Skip to main content Accessibility help
×
Home
Hostname: page-component-747cfc64b6-7hjq6 Total loading time: 0.208 Render date: 2021-06-13T13:00:35.090Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "metricsAbstractViews": false, "figures": true, "newCiteModal": false, "newCitedByModal": true, "newEcommerce": true }

Late time evolution of unforced inviscid two-dimensional turbulence

Published online by Cambridge University Press:  19 October 2009

DAVID G. DRITSCHEL
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
RICHARD K. SCOTT
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
CHARLIE MACASKILL
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
GEORG A. GOTTWALD
Affiliation:
School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia
CHUONG V. TRAN
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
Corresponding
E-mail address:

Abstract

We propose a new unified model for the small, intermediate and large-scale evolution of freely decaying two-dimensional turbulence in the inviscid limit. The new model's centerpiece is a recent theory of vortex self-similarity (Dritschel et al., Phys. Rev. Lett., vol. 101, 2008, no. 094501), applicable to the intermediate range of scales spanned by an expanding population of vortices. This range is predicted to have a steep k−5 energy spectrum. At small scales, this gives way to Batchelor's (Batchelor, Phys. Fluids, vol. 12, 1969, p. 233) k−3 energy spectrum, corresponding to the (forward) enstrophy (mean square vorticity) cascade or, physically, to thinning filamentary debris produced by vortex collisions. This small-scale range carries with it nearly all of the enstrophy but negligible energy. At large scales, the slow growth of the maximum vortex size (~t1/6 in radius) implies a correspondingly slow inverse energy cascade. We argue that this exceedingly slow growth allows the large scales to approach equipartition (Kraichnan, Phys. Fluids, vol. 10, 1967, p. 1417; Fox & Orszag, Phys. Fluids, vol. 12, 1973, p. 169), ultimately leading to a k1 energy spectrum there. Put together, our proposed model has an energy spectrum ℰ(k, t) ∝ t1/3k1 at large scales, together with ℰ(k, t) ∝ t−2/3k−5 over the vortex population, and finally ℰ(k, t) ∝ t−1k−3 over an exponentially widening small-scale range dominated by incoherent filamentary debris.

Support for our model is provided in two parts. First, we address the evolution of large and ultra-large scales (much greater than any vortex) using a novel high-resolution vortex-in-cell simulation. This verifies equipartition, but more importantly allows us to better understand the approach to equipartition. Second, we address the intermediate and small scales by an ensemble of especially high-resolution direct numerical simulations.

Type
Papers
Copyright
Copyright © Cambridge University Press 2009

Access options

Get access to the full version of this content by using one of the access options below.

