MNRAS 000, 1–?? (2015) Preprint 25 October 2021 Compiled using MNRAS LATEX style file v3.0

Helicity inversion in spherical convection as a means forequatorward dynamo wave propagation

Lucia D. V. Duarte,1? Johannes Wicht,2 Matthew K. Browning,1

and Thomas Gastine21Department of Physics and Astronomy, University of Exeter, UK2Max-Planck-Institut fur Sonnensystemforschung, Gottingen, Germany

Accepted XXX. Received YYY; in original form ZZZ

ABSTRACT

We discuss here a purely hydrodynamical mechanism to invert the sign of thekinetic helicity, which plays a key role in determining the direction of propagationof cyclical magnetism in most models of dynamo action by rotating convection. Suchpropagation provides a prominent, and puzzling constraint on dynamo models. Inthe Sun, active regions emerge first at mid-latitudes, then appear nearer the equatorover the course of a cycle, but most previous global-scale dynamo simulations haveexhibited poleward propagation (if they were cyclical at all). Here, we highlight somesimulations in which the direction of propagation of dynamo waves is altered primarilyby an inversion of the kinetic helicity throughout much of the interior, rather than bychanges in the differential rotation. This tends to occur in cases with a low Prandtlnumber and internal heating, in regions where the local density gradient is relativelysmall. We analyse how this inversion arises, and contrast it to the case of convectionthat is either highly columnar (i.e., rapidly rotating) or locally very stratified; in bothof those situations, the typical profile of kinetic helicity (negative throughout most ofthe northern hemisphere) instead prevails.

Key words: convection – dynamo – hydrodynamics – magnetic fields – turbulence– Sun: general – stars: general

1 INTRODUCTION

The systematic equatorward migration of sunspot emer-gence latitudes constitutes one of the most enduring puz-zles of solar physics. Spots appear first at mid-latitudes,then progressively nearer the equator over the course ofa roughly 11-year cycle (e.g., Carrington 1858; Maunder1904; reviews in Ossendrijver 2003, Hathaway & Rightmire2010), constituting the famous “butterfly diagram”. There isnow widespread agreement that the surface magnetism ulti-mately arises from the action of a dynamo within the Sun’selectrically conducting convection zone, but a detailed expla-nation for the equatorward propagation has remained elusive(see, e.g., Moffatt 1978). In many models of the global so-lar dynamo, this propagation is taken to reflect underlyingwave-like behaviour in the generation of sub-surface mag-netism (see, e.g., review in Charbonneau 2010; Priest 2014).

In the classic model developed by Parker (1955) and ex-plored by many other authors since (e.g., Yoshimura 1975;

? E-mail: lduarte@astro.ex.ac.uk

Stix 1976; Gilman 1983), the wave-like behaviour arises fromthe combination of helical turbulence and differential rota-tion, described in mean-field theory by the α-effect and Ω-effect, respectively (e.g., Steenbeck et al. 1966; review inBrandenburg & Subramanian 2005). The former encapsu-lates the production of poloidal magnetic field from toroidal(or vice versa) by convective eddies that rise and twist, whilethe latter describes the generation of toroidal field by linearwinding of a poloidal field by differential rotation. In an αΩ

dynamo, the sign of newly generated toroidal field is deter-mined by the sense of the differential rotation and by the signof the pre-existing poloidal field. The latter in turn dependson the properties of the convective flows. The direction ofpropagation of the dynamo wave is then determined by thelocations where toroidal field, generated by the differentialrotation from stretching of poloidal field loops, cancels or en-hances the pre-existing toroidal field: if the newly generatedfield tends preferentially to cancel pre-existing field near theequator and reinforce it at higher latitudes, the migrationof the field will be poleward. This is encapsulated by thewell-known Parker-Yoshimura sign rule that dynamo waves

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in such models travel in a direction given by s = α∇Ω× ~eφ

(Yoshimura 1975; Stix 1976).

The sign of the α-effect, which partly determines thedirection of propagation of the dynamo wave, is fundamen-tally related to the lack of reflectional symmetry in the flow:α changes sign under transformations from right-handed toleft-handed coordinate systems, and vanishes if the velocityfield is statistically invariant under such parity transforma-tions (see, e.g., Moffatt 1978). In many variants of meanfield theory, assuming certain simplifying features about thevelocity field, α can in turn be related to the kinetic helicityof the flow, u ·ω = u · (∇×u) (with ω the vorticity), often inaddition to other terms involving the current helicity (e.g.,Pouquet et al. 1976; Brandenburg & Subramanian 2005).The sign and spatial variation of the kinetic helicity are thuscrucial for determining the direction of field propagation.

In the past few decades, a wide variety of publishednonlinear dynamo simulations in spherical geometries havebeen shown to exhibit cyclical behaviour (e.g., Gilman 1983;Ghizaru et al. 2010, in the context of stellar astrophysics, orGoudard & Dormy 2008; Schrinner et al. 2011; Simitev &Busse 2012; Gastine et al.; 2012 in the planetary context).However, to the extent that these simulations have exhib-ited systematic latitudinal propagation, this has generallybeen poleward (see, e.g., discussion in Brun et al. 2013), inagreement with the Parker-Yoshimura rule (given the real-ized kinetic helicity and differential rotation) but in conflictwith the observed solar butterfly diagram. More recently, afew groups have published examples of convective dynamoswhose propagation is equatorward: e.g., Kapyla et al. (2012,2013); Augustson et al. (2013); Warnecke et al. (2014). Ineach of these cases, the equatorward migration has been at-tributed largely to features in the differential rotation: e.g.,to regions where the radial Ω gradient is negative (Kapylaet al. 2012; Warnecke et al. 2014), or to non-linear feedbackson the shear (Augustson et al. 2015). The kinetic helicityprofile in these simulations, and with it the purported α-effect, appears to be largely as described in Sec. 2.1 below,and as realized in many previous simulations: it is negativein the northern hemisphere, leading to a positive α-effect.

In this paper, we explore the circumstances under whichthe kinetic helicity, and with it the generation of poloidalmagnetic fields, actually has this spatial distribution. InSec. 2 we review the processes that lead to this helicity distri-bution, and outline a scenario in which it could instead havethe opposite sign throughout much of the spherical domain.In Sec. 3, we carry out non-linear simulations of anelasticconvective dynamos in global spherical shells, and show thatthese can indeed exhibit such “reversed” kinetic helicity pro-files in certain regimes. We demonstrate that simulationsexhibiting this reversal also show systematic equatorwardpropagation of dynamo waves, without any accompanyingchanges in the differential rotation. We analyse the mecha-nisms behind the kinetic helicity reversal more thoroughlyin Sec. 4, and close in Sec. 5 with a summary of our workand a discussion of its possible relevance to the Sun, otherstars, and planets.

Figure 1. Diagram of vorticity generation in columnar convec-

tion, in a rotating spherical shell.

2 REGIMES OF KINETIC HELICITY

2.1 Classical helicity configuration in globaldynamo simulations

The kinetic helicity H, calculated as

H = u ·ω = u · (∇×u), (1)

arises from the twisting and writhing of convective flows asthey rise or fall. It is instructive to consider two regimes ofconvection that have been the subject of particularly widestudy, which we will call “columnar” and “plume-like”. Theformer has been widely studied in the planetary context(see, for example Busse 1970; Busse & Cuong 1977; Olsonet al. 1999; Busse 2002; Aubert 2005; Aubert et al. 2008)and found to dominate in rapidly-rotating systems at lowerRossby numbers. In Boussinesq models, the transition tosuch behaviour has been discussed by Soderlund et al. (2012)and we will address this matter in Sec. 5.2.

Columnar convection consists of dominating nearly 2D(quasi-geostrophic) circular motions (i.e. independent of thez cylindrical coordinate, defined by the rotation axis) with asecondary axial flow induced mainly by the boundaries alongthe columns for a Boussinesq fluid (Busse et al. 1998; Olsonet al. 1999). The 2D vortical motion arises from divergingup flow encountering the top and bottom boundaries, gen-erating clockwise vortical motion. Since the resulting radialvorticity ωr from the diverging up flow is negative in theNorth hemisphere and positive in the Southern (as a resultof the Coriolis force acting on it), the z component of the re-sulting vorticity ωz is negative in both hemispheres. The flowwill then sink toward the equatorial plane and converge tostart rising again, generating now counter-clockwise motion(positive ωz, see Olson et al. 1999, and also Fig. 1). Whilethe divergence of the flow occurs mainly near the bound-aries, convergence can happen anywhere in the bulk and notexclusively at the equator (Fig. 1). According to the Taylor-Proudman constraint (Proudman 1916; Taylor 1922), thelocal axes of vorticity tend to align with the rotation axis z,shaping columns of alternating vorticity sign. As illustratedin Fig. 2, uz is positive in the northern hemisphere and nega-tive in the southern along columns with positive ωz and vice-versa. This means that in columnar convection, the kinetichelicity H is dominated by the cylindrical component in z.

MNRAS 000, 1–?? (2015)

helicity inversion mechanism 3

Figure 2. Diagram of columnar convection inside a rotatingspherical shell.

Thus Hz will always be negative in the northern hemisphere(uz and ωz have opposite sign) and positive in the southern(uz and ωz have the same sign). This helicity organization ishelpful to obtain a large scale dipole field (e.g Olson et al.1999). Note that columnar convection is also present in someplane-layer models (Childress & Soward 1972 and an exam-ple of application in numerical models, Stellmach & Hansen2004), thus it does not necessarily depend on the sphericalgeometry.

A second regime, in which stratification and buoyancyinstead play central roles, also often results in a similar ki-netic helicity profile. In this “plume-like” regime, more akinto the classic Rayleigh-Benard convection problem, convec-tion consists of rising and sinking flow between the inner andouter boundaries, predominantly along the radial direction.We generally expect that this type of convection will replacecolumnar convection in a rotating system when the effect ofrotation is weak compared to buoyancy.

This regime is easily attained near the surface ofstrongly stratified models of stellar and planetary convection(see, e.g., Miesch et al. 2008; Glatzmaier et al. 2009; Gastineet al. 2013; review in Miesch & Toomre 2009), where thedensity gradient is much steeper leading to comparativelylarger Rossby numbers (ratio between inertia and Coriolisforce, e.g. Browning 2008; Gastine & Wicht 2012; Gastineet al. 2013). Furthermore, a higher value of Rossby numbermay also intensify helicity (see Sec. 4.1 below). Accordingto the anelastic continuity equation,

0 = ∇ · (ρu) = ρ∇ ·u︸ ︷︷ ︸incompressible term

+ u ·∇ρ︸ ︷︷ ︸compressible term

, (2)

when the density gradient is dominant (second term on theright side), a parcel rising along r expands due to the rapidlydecreasing density toward the surface (Fig. 4). The Coriolisforce acts on the expanding fluid to generate anticyclonicfluid motion (negative ωr in the northern hemisphere andpositive in the southern, see Glatzmaier 1985). In addition,the rising velocity decreases as a fluid parcel slows down

when approaching the outer boundary, which may also de-crease the effect of the incompressible term of the continuityequation (first term on the right side). For the same rea-son, sinking flow contracts generating cyclones (positive ωrin the northern hemisphere and negative in the southern).Both effects of the rising/sinking flow naturally correlate asnegative helicity, so plume-like convection near the surfacetypically gives a hemispherical North/South helicity patternsimilar to that of columnar convection. Both types of con-vection often co-exist and reinforce each other in stratifiedmodels.

2.2 “Inverted” helicity configuration

In the previous section, we described the most commonlyobserved case of helicity distribution in numerical dynamomodels in rotating spherical shells. In the presence of milderdensity stratification (Nρ .4) and rapid rotation, convectionis often dominated by columnar convection. If on the otherhand buoyancy dominates and the medium is highly strati-fied, plume-like convection (consisting of expanding upflowsand contracting downflows) often prevails. Both lead to akinetic helicity profile that is predominantly negative in thenorthern hemisphere. But what happens under other cir-cumstances?

In the presence of a weak density contrast, roughlyNρ . 1, a rising fluid parcel will tend to contract since itsvelocity is increasing as it starts from the bottom withur = 0 (impenetrable boundaries). More generally, this maybe true even far from the boundaries if ∂ur/∂ r>0. This iseasily understood from the equation of mass conservation(Eq. 2) when assuming that ∇ρ is small. Such small den-sity contrasts typically exist in stratified models in the in-ner 80− 90% of the radius (see Sec. 3 below, particularlyFig. 4). Consequently, the continuity equation becomes ap-proximately the Boussinesq continuity equation of ∇ ·u = 0in the deeper part of the shell. Figure 3 illustrates schemat-ically this behaviour below the dashed line which is a repre-sentation of the inner part of the radius of a spherical shell.In this region, the density gradient is relatively smaller thanabove the dashed line, which represents the outer few per-cent where typically most of the density gradient is located.

In the region of small density gradient below thedashed line of Fig. 3, the Coriolis force acts on the ris-ing+contracting fluid generating positive radial vorticity ωrin the northern hemisphere and negative in the southern(cyclones). In the same region in the same way, sinking flowexpands as it slows down toward the bottom boundary, gen-erating anticyclones. In both cases of rising+contracting andsinking+expanding flows, the resulting sign of helicity is pos-itive in the northern hemisphere and negative in the south-ern. This behaviour is opposite to the outer layer (describedin the previous Section) and it seems only pertinent whencolumnar convection becomes minimal in the bulk.

In this schematic picture, explored in more detail inthe following Sections, the inversion of kinetic helicity is apurely hydrodynamical effect, independent of the magneticfield, though it will ultimately be important to determine thedirection of the poloidal field generated by the α-effect andconsequently the sign of the generated toroidal field, whichexplains the direction of propagation of a dynamo wave inthe Parker-model.

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4 Duarte et al.

Figure 3. 2D portion of a spherical latitudinal section repre-sented in a planar view. The dashed line separates the inner part

of the density gradient where it is mild and the outer part where

the gradient becomes steeper (see Fig. 4). Note that this is merelya sketch, thus is it not done to scale.

