the structure of small-scale magnetic flux tubes

Mon. Not. R. Astron. Soc. 358, 1025–1035 (2005) doi:10.1111/j.1365-2966.2005.08840.x

The structure of small-scale magnetic flux tubes

Robert Cameron1� and David Galloway2�1Max-Planck-Institut fur Sonnensystemforschung,† Max-Planck-Straße 2, D-37191 Katlenburg-Lindau, Germany2School of Mathematics and Statistics, University of Sydney, NSW 2006, Australia

Accepted 2005 January 17. Received 2005 December 20; in original form 2004 August 26

ABSTRACTThree main mechanisms have been described to determine the maximum field strength andstructure of a solar or stellar magnetic flux tube. This paper attempts to relate them to one anotherthrough a series of magnetoconvective calculations. The first process is the balancing of theLorentz force by radial gradients in the buoyancy force. It was first found in the Boussinesqregime, where it was studied in the late 1970s by Galloway, Proctor & Weiss. A similar balancecan occur in the fully compressible case, where we refer to it as quasi-Boussinesq (QB). Thesecond process involves a balance between an outward-directed radial pressure gradient andradial gradients in the buoyancy force outside the tube. This is the mechanism proposed inthe early 1990s by Kerswell & Childress (the KC mechanism). The third mechanism, convectivecollapse (CC), is a process whereby a flux tube can evolve to a high field strength becauseof an instability due to the superadiabaticity of the material within the tube. Until now, ithas been studied using the so-called ‘thin flux tube’ approximation in which convective motionsare ignored even though there is a background superadiabatic density stratification. Here weplace these three mechanisms in a unified framework and explore the transitions betweenthe solutions as various parameters are varied. In particular, we show that the QB solutionsare preferred for a wide range of parameters, whereas CC solutions occur only in very specificcircumstances. In particular, on the Sun, the latter are probably limited to flux tubes with radiiless than approximately 10 km, the turbulent magnetic diffusivity length-scale.

Key words: convection – magnetic fields – MHD.

1 I N T RO D U C T I O N

Small-scale magnetic flux concentrations are seen on the Sun atthe boundaries of granules and supergranules. Similar small-scalemagnetic elements almost certainly exist on other stars as well,where they are presumably important in maintaining stellar chro-mospheres. The formation of these magnetic structures is initiallydriven by kinematic flux expulsion, which drives the magnetic fieldtowards the granular and supergranular boundaries (Parker 1963;Weiss 1964; Clark & Johnson 1967). Once flux accumulates beyonda certain level, the resulting concentrations become dynamically ac-tive. Thereafter they modify the local velocity field, which in turndetermines their internal structure. This paper addresses the issue ofhow the local velocity field is modified and what form the magneticstructures take in consequence.

There are three extant theories that address these questions: con-vective collapse (CC), the Boussinesq analysis of Galloway, Proctor& Weiss (1978) (quasi-Boussinesq or QB in its compressible gen-eralization), and a regime studied by Kerswell & Childress (1992)(KC) that relaxes one of the Boussinesq assumptions. These three

�E-mail: [email protected] (RC); [email protected] (DG)†Formerly the Max-Planck-Institut fur Aeronomie.

theories represent limiting cases of the full problem. The currentstudy is motivated first by the desire to understand the relationbetween the three limiting cases, secondly by recent infrared ob-servations suggesting that a range of field strengths is present on thesolar surface (Lin & Rimmele 1999; Khomenko et al. 2003), andthirdly by the need to understand parameter space before applyingthe theories to other stars. We present computational results thatspan these different limiting regimes, in order to provide a frame-work in which to study the transitions between the various types ofsolution.

The setup we have chosen is similar to that of most recentcompressible magnetoconvection calculations. The most significantsimplifications are that we ignore radiative transfer effects (exceptvia thermal diffusion), and we assume axisymmetry. These assump-tions were also made by Hurlburt & Rucklidge (2000) in a recentstudy of sunspots and pores. The resulting simplifications serve bothto ease computational demands substantially and to keep the physicssimple enough that the main issues remain in sight. Whilst mostfeatures seen on the Sun are not literally axisymmetric, this geom-etry still seems the most appropriate for a general investigation.Non-axisymmetric modes such as interchange instabilities maycause some fragmentation, but if the tube is situated in a convec-tive downdraught, fluid flows will keep regrouping these fragmentsinto a more or less axisymmetric configuration. We have also added

C© 2005 RAS

1026 R. Cameron and D. Galloway

axisymmetric twist components to the problem; this results in someinteresting consequences for the propagation of torsional Alfvenwaves, but we defer the results of this to a later paper and restrictourselves here to the untwisted case.

Section 2 describes in detail the model, the governing equa-tions and the boundary conditions. In Section 3 we first providean overview of the different physical regimes, then in Sections 3.1,3.2 and 3.3 we give a selection of numerical results for the QB, KCand CC cases, respectively, emphasizing the transitions betweenthem. Section 4 discusses the significance of these results, with par-ticular emphasis on the issues of flux tube evacuation and the limitsof applicability for the various types of solution. This is followedby a brief concluding section, where some comments are made onthe range of field strengths that can be expected in small-scale fluxtubes at the solar photosphere.

2 P RO B L E M F O R M U L AT I O N

We solve the equations of fully compressible convection with animposed vertical magnetic field in an axisymmetric cylinder. Ourcoordinates are r in the radial direction and z, which increases withdepth.

Other authors have treated compressible magnetoconvection ina variety of settings, in both two-dimensional and fully three-dimensional geometries (Chan, Sofia & Wolff 1982; Hurlburt &Toomre 1988; Lantz 1995; Blanchflower, Rucklidge & Weiss 1998).The axisymmetric problem has been addressed by Hurlburt &Rucklidge (2000), with a view to explaining the structure of poresand sunspots. Our aim here is to identify the various different phys-ical mechanisms that can govern the structure of a small flux tube,and to show which one is dominant in a given range of parameterspace.