References

Bartello, P. & Warn, T. 1996 Self-similarity of decaying two-dimensional turbulence. J. Fluid Mech. 326, 357372.CrossRefGoogle Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12, 233239.CrossRefGoogle Scholar
Benzi, R., Colella, M., Briscolini, M. & Santangelo, P. 1992 A simple point vortex model for two-dimensional decaying turbulence. Phys. Fluids 4, 1036.CrossRefGoogle Scholar
Benzi, R., Patarnello, S. & Santangelo, P. 1988 Self-similar coherent structures in two-dimensional decaying turbulence. J. Phys. A 21, 12211237.CrossRefGoogle Scholar
Bracco, A., McWilliams, J. C., Murante, G., Provenzale, A. & Weiss, J. B. 2000 Revisiting freely decaying two-dimensional turbulence at millennial resolution. Phys. Fluids 12, 29312941.CrossRefGoogle Scholar
Carnevale, G. F., McWilliams, J. C., Pomeau, Y., Weiss, J. B. & Young, W. R. 1991 Evolution of vortex statistics in two-dimensional turbulence. Phys. Rev. Lett. 66, 27352737.CrossRefGoogle ScholarPubMed
Charney, J. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 10871095.2.0.CO;2>CrossRefGoogle Scholar
Chasnov, J. R. 1997 On the decay of two-dimensional homogeneous turbulence. Phys. Fluids 9, 171180.CrossRefGoogle Scholar
Christiansen, J. P. & Zabusky, N. J. 1973 Instability, coalescence and fission of finite-area vortex structures. J. Fluid Mech. 61, 219243.CrossRefGoogle Scholar
Clercx, H. J. H., Maassen, S. R. & Van Heijst, G. J. F. 1999 Decaying two-dimensional turbulence in square containers with no-slip or stress-free boundaries. Phys. Fluids 11, 611626.CrossRefGoogle Scholar
Davidson, P. A. 2004 Turbulence: An Introduction for Scientists and Engineers. Oxford University Press.Google Scholar
Davidson, P. A. 2007 On the large-scale structure of homogeneous, two-dimensional turbulence. J. Fluid Mech. 580, 431450.CrossRefGoogle Scholar
Dritschel, D. G. & Ambaum, M. H. P. 1997 A contour-advective semi-Lagrangian algorithm for the simulation of fine-scale conservative fields. Quart. J. R. Meteorol. Soc. 123, 10971130.CrossRefGoogle Scholar
Dritschel, D. G. & Scott, R. 2009 On the simulation of nearly inviscid two-dimensional turbulence. J. Comput. Phys. 228, 27072711.CrossRefGoogle Scholar
Dritschel, D. G., Scott, R. K., Macaskill, C., Gottwald, G. A. & Tran, C. V. 2008 Unifying scaling theory for vortex dynamics in two-dimensional turbulence. Phys. Rev. Lett. 101, 094501.CrossRefGoogle ScholarPubMed
Dritschel, D. G., Tran, C. V. & Scott, R. K. 2007 Revisiting Batchelor's theory of two-dimensional turbulence. J. Fluid Mech. 591, 379391.CrossRefGoogle Scholar
Dritschel, D. G. & Zabusky, N. J. 1996 On the nature of vortex interactions and models in unforced nearly-inviscid two-dimensional turbulence. Phys. Fluids 8 (5), 12521256.CrossRefGoogle Scholar
Eyink, G. L. & Spohn, H. 1993 Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence. J. Stat. Phys. 70, 833886.CrossRefGoogle Scholar
Eyink, G. L. & Sreenivasan, K. R. 2006 Onsager and the theory of hydrodynamic turbulence. Rev. Mod. Phys. 78, 87135.CrossRefGoogle Scholar
Fontane, J. & Dritschel, D. G. 2009 The HyperCASL algorithm: a new approach to the numerical simulation of geophysical flows. J. Comput. Phys. 228 (17), 64116425.CrossRefGoogle Scholar
Fox, D. G. & Orszag, S. A. 1973 Inviscid dynamics of two-dimensional turbulence. Phys. Fluids 16 (2), 169171.CrossRefGoogle Scholar
Gill, A. E. 1982 Atmosphere–Ocean Dynamics. Academic.Google Scholar
Joyce, G. & Montgomery, D. 1973 Negative temperature states for the two-dimensional guiding center plasma. J. Plasma Phys. 10, 107121.CrossRefGoogle Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 14171423.CrossRefGoogle Scholar
Laval, J. P., Chavanis, P. H., Dubrulle, B. & Sire, C. 2001 Scaling laws and vortex profiles in 2D decaying turbulence. Phys. Rev. E 63, 065301R.CrossRefGoogle Scholar
Lesieur, M. 2008 Turbulence in Fluids, 4th edn. Springer.CrossRefGoogle Scholar
Lowe, A. J. & Davidson, P. A. 2005 The evolution of freely-decaying, isotropic, two-dimensional turbulence. Eur. J. Mech. B/Fluids 24, 314327.CrossRefGoogle Scholar
McWilliams, J. C. 1984 The emergence of isolated coherent vortices in turbulent flow. J. Fluid Mech. 146, 2143.CrossRefGoogle Scholar
Miller, R. 1990 Statistical mechanics of negative temperature states. Phys. Rev. Lett. 65, 21372140.CrossRefGoogle Scholar
Montgomery, D. & Joyce, G. 1974 Statistical mechanics of negative temperature states. Phys. Fluids 17, 11391145.CrossRefGoogle Scholar
Onsager, L. 1949 Statistical hydrodynamics. Nuovo Cimento 6, 279287.CrossRefGoogle Scholar
Orszag, S. A. 1970 Analytical theories of turbulence. J. Fluid Mech. 41, 363386.CrossRefGoogle Scholar
Ossai, S. & Lesieur, M. 2001 Large-scale energy and pressure dynamics in decaying 2D incompressible isotropic turbulence. J. Turbul. 2, 172205.Google Scholar
Robert, R. 1991 A maximum-entropy principle for two-dimensional perfect fluid dynamics. J. Stat. Phys. 65, 531554.CrossRefGoogle Scholar
Robert, R. & Sommeria, J. 1991 Statistical equilibrium states for two-dimensional flows J. Fluid Mech. 229, 291310.CrossRefGoogle Scholar
Sire, C. & Chavanis, P. H. 2000 Numerical renormalization group of vortex aggregation in 2D decaying turbulence: the role of three-body interactions. Phys. Rev. E 61, 66446653.CrossRefGoogle Scholar
Tabeling, P. 2002 Two-dimensional turbulence: a physicist approach Phys. Rep. 362, 162.CrossRefGoogle Scholar
Tran, C. V. & Dritschel, D. G. 2006 Large-scale dynamics in two-dimensional Euler and surface quasigeostrophic flows. Phys. Fluids 18, 121703.CrossRefGoogle Scholar
Weiss, J. B. & McWilliams, J. C. 1993 Temporal scaling behavior of decaying two-dimensional turbulence. Phys. Fluids 5, 608621.CrossRefGoogle Scholar
13
Cited by

Send article to Kindle

To send this article to your Kindle, first ensure no-reply@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about sending to your Kindle. Find out more about sending to your Kindle.

Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Late time evolution of unforced inviscid two-dimensional turbulence
Available formats
×

Send article to Dropbox

To send this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Dropbox.

Late time evolution of unforced inviscid two-dimensional turbulence
Available formats
×

Send article to Google Drive

To send this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your <service> account. Find out more about sending content to Google Drive.

Late time evolution of unforced inviscid two-dimensional turbulence
Available formats
×
×

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *