Large-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone
aa r X i v : . [ a s t r o - ph . S R ] O c t Astronomy Letters, 2019, Vol. 45, No. 1
Large-Scale Magnetic Field Fragmentation inFlux-Tubes Near the Base of the Solar ConvectionZone
L. L. Kitchatinov*
Abstract
Magnetic quenching of turbulent thermal diffusivity leads to instability of the large-scale field with the productionof spatially isolated regions of enhanced field. This conclusion follows from a linear stability analysis in theframework of mean-field magnetohydrodynamics that allows for thermal diffusivity dependence on the magneticfield. The characteristic growth time of the instability is short compared to the 11-year period of solar activity.The characteristic scale of the increased field regions measures in tens of mega-meters. The instability canproduce magnetic inhomogeneities whose buoyant rise to the solar surface forms the solar active regions. Themagnetic energy of the field concentrations coincides in order of magnitude with the energy of the active regions.
Keywords
Sun: magnetic fields – dynamo – convection
Institute of Solar-Terrestrial Physics, Lermontov Str. 126A, Irkutsk, 664033, Russian Federation * E-mail : [email protected]
1. Introduction
The global magnetic field of the Sun and relatively small-scale fields of its active regions are mutually related. Bab-cock (1961) was probably the first to note that Joy’s law (Haleet al. 1919) for sunspot groups can be the reason for the 11-year cyclic variations of the global field. Estimations basedon sunspot data support the operation of the mechanism en-visaged by Babcock on the Sun (Erofeev 2004; Dasi-Espuiget al. 2010; Kitchatinov & Olemskoy 2011). On the otherhand, magnetic fields of the active regions can be related tothe emergence of global toroidal field fragments – flux-tubes– to the solar surface . This picture is supported by observa-tions of the active regions (see, e.g., Zwaan 1992; Lites et al.1998; Khlystova & Toriumi 2017). Modeling of buoyant flux-tubes reproduces the Joy’s law for the active regions (D’Silva& Choudhuri 1993).The problem however is that rising flux-tube models agreewith observations for sufficiently strong fields of the order of G only (D’Silva & Choudhuri 1993; Caligari et al. 1995;Weber et al. 2011). More specifically, the fields should havethis strength near the base of the convection zone from wherethey start their rise to the solar surface. Solar dynamo mod-els do not show fields of such strength. This is natural: con-vective dynamos cannot amplify the fields to an energy den-sity exceeding the kinetic energy of the field generating flows.The equipartition field, B eq = p πρ u (1) The term ‘flux-tube’ is used here as a short name for the regions ofenhanced mainly azimuthal magnetic field, not for ideal tubes of constantcircular cross-section. ( ρ is density and u is the rmsconvective velocity), reaches itsmaximum strength < ∼ G near the base of the solar convec-tion zone. The fact that most successful current models forthe solar dynamo are based on the Babcock-Leighton mech-anism (Babcock 1961; Leighton 1969) and therefore implic-itly assume strong near-base fields redoubles the problem.This seeming contradiction is usually sidestepped withthe assumption that the mean dynamo-field of several kilo-Gauss consists of isolated flux-tubes of much stronger fields.The mechanism capable of amplifying the field to the strengthof ∼
100 kG was not specified, however. Isolated regions ofrelatively strong fields can result from magnetic buoyancy in-stability (Parker 1979) or can be due to the field expulsionfrom the regions of circular motion (Weiss 1966), however,these do not give an amplification above the equipartitionlevel of Eq. (1). To exceed this level, a more powerful sourceof energy than the energy of convective motions is required,e.g., thermal energy. A promising possibility was noticed byParker (1984): isolated regions of a strong field can resultfrom magnetic suppression of convective heat transport.Parker’s idea was as follows. Magnetic field suppressesconvection. This leads to an increase in super-adiabatic gra-dient and, therefore, to an increase in thermal energy in theconvection zone. Spatial redistribution of the field with itsintermittent concentration in flux-tubes can be ‘energeticallyprofitable’. An increase in magnetic energy from such a re-distribution can be overcompensated by a decrease in thermalenergy due to amplified convective heat transport in the weakfield regions between the tubes.This paper makes a first step in the quantitative analysis arge-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone — 2/7 of this possibility. It concerns a layer with a horizontal mag-netic field, which mimics the large-scale toroidal field of theSun, near the base of the convection zone. The mean-fieldapproach is applied, i.e. convection is accounted for implic-itly by introducing effective (turbulent) transport coefficients.The effective thermal conductivity depends on the strength ofthe magnetic field. A similar approach was formerly appliedto the problem of sunspot equilibrium (Kitchatinov & Mazur2000; Kitchatinov & Olemskoy 2006). In the absence of themagnetic field, the layer is stable (in the framework of themean-field approach, as already stated). In the presence ofthe magnetic field, instability takes place producing isolatedregions of the enhanced field. The paper is confined to thelinear stability analysis. The amplitude of the field ‘bunches’remains therefore uncertain. Computations, however, showthat the increase in thermal energy due to magnetic quench-ing of thermal diffusivity exceeds the magnetic energy. Thefield amplification at the nonlinear stage of the instability can,therefore, be large.