It is also useful to distinguish between the possibilityof deep layers where the kinetic helicity is “reversed” (i.e.,positive in the northern hemisphere), as explored here, fromwell-known boundary effects that would tend to lead to thesame profile within a narrow layer close to the bottom of theconvection zone. It has long been anticipated that such re-versals of kinetic helicity would arise at the base of the solarconvection zone, for example, where convective downflowsbegin to be buoyantly braked and spread (e.g., Brummellet al. 2002). But in simple parametrizations of the solar dy-namo, the layer where this reversal is achieved is typicallytaken to be confined to a region near the bottom of theconvective envelope or below its base (e.g., Charbonneau2010). The convergence of flows to feed plumes that are be-ginning to rise buoyantly from a bottom thermal boundarylayer is also well known (Julien et al. 1996; Nishikawa & Ku-sano 2002; Dube & Charbonneau 2013; Guervilly et al. 2014)and is likewise somewhat distinct from the distributed he-licity profiles described here, though the same basic physicalmechanisms underlie all these. This will be demonstrated inthe following Sections.

3 MODEL

3.1 Numerical model

For this work, we used the code MagIC1 to solve an anelasticversion of the MHD equations, following Gilman & Glatz-maier (1981), Braginsky & Roberts (1995) and Lantz & Fan(1999), in a rotating spherical shell. This code implements adimensionless formulation, where the length scale is the shellthickness d, the time is non-dimensionalized by the viscoustime τν = d2/ν (where ν is the kinematic viscosity) and thetemperature, gravity and density by their values at the outerboundary, respectively To, go and ρo. Lastly, the magneticfield unit is given by

√Ωµλiρo, where Ω is the rotation rate,

µ the magnetic permeability and λi the magnetic diffusivity

1 https://github.com/magic-sph/magic

at the bottom boundary of the domain, ri. The dimensionlessequations are

E(

∂u∂ t

+ u ·∇u)

=−∇pρ−2ez×u +

RaEPr

rro

ser

+1

Pmi ρ(∇×B)×B +

Eρ

∇ ·S,

(3)

∂B∂ t

= ∇× (u×B)− 1Pmi

∇× (λ∇×B), (4)

ρ T(

∂ s∂ t

+ u ·∇s)

=1

Pr∇ · (ρT ∇s)+ ερ

+Pr Di

Ra

[Qν +

1Pm2

i EQ j

].

(5)

∇ · (ρu) = 0, (6)

∇ ·B = 0. (7)

where u, B and s are the velocity, magnetic field and entropyfields, respectively, η is the aspect ratio given by η =ri/ro =0.2 and the traceless rate-of-strain tensor S for homogeneousν is

S = 2ρ

[ei j−

13

δi j∇ ·u], ei j =

12

(∂ui

∂x j+

∂u j

∂xi

), (8)

and δi j is the identity matrix. The viscous and ohmic heatingcontributions are

Qν = 2ρ

[ei je ji−

13

(∇ ·u)2]

(9)

and

Q j = λ (∇×B)2. (10)

The background reference state is defined by the tem-perature gradient dT/dr =−Dig(r) and the background den-sity is derived as ρ(r) = T n, where the tilde corresponds tothe background reference state and n is the polytropic index.The dissipation number Di is expressed as

Di =go dcp To

= 2eNρ/n−1

1 + η, (11)

where Nρ = ln(ri/ro) is the number of density scale heightsand ri and ro are the radii of the inner and outer bound-aries, respectively. In Fig. 4, the solid blue line correspondsto Nρ = 3 and n = 2 and the cyan dot-dashed line to Nρ = 5and n=2. The red dashed line corresponds to a polynomialfit of Jupiter’s interior model (French et al. 2012), where thecorresponding T (r) was obtained from the best correspond-ing n to the data for the background temperature from thesame model. The result is Nρ≈4.9 and n≈2.2.

Finally, the control parameters in Eqs. (3–7) corre-spond to ratios between pairs of terms of the Navier-Stokesequation, namely the Ekman number (viscous over Corio-lis force), the Rayleigh number (measure of the convectivedriving of the system) and the fluid and magnetic Prandtlnumbers,

E =ν

Ωd2 , (12)

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helicity inversion mechanism 5

Figure 4. Background density profiles used in this work. The red

dashed line corresponds to Jupiter’s density profile (French et al.2012), where Nρ ∼ 4.9 in the inner 99% of the shell radius and

fitted by a polytrope of index n∼2.2. The cyan dot-dashed line

shows the background density profile of a polytrope with Nρ = 5and n=2. The blue solid line shows the background density profile

of a polytrope with Nρ =3 and n=2.

Pr =ν

κ, (13)

Pmi =ν

λi, (14)

where κ, ν and λ are the thermal, viscous and magneticdiffusivities, respectively. In our models, we fixed E = 10−4

and we used Pr = 0.1/1.0 and Pmi = 0.0− 20.0 for the ratiosbetween diffusivities. The Rayleigh number is either entropy-based or flux-based as follows

Ra =god3

νκSscale, (15)

where Sscale = ∆s/cp if the boundary conditions of the en-ergy equation are fixed entropy and Sscale = qod/(ρocpκ) forthe models where the boundaries are assumed fixed flux.The quantities qo and cp are the specific entropy flux at theouter boundary and the specific heat capacity at constantpressure, respectively.

Even though the results discussed in this paper werefirst found in our Jupiter models with variable magneticdiffusivity (as reported by Jones 2014), we simplified someof these models by carrying out a small number of hydro-dynamic and magnetic simulations with constant transportproperties along radius, to illustrate our conclusions. Thevariable conductivity profile used in the Jupiter models isdescribed in Gastine et al. (2014).

3.2 Numerical method

The MagIC code uses a pseudo-spectral method to solve theMHD equations described above. Spherical harmonics areused in the horizontal direction (θ ,φ) up to degree and or-der `max and Chebyshev polynomials in the radial directionup to degree Nr (see Wicht 2002, for a more detailed de-scription). The equations are solved in the commonly used

poloidal/toroidal decomposition of the divergence-free fieldsρu and B (Eqs. 6 and 7) as

ρu = ∇×∇× ver + ∇×wer

B = ∇×∇×ber + ∇× t er,(16)

where v and b are the poloidal potentials of ρu and B, re-spectively, while w and t are the toroidal counterparts. Theradial unit vector is represented by er.

The boundary conditions applied in our simulations arethe same for the velocity and magnetic fields. Following pre-vious work which had the purpose of modelling the gas giants(Gastine & Wicht 2012; Gastine et al. 2012; Duarte et al.2013), the inner boundary is assumed no-slip and the innercore is modelled as an electrical conductor. Top boundary isconsidered free-slip and the magnetic field is there matchedto a potential field. The entropy boundary conditions andheating modes vary in our models. Several of our cases as-sume simple bottom heating, i.e. the entropy contrast be-tween the bottom and top boundaries ∆s is fixed and thereis no internal heating in the system, thus the heating enter-ing the system through the bottom boundary exits throughthe top. The other two set-ups for the energy equation con-sidered here account for internal heat sources: in one casewith fixed entropy boundaries and in the other with fixedflux boundary conditions.

