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New type of axisymmetric helical magnetorotational instability in rotating flows with positive shear

We present a new type of axisymmetric magnetorotational instability which is capable of destabilizing viscous and resistive flows with radially increasing angular velocity, i.e., positive shear. This instability works for a wide range of the positive shear when 1. a combination of axial/poloidal and azimuthal/toroidal magnetic fields is applied and 2. the magnetic Prandtl number is not too close to unity. It might have important consequences for the stability of the equatorial parts of the solar tachocline and for the dynamo action in this region.

According to Rayleigh's criterion [1], rotating flows of ideal fluids with radially increasing specific angular momentum are linearly stable. This result has severe astrophysical consequences, implying hydrodynamic stability for the Keplerian rotation of accretion disks. The magnetorotational instability (MRI) [2] is the most likely destabilizing mechanism for these disks, providing outward angular momentum transport necessary for the rapid growth of the respective central objects.
The standard version of MRI (SMRI, with a purely axial/poloidal magnetic field [2]), as well as the azimuthal MRI (AMRI, with a purely azimuthal/toroidal field [3]) and the helical MRI (HMRI, with combined axial and azimuthal fields [4]) have all been extensively studied theoretically and numerically. The two inductionless forms, AMRI and HMRI, have also been obtained in liquid metal experiments [5,6], while the unambiguous experimental demonstration of the inductive SMRI remains elusive, despite promising first results [7].
In contrast to Keplerian-like rotation with increasing angular momentum but decreasing angular velocity, much less attention is devoted to flows with increasing angular velocity. Until recently, it was believed that such flows are very stable, even under magnetic fields. However, for Reynolds numbers Re = O(10 6 ), such flows can yield non-axisymmetric linear instabilities [8]. Apart from this hydrodynamic instability, there is also a special type of AMRI operating for flows with much lower Reynolds number but very strong positive shear [9].
This restriction to very strong shear makes this socalled Super-AMRI unlikely to be astrophysically significant. One of the few positive shear regions is a portion of the tachocline extending ±30 • about the solar equator [10]. However, even there, the shear measured in terms of the Rossby number Ro = r(2Ω) −1 dΩ/dr is only around 0.5, significantly less than the so-called upper Liu limit (ULL) Ro ULL = 2(1 + √ 2) ≈ 4.83 [11] required for Super-AMRI.
Given the general similarity between AMRI and HMRI [12], one might expect a similar result to hold also for Super-HMRI. However, as we report in this Letter, there exists a new type of axisymmetric HMRI which operates in positive shear flows with arbitrary steepness. The only requirements are the presence of both axial and azimuthal field components, and magnetic Prandtl number which is neither zero (the inductionless limit) nor too close to unity.
We start analysis using the Wentzel-Kramers-Brillouin (WKB) short-wavelength formulation of the underlying problem, which is especially useful for understanding the basic features and scaling properties of the instability. The local analysis is then confirmed by global linear stability calculations including appropriate boundary conditions at the inner and outer radii.
Consider a Taylor-Couette flow of an incompressible, conducting fluid between two coaxial cylinders with radii r i and r o . The inner cylinder is stationary, while the outer cylinder rotates at Ω o , creating the angular velocity profile Ω(r) = Ω o (1 − r 2 i /r 2 )/(1 −η 2 ), whereη = r i /r o . The externally imposed helical magnetic field consists of a constant axial component, B 0z , and a radially varying, current-free azimuthal component, B 0φ (r) = βB 0z r o /r, where the parameter β characterizes field's helicity.
In a radially local WKB approximation, we consider the linear stability of this equilibrium to axisymmetric perturbations of the form exp(γt + ik r r + ik z z), where γ is the (complex) frequency and k r and k z are radial and axial wavenumbers. The resulting dispersion relation takes the form of the fourth-order polynomial [12]: with the coefficients a 1 = 2k 2 Re −1 (1 + P m −1 ), If any of its roots has a positive real part, Re(γ) > 0, then the flow is unstable. Here, γ is normalized by Ω o , and the wavenumbers by r −1 o . The other nondimensional parameters are: α = k z /k, where k = k 2 r + k 2 z is the total wavenumber; the Reynolds number and magnetic Reynolds number defined, respectively, as Re = Ω o r 2 o /ν and Rm = Ω o r 2 o /η, and their ratio, the magnetic Prandtl number P m = ν/η = Rm/Re, where ν is viscosity and η resistivity; the Hartmann number Ha = B 0z r o / √ µ 0 ρ 0 νη, measuring the strength of the imposed axial field, where ρ 0 is the (constant) density and µ 0 magnetic permeability. Another relevant number measuring the field strength is the Lundquist number S = Ha · P m 1/2 , which, like Rm, does not involve viscosity.
We focus particularly on positive Rossby numbers, Ro > 0, or positive shear, so that the flow is stable both hydrodynamically according to Rayleigh's criterion, as well as against SMRI with a purely axial field (i.e. when β = 0) [12]. Regarding the dependence on β, as long as β = 0, it follows from the coefficients of Eq. (1) that β can be removed by re-scaling the wavenumbers, Hartmann and Reynolds numbers as k z /β → k z , k/β → k, Re/β 2 → Re, Ha/β → Ha, which does not change these coefficients and hence the eigenfrequencies. Without loss of generality, we therefore set β = 1 in the WKB analysis.
