局所対流ダイナモシミュレーション...政田洋平（神戸大学） 共同研究者：佐野孝好（大阪大学） 局所対流ダイナモシミュレーション

政田洋平（神戸大学）共同研究者：佐野孝好（大阪大学）

局所対流ダイナモシミュレーション- 大局的磁場の周期変動とその物理機構 -

理論懇シンポジウム「計算宇宙物理学の新展開」@つくば国際議場（2012年12月22日-24日）

Introduction & Motivation

太陽物理学最大の未解決問題：太陽ダイナモ問題

ダイナモ理論：太陽黒点の起源を説明するために Larmor (1919) が提唱.

バタフライダイアグラムは、太陽内部のダイナモ過程によって生成された太陽磁場の周期性が太陽表面で顕在化したもの（研究の最終目標）.

P. J. Kapyla et al.: Reynolds stresses and turbulent heat transport 801

Following the treatise of R89, this tensor can be repre-sented by!u!iT

!" = !i j" j , (42)

where !i j is the eddy heat conductivity tensor and " j ="(# jT " g j/cP) the superadiabatic temperature gradient in thedirection j. In the present model geometry the horizontal tem-perature gradients vanish due to the periodic boundaries, andwe must thus limit the discussion to the meridional compo-nents of the tensor. In analogy to the !-coe"cients, Eqs. (35)and (36), we write !i j as

!rr = "!tVV(r, #,#), (43)

!#r = "!tHV(r, #,#), (44)

where !t is the turbulent heat conductivity for which we assumethat !t = $t.

The correlations between velocity and temperature fluctu-ations derived from the calculations can now be used to deter-mine the coe"cients VV and HV with the equations

VV =

#u!zT

!$

V

%z!t, (45)

HV =!u!xT

!"V

%z!t, (46)

where the sign changes for both quantities due to the transfor-mation to spherical coordinates.

5. Results

5.1. Convective structures

To illustrate the general appearance of the structures aris-ing in the present calculations we show a typical snapshotof temperature fluctuations overplotted with the velocity vec-tors from fully developed convection in a moderate rotationcase HICo1-30 in Fig. 3. As in many previous studies we finda convection pattern dominated by broad warm upflows andnarrow cool downflow plumes. Near the top the upflows formcells, separated from each other by a network of downflowstructures at the cell boundaries. The downflow structures re-main coherent over large depths, sometimes even extendingover the whole convectively unstable layer. Generally, the hor-izontal scale of the convective pattern is observed to decreaseas a function of depth. Note that the downflow lanes seen nearthe top tend to form a few well-defined plumes at larger depthsand the regions of upward flow connect and become broader.However, the upflows also tend to become less well-defined, sothat much more small-scale structure is seen as opposed to thelarge and well-defined “granules” near the top. Although thisapparently contradicts the prediction of the mixing length the-ory, on average the upflows occupy a larger area in the deeperlayers in accordance with the basic mixing length concept.

The e$ects of rotation can be studied from Fig. 4 wherewe show horizontal contours of temperature from three di$er-ent calculation sets at four latitudes. In the weak rotation case(set Co01), shown in the uppermost row, the convective cellshave more or less angular shapes, which pattern changes only

0z1

z2

z3

z

z

x

!z

y

2

Fig. 3. Snapshot of temperature fluctuations and velocity vectors fromthe run HICo1-30. At the top surface of the box we show horizontalcontours at the z = z1 plane and at the lower surface at the z = z2 plane.The coordinate system and the direction of the rotation vector areshown in the upper right corner.

slightly as function of latitude. There is a weak tendency of thesizes of the structures to get smaller towards the pole, whiche$ect becomes more pronounced as the rotation is increased(the middle row, set Co1); the size of the structures clearly de-creases as function of increasing latitude. The angular appear-ance of the cells also changes due to the strong vortical down-flows which are generated by the interaction of the convergingflow with the Coriolis force at the cell edges. In the rapid ro-tation case (the bottom row, set Co10) the convective patternis dominated by rather irregular small-scale structures, exceptfor the equatorial case, where the pattern is totally aligned withthe rotation axis, reminiscent of the banana cells seen in someglobal convection models (e.g., Brun & Toomre 2002). Suchan alignment is a generic feature of the convective motions inthe rapid rotation regime, as can be seen from Fig. 5, wherexz-slices of temperature fluctuations overplotted with velocityfield vectors are shown from the run Co7-60.

