A Connection to the work by Friedlander & Vishik (1995)

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Abstract

We perform a local stability analysis of rotational flows in the presence of a constant vertical magnetic field and an azimuthal magnetic field with a general radial dependence. Employing the short-wavelength approximation we develop a unified framework for the investigation of the standard, the helical, and the azimuthal version of the magnetorotational instability, as well as of current-driven kink-type instabilities. Considering the viscous and resistive setup, our main focus is on the case of small magnetic Prandtl numbers which applies, e.g., to liquid metal experiments but also to the colder parts of accretion disks. We show that the inductionless versions of MRI that were previously thought to be restricted to comparably steep rotation profiles extend well to the Keplerian case if only the azimuthal field slightly deviates from its current-free (in the fluid) profile. We find an explicit criterion separating the pure azimuthal inductionless magnetorotational instability from the regime where this instability is mixed with the Tayler instability. We further demonstrate that for particular parameter configurations the azimuthal MRI originates as a result of a dissipation-induced instability of the Chandrasekhar’s equipartition solution of ideal magnetohydrodynamics.

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eurm10 \checkfontmsam10 Local instabilities in magnetized rotational flows]Local instabilities in magnetized rotational flows: A short-wavelength approach O. N. Kirillov, F. Stefani and Y. Fukumoto]O.\lsN.\nsK\lsI\lsR\lsI\lsL\lsL\lsO\lsV1,\nsF.\nsS\lsT\lsE\lsF\lsA\lsN\lsI and Y.\nsF\lsU\lsK\lsU\lsM\lsO\lsT\lsO 2010 \volume650 \pagerange119–126

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1 Introduction

The interaction of rotational flows and magnetic fields is of fundamental importance for many geo- and astrophysical problems (Rüdiger, Kitchatinov & Hollerbach, 2013). On one hand, rotating cosmic bodies, such as planets, stars, and galaxies are known to generate magnetic fields by means of the hydromagnetic dynamo effect. Magnetic fields, in turn, can destabilize rotating flows that would be otherwise hydrodynamically stable. This effect is particularly important for accretion disks in active galactic nuclei, dwarf novae and protoplanetary systems, where it allows for the tremendous enhancement of outward directed angular momentum transport that is necessary to explain the typical mass flow rates onto the respective central objects. Although this magnetorotational instability (MRI), as we call it now, had been discovered already in 1959 by Velikhov (Velikhov, 1959) and then confirmed in 1960 by Chandrasekhar (Chandrasekhar, 1960), it was left to Balbus & Hawley (1991) to point out its relevance for astrophysical accretion processes. Their seminal paper has inspired many investigations related to the action of MRI in active galactic nuclei (Krolik, 1998), X-ray binaries (Done, Gierlinski & Kubota, 2007), protoplanetary disks (Armitage, 2011), stars (Kagan & Wheeler, 2014), and even planetary cores (Petitdemange, Dormy & Balbus, 2008).

An interesting question concerns the non-trivial interplay of the hydromagnetic dynamo effect and magnetically triggered flow instabilities. For a long time, dynamo research had been focussed on how a prescribed flow can produce a magnetic field and, to a lesser extent, on how the self-excitation process saturates when the magnetic field becomes strong enough to act against the source of its own generation. Similarly, most of the early MRI studies have assumed some prescribed magnetic field, e.g., a purely axial or a purely azimuthal field, to assess its capability for triggering instabilities and turbulent angular momentum transport in the flow. Nowadays, however, we witness an increasing interest in treating the dynamo effect and instabilities in magnetized flows in a more self-consistent manner. Combining both processes one can ask for the existence of “self-creating dynamos” (Fuchs, Rädler & Rheinhardt, 1999), i.e. dynamos whose magnetic field triggers, at least partly, the flow structures that are responsible for its self-excitation.

A paradigm of such an essentially non-linear dynamo problem is the case of an accretion disk without any externally applied axial magnetic field. In this case the magnetic field can only be produced in the disk itself, very likely by a periodic MRI dynamo process (Herault et al., 2011) or some sort of an dynamo (Brandenburg et al., 1995), the part of which relies on the turbulent flow structure arising due to the MRI. Such a closed loop of magnetic field self-excitation and MRI has attracted much attention in the past, though with many unsolved questions concerning numerical convergence (Fromang & Papaloizou, 2007), the influence of disk stratification (Shi, Krolik & Hirose, 2010), and the role of boundary conditions for the magnetic field (Käpylä & Korpi, 2011).

