arXiv:1001.1895v1[astro-ph.SR]12 Jan 2010 $^{∗}$Electronic address: gruediger@aip.de, mgellert@aip.de, mschultz@aip.de $^{†}$Electronic address: rh@maths.leeds.ac.uk Dissipative Taylor-Couette flows under the influence of heli cal magnetic fields G. R¨ udiger, M. Gellert, M. Schultz Astrophysikalisches Institut Potsdam, An der Sternwarte 16, D-14482 Potsdam, Germany ∗ R. Hollerbach Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK † (Dated: August 28, 2018) The linear stability of MHD Taylor-Couette flows in axially u nbounded cylinders is considered, for magnetic Prandtl number unity. Magnetic fields varying from purely ax ial to purely azimuthal are imposed, with a general helical field parameterized by β = B$_{φ}$/B $_{z}$. We map out the transition from the standard MRI for β = 0 to the nonaxisymmetric Azimuthal MagnetoRotational Instability (AMRI) for β → ∞ . For finite β , positive and negative wave numbers m , corresponding to right and left spirals, are no longer identical. The transition from β = 0 to β → ∞ includes all the possible forms of MRI with axisymmetric and nonaxisymmetric modes. For the nonaxisymmetric modes, the most unstable mode spirals in the opposite direction to the background field. The standard ( β = 0 ) MRI is axisymmetric for weak fields (including the instabil ity with the lowest Reynolds number) but is nonaxisymmetric for stronger fields. If the azimuthal field is due in part to an axial current flowing through the fluid itself (and not just along the central axis), then it is also unstable to the nonaxisymmetric Tayler instability, which is most effective without rotation. For large β this instability has wavenumber m = 1 , whereas for β ≃ 1 m = 2 is most unstable. The most unstable mode spirals in the same direction as the background field. PACS numbers: 47.20.Ft, 47.20.-k, 47.65.+a as the necessary and sufficient condition for stability of an ideal fluid at rest [4]. Outwardly increasing fields are there - fore unstable. If (3) is violated, the most unstable mode has azimuthal wavenumber m = 1 . The longstanding problem of the generation of turbulence in various hydrodynamically stable situations has found a so- lution in recent years with the MHD shear flow instability, th e so-called magnetorotational instability (MRI), in which the presence of a magnetic field has a destabilizing effect on a differentially rotating flow with outward decreasing angul ar velocity but increasing angular momentum [1, 2]. In the absence of MHD effects, according to the Rayleigh criterion, an ideal flow is stable against axisymmetric pert ur- bations whenever the specific angular momentum increases outward The rich variety of nonaxisymmetric instabilities can be demonstrated by the addition of a differential rotation. In this case even the current-free (within the fluid) profile B$_{φ}$ ∝ 1 /R (which according to (3) is stable for Ω = 0 ) can become un- stable. Even for a differential rotation that by itself would be stable according to 1, the combination of Ω and B$_{φ}$ ∝ 1 /R can be unstable to m = 1 perturbations (see Fig. 2). We have called this phenomenon the Azimuthal MagnetoRotational In- stability (AMRI). It has even been demonstrated that it should be possible to observe the AMRI in laboratory experiments, [5]. Further new phenomena appear if an axial field is added, yielding a spiral, or helical total field. In this case only a s uf- ficient condition for stability against axisymmetric pertu rba- tions is known. In the absence of rotation this is where Ω is the angular velocity, and ( R , φ , z ) are cylindrical coordinates. In the presence of an azimuthal magnetic field B$_{φ}$ , this criterion is modified to where µ 0 is the permeability and ρ the density [3]. Note also that this criterion is both necessary and sufficient for (axi sym- metric) stability. In particular, all ideal flows can thus be destabilized, by azimuthal magnetic fields with the right pr o- files and amplitudes. On the other hand, for nonaxisymmetric modes, one has For the current-free field B$_{φ}$ ∝ 1 /R , only superrotating flows with dΩ / d R > 0 are stable. Indeed, we have demonstrated that dissipative Taylor-Couette flows beyond the Rayleigh limit for centrifugal instability can easily be destabilized by helical magnetic fields with such a current-free azimuthal component [7]. The resulting axisymmetric traveling wave in- stability has become known as the Helical MagnetoRotational Instability (HMRI), and has been obtained in the PROMISE experiment [8, 9]. d dR ( R $^{2}$B2 $_{φ}$) < 0 , (4) R d Ω 2 dR − 1 µ 0 ρR 3 d dR ( RB$_{φ}$ ) 2 > 0 . (5) d d R ( RB 2 $_{φ}$) < 0 (3) 1 R 3 d d R ( R $^{2}$Ω)2 − R µ 0 ρ d d R ( B$_{φ}$ R ) 2 > 0 , (2) d d R ( R $^{2}$Ω)2 > 0 , (1) (see also Eq. (6)). Including rotation, this was extended [6] to I. INTRODUCTION