The equations we use are: the continuity equation,

∂ρ

∂t+ ∇ · (ρu) = 0;

the compressible Navier–Stokes equation including Lorentz andbuoyancy forces,

∂ρu∂t

+ ∇ · (ρuu)

= −∇ P + ρgz + 1

µ0(∇ × B) × B + µ

(∇2u + 1

3∇∇ · u

);

the induction equation,

∂B∂t

= ∇ × (u × B) + η∇2 B;

the heat equation,

ρcv

(∂T

∂t+ u · ∇T

)

= −P∇ · u + K∇2T + viscous heating + ohmic heating;

an equation of state,

P = RρT ;

and the solenoidality condition,

∇ · B = 0.

Boundary conditions are similar to those described in Proctor(1992): the temperature is fixed at the horizontal boundaries z = 0and z = d, no heat or magnetic flux flows across the curved outersurface r = a, and on all boundaries the field is vertical, with no

normal component of velocity and no viscous stress. Most idealizedstudies of both Boussinesq and compressible magnetoconvectionhave used these boundary conditions.

These equations and boundary conditions have a trivial solutionwhere the velocity is zero, the magnetic field is uniform and vertical,the temperature varies linearly from T 1 at the bottom to T 0 at thetop of the cylinder, and the density is a polytrope with index m =gd/[R(T1 − T0)] − 1.

A source of confusion in compressible magnetoconvection is theconsiderable number of different scalings that have been adoptedby various authors, making a comparison of results difficult. Thestandard parameters of incompressible theory, such as the Rayleighnumber, Chandrasekhar number and the various diffusivity ratios,can all be generalized in a number of alternative ways, each being ad-vantageous in different circumstances. Here we use the conventionsadopted by the Cambridge group (Hurlburt & Rucklidge 2000, forexample). Thus the height is scaled with layer depth d, time with thesound traveltime based on the temperature at the top of the cylinderd/

√RT0, density with the density ρ 0 at the top of the layer, tem-

perature with T 0, and the magnetic field with B0, the strength of theimposed vertical magnetic field.

Non-dimensionalizing the equations in this way introduces eightparameters: Q = B2

0 d2/(µ0 µ η), the Chandrasekhar number; �,the ratio of the temperature difference between the bottom and topof the cylinder, T 1 − T 0, to the temperature at the top of the cylinderT 0; m = g/(T 0�) − 1, the polytropic index; σ =µcp/K , the Prandtlnumber; ζ 0 = ηcp/K , the ratio of the Prandtl and magnetic Prandtlnumbers; γ , the ratio of specific heats; K = K/[dρ0cp(RT0)1/2],the dimensionless thermal diffusivity; and A = a/d, the ratio ofthe radius of the cylinder to its height. In this paper we restrict ourattention to the case A = 1 since we are interested in the structure ofthe flux tube near the axis rather than the structure of the surroundingconvective eddies. We also take γ = 5/3.

The non-dimensional equations are then

∂ρ

∂t+ ∇ · (ρu) = 0,

∂ρu∂t

= ∇ · ( −ρuu + Qσζ0 K 2 BB) − ∇(

P + 12 Qσζ0 K 2|B|2)

+ �(m + 1)ρ z + σ K∇ · τ,

∂B∂t

= ∇ × (u × B) + ζ0 K∇2 B,

∂

∂t

{ρ

[T

γ − 1+ 1

2|u|2 − �(m + 1)z

]+ Q

2σζ0 K 2|B|2

}

= −∇ ·{

ρ

[γ T

γ − 1+ 1

2|u|2 − (m + 1)z

]u

}

− ∇ · [Qσζ0 K 2 B × (u × B)] + ∇ · (K∇T )

+ ∇ · {ζ 20 K 3 Qσ [∇ B − (∇ B)T] · B} + ∇ · [Kσu · τ ],

P = ρT ,

τ = − 23 (∇ · u)I + [∇u + (∇u)T]

and

∇ · B = 0,

where z is measured vertically downwards and I is the identity tensor.

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Small-scale magnetic flux tubes 1027

Since ∇ · B = 0 and we are considering only the axisymmetriccase, we can introduce a flux function χ such that

Br = −1

r

(∂χ

∂z

)and Bz = 1

r

(∂χ

∂r

),

resulting in the scalar form for the induction equation,

∂χ

∂t= −(u · ∇)χ + ζ0 K∇ ·

(r 2∇ χ

r 2

).

The temperature of the upper boundary is set at 1 in these non-dimensional units, whilst the bottom boundary has a temperature of1 + �. The remaining boundary conditions are unaffected by thenon-dimensionalization.

The initial condition is taken to be the trivial equilibrium, u =0, B = z, T = 1 + �z and ρ = (1 + �z)m+1, with a slight pertur-bation.

The above choice of non-dimensional parameters was chosen toallow easy comparison with previous research, but it is far fromunique. In particular, the Rayleigh number Ra is often used. Interms of the above parameters, the Rayleigh number, defined usingthe trivial equilibrium at height z = 1/2, is

Ra = (m + 1)2

(1

m + 1− γ − 1

γ

)�2 (1 + �/2)2m−1

σ K 2.

It also turns out to be useful to introduce

T1/2 = 1 + �/2

�(m + 1),

the meaning of which is more clear in dimensional units,

T1/2 = R(T0 + T1)/2

gL,

and the Roberts number q = 1/ζ 0.The problem described above is difficult to solve analytically,

except for special solutions such as the polytropic equilibrium statealready used. Thus numerical solution is necessary. This has beencarried out using a code that is described fully in Cameron (1998).Briefly, it uses a two-step Lax–Wendroff scheme for the advectiveterms (Lax & Wendroff 1960) and a forward-time centred-spacescheme for the diffusive terms. This makes the code somewhat old-fashioned and relatively slow, but quite robust, and adequate for ourpurposes.