5.2. Velocity field characteristics

In Tables 1 and 2 we have calculated some diagnosticsof the velocity field. Comparing the total and fluctuatingrms-velocities (Cols. 6 and 7, respectively), one can note thatboth tend to decrease towards the poles and increase steeplynear the equator for rapid rotation. A similar trend can be seenin the azimuthal and vertical velocity field components (Cols. 9and 10). This is partly due to the fact that the convection ismore e"cient in the equatorial regions compared to the polarones (see Sect. 5.4), partly because the increasing rotational

研究動機：太陽の黒点磁場の起源（Larmorから変わっていない）

428 BROWN ET AL. Vol. 711

(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 1. Convective structures and mean flows in cases D3 and H3. (a) Radial velocity vr in dynamo case D3, shown in global Mollweide projection at 0.95 R!, withupflows light and downflows dark. Poles are at top and bottom and the equator is the thick dashed line. The stellar surface at R! is indicated by the thin surroundingline. (b) Profiles of mean angular velocity !(r, ! ), accompanied in (c) by radial cuts of ! at selected latitudes. A strong differential rotation is established by theconvection. (d) Profiles of meridional circulation, with sense of circulation indicated by color (red counterclockwise, blue clockwise) and streamlines of mass fluxoverlaid. (e)–(h) Companion presentation of fields for hydrodynamic progenitor case H3. The patterns of radial velocity are very similar in both cases. The differentialrotation is much stronger in the hydrodynamic case and the meridional circulations there are somewhat weaker, though their structure remains similar.(A color version of this figure is available in the online journal.)

large and we can achieve modestly high magnetic Reynoldsnumbers even at low Pm. Second, the differential rotationbecomes substantially stronger with both more rapid rotation !0and with lower diffusivities " and #. This global-scale flow is animportant ingredient and reservoir of energy for these dynamos,and the increase in its amplitude means that low Pm dynamoscan still achieve large magnetic Reynolds numbers based on thiszonal flow. Lastly, the critical magnetic Reynolds number fordynamo action likely decreases with increasing kinetic helicity(e.g., Leorat et al. 1981), and helicity generally increases withrotation rate (e.g., Kapyla et al. 2009). Indeed, there are evensuggestions that the presence of a mean shearing flow maylower the critical magnetic Reynolds number (Hughes & Proctor2009), and the strong differential rotation present in these rapidlyrotating suns may serve to lower this threshold for dynamoaction. We find that the rapidly rotating flows considered hereachieve dynamo action at somewhat lower Rm than the modelsof Brun et al. (2004), which rotated at the solar rate.

3. DYNAMOS WITH PERSISTENT MAGNETIC WREATHS

We here explore case D3 which yields fairly persistentwreaths of magnetism in its two hemispheres, though thesedo wax and wane somewhat in strength once established.Examining the properties of this dynamo solution should helpto provide a perspective for the greater variations realized in ourtime-dependent dynamos which will be discussed in a followingpaper.

3.1. Patterns of Convection

The complex and evolving convective structures in our dy-namo cases are substantially similar to the patterns of convectionfound in our hydrodynamic simulations. Our dynamo solutionrotating at three times the solar rate, case D3, is presented inFigure 1, along with its hydrodynamic progenitor, case H3. The

radial velocities shown near the top of the simulated domain(Figures 1(a) and (e)) have broad upflows and narrow down-flows as a consequence of the compressible motions. Near theequator the convection is aligned largely in the north–south di-rection, and these broad fronts sweep through the domain ina prograde fashion. The strongest downflows penetrate to thebottom of the convection zone; the weaker flows are partiallytruncated by the strong zonal flows of differential rotation. Inthe polar regions, the convection is more isotropic and cyclonic.There the networks of downflow lanes surround upflows andboth propagate in a retrograde fashion.