In problems of that kind, a key role is played by the so-called magnetic Prandtl number , i.e. the ratio of the viscosity of the fluid to its magnetic diffusivity (with denoting the magnetic permeability of free space and the conductivity). For the case of an axially applied field, a first systematic study of the -dependence of the growth rate and the angular momentum transport coefficient was carried out by Lesur & Longaretti (2007). The closed-loop MRI-dynamo process, which seems to work well for (Herault et al., 2011), is much more intricate for small values of , as they are typical for the outer parts of accretion disks around black holes and for protoplanetary disks. At least for stratified disks, Oishi & Mac Low (2011) have recently claimed evidence for a critical magnetic Reynolds number in the order of 3000 that is mainly independent of . A somewhat higher value of had been found earlier by Fleming, Stone & Hawley (2000) when considering resistivity while still neglecting viscous effects.

Another paradigm of the interplay of self-excitation and magnetically triggered instabilities is the so-called Tayler-Spruit dynamo as proposed by Spruit (2002). In this particular (and controversially discussed) model of stellar magnetic field generation, the part of the dynamo process (to produce toroidal field from poloidal field) is played, as usual, by the differential rotation, while the part (to produce poloidal from toroidal field) is taken over by the flow structure arising from the kink-type Tayler instability (Tayler, 1973) that sets in when the toroidal field acquires a critical strength to overcome stable stratification.

At small values of , both dynamo and MRI related problems are very hard to treat numerically. This has to do with the fact that both phenomena rely on induction effects which require some finite magnetic Reynolds number. This number is the ratio of magnetic field production by the velocity to magnetic field dissipation due to Joule heating. For a fluid flow with typical size and typical velocity it can be expressed as . The numerical difficulty for small problems arises then from the relation that the hydrodynamic Reynolds number, i.e. , becomes very large, so that extremely fine structures have to be resolved. Furthermore, for MRI problems it is additionally necessary that the magnetic Lundquist number, which is simply a magnetic Reynolds number based on the Alfvén velocity , i.e. , must also be in the order of 1.

A complementary way to study the interaction of rotating flows and magnetic field at small and comparably large is by means of liquid metal experiments. As for the dynamo problem, quite a number of experiments have been carried out (Stefani, Gailitis & Gerbeth, 2008). Up to present, magnetic field self-excitation has been attained in the liquid sodium experiments in Riga (Gailitis et al., 2000), Karlsruhe (Müller & Stieglitz, 2000), and Cadarache (Monchaux et al., 2007). Closely related to these dynamo experiments, some groups have also attempted to explore the standard version of MRI (SMRI), which corresponds to the case that a purely vertical magnetic field is being applied to the flow (Sisan et al., 2004; Nornberg et al., 2010). Recently, the current-driven, kink-type Tayler instability was identified in a liquid metal experiment (Seilmayer et al., 2012), the findings of which were numerically confirmed in the framework of an integro-differential equation approach by Weber et al. (2013).

With view on the peculiarities to do numerics, and experiments, on the standard version of MRI at low , it came as a big surprise when Hollerbach & Rüdiger (2005) showed that the simultaneous application of an axial and an azimuthal magnetic field can change completely the parameter scaling for the onset of MRI. For , the helical MRI (HMRI), as we call it now, was shown to work even in the inductionless limit (Priede, 2011; Kirillov & Stefani, 2011), , and to be governed by the Reynolds number and the Hartmann number , quite in contrast to standard MRI (SMRI) that was known to be governed by and (Ji et al., 2001).

Very soon, however, the enthusiasm about this new inductionless version of MRI cooled down when Liu et al. (2006) showed that HMRI would only work for relatively steep rotation profiles, see also (Ji & Balbus, 2013). Using a short-wavelength approximation, they were able to identify a minimum steepness of the rotation profile , expressed by the Rossby number . This limit, which we will call the lower Liu limit (LLL) in the following, implies that for the inductionless HMRI does not extend to the Keplerian case, characterized by . Interestingly, Liu et al. (2006) found also a second threshold of the Rossby number, which we call the upper Liu limit (ULL), at . This second limit, which predicts a magnetic destabilization of extremely stable flows with strongly increasing angular frequency, has attained nearly no attention up to present, but will play an important role in the present paper.