3.3 Diagnostic parameters

In the following Sections we will describe the flow by oftenreferring to several dimensionless diagnostic parameters tobe consistent with the formulation outlined in the previousSections.

The amplitude of the flow contributions is measured interms of the Rossby numbers Ro. The value of Ro is calcu-lated as

Ro =u

Ωd, with u =

√3

r3o− r3

i

∫ ro

ri

〈u2〉r2 dr , (17)

where u is the rms volume-averaged flow velocity and thetriangular brackets denote the angular average⟨

f⟩

=1

4π

∫π

0

∫ 2π

0f (r,θ ,φ)sinθ dθ dφ . (18)

The local Rossby number has been introduced by Chris-tensen & Aubert (2006) to quantify the relative importanceof the advection term in the Navier-Stokes equation (Eq. 3).Here we consider only the non-axisymmetric part of the flowvelocity um6=0 (i.e. velocity excluding the axisymmetric flowcomponent) to calculate the local convective Rossby num-ber, defined as

Ro`conv =

√1V∫ ro

ri

⟨u2

m6=0⟩

r2 dr

Ω`, (19)

where, ` is a typical flow length scale given by

`(r) =π u2(r)

∑l

l u2l (r)

. (20)

Here ul is the flow contribution of spherical harmonic degreel.

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6 Duarte et al.

The geometry of the surface field is characterized by thedipolarity

fdip =

⟨(Bm=0

l=1)2⟩

⟨∑

l,m≤lmax

(Bm

l)2⟩ , (21)

which measures the relative energy in the axial dipole con-tribution at the outer boundary ro.

Finally, to describe the behaviour of the various com-ponents of the flow, we used correlations between pairs ofvariables. A correlation between two sets of data A and B inthe form of 3D matrices corresponding to the three sphericalcoordinates r,θ ,φ is calculated at each radial level as

corr(A,B)θ ,φ (r) =

=∑θ ,φ

[Aθ ,φ (r)−〈Aθ ,φ (r)〉

][Bθ ,φ (r)−〈Bθ ,φ (r)〉

]√∑θ ,φ

[Aθ ,φ (r)−〈Aθ ,φ (r)〉

]2∑θ ,φ

[Bθ ,φ (r)−〈Bθ ,φ (r)〉

]2 . (22)

The parameters described in this Section are listedalong control parameters in Tab. 1, averaged over at least0.1 viscous time. The values of the Rayleigh number neces-sary for the onset of convection as defined by Jones et al.(2009) (i.e., where all other dimensionless control param-eters are fixed), were obtained for cases without internalheating. These values were used to obtain the supercritical-ity values listed in Tab. 1 at constant Ekman number. In thesame table, simulations without internal heating correspondto ε = 0 (see Eq. 5) and ε 6=0 if internal heating is present.Concerning the entropy boundary conditions correspondingto columns sBC,i (bottom) and sBC,o (top), values 0 and 1indicate fixed entropy and flux, respectively.

4 KINETIC HELICITY REALIZED INNON-LINEAR SIMULATIONS

4.1 Numerical hydro/dynamo models

Figure 5 illustrates the two helicity configurations describedin Secs. 2.1 and Sec. 2.2 for global hydrodynamical anddynamo simulations in a rotating spherical shell, for twodifferent supercriticalities. Contour plots of azimuthally-averaged kinetic helicity are shown here for snapshots,since these helicity patterns are not time-dependent in ourmodels. The top panels show the usual one-layer nega-tive(North)/positive(South) helicity configuration for thecombination of columnar convection and plume-like convec-tion from Sec. 2.1. The bottom plots show the two-layer be-haviour described in Sec. 2.2 and illustrated in Fig. 3. In eachrow, the supercriticality of the models increases from left toright, hence the later onset of convection inside the tangentcylinder TC (cylinder tangent to the inner core boundaryaround the rotation axis) on the left-side panels. As men-tioned in the Introduction, the different sign of helicity (e.g.top and bottom rows of Figure 5) is obtained for modelswhich retain similar differential rotation profiles, as shownin Figure 6. As evident in Fig. 6, the differential rotation issolar-like in the sense of having a faster equator. Model 4does however exhibit some non-geostrophic flow that breaksthe equatorial symmetry (Gastine et al. 2012). This Figure

Figure 5. Azimuthally-averaged contour plots of kinetic helicity.

The top row displays the helicity pattern described in Sec. 2.1

for models 1 (a) and 4 (b) of Tab. 1. The bottom row shows twoexamples of the regime described in Sec. 2.2 for cases 8 (c) and

11 (d) of the same Table.

also shows an additional model (case 5 of Tab. 1) becausewe refer to that simulation later in Sec. 4.2.

Figure 7 shows radial velocity contour plots for threecases displayed in Fig. 5, namely the top row and theright panel of the bottom row. Figure 8 displays models 4and 11 in orthographic projection to illustrate the latitudi-nal/longitudinal distribution of the flow features at the twodepths described in Secs. 2.1 and 2.2. The leftmost panel ofFig. 7 (and top left panel of 5) is distinctly dominated bycolumnar convection. The middle panel has a stronger den-sity gradient in the outer part of the shell as Fig. 4 shows,resulting in a break down of the columns in the outer radius,where convection becomes dominated by radial features (seeFig. 12 of Gastine et al. 2013). Below this outer layer, con-vection remains under a columnar regime, though combinedwith plume-like convection. This set-up corresponds to theone-layer helicity pattern described in Sec. 2.1, where bothtypes of convection co-exist in strongly stratified models.The right panel corresponds to the two-layer helicity pat-tern described in Sec. 2.2, where convection is seemingly notcolumnar any more. In this case, the density gradient usedwas the interior model for Jupiter in Fig 4, which may helpachieving this configuration since, even though the density

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helicity inversion mechanism 7

Figure 6. Azimuthally-averaged contour plots of zonal velocity

uφ . The top row displays model 4 from Tab. 1, the middle rowmodel 5 and the bottom row model 11. The two top rows areexamples of the helicity pattern described in Sec. 2.1 and the

bottom row of Sec. 2.2. The velocity is given in units of Reynoldsnumber, Ro/E (see Eq. 17).

Figure 7. Contour plots of slices of radial velocity for three of

the cases shown in Fig. 5, namely cases 1, 4 and 11 of Tab. 1 from

left to right, corresponding to panels a, b and c, respectively. Thevelocity is given in units of Reynolds number, Ro/E (see Eq. 17).

contrast is also around 5 density scale heights, it now con-centrates most of the gradient in the outer 10−20% of theradius of the shell. However, this is not the only difference,as we will discuss next.

