Dynamo Generated by the Centrifugal Instability

Home , Taylor number

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(section 2). We then study the Taylor-Vortex dynamo with both non-rotating (section 3) and rotating (section 4) outer sphere. Finally, we derive in section 5 a simple model based CENTRIFUGAL INSTABILITY IN SPHERICAL SHELL on symmetry arguments which describes the subcritical nature of the dynamo transition. In the idealized situation where the spheres are replaced by We conclude with a few remarks on the relevance of this two infinitely long rotating cylinders, the ideal laminar Cou- flow for astrophysical and laboratory dynamos. ette solution for the rotation Ω is given by :

A MODEL Ω(r)= A + 2 , (3) 1 r2

We consider an incompressible, electrically conducting Ω 2 Ω 2 2 2 2 2 Ω in which A1 = ( 2r2 1r1)/(r2 r1) and A2 = r1r2( 1 fluid confined between two co-rotating spheres. r and r Ω 2 2 − − − 1 2 2)/(r2 r1). In this case, the Rayleigh criterion predicts are respectively the radius of the inner and outer spheres, and linear stability− if the angular momentum increases outwards Ω , Ω are respectively the angular speed of the inner and the Ω 2 Ω 2 1 2 ( 1r1 2r2). Otherwise, the centrifugal instability is ob- outer spheres. The governing equations are the magnetohy- served≤ when the Reynolds number is large enough, and takes drodynamic (MHD) equations, i.e., the Navier-Stokes equa- the form (at onset) of axisymmetric Taylor-Couette vortices tion coupled to the induction equation. The problem is made contained in the poloidal plane. A similar behavior occurs dimensionless following [16]: this approach allows us to com- in a spherical gap: if the angular momentum decreases out- pare simulations with and without global rotation for a fixed wards along a cylindrical radius, Taylor vortices can be gener- Reynolds number. ated near the equator if the gap is small enough. Note that in ∂ this case, the bifurcation is strongly imperfect, since it appears u 1 Re˜ + (u ∇)u + 2E− ez u = from a background flow already involving a poloidal recircu- ∂t · × (1) lation. Nevertheless, for a sufficiently thin spherical shell, one 1 1 ∇Π + ∆u + E− + Re˜ χ− (∇ B) B, can expect Rayleigh’s criterion to remain valid at least close to − × × the equatorial plane, which translates with our dimensionless ∂B 1 numbers to the condition: ∇ ˜ − ∆ ∂ = (u B)+ RePm B. (2) t × × 1 1 E χ− χ δRe− . (4) The dimensionless parameters are the Ekman number E = ≥ − ∆Ω ν ˜ r2r1 Ω 2 , the Reynolds number Re = ν , the magnetic Prandtl 2r2 number Pm = ν/η and the aspect ratio of the spherical shell Let us first study the case with no global rotation, i.e. when χ ν η = r1/r2, where and are respectively the kinematic Ω2 = 0. At low Reynolds number, the mean flow is equato- viscosity and the magnetic diffusivity. The inner and outer rially symmetric and essentially toroidal, except for a large- spheres are electrical insulators, and no-slip boundary con- scale meridional recirculation between polar and equatorial ditions are used on these boundaries. In all the results re- regions (Fig.1-a ). However, characterisation of the equato- ported in this paper, a thin gap χ = 0.9 is used. Note that rial flow in the narrow-gap spherical shell indicates the exis- the Reynolds number used here as a control parameter (Re˜ ) is tence of a Rayleigh-like stability threshold, above which sev- based on the outer shell radius. In the following, our results eral pairs of counter-rotating, toroidal and axisymmetric Tay- will be presented in terms of a more conventional Reynolds lor vortices (TV) build up on the primary base flow. The new number (Re) based on the gap width δ = (r2 r1)/r2) to al- base state above this threshold corresponds to the saturated − δ r1∆Ω low easier comparison with previous works: Re = ν . The TV concentrated in the vicinity of the equator, as shown in equations (1)-(2) are integrated using the PARODY code [30]. Fig.1 for several Reynolds numbers. This bifurcation has been This code relies on a toroidal/poloidal decomposition of the widely studied in different aspect-ratios by [20–23]. velocity and magnetic fields. It uses spherical harmonics ex- Several purely hydrodynamic simulations have been per- pansion in the latitudinal and azimuthal directions for each formed for Reynolds numbers up to Re˜ = 4500 (Re = 450) in spherical shell, whereas finite differences on a stretched grid order to characterise the successive bifurcations of this spheri- are used in the radial direction. Time integration is performed cal Couette flow. The domain-integrated kinetic energy of the using a Crank-Nicholson scheme for the diffusive terms and poloidal velocity field is plotted against the Reynolds num- an Adams-Bashforth scheme for the nonlinear terms. Typi- ber in Fig.3-a showing an imperfect, supercritical bifurcation cal numerical resolutions involve 64 grid points in the radial at approximately Re = 140. This critical value is in good direction, while retaining 196 spherical harmonics in the lati- agreement with both experimental results by [18] (predicting tudinal direction (lmax = 196), and 96 in the azimuthal direc- Rec = 147 for χ = 0.9) and more recent numerical estimations tion (mmax = 96). This resolution has been proved sufficient [23] (linear extrapolation from their results at χ = 0.877 and 3