3 R E S U LT S

We are attempting to determine how the field strength in a steady-state magnetic flux tube saturates as its flux is increased. The resultswe are about to present are complicated. We therefore begin byproviding a framework in terms of which the details in the followingsubsections can be placed.

Saturation of the magnetic field requires the velocity field tochange (otherwise the problem remains kinematic and the fieldstrength grows linearly with the applied flux). The overview ofour results therefore begins with the time-independent momentumequation:

−∇ · (ρuu) − ∇ P + Qσζ0 K 2 J × B

+ �(m + 1)ρ z + σ K∇ · τ = 0,

We are interested in the underlying balance in this equation. Theterms ∇ · (ρuu) and σ K∇ · τ are important to the details of howthe balance is achieved, but neither term by itself drives a flow or

balances the active forces driving the flux tube into existence. Thisleaves us with three active forces: the buoyancy force, the Lorentzforce and the pressure gradient.

Analytic studies typically balance two of these three terms.Specifically, CC balances the pressure gradient and the Lorentzforce, the KC mechanism balances the buoyancy force and the pres-sure gradient, and the QB mechanism balances the Lorentz andbuoyancy forces. These three mechanisms then are exhaustive oftwo force pairings – they represent all the limiting cases in whichone of the three forces can be ignored. Our numerical calculationsspan these three limiting regimes.

The QB regime has been studied by Galloway, Proctor & Weiss(1977, 1978) and Galloway & Moore (1979), in a similar magne-toconvective setup to ours (restricted, however, to the Boussinesqlimit). In Section 3.1 we describe how their results carry over intothe fully compressible case.

In Section 3.2 we will show how there is a smooth transitionbetween the QB and KC regimes. Our parameter study focuses onthe physics that defines the KC regime: a dynamically importantpressure gradient at the axis near the top of the flux tube. To createsuch a gradient, the pressure of the material carried by the upflowingcentral core of the flux tube must be large compared to the pressureof the ambient material at the top of the cylinder. This is only pos-sible if the ambient temperature at the top is very cool (i.e. T 1/2 −1/2 is small) and the convection is weak (i.e. Ra is only slightlysupercritical); this is discussed more fully in Kerswell & Childress(1992). For these reasons T 1/2 and Ra are the important parametersto vary in order to understand the transition from the KC regime tothe QB regime.

In Section 3.3 we address how in certain instances a bifurcationleads to two branches of solutions for the same parameters. One ofthese corresponds to CC, the other to QB solutions. The importantparameters here reflect the two main assumptions of convectivecollapse. The first is that the flux tube should be thin so that itsradial structure is limited. The tube’s radius is not a parameter we candirectly set; however, assuming the magnetic field strength saturates(which is the case in which we are interested), then varying theChandrasekhar number Q varies the flux tube radius. The secondassumption is that the tube is thermally isolated, and the interestingparameter to vary in this regard is the magnetic Roberts number q.Varying these two parameters allows the transition between the CCand QB regimes to be studied.

The remaining imaginable transition, that from the KC to the CCregime, in fact never occurs. The reason is that the radial pressuregradient is directed inwards for convective collapse, and outwardsfor the KC mechanism. Any transition from CC to KC solutions musttherefore pass through a case where the radial pressure gradient iszero. At this point the force balance must involve only the Lorentzand buoyancy forces, i.e. must be of the QB type. Thus transitionsfrom the KC regime to the convectively collapsed regime all proceedvia the QB regime.

3.1 The quasi-Boussinesq regime

Galloway et al. (1978) and Galloway & Moore (1979) studied thestructure of axisymmetric magnetic flux tubes produced by mag-netoconvection in a Boussinesq fluid. Our calculations are similarto theirs and differ only through full inclusion of the effects due tocompressibility; thus their results form a limiting case of ours.

The effects of including compressibility manifest themselvesmost strongly in the momentum equation, particularly via the pres-sure gradient term. Since the QB regime is defined by a balance

C© 2005 RAS, MNRAS 358, 1025–1035


between the Lorentz force and the concentrating effect of the eddygenerated by the radial gradient in the gravitational force, the pres-sure term is negligible. The results we obtain in the QB regime aretherefore very close to those found in the Boussinesq limit.

Galloway & Moore (1979) found, as we do for QB solutions, thata counter-cell inside the flux tube is created as we leave the kinematicregime. This counter-cell gives rise to a ‘top hat’ magnetic structure.Galloway et al. (1978) found that as the applied magnetic flux isincreased the field strength inside the tube reaches a maximum,which scales as

B2Boussinesq = ρU 2

(ν

η

)1

ln Rm,

and that this maximum is developed leaving the kinematic regimeas the Lorentz force starts to generate the counter-cell. Their paperdiscusses this in detail, and gives a boundary layer analysis thatderives the scaling. We will see later that this result also appliesin the majority of compressible cases, as originally suggested byProctor (1983) and Proctor & Weiss (1984).

3.2 Kerswell and Childress

The Boussinesq approximation has several underlying assumptions(Spiegel & Veronis 1960; Spiegel & Weiss 1982). The essence is thatall departures from uniform total (gas plus magnetic) pressure andtemperature must be small compared to the average. This includesdifferences resulting from the gravitationally stratified density andtemperature maintained between the upper and lower boundaries,and also the perturbations in pressure and temperature associatedwith the convective motions.

Kerswell & Childress (1992) found a regime in which the com-pressibility was important because of the radial pressure gradient.Their boundary layer analysis relaxed one of the assumptions madefor the Boussinesq approximation: they allowed differences in theequilibrium temperatures that were comparable with the magnitudeof the average temperature. They kept the assumptions that differ-ences due to density stratification and convective perturbations weresmall. Under these circumstances they gave the correction for theeffects of compressibility as

B2compressible = min

(1, c2

s /gL)

B2Boussinesq.

