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Paradox of inductionless magnetorotational instabilityPriede, J.; Grants, I.; Gerbeth, G.
The magnetorotational instability (MRI) is sought to be responsible for the fast formation of stars and entire galaxies in accretion disks. The velocity distribution in accretion disks is apparently hydrodynamically stable by the Rayleigh criterion while the viscosity alone is not sufficient to account for the observable accretion rates. However, a hydrodynamically stable velocity profile in the cylindrical Taylor-Couette flow can become unstable in the presence of magnetic field (Velikhov, Sov. Phys. JETP 36, 995, 1959; Balbus and Hawley, Astrophys. J. 376, 214, 1991). In this case, an axial magnetic field provides an additional mechanism of energy exchange between the base flow and perturbations that, however, requires the magnetic Reynolds number to be at least Rm ~ 10. Note that for a liquid metal with the magnetic Prandtl number Pm ~ 10-5 this corresponds to a hydrodynamic Reynolds number Re = Rm/Pm ~ 106. Thus, this instability is hardly observable in the laboratory because any conceivable flow at such Reynolds number would be turbulent. However, it was shown recently (Hollerbach and Rüdiger, Phys. Rev. Lett. 95, 124501, 2005) that MRI can take place in the cylindrical Taylor-Couette flow at Re ~ 103 when the imposed magnetic field is helical. The most surprising fact is that this type of MRI works even in the inductionless limit of Pm = 0 where the critical Reynolds number of the conventional MRI with axial magnetic field diverges as ~ 1/Pm. The induced currents are so weak in this limit that their magnetic field is negligible with respect to the imposed field. Thus, on one hand, the imposed magnetic field does not affect the base flow, which is the only source of energy for the perturbation growth. But on the other hand, flow perturbations are subject to additional damping due to the Ohmic dissipation caused by the induced currents. We show rigorously that, in the limit of Pm=0, the imposed magnetic field increases the energy decay rate of any particular perturbation. On one hand, this means that the energy of any perturbation, which is growing in the presence of magnetic field, grows even faster without the field and vice versa. On the other hand, the flow which is found to be unstable in the presence of magnetic field is certainly known to be stable without the field. This apparent contradiction constitutes the paradox of the inductionless MRI which we address in this study. We consider MRI in the inductionless approximation at Pm=0 that allows us to eliminate the magnetic field and, thus, leads to a considerable simplification of the problem containing only hydrodynamic variables as in the classical Taylor-Couette problem. First, we use a Chebyshev collocation method to calculate the eigenvalue spectrum of the linearised problem. In this way, we confirm that MRI with helical magnetic field indeed works in the inductionless limit. Second, we integrate the linearised equations in time to study the transient behaviour of small amplitude perturbations.
In this way, we show that the energy arguments are correct as well - the energy of an unstable perturbation indeed starts to grow faster when the magnetic field is switched off. However, there is no real contradiction between both facts. The energy grows only for a limited time and then turns to decay in accordance to the linear stability predictions. It is important to stress that the linear stability theory predicts the asymptotic development of an arbitrary small-amplitude perturbation, while the energy stability theory yields the instant growth rate of any particular perturbation but it does not account for the evolution of this perturbation. Thus, although switching off the magnetic field instantly increases the energy growth rate of the most unstable as well as that of any other perturbation, in the same time the critical perturbation ceases to be an eigenmode without the magnetic field.
2nd Int. Symposium "Instabilities and Bifurcations in Fluid Dynamics", 15.-18.08.2006, Copenhagen, Denmark
- Journal of Physics: Conference Series 64(2007), 012011
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