Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics

We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. In the presence of the magnetic field, the hydrodynamically stable flow can demonstrate non-axisymmetric azimuthal magnetorotational instability (AMRI) both in the diffusionless case and in the double-diffusive case with viscous and ohmic dissipation. Performing stability analysis of amplitude transport equations of short-wavelength approximation, we find that the threshold of the diffusionless AMRI via the Hamilton–Hopf bifurcation is a singular limit of the thresholds of the viscous and resistive AMRI corresponding to the dissipative Hopf bifurcation and manifests itself as the Whitney umbrella singular point. A smooth transition between the two types of instabilities is possible only if the magnetic Prandtl number is equal to unity, Pm=1. At a fixed Pm≠1, the threshold of the double-diffusive AMRI is displaced by finite distance in the parameter space with respect to the diffusionless case even in the zero dissipation limit. The complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities, the double-diffusive system reduces to a marginally stable system which is either Hamiltonian or parity–time-symmetric.


Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics
Oleg N. Kirillov  We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. In the presence of the magnetic field, the hydrodynamically stable flow can demonstrate non-axisymmetric azimuthal magnetorotational instability (AMRI) both in the diffusionless case and in the double-diffusive case with viscous and ohmic dissipation. Performing stability analysis of amplitude transport equations of short-wavelength approximation, we find that the threshold of the diffusionless AMRI via the Hamilton-Hopf bifurcation is a singular limit of the thresholds of the viscous and resistive AMRI corresponding to the dissipative Hopf bifurcation and manifests itself as the Whitney umbrella singular point. A smooth transition between the two types of instabilities is possible only if the magnetic Prandtl number is equal to unity, Pm = 1. At a fixed Pm = 1, the threshold of the double-diffusive AMRI is displaced by finite distance in the parameter space with respect to the diffusionless case even in the zero dissipation limit. The complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities, the double-diffusive system reduces to a marginally stable system which is either Hamiltonian or parity-time-symmetric.
2017 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/ by/4.0/, which permits unrestricted use, provided the original author and source are credited.

Introduction
While common sense tends to assign to dissipation the role of a vibration damper, as early as 1879 Kelvin and Tait predicted viscosity-driven instability of Maclaurin's spheroids (proved by Roberts & Stewartson in 1963 [1][2][3]), thus presenting a class of Hamiltonian equilibria, which, although stable in the absence of dissipation, become unstable due to the action of dissipative forces [4,5]. The universality of the dissipation-induced instabilities manifests itself in unexpected links between solid-and fluid mechanics [6][7][8]. For instance, the destabilizing action of viscous dissipation on the negative energy mode of rotation of a particle moving in a rotating cavity [9] selects backward whirling in the rotating frame as an unstable (anticyclonic) motion. Remarkably, this very instability mechanism described by Lamb in 1908 has recently reappeared as a trigger breaking the cyclone-anticyclone vortex symmetry in a rotating fluid in the presence of linear Ekman friction [10].
The onset of the classical Hopf bifurcation in a near-Hamiltonian dissipative system generically does not converge to the onset of the Hamilton-Hopf bifurcation of a Hamiltonian system when dissipation tends to zero [11]. For instance, the onset of secular instability (classical Hopf) of viscous Maclaurin spheroids does not tend to the onset of dynamical instability (Hamilton-Hopf) of inviscid Maclaurin spheroids in the limit of vanishing viscosity [1][2][3]. In meteorology this phenomenon is known as the 'Holopäinen instability mechanism' for a baroclinic flow when waves that are linearly stable in the absence of Ekman friction become dissipatively destabilized in its presence, with the result that the location of the curve of marginal stability is displaced by an order one distance in the parameter space, even if the Ekman number is infinitesimally small [5,[12][13][14][15]. A similar effect in solid mechanics is represented by the 'Ziegler destabilization paradox' [7,[16][17][18][19].
Swaters noticed in [13] that the stability boundary associated with the zero dissipation limit of a dissipative baroclinic instability theory does not collapse to the inviscid result when the Ekman dissipation is replaced by other dissipative mechanisms, e.g. by horizontal turbulent friction, confirming that such a singular limit is generic. However, he also managed to choose a specific dissipative perturbation (in which the dissipation is proportional to the geostrophic potential vorticity) possessing coincidence of the zero dissipation limit of the dissipative marginal stability boundary with the inviscid result [13].
The destabilization by dissipation is especially intriguing when several diffusion mechanisms act simultaneously [2,[20][21][22][23][24]. In this case, 'no simple rule for the effect of introducing small viscosity or diffusivity on flows that are neutral in their absence appears to hold' [25]. In hydrodynamics, a classical example is given by secular instability of the Maclaurin spheroids due to both fluid viscosity and gravitational radiation reaction, where the critical eccentricity of the meridional section of the spheroid depends on the ratio of the two dissipative mechanisms and reaches its maximum, corresponding to the onset of dynamical instability in the ideal system, exactly when this ratio equals 1 [2,22]. In solid mechanics, the generic character of the discontinuity of the instability threshold in the zero dissipation limit was already noticed in the work by Smith [26,27], who found that a viscoelastic shaft rotating in bearings with viscous damping is prone to dissipation-induced instability for almost all ratios of the damping coefficient of the shaft and the damping coefficient of the bearings, except one specific ratio.
