Short wavelength local instabilities of a circular Couette flow with radial temperature gradient
aa r X i v : . [ phy s i c s . f l u - dyn ] D ec This draft was prepared using the LaTeX style file belonging to the Journal of Fluid Mechanics Short wavelength local instabilities of acircular Couette flow with radialtemperature gradient
Oleg N. Kirillov † and Innocent Mutabazi Steklov Mathematical Institute, Russian Academy of Sciences, Gubkina 8, Moscow 119991,Russia Laboratoire Ondes et Milieux Complexes (LOMC), UMR 6294, CNRS-Universit´e du Havre,Normandie Universit´e, B.P. 540, 76058 Le Havre Cedex, France(Received xx; revised xx; accepted xx)
We perform a linearized local stability analysis for short-wavelength perturbations ofa circular Couette flow with the radial temperature gradient. Axisymmetric and non-axisymmetric perturbations are considered and both the thermal diffusivity and thekinematic viscosity of the fluid are taken into account. The effect of the asymmetry ofthe heating both on the centrifugally unstable flows and on the onset of the instabilitiesof the centrifugally stable flows, including the flow with the Keplerian shear profile, isthoroughly investigated. It is found that the inward temperature gradient destabilizes theRayleigh stable flow either via Hopf bifurcation if the liquid is a very good heat conductoror via steady state bifurcation if viscosity prevails over the thermal conductance.
Key words:
1. Introduction
Circular Couette flow of a viscous Newtonian fluid between two coaxial differentiallyrotating cylinders is a canonical system for modeling instabilities leading to spatio-temporal patterns and transition to turbulence in many natural and industrialprocesses. Modern astrophysical applications require understanding of basic instabilitymechanisms in rotating flows with the Keplerian shear profile in accretion- andprotoplanetary disks that are hydrodynamically stable according to the centrifugalRayleigh criterion. Usually, these instabilities are a consequence of additionalfactors such as electrical conductivity of the fluid and magnetic field of thecentral gravitating object (Chandrasekhar 1961; Lifshitz 1987; Friedlander & Vishik1995; Urpin & Brandenburg 1998; Kucherenko & Kryvko 2013; Kirillov et al. et al. et al. † Email address for correspondence: [email protected]
O. N. Kirillov, I. Mutabazi hydrodynamic finite-amplitude nonlinear instabilities (Marcus et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. et al. nstabilities of a circular Couette flow with radial temperature gradient et al. (2015) andprovide analytical expressions for the onset of oscillatory and stationary instabilities,the coordinates of the codimension-2 points, and the Hopf frequency. In the case ofthe sole outer cylinder rotation, solid body rotation, and rotating flow with Keplerianshear we find the destabilizing effect of the inward temperature gradient that leads tothe oscillatory instability at small values of the Prandtl number (Pr <
1) and to thestationary instability at Pr >
2. Equations of motion
We consider the flow of an incompressible viscous fluid of the density ρ , the kinematicviscosity ν and the thermal diffusivity κ in the presence of a radial temperature gradientapplied to a differentially rotating cylindrical annulus. The ratio Pr = ν/κ constitutesthe Prandtl number.The governing equations written in the inertial frame of reference read (Yoshikawa et al. et al. ∇ · u = 0 , (2.1) ∂ u ∂t + u · ∇ u + 1 ρ ∇ p − ν∆ u + αθ g c = 0 , (2.2) ∂θ∂t + u · ∇ θ − κ∆θ = 0 , (2.3)where p is the pressure and θ is the temperature deviation from a reference temperature.The parameter α is the coefficient of thermal expansion.The flow equations are written in cylindrical coordinates ( r, ϕ, z ) in which the velocityfield u = ( u, v, w ) T , where the superscript T denotes transposition. The term u · ∇ u contains the centrifugal-like acceleration g c = ( v /r, , T and the Coriolis-like acceler-ation (0 , uv/r, T coming from the geometry. The Oberbeck-Boussinesq approximation(Chandrasekhar 1961) is used, i.e. we assume small variations of the density with thetemperature only in the centrifugal force term leading to the centrifugal buoyancy whilethe other fluid properties ( ν, κ, α ) are kept constant. This approximation allows oneto keep the flow incompressibility condition and eliminates the acoustic modes from theproblem analysis. The application of the Oberbeck-Boussinesq approximation to rotatingflows can be found in (Lopez et al. g is neglected in order to focusonly on the destabilization effects of the centrifugal buoyancy αθ g c .We assume that the cylindrical annulus has an infinite length. The system of equations(2.1) possesses a stationary axisymmetric solution describing the base flow state: u B =(0 , V ( r ) = rΩ ( r ) , T , p B = P ( r ), and θ B = Θ ( r ). Consider a perturbation of this baseflow state: u = u B + e u , p = p B + e p , and θ = θ B + e θ . Then the equations of the flowperturbations linearized around the base flow state read ∇ · e u = 0 , (2.4) O. N. Kirillov, I. Mutabazi ∂ e u ∂t + u B · ∇ e u + e u · ∇ u B − ν∆ e u + 1 ρ ∇ e p + αθ B e g c + α e θ g cB = 0 , (2.5) ∂ e θ∂t + u B · ∇ e θ + e u · ∇ θ B − κ∆ e θ = 0 , (2.6)with the basic centrifugal gravity g cB = ( rΩ , , T and the perturbative centrifugalgravity e g c = (2 Ω e v, , T so that we can write the centrifugal buoyancy terms as follows αθ B e g c = 2 Ωαθ B e r e Tϕ e u , (2.7) α e θ g cB = α e θrΩ e r , (2.8)where e r and e ϕ are the radial and azimuthal unit vectors, respectively.The gradients of the background fields are ∇ u B = Ω − , ∇ θ B = DΘ , (2.9)where Ro = rDΩ Ω (2.10)is the Rossby number and D = d/dr . Denoting U = ∇ u B , we formulate the linearizedequations of motion in the final form: ∇ · e u = 0 , (2.11) (cid:18) ∂∂t + U + 2 Ωαθ B e r e Tϕ + u B · ∇ (cid:19) e u − ν∆ e u + 1 ρ ∇ e p + αrΩ e r e θ = 0 , (2.12) (cid:18) ∂∂t + u B · ∇ (cid:19) e θ + e u · ∇ θ B − κ∆ e θ = 0 . (2.13)