0.6 0.5 0.4 0.4 0.3 0.2 0.2 0.1 0 0 −0.1 −0.2 −0.2 −0.3 −0.4 −0.4 −0.5

0.7 0.8 0.9 1 0.7 0.8 0.9 Re = 135 Re = 140 Re = 170

FIG. 1: Streamlines of the azimuthally averaged velocity ﬁeld, for different Reynolds number below (Re = 135, i.e. Re˜ = 1350) and above (Re = 140;170, i.e. Re˜ = 1400;1700) the Taylor Vortices threshold, without overall rotation.

χ = 0.84 yields Rec 144). From Rec = 140 up to approxi- TV and the structure of the new base state up to weakly turbu- mately Re = 170, the∼ TV are purely axisymmetric, while sub- lent Reynolds numbers. These results are illustrated in Fig.2, ject at larger Re to secondary hydrodynamic instabilities: then which shows the marginal stability line for pure spherical 1 they break into wavy vortices, characterised by a dominant az- Couette ﬂow in the E− Re plane, showing a non-monotic imuthal mode m = 4 at Re = 170 and m = 5 at Re = 180. As evolution of the onset with− the inverse Ekman number. the ﬂow becomes more chaotic at larger Re, all modes become unstable, thus yielding a complicated azimuthal structure of the Taylor Vortices which become less and less coherent. In addition, the domain of existence of axisymmetric vor-

700 tices signiﬁcantly varies with the overall rotation rate. For 3 E = 10− for example, axisymmetric TV are to be observed 600 Rayleigh line [Ro = 0.2] Ro = 0.5 up to at least Re = 300, where they break into wavy TV 500 Simulations (m = 6), while axisymmetry is lost from Re = 170 already ∞ 400 (m = 4) in the E = case. An extensive hydrodynamic study

Re is far beyond the scope of the present paper ; nevertheless 300 the Fig.1 in [31], although in cylindrical geometry, provides a 200 glimpse into the complexityof the TV phase diagramin spherical Couette ﬂow. 100