2. Problem formulation
We consider a horizontal layer of thickness h near the baseof the convection zone, where the solar dynamo is expectedto produce its strongest fields. Our approach demands thatthe layer be entirely embedded in the convection zone. Thebottom boundary is however placed as close as possible tothe base of this zone. The density ρ = 0 . g/cm , temper-ature T = 2 . × K, and gravity g = 5 × cm /s atthe bottom boundary are therefore taken from the solar struc-ture model for the heliocentric distance r b where the radiativeheat flux F rad = − σT κρ ∂T∂r (2)is only marginally smaller than the total flux: F rad = (1 − ε ) L ⊙ / (4 πr ) , ε ≃ − (cf. Stix 1989). In Eq. (2), σ isthe Stefan-Boltzmann constant, κ is the opacity, and otherstandard notations are used. The opacity is computed withthe OPAL tables . The relative (by mass) content of hydro-gen and heavy elements in this computations where taken X = 0 . and Z = 0 . respectively.Spherical curvature is neglected and the plane layer is un-bounded in horizontal dimensions. The Cartesian coordinatesystem is used with its z = 0 plane being the bottom bound-ary of the layer. The z -axis points upward.The relative deviation of the density and temperature gra-dients from their adiabatic values in the depths of the con-vection zone are small ( < ∼ − ). The lower part of the con-vection zone “lies on essentially the same adiabate” (Gilman https://opalopacity.llnl.gov T ( z ) = T (1 − z/H ) , H = c p T /g,ρ ( z ) = ρ (1 − z/H ) γ − , (3)are therefore neglected. In this equation, c p = 3 . × cgsis the specific heat at constant pressure and γ = c p /c v = 5 / is the adiabaticity index. The deviation from adiabaticitycannot be neglected, however, in the specific entropy S = c v ln( P/ρ γ ) ( P is the pressure) whose gradient is not smallcompared to the (zero) gradient for the adiabatic stratifica-tion.The constant heat flux F = L ⊙ / (4 πr ) = 1 . × erg/(cm s) enters the layer through its bottom. Insidethe layer, the energy is transported by radiation and convec-tion. As already mentioned, the effect of turbulent convection inthe mean-field approach applied is parameterized by the tur-bulent transport coefficients. The characteristic scale of tur-bulent convection near the base of the convection zone is how-ever not small compared to the mean-fields scale. In this case,turbulent transport should be described with non-local (inte-gral) equations. The non-local transport theory is still lack-ing, however, and the local diffusion approximation is usedin this paper in the absence of better possibilities.The magnetic field decreases the transport coefficientsand induces their anisotropy: the transport coefficients for thedirections along and across the field lines differ. This paperneglects