Ohkitani, K. (2012) Asymptotics and numerics of a family of two-dimensional generalized surface quasi-geostrophic equations. Physics of Fluids, 24 (9). 095101. ISSN 1070-6631
Abstract
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 behaves in a more singular fashion, with more effective nonlinearity, when α is increased. Two competing effects are identified: the regularizing effect of a fractional inverse Laplacian (control by conservation) and cancellation by symmetry (nonlinearity depletion). Near α = 0 (complete depletion), the solution behaves in a more singular fashion as α increases. Near α = 2 (maximal control by conservation), the solution behave in a more singular fashion, as α decreases, suggesting that there may be some α in [0, 2] at which the solution behaves in the most singular manner. 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2012 American Institute of Physics. This is an author produced version of a paper subsequently published in Physics of Fluids. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | numerical analysis; pattern formation; variational techniques; vortices |
Dates: |
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Institution: | The University of Sheffield |
Academic Units: | The University of Sheffield > Faculty of Science (Sheffield) > School of Mathematics and Statistics (Sheffield) |
Depositing User: | Symplectic Sheffield |
Date Deposited: | 23 Sep 2016 12:23 |
Last Modified: | 01 Jul 2017 03:23 |
Published Version: | http://dx.doi.org/10.1063/1.4748350 |
Status: | Published |
Publisher: | AIP Publishing |
Refereed: | Yes |
Identification Number: | 10.1063/1.4748350 |
Related URLs: | |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:78709 |