Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-10T13:31:22.532Z Has data issue: false hasContentIssue false
  • English
  • Français

Scale-invariant singularity of the surface quasigeostrophic patch

Published online by Cambridge University Press:  28 January 2019

R. K. Scott Open the ORCID record for R. K. Scott [Opens in a new window]  and
D. G. Dritschel Open the ORCID record for D. G. Dritschel [Opens in a new window]
R. K. Scott*
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, Scotland
D. G. Dritschel
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, Scotland
*
†Email address for correspondence: rks4@st-andrews.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

Numerical simulations of the surface quasigeostrophic patch indicate the development of a scale-invariant singularity of the boundary curvature in finite time, with some evidence of universality across a variety of initial conditions. At the time of singularity, boundary segments are shown to possess an exact and simple analytic form, described by branches of a logarithmic spiral centred on the point of singularity. The angles between the branches depend non-trivially on the shape of the smooth connecting boundary as the singularity is approached, but are independent of the global boundary.

JFM classification

Vortex Flows: Contour dynamics Geophysical and Geological Flows: Quasi-geostrophic flows
Type
JFM Rapids
Information
Journal of Fluid Mechanics , Volume 863 , 25 March 2019 , R2
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bareblatt, G. I. & Zel’dovich, Y. B. 1972 Self-similar solutions as intermediate asymptotics. Annu. Rev. Fluid Mech. 4, 285–312.10.1146/annurev.fl.04.010172.001441Google Scholar
Blumen, W. 1978 Uniform potential vorticity flow. Part I: theory of wave interactions and two-dimensional turbulence. J. Atmos. Sci. 35, 774–783.10.1175/1520-0469(1978)035<0774:UPVFPI>2.0.CO;22.0.CO;2>Google Scholar
Constantin, P. 1995 Nonlinear inviscid incompressible dynamics. Physica D 86, 212–219.Google Scholar
Constantin, P., Majda, A. J. & Tabak, E. 1994 Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar. Nonlinearity 7, 1495–1533.10.1088/0951-7715/7/6/001Google Scholar
Constantin, P., Nie, Q. & Schorghofer, N. 1998 Nonsingular surface quasi-geostrophic flow. Phys. Lett. A 241, 168–172.10.1016/S0375-9601(98)00108-XGoogle Scholar
Cordoba, D. 1998 Nonexistence of simple hyperbolic blow-up for the quasi-geostrophic equation. Ann. Math. 148, 1135–1152.10.2307/121037Google Scholar
Cordoba, D., Fontelos, M. A., Mancho, A. M. & Rodrigo, J. L. 2005 Evidence of singularities for a family of contour dynamics equations. Proc. Natl Acad. Sci. USA 102, 5949–5952.10.1073/pnas.0501977102Google Scholar
Fefferman, C. & Rodrigo, J. L. 2012 Almost sharp fronts for SQG: the limit equations. Commun. Math. Phys. 313, 131–153.10.1007/s00220-012-1486-zGoogle Scholar
Gancedo, F. 2008 Existence for the 𝛼-patch model and the QG sharp front in Sobolev spaces. Adv. Maths 217, 2569–2598.10.1016/j.aim.2007.10.010Google Scholar
Johnson, E. R. 1978 Topographically bound vortices. Geophys. Astrophys. Fluid Dyn. 11, 61–71.10.1080/03091927808242652Google Scholar
Kiselev, A., Ryzhik, L., Yao, Y. & Zlatos̆, A 2016 Finite time singularity for the modified SQG patch equation. Ann. Math. 184, 909–948.10.4007/annals.2016.184.3.7Google Scholar
Lapeyre, G. 2017 Surface qausi-geostrophy. Fluids 2, 7.10.3390/fluids2010007Google Scholar
Mancho, A. M. 2015 Numerical studies on the self-similar collapse of the 𝛼-patches problem. Commun. Nonlinear Sci. Numer. Simul. 26, 152–166.10.1016/j.cnsns.2015.02.009Google Scholar
Ohkitani, K. & Yamada, M. 1997 Inviscid and inviscid-limit behavior of surface quasigeostrophic flow. Phys. Fluids 9, 876–882.10.1063/1.869184Google Scholar
Pierrehumbert, R. T., Held, I. M. & Swanson, K. L. 1994 Spectra of local and nonlocal two-dimensional turbulence. Chaos, Solitons Fractals 4, 1111–1116.10.1016/0960-0779(94)90140-6Google Scholar
Rodrigo, J. L. 2004 The vortex patch problem for the surface quasi-geostrophic equation. Proc. Natl Acad. Sci. USA 101, 2684–2686.10.1073/pnas.0308158101Google Scholar
Rodrigo, J. L. 2005 On the evolution of sharp fronts for the quasi-geostrophic equation. Commun. Pure Appl. Maths 58, 821–866.10.1002/cpa.20059Google Scholar
Scott, R. K. 2011 A scenario for finite-time singularity in the quasigeostrophic equations. J. Fluid Mech. 687, 492–502.10.1017/jfm.2011.377Google Scholar
Scott, R. K. & Dritschel, D. G. 2014 Numerical simulation of a self-similar cascade of filament instabilities in the surface quasigeostrophic system. Phys. Rev. Lett. 112, 144505.10.1103/PhysRevLett.112.144505Google Scholar
Tran, C. V., Dritschel, D. G. & Scott, R. K. 2010 Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence. Phys. Rev. E 81, 01630.Google Scholar
Wu, J. 2007 Existence and uniqueness results for the 2-D dissipative quasi-geostrophic equation. Nonlinear Anal. 67, 3013–3036.10.1016/j.na.2006.09.050Google Scholar