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1.A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers,” C. R. Acad. Sci. URSS 30, 301305 (1941).
2.U. Frisch, Turbulence: The Legacy of A.N. Kolmogorov (Cambridge University Press, 1995).
3. It is also recognized gradually that K41 is flawed in that it is not suitable for higher order statistics. Even through a refinement in 1962 (K62) to depict the intermittency effect, K62 is still flawed. However, K41 works quite well for the second order statistics such as the energy spectrum that is of prime importance in theory and applications. We would be satisfied only with discussion on the energy spectrum and thus not be concerned with the more subtle issue of intermittency.
4.A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics (MIT Press, Cambridge, MA, 1975), Vol. II.
5.G. I. Taylor, “Statistical theory of turbulence. Parts I–IV,” Proc. R. Soc. A 151, 421478 (1935).
6.B. W. Knight and L. Sirovich, “Kolmogorov inertial range for inhomogeneous turbulent flows,” Phys. Rev. Lett. 65, 1356 (1990).
7.R. D. Moser, “Kolmogorov inertial range spectra for inhomogeneous turbulence,” Phys. Fluids 6, 794801 (1994).
8.J. R. Baltzer and R. J. Adrian, “Structure, scaling, and synthesis of proper orthogonal decomposition modes of inhomogeneous turbulence,” Phys. Fluids 23, 015107 (2011).
9.Z. Yoshida and Y. Giga, “Remarks on spectra of operator rot,” Math. Z. 204, 235245 (1990).
10.Z. Yoshida, “Lower bound for the scale reduction in nonlinear dynamo process,” Phys. Rev. Lett. 77, 2722 (1996).
11.N. Ito and Z. Yoshida, “Statistical mechanics of magnetohydrodynamics,” Phys. Rev. E 53, 52005206 (1996).
12.R. Jordan, Z. Yoshida, and N. Ito, “Statistical mechanics of three-dimensional magnetohydrodynamics in a multiply connected domain,” Physica D 114, 251272 (1998).
13.F. Waleffe, “The nature of triad interactions in homogeneous turbulence,” Phys. Fluids A 4, 350363 (1992).
14.F. Waleffe, “Inertial transfers in the helical decomposition,” Phys. Fluids A 5, 677685 (1993).
15.Q. Chen, S. Chen, and G. L. Eyink, “The joint cascade of energy and helicity in three-dimensional turbulence,” Phys. Fluids 15, 361374 (2003).
16.Q. Chen, S. Chen, G. L. Eyink, and D. D. Holm, “Intermittency in the joint cascade of energy and helicity,” Phys. Rev. Lett. 90, 214503 (2003).
17.Q. Chen, S. Chen, G. L. Eyink, and D. D. Holm, “Resonant interactions in rotating homogeneous three-dimensional turbulence,” J. Fluid Mech. 542, 139164 (2005).
18.L. Biferale, S. Musacchio, and F. Toschi, “Inverse energy cascade in three-dimensional isotropic turbulence,” Phys. Rev. Lett. 108, 164501 (2012).
19.H. P. Greenspan, The Theory of Rotating Fluids (Cambridge University Press, 1968).
20.H. E. Moses, “Eigenfunctions of the curl operator, rotationally invariant Helmholtz theorem, and applications to electromagnetic theory and fluid mechanics,” SIAM J. Appl. Math. 21, 114144 (1971).
21.Y.-T. Yang, W.-D. Su, and J.-Z. Wu, “Helical-wave decomposition and applications to channel turbulence with streamwise rotation,” J. Fluid Mech. 662, 91122 (2010).
22.Y.-T. Yang and J.-Z. Wu, “Channel turbulence with spanwise rotation studied using helical wave decomposition,” J. Fluid Mech. 692, 137152 (2012).
23.X. Shan, D. C. Montgomery, and H. Chen, “Nonlinear magnetohydrodynamics by Galerkin-method computation,” Phys. Rev. A 44, 68006818 (1991).