As for the general relation between HMRI and SMRI, two apparently contradicting observations have to be mentioned. On one hand, the numerical results of Hollerbach & Rüdiger (2005) had clearly demonstrated a continuous and monotonic transition between HMRI and SMRI. On the other hand, HMRI was identified by Liu et al. (2006) as a weakly destabilized inertial oscillation, quite in contrast to the SMRI which represents a destabilized slow magneto-Coriolis wave. Only recently, this paradox was resolved by showing that the transition involves a spectral exceptional point at which the inertial wave branch coalesces with the branch of the slow magneto-Coriolis wave (Kirillov & Stefani, 2010).

The significance of the LLL, together with a variety of further predicted parameter dependencies, was experimentally confirmed in the PROMISE facility, a Taylor-Couette cell working with a low liquid metal (Stefani et al., 2006, 2007, 2009). Present experimental work at the same device (Seilmayer et al., 2014) aims at the characterization of the azimuthal MRI (AMRI), a non-axisymmetric “relative” of the axisymmetric HMRI, which dominates at large ratios of to (Hollerbach et al., 2010). However, AMRI as well as inductionless MRI modes with any other azimuthal wavenumber (which may be relevant at small values of ), seem also to be constrained by the LLL as recently shown in a unified treatment of all inductionless versions of MRI by Kirillov et al. (2012).

Actually, it is this apparent failure of HMRI, and AMRI, to apply to Keplerian profiles that has prevented a wider acceptance of those inductionless forms of MRI in the astrophysical community. Given the close proximity of the LLL () and the Keplerian Rossby number (), it is certainly worthwhile to ask whether any physically sensible modification would allow HMRI to extend to Keplerian flows.

Quite early, the validity of the LLL for had been questioned by Rüdiger & Hollerbach (2007). For the convective instability, they found an extension of the LLL to the Keplerian value in global simulations when at least one of the radial boundary conditions was assumed electrically conducting. Later, though, by extending the study to the absolute instability for the traveling HMRI waves, the LLL was vindicated even for such modified electrical boundary conditions by Priede (2011). Kirillov & Stefani (2011) made a second attempt by investigating HMRI for non-zero, but low . For it was found that the essential HMRI mode extends from only to a value , and allows for a maximum Rossby number of which is indeed slightly above the LLL, yet below the Keplerian value.

A third possibility may arise when considering that saturation of MRI could lead to modified flow structures with parts of steeper shear, sandwiched with parts of shallower shear (Umurhan, 2010).

A recent letter (Kirillov & Stefani, 2013) has suggested another way of extending the range of applicability of the inductionless versions of MRI to Keplerian profiles, and beyond. Rather than relying on modified electrical boundary conditions, or on locally steepened profiles, we have evaluated profiles that are flatter than . The main physical idea behind this attempt is the following: assume that in some low- regions, characterized by so that standard MRI is reliably suppressed, is still sufficiently large for inducing significant azimuthal magnetic fields, either from a prevalent axial field or by means of a dynamo process without any prescribed . Note that would only appear in the extreme case of an isolated axial current, while the other extreme case, , would correspond to the case of a homogeneous axial current density in the fluid which is already prone to the kink-type Tayler instability (Seilmayer et al., 2012), even at .

Imagine now a real accretion disk with its complicated conductivity distributions in radial and axial direction. For such real disks a large variety of intermediate profiles between the extreme cases and is well conceivable. Instead of going into those details, one can ask which profiles could make HMRI a viable mechanism for destabilizing Keplerian rotation profiles. By defining an appropriate magnetic Rossby number we showed that the instability extends well beyond the LLL, even reaching when going to . It should be noted that in this extreme case of uniform rotation the only available energy source of the instability is the magnetic field. Going then over into the region of positive in the plane, we found a natural connection with the ULL which was a somewhat mysterious conundrum up to present.

The present paper represents a significant extension of the short letter (Kirillov & Stefani, 2013). In the first instance, we will present a detailed derivation of the dispersion relation for arbitrary azimuthal modes in viscous, resistive rotational flows under the influence of a constant axial and a superposed azimuthal field of arbitrary radial dependence. For this purpose, we employ general short-wavelength asymptotic series following Eckhoff (1981, 1987), Bayly (1988), Lifschitz (1991), Lifschitz & Hameiri (1991), Friedlander & Vishik (1995), Vishik & Friedlander (1998), Hattori & Fukumoto (2003), and Friedlander & Lipton-Lifschitz (2003) as well as a WKB approach of Kirillov & Stefani (2010).