3. Evolution of the localized perturbations
Let ǫ be a small parameter (0 < ǫ ≪ e u = (cid:16) u (0) ( x , t ) + ǫ u (1) ( x , t ) (cid:17) exp (cid:0) iǫ − Φ ( x , t ) (cid:1) + ǫ u ( r ) ( x , t, ǫ ) , (3.1) e θ = (cid:16) θ (0) ( x , t ) + ǫθ (1) ( x , t ) (cid:17) exp (cid:0) iǫ − Φ ( x , t ) (cid:1) + ǫθ ( r ) ( x , t, ǫ ) , (3.2) e p = (cid:16) p (0) ( x , t ) + ǫp (1) ( x , t ) (cid:17) exp (cid:0) iǫ − Φ ( x , t ) (cid:1) + ǫp ( r ) ( x , t, ǫ ) , (3.3)where Φ is generally a complex-valued scalar function that represents the phase of thewave or the eikonal and the remainder terms u ( r ) , θ ( r ) , p ( r ) are assumed to be uniformlybounded in ǫ on any fixed time interval (Eckhoff 1981; Lifschitz 1991; Lifschitz & Hameiri1991, 1993; Lifschitz et al. (cid:0) iǫ − Φ ( x , t ) (cid:1) quickly dieout because of viscosity unless one assumes quadratic dependency of viscosity on thesmall parameter ǫ . Hence, following (Maslov 1986; Dobrokhotov & Shafarevich 1992;Kirillov et al. ν = ǫ e ν and κ = ǫ e κ .Substituting the asymptotic series (3.1) to the incompressibility condition (2.11) andcollecting terms at ǫ − and ǫ , we find ǫ − : u (0) · ∇ Φ = 0 , (3.4) nstabilities of a circular Couette flow with radial temperature gradient ǫ : ∇ · u (0) + i u (1) · ∇ Φ = 0 . (3.5)Similar procedure applied to (2.12) and (2.13) yields the two systems of equations ǫ − : (cid:18) ∂Φ∂t + u B · ∇ Φ ∂Φ∂t + u B · ∇ Φ (cid:19) (cid:18) u (0) θ (0) (cid:19) = − ∇ Φρ (cid:18) p (0) (cid:19) , (3.6) ǫ : i (cid:18) ∂Φ∂t + u B · ∇ Φ ∂Φ∂t + u B · ∇ Φ (cid:19) (cid:18) u (1) θ (1) (cid:19) = − i ∇ Φρ (cid:18) p (1) (cid:19) − (cid:18) ∂∂t + u B · ∇ + U + e ν ( ∇ Φ ) ∂∂t + u B · ∇ + e κ ( ∇ Φ ) (cid:19) (cid:18) u (0) θ (0) (cid:19) − (cid:18) Ωαθ B e r e Tϕ αrΩ e r ( ∇ θ B ) T (cid:19) (cid:18) u (0) θ (0) (cid:19) − ∇ ρ (cid:18) p (0) (cid:19) . (3.7)The amplitudes with the superscript (0) are contained both in the equations (3.6) corre-sponding to the lowest degree of ǫ equal to − ∇ Φ underthe constraint (3.4) we find that for ∇ Φ = 0 p (0) = 0 . (3.8)Under the condition (3.8) the system (3.6) has a nontrivial solution if the determinantof the 4 × ∂Φ∂t + u B · ∇ Φ = 0 (3.9)with the initial data: Φ ( x ,
0) = Φ ( x ).Taking the gradient of (3.9) yields the eikonal equation (Lifschitz & Hameiri 1991) (cid:18) ∂∂t + u B · ∇ (cid:19) ∇ Φ + ∇ u B · ∇ Φ = 0 (3.10)with the initial condition ∇ Φ ( x ,
0) = ∇ Φ ( x ). In the notation k = ∇ Φ and ddt = ∂∂t + u B · ∇ the eikonal equation (3.10) is (Lifschitz 1991) d k dt = −∇ u B · k = −U T k , (3.11)where U T denotes the transposed 3 × U (Eckhardt & Yao 1995).Relations (3.8) and (3.9) allow us to reduce the system (3.7) to (cid:18) ddt + U + e ν ( ∇ Φ ) + 2 αΩθ B e r e Tϕ (cid:19) u (0) + αrΩ e r θ (0) = − i ∇ Φρ p (1) , (3.12)( ∇ θ B ) T u (0) + (cid:18) ddt + e κ ( ∇ Φ ) (cid:19) θ (0) = 0 . (3.13)Multiplying the equation (3.12) with ∇ Φ from the left, we isolate the pressure term p (1) = iρ ∇ Φ ( ∇ Φ ) · (cid:18)(cid:20) ddt + U + 2 αΩθ B e r e Tϕ (cid:21) u (0) + αrΩ e r θ (0) (cid:19) . (3.14) O. N. Kirillov, I. Mutabazi
Taking into account the identity (Lifschitz & Hameiri 1991; Kirillov et al. ddt ( ∇ Φ · u (0) ) = d ∇ Φdt · u (0) + ∇ Φ · d u (0) dt = 0 (3.15)we modify (3.14) in the following way p (1) = iρ ∇ Φ ( ∇ Φ ) · (cid:16) ( U + 2 αΩθ B e r e Tϕ ) u (0) + αrΩ e r θ (0) (cid:17) − iρ ∇ Φ ) d ∇ Φdt · u (0) . (3.16)Using now (3.11) we re-write (3.16) in terms of the wave vector k p (1) = 2 iραΩθ B k T e r e Tϕ | k | u (0) + iραrΩ k T e r | k | θ (0) + 2 iρ k T U| k | u (0) . (3.17)Substituting (3.17) into (3.12) we finally arrive at the transport equations for the leading-order amplitudes u (0) and θ (0) in the expansions (3.1) and (3.2): d u (0) dt = − e ν | k | u (0) − (cid:18) I − kk T | k | (cid:19) U u (0) − αΩθ B (cid:18) I − kk T | k | (cid:19) e r e Tϕ u (0) − αrΩ (cid:18) I − kk T | k | (cid:19) e r θ (0) , (3.18) dθ (0) dt = − e κ | k | θ (0) − ( ∇ θ B ) T u (0) (3.19)with the initial data u (0) ( x ,
0) = u (0)0 ( x ) and θ (0) ( x ,
0) = θ (0)0 ( x ).Note that the leading order terms dominate the solution (3.1) and (3.2) for a sufficientlylong time, provided that ǫ is small enough (Lifschitz 1991; Lifschitz et al. x at theinitial moment t = 0 : d x dt = u B , x (0) = x . (3.20)Then, the eikonal equation (3.11) can be interpreted as an ordinary differential equationdescribing the evolution of the wave vector k ( t ) = ∇ Φ ( x ( t ) , t ) along the stream line(3.20) with the initial condition k (0) = k = ∇ Φ ( x ). Consequently, the transportequations (3.18) and (3.19) can also be treated as the system of ODEs along the streamlines of the flow for the amplitudes u (0) ( x ( t ) , t ) and θ (0) ( x ( t ) , t ) with the initial data u (0) (0) = u (0)0 ( x ) and θ (0) (0) = θ (0)0 ( x ). Therefore, the characteristic equations (3.20),(3.11), (3.18), and (3.19) describe motion of the envelope of a perturbation localized atthe initial moment of time at x along with the particles in the fluid flow that passthrough x at t = 0.
