Ohkitani, K. (2011) Growth rate analysis of scalar gradients in generalized surface quasigeostrophic equations of ideal fluids. Physical Review E, 83 (3). 036317. ISSN 1539-3755
Abstract
The growth rates of scalar gradients are studied numerically and analytically in a family of two-dimensional (2D) incompressible fluid equations related to the surface quasigeostrophic (SQG) equation. The active scalar is related to the stream function ψ by θ=(−△)α/2ψ (0⩽α⩽2). A notable difference is observed in a comparison of the instantaneous growth rates in Lp and in L∞ norms, depending on the stage of the time evolution. The crux is the phase-shift effect of singular integral operators, which displaces the peak location of the scalar gradient from that of the strain rate. On this basis, a method of detecting such a dislocation is proposed in view of the importance of their coalescence needed for a possible blow-up. Moreover, it is found in the long-time evolution that a solution of the SQG equation (whose regularity is not known) is less singular than that of the 2D Euler equations (known to be regular) on the time interval covered by this computation. This consistently expands an earlier observation by Majda and Tabak [Physica D 98, 515 (1996).] in some detail. A 1D model problem is discussed to illustrate the present method, and extensions to the 3D case are also are briefly discussed.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2011 American Physical Society. This is an author produced version of a paper subsequently published in Physical Review E. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | One-Dimensional Model; Turbulence; Flow |
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 12:03 |
Last Modified: | 02 Jul 2017 02:23 |
Published Version: | http://dx.doi.org/10.1103/PhysRevE.83.036317 |
Status: | Published |
Publisher: | American Physical Society |
Refereed: | Yes |
Identification Number: | 10.1103/PhysRevE.83.036317 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:78707 |