24.P. D. Mininni and D. C. Montgomery, “Magnetohydrodynamic activity inside a sphere,” Phys. Fluids 18, 116602 (2006).
25.Z.-J. Liao and W.-D. Su, “A Galerkin spectral method based on helical-wave decomposition for the incompressible Navier–Stokes equations,” Int. J. Numer. Methods Fluids (published online 2015).
26.S. Chandrasekhar and P. C. Kendall, “On the force-free magnetic fields,” Astrophys. J. 126, 457460 (1957).
27.J. Cantarella, D. DeTurck, H. Gluck, and M. Teytel, “The spectrum of the curl operator on spherically symmetric domains,” Phys. Plasma 7, 27662775 (2000).
28.A. M. Obukhov, “On the distribution of energy in the spectrum of turbulent flow,” Izv. Akad. Nauk. SSSR, Ser. Geogr. Geofiz. 5, 453466 (1941).
29.T. Gotoh, D. Fukayama, and T. Nakano, “Velocity field statistics in homogeneous steady turbulence obtained using a high-resolution direct numerical simulation,” Phys. Fluids 14, 10651081 (2002).
30. Since a perfect plateau does not exist in a plotted compensated spectrum, to discern a definite inertial range and thus to determine the constant CK with no dispute are often difficult. The inertial range in our case is taken as 0.038 ≤ λη ≤ 0.062, obviously smaller than that in Ref. 29. The reason is the lowest wavenumber in the sphere is about 4.49 times high than that in the periodic box. If the same fitting range was taken for the case in Ref. 29, CK would be 1.81 rather than 1.64.
31.J. Schumacher, “Sub-Kolmogorov-scale fluctuations in fluid turbulence,” Europhys. Lett. 80, 54001 (2007).
32.S. G. Saddoughi and S. V. Veeravalli, “Local isotropy in turbulent boundary layers at high Reynolds number,” J. Fluid Mech. 268, 333372 (1994).
33.T. Ishihara, Y. Kaneda, M. Yokokawa, K. Itakura, and A. Uno, “Energy spectrum in the near dissipation range of high resolution direct numerical simulation of turbulence,” J. Phys. Soc. Jpn. 74, 14641471 (2005).
34.R. H. Kraichnan, “The structure of isotropic turbulence at very high Reynolds numbers,” J. Fluid Mech. 5, 497543 (1959).
35.C. Foias, O. Manley, and L. Sirovich, “Empirical and stokes eigenfunctions and the far-dissipative turbulent spectrum,” Phys. Fluids A 2, 464467 (1990).
36. Recalling that ψk vanishes at the poles of the sphere, there is no singularities here. It is important to note that, for most polar angles, varies only slowly.
37.L. Sirovich and B. W. Knight, “The eigenfunction problem in higher dimensions: Asymptotic theory,” Proc. Natl. Acad. Sci. U. S. A 82, 8275 (1985).
38.B. W. Knight and L. Sirovich, “The eigenfunction problem in higher dimensions: Exact results,” Proc. Natl. Acad. Sci. U. S. A 83, 527 (1986).
39. Evidently, the energy spectrum can be altered due to the presence of strong rotation, magnetic fields, buoyancy, etc. See, e.g., P. D. Mininni and A. Pouquet, Phys. Fluids 22, 035105 (2010). Here, we intend to exclude these body forces with the emphasis on the underlying roles of geometry of the domain and the boundary conditions.

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We relate the justification of Kolmogorov’s hypotheses on the local isotropy and small-scale universality in real turbulent flows to an observed universality of basis independence for the global energy spectrum and energy flux of small-scale turbulence. To readily examine the small-scale universality, an approach is suggested that investigates the global energy spectrum in a general spectral space for which the nonlinear interscale interaction may not be Fourier-triadic. Specific verifications are performed based on direct numerical simulations of turbulence in a spherical geometry and reexaminations of several existing results for turbulent channel flows.


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