Mean-field equations for weakly nonlinear two-scale perturbations of forced hydromagnetic convection in a rotating layer

Geophysical and Astrophysical Fluid Dynamics

Volumn 102, Issue 5, 2008, Pages 489-540

  Zheligovsky, Vladislav ^a  

^a INSTITUTE OF EARTHQUAKE PREDICTION THEORY AND MATHEMATICAL GEOPHYSICS   (Russian Federation)

Author keywords

Effect; Amplitude equation; Eddy diffusivity; Homogenisation; Hydromagnetic convection; Mean field equations; Weakly nonlinear stability

Indexed keywords

CONTROL NONLINEARITIES; CONTROL THEORY; EQUATIONS OF STATE; LINEAR EQUATIONS; MATHEMATICAL OPERATORS; PERTURBATION TECHNIQUES; ROTATION; STANDARDS;

AMPLITUDE EQUATION; AUXILIARY PROBLEMS; CUBIC NON LINEARITIES; EDDY DIFFUSIVITY; EDDY-ADVECTION; HOMOGENISATION; HYDROMAGNETIC CONVECTION; LINEAR OPERATORS; MEAN-FIELD EQUATIONS; NON LOCAL; PSEUDO-DIFFERENTIAL OPERATORS; SECOND ORDERS; SPATIAL SYMMETRIES; STANDARD EQUATIONS; SYMMETRY-BREAKING BIFURCATIONS; TEMPORAL SCALING; WEAKLY NONLINEAR STABILITY;

NONLINEAR EQUATIONS;

AMPLITUDE; BIFURCATION; EDDY COVARIANCE; FLUID DYNAMICS; HOMOGENEITY; HOMOGENEOUS MEDIUM; NONLINEARITY; SPATIOTEMPORAL ANALYSIS; THERMAL CONVECTION;

EID: 52749092568     PISSN: 03091929     EISSN: 10290419     Source Type: Journal    
DOI: 10.1080/03091920802137573     Document Type: Article

References (69)

