Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents

Shvydkoy, Roman; Latushkin, Yuri

doi:10.1007/s00021-004-0114-x

Cited by 17 publications

(20 citation statements)

References 21 publications

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“…A complete description of the spectrum obtained in papers [46,47] reveals the opposite inclusion also. Thus, the following theorem holds.…”

Section: -D Casementioning

confidence: 87%

“…A topological argument shows that the condition on orbits is satisfied if u ≡ 0 and u has at least two distinct stagnation points [46]. If, on the contrary, all trajectories have uniformly bounded periods, the spectra are still included into the unit circle and imaginary axis respectively, and hence, they do not intersect the unstable region.…”

Section: -D Casementioning

confidence: 99%

“…Indeed, if Lv ∼ λv, then e In Section 4.1 we indicated that the opposite inclusion is valid for the 2 dimensional Euler semigroup. We say that such a semigroup has the Spectral Mapping Property, or satisfies the Spectral Mapping Theorem, (46) e…”

Section: 5mentioning

confidence: 99%

“…In the classical theory, to order to prove (46) or at least (47), one starts with a sequence {f n } of approximate eigenfunctions for the semigroup corresponding to an eigenvalue e tλ , so that…”

Section: 5mentioning

confidence: 99%

See 3 more Smart Citations

On recent developments in the spectral problem for the linearized Euler equation

Shvydkoy¹,

Friedlander²

2005

Nonlinear Partial Differential Equations and Related Analysis

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“…A complete description of the spectrum obtained in papers [46,47] reveals the opposite inclusion also. Thus, the following theorem holds.…”

Section: -D Casementioning

confidence: 87%

Section: -D Casementioning

confidence: 99%

Section: 5mentioning

confidence: 99%

Section: 5mentioning

confidence: 99%

See 2 more Smart Citations

On recent developments in the spectral problem for the linearized Euler equation

Shvydkoy¹,

Friedlander²

2005

Nonlinear Partial Differential Equations and Related Analysis

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“…After substituting into (19), and cancelling the low order terms along with the exponents, we obtain…”

Section: The Spectrum Of the Sqg Equationmentioning

confidence: 99%

The Unstable Spectrum of the Surface Quasi-Geostrophic Equation

Friedlander

Shvydkoy

2005

J. math. fluid mech.

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Abstract. We study the unstable spectrum of an equation that arises in geophysical fluid dynamics known as the surface quasi-geostrophic equation. In general the spectrum is the union of discrete eigenvalues and an essential spectrum. We demonstrate the existence of unstable eigenvalues in a particular example. We examine the spectra of the semigroup and the evolution operator. We exhibit the structure of these spectra for general flows and prove that a spectral mapping theorem holds. We observe that the spectral properties of the SQG equation are closely analogous to those of the two dimensional Euler equation .

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Linear instability of ideal flows on a sphere

Skiba

2008

Math Methods in App Sciences

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SUMMARYA unified approach to the normal mode instability study of steady solutions to the vorticity equation governing the motion of an ideal incompressible fluid on a rotating sphere is considered. The four types of well-known solutions are considered, namely, the Legendre-polynomial (LP) flows, Rossby-Haurwitz (RH) waves, Wu-Verkley (WV) waves and modons. A conservation law for disturbances to each solution is derived and used to obtain a necessary condition for its exponential instability. By these conditions, Fjörtoft's (Tellus 1953; 5:225-230) average spectral number of the amplitude of an unstable mode must be equal to a special value. In the case of LP flows or RH waves, this value is related only with the basic flow degree. For the WV waves and modons, it depends both on the basic flow degree and on the spectral distribution of the mode energy in the inner and outer regions of the flow. Peculiarities of the instability conditions for different types of modons are discussed. The new instability conditions specify the spectral structure of growing disturbances localizing them in the phase space. For the LP flows, this condition complements the well-known Rayleigh-Kuo and Fjörtoft conditions related to the zonal flow profile. Some analytical and numerical examples are considered. The maximum growth rate of unstable modes is also estimated, and the orthogonality of any unstable, decaying and non-stationary mode to the basic flow is shown in the energy inner product.The analytical instability results obtained here can also be applied for testing the accuracy of computational programs and algorithms used for the numerical stability study. It should be stressed that Fjörtoft's spectral number appearing both in the instability conditions and in the maximum growth rate estimates is the parameter of paramount importance in the linear instability problem of ideal flows on a sphere.

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Essential Spectrum of the Linearized 2D Euler Equation and Lyapunov–Oseledets Exponents

Cited by 17 publications

References 21 publications

On recent developments in the spectral problem for the linearized Euler equation

On recent developments in the spectral problem for the linearized Euler equation

The Unstable Spectrum of the Surface Quasi-Geostrophic Equation

Linear instability of ideal flows on a sphere

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