Skip to main content
SpringerLink
Log in

On Magnetic Inhibition Theory in Non-resistive Magnetohydrodynamic Fluids

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We investigate why the non-slip boundary condition for the velocity, imposed in the direction of impressed magnetic fields, can contribute to the magnetic inhibition effect based on the magnetic Rayleigh–Taylor (abbr. NMRT) problem in nonhomogeneous incompressible non-resistive magnetohydrodynamic (abbr. MHD) fluids. Exploiting an infinitesimal method in Lagrangian coordinates, the idea of (equivalent) magnetic tension, and the differential version of magnetic flux conservation, we give an explanation of a physical mechanism for the magnetic inhibition phenomenon in a non-resistive MHD fluid. Moreover, we find that the magnetic energy in the non-resistive MHD fluid depends on the displacement of fluid particles, and thus can be regarded as elastic potential energy. Motivated by this observation, we further use the well-known minimum potential energy principle to explain the physical meaning of the stability/instability criteria in the NMRT problem. As a result of the analysis, we further extend the results on the NMRT problem to the stratified MHD fluid case. We point out that our magnetic inhibition theory can be used to explain the inhibition phenomenon of other flow instabilities, such as thermal instability, magnetic buoyancy instability, and so on, by impressed magnetic fields in non-resistive MHD fluids.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Abidi, H.: Zhang, P: On the global solution of a 3-D MHD system with initial data near equilibrium. Commun. Pure Appl. Math. 70, 1509–1561 (2017)

    Article  MATH  Google Scholar 

  2. Adams, R.A., John, J.F.F.: Sobolev Space. Academic, New York (2005)

    Google Scholar 

  3. Bittencourt, J.A.: Fundamentals of Plasma Physics, 3rd edn. Springer, New York (2004)

    Book  MATH  Google Scholar 

  4. Chandrasekhar, S.: On the inhibition of convection by a magnetic field. Philos. Mag. 43, 501–532 (1952)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chandrasekhar, S.: On the inhibition of convection by a magnetic field. II. Phil. Mag. Sc. Ser. 45, 1177–1191 (1954)

    Article  MathSciNet  Google Scholar 

  6. Chandrasekhar, S.: Hydrodynamic and Hydromagnetic Stability. The International Series of Monographs on Physics. Clarendon Press, Oxford (1961)

    MATH  Google Scholar 

  7. Drazin, P.G., Reid, W.H.: Hydrodynamic Stability, 2nd edn. University Press, Cambridge (2004)

    Book  MATH  Google Scholar 

  8. Fälthammar, C.G.: Comments on the motion of magnetic field lines. Am. J. Phys. 74, 454–455 (2006)

    Article  ADS  Google Scholar 

  9. Galdi, G.: Nonlinear stability of the magnetic Bénard problem via a generalized energy method. Arch. Rational Mech. Anal. 62, 167–186 (1985)

    Article  MATH  Google Scholar 

  10. Galdi, G., Padula, M.: New contributions to nonlinear stability of the magnetic Bénard problem. Proceedings of the Third German–Italian Symposium Applications of Mathematics in Industry and Technology, Vieweg & Teubner Verlag, Wiesbaden, 166–178, 1989

  11. Galdi, G., Padula, M.: Further results in the nonlinear stability of the magnetic Bénard problem. Mathematical Aspects of Fluid and Plasma Dynamics, Springer, Berlin, 140–151, 1991

  12. Grafakos, L.: Classical fourier analysis, 2nd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  13. Hu, X.P.: Global existence for two dimensional compressible magnetohydrodynamic flows with zero magnetic diffusivity. arXiv:1405.0274v1 2014

  14. Hu, X.P., Lin, F.H.: Global existence for two dimensional incompressible magnetohydrodynamic flows with zero magnetic diffusivity. arXiv:1405.0082v1, 2014

  15. Jiang, F.: On effects of viscosity and magnetic fields on the largest growth rate of linear Rayleigh-Taylor instability. Journal of Mathematical Physics 57(11), 111503 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  16. Jiang, F., Jiang, S.: Instability of the abstract Rayleigh–Taylor problem and applications. arXiv:1811.11610, 2018

  17. Jiang, F., Jiang, S.: On the dynamical stability and instability of Parker problem. Physica D (2019). https://doi.org/10.1016/j.physd.2018.11.004

    MathSciNet  Google Scholar 

  18. Jiang, F., Jiang, S.: Nonlinear stability and instablity in Rayleigh-Taylor problem of stratisfied compressible MHD fluids. Calc. Var. 58(1), 29 (2019)

    Article  MATH  Google Scholar 

  19. Jiang, F., Jiang, S.: On the stabilizing effect of the magnetic field in the magnetic Rayleigh-Taylor problem. SIAM J. Math. Anal. 50, 491–540 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jiang, F., Jiang, S., Wang, Y.J.: On the Rayleigh-Taylor instability for the incompressible viscous magnetohydrodynamic equations. Commun. Partial Differential Equations 39, 399–438 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kestelman, H.: Mappings with non-vanishing Jacobian. The American Mathematical Monthly 78, 662–663 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  22. Lee, J.M.: Introduction to Smooth Manifolds. Springer, New York (2013)

