A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow

Majda, Andrew J.; Tabak, Esteban G.

doi:10.1016/0167-2789(96)00114-5

Cited by 97 publications

(63 citation statements)

References 11 publications

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“…A more careful simulation that was conducted in [6,7] revealed the absence of such singularity and attributed the observation in the simualtion of [12] to a growth of the type e e t . Indeed it was proven rigorously in [8] that the scenario of blow up suggested in [12] is not possible.…”

Section: Introductionmentioning

confidence: 99%

“…However, it has also been the focus of interesting mathematical work [3,4,5,12], since the evolution of ∇θ resembles the evolution of the vorticity in the 3D Euler equations. This is despite the 2D nature of the equation, and the misleading impression that θ evolves like the vorticity in the 2D Euler equations.…”

Section: Introductionmentioning

confidence: 99%

“…This is despite the 2D nature of the equation, and the misleading impression that θ evolves like the vorticity in the 2D Euler equations. Preliminary numerical simulations conducted in [12] revealed that the SQG equations (1) with smooth initial data develop sharp fronts in the level contours of θ and conjectured the possibility of formation of a finite time singularity in ∇θ. A more careful simulation that was conducted in [6,7] revealed the absence of such singularity and attributed the observation in the simualtion of [12] to a growth of the type e e t .…”

Section: Introductionmentioning

confidence: 99%

“…Preliminary numerical simulations conducted in [12] revealed that the SQG equations (1) with smooth initial data develop sharp fronts in the level contours of θ and conjectured the possibility of formation of a finite time singularity in ∇θ. A more careful simulation that was conducted in [6,7] revealed the absence of such singularity and attributed the observation in the simualtion of [12] to a growth of the type e e t . Indeed it was proven rigorously in [8] that the scenario of blow up suggested in [12] is not possible.…”

Section: Introductionmentioning

confidence: 99%

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An inviscid regularization for the surface quasi‐geostrophic equation

Khouider

Titi

2007

Comm Pure Appl Math

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Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations. We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter α tends to zero, for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite time blow up testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times. Moreover, much like the original problem, the inviscid regularization admits a maximum principle.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

An inviscid regularization for the surface quasi‐geostrophic equation

Khouider

Titi

2007

Comm Pure Appl Math

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“…Since then, lots of studies have been made both mathematically and computationally, e.g. [3][4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations

Ohkitani

2012

Physics of Fluids

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We study the generalised 2D surface quasi-geostrophic (SQG) equation, where the active scalar is given by a fractional power α of Laplacian applied to the stream function. This includes the 2D SQG and Euler equations as special cases.Using Poincaré's successive approximation to higher α-derivatives of the active scalar, we derive a variational equation for describing perturbations in the generalized SQG equation. In particular, in the limit α → 0, an asymptotic equation is derived on a stretched time variable τ = αt, which unifies equations in the family near α = 0. The successive approximation is also discussed at the other extreme of the 2D Euler limit α = 2-0. Numerical experiments are presented for both limits.We consider whether the solution becomes more singular or regular with increasing α. Two competing effects are identified: the regularizing effect of inverse "Laplacian" and cancellation by symmetry (nonlinearity depletion). Near α = 0 (complete depletion) as α increases the solution becomes more singular. Near α = 2 (maximal smoothing effect of inverse Laplacian) as α decreases the solution becomes more singular, suggesting that there may be some α in [0,1] at which the solution is most singular.We also present some numerical results of the family for α = 0.5, 1 and 1.5. On the original time t, the H 1 norm of θ generally grows more rapidly with increasing α. However, on the new time τ , this order is reversed. On the other hand, contour patterns for different α appear to be similar at fixed τ , even though the norms are markedly different in magnitude. Finally, point-vortex systems for the generalized SQG family are discussed to shed light on the above problems of time scale.

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Ekman layers of rotating fluids: The case of general initial data

Masmoudi

2000

Comm. Pure Appl. Math.

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A two-dimensional model for quasigeostrophic flow: comparison with the two-dimensional Euler flow

Cited by 97 publications

References 11 publications

An inviscid regularization for the surface quasi‐geostrophic equation

An inviscid regularization for the surface quasi‐geostrophic equation

Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations

Ekman layers of rotating fluids: The case of general initial data

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