Perhaps more significantly, the cases in Figs. 5–8 thatshow “inverted” helicity (models 8 and 11) also differ fromthe other cases displayed here in these Figures by adoptinga lower value of Pr and a different mode of heating. WhilePr = 1.0 in the top row panels of Fig. 5, as in most of ourprevious work (for example, Gastine & Wicht 2012; Gastineet al. 2012; Yadav et al. 2013; Duarte et al. 2013), the casesin the bottom row have a lower value of Pr =0.1. The effectof lowering the Prandtl number has been thoroughly stud-ied by Simitev & Busse (2005); Sreenivasan & Jones (2006),where their main conclusion was the significant increase ofthe role of inertia when decreasing Pr by one order of mag-nitude or more from unity. When inertia enters the forcebalance at a significant degree, the columnar convection con-straint weakens in the bulk, as discussed by Christensen &Aubert (2006). The change of heating mode in conjunctionwith fixed flux boundaries is also crucial. An internal heatingsource has been shown to spread convection in the domain,thus also weakening convection in the bulk as it tends to de-tach the deeper convective motions from the inner boundary(Hori et al. 2010). With fixed entropy boundary conditions,on the other hand, changes in the internal heating mode didnot appear to change the outcome in our simulations (seehowever Jones 2014, for a possible counter-example).

Lowering the value of Pr appears to be the most impor-tant factor in changing the helicity pattern due to its relationto the amount of inertia in the system, with a tendency topromote plume-like convection. When applying only a dif-ferent heating mode or a different density profile or both, wesaw very little difference in the final results of our models asthe one-layer helicity configuration remains dominant. How-ever, lowering Pr alone is not sufficient either. In Fig. 9,we show two cases with the same lower Prandtl numberPr=0.1, though convection in the left panel is driven by bot-tom heating and on right by internal heating proportionalto the background density profile (Eq. 5). The left panelshows that indeed the inversion to attain the helicity regimeof Sec. 2.2 is not complete unless we combine the lower Prwith internal heating.

MNRAS 000, 1–?? (2015)

8 Duarte et al.

Figure 8. Contour plots of radial velocity ur for the two models on the right column of Fig. 5, namely models 4 (panels a and c) and 11(panels b and d) of Tab. 1. Panels a and b correspond to the inner radial level of r =0.7ro and c and d to r =0.97ro. The latitude circlesand meridians (black dashed lines) are placed 60 apart. The velocity is given in units of Reynolds number, Ro/E (see Eq. 17).

Figure 9. Azimuthally-averaged countour plots of kinetic helicity

for 2 cases with Pr=0.1. The heating mode of the model in panel

a is bottom heating (model 15 from Tab. 1) and on b is internalheating (model 11 from Tab. 1).

In conclusion, the combination of the three effects (highNρ , low Pr and internal heating) is important to achieve therequired degree of non-columnar/plume-like convection inthe bulk of the shell. The requirement of high Nρ is harderto constrain and appears to be inconsistent with the require-ment for negligible density contrast in the interior. This ap-parent contradiction is because the higher the overall densitycontrast, the closer we get to a negligible density gradient inthe bulk. Even though this does not eliminate the possibil-ity of Sec. 2.2 also occurring in a weakly stratified or evenBoussinesq model, cases 31 and 32 in Tab. 1 show a helicitypattern very similar to the plane layer configuration, witha symmetry about half of the height (Julien et al. 1996), orhalf the radius in the case of a spherical shell. How exactlythis symmetry about the mid-plane is broken by the com-bination of density stratification and internal heating, andwhether the “inverted” helicity layers studied here requireboth these elements or are conceptually unrelated, requiresfurther study.

Kapyla et al. (2013) has reported a relation between therotation rate and the degree of stratification for the onset ofoscillatory solutions, which we did not explore here: they findthat at higher Nρ , lower Ro is necessary to find oscillatory

solutions. They also argued that equatorward wave solutionsoccur only at larger Nρ , whereas poleward propagation isfound instead at milder density contrasts.

4.2 Dynamo waves

In Fig. 10 we show a few examples of butterfly diagrams con-structed from our models, to illustrate the consequence ofthe different helicity patterns described above, in rough ac-cordance with the Parker model. The top left panel shows aprevious simulation from our previous Jupiter models whichhas a poleward-propagating dynamo wave for comparison(see Gastine et al. 2012; Duarte et al. 2013). The otherthree panels exemplify the variety of equatorward propagat-ing waves found in the models described here. In some cases,the equatorward part is confined to a lower latitudinal bandwith weak poleward propagation at high latitudes, likely dueto the lack of convection inside the TC at lower supercritical-ity (see examples of Fig. 5). Nonetheless, this higher-latitudefeature is common in most models and a similar higher lat-itude feature has also been observed in the Sun (Hathaway& Upton 2014).

Figure 11 shows two butterfly diagrams for the samemodel. The toroidal field is shown on the left panel at thesame depth as in Fig. 10 and the poloidal field is shown atthe surface in the right panel. These two panels demonstratethat the wave-like motion is also observed at the surface ofour models, which would result in a Sun-like butterfly dia-gram. Following Gilman (1983), we also plotted here the ax-isymmetric toroidal components of the kinetic (dashed line)and magnetic (solid line) energies, normalized by the to-tal kinetic and magnetic energies, respectively. As Gilman(1983) reported, it is possible to identify the imprint of thewave in the kinetic and magnetic energy time series.

A detailed study of dynamo cases is out of the scope ofthis paper, since we intended to focus simply on the mecha-nisms that alter the kinetic helicity and with it the directionof migration. Aspects related to the magnetic field will befurther analysed in future work. For some of these cases, thedynamo wave is not stable, eventually stabilizing in an oc-tupolar solution, which may be related to the use of variabletransport properties from our work on Jupiter models.

MNRAS 000, 1–?? (2015)

helicity inversion mechanism 9

Figure 10. Four examples of models from our database that illustrate the effect of the helicity regime in the direction of propagation

of the dynamo wave. The time-evolution of the toroidal component of the magnetic field is shown averaged over longitude. The time is

given in viscous time τν . The cases in correspond to models 5, 17, 9 and 11 of Tab. 1 in clockwise direction, starting from the top leftpanel.

Figure 11. Illustration of the wave at 80% of the radius (left panel) and the resulting signature in the surface field (right panel), formodel 18 of Tab. 1.

MNRAS 000, 1–?? (2015)

10 Duarte et al.

5 ANALYSIS AND INTERPRETATION

5.1 Correlations to describe flow behaviour

According to the idealized scenario described in Sec. 2.1,columnar convection often dominates in the inner part ofthe shell, where the ‘rising/sinking’ component of the flowuz (along a column and away from the equator) is anti-symmetric about the equator but (since the column rotatesin the same direction in both hemispheres) ωz is symmet-ric. Converting between cylindrical and spherical coordinatesystems,

uz = ur cosθ −uθ sinθ ⇒ ur = us sinθ + uz cosθ , (23)

where s represents here the cylindrical radius, we see thatthe radial component of the velocity ur is symmetric overthe equator, while ωr is anti-symmetric. The radial quan-tities are particularly relevant in strongly stratified mod-els, where convection becomes plume-like thus the preferred’rising/sinking’ direction is along spherical radius. At eachradial level, divergence and convergence of the flow is repre-sented by the horizontal divergence of the flow (in sphericalcoordinates)

∇h ·uh =1

r sinθ

∂

∂θ(uθ sinθ)+

1r sinθ

∂uφ

∂θ. (24)

A similar divergence can be adapted to columnar convection,by assuming a horizontal plane defined by the cylindricalcoordinates s,φ at a fixed height z, thus replacing uh = f (θ ,φ)by uhz = f (s,φ) and similarly ∇h by ∇hz.