Second, we will discuss in much more detail the stability map in the -plane in the inductionless case of vanishing magnetic Prandtl number. For various limits we will discuss a number of strict results concerning the stability threshold and the growth rates. Special focus will be laid on the role that is played by the line , and by the point in particular.

Third, we will elaborate the dependence of the instability on the azimuthal wavenumber and on the ratio of the axial and radial wavenumbers and prove that the pattern of instability domains in the case of very small, but finite , is governed by a periodic band structure found in the inductionless limit.

Next, we will establish a connection between dissipation-induced destabilization of Chandrasekhar’s equipartition solution (a special solution for which the fluid velocity is parallel to the direction of the magnetic field and magnetic and kinetic energies are finite and equal (Chandrasekhar, 1956, 1961)) and the azimuthal MRI, and we will explore the links between the Tayler instability and AMRI.

Last, but not least, we will delineate some possible astrophysical and experimental consequences of our findings, although a comprehensive discussions of the corresponding details must be left for future work.

2 Mathematical setting

2.1 Non-linear equations

The standard set of non-linear equations of dissipative incompressible magnetohydrodynamics consists of the Navier-Stokes equation for the fluid velocity and of the induction equation for the magnetic field

(1)

where is the total pressure, is the hydrodynamic pressure, the density, the kinematic viscosity, the magnetic diffusivity, the conductivity of the fluid, and the magnetic permeability of free space. Additionally, the mass continuity equation for incompressible flows and the solenoidal condition for the magnetic induction yield

(2)

2.2 Steady state

We consider the rotational fluid flow in the gap between the radii and , with an imposed magnetic field sustained by electrical currents. Introducing the cylindrical coordinates we consider the stability of a steady-state background liquid flow with the angular velocity profile in helical background magnetic field (a magnetized Taylor-Couette (TC) flow)

(3)

Note that if the azimuthal component is produced by an axial current confined to , then

(4)

Introducing the hydrodynamic Rossby number by means of the relation

(5)

we find that the solid body rotation corresponds to , the Keplerian rotation to , whereas the velocity profile corresponds to . Note that although Keplerian rotation is not possible globally in TC-flow, an approximate Keplerian profile can locally be achieved.

Similarly, following Kirillov & Stefani (2013) we define the magnetic Rossby number

(6)

results from a linear dependence of the magnetic field on the radius, , as it would be produced by a homogeneous axial current in the fluid. corresponds to the radial dependence given by Eq. (4). Note that the logarithmic derivatives and introduced in the work by Ogilvie & Pringle (1996) are nothing else but the doubled hydrodynamic and magnetic Rossby numbers: , .

2.3 Linearization with respect to non-axisymmetric perturbations

To describe natural oscillations in the neighborhood of the magnetized Taylor-Couette flow we linearize equations (2.1)-(2) in the vicinity of the stationary solution (3) assuming general perturbations , , and and leaving only the terms of first order with respect to the primed quantities:

(7)

where the perturbations fulfil the constraints

(8)

Note that by adding/subtracting the second of equations (2.3) divided by to/from the first one yields the linearized equations in a more symmetrical Elsasser form; in the ideal MHD case they were derived, e.g. by Friedlander & Vishik (1995).

Introducing the gradients of the background fields represented by the two matrices

(9)

we write the linearized equations of motion in the form

(10)

3 Geometrical optics equations

Following Eckhoff (1981), Lifschitz (1989, 1991) and Friedlander & Lipton-Lifschitz (2003), we seek for solutions of the linearized equations (2.3) in terms of the formal asymptotic series with respect to the small parameter , :

(11)

where is a vector of coordinates, is a real-valued scalar function that represents the ‘fast’ phase of oscillations, and , , and , are ‘slow’ complex-valued amplitudes with the index denoting residual terms.