0 0 2000 4000 6000 8000 10000 12000 14000 16000 E−1

Below the TV threshold, the kinetic energy contained in FIG. 2: Marginal stability curve for the destabilization of laminar the two meridional recirculation cells increases quasi-linearly Couette flow in a spherical gap, for the purely hydrodynamic problem. Red line indicate Rayleigh’s criterion for infinite cylinders (4). with Re as can be seen in Fig.3-a. We therefore approximate this contribution with a linear fit and subtract it from the total kinetic energy associated with poloidal motion close to the TV threshold. In Fig.3-b we present this last quantity as a We now consider the case where, adding overall rotation function of the supercriticality coefficient Ta/Ta , where Ta to the system, the Coriolis force is taken into account in the c is the Taylor number (Tac the TV critical Taylor number), as momentum equations (expressed in the frame of reference of Ω2 2 4 4 (1 Ω2/Ω1)(1 χ Ω2/Ω1) defined in [32]: Ta 1 R − − − . This has the outer sphere). For different Ekman numbers ranging from = ν2 1 (1 χ2)2 2 4 − E = 10− upto E = 10− , we systematically performed purely the advantage of providing a single dimensionless parameter hydrodynamic simulations in order to determine the onset of to describe the TV bifurcation, for all overall rotation rates. 4

−3

x 10 4

3.5 E = ∞ E = 10−2 3 −5 E = 10−3 10 −4 2.5 E = 5.10

PV 2 E B 1.5 E −3 −10 x 10 10 1.5 1 Re = 400, Pm = 1.5 1 0.5 0.5 0 Re = 400, Pm = 1.5 50 100 150 200 250 300 350 400 450 500 0 Re −15 Re = 250, Pm = 2.5 0 2000 4000 6000 8000 10000 12000 10 Re = 160, Pm = 4.0 −4 x 10 6 0 500 1000 1500 2000 2500 time 5 E = ∞ 4 E = 10−2 FIG. 4: Time series of the domain-integrated magnetic energy for E = 10−3 −4 Re = 160,Rm = 640 , Re = 250,Rm = 625 and Re = 400,Rm = 3 E = 5.10 TV {600 . For larger Re,} the{ magnetic ﬁeld exhibits} intermittent{ bursts E 2 near} the dynamo threshold (inset).

−1 0 0.5 1 1.5 2 2.5 3 Ta/Ta c riodic oscillations around a well defined value. As Re is increased, the magnetic field exhibits more and more chaotic FIG. 3: Top: Evolution of the domain-integrated kinetic energy asso- behavior under the effect of turbulent fluctuations in the TV. ciated with poloidal motion (EPV ) as a function of the Reynolds num- Finally, for larger Re, the magnetic field exhibits intermittent 2 3 ber, for four different Ekman numbers (E = ∞, E = 10− , E = 10− 4 growth with short, noisy saturation phases when close to the and E = 5.10− ). Bottom: Kinetic energy contained in the Taylor dynamo onset (see inset of Fig.4). Vortices (ETV , see text) as a function of the distance Ta/Tac to the TV onset, for the same Ekman numbers (retaining only weakly supercritical points).

TAYLOR VORTEX DYNAMO AT INFINITE EKMAN NUMBER

In this section, we first assume the outer sphere to be at rest (Ω2 = 0) and investigate the possibility for a dynamo instability to arise in the absence of Coriolis force. Solving now for the full set of MHD equations (1-2), the flow is found to sustain a dynamo in the equatorial region as soon as the bifurcation for axisymmetric TV is reached. For sufficiently high values of the magnetic Prandtl number Pm, a seed magnetic field experiences an exponential growth followed by a saturated phase as the feedback of the Lorentz force modifies the base flow. Fig.4 shows the time evolution of the magnetic energy for different values of the Reynolds number. First, one can see that the kinematic growth rate of the magnetic energy increases with the fluid Reynolds number, although the magnetic Reynolds number Rm = PmRe slightly decreases. Note that this behavior is quite different from what is observed in the cylindrical geometry, where the turbulent fluctuations de- FIG. 5: The m 1 dynamo at Re 150, Pm 7. Top, from Left crease the growth rate of the magnetic field, such that dynamo = = = to Right: Meridonal maps of radial velocity Vr, azimuthal velocity action is inhibited at large Re (see Fig.4 in [29]). Vφ , and latitudinal magnetic field Bθ , Bottom, from Left to Right: The dynamics in the saturated phase also strongly depends Non-axisymmetric part of the radial components of V and B, in the on the level of turbulent fluctuations. At small Re, both the equatorial plane. magnetic energy and the kinetic energy undergo smooth pe- 5

