Reduction in the Research of Large-Scale Dynamics with Allowance of the Effects of Magnetic Field Diffusion
74 33
Keywords:
Ideal fluid dynamic problems, magnetohydrodynamic equations, reduction of vector equations to scalar equations, analytical method, diffusions of magnetic fieldAbstract
A system of nonlinear partial differential equations is considered that models
perturbations in a layer of an ideal electrically conducting rotating fluid bounded by spatially and
temporally varying surfaces with allowance for inertial forces and diffusions of magnetic field. The
system is reduced to a scalar equation. The solvability of initial boundary value problems arising
in the theory of waves in conducting rotating fluids can be established by analyzing this equation.
Solutions to the scalar equation are constructed that describe small-amplitude wave propagation in
an infinite horizontal layer and a long narrow channel.
References
Smith, T.F., Waterman, M.S.Alfven, H. and Falthammer, C.-G.: Cosmical Electrodynamics. Oxford Univ.
Press. (1963)
2. F. H. Busse: A Model of the Geodynamo. In: Geophys. J. R. Asron. Soc., vol. 42, pp. 437–459. (1975)
3. F. H. Busse: Generation of Planetary Magnetism by Convection. In: Phys. Earth Planet. Inter., vol. 12, pp.
350–358. (1976)
4. Zhang, K.-K. and Busse, F. H.: Finite Amplitude Convection and Magnetic Field Generation in a Rotating
Spherical Shell. In: Geophys. Astrophys. Fluid Dyn., vol. 44, pp. 33–54. ( 1988)
5. Zhang, K.-K. and Busse, F. H.: Convection Driven Magnetohydrodynamic Dynamos. In Rotating Spherical
Shell. In: Geophys. Astrophys. Fluid Dyn., vol. 49, pp. 97–116. (1989)
6. Zhang, K.-K. and Busse, F. H.: Generation of Magnetic Fields by Convection. in a Rotating Spherical Fluid
Shell of Infinite Prandtl Number. In: Phys. Earth Planet. Inter., vol. 59, pp. 208–222. (1990)
7. I. A. Eltayeb: Hydromagnetic Convection in a Rapidly Rotating Fluid Layer. In Proc. R. Soc., London, vol.
A 326, pp. 229–254. (1972)
8. I. A. Eltayeb: Overstable Hydromagnetic Convection in a Rapidly Rotating Fluid Layer. In: J. Fluid Mech.,
vol. 71, pp. 161–179. (1975)
9. Kholodova, S. E. and Peregudin, S. I.: Modelling and Analysis of Streams ana Waves in Liquid and Loosy
Mediums [in Russian. Saint Petesburg State University. Saint Petersburg. (2009)
10. Peregudin, S. I. and Kholodova, S. E.: Dynamics of a Rotating Layer of an Ideal Electrically Conducting
Incompressible Inhomogeneous Fluid in an Equatorial Region. In: Computational Mathematics and
Mathematical Physics, vol. 50. pp. 1871–1885. (2010)
11. Peregudin, S. I. and Kholodova, S. E.: Specific features of propagation of unsteady waves in a rotating spherical
layer of an ideal incompressible stratified electroconducting fluid in the equatorial latitude belt. In: Journal
of Applied Mechanics and Technical Physics, vol. 52. pp. 193–199. (2011)
12. Peregudin, S. I. and Kholodova, S. E.: Waves in a rotating layer of an ideal electrically conducting
incompressible fluid with allowance effects of diffusion of magnetic field. In: 2014 20th International Workshop
on Beam Dynamics and Optimization, BDO 2014; St. Petersburg; Russian Federation; 30 June 2014 through
4 July 2014. pp. 127–129. (2014)
Press. (1963)
2. F. H. Busse: A Model of the Geodynamo. In: Geophys. J. R. Asron. Soc., vol. 42, pp. 437–459. (1975)
3. F. H. Busse: Generation of Planetary Magnetism by Convection. In: Phys. Earth Planet. Inter., vol. 12, pp.
350–358. (1976)
4. Zhang, K.-K. and Busse, F. H.: Finite Amplitude Convection and Magnetic Field Generation in a Rotating
Spherical Shell. In: Geophys. Astrophys. Fluid Dyn., vol. 44, pp. 33–54. ( 1988)
5. Zhang, K.-K. and Busse, F. H.: Convection Driven Magnetohydrodynamic Dynamos. In Rotating Spherical
Shell. In: Geophys. Astrophys. Fluid Dyn., vol. 49, pp. 97–116. (1989)
6. Zhang, K.-K. and Busse, F. H.: Generation of Magnetic Fields by Convection. in a Rotating Spherical Fluid
Shell of Infinite Prandtl Number. In: Phys. Earth Planet. Inter., vol. 59, pp. 208–222. (1990)
7. I. A. Eltayeb: Hydromagnetic Convection in a Rapidly Rotating Fluid Layer. In Proc. R. Soc., London, vol.
A 326, pp. 229–254. (1972)
8. I. A. Eltayeb: Overstable Hydromagnetic Convection in a Rapidly Rotating Fluid Layer. In: J. Fluid Mech.,
vol. 71, pp. 161–179. (1975)
9. Kholodova, S. E. and Peregudin, S. I.: Modelling and Analysis of Streams ana Waves in Liquid and Loosy
Mediums [in Russian. Saint Petesburg State University. Saint Petersburg. (2009)
10. Peregudin, S. I. and Kholodova, S. E.: Dynamics of a Rotating Layer of an Ideal Electrically Conducting
Incompressible Inhomogeneous Fluid in an Equatorial Region. In: Computational Mathematics and
Mathematical Physics, vol. 50. pp. 1871–1885. (2010)
11. Peregudin, S. I. and Kholodova, S. E.: Specific features of propagation of unsteady waves in a rotating spherical
layer of an ideal incompressible stratified electroconducting fluid in the equatorial latitude belt. In: Journal
of Applied Mechanics and Technical Physics, vol. 52. pp. 193–199. (2011)
12. Peregudin, S. I. and Kholodova, S. E.: Waves in a rotating layer of an ideal electrically conducting
incompressible fluid with allowance effects of diffusion of magnetic field. In: 2014 20th International Workshop
on Beam Dynamics and Optimization, BDO 2014; St. Petersburg; Russian Federation; 30 June 2014 through
4 July 2014. pp. 127–129. (2014)
Downloads
How to Cite
Peregudin, S., & Kholodova, S. (2018). Reduction in the Research of Large-Scale Dynamics with Allowance of the Effects of Magnetic Field Diffusion. Journal of Mathematics, Mechanics and Computer Science, 86(3), 51–57. Retrieved from https://bm.kaznu.kz/index.php/kaznu/article/view/411
Issue
Section
Mathematical modeling of technological processes