4. Dispersion relation and its parameterizations
Following the procedure described in (Friedlander & Vishik 1995) we write the eikonalequation (3.11) in cylindrical coordinates r, ϕ, z∂ k ∂t = − Ω Ro 00 0 00 0 0 k ( t ) (4.1) nstabilities of a circular Couette flow with radial temperature gradient k r = const , k ϕ = 0, and k z = const (Friedlander & Vishik 1995;Eckhardt & Yao 1995). With this, we can write the amplitude transport equations (3.18)in the coordinate form as (cid:20) ∂∂t + Ω ∂∂ϕ + e ν | k | (cid:21) u (0) r = β h Ω (1 − αθ B ) u (0) ϕ − αrΩ θ (0) i , (4.2) (cid:20) ∂∂t + Ω ∂∂ϕ + e ν | k | (cid:21) u (0) ϕ = − Ω (1 + Ro) u (0) r , (4.3) (cid:20) ∂∂t + Ω ∂∂ϕ + e κ | k | (cid:21) θ (0) = − DΘu (0) r , (4.4) (cid:20) ∂∂t + Ω ∂∂ϕ + e ν | k | (cid:21) u (0) z = − β k r k z h Ω (1 − αθ B ) u (0) ϕ − αrΩ θ (0) i , (4.5)where β = k z | k | − and | k | = p k r + k z .We see that the equations (4.2), (4.3), and (4.4) form a closed sub-system withrespect to u (0) r , u (0) ϕ , and θ (0) . Indeed, the transformation u (0) z → ( − k r /k z ) u (0) r makesthe equation (4.5) equivalent to (4.2). Following (Friedlander & Vishik 1995) we willlook for a solution to this sub-system in the modal form as θ (0) = ˆ θe λt + imϕ , and writethe amplitude equations in the matrix form Hξ = λξ , where ξ = (ˆ u r , ˆ u ϕ , ˆ θ ) T is theeigenvector associated with the eigenvalue λ of the matrix † H = − imΩ − e ν | k | Ωβ (1 − αΘ ) − αrΩ β − Ω (1 + Ro) − imΩ − e ν | k | − DΘ − imΩ − e κ | k | . (4.6)The solvability condition for a non-trivial solution yields the dispersion relation p ( λ ) = det( H − λI ) = 0 , (4.7)where I is the 3 × ‡ .Let us introduce the nondimensional parametersTa = βΩ e ν | k | , ˆ Θ = Θ∆T , γ a = α∆T, Rt = rDΘ Θ , n = mβ , s = λβΩ , (4.8)where Ta is the Taylor number, Rt is the thermal analog of the Rossby number and itmeasures the local slope of the temperature profile ¶ , ∆T is the temperature differenceimposed at the cylindrical surfaces bounding the flow, ˆ Θ represents the dimensionlesstemperature, n is a modified azimuthal wavenumber, and s is the dimensionless eigen-value. Introducing the diagonal matrix R = diag(1 , , ∆T /r ), we transform (4.6) into R − HR = βΩ − in − β (1 − γ a ˆ Θ ) − γ a βΩ − β (1 + Ro) − in − − Θ Rt βΩ − in − (4.9) † Cf. Eq. (A 19) that is obtained in the narrow gap approximation. ‡ Note that a similar dispersion relation was derived also in Economides & Moir (1980) inthe case when the radial acceleration g does not depend on r (see Appendix). In our case g ( r ) = g cB = ( rΩ , , T . ¶ not to be confused with the thermal Rossby number used in Atmospheric Physics (Lappa2012). O. N. Kirillov, I. Mutabazi
Rotation case Ro Pr S Pr H Inner cylinder − − η − η ln η − η ( γ a − ) − h η ln η − η ( γ a − ) i − Outer cylinder η − η η − η ( γ a − ) − h − η − η ( γ a − ) i − Keplerian − √ η + η (1+ √ η ) √ η ln η (1+ √ η ) ( γ a − ) − h − √ η ln η (1+ √ η ) ( γ a − ) i − Solid body 0 4ln η ( γ a − ) − h − η ( γ a − ) i − Table 1.
Rossby numbers for different rotating cases of the Couette-Taylor system accordingto Eq. (4.16) and the corresponding asymptotic values (6.4) that bound the intervals of Pr atwhich stationary (Pr S ) and oscillatory (Pr H ) modes are admissible. and re-write the dispersion relation (4.7) in the equivalent formdet( R − HR − sβΩI ) = − β Ω q ( s ) = 0 , (4.10)where q ( s ) is a third degree polynomial with complex coefficients that do not contain theparameters β and Ω and depend entirely on the dimensionless control parameters Ta,Pr, ˆ Θ , γ a , n and on the logarithmic derivatives of the velocity and temperature profilesof the base flow represented by the dimensionless hydrodynamic and thermal Rossbynumbers Ro and Rt, respectively.Introducing the new spectral variable χ = s + in into the complex dispersion relation q ( s ) = 0, we transform it into a real third degree polynomial equation in χ : χ + a χ + a χ + a = 0 . (4.11)The real coefficients a i of the characteristic polynomial (4.11) depend on the dimension-less flow parameters as follows: a = 1Ta " + 4(1 + Ro)(1 − γ a ˆ Θ )Pr − γ a ˆ Θ Rt , (4.12) a = 4(1 + Ro)(1 − γ a ˆ Θ ) − γ a ˆ Θ Rt + Pr + 2PrTa , (4.13) a = 2Pr + 1TaPr . (4.14)The set of dimensionless parameters (4.8) is convenient for applications in astrophysicalgasdynamics (Acheson & Gibbons 1978; Balbus & Potter 2016). In order to facilitatecomparison with the experimental data we need to express these parameters in terms ofthe realistic Taylor-Couette system. In this case, the base flow state confined in the gapbetween two coaxial cylinders of radii a and b rotating with the angular frequencies Ω a and Ω b , respectively, is given by (Chandrasekhar 1961; Meyer et al. Ω = µ − η − η + 1 − µ − η η (1 − η ) r , ˆ Θ = ln(1 − η ) r ln η , (4.15)where η = a/b < µ = Ω b /Ω a is the rotation ratio. The angularvelocity is measured in the units of Ω a and the radial distance in the units of d = b − a .Evaluating the parameters (2.10), (4.8), and (4.15) at the geometric mean radius r g = nstabilities of a circular Couette flow with radial temperature gradient √ η/ (1 − η ) (as proposed by Dubrulle et al. (2005)), we find † ˆ Θ = 12 , Rt = 1ln η ,