However, they took γ = 1 so that c2s = T . In our model, where we

have taken γ = 5/3, the corresponding correction is

B2compressible = min(1, RT /gL)B2

Boussinesq.

The T or c2s which appear above are necessarily ambiguous in

the KC model since it is a one-dimensional boundary layer analysisin the radial direction, and treats only superficially the variationin the vertical. However, the appearance of L in the denominatorstrongly argues in favour of taking the average temperature of theinitial equilibrium state in the numerator of the preceding formula.Hence we recast the prediction of Kerswell & Childress (1992) intothe form

B2compressible = min

[1,

1 + �/2

(m + 1)L

]B2

Boussinesq,

which is plotted against T 1/2 in Fig. 1 as a solid line. Fig. 2 is anenlargement of the left-hand side of Fig. 1 where the deviation fromBoussinesq behaviour is predicted.

Note that the plots only extend down to

T1/2 = 1 + �/2

�(m + 1)= 0.5

B /(KE ln(Rm))

normalized by Boussinesq values

2 max

Figure 1. The ratio between the kinetic and magnetic energy densities(normalized by the corresponding Boussinesq energies). The comparisonbetween the theoretical prediction of Kerswell & Childress (1992) and thenumerical calculation with Ra = 2875 is quite good, given that the theoryapproximates the transition with two straight lines. As Ra is increased, thedeparture from Boussinesq behaviour becomes less significant.

B /(KE ln(Rm))

normalized by Boussinesq values

2 max

Figure 2. An enlargement of the left part of Fig. 1.

because we can rewrite it as

T1/2 = 1/� + 1/2

m + 1,

which has a minimum value as � → ∞ when T 1/2 = (1/2)/(m + 1). Since the correction described by Kerswell & Childress(1992) is only applicable when m is small (so as to make the result-ing equilibrium density almost uniform), it follows that the relevantminimum value of T 1/2 is 1/2, implying that the correction is alwaysgoing to be less than a factor of 2.

To reproduce the KC mechanism, we performed a series of cal-culations with Ra = 2875, Q = 80, q = 8, m = 0.05, σ = 0.5and γ = 5/3. We fixed Ra rather than K , as this allows us to com-pare solutions where the strength of the convective instability issimilar. These parameters values were chosen to meet the assump-tions adopted by Kerswell & Childress (1992). In particular we tookm = 0.05 so that the equilibrium density stratification is small, andRa = 2875 so that the convective flows are weak and the subsequent

C© 2005 RAS, MNRAS 358, 1025–1035


Figure 3. The format in which individual solutions are shown. The stream-lines shown in the lower left-hand corner are superimposed on a grey-scaleimage of the kinetic energy density. Black indicates low values, white high.

density and temperature perturbations are small. The remaining pa-rameter, T 1/2/(m + 1), was varied through 0.55, 0.6, 0.8, 1.0, 2.0,5.0 and 10.0, which allowed us to explore a range from the almostBoussinesq case T 1/2 = 10 (where � = 0.1 so that the temperaturedifference is small compared to the average temperature) nearlydown to the limiting case T 1/2 = 0.5.

The results of our numerical calculations are shown as circles inFigs 1 and 2. The match with Kerswell & Childress (1992) is quitegood, with the most obvious discrepancies being explained by thesimplicity of their modelling. To contrast the behaviour for smalland large T 1/2, Figs 4 and 5 show solutions typical of the QB andKC regimes respectively; the defining layout for these and similardiagrams is shown in Fig. 3. In Figs 4 and 5 the magnetic field isconcentrated in a tube with a radius of about 0.3 centred on the axis.The streamlines and temperature isolines look similar, although thetemperature varies from 9.5 to 10.5 in the QB case and from 0.1 to1.1 in the KC case. The grey-scale images of the density perturbationshow the largest difference between the two cases: in the QB casethe density at the very top of the box looks almost uniform, but inthe KC case there is a large negative density perturbation inside thetube.

The details can be seen more clearly in Fig. 6 for a QB case andin Fig. 7 for a KC case. In Fig. 6 we see that the radial Lorentzand pressure forces balance one another. This is expected, since themajor component to the radial component of the Lorentz force is itsmagnetic pressure, and in the Boussinesq limit the total pressure isconstant (because sound waves lead to a uniform pressure on veryshort time-scales).

Still looking at Fig. 6, we see that the gravitational force is almostconstant inside the tube, and has a gradient from approximately r =0.4 to 1. This gradient is not constant with height, and thus producesa torque; this is what drives the outer eddy. In the Boussinesq theorythe flux tube reaches an equilibrium when the resulting vorticity isbalanced by that generated by the magnetic tension near the axis(Galloway et al. 1978).

Fig. 7 shows, for a KC case, that although the pressure and mag-netic pressures look similar to the QB case, the density perturbation

Figure 4. The steady-state solution for T = 10, Ra = 2875, Q = 80, q =8, σ = 0.5 and m = 0.05: a QB result.

Figure 5. The steady-state solution for T = 0.6, Ra = 2875, Q = 80, q =8, σ = 0.5 and m = 0.05. Here, as will be shown in Fig. 7, the radial pressuregradient determines the flux tube structure.

is significantly different; the material inside the flux tube is muchless dense than the material adjacent to it, so that the flux tube ismagnetically buoyant. This mass deficit, which is most pronouncedat the top of the tube, leads to a force that tries to drive mass upthe tube. This force exists only within the tube and thus produces anet torque that acts to stop further contraction of the flux tube. This

C© 2005 RAS, MNRAS 358, 1025–1035


Figure 6. Radial profile of some force components average over the top 10per cent of the box for the QB case of Fig. 4. Shown are the radial componentof the Lorentz force (thick solid line), the radial component of the pressureforce (dotted line), the radial component of −U × ∇ × U (thin dot-dashedline), and the vertical gravitational force, with its value at the outer wallsubtracted (thick dashed line). We clearly see that the radial component ofthe Lorentz force is largely balanced by the pressure force. The gravitationalforce shows little variation inside the tube. Its gradient, from about r = 0.4 to1, drives the external convection cell, generating vorticity, which is balancedat the edge of the flux tube by the vorticity driven by the Lorentz force.