The convection establishes a prominent differential rotationprofile by redistributing angular momentum and entropy, build-ing gradients in latitude of angular velocity and temperature.Figures 1(b) and (f) show the mean angular velocity !(r, ! ) forcases D3 and H3, revealing a solar-like structure with a prograde(fast) equator and retrograde (slow) pole. Figures 1(c) and (g)present in turn radial cuts of ! at selected latitudes, which areuseful as we consider the angular velocity patterns realized herewith faster rotation. These !(r, ! ) profiles are averaged in az-imuth (longitude) and time over a period of roughly 200 days.Contours of constant angular velocity are aligned nearly oncylinders, influenced by the Taylor–Proudman theorem.

In the Sun, helioseismology has revealed that the contours ofangular velocity are aligned almost on radial lines rather thanon cylinders. The tilt of ! contours in the Sun may be duein part to the thermal structure of the solar tachocline, as firstfound in the mean-field models of Rempel (2005) and then in 3Dsimulations of global-scale convection by Miesch et al. (2006).In those computations, it was realized that introducing a weaklatitudinal gradient of entropy at the base of the convection zone,consistent with a thermal wind balance in a tachocline of shear,can serve to tilt the ! contours toward a more radial alignmentwithout significantly changing either the overall ! contrast withlatitude or the convective patterns. Ballot et al. (2007) explored

■ 全球グローバル計算: ■ 局所計算:

Brown et al. 2010

(Cattaneo+, Brandenburg+, Tobias+....)(Brun, Miesch +, Charbonneau+, Kapyla+...)

Kapyla et al. 2004.. ..

■ 太陽ダイナモシミュレーションの長年の課題： どこで、いかにして黒点磁場が作られているのか?

= 大局的かつコヒーレント, そして周期変動性を持つ

- 乱流磁場は全球計算でも局所計算でも「比較的簡単に」生成・維持される.

- 太陽対流層（乱流磁場の嵐）の中で、大局的磁場がどのように生まれるのか？

Numerical Setting

計算設定：Local Cartesian Box

x

z

y

Ω

対流層

オーバーシュート層

(m=1)

冷却層（対流安定）

z (=-r)

x (=θ)y (=φ)

(cf., Brandenburg et al. 1996)

重力

Ω

θ

z0

z1

z2

z3

2L

4L

4L

■ 初期設定：静水圧平衡な区分的三層ポリトロープ構造 （Top: 冷却層 + Mid :対流層 + Bottom :対流安定層）

■ 太陽内部の局所領域をカーテシアンボックスでモデル化 (modified f-plane)

■特徴: 回転軸の向き (コリオリ力の働く方向) がコントロールパラメター

θ

■ 圧縮性MHD方程式（回転座標系）

(m=3: 対流安定)

P. J. Kapyla et al.: Reynolds stresses and turbulent heat transport 795

z 0z 1

z 2

z 3

cooling layer

x y

overshoot layer

!

"

g

z

x y

z

radiative interior CZ

convection zone

Fig. 1. Sketch of the model setup. Left: the coordinate system is set up so that x-axis points from south to north, y-axis from west to east,and z-axis radially inwards. The angle ! is the latitude, i.e. the angular distance of the centre of the box from the equator. Right: the domainis divided into three parts, an upper cooling layer, the convectively unstable layer, and the lower overshoot region. The coordinates z0 and z3

denote the upper and lower boundaries of the box, whereas z1 and z2 are the boundaries between stable and unstable layers.

y, and z correspond to the south-north, west-east, and radiallyinward directions, respectively. When analysing the results itis important to keep in mind that our (x, y, z) corresponds to(!!, ",!r) in spherical polar coordinates, where ! is the colat-itude. The angular velocity as function of latitude can now bewritten as ! = "(cos!ex ! sin!ez). We solve a set of MHD-equations