Ro = − µ − η (1 − η )( η + µ ) . (4.16)Relations (4.16) allow us to interpret the local Rossby numbers Ro and Rt in termsof the radius and velocity ratios η and µ of the realistic Taylor-Couette cell, see Table 1containing the four different rotation cases: The sole rotation of the inner cylinder ( µ = 0),the Keplerian rotation ( µ = η / ), the sole rotation of the outer cylinder ( µ → ∞ ), andthe solid body rotation ( µ = 1). In particular, Rt < < η < ˆ Θ
5. Local instabilities
Diffusionless instabilities
In the case when the viscosity and thermal diffusivity are both set to zero in (4.6) or,equivalently, Ta → ∞ in (4.9), the dispersion relation (4.11) factorizes to χ (cid:16) χ + 4(1 + Ro)(1 − γ a ˆ Θ ) − γ a ˆ Θ Rt (cid:17) = 0 . (5.1)The root χ = 0 corresponds to the pure imaginary eigenvalue λ = βΩs = − imΩ describing a stable rotating wave with the local frequency Ω . Similarly, the flow admitsstable inertial waves with frequencies given by ωΩ = − m ± β q − γ a ˆ Θ ) − γ a ˆ Θ Rt , (5.2)if the radicand in (5.2) is positive:4(1 + Ro)(1 − γ a ˆ Θ ) − γ a ˆ Θ Rt > . (5.3)These inertial waves result from the combined effects of the rotation and temperaturegradient. While the term γ a ˆ Θ is very weak in the Boussinesq approximation, the term γ a ˆ Θ Rt can become significant depending on the steepness of the temperature profile. Inthe isothermal case ( γ a = 0), we retrieve the frequency of the Kelvin waves in rotatingflows.The inequality (5.3) contains the generalized Rayleigh criterion derived in (Meyer et al. et al. (2015) found the criterion for the stability against axisymmetric perturba-tions in the diffusionless case in terms of the generalized Rayleigh discriminant Ψ ( r ): Ψ ( r ) > , Ψ ( r ) = Φ − γ a ˆ ΘΦ + d ˆ Θdr V r ! , Φ ( r ) = 1 r d ( rV ) dr . (5.4)Expressing the classical hydrodynamic Rayleigh discriminant Φ ( r ) via the hydrody-namic Rossby number as Φ = 4 Ω (1 + Ro), we obtain Ψ = Ω n − γ a ˆ Θ [Rt + 2(1 + Ro)] o . (5.5) † In the case of counter-rotating cylinders, the hydrodynamic Rossby number diverges at µ = − η . For µ − η , one may choose the geometric mean radius of the potentially unstablezone. O. N. Kirillov, I. Mutabazi
Requiring
Ψ >
Ψ < − γ a ˆ Θ ) − γ a ˆ Θ Rt < diffusionless Goldreich-Schubert-Fricke (GSF) instability (Acheson & Gibbons 1978).5.2. Enhancement of stability by viscosity and thermal diffusivity at
Pr = 1In the case when Pr = 1 (i.e. when the viscous and thermal diffusion timescales areequal) the dispersion relation (4.11) can also be factorized: (cid:18) χ + 1Ta (cid:19) "(cid:18) χ + 1Ta (cid:19) + 4(1 − γ a ˆ Θ )(1 + Ro) − γ a ˆ Θ Rt = 0 . (5.6)By this reason, the roots of the dispersion relation (5.6) can be found explicitly: s , = − − in ± i q − γ a ˆ Θ ) − γ a ˆ Θ Rt , (5.7) s = − − in. (5.8)The mode with the eigenvalue s is a pure hydrodynamic mode as it does not containthe temperature gradient, it is damped by the viscosity. The modes corresponding to theeigenvalue s , contain both the hydrodynamic and thermal effects. One of the modes isalways damped, the other one should be excited when its growth rate vanishes, i.e. when4(1 + Ro)(1 − γ a ˆ Θ ) − γ a ˆ Θ Rt + 1Ta = 0 . (5.9)The double-diffusive stability criterion given by the requirement for the left hand sideof Eq. (5.9) to be positive suggests that at Pr = 1 the viscosity and thermal diffusivityenlarge the stability domain of the diffusionless system. Furthermore, in the limit Ta → ∞ the roots (5.7) and (5.8) reduce to the roots of the diffusionless dispersion relation (5.1),whereas the double-diffusive stability criterion reduces to the diffusionless criterion (5.3).Note that the limiting procedure that first makes the diffusion coefficients of a dis-sipative system with two diffusion mechanisms equal and then tends them to zerotypically yields a correct diffusionless stability criterion in many double-diffusive systemsof hydrodynamics and magnetohydrodynamics (Kirillov et al. Double-diffusive flow stability criteria at arbitrary
PrWe derive stability conditions of the base flow at arbitrary Pr by applying the Lienard-Chipart stability criterion (Kirillov 2013) to the real polynomial (4.11) of degree 3 in χ : a > , a > , − a ( a − a a ) > . (5.10)The second inequality is always fulfilled whereas the first one yields:4(1 + Ro)(1 − γ a ˆ Θ ) − γ a ˆ Θ RtPr + 1Ta > . (5.11)The condition (5.11) is a modified Rayleigh criterion that takes into account both thekinematic viscosity and the thermal diffusion. Writing it as1 + Ro > γ a ˆ Θ RtPr4(1 − γ a ˆ Θ ) − (1 − γ a ˆ Θ ) , nstabilities of a circular Couette flow with radial temperature gradient < ˆ Θ
1, the kinematic viscosity and the outwardheating ( γ a >
0) are stabilizing and the inward heating ( γ a <
0) is destabilizing withrespect to the Rayleigh criterion (1 + Ro >
0) for the ideal incompressible fluid.At Pr = 1 the condition (5.11) reduces to the criterion (5.9). Furthermore, the limitTa → ∞ applied to (5.11) yields the diffusionless criterion (5.3) only if Pr = 1.The condition a = 0 yields the control parameter Ta as a function of the other controlparameters: Ta Sc = 1 q γ a ˆ Θ RtPr − − γ a ˆ Θ ) . (5.12)Since at a = 0 the dispersion equation (4.11) has the root χ = 0, corresponding to s = − in , the relation (5.12) is the threshold of the double-diffusive Goldreich-Schubert-Frickeinstability (Acheson & Gibbons 1978), which is a stationary axisymmetric instability inthe case when n = 0.The third inequality of (5.10) yields − (cid:18) γ a ˆ Θ (Pr + 1)Rt − − γ a ˆ Θ )(1 + Ro) − (Pr + 1) PrTa (cid:19) > , (5.13)from which the threshold of the oscillatory instability through a Hopf bifurcation natu-rally follows: Ta Hc = Pr + 1 q Pr(Pr + 1) γ a ˆ Θ Rt − (1 − γ a ˆ Θ )(1 + Ro) . (5.14)Indeed, from the Vieta’s formulas (Vinberg 2003) χ + χ + χ = − a , χ χ + χ χ + χ χ = a , χ χ χ = − a . (5.15)it follows that at Ta = Ta Hc the characteristic equation (4.11) admits two imaginary roots χ , = ± i ¯ ω c , corresponding to s = − in ± i ¯ ω c where n can be either zero or not, and onereal root χ = − a . The real root corresponds to a damped mode while the imaginaryroots yield the Hopf frequency ¯ ω c given by¯ ω c = q − γ a ˆ Θ ) − γ a ˆ Θ RtPr(Pr + 1)Pr + 1 (5.16)and describe a marginal oscillatory mode. The total frequency of this marginal mode isgiven by ω c = − βΩ ( n ± ¯ ω c ). The marginal value Ta Hc corresponds to a Hopf bifurcationof the base flow to a state oscillating with a Hopf frequency ¯ ω c . For n = 0, the marginalmode is an oscillatory axisymmetric mode while for n = 0, the marginal mode is anoscillatory non-axisymmetric mode.5.4. Codimension-2 points in the case of the Rayleigh-unstable flows
In the parameter space, the boundary of the stationary instability (5.12), correspondingto a simple zero root, and the boundary of the Hopf bifurcation (5.14), corresponding toa pair of imaginary roots, may have common codimension-2 points with the coordinates(Pr ∗ , Ta ∗ ) given for Ro + 1 < γ a > ∗ = −
12 + 12 s − γ a ˆ Θ ) γ a ˆ Θ (1 + Ro)Rt , (5.17)2 O. N. Kirillov, I. Mutabazi
Rot. Ta Sc Ta Hc ω c /Ω Inn. h − γ a ) η − η + γ a Prln η i − h − γ a ) η − η + γ a η i − β h η ( γ a − − η − γ a Pr(1+Pr)2ln η i Ray. h γ a Prln η i − Sol. h γ a − γ a Prln η i − h γ a − γ a η i − β h − γ a ) − γ a Pr(1+Pr)2ln η i Kep. h √ η (2 − γ a )(1+ √ η ) + γ a Prln η i − h √ η (2 − γ a )(1+ √ η ) + γ a η i − β h √ η ( γ a − √ η ) − γ a Pr(1+Pr)2ln η i Out. h − γ a ) η − + γ a Prln η i − h γ a − − η + γ a η i − β h − γ a )1 − η − γ a Pr(1+Pr)2ln η i Table 2.
Thresholds of the critical modes and the critical Hopf frequency for different rotationregimes in the Couette-Taylor system: (Inn.) inner cylinder rotating, (Ray.) Rayleigh line, (Sol.)solid body rotation, (Kep.) Keplerian rotation, (Out.) outer cylinder rotating. Ta ∗ = 1 q γ a ˆ Θ RtPr ∗ − − γ a ˆ Θ )(1 + Ro) . (5.18)5.5. The Rayleigh line
In the particular case Ro = −
1, corresponding to the Rayleigh line, the roots of thedispersion relation are found explicitly for an arbitrary Pr: χ , = − Pr + 12TaPr ± r Θγ a + (Pr − Pr , χ = − . One of the first two roots vanishes at the threshold of stationary instability Ta = Ta Sc ,where ( γ a < , Rt <
0) Ta Sc = 1 q γ a ˆ Θ RtPr . Thus, at the Rayleigh line the stationary instability is possible for the inward heatingonly. When the heating is outward and Rt <
0, both the criterion (5.11) and (5.13) isfulfilled and the flow is stable.
6. Stationary and oscillatory modes
The relations (5.12) and (5.14) contain besides Pr, the group of parameters Ro, γ a ˆ Θ and γ a ˆ Θ Rt, Rt <
0. Therefore, the conditions of the occurrence of the stationary oroscillatory modes can be analyzed in terms of these parameters.Let us define the Brunt-V¨ais¨al¨a frequency N as (Meyer et al. † N = 1 ρ ∇ ρ · g cB = 1 ρ dρdr V r . Taking into account that ρ ( r ) = ρ (1 − αθ ( r )) in the Boussinesq approximation, weconclude that N = − γ a ˆ Θ Rt Ω .In the diffusionless case, for Ro < −
1, the flow is Rayleigh unstable, i.e. the radialstratification of the square of the angular momentum is negative; for Ro > −
1, the † These authors have defined N with the opposite sign. nstabilities of a circular Couette flow with radial temperature gradient Figure 1.
Rotating inner cylinder with µ = 0, η = 0 .
99. (Left) Stability diagram at γ a = 0 . ∗ ≈ . , Ta ∗ / Ta ≈ . χ : χ < χ > χ = 0 at the codimension-2 point. (Center) The critical values ofTa / Ta versus γ a at Pr ≈ .
55 with the codimension-2 point ( γ a = 0 . , Ta / Ta ≈ . χ ) at Pr ≈ .
55 and(red) γ a = − . γ a = 0, (green) γ a = 0 . γ a = 0 . flow is Rayleigh stable (Chandrasekhar 1961). The centrifugal acceleration is orientedaway from the rotation axis, implying that the outward heating ( γ a >
0) correspondsto a stable stratification of the density (N >
0) while the inward heating ( γ a < < becomes significant. These inertialwaves, observed in numerical simulations, have been recently reported in (Meyer et al. Stationary modes
Taking into account the relations (4.16) we write the threshold for the onset of thestationary instability (5.12) in terms of the parameters of the Couette-Taylor flow r S = Ta Sc Ta = (cid:20) − γ a γ a Prln η (Ta ) (cid:21) − / , (6.1)where we have introduced the threshold for the isothermal flow as followsTa Sc ( γ a = 0) = 12 r − ≡ Ta . (6.2)For the circular Couette flow, the quantity (Ta ) is positive when Ro + 1 < µ < η ), i.e. in the Rayleigh unstable zone and (Ta ) < >
0, i.e. in theRayleigh stable zone (Chandrasekhar 1961). The explicit expressions for Ta Sc calculatedfor different rotating regimes of the Couette-Taylor flow are given in Table 2.In Figure 1 we plot the threshold (6.1) in the Rayleigh-unstable case of the innercylinder rotation with µ = 0 and Ro + 1 = − η/ (1 − η ) <
0, see Table 1. We see that theratio r S <
1, i.e. there is a destabilization, for the inward heating ( γ a < r S > O. N. Kirillov, I. Mutabazi
Figure 2.
Keplerian case with µ = η / and η = 0 .
99: (Upper left) Variation of thethreshold of the stationary instability and Hopf bifurcation with Pr at γ a < H ≈ . H ≈ .
98, Pr S ≈ .
01 and Pr S ≈ . S = const. inthe ( η, γ a )–plane. (Lower row) Variation of the frequencies (Im χ ) and the growth rates (Re χ )with Pr at Ta = 6 and γ a = − .
01 (the inward temperature gradient). The green lines correspondto oscillatory modes, and the red lines to the stationary modes. for the outward heating ( γ a > b S = − γ a Pr / ln η , where | γ a | ≪ et al. S = b Sη/ (1 − η ).The stationary modes can exist if the radicand in (5.12) is positive, this yields acondition on Pr to be satisfied. In the case of the Rayleigh unstable flow, i.e. when1 + Ro <
0, the stationary modes exist for any value of Pr when γ a < Pr < Pr S when γ a >
0, where the critical value of the Prandtl number isPr S = 2(1 + Ro)(1 − γ a ˆ Θ ) γ a ˆ Θ Rt . (6.3)The values of Pr S calculated in terms of the parameters (4.16) for different rotatingcases of the Couette-Taylor system are given in Table 1. In the case of the rotating inner nstabilities of a circular Couette flow with radial temperature gradient Figure 3. (Upper left) Solid body rotation with µ = 1 and η = 0 .
99. Thresholds of the stationaryinstability and Hopf bifurcation with the asymptotic values of Pr marked by dot-dashed lines:Pr H ≈ . H ≈ . S ≈ . S ≈ . µ = ∞ and η = 0 .
99: Thresholds of the stationary instability and Hopf bifurcationwith the asymptotic values of the Prandtl number: Pr H ≈ . S ≈ .
02. (Lowerrow) Variation of the (left) frequencies and (right) growth rates with Pr for the outer cylinderrotation ( µ = ∞ , η = 0 .
99) at given Ta = 1000 and γ a = − .
01. The green lines correspond tothe oscillatory modes and the red one corresponds to a (damped) stationary mode. cylinder shown in the left panel of Figure 1 the asymptotic value (6.3) for the threshold(6.1) is Pr S ≈ > γ a <
0) at Pr > Pr S . Figure 2 shows the variationof the threshold with Pr for the Keplerian rotation in the small gap case η = 0 .
99; thecorresponding stability diagrams for the cases of the solid body rotation and the outercylinder rotation are shown in Figure 3.Note that for the Rayleigh unstable flows the stationary modes exist even in the limitof the vanishing Prandtl number (Pr → Sc = 1 q − − γ a ˆ Θ )that does not depend on the sign of γ a .6 O. N. Kirillov, I. Mutabazi
Figure 4.
Rotating inner cylinder with µ = 0 and η = 0 .
99. The critical frequency ω c atthe onset of (red) stationary instability and (green) Hopf bifurcation in the units of βΩ (left)versus Pr at γ a = 0 . γ a according to Eq. (6.5). The limiting frequency(6.10) at γ a = 0 . ω ∞ βΩ ≈ . ω c as a functionof the Brunt-V¨ais¨al¨a frequency N according to Eq. (6.9) and Eq. (6.11). The critical frequencyvanishes at N c /Ω ≈ . c /Ω ≈ .
28 for Pr = 100, and at N c /Ω ≈ . For the Rayleigh-stable flows (1 + Ro >
0) the threshold of the stationary modes(5.12) decreases towards zero at γ a < → ∞ ), i.e. for highly viscous and poorly heat-conducting fluids, see Figure 2 and 3.6.2. Hopf bifurcation
In terms of the parameters (4.16) and (6.2) the threshold for the onset of oscillatoryinstability (5.14) becomes r H = Ta Hc Ta = Pr + 1Pr (cid:20) − γ a γ a η Pr + 1Pr (Ta ) (cid:21) − / (6.4)and the Hopf frequency of the oscillatory modes (5.16) acquires the form ω c Ω = β Pr + 1 1Ta s − γ a − γ a Pr(Pr + 1)2ln η (Ta ) . (6.5)The left and the central panels of Figure 1 illustrate the variation of the threshold ofthe Hopf bifurcation with Pr and γ a for the circular Couette flow with the inner cylinderrotating alone. The variation of the Hopf frequency with Pr and γ a in this rotating caseis shown in Figure 4. Similarly to the stationary instability, both the threshold (6.4) andthe frequency (6.5) of the oscillatory modes depend explicitly on the Prandtl number Prand on the temperature drop parameter b S in agreement with the numerical calculationsof (Meyer et al. γ a > r H > r S >
1, i.e. the outward heating is stabilizingfor both stationary and oscillatory modes with respect to the isothermal flow. The growthrates shown in the right panel of Figure 1 demonstrate that the complex roots do notexist at γ a
0; they appear at γ a > nstabilities of a circular Couette flow with radial temperature gradient ∗ = −
12 + 12 s − − γ a γ a ln η (Ta ) , (6.6)Ta ∗ = Ta " − γ a γ a Pr ∗ (Ta ) ln η − / . (6.7)At the codimension-2 point one of the roots of the dispersion relation is double zeroand the other one is simple real and negative, corresponding to a damped mode, as thegrowth rates demonstrate in the right panel of Figure 1.The threshold for the excitation of oscillatory modes and the corresponding criticalfrequency of the oscillatory modes for both the Rayleigh-stable and the Rayleigh-unstablerotation regimes in the Couette-Taylor system are summarized in Table 2. For theRayleigh unstable flows with the outward heating the oscillatory instability actuallyoccurs at Pr > Pr ∗ , where Pr ∗ is given by Eq. (5.17) or Eq. (6.6), see Figure 1.Figure 2 and 3 show the boundaries of stationary instability and Hopf bifurcation inthe (Pr , Ta) - plane for the Rayleigh-stable flows such as Keplerian rotation, solid-bodyrotation and the case of the outer cylinder rotating. We see also that the frequency ofthe oscillatory instability is increasing while approaching the outer cylinder rotating casewhereas the growth rate of the oscillatory instability at low Pr is decreasing.All these Rayleigh stable flows (1 + Ro > µ > η ) with the inward heating( γ a <
0) share the same property. Namely, the oscillatory modes are excited when thermaldiffusion dominates over the fluid viscosity0 < Pr < Pr H = γ a ˆ Θ Rt γ a ˆ Θ Rt − − γ a ˆ Θ ) < . (6.8)This result follows from the observation that the Hopf bifurcation from the base flow canoccur if the radicand in (5.14) is positive. In contrast, the stationary modes are excitedin the Rayleigh-stable flow by the inward heating when Pr > Pr S >
1, where Pr S is givenin (6.3) and Table 1.Therefore, for the Rayleigh stable flows the inward radial temperature gradient destabi-lizes the flow depending on the value of Pr. The oscillatory instability occurs at 0 < Pr < >
1. At Pr = 1 the flows remain stable,becoming unstable only when Pr = 1 which is a characteristic feature of the double-diffusive instabilities (Acheson & Gibbons 1978; Kirillov & Verhulst 2010; Kirillov 2013).As Figure 2 and 3 demonstrate, when Pr →
0, the threshold of the oscillatory modes(5.14) occurring in the Rayleigh stable flows becomes infinite (Ta Hc → ∞ ) while theirfrequency ¯ ω c → q (1 + Ro)(1 − γ a ˆ Θ ) depends weakly on the heating parameter γ a .The opposite limit, when Pr → ∞ , is applicable to the threshold of the Hopf bi-furcation in the Rayleigh unstable flows with the outward heating ( γ a > Hc → [ γ a ˆ Θ Rt − − γ a ˆ Θ )(1 + Ro)] − / . The corresponding Hopf frequency becomesindependent on the rotation rate: ¯ ω ∞ = q − γ a ˆ Θ Rt as is visible in the left panel ofFigure 4. The central panel of the same Figure shows that the Hopf frequency is bounded(0 < ¯ ω c < ¯ ω ∞ ) at any fixed Pr ∗ < Pr < ∞ .The frequency ¯ ω ∞ is fixed by the Brunt-V¨ais¨al¨a frequency N in the case of stablestratification of the fluid density ( γ a >
0, Rt < O. N. Kirillov, I. Mutabazi the Brunt-V¨ais¨al¨a frequency as¯ ω c = s N Ω PrPr + 1 + 4(1 + Ro)(1 − γ a ˆ Θ )(Pr + 1) . (6.9)Hence, in the limit of Pr → ∞ we have¯ ω ∞ = 1 √ Ω = r − γ a η , (6.10)which, in particular, yields another expression for the Brunt-V¨ais¨al¨a frequency in thecase of the Rayleigh unstable flows with the outward heating ( γ a > = − γ a ln η Ω . (6.11)The right panel of Figure 4 shows that all the critical frequencies tend asymptoticallyto the line (6.10) as N /Ω → ∞ . With the decrease in the ratio N /Ω the critical frequencyvanishes at the characteristic value of the Brunt-V¨ais¨al¨a frequency N c given byN c Ω = − − γ a ˆ Θ )Pr(Pr + 1) . (6.12)The characteristic Brunt-V¨ais¨al¨a frequency N c determines the stratification that is re-quired to excite the inertial waves by the rotation. Since N c decreases with the increasein Pr, Figure 4, we conclude that the inertial waves are easier to excite in the Rayleigh-unstable flow by the outward radial temperature gradient in the case of highly viscousfluids than in the case of the weakly viscous fluids.
7. Discussion
The short wavelength approximation has been used to investigate the stability ofthe Taylor-Couette flow with a radial buoyancy force induced by the coupling betweenradial temperature gradient and the centrifugal force in the Boussinesq approximation.A characteristic polynomial equation has been analyzed and marginal stability branchesof stationary and oscillatory modes have been found analytically. The present methodallows one to find the explicit dependence of the marginal state with the main controlparameters: Pr, Ro, γ a , and Rt.The reader should be aware that for the validity of the Boussinesq approximation, thethermal expansion parameter must be very small i.e. | γ a | ≪ γ a before 1 in all expressions containing 1 − γ a /
2. In fact, in our computations,the largest value of | γ a | was | γ a | = 0 .
01 for fluids with the largest expansion coefficientfound in literature to which a maximum of temperature difference applied should notexceed ∆T max = 5 ◦ C .From the above results computed at the geometric mean radius, for each value ofrotating case of the circular Couette flow (i.e. for a given value of µ ), it is possibleto determine the critical modes and their threshold Ta c as a function of different flowparameters η, γ a , and Pr together with the critical frequency for oscillatory modes. Wehave determined the critical parameters for the particular case of specific interest: theinner cylinder sole rotation ( µ = 0), the outer cylinder sole rotation ( µ → ∞ ), the solidbody rotation ( µ = 1) and the Keplerian rotation ( µ = η / ).For the inner cylinder sole rotation, we have retrieved and extended the results oflinear stability analysis (Meyer et al. nstabilities of a circular Couette flow with radial temperature gradient γ a = 0) is increasing with both Prand η . In fact, one gets d Ta dγ a ( γ a = 0) = Ta (cid:18) − ) ln η (cid:19) . (7.1)To our best knowledge, the Rayleigh stable flows (except the solid body rotation) witha radial temperature gradient have never been treated analytically in such a detail so farand the results presented in this work offer new predictions of convective instabilities. Inthe case of the outer cylinder sole rotation, in the case of the solid body rotation, andin the Keplerian regime the inward heating is destabilizing and leads to the oscillatoryinstability via Hopf bifurcation at 0 < Pr < Pr H < > Pr S > γ a and η (Table 1). Thevalues of Pr for which the oscillatory instability is observed are comparable to that ofthe astrophysical flows for which Pr ≈ .
02. At the Rayleigh line the flow is unstablefor inward heating. The present model is based on the short wavelength approximationin the axial direction, so, strictly speaking, it cannot be applied as such to the case ofthe solid body rotation where it is known (Auer et al. n = 0 , k = 0).In the Appendix we have compared the present results with those derived by Econo-mides and Moir (Economides & Moir 1980) for the Taylor-Couette flow with a centripetalacceleration and a radial temperature gradient. Lopez et al. (2013) have investigated thestability of the rotating cylindrical annulus with a negative radial temperature gradientand in the presence of the gravity g . While these authors suggested that quasi-Keplerianflows may be stable for weak stratification in the radial direction, our results show thatlaminar quasi-Keplerian flows may be destabilized by the centrifugal buoyancy, leadingto oscillatory modes for values of Pr relevant to accretion-disc problem, although we haveconsidered a model that has solid radial boundaries.
8. Conclusion
The short wavelength approximation method has been applied to the linear stability ofthe circular Couette flow with a centrifugal buoyancy induced by the radial temperaturegradient in the absence of the natural gravity. The marginal states (stationary andoscillatory modes) have been determined analytically and the effects of the differentparameters of the problem on the flow stability have been analyzed in detail. Thecentrifugal buoyancy enhances the instability of Rayleigh unstable flows in the inwardheating and in the outward heating it induces oscillatory modes the frequency of whichcan be compared with the Brunt-V¨ais¨ala frequency in the limit of large Pr values. In theinward heating it also induces instability in Rayleigh stable flows in the form of stationarymodes for large values of Pr or oscillatory modes for small values of Pr. The present studymay serve as a theoretical guideline to linear stability analysis and to direct numericalsimulations (DNS) of the flow.
Acknowledgments
O.N.K. has benefited from the CNRS grant for visiting senior scholars and acknowl-edges financial support from the ERC Advanced Grant “Instabilities and nonlocal mul-tiscale modelling of materials” FP7-PEOPLE-IDEAS-ERC-2013-AdG (2014-2019). I.M.is thankful to A. Meyer and Y. Harunori and O.N.K. is thankful to L. Tuckerman forfruitful exchanges on this work. I.M. acknowledges the support from the French National0
O. N. Kirillov, I. Mutabazi
Research Agency (ANR) through the program Investissements d’Avenir (No. ANR-10LABX-09-01), LABEX EMC (project TUVECO) and from the CPER Normandie. Appendix A. Appendix. Link to Economides & Moir (1980)
In the Appendix we would like to compare our short-wavelength equations with thematrix (4.6) to the results by Economides and Moir (Economides & Moir 1980) derivedin the narrow gap approximation.Let d = R − R ≪ R is the size of the gap between the cylinders in the Couette cell.Then, we can write x = ( r − R ) /d, µ = Ω /Ω , Ω ( x ) = 1 − (1 − µ ) x, x ∈ [0 , . (A 1)Denoting D = d/dx so that d/dr = d − d/dx we define the operators L = D − ( k z d ) − σ − ik √ T Ω ( x ) ,M = D − ( k z d ) − σ Pr − ik Pr √ T Ω ( x ) , (A 2)where Pr = ν/κ is the Prandtl number and σ = ωd ν , k = m r − Ω A , T = − AΩ d ν , ¯ β = T − T R − R . (A 3)The temporal eigenvalue is denoted by ω and the azimuthal wavenumber by m . Theparameter A is a coefficient in the expression for the background circular Couette flow rΩ ( r ) = Ar + Br (A 4)and can be expressed via the Rossby number Ro as Kirillov et al. (2014): A = Ω (1 + Ro) . (A 5)Then, the system of the narrow gap equations derived in Economides & Moir (1980)is L ( D − ( k z d ) ) u ′ = − ( k z d ) T Ω ( x ) v ′ + ( k z d ) Rθ ′ ,Lv ′ = u ′ ,M θ ′ = u ′ , (A 6)where (with g denoting constant gravity acceleration and α the coefficient of thermalexpansion) R = gα ¯ βd νκ , u ′ = 2 AdδνΩ u, v ′ = vR Ω , θ ′ = 2 Aκθ ¯ βνΩ R , δ = dR . (A 7)For any w ∼ exp( ik r r ), we have Lw = − d ν (cid:2) ν ( k r + k z ) + ω + imΩ Ω ( x ) (cid:3) w = − d ν (cid:2) ν | k | + ω + imΩ Ω ( x ) (cid:3) w = − d ν [ ω ν + ω + imΩ Ω ( x )] w. (A 8) nstabilities of a circular Couette flow with radial temperature gradient u, v ∼ exp( ik r r ) the first of equations (A 6) becomes − d ν [ ω ν + ω + imΩ Ω ( x )] ( − d | k | ) 2 AdδνΩ u = − ( k z d ) ( − AΩ d ) Ω ( x ) ν R Ω v + ( k z d ) Agαd ν Ω R θ. (A 9)Simplifying the above expression, we find[ ω ν + ω + imΩ Ω ( x )] u = k z | k | Ω Ω ( x ) dδR v + k z | k | dδR gαθ (A 10)and finally [ ω ν + ω + imΩ Ω ( x )] u = k z | k | Ω Ω ( x ) v + k z | k | gαθ, (A 11)where ω ν = ν | k | .Analogously, the second of equations (A 6) becomes − d νR Ω [ ω ν + ω + imΩ Ω ( x )] v = 2 AdδνΩ u. (A 12)Simplifying it, we get [ ω ν + ω + imΩ Ω ( x )] v = − Au (A 13)and, finally [ ω ν + ω + imΩ Ω ( x )] v = − Ω (1 + Ro) u. (A 14)Now, for the third of equations (A 6) we find h − k r d − ( k z d ) − σ Pr − ik Pr √ T Ω ( x ) i Aκ ¯ βνΩ R θ = 2 AdδνΩ u, (A 15)and equivalently, (cid:2) − κk r − κk z − ω − imΩ Ω ( x ) (cid:3) d ¯ βνR θ = δν u, (A 16)so that finally [ ω κ + ω + imΩ Ω ( x )] θ = − ¯ βu, (A 17)where ω κ = κ | k | .We can denote Ω = Ω Ω ( x ) and write the three equations in the matrix form ω ν + ω + imΩ − k z | k | Ω − k z | k | gα Ω (1 + Ro) ω ν + ω + imΩ β ω κ + ω + imΩ uvθ = 0 . (A 18)In fact, ¯ β = T − T R − R is a ‘derivative’ of the temperature with respect to radius and inlocal approximation we can replace it with Θ ′ = dθ /dr . Then, we can denote β = k z | k | and write ω ν + ω + imΩ − Ωβ − αgβ Ω (1 + Ro) ω ν + ω + imΩ Θ ′ ω κ + ω + imΩ uvθ = 0 . (A 19)For the stationary axisymmetric instability the Bilharz criterion (Bilharz 1944) applied2 O. N. Kirillov, I. Mutabazi to the characteristic polynomial of the matrix in the left hand side of equation (A 19)yields: 4(Ro + 1) + Pr gΩ Θαr + 1Ta > g = Ω r , and taking into account that the sign of α in (Economides & Moir 1980)is opposite to the sign of α in (2.1), we reduce the above equation to the form4(Ro + 1) − Θγ a + 1Ta > , (A 21)which differs from the condition (5.11) only by the factor 1 − γ a ˆ Θ at the first term.The reason of this discrepancy is that in Economides & Moir (1980) g is assumed tobe a constant radial gravitational acceleration whereas in Meyer et al. (2015) it is thecentrifugal acceleration that depends on radius. Hence, substitution of g = Ω r into Eq.(A 20) is not justified. On the other hand, the factor 1 − γ a ˆ Θ is small in the Boussinesqapproximation and consequently the equations (5.11) and (A 21) can be considered asequivalent.Taking into account that in the Boussinesq approximation the squared Brunt-V¨ais¨al¨afrequency is N = − γ a ˆ Θ Rt Ω , we transform (A 21) as4(Ro + 1) + Pr N Ω + 1Ta > . (A 22)With the opposite sign the inequality (A 22) is the onset of the double-diffusive Goldreich-Schubert-Fricke instability in the form derived in (Acheson & Gibbons 1978). REFERENCESAcheson, D. J. & Gibbons, M. P.
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