1. Anufriev, AP and Braginsky, SI (1975) Influence of irregularities of the boundary of the Earth's core on the velocity of the liquid and on the magnetic field. Geomagn. Aeron., 15, pp. 754-757.
2. Anufriev, AP and Braginsky, SI (1977) Influence of irregularities of the boundary of the Earth's core on the velocity of the liquid and on the magnetic field. II. Geomagn. Aeron., 17, pp. 78-82.
3. Anufriev, AP and Braginsky, SI (1977) Effect of irregularities of the boundary of the Earth's core on the speed of the fluid and on the magnetic field. III. Geomagn. Aeron., 17, pp. 492-496.
4. Baptista, M., Gama, SMA and Zheligovsky, V. (2007) Eddy diffusivity in convective hydromagnetic systems. Eur. Phys. J. B, 60, pp. 337-352.
5. Bassom, AP and Zhang, K. (1994) Strongly non-linear convection cells in a rapidly rotating fluid layer. Geophys. Astrophys. Fluid Dyn., 76, pp. 223-238.
6. Bensoussan, A., Lions, J-L and Papanicolaou, G. (1978) Asymptotic analysis for periodic structures, p. 700. Amsterdam, North Holland
7. Biferale, L., Crisanti, A., Vergassola, M. and Vulpiani, A. (1995) Eddy viscosity in scalar transport. Phys. Fluids, 7:11, pp. 2725-2734.
8. Bodenschatz, E., Pesch, W. and Ahlers, G. (2000) Recent developments in Rayleigh-Benard convection. Ann. Rev. Fluid Mech., 32, pp. 709-778.
9. Bowin, C. (1986) Topography at the core-mantle boundary. Geophys. Res. Lett., 13, pp. 1513-1516.
10. Braginsky, SI (1964) Self excitation of a magnetic field during the motion of a highly conducting fluid. JETP, 47, pp. 1084-1098.
11. Braginsky, SI (1964) Theory of the hydromagnetic dynamo. JETP, 47, pp. 2178-2193.
12. Braginsky, SI (1964) Kinematic models of the Earth's hydromagnetic dynamo. Geomagn. Aeron., 4:4, pp. 732-747.
13. Braginsky, SI (1964) Magnetohydrodynamics of the Earth's core. Geomagn. Aeron., 4:5, pp. 898-916.
14. Braginsky, SI (1967) Magnetic waves in the Earth's core. Geomagn. Aeron., 7:6, pp. 1050-1060.
15. Braginsky, SI (1975) An almost axially symmetric model of the hydromagnetic dynamo of the Earth, I. Geomagn. Aeron., 15:1, pp. 149-156.
16. Cioranescu, D. and Donato, P. (1999) An introduction to homogenization, p. 262. Oxford Univ. Press
17. Cross, MC and Newell, AC (1984) Convection patterns in large aspect ratio systems. Physica D, 10, pp. 299-328.
18. Dubrulle, B. and Frisch, U. (1991) Eddy viscosity of parity-invariant flow. Phys. Rev. A, 43, pp. 5355-5364.
19. E, W and Shu, C-W (1993) Effective equations and the inverse cascade theory for Kolmogorov flows. Phys. Fluids A, 5, pp. 998-1010.
20. Frisch, U. (1995) Turbulence. The Legacy of A.N. Kolmogorov, Cambridge Univ. Press, New York
21. Frisch, U., Legras, B. and Villone, B. (1996) Large-scale Kolmogorov flow on the beta-plane and resonant wave interactions. Physica D, 94, pp. 36-56.
22. Gama, S. and Chaves, M. (2000) Time evolution of the eddy viscosity in two-dimensional Navier-Stokes flow. Phys. Rev. Lett., 61, pp. 2118-2120.
23. Gama, S., Vergassola, M. and Frisch, U. (1994) Negative eddy viscosity in isotropically forced two-dimensional flow: Linear and non-linear dynamics. J. Fluid Mech., 260, pp. 95-126.
24. Guckenheimer, J. and Holmes, P. (1990) Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, p. 453. Springer-Verlag, New York
25. Jikov, VV, Kozlov, SM and Oleinik, OA (1994) Homogenization of differential operators and integral functionals, p. 570. Springer-Verlag, Berlin
26. Julien, K. and Knobloch, E. (1997) Fully non-linear oscillatory convection in a rotating layer. Phys. Fluids, 9, pp. 1906-1913.
27. Julien, K. and Knobloch, E. (1998) Strongly non-linear convection cells in a rapidly rotating layer: The tilted f-plane. J. Fluid Mech., 360, pp. 141-178.
28. Julien, K., Knobloch, E. and Werne, J. (1998) A new class of equations for rotationally constrained flows. Theoret. Comput. Fluid Dyn., 11, pp. 251-261.
29. Julien, K. and Knobloch, E. (1999) Fully non-linear three-dimensional convection in a rapidly rotating layer. Phys. Fluids, 11:6, pp. 1469-1483.
30. Julien, K., Knobloch, E. and Tobias, SM (1999) Strongly non-linear magnetoconvection in three dimensions. Physica D, 128, pp. 105-129.
31. Julien, K., Knobloch, E. and Tobias, SM (2000) Strongly non-linear magnetoconvection in the presence of oblique fields. J. Fluid Mech., 410, pp. 285-322.
32. Julien, K., Knobloch, E. and Tobias, SM (2003) Highly supercritical convection in strong magnetic fields. Advances in non-linear dynamos, pp. 195-223. Taylor and Francis, London
33. Lanotte, A., Noullez, A., Vergassola, M. and Wirth, A. (1999) Large-scale dynamo by negative magnetic eddy diffusivities. Geophys. Astrophys. Fluid Dyn., 91, pp. 131-146.
34. Liusternik, LA and Sobolev, VJ (1961) Elements of functional analysis, Frederick Ungar Publ. Co, New York
35. Majda, AJ and Kramer, PR (1999) Simplified models for turbulent diffusion: Theory, numerical modelling, and physical phenomena. Phys. Rep., 314, pp. 237-574.