In hydrodynamics and magnetohydrodynamics (MHD) the ratio of damping coefficients corresponding to different dissipative mechanisms is traditionally called the Prandtl number. For example, the Prandtl number, Pr = ν/κ, measures the relative strength of the diffusion of vorticity represented in the Navier-Stokes equations by the kinematic viscosity coefficient ν and thermal diffusion with the coefficient of thermal diffusivity κ [28,29]. The magnetic Prandtl number, Pm = ν/η, is the ratio of the coefficients of the kinematic viscosity and ohmic diffusion, η [28][29][30].
To get an idea of the key role of the Prandtl numbers in the correspondence between stability criteria in the diffusionless and the double-diffusive case, let us consider the Rayleigh centrifugal instability criterion and its extensions.  [30] predicts a stationary axisymmetric instability of an ideal incompressible Newtonian fluid, differentially rotating with the radially varying angular velocity where Ro is the fluid Rossby number and ∂ r = ∂/∂r. For a viscous fluid, the Rayleigh criterion (1.1) is modified as follows [34]: and reduces to the diffusionless criterion (1.1) as the Reynolds number, Re → ∞.
In the general multiple-diffusive case, the existence of such a direct correspondence between the diffusionless and diffusive stability criteria is not evident. In many cases, however, the reduction of the double-diffusive instability criteria to the diffusionless ones can be achieved by setting the corresponding Prandtl number to a specific value, e.g. to 1, and then tending diffusivities to zero (or, equivalently, the corresponding Reynolds numbers to infinity) [22].
For example, the stationary axisymmetric instability known as the double-diffusive Goldreich-Schubert-Fricke (GSF) instability [28,29,35] develops in a rotating viscous and thermally conducting fluid when the extended Rayleigh criterion is fulfilled [28]: where N is the Brunt-Väisälä frequency 1 [36], and p is the pressure of the fluid, ρ the density, γ the adiabatic index and g the radial acceleration. When dissipative effects are absent, ν = 0, κ = 0, the diffusionless GSF instability occurs for [28] 4(Ro Evidently, Pr = 1 is the only value at which the criterion (1.4) reduces to (1.5) in the limit Re → ∞. Similarly, Michael's criterion of ideal MHD [37] predicts stationary axisymmetric instability caused by an azimuthal magnetic field for a rotating flow of a non-viscous incompressible Newtonian fluid that is a perfect electrical conductor if [37] Ro where Rb is the magnetic Rossby number [38], and ω A φ is the Alfvén angular velocity related to the magnitude of the magnetic field [39]. Again, the diffusionless Michael's criterion (1.6) follows in the limit of Re → ∞ from its double-diffusive  [28,43] Ro + 1 − Pm In particular, Michael's criterion for both the diffusionless and the double-diffusive problem predicts stability with respect to axisymmetric perturbations for the rotating flow and the azimuthal magnetic field that satisfy the following constraints: (1.9) In 1956, Chandrasekhar [44] observed that the properties (1.9) correspond to an exact steady solution of the MHD equations for an incompressible fluid in the ideal case, i.e. when ν = 0 and η = 0. For this solution, the total pressure of the fluid and the magnetic field are constant, the fluid velocity at every point is parallel to the direction of the magnetic field at that point and the Alfvén angular velocity is equal to the angular velocity of the fluid, which implies equality of the densities of the fluid magnetic and kinetic energies. This energy equipartition solution of the ideal MHD was proved by Chandrasekhar [44] to be marginally stable against general perturbations. 3 To illustrate stability of the equipartition solution (1.9) with respect to non-axisymmetric perturbations, we substitute it into the following criterion of destabilization of a hydrodynamically stable rotating flow of an inviscid and perfectly conducting fluid by an azimuthal magnetic field: where m 1 is the azimuthal wavenumber and Ro < 0 [28,39,47]. The criterion (1.10) is valid in the limit of infinitely large axial and azimuthal wavenumbers of the perturbation. Naturally, the solution (1.9) violates (1.10) already at m ≥ 2, thus confirming the Chandrasekhar theorem [44]. Recently, Bogoyavlenskij [48] discovered that viscous and resistive incompressible MHD equations possess exact unsteady equipartition solutions with finite and equal kinetic and magnetic energies when the fluid velocity and the magnetic field are collinear and the kinematic viscosity ν is equal to the magnetic diffusivity η, i.e. when Pm = 1. Under the constraint Pm = 1, the Bogoyavlenskij unsteady equipartition solutions turn into the ideal and steady Chandrasekhar equipartition equilibria when ν = η → 0 [48].
One could expect, that in double-diffusive MHD, the remarkable stability of the Chandrasekhar energy equipartition solution is preserved under the constraint Pm = 1. As soon as the constraint is violated, one could anticipate a dissipation-induced instability of the equipartition solution. For instance, recent analytical works [38,43] (1.11) The above inequality predicts the onset of the azimuthal magnetorotational instability (AMRI) even in the case of the Keplerian rotating flow with Ro = − 3 4 when Rb > −25/32 [38]. In particular, (1.11) implies destabilization of the Chandrasekhar equipartition solution, whose susceptibility to the double-diffusive AMRI at Pm 1 has been confirmed numerically in [53,54].
According to the group-theoretical argument by Julien & Knobloch [55], AMRI is an oscillatory instability with a non-zero azimuthal wavenumber, which is most likely to develop in the presence of the azimuthal magnetic field [49,56]. Hence, its onset in the double-diffusion case is characterized by the classical Hopf bifurcation, at which simple eigenvalues cross the imaginary axis in the complex plane. On the other hand, the equations of the diffusionless MHD can be written in Hamiltonian form [57]. For this reason, the stable oscillatory non-axisymmetric modes in the ideal MHD case can carry both positive and negative energy; their interaction yields the Hamilton-Hopf bifurcation at the onset of the non-axisymmetric oscillatory instabilities [58].
In this study, we perform a local stability analysis of a circular Couette-Taylor flow of a viscous and electrically conducting fluid in an azimuthal magnetic field of arbitrary radial dependence. We obtain a unifying geometric picture that naturally connects the diffusionless and doublediffusive AMRI in low-and high-Pm regimes in the spirit of the singularity theory approach by Bottema [17], Arnold [59] and Langford [11] on generic singularities in the multiparameter families of matrices, which is especially efficient when combined with the perturbation of multiple eigenvalues, index theory and exploitation of the fundamental symmetries of the ideal system [6,8,[60][61][62].
After a brief re-derivation of the already known equations of the system in the shortwavelength approximation, we write the corresponding algebraic eigenvalue problem, which determines the dispersion relation, as a non-Hamiltonian perturbation of a Hamiltonian eigenvalue problem. The latter yields the dispersion relation of the ideal system. This allows us to investigate systematically the singular limit of the onset of the oscillatory AMRI due to the classical Hopf bifurcation at arbitrary Pm when viscous and resistive terms tend to zero.
In the frame of the local stability analysis, we show that the threshold of the double-diffusive AMRI tends to the threshold of the diffusionless AMRI only at Pm = 1 as the Reynolds numbers tend to infinity and find the Whitney umbrella singularity on the neutral stability surface that dictates this specific choice of Pm. We classify the stable oscillatory modes involved in the Hamilton-Hopf bifurcation by their Krein (or energy) sign. Then, we explicitly demonstrate by means of the perturbation theory for eigenvalues that when viscosity and ohmic diffusivity are weak (and even infinitesimally small), the dominance of viscosity destroys the stability of the negative energy mode at Pm > 1, whereas the dominance of ohmic diffusivity destabilizes the positive energy mode at Pm < 1 (including the inductionless case Pm = 0) in the close vicinity of the Hamilton-Hopf bifurcation. However, when the fluid Rossby number exceeds some critical value, the destabilization is possible only at finite values of Reynolds numbers and is accompanied by a transfer of instability between negative-and positive-energy modes that occurs due to the presence of complex exceptional points in the spectrum. This clarifies the reasons for instability of Chandrasekhar's equipartition solution and its extensions at both low and high Pm.

Transport equation for amplitudes and its dispersion relation
coupled with the induction equation for magnetic field B [36,43]: In equations (2.1), the total pressure is defined by P = p + B 2 /2μ 0 , where p is the hydrodynamic pressure, ρ = const. the density, ν = const. the kinematic viscosity, η = (μ 0 σ ) −1 the magnetic diffusivity, σ = const. the conductivity of the fluid and μ 0 is the magnetic permeability of free space. In addition, the incompressible flow and the solenoidal magnetic field fulfil the constraints V · u = 0 and V · B = 0.
It is well known that, for a flow differentially rotating in a gap between the radii r 1 and r 2 > r 1 , equations (2.1) and (2.2) possess a steady solution of the general form [53,63] u 0 (r) = r Ω(r) e φ , p = p 0 (r) and B 0 (r) in the cylindrical coordinate system (r, φ, z). In the magnetized circular Couette-Taylor flow (2.3), the angular velocity profile Ω(r) and the azimuthal magnetic field B 0 φ (r) are arbitrary functions of the radial coordinate r satisfying boundary conditions for an inviscid and non-resistive fluid [53,63]. For a viscous and resistive fluid, the angular velocity has the form Ω(r) = a + br −2 , while the expression for the magnetic field is given by B 0 φ (r) = cr + dr −1 with the coefficients determined from boundary conditions [53,63]. In the frame of the local linear stability analysis of the flow (2.3) that will be performed in the following, boundary conditions are ignored and the steady state of the double diffusive system is also the steady state of the diffusionless system.
In 1956, Chandrasekhar [44] observed that for the exact stationary solution (2.3) of equations (2.1) and (2.2) with Ω = B 0 φ /(r √ ρμ 0 ) and P = const. in the ideal case, i.e. when ν = 0 and η = 0, the kinetic and magnetic energies are in equipartition, ρ(Ωr) 2 /2 = (B 0 φ ) 2 /(2μ 0 ), and Ro = Rb = −1. The latter equality follows from the condition of constant total pressure and from the fact that, in the steady state, the centrifugal acceleration of the background flow is compensated by the pressure gradient, rΩ 2 = (1/ρ)∂ r p 0 [43]. Note that Ro = −1 corresponds to the velocity profile Ω(r) ∼ r −2 , whereas Rb = −1 corresponds to the magnetic field produced by an axial current I isolated from the fluid [43,49,53]: B 0 φ (r) = μ 0 I/(2π r). Linearizing equations (2.1) and (2.2) in the vicinity of the stationary solution (2.3) by assuming general perturbations u = u 0 + u , p = p 0 + p and B = B 0 + B , leaving only the terms of first order with respect to the primed quantities, and introducing the gradients of the background fields represented by the two 3 × 3 matrices we arrive at the linearized system of MHD [38,43,64] ⎛ where the perturbations fulfil the constraints

(b) Derivation of the amplitude transport equations
Let be a small parameter (0 < 1). We seek solutions of the linearized equations (2.5) in the form of asymptotic expansions with respect to the small parameter [65]: x is a vector of coordinates, Φ represents the phase of the wave or the eikonal, and u (j) , B (j) and p (j) , j = 0, 1, r, are complex-valued amplitudes. The index r denotes the remainder terms that are assumed to be uniformly bounded in on any fixed time interval [66,67]. Maslov [68] observed that high-frequency oscillations exp(i −1 Φ(x, t)) quickly die out because of viscosity unless one assumes a quadratic dependency of viscosity on the small parameter . Following [34,68,69], we assume that ν = 2ν and η = 2η .
Substituting expansions (2.7) in (2.5) and collecting terms at −1 and 0 , we find [43] −1 : The solenoidality conditions (2.6) yield Taking the dot product of the first of the equations in system (2.8) with VΦ under the constraints (2.10), we find that, for VΦ = 0, Under condition (2.11), equation (2.8) has a non-trivial solution if the determinant of the 6 × 6 matrix in its left-hand side vanishes. This gives us two characteristic roots corresponding to the two Alfvén waves [64,65] that yield the following two Hamilton-Jacobi equations: (2.12) The characteristic roots −u 0 ± B 0 / √ ρμ 0 · VΦ are triple and semi-simple and degenerate into a semi-simple characteristic root of multiplicity 6 on the surface [64,65] B 0 · VΦ = 0. Using relations (2.11), (2.13) and (2.14), we simplify equations (2.9): Eliminating pressure in the first of equations (2.15) via multiplication of it by VΦ and taking into account the constraints (2.10), then using the identities and, finally, defining k = VΦ, we write the transport equations for the amplitudes (2.15) as and where I is the 3 × 3 identity matrix. From phase equation (2.16), we deduce that Equations (2.17) and (2.18) are valid under the assumption that condition (2.13) is fulfilled. Local partial differential equations (2.17) are fully equivalent to the transport equations of [8,43]. In the case of the ideal MHD when viscosity and resistivity are zero, equations (2.17) exactly coincide with those of the work [64] and are fully equivalent to the transport equations derived in [70]. In the absence of the magnetic field, these equations are reduced to that of the work [34] that considered stability of the viscous Couette-Taylor flow.
Note that the leading-order terms dominate solution (2.7) for a sufficiently long time, provided that is small enough [66,67], which reduces analysis of instabilities to the investigation of the growth rates of solutions of transport equations (2.17).
According to [34,70], in order to study physically relevant and potentially unstable modes, we have to choose bounded and asymptotically non-decaying solutions of system (2.18). These correspond to k φ ≡ 0, and k R and k z time-independent. Note that this solution is compatible with the constraint B 0 · k = 0 following from (2.13).

(c) Dispersion relation of the double-diffusive amplitude equations
Define α = k z |k| −1 , |k| 2 = k 2 R + k 2 z and introduce the Alfvén angular velocity, the viscous and resistive frequencies, and the hydrodynamic and magnetic Reynolds numbers [43]: In particular, Rm = Re Pm. Looking for a solution to equations (2.17) in the modal form [70]: u (0) =û e αΩλt+imφ , B (0) = √ ρμ 0B e αΩλt+imφ , we write the amplitude equations in the matrix form [38,43,71] The ratio n = m/α is the modified azimuthal wavenumber and S = ω A φ /Ω is the Alfvén angular velocity in the units of Ω. Let us introduce a Hermitian matrix and define an indefinite inner product in C 4 as [x, y] =ȳ T Gx [8,72] and a standard inner product as (x, y) =ȳ T x. The matrix H 0 = −iGA 0 is Hermitian too: Consequently, the eigenvalue problem A 0 z = λz can be written in the Hamiltonian form with the Hamiltonian H 0 [8,72,73]: The fundamental symmetry where the overbar denotes complex conjugation, implies the symmetry of the spectrum of the matrix A 0 with respect to the imaginary axis [8,72]. The full eigenvalue problem (2.20) is thus a dissipative perturbation of the Hamiltonian eigenvalue problem (2.24): where H 1 = −iGA 1 is a complex non-Hermitian matrix:  [8,72]. Denote by p the vector of all parameters of the matrix H 0 : p = (S, Ro, Rb, n) T ∈ R 4 . Let at p = p 0 the matrix H 0 = H(p 0 ) have a double imaginary eigenvalue λ = iω 0 (ω 0 ≥ 0) with the Jordan chain consisting of the eigenvector z 0 and the associated vector z 1 that satisfy the following equations [8,72]: Transposing these equations and applying the complex conjugation yields As a consequence,z T 0 Gz 0 = 0 andz T 1 Gz 0 +z T 0 Gz 1 = 0 or, in the other notation, Varying parameters along a curve p = p(ε) (p(0) = p 0 ), where ε is a real parameter, and assuming the Newton-Puiseux expansions for the double eigenvalue iω 0 and its eigenvector in powers of ε 1/2 when |ε| is small, we find [8] λ ± = iω 0 ± iω 1 ε 1/2 + o(ε 1/2 ) and z ± = z 0 ± iω 1 z 1 ε 1/2 + o(ε 1/2 ), (3.4) with Taking into account thatz T 0 Hz 0 is real andz T 1 Gz 0 is imaginary, we assume that ω 1 > 0, which is a reasonable assumption in view of the fact that ω 0 > 0 and |ε| is small. Then, for ε > 0, the double eigenvalue iω 0 splits into two pure imaginary ones λ ± = iω 0 ± iω 1 √ ε (stability). When ε < 0, the splitting yields a pair of complex eigenvalues with real parts of different sign (instability). Therefore, varying parameters along a curve p(ε), we have a linear Hamilton-Hopf bifurcation at the point p 0 , which is a regular point of the boundary between the domains of stability and oscillatory instability. The path p(ε) crosses the stability boundary at the point p 0 . Calculating the indefinite inner product for the perturbed eigenvectors z ± at ε > 0, we find [8] [z + , Therefore, the simple imaginary eigenvalue λ + with the eigenvector u + has the Krein sign which is opposite to the Krein sign of the eigenvalue λ − with the eigenvector u − . With decreasing ε > 0, the imaginary eigenvalues λ + and λ − with opposite Krein signs move towards each other along the imaginary axis until at ε = 0 (i.e. at p = p 0 ) they merge and form the double imaginary eigenvalue iω 0 , which further splits into two complex eigenvalues when ε takes negative values. The opposite Krein signs is a necessary and sufficient condition for the imaginary eigenvalues participating in the merging to leave the imaginary axis [6,72,73]. Below we demonstrate the Krein collision at the onset of the diffusionless AMRI by calculating the roots of the dispersion relation both analytically and numerically.
If δ = 0, i.e. Ro = RbS 2 , then equation (3.7) simplifies and its roots are [43] and The eigenvalues λ 1,2,3,4 are imaginary and simple for all 0 < n ≤ 2 if 0 ≤ S < 1. The equality S = 1 implies Ro = Rb and the existence of a double zero eigenvalue which is semi-simple at all 0 ≤ n ≤ 2 except n = 1 where it has a Jordan block of order 2; the other two eigenvalue branches are formed by simple imaginary eigenvalues (marginal stability). At S > 1, complex eigenvalues originate (oscillatory instability) if At the boundary of domain (3.9), the eigenvalues are double imaginary with a Jordan block.
Let us now assume that S = 1. Then, inequality (3.10) takes the form and the dispersion relation at S = 1 factorizes as follows: The equality in (3.11) corresponds to the transition from marginal stability to oscillatory instability via the linear Hamilton-Hopf bifurcation (figure 1). At the marginal stability curve with S = 1, one of the eigenvalues λ is always zero and simple, another one is simple and imaginary, and the last two form a double and imaginary eigenvalue with the Jordan block. At S = 1 and Rb = −1, the critical value of the fluid Rossby number follows from (3.11) and is equal to (3.15) Naturally, such explicit expressions for double imaginary eigenvalues can be obtained with the use of (3.13) and (3.14) for any other value of n. The choice of n does not influence the qualitative picture of eigenvalue interaction shown in figure 2. The value n = √ 2 is known to be optimal in several respects [43,56,74], which will be discussed further in the text.
Notice that the eigenvalues λ + and λ − of Chandrasekhar's equipartition solution have the opposite Krein signs:

Dissipation-induced instabilities of the double-diffusive system (a) Dissipative perturbation of simple imaginary eigenvalues
The complex non-Hermitian matrix of the dissipative perturbation can be decomposed into its Hermitian and anti-Hermitian components: At large Rm, an increment δλ to a simple imaginary eigenvalue λ with an eigenvector z is given by a standard perturbation theory [4,8,60,61] as By contrast, the increment δλ A = i((H A 1 z, z)/[z, z]) is real. For instance, the eigenvalues λ + and λ − of Chandrasekhar's equipartition solution acquire the following increments: where h is the harmonic mean of the two Reynolds numbers.

(b) Weak ohmic diffusion destabilizes positive energy waves at low Pm
In the close vicinity of the critical Rossby number of the Hamilton-Hopf bifurcation Ro c ≈ −1.07855, the real increment δλ A to imaginary eigenvalues λ 1 with negative Krein sign and λ 2 with positive Krein sign are shown in figure 3b for fixed Rm = 10 3 and varying Pm (the fluid Reynolds number is calculated as Re = Rm/Pm). The eigenvalues with the negative Krein sign become dissipatively destabilized when Pm > 1, i.e. when the losses due to viscosity of the fluid exceed the ohmic losses (cf. [40]). Remarkably, the eigenvalues with the positive Krein sign can also acquire positive growth rates. However, this happens at Pm < 1 when the electrical resistivity prevails over the kinematic viscosity. Indeed, the destabilizing influence of the kinematic viscosity of the fluid on negative energy waves is well known in hydrodynamics [6,20,25,40], which therefore places the dissipation-induced instability at Pm > 1 and |Ro − Ro c | 1 into an established context. The destabilization of positive energy modes was noticed in the context of solid mechanics, in particular, in gyroscopic systems with damping and non-conservative positional (or circulatory, or curl [75]) forces in [  Radiative dissipation due to emission of electromagnetic, acoustic and gravitational waves is a well-known reason for instability of modes of positive energy in hydrodynamics and plasma physics [2,22,23,76]. To the best of our knowledge, the dissipative destabilization of the positive energy modes due to ohmic losses has not been previously reported in MHD. The interval of negative real increments in figure 3b decreases with the decrease in deviation from the critical value of the Rossby number at the Hamilton-Hopf bifurcation, i.e. as Ro = Ro − Ro c tends to zero. When Ro = 0, the stable interval reduces to the single value: Pm = 1. Hence, weak ohmic diffusion (weak kinematic viscosity) destabilizes positive (negative) energy waves at Pm < 1 (Pm > 1) if |Ro − Ro c | is sufficiently small.
(c) Diffusionless and double-diffusive criteria are connected at Pm = 1 We complement the sensitivity analysis of eigenvalues of the diffusionless Hamiltonian eigenvalue problem with respect to a double-diffusive perturbation with the direct computation of the stability boundaries based on the algebraic Bilharz stability criterion. The Bilharz criterion [77] guarantees localization of all the roots of a complex polynomial of degree n to the left of the imaginary axis in the complex plane, provided that all principal minors of even order of the 2n × 2n Bilharz matrix composed of the real and imaginary parts of the coefficients of the polynomial are positive [8].
Applying the Bilharz criterion to the characteristic polynomial of the eigenvalue problem Approaching the origin along the ray Pm = 1 means letting the Reynolds numbers tend to infinity with their ratio being kept equal to unity. Figure 4b demonstrates that, in the limit Re = Rm → ∞, the neutral stability curve of the double-diffusive system approaches the threshold of instability of the diffusionless system from below. The instability domain of the doublediffusive system always remains smaller than in the diffusionless case. As a consequence, the Chandrasekhar equipartition solution (Ro = Rb = −1, S = 1), being stable in the diffusionless case, remains stable at Pm = 1 no matter what the value of the Reynolds numbers is ( figure 4b).
Indeed, in the case when Ro = RbS 2 and Re = Rm, the roots of the characteristic polynomial of the eigenvalue problem (2.26) can be found explicitly and The eigenvalues (4.4) are just the eigenvalues (3.8) that are shifted by dissipation to the left in the complex plane (asymptotic stability). This fact agrees perfectly with the result of Bogoyavlenskij [48], who found at Pm = 1 exact unsteady energy equipartition solutions of the viscous and resistive incompressible MHD equations that relax with the growth rate equal to −1/Re = −1/Rm < 0 to the ideal and steady Chandrasekhar equipartition equilibria [44]. Note also that even earlier Lerner and Knobloch reported a 'cooperative, accelerated decay' of solutions at Pm = 1 in the study of stability of the magnetized plane Couette flow [33].
Well known is a similar result on the secular instability of the Maclaurin spheroids due to both fluid viscosity and gravitational radiation reaction 5 when the value of the critical eccentricity of the meridional section at the onset of instability in the ideal case is attained only when the ratio of the two dissipation mechanisms is exactly 1 [22,76]. The unfolding of the eigenvalue crossing into the avoided crossing can happen in two different ways depending on the sign of Pm − 1. At Pm < 1 (Pm > 1), the complex eigenvalues stemming from the imaginary eigenvalues of the diffusionless system with positive (negative) Krein sign form a branch that bends to the right and crosses the imaginary axis at some Ro(Re, Rm) = Ro c (figure 5a, cf. [2]). The critical values Ro(Re, Rm) of the double-diffusive system live on the surface in the (Re −1 , Rm −1 , Ro)-space that has a self-intersection along the Ro-axis (figure 5b). The angle of the self-intersection tends to zero as Ro → Ro c and at the point (0, 0, Ro c ) the surface has a singularity known as the Whitney umbrella 6 [7,11,59]. In the vicinity of the Ro-axis, the instability threshold is effectively a ruled surface [17], where the slope of each ruler is determined by Pm. Letting the Reynolds numbers tend to infinity while keeping the magnetic Prandtl number fixed means that the Ro-axis is approached in the (Re −1 , Rm −1 , Ro)-space along a ruler corresponding to this value of Pm. Generically, for all values of Pm except Pm = 1, a ruler leads to a limiting value of Ro that exceeds Ro c and thus extends the instability interval of the fluid Rossby numbers with respect to that of the diffusionless system, as is visible in figures 5b and 6a. The plane Pm = 1 divides the neutral stability surface in the vicinity of Ro = Ro c into two parts corresponding to positive energy modes destabilized by the dominating ohmic diffusion at Pm < 1 and to negative energy modes destabilized by the dominating fluid viscosity at Pm > 1 (figure 5b). The ray determined by the conditions Re = Rm > 0, Ro = Ro c belongs to the stability domain of the double-diffusive system and contains exceptional points (4.3) that determine 7 behaviour of eigenvalues shown in figure 5a. Figure 6a shows that, at a fixed Pm = 1, the critical value of Ro at the onset of the doublediffusive AMRI is displaced by an order one distance along the Ro-axis with respect to the critical value Ro c of the diffusionless case, when both viscous and ohmic diffusion tend to zero. This effect does not depend on the choice of n ( figure 6b). Indeed, the critical values of Ro in the limit of vanishing dissipation at a fixed Pm and S = 1 and Rb = −1 satisfy the following equation: (Pm + 1) 3 n 2 (3Ro 2 − 4n 2 − 14Ro − 9) + 4(PmRo − 3Ro − 4)(2PmRo + 3Pm + 1) 2 = 16(Pm + 1)Pm(Ro + 1) 2 n 2 . (4.5) Using this equation, one can easily check analytically that the critical Ro has its minimum at Pm = 1, independent of the choice of n (figure 6c, cf. [22]). Nevertheless, the displacement is rather small if Pm ∈ [0, 1], with the maximum attained at Pm = 0 where the diffusionless limit
We see that there exists a critical value of the magnetic Prandtl number Pm l < 1 such that, at Pm ∈ [0, Pm l ], the Chandrasekhar equipartition solution with Rb = Ro = −1, and S = 1 is destabilized by dissipation when viscosity is sufficiently small and ohmic diffusion is sufficiently large. By contrast, at Ro − Ro c 1 the marginally stable diffusionless system can be destabilized at Pm < 1 when both viscosity and resistivity are infinitesimally small (figure 4a).
To understand how these instabilities are related to each other, we plot the neutral stability curves in the plane of inverse Reynolds numbers Re −1 , Rm −1 ∈ [−0.5, 1] for Ro ∈ [Ro c , −1] (figure 7a). Although negative Reynolds numbers have no physical meaning, it is instructive to extend the neutral stability curves to the corresponding region of the parameter plane. At Ro = Ro c , the stability domain is inside the area bounded by a curve having a cuspidal singularity at the origin with the tangent line at the cuspidal point specified by the condition Pm = 1; this geometry yields destabilization by infinitesimally small dissipation at all Pm = 1.
As soon as Ro departs from Ro c , the cusp at the origin transforms into a self-intersection, the angle of which increases with the increase in Ro and becomes equal to π at Ro = −1. For this reason, at Ro close to −1 the neutral stability curve partially belongs to the region of negative Reynolds numbers which makes destabilization by infinitesimally small dissipation impossible for all Pm > 0. In particular, at S = 1 and vanishing viscosity the ohmic diffusion is stabilizing in the interval 0 < Rm −1 < Rm −1 * when Ro > Ro Rm := 6Rb 2 − Rbn 2 + 6n 2 − 2(n 2 − 3Rb) Rb 2 + 3n 2 3n 2 . (iii) Transfer of instability between modes when Pm significantly deviates from 1 At both exceptional points, there exists a pair of simple imaginary eigenvalues and a double imaginary eigenvalue λ with a Jordan block: A segment of the anti-diagonal between the exceptional points is a part of the stability boundary at Ro = −1 and all the eigenvalues at the points of this segment are imaginary. We see that the domain of asymptotic stability at Ro = −1 extends to the region of negative Reynolds numbers and that, at the constraint Rm = −Re, the double-diffusive system has imaginary spectrum on the interval between the two exceptional points. If we interpret the negative dissipation as an energy gain, then, formally, we could say that, at Rm = −Re, the energy gain is compensated by the energy loss. Non-Hermitian systems in which gain and loss are balanced are known as parity-time (PT) symmetric systems [62,80]. The interval of marginal stability of the PT-symmetric system forms a self-intersection singularity on the stability boundary of a general dissipative system with the Whitney umbrella singularities at the exceptional points corresponding to double imaginary eigenvalues [8,62]. Therefore, the neutral stability surface of our double-diffusive system contains the interval of self-intersection on the Ro-axis (Ro > Ro c ) that is orthogonal at Ro = −1 to the interval of the anti-diagonal with the slope Pm = −1 confined between the two exceptional points. At the exceptional points of this interval and at the exceptional point on the Ro-axis at Ro = Ro c , the neutral stability surface in the (Rm −1 , Re −1 , Ro)-space has three Whitney umbrella singularities. The singularities 'hidden' in the region of negative Reynolds numbers are responsible for the separation of domains of AMRI due to weak or strong dissipation.
It turns out that this separation is not only quantitative but also qualitative, as comparison of the movement of eigenvalues demonstrates at fixed Re = 1000 and Rm = 500 in figure 5a and at Re = 1000 and Rm ≈ 1.789 in figure 7b. In both cases, Pm < 1. However, in the case of Pm = 0.5, it is the branch with lower negative frequencies corresponding to the perturbed imaginary eigenvalues with positive Krein sign of the diffusionless Hamiltonian system that becomes unstable due to prevailing ohmic diffusion. By contrast, at much smaller Pm ≈ 0.001789 the instability moves to a branch with higher negative frequencies that can be seen as stemming from the imaginary eigenvalues with negative Krein sign of the diffusionless Hamiltonian system. Keeping Re = 1000 and slightly increasing the magnetic Reynolds number to Rm ≈ 2.095, we see at Ro = −1 the crossing of the eigenvalue branches at the double eigenvalue λ EP − ≈ −i0.5086 − 0.2391. The crossing transforms into another avoided crossing when Rm = 2.5. At Rm = 2.5, again, it is the branch corresponding to higher negative frequencies (positive Krein sign) that is destabilized by dissipation (figure 7b).
In fact, when Re = 1000 is given, the branch corresponding to the unperturbed imaginary eigenvalues with positive Krein sign is destabilized by dissipation when the magnetic Reynolds number decreases from Rm = 1000 (Pm = 1) to Rm ≈ 2.095 (Pm ≈ 0.002095). As soon as Rm < 2.095 (Pm < 0.002095), the instability is transferred to a branch corresponding to the unperturbed imaginary eigenvalues with negative Krein sign. The reason is the existence of a set in the stability domain corresponding to double complex eigenvalues. This set exists at Ro = −1 and consists of the two straight lines   We see that the three Whitney umbrella points and, related to them, three lines of double complex eigenvalues (marked in figure 7a as EP ± and EP 0 ) actually control the dissipationinduced destabilization, acting as switches of unstable modes. The singular geometry of the neutral stability surface guides the limiting scenarios and connection of the double-diffusive system to a Hamiltonian or to a PT-symmetric one.