−4 x 10 For Re = 150, Fig.5-top shows velocity components Vr, Vφ , 10 −3 x 10 and magnetic component Bθ in a meridional cut (φ = 0) , 3 8 whereas Fig.5-bottom shows non-axisymmetric radial com- 2 θ ponent of velocity and magnetic ﬁelds in the midplane = 0. 6 1 failed dynamo This ﬁgure illustrates the non-dipolar character of the dynamo dynamo

B 0 4 data3 field in which the dominant mode near the onset is an az- E 2 3 4 5 data4 imuthal mode m = 1, which in turn sustains a m = 2 struc- data5 2 data6 ture in the velocity field through the action of the Lorentz data7 data8 0 force. The longitudinal structure of the magnetic field is es- data9 sentially determined by those of the forcing TV: while always data10 −2 2 2.5 3 3.5 subharmonic in the cylindrical geometry [29], the spherical Pm

Taylor-Couette dynamo displays both harmonic and subhar- −4 x 10 monic features in the vertical direction. 6.3

The bifurcation diagram of Fig.6-a displays the evolution of 6.25 dynamo the time-averaged, integrated magnetic energy after saturation failed dynamo −4 6.2 x 10 dynamo of the dynamo, as a function of the magnetic Prandtl number 6.4 data4 at a ﬁxed value of the Reynolds number Re = 160. Fig.6- data5 PV 6.15 6.2

E data6 (b) shows the corresponding time-averaged, integrated kinetic data7 6.1 6 failed dynamo energy contained in poloidal motion after saturation. The bi- dynamo furcation is clearly subcritical and displays well-deﬁned hys- 6.05 2 3 4 5 teresis near the dynamo threshold, the amplitude of the TV

6 weakening in concert with the strengthening dynamo as the 2 2.5 3 3.5 flow transfers kinetic energy to the magnetic field. It is in- Pm teresting to note that the hysteresis is smaller close to the TV onset Re and tends to a supercritical transition in this case. FIG. 6: Bifurcation diagram of domain-integrated magnetic (top) and c ∞ As Re is increased, the magnetic dominant mode switches to poloidal kinetic (bottom) energy vs Pm, for Re = 160 and E = . For these parameters, the dynamo bifurcation is clearly subcritical. m = 2 (for Re 160), triggering a m = 4 component in the velocity field. ≥ In Fig.7, we report the marginal stability curve for dynamo action in the (Rm Re) parameter space. First, the critical 1000 − magnetic Reynolds number Rmc for magnetic field amplifi- 900 cation clearly diverges near the hydrodynamic TV onset at 800 Rec = 140 , meaning that the threshold for dynamo action is 700 essentially determined by the onset of TV, without which no magnetic field can be sustained by the flow. Note however 600 Rm that this Rmc diverges only very close to Rec, but reaches its 500 min minimum value immediately above this point (Rec 170). 