36. Matthews, PC (1999) Asymptotic solutions for non-linear magnetoconvection. J. Fluid Mech., 387, pp. 397-409.
37. Merrill, RT, McEllhiny, MW and McFadden, PhL (1996) The magnetic field of the Earth. Paleomagnetism, the core and the deep mantle, p. 527. Academic Press, San Diego
38. Murakami, Y., Murakami, M. and Gotoh, K. (1995) Three-dimensional negative eddy viscosity effect on the onset of instability in some planar flows. Phys. Rev. E, 51:5, pp. 5128-5131.
39. Nepomnyashchy, AA (1976) On the stability of the secondary flow of a viscous fluid in an infinite domain. Appl. Math. Mech., 40, pp. 886-891.
40. Newell, AC (1983) Two-dimensional convection patterns in large aspect ratio systems. Nonlinear Partial Differential Equations in Applied Science, pp. 202-231. Amsterdam, North-Holland
41. Newell, AC, Passot, T. and Lega, J. (1993) Order parameter equations for patterns. Ann. Rev. Fluid Mech., 25, pp. 399-453.
42. Newell, AC, Passot, T., Bowman, C., Ercolani, N. and Indik, R. (1996) Defects are weak and self-dual solutions of the Cross-Newell phase diffusion equation for natural patterns. Physica D, 97, pp. 185-205.
43. Newell, AC, Passot, T. and Souli, M. (1990) Convection at finite Rayleigh numbers in large-aspect-ratio containers. Phys. Rev. Lett., 64:20, pp. 2378-2381.
44. Newell, AC, Passot, T. and Souli, M. (1990) The phase diffusion and mean drift equations for convection at finite Rayleigh numbers in large containers. J. Fluid Mech., 220, pp. 187-552.
45. Novikov, A. (2004) Eddy viscosity of cellular flows by upscaling. J. Comp. Phys., 195, pp. 341-354.
46. Novikov, A. and Papanicolaou, G. (2001) Eddy viscosity of cellular flows. J. Fluid Mech., 446, pp. 173-198.
47. Oleinik, OA, Shamaev, AS and Yosifian, GA (1992) Mathematical problems in elasticity and homogenization, p. 398. Elsevier Science Publishers, Amsterdam
48. Podvigina, OM (2006) Magnetic field generation by convective flows in a plane layer. Eur. Phys. J. B, 50, pp. 639-652.
49. Polyanin, AD (2001) A handbook of linear equations of mathematical physics, p. 576. Fizmatlit, Moscow
50. Ponty, Y., Passot, T. and Sulem, PL (1997) Pattern dynamics in rotating convection at finite Prandtl number. Phys. Rev. E, 56:4, pp. 4162-4178.
51. Ponty, Y., Gilbert, AD and Soward, AM (2001) Kinematic dynamo action in flows driven by shear and convection. J. Fluid Mech., 435, pp. 261-287.
52. Ponty, Y., Gilbert, AD and Soward, AM (2001) Dynamo action due to Ekman layer instability. Dynamo and dynamics, a mathematical challenge, pp. 75-82. Kluwer, Boston
53. Ponty, Y., Gilbert, AD and Soward, AM (2003) The onset of thermal convection in Ekman-Couette shear flow with oblique rotation. J. Fluid Mech., 487, pp. 91-123.
54. Rotvig, J. and Jones, CA (2002) Rotating convection driven dynamos at low Ekman number. Phys. Rev. E, 66:56308, p. 15.
55. She, ZS (1987) Metastability and vortex pairing in the Kolmogorov flow. Phys. Lett. A, 124, pp. 161-164.
56. Sivashinsky, GI and Yakhot, V. (1985) Negative viscosity effect in large-scale flows. Phys. Fluids, 28, pp. 1040-1042.
57. Sivashinsky, GI and Frenkel, AL (1992) On negative eddy viscosity under conditions of isotropy. Phys. Fluids A, 4:8, pp. 1608-1610.
58. Soward, AM (1972) A kinematic theory of large magnetic Reynolds number dynamos. Phil. Trans. Roy. Soc. A, 272, pp. 431-462.
59. Soward, AM (1974) A convection driven dynamo I. The weak field case. Phil. Trans. Roy. Soc. A, 275, pp. 611-651.
60. Starr, VP (1968) Physics of negative viscosity phenomena, McGraw-Hill, New York
61. Vergassola, M. and Avellaneda, M. (1997) Scalar transport in compressible flow. Physica D, 106, pp. 148-166.
62. Wirth, A. (1994) Complex eddy-viscosity: A three-dimensional effect. Physica D, 76, pp. 312-317.
63. Wirth, A., Gama, S. and Frisch, U. (1995) Eddy viscosity of three-dimensional flow. J. Fluid Mech., 288, pp. 249-264.
64. Zheligovsky, VA (2003) On the linear stability of spatially periodic steady magnetohydrodynamic systems with respect to long-period perturbations. Izvestiya, Physics of the Solid Earth, 39:5, pp. 409-418.
65. Zheligovsky, VA (2005) Convective plan-form two-scale dynamos in a plane layer. Geophys. Astrophys. Fluid Dyn., 99, pp. 151-175.
66. Zheligovsky, VA (2006) Weakly non-linear stability of centrally symmetric magnetohydrodynamic systems to perturbations involving large-scales. Izvestiya, Physics of the Solid Earth, 42:3, pp. 244-253.
67. Zheligovsky, VA (2006) Weakly non-linear stability to large-scale perturbations in convective magnetohydrodynamic systems without the α-effect. Izvestiya, Physics of the Solid Earth, 42:12, pp. 1051-1068.
68. Zheligovsky, VA and Podvigina, OM (2003) Generation of multiscale magnetic field by parity-invariant time-periodic flows. Geophys. Astrophys. Fluid Dyn., 97, pp. 225-248.
69. Zheligovsky, VA, Podvigina, OM and Frisch, U. (2001) Dynamo effect in parity-invariant flow with large and moderate separation of scales. Geophys. Astrophys. Fluid Dyn., 95, pp. 227-268.