In the inductionless limit, P m → 0, the roots of Eq. (1) can be found analytically [12]. For positive and relatively large Ro > Ro ULL , one of the roots always has a positive real part, where which corresponds to the growth rate of HMRI operating at positive shear, i.e. Super-HMRI. Our main goal here though is to reveal that apart from this Super-HMRI at large positive shear, Eq. (1) also yields a completely new type of dissipation-induced, or double-diffusive instability at finite P m. Figure 1a shows the growth rate Re(γ) as a function of the axial wavenumber, determined from a numerical solution of Eq. (1) at finite but very small P m = 10 −6 , together with solution (2) in the inductionless limit, for fixed Ha and Re. For the Rossby number we take values lower, Ro = 1.5, 2, and higher, Ro = 6, than Ro ULL . Two distinct instability regimes are clearly seen. The first one is concentrated at small k z and exists at finite P m both for Ro < Ro ULL and Ro > Ro ULL , i.e. it is independent of the Liu limit, but disappears for P m → 0 with fixed Hartmann and Reynolds numbers. In contrast, the second one, Super-HMRI, is concentrated at larger k z , exists only for Ro > Ro ULL , and approaches the inductionless solution as P m → 0.
The first instability branch represents a new positive shear, dissipation-induced mode, which appears to require the presence of both finite viscosity and resistivity. As we will see below though, it does not operate in the immediate vicinity of P m = 1. That is, it is double-diffusive in nature, operating for both small and large P m, but not for P m = O(1). Just as all previous MRI variants, this one also derives its energy solely from the shear, since the imposed field is current-free, thereby eliminating current-driven instabilities. Energy is drawn from the background flow rΩ(r) to the growing perturbations due to the coupling between meridional circulation and azimuthal field perturbations brought about by the imposed azimuthal field, a mechanism also underlying HMRI [4,11]. For this reason, we label this new instability a double-diffusive HMRI at positive shear.
The main goal of this Letter is to describe the properties of this instability and point out an important astrophysical situation where it could apply. Super-HMRI, existing for Ro > Ro ULL and persisting even in the inductionless limit P m → 0 [11,12], is also a new and interesting instability, but will not be considered further here.
To explore the behavior of the novel instability further, we first vary Hartmann and Reynolds numbers. For given values of these numbers, the growth rate has been maximized over α and k z , and plotted as a function of Ha and Re in Fig. 1b at P m = 10 −6 and Ro = 1.5 < Ro ULL (where Super-HMRI is absent). The most unstable region is quite localized, with the growth rate decreasing for both small and large Ha and Re, implying that the instability relies on finite viscosity and resistivity, i.e. it is indeed double-diffusive. The overall pattern in (Ha, Re)plane does not change qualitatively at other P m and Ro; the most unstable region always remains localized, and shifts to larger Ha and Re with decreasing P m. At given  Fig. 1b; these values can be rescaled to other β as noted above. We next consider variation with P m again at fixed Ro = 1.5 < Ro ULL , so that Super-HMRI is excluded. Figure 2 shows the growth rate γ m , maximized over k z , Ha and Re, as a function of P m. The associated Ha m and Re m at which this maximum growth is achieved are also shown. We see that for P m 10 −2 , the growth rate is practically constant, γ m = 0.043, while Ha m and Re m increase with decreasing P m as power-laws, with Ha m ∝ P m −1/2 and Re m ∝ P m −1 . That is, in this small-P m regime this new instability is more usefully described in terms of the Lundquist and magnetic Reynolds numbers, since these scale as S m = Ha m · P m 1/2 = 0.7, Rm m = Re m · P m = 0.091, and are thus independent of P m. These scalings with S and Rm, being independent of P m at P m → 0, are identical to those of SMRI, and imply that this new instability also does not exist in the inductionless limit, which would require S, Rm → 0 if Ha and Re remain finite.
With increasing P m, beyond P m ∼ 0.01, γ m rapidly decreases, and eventually the instability vanishes at the first critical value P m c1 = 0.223, with corresponding Ha m = 2.018, Re m = 0.071, and critical axial wavenumber k zm = 0.0024. The instability reappears for larger P m at the second critical value P m c2 = 4.46, with Ha m = 2, Re m = 0.046, and k zm = 0.005, comparable to those at P m c1 . Further increasing P m, for P m 10, γ m eventually levels off to a constant value of 0.29. The corresponding Hartmann and Reynolds numbers again follow power-law scalings, now Ha m ∝ P m 1/3 and Re m ∝ P m −1/4 .  Note that while we have only presented Ro = 1.5 case in detail here, other values yield qualitatively similar behavior and scalings of γ m , Ha m , Re m . This new instability therefore exists over a broad range of magnetic Prandtl numbers, provided the viscosity ν and resistivity η are such that the immediate neighborhood of P m = 1 is avoided, and instead P m < P m c1 < 1 or P m > P m c2 > 1. Figure 3a shows the unstable regions in (Ro, P m)plane. For all Ro, the two bounding curves where γ m = 0 satisfy P m c1 < 1 and P m c2 > 1, and are further related by P m c1 P m c2 = 1, as seen in Fig. 3b. For high shear (Ro → ∞), the stable strip around P m = 1 becomes increasingly narrow, so most P m values are unstable, whereas for Ro → 0, the stable strip widens to include all P m. This is readily understood by noting that if shear is the only energy source (just as it is for SMRI, AMRI, and HMRI), then clearly there can be no instability at all for Ro = 0, corresponding to solid-body rotation. Note finally how P m c1 and P m c2 stability curves are completely unaffected by the Liu limit at Ro ULL = 4.83, and the instability continues to exist even for Ro < Ro ULL .