400 This is similar to what is observed for convective dynamos,∼ 300 for which convection instability is necessary, or to the results obtained by [16], who demonstrated in a large-gap geometry 200 100 that the single pair of equatorially symmetric eddies due to 150 200 250 300 350 400 450 500 secondary meridional circulation could not, as such, sustain a Re dynamo in spherical Couette flow. The most favorable window for dynamo action (in terms of critical Rm) seems to lie FIG. 7: Magnetic Reynolds number Rm versus Re, without overall in the Re = 170 180 range. Interestingly, this window also rotation of the system. Empty circles denote dynamo runs (whether corresponds to the− regime of wavy, but still laminar Taylor subcritical or not); filled circles: failed dynamos, and squares: intermittent bursts. Note the divergence at low Re and the constant critical Vortices. As we will see in section 4, this last feature is essen- Rm at large Re. tial for the comprehension of the dynamo bifurcation. It is remarkable that the critical magnetic Reynolds number for dynamo action Rmc = PmcRe seems to saturate above Re = 350. Indeed, the key element for the spherical Taylor- is observed in large gap spherical Couette dynamo or weakly Couette dynamo is the coherence of the TV, crucial in sus- rotating convection dynamos, for which a critical magnetic taining the magnetic field. In the cylindrical case (see [29]), Prandlt number Pmc is obtained at large Re, rather than our further increase in Reynolds number results in a loss of coher- critical Rm. ence in the TV as the flow developschaotic features, which re- Numerical resolution requirementsdid not allow us to reach sults in a loss of efficiency for the dynamo. A similar behavior values of Re above 450 (corresponding to our control param- 6 eter Re˜ = 4500). However, recent work by [23] on the hydro- the weakly turbulent regime is reached, the dynamo threshold dynamics of spherical Couette flow in relatively thin gaps up rapildy rises to a higher value, which seems nearly constant to large Reynolds number (corresponding to Re 850 with in the infinite Ekman number case. (We could not investigate our definition) shows a re-laminarisation of the TV.∼ sufficiently high Re˜ regime to confirm this last observation for smaller Ekman numbers). Close to the TV threshold or in the turbulent regime, over- EFFECT OF GLOBAL ROTATION ON THE DYNAMO all rotation of the system does not particularly help dynamo action. However, at reasonable distance from the TV onset, In this section, we now investigate the effect of the rota- the linear, critical Rm can be decreased down to considerably tion of the outer sphere on the dynamo, which is equivalent lower values (by nearly a factor 3) when the system is rapidly to study the effect of global rotation on the Taylor-Vortex dy- rotating. It is interesting to compare the flow realizing the namo described in the previous section. We have computed minimum Rmc for each overall rotation rate investigated here. hydrodynamic and MHD simulations for different values of Interestingly, the minimal dynamo onset is systematically 1 2 3 4 E = 10− ;10− ;10− ;5.10− in order to obtain both TV obtained in the vicinity of the hydrodynamic instability to { } 2 onset and dynamo threshold. wavy vortices: for E = ∞ and E = 10− , the dynamo is characterised by a mB = 2 azimuthal wavenumber and exhibits a