Figure. 12 shows several correlations between differentproperties of the rising/sinking flow that dominate convec-tion, only in the radial direction for a first analysis. Themodel of the panels of the left column represents the helic-ity set-up described in Sec. 2.1 and the model of the rightcolumn corresponds to the set-up described in Sec. 2.2. Fo-cusing mainly on the right column, since we expect convec-tion to be non-columnar here, in the top panel we see thatthe rising velocity ur correlates negatively with the horizon-tal divergence throughout most of the radius as expected,i.e. as the flow rises it contracts and when it sinks it di-verges, as described in Sec. 2.2. The picture naturally in-verts in the outer 10% of the layer, where the density gra-dient dominates, causing a positive correlation between thetwo quantities. The second-row panel on the right side showsthe expected role of the Coriolis force acting on the diverg-ing/converging flow to generate negative/positive ωr in thenorthern hemisphere (grey line) and positive/negative in thesouthern (black line). The third panel correlates directly urand ωr, clearly showing that rising/sinking flow in the outer∼ 10% of the shell gives negative/positive vorticity, whilethe picture inverts in the bulk. This panel also translatesdirectly into the preferred sign of kinetic helicity, negativenear the surface but positive below ∼90% (Sec. 2.2 scenario).Finally, the bottom panel displays the correlation betweenur and the acceleration or deceleration of the fluid, since inthe continuity equation it is dur

dr that is ultimately linked tohorizontal divergence/convergence (if the compressible termis negligible). This shows that the flow is consistently speed-ing up as it rises as well as slowing down when it sinks inmost of the shell, which explains the tendency to contractas it rises.

The left column of Fig. 12 displays a model which has

been shown in the previous Section to be mostly dominatedby columnar convection in the bulk, resulting in the helicitypattern described in Sec. 2.1. As a result, the radial correla-tions displayed in the left column of Fig. 12 show some in-consistencies and asymmetries between the two hemispheres,while the right side panels of the same Figure show almostperfect symmetry. This occurs because in columnar convec-tion, such quantities are better expressed in terms of thecylindrical z coordinate. Thus Fig. 12 shows two examplesof changes in the flow regime as the rotational constraint isvaried, as also seen in the values of convective local Rossbynumber: Ro`conv =0.276 and Ro`conv =0.423 for the models onthe left and right columns, respectively.

Figure 13 shows the same left column of Fig. 12 and theright column represents equivalent calculations for the samemodel along z, as described in the beginning of this Section.There is some asymmetry in both columns, since there is asmall superposition of the two convective regimes, but theright column more clearly illustrates the behaviour of theflow. In the right top panel, when uz is positive (away fromthe equator and toward the upper boundary) in the north-ern hemisphere, the flow will diverge at the top of a columnwhich is seen by tracking the grey line. The inner part is anexception though, likely due to either a superposition withthe radial component or to the influence of convection insidethe TC (not explored here) since the correlations shown areaveraged over horizontal spherical coordinates (see captionof Fig. 12). An asymmetry is also perceptible in the secondrow, though the cylindrical coordinate system plays the ma-jor role in contributing the positive peak near the bottomof the layer (0.3− 0.4ro) in the radial correlations betweenωr and ∇huh in the northern hemisphere (grey line). As il-lustrated above in Fig. 1, the flow diverges mainly at theextremities of a column with negative ωz, where it encoun-ters the bottom and top boundaries, but it converges alongits length in most of the bulk. This explains the positivepeak of corr(ωz,∇hzuhz) in the bottom half of the shell thatdirectly affects the sign of corr(ωr,∇huh) through Eq. 23, ap-pearing to result in an inconsistent behaviour of the Coriolisforce in the radial component. It’s the cylindrical calculationon the right panel of the second row that is most relevantfor this part of the shell. The top four panels show this per-sistent asymmetry between the two hemispheres as a directconsequence of the different symmetries between ur/uz andωr/ωz. This asymmetry is also seen in the bottom four pan-els, though more camouflaged. In particular the third rowonce again shows the clear resulting sign of kinetic helic-ity in the majority of the shell, consistent with the set-updescribed in Sec. 2.1.

Figure 14 shows a segment of the near-surface maps ofdifferent properties of the flow to illustrate the typical be-haviour of the flows at the outer radii common for moststrongly stratified models. In the northern hemisphere, theflow rises (first panel from the top), expands as the densitydecreases (horizontal divergence of the horizontal velocity∇h · uh, in the third panel) and rotates clockwise (negativeωr in the second panel, i.e. anti-cyclones – see Sec. 2.1). Thesame happens when the flow sinks as it is promptly com-pressed in the filaments seen in these three panels aroundthe sources represented by the horizontal divergence of thehorizontal flow ∇h ·uh.

Figure 15 illustrates the alternative convection mech-

MNRAS 000, 1–?? (2015)

helicity inversion mechanism 11

Figure 12. Correlations between several properties of the flow along the radial direction, namely rising/sinking flow ur, horizontal

divergence ∇h ·uh, radial vorticity ωr and acceleration/deceleration ∂rur. The left column corresponds to model 4 in Tab. 1 and the rightcolumn to model 11. All correlations were calculated at each grid point according to Eq. 22, averaged over the horizontal spherical

coordinates θ ,φ at each radial level r and over time from ∼100 snapshots distributed over an interval of ∼20% of a viscous time.

anism in the inner part of the shell described in Sec. 2.2,at a depth of 40% below the surface of the model. The dif-ferent dynamics shown above in Fig. 12 by the inversion ofsign in the correlations of the third row, can also be seenhere as positive ur in the top panel of Fig. 15 now correlateswith negative horizontal divergence (middle panel) and thusnegative ωr (cyclones, see Sec. 2.2) due to the action of theCoriolis force. However, these linkages are less clear than inFigure 14, or the idealised description of Sec. 2.2, partly be-cause of the complexity of the flow field and because a varietyof processes contribute at some level to vorticity generation.On average, though, as Fig. 12 shows, upflows in this regionare well correlated with horizontally convergent flows.

5.2 Columnarity

Contour plots of radial velocity shown in Fig. 7 of the pre-vious Section show the predominance of the columnar con-vection regime in the bulk of a spherical shell, as suggestedin Sec. 2.2. Near the surface both set-ups of Secs. 2.1 and2.2 are identical, except for a slight difference in the scaleof the convection, since lower value of Pr is known to pro-duce larger convective flow scales (Jones et al. 2009). Atdeeper radial levels as discussed above, columnar convec-tion tends to remain the dominant type of convection, whilestill predominantly plume-like near the surface. Soderlundet al. (2012) defined a parameter they called “Columnarity”,meant to determine the degree at which the flow is organizedin convection columns. They defined this parameter from theratio of the integral of ωz in the cylindrical z-direction and

MNRAS 000, 1–?? (2015)

12 Duarte et al.

Figure 13. Correlations between several properties of the flow along the radial direction on the left column (similarly to Fig. 12) and

along the cylindrical z coordinate on the right column. Both columns correspond to model 4 in Tab. 1. All correlations were calculatedat each grid point according to Eq. 22, averaged over the horizontal spherical coordinates θ ,φ at each radial level r and over time from

102 snapshots distributed over an interval of ∼20% of a viscous time.

the integral in z of the total RMS vorticity ω of the flow:

Cωz =

∑s,φ|〈ω ′ · z〉z|

∑s,φ〈|ω ′|〉z

, (25)

where primes mean that vorticity is calculated from the non-axisymmetric part of the velocity field.