#A#t= u " B ! $µ0 J , (1)

# ln %#t

= !(u · #) ln % + # · u, (2)

#u#t= !(u · #)u ! 1

%#p ! 2! " u + g

+1%

J " B +1%# · &, (3)

#e#t= !(u · #)e ! p

%(# · u) +

1%# · ('%#e)

+#visc + #Joule ! #cool, (4)

where A is the vector potential, u the velocity, B = # " Athe magnetic field, J = # " B/µ0 the current density, % themass density, p the pressure, g = gez the (constant) gravity,and e = cVT the internal energy per unit mass. $ and ( arethe magnetic di$usivity and kinematic viscosity, respectively,µ0 the vacuum permeability, and ' the thermal di$usivity. Theequation of state is that of an ideal gas

p = %e() ! 1), (5)

where the ratio of the specific heats ) = cP/cV = 5/3. & =2%(S is the stress tensor where

S i j =12

!#ui

#x j+#u j

#xi! 2

3*i j# · u

". (6)

The terms responsible for viscous and Joule heating can bewritten as

#visc = 2(S i j#ui

#x j, (7)

#Joule =$µ0

%J2. (8)

We use a narrow cooling layer on top of the convection zone,cooled with a term

#cool =1

tcoolf (z)(e ! e0), (9)

where tcool is a cooling time, chosen to be short enough forthe upper boundary to stay isothermal, f (z) a function whichvanishes everywhere else but in the interval z0 $ z < z1, ande0 = e(z0) the value of internal energy at the top of the box.This parametrisation mimics the radiative losses at the stellarsurface and, although still rather simple in comparison to thereal surface layers of stars, works as a more realistic boundarycondition and stabilises the numerics better than just imposinga constant temperature at the boundary.

We adopt periodic boundary conditions in the horizontaldirections, and closed stress free boundaries at the top and bot-tom. The temperature is kept fixed at the top of the box and aconstant heat flux is applied at the bottom

#ux

#z=#uy#z= uz = 0 at z = z0, z3; (10)

e(z0) = e0, (11)#e#z

####z3=

g

() ! 1) (m3 + 1), (12)

where m3 is the polytropic index of the lower overshoot layer.

Ω

θ

緯度に依存したコリオリ力の変化を計算ボックスの中に取り込むことができる.

This model can simulate the spherical solar dynamo site by changing the rotation axis.

計算設定：Local Cartesian Box

x

z

y

Ω

(m=1) z (=-r)

x (=θ)y (=φ)Ω

■ 計算コード：Godunov CMoC-CT

■ 解像度：Nx×Ny×Nz = 256×256×128　

θ

z0

z1

z2

z3

z0 = -0.15,z1 = 0.0,z2 = 1.0,z3 = 0.85

2L

4L

4L

θ

■ Horizontal B.C.: Periodic B.C.

■ Vertical B.C.: - stress free B.C. for V at z0 and z3

- open field B.C. for B @ z0

perfect conductor B.C. for B @z3

- T = const. @z0

∂T/∂z = const. @z3

■ コントロールパラメター：　拡散係数：ν, κ, η (fixed here) 成層の強さ：ξ (fixed)　緯度：θ　角速度：Ω

(c.f., Sano et al. 1998)

対流層(m=1)

冷却層（対流安定）

重力

計算設定：Local Cartesian Box

P. J. Kapyla et al.: Reynolds stresses and turbulent heat transport 795

z 0z 1

z 2

z 3

cooling layer

x y

overshoot layer

!