minimum onset for Ta/Tac 1.5, while wavy vortices mV = 1200 ∼ ∞ 2mB = 4 are hydrodynamically unstable for Ta/Tac 1.35. E = 3 ∼ −2 1000 E = 10 Similarly, for E = 10− , we observe a minimum onset at E = 10−3 Ta/Tac 12, with the most unstable mode mB = 3, very close −4 ∼ 800 E = 5.10 to the onset for waviness mV = 2mB = 6 at Ta/Tac 11. ∼

c From Fig.8 it is clear that the apparently favorable inﬂu- 600

Rm ence of overall rotation on dynamo action is not simply de-

400 termined by the distance to the TV onset. Moreover, the strength of the hydrodynamic forcing by the TV does not ex- 200 plain the discrepancy in the minimal thresholds. We com- e puted an effective magnetic Reynolds number Rmc, based on 0 0 5 10 15 20 the gap width and the TV forcing velocity, which allows us to Ta/Ta c distinguish between the contribution of the large-scale, non- dynamo Couette flow and the rôle of the equatorial TV forc- FIG. 8: Critical Rm as a function of Ta/Tac (where Tac is the TV ing responsible for the dynamo action at scale δ. We find critical Taylor number), for different values of the Ekman number e 3 Rmc = 49.8 for E = 10− ,Ta/Tac = 11.92 against, for ex- E. Empty markers denote laminar, axisymmetric TV (in the purely e { 2 } ample, Rm = 107.7for E = 10 ,Ta/Tac = 1.59 . Thisex- hydrodynamic regime). Markers filled in black denote laminar, wavy c { − } TV and markers filled in grey correspond to turbulent flows. ample illustrates well that overall rotation is capable of effec- tively favouring dynamo action, independently from the hydrodynamic forcing strength. We first compare in Fig.8 the dynamo threshold - in terms of the critical magnetic Reynolds number Rm - for various global rotation rates, as a function of the supercriticality coef- LOW-DIMENSIONAL MODEL FOR THE SUBCRITICAL DYNAMO BIFURCATION ficient Ta/Tac (for TV onset). Note that our marginal curve corresponds to the linear instability, i.e. critical Pm is obtained by interpolation of time-averaged kinematic growth rate of the We now present a simple model aiming to explain how a total magnetic energy computed close to the dynamo onset. subcritical dynamo bifurcation can be generated in such sys- For all the parameters explored so far, the dynamo thresh- tem, and why the minimum threshold is systematically ob- old diverges very sharply close to the onset of TV, such that tained close to the transition to wavy Taylor vortices. For the loss of TV vortices is always accompanied with the loss the parameter Re,Rm explored in these simulations, the of dynamo action, similarly to the non-rotating case (Fig.7). system is always{ close} to both the dynamo transition and Although there are some uncertainties related to the estima- the wavy vortex bifurcation, leading to a codimension-two tion of both the critical Re for TV onset and the critical Pm bifurcation. We denote by A(t) the complex amplitude of for dynamo, the following picture emerges from the simula- the non-axisymmetric magnetic field with wavenumber mB, tions: for all the overall rotation rates investigated here, the its phase describing the angle of the magnetic mode in the dynamo threshold decreases as the Taylor number is gradually equatorial plane. This non-axisymmetric mode generates a increased from its critical value Tac, while still in the lami- non-axisymmetric velocity field of complex amplitude V(t) nar, axisymmetric TV regime. It further drops as the hydro- through the Lorentz force, which depends quadratically on the dynamic base state switches from axisymmetric to wavy TV, magnetic field. When writing amplitude equations for these reaching its minimum value in this new regime. Then, once two modes, symmetry requirements constrain the form of the 7 equations: A A (symmetry of the induction equation) and

iΨ →− 2iΨ 1.5 A Ae , V Ve (rotational invariance about the rotation 0 λ = 1.0 axis),→ which yields:→ 1 −0.2 subcritical σ 0.5 s c 2 2 r ˙ λ σ 0 V = V + A V V, (5) linear −0.4 c −1.2 −1 −0.8 −0.6 −0.4 −0.2 − | | σ

2 σ A˙ = σA +VA¯ A A, (6)

− | | −0.6 1 2 ¯ The quadratic terms A and VA, which respectively repre- −0.8 0.5 sent the Lorentz force and the induction term of MHD equa- λ = 0.1 0

−1 −0.5 −0.4 −0.3 −0.2 −0.1 tions, describe the coupling between the dynamo mode with σ

