Instabilities in a Non-Uniformly Rotating Medium with Stratification of the Temperature in an External Uniform Magnetic Field
Abstract
In this paper the stability of the non-uniformly rotating cylindrical plasma in the axial uniform magnetic field with the vertical temperature gradient is investigated. In the approximation of geometrical optics a dispersion equation for small axisymmetric perturbations is obtained with the effects of viscosity, ohmic and heat conductive dissipation taken into account. The stability criteria for azimuthal plasma flows are obtained in the presence of the vertical temperature gradient and the constant magnetic field. The Rayleigh-Benard problem for stationary convection in the non-uniformly rotating layer of the electrically conducting fluid in the axial uniform magnetic field is considered. In the linear theory of stationary convection the critical value of the Rayleigh number subject to the profile of the inhomogeneous rotation (Rossby number) is obtained. It is shown that the negative values of the Rossby number have a destabilizing effect, since the critical Rayleigh number becomes smaller, than in the case of the uniform rotation , or of the rotation with positive Rossby numbers . To describe the nonlinear convective phenomena the local Cartesian coordinate system was used, where the inhomogeneous rotation of the fluid layer was represented as the rotation with a constant angular velocity and azimuthal shear with linear dependence on the coordinate. As a result of applying the method of perturbation theory for the small parameter of supercriticality of the stationary Rayleigh number a nonlinear Ginzburg-Landau equation was obtaned. This equation describes the evolution of the finite amplitude of perturbations by utilizing the solution of the Ginzburg-Landau equation. It is shown that the weakly nonlinear convection based on the equations of the six-mode Lorentz model transforms into the identical Ginzburg-Landau equation. By utilizing the solution of the Ginzburg-Landau equation, we determined the dynamics of unsteady heat transfer for various profiles of the angular velocity of the rotation of electrically conductive fluid.
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References
S. Chandrasekhar, Hydrodynamics and Hydromagnetic Stability (Oxford Uni. Press, London, 1961), p. 652.
G.Z. Gershuni and E.M. Zhukhovitckii, Convective Stability of Incompressible Fluids (Nauka, Moscow, 1972), p. 392 (in Russian)
A.V. Getling, Rayleigh-Benard Convection: Structures and Dynamics (URSS, Moscow, 1999), p. 235. (in Russian)
Chris A. Jones, Dynamo theory (University of Leeds, UK, 2007).
P.H. Roberts and G.A. Glatzmaier, Geophys. Astrophys. Fluid Dynam. 94(1), 47-84 (2001).
A. Tur and V.Yanovsky, Coherent Vortex Structures in Fluids and Plasmas (Springer, New York, 2017). p. 253.
I.A. Eltayeb, Proc. R. Soc. Lond. A. 326, 229-254 (1972), https://doi.org/10.1098/rspa.1972.0007.
I.A. Eltayeb, J. Fluid Mech. 71(1), 161–179 (1975), https://doi.org/10.1017/S0022112075002480.
R. Avila and A. Cabello, Mathematical Problems in Engineering, 2013, 1-15 (2013), https://doi.org/10.1155/2013/236901.
W.V.R. Malkus and G. Veronis, J. Fluid Mech. 4(3), 225-260 (1958), https://doi.org/10.1017/S0022112058000410.
J.K. Bhattacharjee, J. Phy. A: Math. Gen. 22(24), L1135-L1189 (1989), https://doi.org/10.1088/0305-4470/22/24/001.
J.K. Bhattacharjee, Phy. Rev. A. 41, 5491-5494 (1990), https://doi.org/10.1103/PhysRevA.41.5491.
B.S., Bhadauria and P. Kiran, Ain Shams Eng. J. 5(4), 1287-1297 (2015), https://doi.org/10.1016/j.asej.2014.05.005.
R. Ramya, E.J. Shelin and G.K. Sangeetha, International Journal of Mathematics Trends and Technology, 54(6), 477-484 (2018), https://doi.org/10.14445/22315373/IJMTT-V54P558.
P. Kiran, Ain Shams Eng. J. 7(2), 639-651 (2016), https://doi.org/10.1016/j.asej.2015.06.005.
P.G. Siddheshwar, B.S. Bhadauria and A. Srivastava, Transp. Porous Media, 91(2), 585- 604 (2012), https://doi.org/10.1007/s11242-011-9861-3.
B.S. Bhadauria, P.G. Siddheshwar, J. Kumar and O.P. Suthar, Trans. Porous Med. 73(3), 633-647 (2012), https://doi.org/10.1007/s11242-011-9925-4.
P.G. Siddheshwar, B.S. Bhadauria, Pankaj Mishra and A.K. Srivastava, Int. J. Non Linear Mech. 47, 418-425 (2012), https://doi.org/10.1016/j.ijnonlinmec.2011.06.006.
B.S. Bhadauria and P. Kiran, J. Appl. Fluid Mech. 8(4), 735-746 (2015), https://doi.org/10.18869/acadpub.jafm.73.238.22740.
B.S. Bhadauria and P. Kiran, Transp. Porous Media. 100, 279-295 (2013), https://doi.org/10.1007/s11242-013-0216-0.
B.S. Bhadauria, P. Kiran, Phys. Scr. 89(9), 095209 (2014), https://doi.org/10.1088/0031-8949/89/9/095209.
S. Aniss, M. Belhaq and M. Souhar, J. Heat Transfer, 123(3), 428-433 (2001), https://doi.org/10.1115/1.1370501.
S. Chandrasekhar, On the stability of the simplest solution of the equations of hydromagnetics. // Proc. Natl Acad. Sci. USA, 42(5), 273-276 (1956), https://doi.org/10.1073/pnas.42.5.273.