This parameter works well for Boussinesq models, whereconvection onsets and remains attached to the inner bound-ary, unless the supercriticality is large enough (typicallyabove 50-100 times supercritical). We saw that even thoughthis criterion works fairly well for mildly stratified models(e.g. N.3), it is not descriptive enough for strongly strati-fied models. However, the previous Sections suggest that themost likely reason for this is that even though the flow is stillcolumnar in the bulk, it is not in the outer ∼10% of the ra-

dius. This will naturally affect the total value of Cωz sincethe integration includes this outer part of the shell whereconvection is not columnar. As a result, the value of Cωz isnearly the same for models with different helicity patterns.We attempted to calculate Cωz below deeper radii to removethe major effect of the outer part and thus find a way to stilluse this parameter to distinguish one-layer helicity modelsfrom two-layer. The result is shown in Fig. 16 for severalof our models, obtained from snapshots. By excluding theouter part, it appears indeed possible to separate the tworegimes using Cωz, i.e. to separate the poleward-propagatingwave models from the equatorward ones. The two differ-ent rotational regimes represented by the grey/black linesare also characterized by a generally higher value of Ro`, asmentioned in previous Section.

MNRAS 000, 1–?? (2015)

helicity inversion mechanism 13

Figure 14. Properties of the flow in the upper layer of stronger

density gradient in the northern hemisphere, in a segment ex-tracted from the right bottom panel of Fig. 5 (model 11 of Tab. 1).

The radial level corresponds to the outer white dashed line in the

same Figure (r=0.97ro). The dotted black meridians and latitudecircles are 3 degrees apart.

6 DISCUSSION AND APPLICATION TOSTARS

In many classic theories of stellar and planetary dynamos,the direction of propagation of dynamo “waves” is deter-mined partly by the differential rotation and partly by thekinetic helicity (e.g., Parker 1955; Steenbeck et al. 1966;Moffatt 1978). In broad accordance with this expectation,many prior dynamo simulations that possessed some formof cyclical behavior have exhibited poleward propagation offields (Gilman 1983; Ghizaru et al. 2010; Goudard & Dormy2008; Schrinner et al. 2011; Simitev & Busse 2012; Gastineet al. 2012; Duarte et al. 2013), in keeping with the realizedprofiles of helicity and differential rotation but in contrastto what is seen in the Sun. Notable exceptions in the stel-lar context include Kapyla et al. (2013); Augustson et al.(2013), and Warnecke et al. (2014), who attributed equator-ward propagation in the simulations primarily to changes inthe differential rotation profile. In those simulations, as inprior ones exhibiting poleward propagation, the kinetic he-licity of the flows appears to have remained predominantlynegative in the northern hemisphere (positive in the south-ern hemisphere), in accord with theoretical expectations forboth columnar and highly stratified convection (as summa-rized in Sec. 2.1). In the planetary context, models withequatorward propagating dynamo waves in a set-up closerto ours were reported by Jones (2014) for“failed”Jupiter-like

Figure 15. Properties of the flow in the lower layer of milder

density gradient and inverted helicity in the southern hemispherefor model 16 of Tab. 1. The radial level now corresponds to r =

0.6ro.

Figure 16. Columnarity (Soderlund et al. 2012) as a function of

the outer radius used as the top boundary for the integration/sumin Eq. 25. The black lines correspond to models 1, 2, 3, 4, 5, 6,22, 23, 24, 25, 26, 27, 28, 29, 30 of Tab. 1, which have the helicity

pattern from Sec. 2.1 and poleward propagation in the case ofmagnetic cases. The grey lines correspond to models 7, 8, 9, 10, 11,

12, 16, 17, 18, 19, 20, which have the helicity pattern described inSec. 2.2 and equatorward propagation if magnetic. The remainingcases of Tab. 1 do not have a clear helicity pattern nor a preferred

direction of propagation.

MNRAS 000, 1–?? (2015)

14 Duarte et al.

simulations, carried out at lower values of Prandtl numberand always driven by internal heating.

We have examined here whether the kinetic helicity inglobal-scale convection simulations of stars and planets mustnecessarily accord with the classic expectation describedabove (and in Sec. 2.1), or whether other self-consistent pro-files are possible. We have demonstrated that in some casesequatorward migration of magnetism can arise not from un-usual differential rotation profiles, but from the realization ofa kinetic helicity profile that is the opposite to that encoun-tered in many prior simulations. We have focused here onthe mechanisms by which this helicity “inversion” is accom-plished, while deferring a detailed study of the properties ofdynamo solutions to later work.

Our analysis indicates that the “classic” helicity con-figuration commonly results when convection is primarilycolumnar (as often occurs in rapidly rotating cases withsmall to moderate density contrasts), when no internal heat-ing occurs, and/or when the Prandtl number is unity orgreater (as adopted in many prior simulations). But bychanging a combination of these factors, we have demon-strated that the helicity in the bulk of the fluid mayswitch sign, becoming positive (in the northern hemisphere)throughout much of the rotating domain (rather than justin narrow boundary layers). This is due in part to the pro-motion of a second, deeper layer below the outermost re-gions with a stronger density gradient. In this deep layer,convective flows are neither columnar nor dominated by ex-pansion/contraction associated with the density gradient.The resulting “plume-like” flows there tend to have upflowsthat are associated with converging horizontal flows (ratherthan divergent ones, as realized in more strongly stratifiedregions), which in combination with Coriolis forces leads tocyclonic vorticity and positive kinetic helicity. Lowering thefluid Prandtl number promotes the disruption of convectivecolumns due to the first-order effect of inertia on the forcebalance, but additionally the presence of internal heating inthe system and a mild density stratification in the deep inte-rior help to finalize a stable inversion of the helicity pattern.

