Abstract
Based on expansion terms of the Beltrami-flow type, we use multiscale methods to effectively construct an asymptotic expansion at large Reynolds numbers R for the long-wavelength perturbation of the nonstationary anisotropic helical solution of the force-free Navier—Stokes equation (the Trkal solution). We prove that the systematic asymptotic procedure can be implemented only in the case where the scaling parameter is R1/2. Projections of quasistationary large-scale streamlines on a plane orthogonal to the anisotropy direction turn out to be the gradient lines of the energy density determined by the initial conditions for two modulated anisotropic Beltrami flows (modulated as a result of scaling) with the same eigenvalues of the curl operator. The three-dimensional streamlines and the curl lines, not coinciding, fill invariant vorticity tubes inside which the velocity and vorticity vectors are collinear up to terms of the order of 1/R.
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References
E. Levich and A. Tsinober, Phys. Lett. A, 93, 293–297 (1983).
E. Levich and E. Tzvetkov, Phys. Lett. A, 100, 53–56 (1984).
E. Levich, Old and New Concepts Phys., 6, 239–457 (2009).
A. Tsinober and E. Levích, Phys. Lett. A, 99, 321–324 (1983).
H. K. Moffatt, J. Fluid Mech., 159, 359–378 (1985).
H. K. Moffatt, J. Fluid Mech., 166, 359–378 (1986).
H. K. Moffatt, “The topology of scalar fields in 2D and 3D turbulence,” in: IUTAM Symposium on Geometry and Statistics of Turbulence (Fluid Mech. Appl., Vol. 59, T. Kambe, T. Nakano, and T. Miyauchi, eds.), Kluwer, Dordrecht (2001), pp. 13–22.
V. Trkal, Časopis št. Mat., 48, 302–311 (1919).
Ya. Andreopoulos, Russ. J. Electrochemistry, 44, 390–396 (2008).
Y. Choi, B.-G. Kim, and C Lee, Phys. Rev. E, 80, 017301 (2009).
M. S. Lilley, S. Lovejoy, K. Strawbridge, D. Schertzer, and A. Radkevich, Quart. J. Roy. Meteor. Soc, 134, 301–315 (2008).
S. Lovejoy, A. Tuck, S. Hovde, and D. Schertzer, Geophys. Res. Lett., 34, L15802 (2007).
S. Lovejoy, D. Schertzer, M. Lilley, K. Strawbridge, and A. Radkevich, Quart. J. Roy. Meteor. Soc, 134, 277–300 (2008).
R D. Mininni, A. Alexakis, and A. Pouquet, Phys. Rev. E, 74, 016303 (2006); arXiv:physics/0602148v2 (2006).
P. D. Mininni, A. Alexakis, and A. Pouquet, Phys. Rev. E, 77, 036306 (2008); arXiv:0709.1939vl [physics.fludyn] (2007).
J. Molinari and D. VoUaro, Monthly Weather Rev., 136, 4355–4372 (2008).
Annick Pouquet, “Figures and videos,” http://www.image.ucar.edu/~pouquet/Figs.html.
A. Radkevich, S. Lovejoy, K. Strawbridge, D. Schertzer, and M. Lilley, Quart. J. Roy. Meteor. Soc, 134, 317–335 (2008).
G. I. Sivashinsky, Phys. D, 17, 243–255 (1985).
A. Libin and G. I. Sivashinsky, Quart. Appl. Math., 48, 611–623 (1990).
A. Libin and G. I. Sivashinsky, Phys. Lett. A, 144, 172–178 (1990).
L. Shtilman and G. Sivashinsky, J. de Physique, 47, 1137–1140 (1986).
V. Yakhot and G. Sivashinsky, Phys. Rev. A, 35, 815–820 (1987).
A. Libin, G. I. Sivashinsky, and E. Levich, Phys. Fluids, 30, 2984–2986 (1987).
L. Polterovich, Private communication (2009).
V. I. Arnold, Proc. Steklov Inst. Math., 258, 3–12 (2007).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 165, No. 2, pp. 350–369, November, 2010.
An erratum to this article can be found at http://dx.doi.org/10.1007/s11232-011-0011-4
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Libin, A.S. Large-scale structures as gradient lines: The case of the trkal flow. Theor Math Phys 165, 1534–1551 (2010). https://doi.org/10.1007/s11232-010-0128-x
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DOI: https://doi.org/10.1007/s11232-010-0128-x