the anisotropy and quenching of the viscosity andmagnetic diffusivity. Multiple simplifications and approxima-tions are unavoidable in the complicated problem. Otherwise,the physics of the results are difficult to interpret.The expected flux-tube formation is related to magneticquenching of the thermal diffusivity. The heat transport equa-tion ρT (cid:18) ∂S∂t + u · ∇ S (cid:19) = ∇ · (cid:16) ρT χ ∇ S − F rad (cid:17) , (4)therefore, keeps the dependence of turbulent diffusivity χ onthe magnetic field: χ = χ T φ ( β ) , (5)where χ T is the thermal diffusivity in the absence of the mag-netic field and β = B/B eq is the ratio of the field strength toits equipartition value (1). The function φ ( β ) = 38 β (cid:18) β − β + 1 + β + 1 β arctg( β ) (cid:19) (6)for the dependence is taken from the quasi-linear theory (Kit-chatinov et al. 1994).The relation u = − ℓ g c p ∂S∂z of the mixing-length theoryis used to estimate the thermal diffusivity χ T = ℓu/ for arge-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone — 3/7Figure 1. Equipartition field (leftpanel) and turbulentdiffusivity (right) versus height z above the base of theconvection zone.the non-magnetic case ( ℓ = α MLT H p is the mixing-lengthproportional to the pressure scale height H p = − P (cid:0) d P d z (cid:1) − ).The steady solution of the equation (4) then gives χ T = α / MLT ( c p − c v ) Tg (cid:18) ( γ − δF γρ (cid:19) / ,B eq = √ πρ / (cid:18) α MLT γ − γ δF (cid:19) / , (7)where δF = F − F rad is the convective heat flux.Figure 1 shows the dependence of χ T and B eq of Eq. (7)on the height z above the bottom of the convection zone forthe case of the layer thickness h = 40 Mm and α MLT = 0 . .At the height of 40 Mm, the diffusivity reaches its maximumvalue and decreases both downwards and upwards from thisheight. The choice of the α MLT -value will be explained later.The induction equation for the large-scale field, ∂ B ∂t = ∇ × (cid:0) v × B − √ η T ∇ × ( √ η T B ) (cid:1) , (8)accounts for the diamagnetic pumping with the effective ve-locity v dia = − ∇ η T / (Zeldovich 1957; Krause and R¨adler1980). The motion equation with turbulent viscosity ν T reads ρ ∂v i ∂t + ρv j ∇ j v i = 14 π ∇ j (cid:0) B i B j − δ ij B / (cid:1) + ∇ j ρν T (cid:18) ∇ j v i + ∇ i v j − δ ij ( ∇ · v ) (cid:19) − ∇ i P + ρg i , (9)where repetition of subscripts signifies summation.The Prandtl number Pr = ν T /χ T and the magnetic Prandtlnumber Pm = ν T /η T for turbulent convection are of order one(Yousef et al. 2003). The quasi-linear theory gives equal val-ues Pr = Pm =0.8 for these numbers (Kitchatinov et al. 1994),which are used in what follows.The magnetic field in the steady background state of thestability analysis is prescribed with its given value on the top Figure 2.