In summary, to guarantee plume-like convection in thebulk of a spherical domain, the three effects required in ourmodels were strong density stratification (with most of thedensity gradient confined to the outermost region), lowerfluid Prandtl number, and a combination of internal heatingand fixed flux boundary conditions to weaken convection inthe bulk. The combination of the last two effects appearsto be particularly essential in our cases. We did not carryout an extensive study of the relative thickness of the twohelicity layers in each hemisphere, its dependence on controlparameters (including the fluid Prandtl number) or the tran-sitional Pr needed to reverse the sign of helicity. Because Pr isalso a function of many other control parameters (includingEkman and Rayleigh numbers and different boundary con-ditions or heating modes), we consider this to be beyond thescope of this paper. However, a few preliminary simulationssuggest that the size of the outer layer relative to the size ofthe domain – in which the helicity remains negative in thenorthern hemisphere – decreases significantly with Ekmannumber. On the other hand, increasing the Rayleigh num-ber (to approach a Rossby number of order unity) tends toact in the opposite direction, decreasing the extent of the re-gion of inverted helicity. Though the simulations considered

here are all unstably stratified throughout their interiors, thefraction of energy carried by the convection is small in somecases (primarily because of the low value of Pr adopted); weexpect this fraction to grow with Ra, and this may furtherinfluence the size of the “inverted” helicity region. It is notyet clear how these effects would combine to determine thehelicity profile at much lower E and much higher Ra, but weintend to examine this in future work.

It is not entirely clear whether the “inverted” kinetichelicity profiles explored here, and the accompanying equa-torward migration of magnetic fields, are likely to be real-ized in stellar or planetary interiors. Some of the factors wehave identified as contributing to these profiles are reliablypresent: for example, Pr 1 in many astrophysical plas-mas (including the Sun; see, e.g., Miesch 2005). Likewise,the condition that density stratification be comparativelyweak throughout part of the interior, so that the expan-sion/contraction of rising/sinking parcels (and the vorticalhorizontal flows associated with this) do not utterly domi-nate the production of kinetic helicity, is satisfied in manystellar and planetary convection zones. Even in the Sun,the density scale height at the base of the convection zoneis comparable to the depth below the surface, decreasingrapidly only nearer the photosphere. (Near the photosphere,it seems safe to assume that the expansion/contraction ofrising/falling convective cells will dominate over other ef-fects, leading to kinetic helicity that is negative in the north-ern hemisphere.) On the other hand, we have found that ex-tended internal heating is also important for giving rise tothe “inverted” helicity profiles; in its absence (i.e., in casesheated solely from below), both mixing length theory andour simulations suggest that (in regions where the densitystratification is weak) the velocity should not increase signif-icantly with radius outside of a narrow boundary layer. Thisin turn often leads to the “classic” helicity profile except inthe boundary layer. In essence, the presence of extended in-ternal heating allows a phenomenon that might otherwise beconfined to a narrow boundary layer to persist throughoutmuch of the fluid. Because of this, we suspect that our re-sults may be more relevant to objects like brown dwarfs andvery low mass stars, in which internal energy generation byfusion or gravitational contraction extends over a large frac-tion of the interior (e.g., Chabrier & Baraffe 1997), than tostars like the Sun, in which the luminosity that must be car-ried by convection is roughly constant across the convectiveenvelope. Even in the latter case, however, it is conceivablethat non-standard models in which heat transport is domi-nated by cooling from the top boundary (i.e., “entropy rain”,as studied in Brandenburg 2015) might lead to kinetic helic-ity profiles resembling those here. We defer a more detailedexploration of these possibilities, and their consequences forstellar and planetary dynamos, to future work.

MNRAS 000, 1–?? (2015)

helicity inversion mechanism 15

MNRAS 000, 1–?? (2015)

16 Duarte et al.

Tab

le1:S

um

mary

of

most

lyti

me-

aver

aged

resu

lts,

for

E=

10−

4 .T

he

sup

ercr

it-

icality

valu

esR

a/R

a cr

corr

esp

on

dto

the

defi

nit

ion

ap

plied

ton

on

-dim

ensi

on

al

form

ula

tion

(see

Jon

eset

al.

2009,fo

ra

full

des

crip

tion

).T

he

qu

anti

ties

Cω

z,to

tal

and

Cω

z,90

%co

rres

pon

dto

snap

shots

of

the

valu

esof

colu

mn

ari

tyd

eter

min

edfo

rth

ew

hole

shel

land

bel

ow

90%

of

the

tota

lra

diu

s,re

spec

tivel

y.1

Ra c

r=

4.64

8×

106

2R

a cr=

5.37

2×

106

3R

a cr=

2.97

5×

107

4R

a cr=

5.41

2×

106

∗ Model

sw

her

eλ

isa

fun

ctio

nof

radiu

s(s

eeG

ast

ine

etal.

2014).

Model

Nρ

nR

aP

rP

mi

εs B

C,i

s BC,o

Eki

nE

mag

f dip

SDdi

pC

ωz,

tota

lC

ωz,

90%

Ro

Ro `

,con

v

13.0

2.0

1.1×

107(

1)1.0

2.0

0.0

00

7.94×

104

4.45×

104

2.86×

10−

34.

92×

10−

30.

485

0.50

60.

0078

0.05

7

23.0

2.0

2.0×

107(

1)1.0

2.0

0.0

00

4.88×

105

3.39×

105

1.17×

10−

26.

27×

10−

20.

321

0.35

10.

0186

0.14

3

35.0

2.0

4.0×

107(

2)1.0

2.0

0.0

00

1.07×

106

9.02×

105

3.18×

10−

31.

31×

10−

20.

275

0.36

70.

0184

0.23

1

45.0

2.0

5.0×

107(

2)1.0

2.0

0.0

00

1.65×

106

1.30×

106

2.89×

10−

33.

17×

10−

20.

244

0.36

70.

0217

0.27

6

5∗

4.9

2.2

2.2×

108(

3)1.0

1.0

0.0

00

4.46×

106

7.68×

105

3.02×

10−

12.

61×

10−

10.

252

0.33

00.

0275

0.27

6

64.9

2.2

4.0×

108(

3)1.0

2.0

0.0

00

8.34×

106

3.16×

106

2.91×

10−

32.

31×

10−

20.

208

0.29

30.

0331

0.40

07

4.9

2.2

2.6×

109

0.1

0.0

0.52

81

14.

76×

106

0.0

−−

0.22

70.

172

0.03

220.

234

84.9

2.2

2.6×

109

0.1

4.0

0.52

81

13.

67×

106

6.39×

105

1.55×

10−

41.

90×

10−

20.

229

0.19

40.

0284

0.22

59

4.9

2.2

3.0×

109

0.1

2.0

0.52

81

17.

91×

106

1.25×

104

2.22×

10−

42.

94×

10−

20.

217

0.18

40.

0384

0.30

810

4.9

2.2

4.0×

109

0.1

0.0

0.52

81

11.

94×

107

0.0

−−

0.21

30.

230

0.05

840.

458

11

4.9

2.2

4.0×

109

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9

MNRAS 000, 1–?? (2015)

helicity inversion mechanism 17

ACKNOWLEDGEMENTS

This work was supported in part by the European Re-search Council under ERC grant agreement no 337705(CHASM) and by a Consolidated Grant from the UK STFC(ST/J001627/1). Most of the computations for this paperwere carried out in the GWDG computer facilities in Got-tingen and some were performed on the University of Exetersupercomputer, a DiRAC Facility jointly funded by STFC,the Large Facilities Capital Fund of BIS and the Universityof Exeter.

The authors would like to thank Rakesh Yadav for use-ful discussions and help with the display of Figures and thereviewer for the input which contributed in improving ourpaper.

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