"

g

z

x y

z

radiative interior CZ

convection zone

Fig. 1. Sketch of the model setup. Left: the coordinate system is set up so that x-axis points from south to north, y-axis from west to east,and z-axis radially inwards. The angle ! is the latitude, i.e. the angular distance of the centre of the box from the equator. Right: the domainis divided into three parts, an upper cooling layer, the convectively unstable layer, and the lower overshoot region. The coordinates z0 and z3

denote the upper and lower boundaries of the box, whereas z1 and z2 are the boundaries between stable and unstable layers.

y, and z correspond to the south-north, west-east, and radiallyinward directions, respectively. When analysing the results itis important to keep in mind that our (x, y, z) corresponds to(!!, ",!r) in spherical polar coordinates, where ! is the colat-itude. The angular velocity as function of latitude can now bewritten as ! = "(cos!ex ! sin!ez). We solve a set of MHD-equations

#A#t= u " B ! $µ0 J , (1)

# ln %#t

= !(u · #) ln % + # · u, (2)

#u#t= !(u · #)u ! 1

%#p ! 2! " u + g

+1%

J " B +1%# · &, (3)

#e#t= !(u · #)e ! p

%(# · u) +

1%# · ('%#e)

+#visc + #Joule ! #cool, (4)

where A is the vector potential, u the velocity, B = # " Athe magnetic field, J = # " B/µ0 the current density, % themass density, p the pressure, g = gez the (constant) gravity,and e = cVT the internal energy per unit mass. $ and ( arethe magnetic di$usivity and kinematic viscosity, respectively,µ0 the vacuum permeability, and ' the thermal di$usivity. Theequation of state is that of an ideal gas

p = %e() ! 1), (5)

where the ratio of the specific heats ) = cP/cV = 5/3. & =2%(S is the stress tensor where

S i j =12

!#ui

#x j+#u j

#xi! 2

3*i j# · u

". (6)

The terms responsible for viscous and Joule heating can bewritten as

#visc = 2(S i j#ui

#x j, (7)

#Joule =$µ0

%J2. (8)

We use a narrow cooling layer on top of the convection zone,cooled with a term

#cool =1

tcoolf (z)(e ! e0), (9)

where tcool is a cooling time, chosen to be short enough forthe upper boundary to stay isothermal, f (z) a function whichvanishes everywhere else but in the interval z0 $ z < z1, ande0 = e(z0) the value of internal energy at the top of the box.This parametrisation mimics the radiative losses at the stellarsurface and, although still rather simple in comparison to thereal surface layers of stars, works as a more realistic boundarycondition and stabilises the numerics better than just imposinga constant temperature at the boundary.

We adopt periodic boundary conditions in the horizontaldirections, and closed stress free boundaries at the top and bot-tom. The temperature is kept fixed at the top of the box and aconstant heat flux is applied at the bottom

#ux

#z=#uy#z= uz = 0 at z = z0, z3; (10)

e(z0) = e0, (11)#e#z

####z3=

g

() ! 1) (m3 + 1), (12)

where m3 is the polytropic index of the lower overshoot layer.

オーバーシュート層 (m=3: 対流安定)

Numerical Results

■ Kapyla et al. 2009が「局所計算での」大局的磁場生成を初めて報告.

■ 大局的磁場が周期変動を示す（Kapyla et al. 2011）.

■ Kapyal et al.は θ=90°の極域にしか注目していない.

■ 大局的磁場の生成・周期変動メカニズムに対する十分な説明は無い.

局所対流ダイナモシミュレーション：Current Status

November 30, 2011 1:18 Geophysical and Astrophysical Fluid Dynamics paper

Oscillatory large-scale dynamos from Cartesian convection simulations 7

(a) (b)

Figure 2. Same as Fig. 1 but for oscillatory !-shear dynamo Run A2 (a) and !2 dynamo Run C1 (b).

(a) (b)

Figure 3. Phase diagrams for the same runs as in Fig. 2.

functions of z whereas in the more rapidly rotating Run C2 the large-scale fields depend also on x and y.Furthermore, the oscillatory nature of the solution is not so clear. Figure 3(b) shows the phase diagramof the horizontal components of the large-scale field in Run C1. There is a phase shift of !/2.The saturation level of the dynamo is sensitive to the magnetic Reynolds number. Decreasing Rm from

66 to 39 by doubling the value of ", decreases the saturation field strength by a factor of three (Run C1b).Another doubling of " shuts the dynamo o! (Run C1c).Our standard setup in the present paper is to use perfect conductor boundaries at the bottom and

vertical field conditions at the top. Changing also the lower boundary to vertical field conditions producesno discernible di!erence in the solution (Run C1d). However, imposing perfect conductor conditions onboth boundaries decreases the saturation strength to less than a half of the standard setup and decreasesthe fraction of the large-scale field (Run C1e). We have not, however, studied the Rm-dependence of thesaturation field strength in this case (cf. Kapyla et al. 2010a).