Figure 7. As for Fig. 6 except for the KC case of Fig. 5. The gravitationalforce now shows large radial variations associated with the flux tube, aspredicted by Kerswell & Childress (1992). The radial components of theLorentz and pressure forces are still almost in balance, but the low temper-ature (T 1/2 − 1/2 = 0.1) at the top of the cylinder implies that the pressureperturbation now corresponds to a large density perturbation. The densitygradient produces a radial gradient of the gravitational force. The propertiesof the vertical temperature stratification and the vertical pressure gradientslead the gravitational force to drive vorticity, causing the flux tube to equi-librate at lower field strengths than in the QB case. This is the mechanismthat limits the field strength in the KC regime.

alternative possibility for balancing the vorticity is that describedby Kerswell & Childress (1992). In the Boussinesq limit the mag-netic buoyancy is negligible compared to the convective buoyancy.However, as we move away from the Boussinesq case the magneticand convective buoyancies become comparable.

To see how the KC mechanism applies in our calculations,we need to know how various quantities scale with �, our non-dimensional measure of the temperature difference between the topand bottom boundaries. The aim is to obtain first the scaling of the

ratio of the two sources of buoyancy, and then the scaling of theratio of maximum magnetic to kinetic energy densities (ME/KE).We keep Ra, Q, σ , ζ 0, m and γ fixed, which means K and T 1/2

will vary with �. We assume that ζ 0 = 1/q � 1 so that the tube isstrongly thermally coupled to the surrounding material, and that Rais only slightly above its critical value. The first assumption can beseen to hold in Figs 4 and 5 – there is almost no horizontal variationof the temperature across the flux tube, which is at the same temper-ature as the material immediately outside it; the second assumptionmeans the convection is weak. We take a small value of m so theunperturbed density is almost uniform and equal to 1. This corre-sponds to the Boussinesq requirement that density perturbations areonly important in the buoyancy term.

We now turn to the scalings. First, from the momentum equa-tion, the magnetic pressure is Qσζ0 K 2 B2/2, where B is the fieldstrength in the flux tube. Since Q, σ and ζ 0 are fixed, this scalesas K 2 B2. Next, �P, the pressure difference between the plasmainside the tube and that immediately adjacent, balances the mag-netic pressure (Figs 6 and 7), and so must also scale like K 2 B2.Proceeding from this we obtain the scaling for the density perturba-tion using the equation of state: �P = (�ρ) T + ρ(�T ) + (�ρ)(�T ). Also, since ζ 0 � 1, the difference between the tempera-ture inside the tube and that adjacent to the tube, �T , is small. So�P = (�ρ)T , where T is a characteristic non-dimensional temper-ature of the tube, which will be specified below. It then follows that�ρ ∼ K 2 B2/T , which is the density perturbation associated withthe magnetic buoyancy.

We now use the fact that at fixed and only mildly supercriticalRayleigh number the strength of the typical convective velocity Uscales like K (this can be seen, for example, by comparing theadvective and diffusive terms in the energy equation). The kineticenergy KE = ρ U 2/2 then scales like K 2, since the density outsidethe tube is of order 1. The scaling of the magnetic energy den-sity ME can be obtained directly from the energy equation; it isME = (Qσζ0 K 2/2)B2. The ratio of the magnetic to kinetic energydensities therefore scales as ME/KE ∼ B2.

The associated gravitational force can be read off from themomentum equation as �(m + 1)�ρ and therefore scales like�K 2(ME/KE)/T . Since the Rayleigh number is fixed, the grav-itational force associated with convection scales like the product ofthe viscous and thermal diffusivities: for fixed Prandtl number σ , ittherefore scales like K 2. Hence the ratio of the buoyancy associatedwith magnetic pressure to the buoyancy associated with convectionscales as (�/T ) ME/KE.

The buoyancy forces are important because they imply torques,which indirectly determine the strength of the flux tube. Since thesetorques act over the whole height of the cylinder, an appropriate T isthe average temperature of the flux tube. For weak convection thisis just the average of the values at the upper and lower boundaries,T =1+�/2. The above ratio then scales as [�/(1+�/2)] ME/KE.For small m, this is just (1/T 1/2) ME/KE.

In the Boussinesq limit T 1/2 � 1, this formula shows that thebuoyancy associated with the flux tube is unimportant and we arein the QB regime. As we move away from the Boussinesq regimetowards the limiting case T 1/2 = 1/2, the buoyancy associated withthe magnetic pressure becomes larger than that associated with theconvection. Too great an excess would be unphysical because theconvective buoyancy is the reason the flux tube is there in the firstplace, whereas the magnetic buoyancy tries to disperse it. To avoidsuch behaviour, the magnetic energy density saturates as T 1/2 fallsto 1/2. The behaviour in the KC regime therefore satisfies (1/T 1/2)ME/KE � 1; for T 1/2 ≈ 1/2, the ratio ME/KE scales like T 1/2.

C© 2005 RAS, MNRAS 358, 1025–1035


Figure 8. The steady-state solution for T = 0.6, Ra = 6250, Q = 80, q = 8,σ = 0.5 and m = 0.05. At this higher Ra the radial pressure gradient becomesless significant and the solution’s magnetic field reverts to its Boussinesqform.

Between T 1/2 = 1/2 and T 1/2 → ∞, the behaviour changes fromscaling like T 1/2 to being constant. If we are not too concerned withthe transition, which occurs when T 1/2 ∼ 1, we can write ME/KE =min(T 1/2, 1) ME/KEBoussinesq, our version of the result of Kerswell& Childress (1992). In considering the match between the theoryand the calculations in Figs 1 and 2, one should bear in mind that theeffectiveness of the magnetic pressure in limiting the field strengthvaries continuously. The reason for believing the KC theory is validis the close match between the theory and calculations near T 1/2 =1/2 in the lowest Rayleigh number case where the conditions forthe theory are best satisfied.

The question naturally arises as to what happens when additionalassumptions of the Boussinesq approximation are relaxed. At highervalues of Ra, corresponding to the two upper curves in Figs 1 and2, the greater vigour of the convection creates larger perturbationsand velocities, and we move out of the region where the KC approx-imations are valid. For Ra = 12 500, the lowest values of T 1/2, 0.55and 0.6, cannot be followed to a steady state because the motionsnear the top of the domain become sonic and the tube becomes evac-uated (to less than 0.5 per cent of the surrounding densities). Thesolutions we could follow have maximum field strengths that moveback towards the values of the Boussinesq limit (see Fig. 8). Thepossibility of strong evacuation without significant departures fromthe field strengths of the QB theory was predicted by Proctor (1983).The low density of the tube here is an instance of flux tube evacua-tion in the absence of convective collapse (examples of which willbe provided below). Later we shall also see that convective collapsecan occur in the absence of flux tube evacuation.

3.3 Convective collapse

The traditional convective collapse picture starts with a flux tubewhere the internal energy density is much higher than the magnetic

and kinetic energy densities. The tube is in equilibrium, with theplasma hydrostatically stratified. In the absence of a magnetic field,this stratification is convectively unstable. The essence of convec-tive collapse was recognized by Parker (1978) and Spruit (1979):even in the presence of magnetic fields this equilibrium can still beunstable – any purely vertical motions in the tube can be amplifiedby the ‘convective’ instability. In particular, downward motions willgrow in amplitude and drain material from the flux tube, which willconsequently decrease in radius, causing the magnetic field strengthto rise.

The literature on convective collapse is substantial, and we donot intend to review it here: we restrict our attention to a singleresult that is needed in order to understand the convective collapsemechanism as it appears in our magnetoconvection calculations.Venkatakrishnan (1986) and Schussler (1986) discuss what happenswhen the flux tube’s radius becomes comparable to the thermal dif-fusive length-scale. In that case, thermal energy can freely enterthe flux tube and heat the plasma. Horizontal pressure balance thenrequires a decrease in density. This produces buoyancy, which canhalt the downward flow, stabilizing the tube against convective col-lapse. The length-scale associated with thermal diffusivity can beestimated as (κ/UL)1/2L. In the kinematic regime the radius of thetube scales like (η/UL)1/2L, so the thermal isolation of the tube de-pends on the ratio of κ to η, our parameter q. At high values of q,convective collapse is expected to be inhibited, while at low valuesit will occur.

We have therefore performed a series of numerical experimentswith Ra = 12 500, m = 0.05, σ = 1.25, γ = 5/3 and T 1/2(m + 1) =2.0. Since the expected transition between QB and CC solutionsoccurs when q is of the order of 1, we have considered the set ofq values 1.6, 0.8, 0.4, 0.3, 0.2 and 0.16. The effects of convectivecollapse are expected at the smaller values. For each q, Q was variedfrom 5 to 100 as indicated in Fig. 9. Runs were performed iteratively,beginning with small Q, following the evolution until it reached asteady state, and using the solution to generate an initial conditionfor the next value of Q. We looked for multiple steady states bystarting the calculation afresh at different values of Q. Once a newbranch of solutions was found, we proceeded iteratively to map it.This is clearly not an exhaustive search, but it managed to find twobranches of solutions for q = 0.16 and q = 0.2.

0

0.5

1

1.5

2

0 20 40 60 80 100

(total ME)/(total KE)

Q

q=0.16 q=0.2 q=0.3q=0.4 q=0.8 q=1.6

Figure 9. The ratio of the total magnetic energy to the total kinetic energy,as a function of the Chandrasekhar number Q, for different values of theRoberts number q. Bifurcations can be seen for q = 0.16 and q = 0.2, theupper branches corresponding to CC, the lower branches to QB.

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For parameters that are nearly Boussinesq, there is no significantpreference for either upflows or downflows near the axis (as is tobe expected since the Boussinesq case has a symmetry with respectto the transformation z → −z). As the temperature stratificationbecomes more important, we leave the Boussinesq limit and havefound, for our parameter choices, an increasing preference for up-flow near the axis. That is, if we begin from random initial conditionswe find more solutions with a central upflow than downflow. How-ever, both types of solution exist and are stable for all parameterswe have tested. This issue is not directly relevant to the questionsof the relationship between the various regimes and is not pursuedfurther in this paper. Since both the KC and CC behaviours can beexpected to occur preferentially in cases when there is a downflowat the axis, we have restricted our attention to that case. In addition,we only discuss steady-state solutions, and ignore the time develop-ment of flux expulsion and concentration. Thus we are investigatingthe end-product of convective collapse, or QB flux tube formation,rather than how a particular initial condition evolves.

The results from the above series of runs are presented in Figs 9to 14. We begin by noting that for q > 0.2 the curves in Fig. 9 allhave a similar behaviour: near Q = 0 the ratio increases linearlywith Q (this is the kinematic regime). The curves become flatter ata value of Q between 5 and 30 (with the turnover occurring morequickly for larger q), and increase only slowly thereafter. This typeof behaviour is that expected from Galloway & Moore (1979), andit is easy to identify this as the QB branch.

The solutions for q = 0.2 and q = 0.16 are markedly different.For both these values there are two branches of solutions for a rangeof Q. The lower one corresponds to the QB solution seen for q > 0.2.The new branch has stronger magnetic fields, with a ratio betweenthe total magnetic and kinetic energies that depends linearly on Q.Fig. 11 shows that the two types of solutions have similar kineticenergies, and taken with Fig. 10 shows that the ratio remains linearbecause B continues to grow, not because the kinetic energy falls.

The constant linear relation between the magnetic and kinetic en-ergies extends down to small Q, and indicates that this branch ofsolutions is in fact kinematic. An examination of the radius multi-plied by its kinematic scaling Rm−1/2, shown in Fig. 12, confirmsthis view. It shows that the radius of the tube radius scales as Rm−1/2

for the whole range of convectively collapsed solutions. (Here wehave calculated Rm using a typical speed obtained by dividing theintegrated kinetic energy by the total mass, multiplying by 2 and

0

0.005

0.01

0.015

0.02

0.025

0.03

0 20 40 60 80 100

total ME

Q

q=0.16 q=0.2 q=0.3q=0.4 q=0.8 q=1.6

Figure 10. As for Fig. 9 except that here we plot the total magnetic energy.We see that for the same value of Q and q there is a large change in themagnetic energy between the collapsed and non-collapsed solutions.

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 20 40 60 80 100

total KE

Q

q=0.16 q=0.2 q=0.3q=0.4 q=0.8 q=1.6

Figure 11. Together with Fig. 10, this plot of the total kinetic energy showsthat the differences in the ratio of magnetic to kinetic energies between thecollapsed and non-collapsed states (seen in Fig. 9) are mainly due to a changein the magnetic energy. The total kinetic energy varies only slightly betweenthe collapsed and non-collapsed branches.

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 20 40 60 80 100

R(top,50%) *sqrt(Rm)

Q

q=0.16 q=0.2 q=0.3q=0.4 q=0.8 q=1.6

Figure 12. Radius of the flux tube (containing 50 per cent of the flux)at the top of the cylinder, multiplied by the square root of the magneticReynolds number as a function of q and Q. Note that the normalized tuberadius remains constant for the collapsed tubes even at high fluxes.

taking the square root.) This Rm−1/2 scaling corresponds to fluxtubes that are as thin as is allowed by the induction equation: con-vective collapse can produce flux tubes no thinner this.

For Q = 0.16 the ratio between the integrated magnetic and ki-netic energies exceeds 1 in Fig. 9. Clearly this will happen if thevalue of Q is so high that no other outcome is possible, but that isnot the case here since there is an available equilibrium with lessmagnetic energy, on the QB branch. The fact that this solution stillbehaves kinematically implies that we are looking not at a balancebetween the advection of vorticity and the Lorentz term but at abalance between the inward-directed radial pressure force and theLorentz force. This is confirmed by Fig. 13, where we show thespatial structure of the solutions. There is indeed a large tempera-ture gradient across the tube, generating an inward-directed pressuregradient. This radial pressure gradient is the result of the fact thatthe tube is thin and thermally isolated. These are precisely the con-ditions required for convective collapse.

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Figure 13. The steady-state solutions for Ra = 12 500, m = 0.05, σ = 1.25, γ = 5/3, T 1/2 = 2/1.05 and q = 0.2, for various values of Q. There is a bifurcationnear Q = 37.5. The branch shown on the right corresponds to CC solutions, whilst the branch on the left corresponds to solutions of the QB type.

There are two differences between the convectively collapsedbranches for q = 0.16 and q = 0.2. The first is that the branch forq = 0.2 ends abruptly near Q = 47.5. This corresponds to the sat-uration of the balance between the Lorentz force and the radialpressure gradient. As saturation occurs, the tube wants to leavethe kinematic regime and expand; however, expansion meansthat the tube becomes broad enough to begin developing internal

structure. Apparently the evolution of internal structure is sudden(as Q is varied). For comparison, Fig. 14 shows the smooth depen-dence of the counter-cell on Q for (the case q = 0.4). The seconddifference is that for q = 0.16 there is a discontinuous jump betweenthe two branches.

We now discuss in more detail the conditions under which convec-tive collapse occurs. The mechanism is driven by the superadiabatic

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Figure 14. For Ra = 12 500, m = 0.05, σ = 1.25, γ = 5/3, T 1/2 = 2/1.05and q = 0.4, no bifurcation occurs as Q is increased. In this case the thermalboundary layer is thicker than the magnetic one and hence the inside of thetube is poorly insulated: convective collapse does not occur.

instability within the flux tube. The effectiveness of the instabilitydepends on the extent to which the flux tube is thermally isolatedfrom the plasma outside the flux tube (Parker 1978; Venkatakrishnan1986). It is helpful to look at the various radial length-scales.

Consider a characteristic velocity U (for example, the observedvelocities of the convective eddies in the quiet Sun, or the rms ve-locity in our calculations) and a characteristic global length-scale L(for example, the depth of granules or the size of our computationaldomain). Then the magnetic, viscous and thermal diffusivities, η, ν

and κ , define length-scales

lmagnetic = (U L/η)−1/2 L,

lviscous = (U L/ν)−1/2 L,

lthermal = (U L/κ)−1/2 L.

An additional length-scale is defined by the total flux, F, and themaximum field strength in the tube, Bmax:

lflux = [F/(πBmax)]1/2.

The value of Bmax cannot be prescribed beforehand but is a result ofthe calculation.

Provided that mass inflow into the flux tube is substantially im-peded by the magnetic field, thermal insulation will occur wheneverl thermal < l magnetic, since the latter is always a minimum bound forthe size of a flux tube. Earlier, we varied q so that we spanned thetransition from l thermal > l magnetic to l thermal < l magnetic. We indeedsaw a transition from the QB to CC solutions. High values of theratio between the total magnetic energy and the total kinetic energyare possible in this regime because the magnetic energy density inthe tube is directly balanced by the lowered internal energy densitythere.

We also varied Q in order to explore the transition from l flux =l magnetic (corresponding to the kinematic case) to l flux > l magnetic.For q = 0.2 we saw an abrupt increase in the flux tube radius atabout Q = 47.5 where the solution moves on to the QB branch.This is accompanied by the breaking of the explicit assumption inconvective collapse models that the tube has little internal structure:QB solutions have an elongated convective cell within the tube.

The assumption of no internal structure is justified when the ra-dius of the flux tube is less than or equal to at least one diffusivelength-scale. However, as soon as lflux exceeds all three of them, itis difficult to see why internal structure will not develop. Certainly,our calculations all show such structure as the solutions drop back tothe QB branch. The thin flux tube approximation, which uses onlyone or two terms of an expansion in terms of powers of r, must thencease to apply.

As well as providing the theory of the KC regime discussed earlier,Kerswell & Childress (1992) also identify another effect, which theycall ‘runaway cooling’. A comparison of this with our convectivecollapse branch would be interesting, but is rendered difficult bythe non-isotropic diffusivities used by Kerswell and Childress. Allwe can do is comment that their mechanism is possibly related towhat we see in our calculations.

4 C O N C L U S I O N

A number of issues are raised by the above results, which we nowdeal with in turn.

We have shown that evacuation is a concept that is independentof convective collapse. As predicted by Proctor (1983), and seenin our calculations, evacuated flux tubes can occur without con-vective collapse. This will be the typical case when q > 1 and the

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fields predicted by Galloway et al. (1978) become comparable to theplasma pressure. Conversely, convective collapse does not requiresignificant evacuation of the flux tubes, the collapse coming to anend when the thermal structure inside the tube is stabilized. Thiscan happen through radial diffusion of energy into the flux tube,through a redistribution of the energy so that the temperature gradi-ent becomes adiabatic, or through mass crossing the field lines viamagnetic diffusion. Any of these effects can take place well beforethe tube becomes evacuated.

Turning to the three possible regimes, these can be understoodby reference to the driving terms in the momentum equation, i.e.the Lorentz, pressure and gravitational forces. We can classifythe types of solution we have found by looking at the three pos-sible pairings of these terms. In the QB regime the dominant termsare the Lorentz and gravitational forces: there is a balance betweenthe production of vorticity by the convective buoyancy and theLorentz force (mediated through viscosity). In the CC regime thedominant terms are plasma pressure and Lorentz force: here the bal-ance is between the radial gas and magnetic pressures. Lastly in theKC regime the dominant terms are the plasma pressure and gravi-tational force: here the match is between the generation of vorticityby the buoyancy associated with the pressure deficit in the tube andthat of the exterior convection.

The QB type of behaviour occurs over the broadest range ofparameters: overlapping with the CC regime and having a smoothtransition to the KC solutions. The predicted field strengths dependon the ratio of the viscous and magnetic diffusivities. It is a vexedissue whether one should use laminar or turbulent values, but thelatter possibility seems more reasonable than the former. In that case,the diffusivities are likely to be of the same order of magnitude, andthe maximum magnetic energy density will be comparable to themean kinetic energy density of the convective motions (Gallowayet al. 1977).

Kerswell & Childress (1992) introduce a regime where the radialpressure gradient limits the intensification of magnetic flux tubes.This mechanism is quite delicate: our numerical experiments con-firm its existence but only for weak convection, mild density strat-ification and low surface temperatures. For this reason the effect isunlikely to operate in the Sun, but might be relevant for other classesof stars that have very weak convection, e.g. possibly A-type stars.

The region of parameter space in which the plasma pressure andLorentz forces balance is also limited: we require the plasma pres-sure force, created by the existence of the tube and its internal ther-modynamics, to drive the field strength to high values. This requiresthat the material inside the tube be thermally insulated from thesurrounding material. Hence the thermal diffusion length-scale isrequired to be significantly smaller than the effective tube radius.This condition places constraints on the various parameters (particu-larly q and Q). Much more restrictively, there should be little internalstructure within the flux tube. This condition is violated in our calcu-lations as soon as the flux tube begins to grow beyond its kinematicradius. Further we can see no obvious mechanisms on the Sun orother stars to enforce this uniformity. Our calculations thus suggestthat, on the Sun, the CC mechanism is limited to flux tubes withradii less than about 10 km, this being an estimate (Schussler 1986)of the turbulent magnetic diffusion length-scale. Observed magneticfeatures with kilogauss field strengths and length-scales of severalhundred kilometres presumably then consist of many unresolvedflux tubes. This corresponds to the MISMA (microstructured mag-netic atmosphere) hypothesis (Sanchez Almeida et al. 1996), which

was originally developed as an interpretation of observations ofStokes V profiles.

Thus for the quiet Sun we expect the following. Any magneticelements with field strengths comparable to that of equipartition withthe gas pressure will be structured down to diffusive length-scalesof perhaps 10 km. More uniform elements larger than this will havelower field strengths given by the GPW formula. In the latter caseissues of 3D stability arise which might break the flux tube up intosmaller, more intense bundles. At least, in many cases, kilogaussfields are observed, close to equipartition with the gas pressure andwith substantial fluxes and areas. In our picture, this is only possibleif the magnetic elements are in a MISMA state, consisting of manysmall tubes, most of which are in the CC state.

AC K N OW L E D G M E N T S

We are grateful for helpful conversations with Mike Proctor andNigel Weiss, as well as for useful suggestions from an anonymousreferee. DG appreciates the hospitality of the Isaac Newton Institutefor Mathematical Sciences, Cambridge, UK, where some of thiswork was carried out.

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