On the double cascades of energy and enstrophy in two-dimensional turbulence. Part 1. Theoretical formulation

Discrete and Continuous Dynamical Systems - Series B

Volumn 5, Issue 1, 2005, Pages 79-102

  Gkioulekas, Eleftherios ^a   Tung, Ka Kit ^a  

^a UNIVERSITY OF WASHINGTON   (United States)

Author keywords

2d turbulence; Energy; Enstrophy; QG turbulence

Indexed keywords

EID: 15844416308     PISSN: 15313492     EISSN: None     Source Type: Journal    
DOI: 10.3934/dcdsb.2005.5.79     Document Type: Article

References (46)

1. G.K. Batchelor, Kolmogorov's theory of locally isotropic turbulence, Proc. Camb. Phil. Soc., 43 (1947), 533-559.
2. _, The theory of homogeneous turbulence, Cambridge University Press, Cambridge, 1953.
3. _, Computation of the energy spectrum in homogeneous, two dimensional turbulence, Phys. Fluids Suppl., II 12 (1969), 233-239.
4. V. Belinicher, V.S. L'vov, A. Pomyalov, and I. Procaccia, Computing the scaling exponents in fluid turbulence from first principles: Demonstration of multiscaling, J. Stat. Phys., 93 (1998), 797.
5. V. Belinicher, V.S. L'vov, and I. Procaccia, A new approach to computing the scaling exponents in fluid turbulence from first principles, Physica A, 254 (1998), 215-230.
6. J.C. Bowman, On inertial-range scaling laws, J. Fluid. Mech., 306 (1996), 167-181.
7. A. Einstein, Method for the determination of the statistical values of observations concerning quantities subject to irregular fluctuations, IEEE ASSP Magazine, 4 (4) (1987), 6.
8. G.L Eyink, Exact results on stationary turbulence in two dimensions: Consequences of vorticity conservation, Physica D, 91 (1996), 97-142.
9. G. Falkovich and V. Lebedev, Non-local vorticity cascade in two dimensions, Phys. Rev. E, 423 (1994), 1800-1803.
10. _, Universal direct cascade in two dimensional turbulence, Phys. Rev. E, 50 (1994), 3883-3899.
11. U. Frisch, Prom global scaling, a la Kolmogorov, to local multifractal scaling in fully developed turbulence, Proc. R. Soc. Lond. A, 434 (1991), 89-99.
12. _, Turbulence: The legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995.
13. W. A. Gardner, Introduction to Einstein's contribution to time-series analysis, IEEE ASSP Magazine, 4 (4) (1987), 4-5.
14. M.J. Giles, Anomalous scaling in homogeneous isotropic turbulence, J. Phys. A, 34 (2001), 4389-4435.
15. E. Gkioulekas and K.K. Tung, On the double cascades of energy and enstrophy in two dimensional turbulence. Part 2. Approach to the KLB limit and interpretation of experimental evidence, Discrete and Continuous Dynamical Systems Series B, 5 (2005), 103-124.
16. A. Khinchin, Correlation theory of stationary stochastic processes, Math Ann., 109 (1934), 604-615.
17. A.N. Kolmogorov, Dissipation of energy in the locally isotropic turbulence, Dokl. Akad. Nauk. SSSR, 32 (1941), 16-18.
18. _, The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers, Dokl. Akad. Nauk. SSSR, 30 (1941), 301-305.
19. R.H. Kraichnan, Inertial ranges in two dimensional turbulence, Phys. Fluids, 10 (1967), 1417-1423.
20. _, An almost Markovian Galilean invariant turbulence model, J. Fluid. Mech., 47 (1971), 513-524.
21. _, Inertial range transfer in two and three dimensional turbulence, J. Fluid. Mech., 47 1971), 525-535.
22. C.E. Leith, Diffusion approximation for two dimensional turbulence, Phys. Fluids, 11 (1968), 671-673.
23. D.K. Lilly, Two dimensional turbulence generated by energy sources at two scales, J. Atmos. Sci., 46 (1989), 2026-2030.
24. E. Lindborg and K. Alvelius, The kinetic energy spectrum of two dimensional enstrophy turbulence cascade, Phys. Fluids, 12 (2000), 945-947.
25. V.S. L'vov and I. Procaccia, Cornerstones of a theory of anomalous scaling in turbulence, Physica Scripta, T64 (1996), 131-135.
26. _, Exact resummations in the theory of hydrodynamic turbulence. III. Scenarios for anomalous scaling and intermittency, Phys. Rev. E, 53 (1996), 3468-3490.
27. _, Fusion rules in turbulent systems with flux equilibrium, Phys. Rev. Letters, 76 (1996), 2898-2901.
28. _, Towards a nonperturbative theory of hydrodynamic turbulence: fusion rules, exact bridge relations, and anomalous viscous scaling functions, Phys. Rev. E, 54 (1996), 6268-6284.
29. _, Viscous lengths in hydrodynamic turbulence are anomalous scaling functions, Phys. Rev. Letters, 77 (1996), 3541-3544.
30. _, Computing the scaling exponents in fluid turbulence from first principles: The formal setup, Physica A, 257 (1998), 165-196.
31. _, Analytic calculation of the anomalous exponents in turbulence: using the fusion rules to flush out a small parameter, Phys. Rev. E, 62 (2000), 8037-8057.
32. H.K. Moffatt, Advances in Turbulence, (G.Comte-Bellot and J. Mathieu, eds.), Springer-Verlag, Berlin, 1986, p. 284.
33. S. Panchev, Random functions and turbulence, Pergamon press, New York, 1971.
34. J. Pedlosky, Geostrophic turbulence, Springer-Verlag, New York, 1979.
35. A.M. Polyakov, The theory of turbulence in two dimensions, Nuclear Physics B, 396 (1993), 367-385.
36. A. Pouquet, M. Lesieur, J.C. Andre, and C. Basdevant, Evolution of high reynolds number two-dimensional turbulence, J. Fluid. Mech., 75 (1975), 305-319.
37. M. Rivera and X.L. Wu, External dissipation in driven two dimensional turbulence, Phys. Rev. Letters, 85 (2000), 976-979.
38. P.G. Saffman, On the spectrum and decay of random two dimensional vorticity distributions at large reynolds number, Stud. Appl. Math., 50 (1971), 377.
39. -5/3 energy spectrum of the atmospheric turbulence: quasi-geostrophic two level model simulation, J. Atmos. Sci., 61 (2004), 937-942
40. G.I. Taylor, The spectrum of turbulence, Proc. Roy. Soc. A, 164 (1938), 476-490.
41. C.V. Tran and J.C. Bowman, Robustness of the inverse cascade in two dimensional turbulence, Phys. Rev. E, 69 (2004), 036303
42. _, On the dual cascade in two dimensional turbulence, Physica D, 176 (2003), 242-255.
43. C.V. Tran and T.G. Shepherd, Constraints on the spectral distribution of energy and enstrophy dissipation in forced two dimensional turbulence, Physica D, 165 (2002), 199-212.
44. K.K. Tung and W.W. Orlando, On the differences between 2d and qg turbulence, Discrete and Continuous Dynamical Systems Series B, 3 (2003), 145-162.
45. N. Wiener, Generalized harmonic analysis, Acta Math., 55 (1930), 117-258.
46. A.M. Yaglom, Einstein's 1914 paper on the theory of irregularly fluctuating series of observations, IEEE ASSP Magazine, 4 (4) (1987), 7-11.