Skip to main content Accessibility help
×
Home

Dimensionality, secondary flows and helicity in low-Rm MHD vortices

  • Nathaniel T. Baker (a1) (a2) (a3), Alban Pothérat (a1) and Laurent Davoust (a2)

Abstract

In this paper, we examine the dimensionality of a single electrically driven vortex bounded by two no-slip and perfectly insulating horizontal walls a distance $h$ apart. The study was performed in the weakly inertial limit by means of an asymptotic expansion, which is valid for any Hartmann number. We show that the dimensionality of the leading order can be fully described using the single parameter $l_{z}^{{\it\nu}}/h$ , where $l_{z}^{{\it\nu}}$ represents the distance over which the Lorentz force is able to act before being balanced by viscous dissipation. The base flow happens to introduce inertial recirculations in the meridional plane at the first order, which are shown to follow two radically different mechanisms: inverse Ekman pumping driven by a vertical pressure gradient along the axis of the vortex, or direct Ekman pumping driven by a radial pressure gradient in the Hartmann boundary layers. We demonstrate that when the base flow is quasi-2D, the relative importance of direct and inverse pumping is solely determined by the aspect ratio ${\it\eta}/h$ , where ${\it\eta}$ refers to the width of the vortex. Of the two mechanisms, only inverse pumping appears to act as a significant source of helicity.

Copyright

Corresponding author

Email address for correspondence: nathaniel.baker@creta.cnrs.fr

References

Hide All
Akkermans, R. A. D., Cieslik, A. R., Kamp, L. P. J., Trieling, R. R., Clercx, H. J. H. & Van Heijst, G. J. F. 2008 The three-dimensional structure of an electromagnetically generated dipolar vortex in a shallow fluid layer. Phys. Fluids 20, 116601.
Clercx, H. J. H. & Van Heijst, G. 2009 Two-dimensional Navier–Stokes turbulence in bounded domains. Appl. Mech. Rev. 62, 125.
Davidson, P. A. 2014 The dynamics and scaling laws of planetary dynamos driven by inertial waves. Geophys. J. Intl 198 (3), 18321847.
Davoust, L., Achard, J.-L. & Drazek, L. 2015 Low-to-moderate Reynolds number swirling flow in an annular channel with a rotating end wall. Phys. Rev. E 91, 023019.
Deusebio, E. & Lindborg, E. 2014 Helicity in the Ekman boundary layer. J. Fluid Mech. 755, 654671.
Ekman, V. W. 1905 On the influence of the Earth’s rotation on ocean currents. Ark. Mat. Astron. Fys. 2, 153.
Gilbert, A. D., Frisch, U. & Pouquet, A. 1988 Helicity is unnecessary for alpha effect dynamos, but it helps. Geophys. Astrophys. Fluid Dyn. 42 (1–2), 151161.
Kalis, Kh. E. & Kolesnikov, Yu. B. 1980 Numerical study of a single vortex of a viscous incompressible electrically conducting fluid in a homogeneous axial magnetic field. Magnetohydrodynamics 16, 155158.
Klein, R. & Pothérat, A. 2010 Appearance of three-dimensionality in wall bounded MHD flows. Phys. Rev. Lett. 104 (3), 034502.
Kornet, K. & Pothérat, A. 2015 A method for spectral DNS of low Rm channel flows based on the least dissipative modes. J. Comput. Phys. 298, 266279.
Lindborg, E. 1999 Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech. 388, 259288.
Messadek, K. & Moreau, R. 2002 An experimental investigation of MHD quasi-two-dimensional turbulent shear flows. J. Fluid Mech. 456, 137159.
Pothérat, A. & Klein, R. 2014 Why, how and when MHD turbulence at low Rm becomes three-dimensional. J. Fluid Mech. 761, 168205.
Pothérat, A., Rubiconi, F., Charles, Y. & Dousset, V. 2013 Direct and inverse pumping in flows with homogeneous and non-homogeneous swirl. Eur. Phys. J. E 36 (8), 94.
Pothérat, A., Sommeria, J. & Moreau, R. 2000 An effective two-dimensional model for MHD flows with transverse magnetic field. J. Fluid Mech. 424, 75100.
Pothérat, A., Sommeria, J. & Moreau, R. 2002 Effective boundary conditions for magnetohydrodynamic flows with thin Hartmann layers. Phys. Fluids 14 (1), 403410.
Roberts, P. H. 1967 Introduction to Magnetohydrodynamics. Longmans.
Satijn, M. P., Cense, A. W., Verzicco, H., Clercx, H. J. H. & van Heijst, G. J. F. 2001 Three-dimensional structure and decay properties of vortices in shallow fluid layers. Phys. Fluids 13 (7), 19321945.
Shats, M., Byrne, D. & Xia, H. 2010 Turbulence decay rate as a measure of flow dimensionality. Phys. Rev. Lett. 105, 264501.
Smith, D. M. 1991 Algorithm 693: a Fortran package for floating-point multiple-precision arithmetic. ACM Trans. Math. Softw. 17 (2), 273283.
Sommeria, J. 1988 Electrically driven vortices in a strong magnetic field. J. Fluid Mech. 189, 553569.
Sommeria, J. & Moreau, R. 1982 Why, how and when MHD turbulence becomes two-dimensional. J. Fluid Mech. 118, 507518.
Tabeling, P. 2002 Two-dimensional turbulence: a physicist approach. Phys. Rep. 362 (1), 162.
MathJax
MathJax is a JavaScript display engine for mathematics. For more information see http://www.mathjax.org.

JFM classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed