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Nonlinear transfer and spectral distribution of energy in \(\alpha\) turbulence. (English) Zbl 1054.76529
Summary: Two-dimensional turbulence governed by the so-called \(\alpha\) turbulence equations, which include the surface quasi-geostrophic equation \((\alpha=1)\), the Navier–Stokes system \((\alpha=2)\), and the governing equation for a shallow flow on a rotating domain driven by a uniform internal heating \((\alpha=3)\), is studied here in both the unbounded and doubly periodic domains. This family of equations conserves two inviscid invariants (energy and enstrophy in the Navier–Stokes case), the dynamics of which are believed to undergo a dual cascade. It is shown that an inverse cascade can exist in the absence of a direct cascade and that the latter is possible only when the inverse transfer rate of the inverse-cascading quantity approaches its own injection rate. Constraints on the spectral exponents in the wavenumber ranges lower and higher than the injection range are derived. For Navier–Stokes turbulence with moderate Reynolds numbers, the realization of an inverse energy cascade in the complete absence of a direct enstrophy cascade is confirmed by numerical simulations.

76F02 Fundamentals of turbulence
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[1] Baroud, C.N.; Plapp, B.B.; She, Z.-S.; Swinney, H.L., Anomalous self-similarity in a turbulent rapidly rotating fluid, Phys. rev. lett., 88, 114501, (2002)
[2] Batchelor, G.K., Computation of the energy spectrum in homogeneous two-dimensional turbulence, Phys. fluids, 12, II, 233-239, (1969) · Zbl 0217.25801
[3] Blumen, W., Uniform potential vorticity flow. part I. theory of wave interactions and two-dimensional turbulence, J. atmos. sci., 35, 774-783, (1978)
[4] Boffetta, G.; Celani, A.; Vergassola, M., Inverse energy cascade in two-dimensional turbulence: deviations from Gaussian behaviour, Phys. rev. E, 61, R29-R32, (2000)
[5] Borue, V., Inverse energy cascade in stationary two-dimensional homogeneous turbulence, Phys. rev. lett., 72, 1475-1478, (1994)
[6] Charney, J.G., On the scale of atmospheric motions, Geof. publ., 17, 3-17, (1948)
[7] Constantin, P., Geometric statistics in turbulence, SIAM rev., 36, 73-98, (1994) · Zbl 0803.35106
[8] Constantin, P., Energy spectrum of quasigeostrophic turbulence, Phys. rev. lett., 89, 184501, (2002)
[9] Constantin, P.; Majda, A.J.; Tabak, E.G., Singular front formation in a model for quasigeostrophic flow, Phys. fluids, 6, 9-11, (1994) · Zbl 0826.76014
[10] Fjørtoft, R., On the changes in spectral distribution of kinetic energy for twodimensional nondivergent flow, Tellus, 5, 225-230, (1953)
[11] U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov, Cambridge University Press, Cambridge, 1995. · Zbl 0832.76001
[12] Frisch, U.; Sulem, P.L., Numerical simulation of the inverse cascade in two-dimensional turbulence, Phys. fluids, 27, 1921-1923, (1984) · Zbl 0546.76082
[13] Hasegawa, A.; Mima, K., Pseudo-three-dimensional turbulence in magnetized nonuniform plasma, Phys. fluids, 21, 87-92, (1978) · Zbl 0374.76046
[14] Hasegawa, A.; Maclennan, C.G.; Kodama, Y., Nonlinear behavior and turbulence spectra of drift waves and Rossby waves, Phys. fluids, 22, 2122-2129, (1979) · Zbl 0424.76095
[15] Held, I.M.; Pierrehumbert, R.T.; Garner, S.T.; Swanson, K.L., Surface quasi-geostrophic dynamics, J. fluid mech., 282, 1-20, (1995) · Zbl 0832.76012
[16] Kraichnan, R.H., Inertial ranges in two-dimensional turbulence, Phys. fluids, 10, 1417-1423, (1967)
[17] Kraichnan, R.H., Inertial-range transfer in two- and three-dimensional turbulence, J. fluid mech., 47, 525-535, (1971) · Zbl 0224.76053
[18] Leith, C.E., Diffusion approximation for two-dimensional turbulence, Phys. fluids, 11, 671-673, (1968) · Zbl 0182.29203
[19] Maltrud, M.E.; Vallis, G.K., Energy spectra and coherent structures in forced two-dimensional and beta-plane turbulence, J. fluid mech., 228, 321-342, (1991)
[20] H.K. Moffatt, Advances in Turbulence, Springer-Verlag, Berlin, 1986.
[21] Ohkitani, K.; Yamada, M., Inviscid and inviscid-limit behavior of a surface quasigeostrophic flow, Phys. fluid, 9, 876-882, (1997) · Zbl 1185.76841
[22] J. Pedlosky, Geophysical Fluid Dynamics, 2nd ed., Springer-Verlag, New York, 1987. · Zbl 0713.76005
[23] Pierrehumbert, R.T.; Held, I.M.; Swanson, K.L., Spectra of local and nonlocal two-dimensional turbulence, Chaos solitons fract., 4, 1111-1116, (1994) · Zbl 0823.76034
[24] Saffman, P.G., On the spectrum and decay of random two-dimensional vorticity distribution of large Reynolds number, Stud. appl. math., 50, 377-383, (1971) · Zbl 0237.76029
[25] Schorghofer, N., Universality of probability distributions among two-dimensional turbulence flows, Phys. rev. E, 61, 6568-6571, (2000)
[26] Schorghofer, N., Energy spectra of steady two-dimensional turbulence flows, Phys. rev. E, 61, 6572-6577, (2000)
[27] Shepherd, T.G., Rossby waves and two-dimensional turbulence in a large-scale zonal jet, J. fluid mech., 183, 467-509, (1987) · Zbl 0639.76068
[28] Smith, K.S.; Boccaletti, G.; Henning, C.C.; Marinov, I.; Tam, C.Y.; Held, I.M.; Vallis, G.K., Turbulent diffusion in the geostrophic inverse cascade, J. fluid mech., 469, 13-48, (2002) · Zbl 1152.76402
[29] Sulem, P.L.; Frisch, U., Bounds on energy flux for finite energy turbulence, J. fluid mech., 72, 417-423, (1975) · Zbl 0346.76041
[30] Tran, C.V.; Bowman, J.C., On the dual cascade in two-dimensional turbulence, Physica D, 176, 242-255, (2003) · Zbl 1086.76518
[31] Tran, C.V.; Bowman, J.C., Energy budgets in charney – hasegawa – mima and surface quasigeostrophic turbulence, Phys. rev. E, 68, 036304, (2003)
[32] C.V. Tran, J.C. Bowman, Robustness of the inverse cascade in two-dimensional turbulence, Phys. Rev. E (2003), accepted.
[33] Tran, C.V.; Shepherd, T.G., Constraints on the spectral distribution of energy and enstrophy dissipation in forced two-dimensional turbulence, Physica D, 165, 199-212, (2002) · Zbl 1065.76124
[34] Tung, K.K.; Orlando, W.W., On the differences between 2D and QG turbulence, Discrete contin. dyn. syst. ser. B, 3, 2, 145-162, (2003) · Zbl 1388.86015
[35] Weinstein, S.; Olson, P.; Yuen, D., Time-dependent large aspect-ratio thermal-convection in the earths mantle, Geophys. astrophys. fluid dyn., 47, 157-197, (1989)
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