E.P. Velikhov, Stability of an ideally conducting fluid flowing between cylinders rotating in a magnetic field, JETP, 36, 1398-1404 (1959). (in Russian)
S.A. Balbus and J.F. Hawley, Astrophys. J. 376, 214-222 (1991), https://doi.org/10.1086/170270.
C. Nipoti and L. Posti, (2012), https://arxiv.org/pdf/1206.3890.pdf.
V.P. Lakhin and V.I. Ilgisonis, On the Influence of Dissipative Effects on Instabilities of Differentially-Rotating Plasmas, JETP. 137(4), 783-788 (2010). (in Russian)
O.N. Kirillov and F. Stefani, Proceedings of the International Astronomical Union. 8, 233-234 (2012), https://doi.org/10.1017/S1743921312019771.
O.N. Kirillov, F. Stefani and Y. Fukumoto, J. Fluid Mech. 760, 591- 633 (2014), https://doi.org/10.1017/jfm.2014.614.
G. Rüdiger, R. Hollerbach and L.L. Kitchatinov, Magnetic Processes in Astrophysics: Theory, Simulations, Experiment.- (Wiley-VCH, 2013). p.356, https://doi.org/10.1002/9783527648924.
A.M. Soward, Phys. Earth Planet Int. 20(2-4), 134-151 (1979), https://doi.org/10.1016/0031-9201(79)90036-0.
S. Childress and A.M. Soward, Phys. Rev. Lett. 29, 837-839 (1972), https://doi.org/10.1103/PhysRevLett.29.837.
F.H. Busse, Phys. Earth Planet. Int. 12,. 350-358 (1976), https://doi.org/10.1016/0031-9201(76)90030-3.
F. Busse and F. Finocchi, Physics of The Earth and Planetary Interiors. 80(1-2), 13-23 (1993), https://doi.org/10.1016/0031-9201(93)90069-L.
E. Kurt, F.H. Busse and W. Pesch, Theoret. Comput. Fluid Dynamics, 18, 251-263 (2004), https://doi.org/10.1007/s00162-004-0132-6.
M.I. Kopp, A.V. Tur and V.V. Yanovsky, Problems of Atomic Science and Technology, 4(116), 230-234 (2018), https://arxiv.org/pdf/1805.11894.pdf.
V.P. Maslov and M.V. Fedoriuk, Квазиклассическое приближение для уравнений квантовой механики [Quasi-Classical Approximation for Quantum Mechanics Equations], (Nauka, Moscow, 1976), p. 296. (in Russian)
A.B. Mikhailovskii, Теория плазменных неустоичивостей. T.2. Неустойчивости неоднородной плазмы [Theory of Plasma Instabilities V.2, Inhomogeneous plasma instabilities], (Atomizdat, Moscow, 1971). p. 312. (in Russian)
D.A. Shalybkov, Гидродинамическая и гидромагнитная устойчивость течения Куэтта [Hydrodynamic and hydromagnetic stability of the Couette flow], Usp. Fiz. Nauk, 179(9), 971-993 (2009), https://doi.org/10.3367/UFNe.0179.200909d.0971. (in Russian)
R. Narayan, E. Quataert, I.V. Igumenshchev and M.A. Abramowicz, Astrophys. J. 577, 295-301 (2002), https://doi.org/10.1086/342159.
H. Ji, J. Goodman and A. Kageyama, Mon. Not. R. Astron. Soc. 325, L1-L5 (2001), https://doi.org/10.1046/j.1365-8711.2001.04647.x.
K. Noguchi, V.I. Pariev, S.A. Colgate, H.F Beckley and J. Nordhaus, Astrophys. J. 575, 1151-1162 (2002), https://doi.org/10.1086/341502.
E.P. Velikhov, A.A. Ivanov, V.P. Lakhin, and K.S Serebrennikov, Phys. Letters A. 356, 357-365 (2006), https://doi.org/10.1016/j.physleta.2006.03.073.
F.R. Gantmakher, Лекции по аналитической механике [Lectures in analytical mechanics], (Fizmatlit, Moscow, 2005), p. 264. (in Russian)
Moffat G. Возбуждение магнитного поля в проводящей среде [Magnetic Field Generation in Electrically Conducting Fluids], (Mir, Moscow, 1980), p. 343p. (in Russian)
G. Rudiger and M. Kuker, (2016), https://arxiv.org/pdf/1601.03877.pdf.
F. Krauze and K.H. Redler, Магнитная гидродинамика средних полей и теория динамо [Mean-Field Magnetohydrodynamics and Dynamo Theory], (Mir, Moscow, 1984), p. 314p. (in Russian)
A.V. Getling, Astronomy Reports, 45, 569-576 (2001), https://doi.org/10.1134/1.1383816.
A.V. Getling, Astronomy Reports, 56, 395-402 (2012), https://doi.org/10.1134/S106377291.
A.V. Getling, Solar Physics, 239, 93–111 (2006), https://doi.org/10.1007/s11207-006-0231-1.
P. Goldreich and D. Lynden-Bell, Mon. Not. R. Astron. Soc. 130(2), 125-158 (1965), https://doi.org/10.1093/mnras/130.2.125.
E. Knobloch and K. Jullien, Physics of Fluids, 17(9), 094106 (2005), https://doi.org/10.1063/1.2047592.
R. Haberman, Elementary Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 4th ed. (Pearson/Prentice Hall, N.J., 2004), p. 769.
P.G. Siddheshwar and C. Kanchana, Int. J. Mech. Sci. 131, 1061–1072 (2017), https://doi.org/10.1016/j.ijmecsci.2017.07.050.
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