4 Conclusions

We have presented results from simulations of turbulent magnetized convection both with an imposedshear flow using the shearing box approximation (Sets A and B) and in rigidly rotating cases (Set C). Inaccordance with previous results, we find the generation of dynamically important large-scale magnetic

November 30, 2011 1:18 Geophysical and Astrophysical Fluid Dynamics paper

Oscillatory large-scale dynamos from Cartesian convection simulations 7

(a) (b)

Figure 2. Same as Fig. 1 but for oscillatory !-shear dynamo Run A2 (a) and !2 dynamo Run C1 (b).

(a) (b)

Figure 3. Phase diagrams for the same runs as in Fig. 2.

functions of z whereas in the more rapidly rotating Run C2 the large-scale fields depend also on x and y.Furthermore, the oscillatory nature of the solution is not so clear. Figure 3(b) shows the phase diagramof the horizontal components of the large-scale field in Run C1. There is a phase shift of !/2.The saturation level of the dynamo is sensitive to the magnetic Reynolds number. Decreasing Rm from

66 to 39 by doubling the value of ", decreases the saturation field strength by a factor of three (Run C1b).Another doubling of " shuts the dynamo o! (Run C1c).Our standard setup in the present paper is to use perfect conductor boundaries at the bottom and

vertical field conditions at the top. Changing also the lower boundary to vertical field conditions producesno discernible di!erence in the solution (Run C1d). However, imposing perfect conductor conditions onboth boundaries decreases the saturation strength to less than a half of the standard setup and decreasesthe fraction of the large-scale field (Run C1e). We have not, however, studied the Rm-dependence of thesaturation field strength in this case (cf. Kapyla et al. 2010a).

4 Conclusions

We have presented results from simulations of turbulent magnetized convection both with an imposedshear flow using the shearing box approximation (Sets A and B) and in rigidly rotating cases (Set C). Inaccordance with previous results, we find the generation of dynamically important large-scale magnetic

1158 KAPYLA, KORPI, & BRANDENBURG Vol. 697

the amplitudes of the three smallest wavenumbers are shown asfunctions of rotation. Substantial large-scale fields are observedonly for the two largest values of Co; for Co ! 1.5 the runsshow very weak large-scale contributions. This is also visiblein the two-dimensional power spectra, taken from the middle ofthe convectively unstable layer, see Figure 9. The most rapidlyrotating case (Run B6) is the only one showing clear signs oflarge-scale fields, in accordance with the fact that the large-scalefield is only periodically present in Run B5. The velocity spec-tra in the saturated state are similar to those in the kinematicphase.

In comparison to earlier studies of rotating convection (e.g.,Cattaneo & Hughes 2006; Tobias et al. 2008), we note that it ischaracteristic of these studies that when the Rayleigh number isincreased, the Taylor number is kept constant. Increasing Ra inthese models generates a larger urms and this inevitably meansthat the Coriolis number decreases as the Rayleigh number isincreased, i.e., for a fixed Ta the rotational influence is large for asmall Rayleigh number and vice versa. For example, in the paperof Cattaneo & Hughes (2006), the smallest Rayleigh numbersin combination with Ta = 5 ! 105 gives a Coriolis numbercomparable to our largest values. However, these simulationsdo not exhibit dynamo action due to a too low Rm, whosevalue also depends on urms. For their highest Rayleigh numbercase, however, Rm is large enough for dynamo excitation butthe Coriolis number is smaller by approximately an orderof magnitude and no large-scale fields are observed. Similararguments apply to the simulations of Tobias et al. (2008).

