear, quasilinear, nonlocal asymptotic equation for weakly nonlinear solutions. We show that exactly the same asymptotic equation describes surface waves on a planar discontinuity in vorticity in two-dimensional inviscid, incompressible fluid flows. Thus, the BurgersHilbert equation provides an effective equation for these waves. We describe the Hamiltonian structure of the Burgers-Hilbert and asymptotic equations, and show that the asymptotic equation may be also be derived by means of a near-identity transformation. We derive a semi-classical approximation of the asymptotic equation, and show that spatially periodic, harmonic traveling waves are linearly and modulationaly stable. Numerical solutions of the Burgers-Hilbert and asymptotic equations are in excellent agreement in the appropriate regime. In particular, the lifespan of small-amplitude smooth solutions of the Burgers-Hilbert equation is given by the cubically nonlinear timescale predicted by the asymptotic equation.

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Section: Surface Waves On a Vorticity Discontinuitymentioning

confidence: 58%

Section: The Burgers-hilbert Equationmentioning

confidence: 99%

“…introduced in [4] as a one-dimensional model for vortex stretching, includes a Hilbert transform in the nonlinearity, as does the equation…”

Section: The Burgers-hilbert Equationmentioning

confidence: 99%

“…In this section, we describe the formal Hamiltonian structure of the asymptotic equation (4), and derive equivalent spectral and real forms.…”

Section: The Asymptotic Equationmentioning

confidence: 99%

“…To derive a semi-classical approximation for (4) it is convenient to proceed informally. The same results can be obtained by the explicit introduction of a small parameter.…”

Section: Semi-classical Solutionsmentioning

confidence: 99%

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