Dependence of the magnetic energy (11) (dashedline) and increment in the thermal energy (12) (fullline) onthe field strength B on the top boundary.boundary. The field strength B on the top boundary is aparameter of the model. As rotation is not included, all hor-izontal directions are equivalent. The background field B isassumed to point along the y -axis. Equations (8) and (4) givethe distributions of the magnetic field and the entropy gradi-ent for the background state, B ( z ) = B (cid:18) η T ( h ) η T ( z ) (cid:19) / ,∂S ∂z = − δFρT χ T φ ( β ) , (10)and Eq. (9) allows the mean flow to be absent, v = 0 . Onlythe gradient of the entropy - not the entropy value - is requiredfor the stability analysis. Nevertheless, we fix the value withthe boundary condition S ( h ) = 0 .It is remarkable that the surface density of magnetic en-ergy, i.e. the height-integrated energy density W B = 18 π h Z B ( z )d z, (11)is smaller than the magnetically induced increment in thethermal energy W T = h Z ρT δS d z. (12)In this equation, δS is the increment in the specific entropyinduced by the magnetic field, i.e. the difference in the valuesof S between the cases of finite and zero magnetic fields. De-pendencies of the energies of Eqs. (11) and (12) on the fieldstrength B at the top boundary are shown in Fig. 2. The in-crement in thermal energy is about ten times larger than the arge-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone — 4/7 magnetic energy. Therefore, the field fragmentation in flux-tubes (Parker 1984) can indeed be ‘energetically profitable’. The continuity equation for the depths of the solar convec-tion zone can be written in the inelastic approximation (see,e.g., Gilman & Glatzmeier 1981), ∇ · ( ρ v ) = 0 . For analy-sis of the stability of the above-defined equilibrium to smalldisturbances, it is convenient to split the disturbances of themagnetic field b and momentum density ρ v into their toroidaland poloidal parts: b = ∇ × (ˆ z T ′ + ∇ × (ˆ z P ′ )) , v = 1 ρ ∇ × (ˆ z W + ∇ × (ˆ z V )) , (13)where ˆ z is the unit vector along the z -axis and dashes inthe notations for the toroidal ( T ′ ) and poloidal ( P ′ ) field po-tentials distinguish them from the notations for temperatureand pressure. Equations (13) are introduced by analogy withthe stability problems in spherical geometry (Chandrasekhar1961, p.622) to ensure the divergence-free of the magneticand flow disturbances.Linearization of equations (4), (8) and (9) in small distur-bances gives a system of five equations for the linear stabilityproblem: four equations for the poloidal and toroidal compo-nents of the magnetic field and flow and an equation for theentropy disturbances. Coefficients in these equations do notdepend on x and y . The wave-type dependence exp(i k x +i k y ) on these coordinates can, therefore, be prescribed.The linearized entropy equation, ∂S∂t = i ρT ∂∂z (cid:18) ρT φ ′ ( β ) χ T B eq d S d z (cid:18) k ∂P ′ ∂z − k T ′ (cid:19)(cid:19) + 1 ρT ∂∂z (cid:18) ρT χ T φ ( β ) ∂S∂z (cid:19) − k χ T φ ( β ) S − k ρ ∂S ∂z V, (14)includes the contribution of magnetic disturbances (the firstterm on the right-hand side). This contribution distinguishesthe problem at hand from the standard analysis of thermalconvection. This new contribution comes from the depen-dence of the effective thermal diffusivity (5) on the magneticfield. It affects the solution of the problem considerably. Inthe Eq. (14), k = k + k and φ ′ signifies the derivative ofthe function (6).The divergence-free of the flow (13) demands the poten-tial part of Eq. (9) to be filtered-out. Equation (9) is curledfor this purpose. The z -component of the resulting equationgoverns the toroidal flow. The poloidal flow equation is the z -component of the motion equation curled twice. Neglect-ing disturbances in pressure to exclude magnetic buoyancyinstability (Acheson & Gibbons 1978), the gravity term can be transformed as follows ˆ z · ( ∇ × ( ∇ × ρ g )) = − ρgc p (ˆ z × ∇ ) · (ˆ z × ∇ ) S = − ρgc p ∆ S, (15)where ∆ = ∂ /∂x + ∂ /∂y is the 2D Laplacian. Thisleads to the following equation for the poloidal flow: ∂∂t (cid:18)(cid:18) ∂ ∂z − k (cid:19) V (cid:19) = − ρgc p S + i k π (cid:18) B ∂ P ′ ∂z − ∂ B∂z P ′ − k BP ′ (cid:19) + (cid:18) ∂ ∂z − k (cid:19) (cid:18) ρν T ∂∂z ρ ∂V∂z − ν T k V (cid:19) + 2 k (cid:18) ρ ∂ ( ρν T ) ∂z − ρ ∂ ( ρν T ) ∂z ∂ρ∂z (cid:19) V. (16)Derivation of other equations of the full system does not pre-sent difficulties and requires no comment. These equationsare therefore omitted.Conditions at the bottom boundary assume an interfacewith a superconductor beneath, zero