    MATH  Google Scholar 

  23. Liang, Z.X., Liang, Y.: Significance of polarization charges and isomagnetic surface in magnetohydrodynamics. PLoS One 10(8), e0136936 (2015)

    Article  Google Scholar 

  24. Lin, F.H., Zhang, P.: Global small solutions to an MHD-type system: the three-dimensional case. Commun. Pure Appl. Math. 67(4), 531–580 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lin, F.H., Xu, L., Zhang, P.: Global small solutions of 2-D incompressible MHD system. J. Differential Equations 259, 54405485 (2015)

    MathSciNet  Google Scholar 

  26. Majda, A.J., Bertozzi, A.L.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  27. Meisters, G.H., Olech, C.: Locally one-to-one mappings and a classical theorem on schlicht functions. Duke Math. J. 30, 63–80 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  28. Mulones, G., Rionero, S.: Necessary and sufficient conditions for nonlinear stability in the magnetic Bénard problem. Arch. Rational Mech. Anal. 166, 197–218 (2003)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  29. Nakagawa, Y.: An experiment on the inhibition of thermal convection by a magnetic field. Nature 175, 417–419 (1955)

    Article  ADS  Google Scholar 

  30. Nakagawa, Y.: Experiments on the inhibition of thermal convection by a magnetic field. Proc. Royal Soc. (London) A 240, 108–113, 1957

  31. Novotnỳ, A., Straškraba, I.: Introduction to the Mathematical Theory of Compressible Flow. Oxford University Press, Oxford (2004)

    MATH  Google Scholar 

  32. Rudin, W.: Principles of Mathematical Analysis, 3rd edn. China Machine Press, Beijing (2004)

    MATH  Google Scholar 

  33. Pan, R.H., Zhou, Y., Zhu, Y.: Global classical solutions of three dimensional viscous MHD system without magnetic diffusion on periodic boxes. Arch. Ration. Mech. Anal. 227(2), 637–662 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  34. Ren, X.X., Wu, J.H., Xiang, Z.Y., Zhang, Z.F.: Global existence and decay of smooth solution for the 2-DMHD equations without magnetic diffusion. J. Funct. Anal. 267, 503–541 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ren, X.X., Xiang, Z.Y., Zhang, Z.F.: Global existence and decay of smooth solutions for the 3-D MHD-type equations without magnetic diffusion. Scientia Sinica Mathematica 59(10), 1–26 (2016)

    MathSciNet  MATH  Google Scholar 

  36. Shibata, K., Matsumoto, R.: Formation of giant molecular clouds and helical magnetic fields by the parker instablity. Nature 353, 633–635 (1991)

    Article  ADS  Google Scholar 

  37. Stern, D.P.: The motion of magnetic field lines. Space Science Reviews 6, 147–173 (1966)

    Article  ADS  Google Scholar 

  38. Tan, Z., Wang, Y.J.: Global well-posedness of an initial-boundary value problem for viscous non-resistive MHD systems. SIAM Journal on Mathematical Analysis 50(1), 1432–1470 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  39. Thompson, W.B.: Thermal convection in a magnetic field. Phil. Mag. Ser. 7(42), 1417–1432 (1951)

    Article  MathSciNet  MATH  Google Scholar 

  40. Xu, L., Zhang, P.: Global small solutions to three-dimensional incompressible magnetohydrodynamical system. SIAM J. Math. Anal. 47, 26–65 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  41. Wang, Y.J.: Critical magnetic number in the MHD Rayleigh-Taylor instability. Journal of Math. Phys. 53, 073701 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  42. Wang, Y.J.: Sharp nonlinear stability criterion of viscous non-resistive MHD internal waves in 3D. Arch. Ration. Mech. Anal. 231, 1675–1743 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Water, W.: Ordinary Differential Equations. Springer, New York (1998)

    Book  Google Scholar 

Download references

Acknowledgements

The authors thank Dr. Weicheng Zhan for pointing out Lemmas 4.1 and 4.5, and Prof. C.H. Arthur Cheng for kindly discussing the global existence thereon of inverse functions. The authors also thank the anonymous referee for invaluable suggestions, which improve the presentation of this paper.

Author information

Authors and Affiliations

  1. College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350108, China

    Fei Jiang

  2. Institute of Applied Physics and Computational Mathematics, P.O. Box 8009, Beijing, 100088, China

    Song Jiang

Authors
  1. Fei Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Song Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Fei Jiang.

Additional information

Communicated by F. Lin

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

The research of Fei Jiang was supported by the NSFC (Grant No. 11671086) and the NSF of Fujian Province of China (Grant No. 2016J06001), and the research of Song Jiang by the NSFC (Grant Nos. 11631008 and 11371065)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jiang, F., Jiang, S. On Magnetic Inhibition Theory in Non-resistive Magnetohydrodynamic Fluids. Arch Rational Mech Anal 233, 749–798 (2019). https://doi.org/10.1007/s00205-019-01367-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-019-01367-8

Search

Navigation