We find that the large-scale dynamo is excited for all boxsizes for the most rapidly rotating case explored in the presentstudy, as is evident from the spectra shown in Figure 10. Fromthe spectra it would seem that an increasing amount of energyis in the large scales as the box size increases. The growth rateof the total field does not show any clear trend with the systemsize: the largest departure from a constant growth rate is thesomewhat lower value for Run B6 with the intermediate boxsize (see Figure 11).

Figure 11 shows the sums of the Fourier amplitudes of thethree smallest wavenumbers as functions of the system size.For the smallest box (Run A6), the large-scale field is moreconcentrated on the k/k1 = 0 contribution, whereas in Run B6with LH/d = 4 the amplitudes for k/k1 = 0 and 1 are similar.For the largest domain size the k = 0 mode is significantlyweaker than the k/k1 = 1 and 2 modes. These results suggest

Figure 6. Root-mean-square total magnetic field as a function of time for fourmagnetic Reynolds numbers for Runs A6–A9.

that for the present parameters the maximum size of the large-scale structures is somewhere in the range 2 < Lmax/d < 8.

Comparing the saturation level of the large-scale magneticfield in the small box Runs A7, A6, and A10 shows a significantdecrease in the m = 0 component in Run A10 whereas thestrength of the m = 1 mode is only mildly affected. On theother hand, comparing Runs B6 and D1 with a larger domainsize shows again a decreasing m = 0 contribution in Run D1 buta two times larger m = 1 component. However, these numbersshould be taken only as a rough guide because the large-scalecontribution to the magnetic field shows large fluctuations andthe higher Rm runs are fairly short. Taken at face value, theresults would seem to suggest that the strength of the m = 0mode decreases with increasing Rm and that the m = 1 moderemains unaffected or that it can even increase. We note that thehighest Rm runs also have larger fluid Reynolds and Rayleighnumbers which means that also the flow is more turbulentin those cases which could affect the dynamo and thus thesaturation level of the large-scale field.

Although we have used open (VF) boundary conditions thatdo permit magnetic helicity fluxes, such fluxes may not actuallyoccur unless they are driven toward the boundaries by internalmagnetic helicity fluxes. One such flux is the Vishniac & Cho(2001) flux, but it requires shear which is absent in our case.Other fluxes are possible (Subramanian & Brandenburg 2006),but we do not know how efficient they are in our model. It is

Figure 5. Magnetic field component Bx for Run D1 in the kinematic (left panel) and saturated (right) states. The sides of the box show the periphery of thedomain whereas the top and bottom panels show the field from the top (z = d) and bottom (z = 0) of the convectively unstable layer, respectively. See alsohttp://www.helsinki.fi/"kapyla/movies.html.(A color version of this figure is available in the online journal.)

■ Kapyla, Korpi & Brandenburg 2009 ■ Kapyla, Mantere & Brandenburg 2011.. .. .. ..

1) 大局的磁場は他の緯度でも生成されるのか ?

2) 大局的磁場の生成・周期変動の物理機構は ?

※ 2つの疑問

Re ~ 100Rm ~ 400

Pr = 1.4Pm = 4.0Ra = 3.9x106

Ω = 0.4

We fix control parameters in the following:

Convective Dynamo Simulation(By field at the saturated state is visualized by 3D volume rendering)

Blue : negative By componentsRed : positive By components

(Nx ,Ny ,Nz) = (256,256,128)

Ro ~ 0.025

(Polar region model with θ＝90°)

(Lx ,Ly ,Lconv) = (4,4,1)

C.Z.overshoot zone

By changing the latitude θ, we study the generation and cyclic variation mechanisms of LMFs.