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 wavenumber mB and the wavy component mV = 2mB of the λ flow. λ represents the linear growth rate (or equivalently the distance from onset) of the wavy mode in a non-magnetized FIG. 9: Stability diagram in the λ σ parameter space, where both σ − situation (A = 0), whereas is the linear growth rate of the subcritical and linear thresholds σc are represented for each λ. In- dynamo in absence of wavy vortices mV . Note that the ef- sets: Subcritical (respectively, supercritical) bifurcation diagrams for fect of axisymmetric TVs, main source of the dynamo, is im- λ = 0.1 (respectively, λ = 1.0). plicitly described by σ. At leading order, we have therefore λ W σ W Re Rec and Rm Rmc, where Rmc and Rec are respectively∼ − the onset for∼ dynamo− action in the absence of waviness in the flow and the onset for resonant wavy vortices (in 0.2 m = 1 m = 2 B B the absence of magnetic field). The above model is the normal 0.15 formof the well known1:2 resonanceandhas beenstudied in 0.1 various situations ([33, 34]). Let us write A reiθ , V seiφ , = = 0.05 and ψ = φ 2θ and look for steady solution (˙r = s˙ = ψ˙ = 0). s 0 − λ The model has two trivial solutions : s = 0,r = 0 and the pure hydrodynamic mode s = √λ,r ={ 0 . In addition,} the −0.05 { } m = 2 mixed modes of the model are given by the equations: −0.1 m = 3 m = 4 −0.15 s = r2 σ , (7) m = 5 − −0.2 3 2 2 4 6 1500 1550 1600 1650 1700 1750 1800 0 = λσ σ + 3σ λ 1 r 3σr + r . (8) Re − − − − This low-dimensional model provides a simple explanation FIG. 10: Growth rates (λS) in the simulations of the hydrodynamic for the subcritical nature of the dynamo. Let us consider wavy modes m = 2 5 at E = ∞. For comparison, the magnetic mode − λ < 0, i.e. hydrodynamically damped wavy vortices: for mB selected at the dynamo onset is indicated. As can be seen in Fig.7, σ > 0, the solution A,V = 0 is linearly unstable and bifur- the minimal Rmc lies in the vicinity of Re 170, corresponding to ∼ cates to a mixed mode A = 0, associated to a velocity mode the hydrodynamic bifurcation of wavy vortices. V = 0 due to the action of6 the Lorentz force. Far away from the6 onset of wavy vortices λ 1, it is easy to show that the equation for A reduces to a| quadratic| ≫ equation, meaning that the dynamo bifurcation is supercritical. This model shares several similarities with the equations On the contrary, very close to the onset for hydrodynamic derivedin [35], who recently used a dynamicalsystem to show wavyness, λ 1 leads to a subcritical transition: the mixed that a competition between an hydrodynamic instability and mode can be| stabilized| ≪ and is therefore bistable with A,V = 0 a growing magnetic mode can lead to a subcritical dynamo. for σ < 0. This mechanism is illustrated in Fig.9, where the Note however that the 1:2 resonance underlying the present model (5)-(6) has been integrated in time for various values of approach directly constrain the symmetry of the modes that σ and λ, and for variousinitial conditions(but such that ψ = 0 can be involved in the bifurcation, and the quadratic order of is preferred over ψ = π). As λ increases from large, negative the coupling terms provides a direct identification to the orig- values (where the dynamo bifurcation is supercritical) to zero, inal MHD equations from which the model is derived. the subcritical threshold for the dynamo lowers, resulting in a This scenario is in excellent agreement with our simula- widening bistability region. Note that in the vicinity of λ = 0, tions, as illustrated in Fig.10 at E = ∞ (to be compared also subcritical bifurcation is obtained independently of the sign with the results presented in Fig.7): the lower values for the of λ. Then, for λ > 0, as the hydrodynamic mode becomes dynamo threshold (characterised by mB = 2) are reached in more and more linearly unstable, the linear dynamo threshold the vicinity of the onset of the resonant mode (here mV = 4). drops faster than the subcritical threshold, which results in re- The same mechanism is observed at smaller Ekman numbers, 2 covering a supercritical bifurcation at large λ (see upper inset with again a 2:4 resonance at E = 10− , and a 3:6 resonance 3 4 in Fig.9). at E = 10− . The dynamos we investigated at E = 5.10− 8 did not appear to lie in the vicinity of a wavy bifurcation, and the equatorial region, at the Taylor-vortices scale. Without we did not observe either the associated, sudden decrease in overall rotation of the system, the dynamo bifurcation is found dynamo threshold that we find for other Ekman numbers. It to be subcritical and the critical magnetic Reynolds number is also remarkable that the dynamo selects a mB = 1 mode at seems to reach a constant value in the flow turbulent regime, Re = 150,E = ∞ where the least stable velocity mode is the where the magnetic field experiences intermittent bursts near resonant mode mV = 2 = 2mB. Our previous observation that the dynamo onset. Sufficiently far from the onset for centrifu- the bistability region seems to tighten toward the TV onset is gal instability, the effect of overall rotation strongly reduces also well consistent with the recovery of a supercritical tran- the dynamo threshold. sition for λ 1. This decrease of the dynamo onset together with the sub- | | ≫ critical nature of the bifurcation is understood in the frame-

0 10 work of a simple model: close to the hydrodynamic onset E m = 6 for wavy vortices, the Lorentz force produces a quadratic V −5 10 resonant interaction between the dynamo magnetic ﬁeld of wavenumber m and the velocity mode of wavenumber 2m. In −10 10 consequence, although the axisymmetric Taylor vortices con-