Following Landman & Saffman (1987), Lifschitz (1991), Dobrokhotov & Shafarevich (1992), and Eckhardt & Yao (1995) we assume further in the text that and and introduce the derivative along the fluid stream lines

(12)

Substituting expansions (3) into equations (2.3), taking into account the identity

(13)

as well as the relation

(14)

collecting terms at and we arrive at the system of local partial differential equations (cf. Friedlander & Vishik (1995))

(15)

From the solenoidality conditions (8) it follows that

(16)

Following Lifschitz (1991) we take the dot product of the first two of the equations (15) with , , and, in view of the constraints (3), arrive at the system

(17)

that has for , , and a unique solution

(18)

With the use of the relations (18) we simplify the last two of the equations (15) as

(19)

Multiplying the first of Eqs. (3) by from the left and then dividing both parts by we find an expression for . Substituting it back to the right hand side of the first of Eqs. (3) and taking into account the constraints (3), we eliminate the pressure terms and transform this equation into

(20)

quite in accordance with the standard procedure described, e.g., in Lifschitz (1991) and Vishik & Friedlander (1998). Differentiating the first of the identities (3) yields

(21)

Using the identity (21), we re-write Eq. (3) as follows

Now we take the gradient of the identity :

(22)

Denoting , we deduce from the phase equation (3) that

(23)

Hence, the transport equations for the amplitudes (3) take the final form

(24)

where is a identity matrix.

Eqs. (3) are local partial differential equations analogous to those derived (in terms of Elsasser variables ) by Friedlander & Vishik (1995) who considered non-axisymmetric perturbation of a rotating flow of an ideal incompressible fluid which is a perfect electrical conductor in the presence of the azimuthal magnetic field of arbitrary radial and axial dependency, see Appendix A.

In the absence of the magnetic field these equations can be treated as ordinary differential equations with respect to the convective derivative (12) and thus are reduced to that of Eckhardt & Yao (1995) who considered stability of the viscous TC-flow. Note that the same form of the transport equations (with the different matrix ) appears in the studies of elliptical instability by Landman & Saffman (1987) and of three-dimensional local instabilities of more general viscous and inviscid basic flows by Lifschitz (1991), Lifschitz & Hameiri (1991), and Dobrokhotov & Shafarevich (1992).

4 Dispersion relation of AMRI

In case when the axial field is absent, i.e. , we can derive the dispersion relation of AMRI directly from the transport equations (3) according to the procedure described in Friedlander & Vishik (1995).

Let the orthogonal unit vectors , , and form a basis in a cylindrical coordinate system moving along the fluid trajectory. With , , and with the matrix from (2.3), we find

(25)

Hence, the equation (23) in the coordinate form is

(26)

According to Eckhardt & Yao (1995) and Friedlander & Vishik (1995), in order to study physically relevant and potentially unstable modes we have to choose bounded and asymptotically non-decaying solutions of the system (26). These correspond to and and time-independent. Note that when this solution is compatible with the constraint following from (18).

Denoting , where , we find that we write the local partial differential equations (3) for the amplitudes in the coordinate representation. Analogously to Friedlander & Vishik (1995), we single out the equations for the radial and azimuthal components of the fluid velocity and magnetic field by using the orthogonality condition that follows from (3):

(27)

According to Friedlander & Vishik (1995) a natural step is to seek for a solution to Eqs. (4) in the modal form , in order to end up with the dispersion relation for the transport equations in case of AMRI:

(28)

Introducing the viscous and resistive frequencies and the angular velocity (Ogilvie & Pringle, 1996) corresponding to the azimuthal magnetic field:

(29)

so that is simply

(30)

we write the amplitude equations (4) as , where and

(31)

In the case when and the coefficients of the characteristic polynomial of the matrix (31) exactly coincide with the dispersion relation of ideal AMRI derived by Friedlander & Vishik (1995), see Appendix A for the detailed comparison.

5 Dispersion relation of HMRI

It is instructive to derive the dispersion relation of HMRI by another approach similar to that of Kirillov & Stefani (2010).

We write the linearized equations (2.3) in cylindrical coordinates and assume that , , and . This yields a set of equations containing only functions of and radial derivatives:

(32)
(33)

where . The same operation applied to (8) yields

(34)

Note that the third equation of (5) is already separated from the others.

Substituting and into (5) and assuming that is fixed for a local analysis, we introduce into the resulting equations the parameters

(35)

and finally set in these re-parameterized equations , where is a small parameter. After expanding the result in the powers of , we find that the first two of equations (5) take the form

(36)

Next, we express the pressure term from the third of equations (5) as follows:

(37)

Analogously, finding and from (34) we substitute these terms together with the pressure term (37) into the first two of equations (5), assume in the result that and and take into account expressions (35). Setting then in the equations and expanding them in , we obtain

(38)

The equations (38) and (36) yield a system with and (cf. Kirillov & Stefani (2013); Squire & Bhattacharjee (2014))