surface stress, zero nor-mal components of the magnetic field and velocity, and theabsence of disturbances in the heat flux, ∂∂z ( √ η T T ′ ) = ∂∂z (cid:18) Wρ (cid:19) = ∂S∂z = P ′ = V = 0 , ( z = 0) . (17)All the disturbances were put to zero on the (artificial) surfaceboundary at z = h .Equations for the disturbances were solved numericallywith finite-difference representation of the derivatives in z .Inhomogeneity of solutions near the bottom boundary can besharp. A non-uniform grid with higher density of grid-pointsnear the bottom was, therefore, applied: z = 0 , z i = h (cid:18) − cos (cid:18) π i − / N − (cid:19)(cid:19) , ≤ i ≤ N, (18)where N is the grid-point number. The results to follow wereobtained with N = 52 . Trial computations with larger N gave practically the same results.Searching for the dependence of small disturbances ontime in the exponential form, exp( σt ) , leads to the eigen-value problem (from now on, σ is the eigenvalue of the linearstability problem). A positive real part ℜ ( σ ) > means aninstability.Obviously, an instability can emerge even without themagnetic field if too small turbulent transport coefficients areprescribed (Tuominen et al. 1994). The smaller the turbulentdiffusivity, the larger the entropy gradient (superadiabaticity)in the background state. The usual convective instability de-velops for sufficiently small diffusivity. The problem formu-lation then loses consistency because the turbulent transport arge-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone — 5/7Figure 3. Toppanel: growth rates of bending (full line) andinterchange (dashed) modes in dependence on B . Bottompanel: The wave lengths of bending and interchangedisturbances corresponding to the maximum growth rates ofthe upper panel.coefficients no longer parameterise the convective turbulenceadequately. The value of the turbulent diffusivity (7) is con-trolled by the α MLT -parameter. α MLT = 0 . is the marginal value for the onset of con-vective instability. The slightly supercritical value α MLT =0 . is used in the computations to follow. Such a choice wasjustified in a preceding paper (Kitchatinov & Mazur 2000).The relatively small value of α MLT = 0 . is related to theconsideration of the deep near-bottom region of the convec-tion zone. For higher regions, the critical values of α MLT arelarger. α MLT should probably decrease with depth in a morerealistic model.It can be seen from the equations for small perturbationsthat the eigenvalues do not change with an inversion of signof the wave number k or k or both. We therefore considerthe positive wave numbers only.
3. Results and discussion
Eigenvalues of all unstable modes are real (change of stabil-ity). In no case do the largest growth rates belong to themodes whose components k and k of the wave vector dif-fer from zero simultaneously. Depending on the value of B ,the disturbances with the wave vector along either the x or Figure 4.
The same as in Fig. 3 but with B eq decreased threetimes compared to Fig. 1. y axis show the most rapid growth. Only these two casesare therefore discussed. The disturbances with k = 0 and k = 0 deform (bend) the field lines. These disturbances willbe called ‘bending modes’. In the case of k = 0 and k = 0 the field lines are interchanged without bending. Such distur-bances will be called ‘interchange modes’.The magnetic field can oppose flow bending the field lines.It can therefore be expected that magnetic quenching of tur-bulent diffusivity leads to an instability of interchange modeswhich do not bend the lines. Figure 3, however, shows closegrowth rates for bending and interchange modes. For weakfields, bending modes grow even faster. This is a conse-quence of the magnetic suppression of the thermal diffusiv-ity. If the suppression is neglected (the first term on the right-hand side of Eq. (14) is dropped), interchange mode domi-nates for any B .Numerical experiments by Karak et al. (2014) have shownthat equations (5) and (6) reproduce satisfactorily the diffu-sivity quenching but only if B eq is defined not for the origi-nal turbulence, which would take place in the absence of themagnetic field, but for the actual magnetized flow. This isequivalent to a reduced value of B eq compared to our esti-mations. The computations were repeated with B eq reducedthree times compared to Fig. 1. The results of these compu-tations are shown in Fig. 4. The bending modes are dominat-ing now in a wider range of B values and the growth ratesincrease considerably. The difference between thermal and arge-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone — 6/7Figure 5. Fromtoptobottom: isolines of magneticdisturbances, stream-lines of the flow, and the entropydisturbances isolines for the most rapidly growinginterchange mode for B = 1000 G. The full (dashed) linesshow positive (negative) levels and clockwise(anti-clockwise) circulation.magnetic energies also increases compared to Fig. 2. Thestructure of unstable disturbances, which will be discussedshortly, depends weakly on the definition of B eq . The dis-cussion to follow refers to the definition of B eq of the Eq. (7)and Fig. 1.Figure 5 shows the structure of the most rapidly grow-ing interchange mode for B = 1 kG. The magnetic fielddisturbances are concentrated in the lower part of the layer.The field amplifications (positive disturbances) are connectedwith the converging horizontal flows. The upward flows risefrom relatively hot regions of positive entropy disturbances.The most rapidly growing bending mode is shown in Fig. 6.The field structure resulting from this unstable mode is shownin Fig. 7 as a superposition of the background field and its dis-turbance with an amplitude of about 30% of the backgroundfield. In contrast with the interchange mode of Fig. 5, inho-mogeneity along the y -direction of the background field isnow present. Horizontal flows along the background fieldlines do not disturb the field. Field amplification near thebase of the layer occurs in the regions of downflow.An interchange of the field lines is unlikely to produce aconsiderable field amplification. The bending mode of Figs. 6 Figure 6.
Fromtoptobottom: field lines of magneticdisturbances, stream-lines of the flow, and isolines of theentropy disturbances of the bending mode for B = 1000 G.The full (dashed) lines show positive (negative) levels andclockwise (anti-clockwise) circulation.and 7 can be more relevant to the hypothetical formation ofstrong field regions. In this case, redistribution of the fluidalong the field lines can change the field strength consid-erably. However, the bending disturbances do not produceflux-tubes: they are homogeneous along the x -axis normal tothe background field. Increased field regions of finite dimen-sions in both horizontal directions can result from a superpo-sition of the bending and interchange disturbances (similarto the B´ernard cells formation by a superposition of the lin-ear modes of thermal convection with different directions ofhorizontal wave vectors; Chandrasekhar 1961, p.47).Comparison of the magnetic and thermal energies of Fig. 2suggests that the field amplification in unstable disturbancescan be considerable. The amount of the amplification canbe evaluated with nonlinear computations only. Some order-of-magnitude estimations are nevertheless possible from thelinear computations. Figures 5 and 6 show the characteris-tic scales of the field amplification regions in horizontal di-mensions L x ≈ L y ≈ Mm. The disturbances are local-ized near the base of the convection zone. Their vertical size L z ≈ Mm. The magnetic energy W M ≈ B L x L z / (8 π ) arge-Scale Magnetic Field Fragmentation in Flux-Tubes Near the Base of the Solar Convection Zone — 7/7Figure 7. Superposition of the background magnetic fieldand magnetic disturbances of the bending mode of Fig. 6.Amplitude of the disturbance is about 30% of thebackground field.in this region can be estimated as W M ≈ (cid:18) B (cid:19) erg , (19)where B is the characteristic strength of the background field.The rough estimation (19) seems to be the first attempt at con-necting the large-scale fields of the solar dynamo with param-eters of the active regions. The estimation however agrees inorder of magnitude with the magnetic energy of the activeregions (Sun et al. 2012; Livshits et al. 2015).Dynamo models give toroidal fields of several kilo-Gaussnear the base of the solar convection zone. This field shouldbe amplified tens times by the presumed instability for itsrise to the solar surface to fit the observational properties ofthe active regions (D’Silva & Choudhuri 1993). The fieldstrength in sunspots suggests that the flux-tube expansion inthe course of the rise reduces the field strength again to sev-eral kilo-Gauss keeping the estimation (19) for the magneticenergy. Figures 3 and 4 show that the characteristic time ofseveral months for the instability is short compared to the 11-year period of the solar cycle. Acknowledgments
This work was supported by budgetary funding of Basic Re-search program II.16 and by the Russian Foundation for Ba-sic Research (project 17-02-00016).
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