E trol the magnetic ﬁeld generation, the waviness of the ﬂow −15 E 10 m = 3 B can stabilize the dynamo solution well below its linear onset. Interestingly, this effect is observed even below the hydrody- −20 10 namic onset of wavy vortices. When the wavy resonant mode

−25 is linearly unstable, the kinematic dynamo onset can also be 10 0 200 400 600 800 1000 1200 1400 time significantly decreased. It is now worth discussing the relevance of this scenario for real systems. Although it is known to play little role for the FIG. 11: Time-series of the magnetic energy in the Fourier mode Earth’s dynamo, differential rotation may be very important mB = 3 (dashed, black) and of the velocity mode mV = 6 (blue), in other astrophysical objects. For instance, observations and for E 10 3, Ta Ta 11 92, Pm 1 25. Both the dynamics and = − / c = . = . theoretical models [36] suggest that surface zonal flows ob- the structure of the dynamo are determined by the strong correlation between the two modes, as predicted by the model. served on the jovian planets can extend below the observable surface layers. Recent observations of stellar interiors have re- vealed, by means of asteroseismic techniques, the existenceof Another confirmation of this scenario is given by the time internal, radial differential rotation in red giants or subgiants 3 series shown in Fig.11 for E = 10− ;Ta/Tac = 11.92;Pm = (see e.g. [37],[38]), with a ratio of the core/envelope angular 1.25 : here, the system lies{ relatively far from the dynamo velocities in the 2.5-20 range. } onset (Pm = 2.5Pmc) and the magnetic field is characterised If the angular momentum sufficiently decreases outwards by several unstable Fourier modes. However in the kinematic near the equator in these different systems, the Taylor-vortex phase, the dynamo clearly selects the magnetic mode (mB = 3) dynamo described in this paper may play a role in the which is resonant with the hydrodynamically unstable mode generation of their magnetic field. This scenario may be mV = 6. The most unstable mode mB = 3 then rapidly sat- particularly relevant for explaining magnetic fields that urates, and the non-linear phase is characterised by coupled exhibit non-dipolar geometries, unlike those predicted for oscillations between these two resonant modes. Interestingly, convection-driven flows. this simulation eventually exhibits a secondary transition at large time, in which the magnetic field switches to mB = 2, Such spherical Taylor-Couette dynamo also opens interest- and is therefore associated to a bifurcation of the velocity field ing perpectives for experimental flows: MHD spherical Cou- to the resonant mode mV = 4. ette experiments([39, 40]) are currently based on aspect-ratios similar to the Earth’s core, thus forbidding the occurence of the centrifugal instability. Although the smallest Rm reported CONCLUSIONS in this paper (Rmmin 150) is still very challenging from an experimental point of∼ view, one may imagine to use a smaller Previous studies have shown that laminar spherical Cou- gap in these experiments, more prone to the generation of Tay- ette flow fails to sustain dynamo action, which can only build lor vortices in the equatorial plane. To that end, future work on a primary hydrodynamic instability of the equatorial jet should focus on the search for Rm smaller than the ones re- or the Stewarston quasi-geostrophic layer. Although most of ported in this paper, by means of a more systematic study of the interest in spherical dynamos is with convection-driven this new spherical TC dynamo, particularly regarding the role flows, we demonstrate here that centrifugal instability is a pos- of Re and of the gap between the spheres. sible candidate for magnetic field amplification in a spheri- Finally, the dynamical system (5)-(6) describing the genera- cal shell. The toroidal, counter-rotating vortices of spherical tion of a subcritical dynamo is mainly based on symmetry ar- Taylor-Couette flow maintain a non-axisymmetric dynamo in guments and on the quadratic form of the Lorentz force. One 9 can therefore expect it to be quite general and may be relevant [19] WIMMER, M. 1976 Experiments on a viscous fluid flow be- beyond the present paper: it would be interesting to see if a tween concentric rotating spheres, J. 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