Following this local WKB analysis, we confirm the results by a global analysis, including also boundary conditions both for the flow and magnetic field. The problem reduces to a one-dimensional linear eigenvalue problem from [4], with the only difference being that now the imposed rotation profile Ω(r) corresponds to a rotating outer and stationary inner cylinder. The boundary conditions are no-slip for the flow and perfectly conducting for the field. As in [4], the radial structure of the quantities are expanded in Chebyshev polynomials, typically up to N = 30 − 40. The governing equations and boundary conditions then reduce to a large (4N × 4N ) matrix eigenvalue problem, with the eigenvalues being the growth/decay rates of the eigenmodes.
Having in mind possible detection of this new type of HMRI in the upcoming state-of-the-art liquid sodium Taylor-Couette experiments planned within the DRES-DYN project [13], we here fix the ratio of the cylinders' radii toη = 0.5. A more comprehensive global linear stability analysis, addressing the dependence of growth rate on the flow parameters (η, β, Re, Ha, P m) will be presented later. We also plan to explore the effects of boundary conditions, perfectly conducting versus insulating, since they can lead to different onset criteria and growth rates of magnetorotational instabilities in Taylor-Couette flows [14]. Figure 4 shows the growth rate Re(γ) as a function of the wavenumber k z for both small and large P m, and clearly demonstrates that this new type of doublediffusive HMRI at positive shear also exists in the global analysis. For β = 1, the instability is concentrated mainly at small k z ∼ 10 −2 − 1 for P m ≪ 1, in agreement with the local analysis (Fig. 1a), and shifts to larger k z 10 for P m ≫ 1. Note also that here the solutions explicitly depend on β, since the radial coordinate appears in the governing equations [4], unlike in the local analysis, where β effectively scales out. However, as seen in Fig. 4, qualitatively the expected scalings with β are still followed, with Ha and Re (or S and Rm), as well as k z,m , increasing with β.
Note also in Fig. 4a how the three solid curves, corresponding to increasingly small P m, converge in this limit. This is consistent with the previous local result that the relevant parameters in this case are the Lundquist, S, and magnetic Reynolds, Rm, numbers, since they are kept fixed for these three curves. Figure 5 focuses further on this small-P m limit, and shows how the maximum growth rate (i.e., maximized over k z ) varies with S and Rm, at fixed β = 1. Just as in Fig. 1b for the local analysis, the instability is again localized in (S, Rm)-plane, with the overall maximum growth rate, γ m = 4.8 · 10 −3 , occurring at S m = 5.2 and Rm m = 25 (Ha m = 5.2 · 10 3 and Re m = 2.5 · 10 7 ). The minimum, or critical values for the instability to first appear are S c ≈ 0.3 and Rm c ≈ 0.9. These values are well within the capabilities of the new Taylor-Couette device at HZDR [13], offering prospects for experimental realization of this new doublediffusive, positive shear HMRI. Although the local and global results are qualitatively quite similar, there are quantitative differences, with the local analysis yielding somewhat lower values for the Hartmann and Reynolds numbers corresponding to the maximal growth rate. Such differences could be due to the finite sized gap (η = 0.5) and associated variations with r in the global analysis, whereas the local analysis strictly applies in the narrow gap limit,η → 1, where such variations can be neglected. Having a finite distance between the inner and outer cylinders also excludes very small radial wavenumbers (α → 1), which are formally present in the WKB analysis, and correspond to larger growth rates (see Figs. 1 and 2, both having α = 1).
In summary, we have uncovered and analysed a new type of double-diffusive HMRI which exists in rotating flows with arbitrary positive shear, including 0 < Ro < Ro ULL where magnetorotational instabilities were previously unknown. The only prerequisites are that P m = 1, and the imposed magnetic field must include both axial and azimuthal components. Both these conditions are satisfied in the equatorial regions of the solar tachocline, making it a likely astrophysical domain of applicability of this new type of HMRI.
This instability could also resurrect the idea of a subcritical solar dynamo. In particular, its axisymmetric (m = 0) nature can help to overcome the difficulties that have been identified [15] in getting the so-called Tayler-Spruit dynamo [16] to form a closed dynamo loop from the combination of the m = 1 Tayler instability and the m = 0 Ω-effect.
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sk lodowska-Curie Grant Agreement No. 795158. GM also acknowledges support from the Alexander von Humboldt Foundation (Germany).