(iv) Connection between the lower and upper Liu limits at Pm 1 Let us keep Re = 1000 and allow the magnetic Reynolds number to decrease beyond the critical value Rm EP − ≈ 2.095. During this process, the pattern of interacting eigenvalues remains qualitatively the same (cf. figures 7b and 8a). However, an important new feature appears as the magnetic Prandtl number approaches the inductionless limit Pm = 0. Indeed, at Re = 1000 and Rm = 0.01 corresponding to Pm = 10 −5 , one and the same eigenvalue branch has unstable parts both at Ro < 0 and at Ro > 0 (figure 8a). This is in striking contrast to the case of moderately small magnetic Prandtl numbers shown in figure 7b or to the diffusionless case when the instability occurs only at Ro < 0.
The Bilharz criterion reveals two regions of instability in the (S, Ro)-plane for Rb = −1, n = √ 2 and Re = 1000 and Rm = 0.01 (figure 8b). The first one exists at Ro < 2 − 2 √ 2 < 0 and the second one at Ro > 2 + 2 √ 2 > 0. In the gap between the lower Liu limit (2 − 2 √ 2) and the upper Liu limit (2 + 2 √ 2), the system is stable [43,79]. Both Liu limits are attained when Re → ∞ and Rm → 0. If the double-diffusive instability domain at Ro < 0 can be considered as a deformation of the instability domain of the diffusionless system, the instability of the magnetized circular Couette-Taylor flow in superrotation [74] at Ro > 0 turns out to exist only in the presence of dissipation. Remarkably, the two seemingly different instabilities are caused by the eigenvalues living on a single eigenvalue branch in the complex plane (figure 8a).
The oscillatory instability at Pm 1 of a circular Couette-Taylor flow in an azimuthal magnetic field with Rb = −1 and Ro < 2 − 2 √ 2, i.e. the AMRI, has already been observed in recent experiments with liquid metals [49]. We therefore identify the observed inductionless AMRI at Pm 1 as simply the manifestation of a dissipation-induced instability of waves of negative energy of the diffusionless system caused by the prevailing ohmic diffusion. In particular, at Ro = Rb = −1 and S = 1 the inductionless AMRI is the dissipation-induced instability of the Chandrasekhar equipartition solution.

Conclusion
We have studied AMRI of a circular Couette-Taylor flow of an incompressible electrically conducting Newtonian fluid in the presence of an azimuthal magnetic field of arbitrary radial dependence. With the use of geometrical optics asymptotic solutions, we have reduced the problem to the analysis of the dispersion relation of the transport equation for the amplitude of a localized perturbation. We have represented the corresponding matrix eigenvalue problem in the form of a Hamiltonian diffusionless system perturbed by ohmic diffusion and fluid viscosity. We have established that the diffusionless AMRI corresponds to the Krein collision of simple imaginary eigenvalues with the opposite Krein (or energy) sign and have derived an analytic expression for the instability threshold of the diffusionless system using the discriminant of the complex polynomial dispersion relation. We have demonstrated that the threshold of the doublediffusive AMRI with equal viscosity and electrical resistivity (Pm = 1) smoothly converges to the threshold of the diffusionless AMRI in the limit of the infinitesimally small dissipation, and this result does not change when other parameters are varied.
In contrast with the case when the coefficients of viscosity and resistivity are equal, the prevalence of resistivity over viscosity or vice versa causes the AMRI in the parameter regions where the diffusionless AMRI is prohibited, for instance, in the case of super rotating flows. In particular, non-equal and finite viscosity and resistivity destabilize the celebrated Chandrasekhar energy equipartition solution. Analysing the neutral stability surface of the double-diffusive system, we have found that: -marginally stable Hamiltonian equilibria of the diffusionless system form an edge on the neutral stability surface of the double-diffusive system that ends up with the Whitney umbrella singular point at the onset of the Hamilton-Hopf bifurcation; -another edge with the two Whitney umbrella singular points at its ends corresponds to marginally stable double-diffusive systems with the balanced energy gain and loss (PTsymmetric systems); -three codimension-2 sets corresponding to complex double-degenerate eigenvalues with Jordan blocks (exceptional points) stem from each of the Whitney umbrella singularities and live in the stability domain of the double-diffusive system; -the sets of exceptional points control transfer of instability between modes of positive and negative energy, whereas the Whitney umbrellas govern the limiting scenarios for the instability thresholds including the case of vanishing dissipation; -AMRI can be interpreted as an instability of the Chandrasekhar equipartition solution induced by finite dissipation when either Pm ∈ [0, 1) is sufficiently small or Pm ∈ (1, ∞) is sufficiently large; -inductionless AMRI occurring both at Ro < 0 and Ro > 0 when Pm 1 is caused by the eigenvalues of the one and the same branch stemming from the negative energy modes of the diffusionless system, as in the classical dissipation-induced instability.
Data accessibility. This paper has no additional data. Competing interests. I declare I have no competing interests.