Generation of LMFs in the bottom of C.Z. and their reversals can be observed in this movie.

seed: random field

Cray XT4 105 node・time

対流／磁気エネルギーの典型的な時間進化

10!8

10!7

10!6

10!5

10!4

10!3

10!2

0 50 100 150 200 250

!Ekin",

!Em

ag"

t/tcv

■ Emagが増幅されるかどうかは解像度にも依存（～20 grids / 対流渦）.■ Ekinに比べてEmagの飽和時間は長い（約5倍）.■ Emagは時間線形に成長するフェーズを経て飽和 [O(10)tcv後].■ 飽和フェーズではEmag/Ekin ~ 30％. 初期磁場構造にはほとんど依存しない.

0

2

4

6

8

10

12

14

16(!10!5)

0 50 100 150 200 250

Em

ag

t/tcv

Emag

Ekin

1283

Emag ∝ t

(642×128でもconvergeする)

θ=60°

y(=φ)

x (=θ)

θ=90° [pole]

y(=φ)

x (=θ)

対流構造の緯度依存性（Vzの水平面分布＠表面）

θ=0° [E.P.]

y(=φ)

x (=θ)

θ=30°

y(=φ)

x (=θ)

θ=90°θ=60°

θ=30°θ=

0°

■ narrow downflow lane　 ＋ broad upflow cell　

■ 低緯度域で対流セルは 緯度方向へ伸張, 高緯度 域では網目状構造.

■ 高速回転ほど対流セルの 大きさは収縮.

コリオリ力の効果

圧縮性対流の重要な性質：Buoyancy Braking

Vz

ωz

(z軸は動径方向負の向き)

ΩΩ上昇流 = diverging flow 下降流 = converging flow

δT > 0δP > 0

δT < 0δP < 0

ambient

ambient

ωz > 0 ωz < 0

Coriolis force induces C.W. motion

Coriolis force induces C.C.W. motion

(z軸は動径方向負の向き)

Vz

Side View

Top View

←下降流→

←上昇流→

←渦度正→

←渦度負→

10!7

10!6

10!5

10!4

10!3

10!2

0 20 40 60 80 100 120 140 160 180 200

Ekin

,E

mag

t/tturn

θ=60°

対流安定層対流層

大局的磁場の周期的反転. 周期は低緯度の方が長い.

1.5×10-2

-1.5×10-2

1.0×10-2

-1.0×10-2

0.0

Latitu

din

al Field

■ snapshot when t/trot = 330

-z(=

r)

-z(=

r)

y(=φ)y(=φ)

-0.5×10-2

0.5×10-2

8×10-3

-8×10-2

4×10-3

-4×10-3

0.0

Latitu

din

al Field

対流安定層対流層

θ=90° [pole]

■ snapshot when t/trot = 250

z 3-z

z 3-z

0.20.40.60.81.01.21.41.61.82.0

0.20.40.60.81.01.21.41.61.82.0

ダイナモ機構を理解する鍵：対流のup-down asymmetry?

■ 渦度が正の領域（上昇流）での磁束密度の時間進化（Spはωz >0の領域の面積）：

SUBMITTED TO APJ LETTER, PREPRINT TYPESET USING LATEXSTYLEPreprint typeset using LATEX style emulateapj v. 10/09/06

CYCLIC LARGE-SCALE MAGNETIC FIELDS AND BANDED VORTEX SHEETSBY TURBULENT PENETRATIVE CONVECTION

YOUHEI MASADA1, AND TAKAYOSHI SANO2

Submitted to ApJ Letter, Preprint typeset using LATEXstyle

ABSTRACTBy means of local Cartesian simulation of rotating penetrative convection, we study storage, amplification

and cyclic variation mechanisms of large-scale magnetic fields. Our numerical model delivers a well-regulatedcyclic magnetic activity with relatively short cycle period. The strong mean horizontal fields, reaching a halfof equipartition strength when maxima, are stored preferentially in bottom convection zone, and show time-reversal of their polarities. The field reversal interface then propagates toward upper convection zone. Thebroken symmetry between x- and y-directions yields different time evolutions of Bx and By field components.It is characteristic in our model that statistically stationary, horizontally-banded, vortex sheets with alternatingvortities are developed and maintained in a thin layer covering bottom convection and top overshoot zoneswhere magnetic cycles are initiated. The magnetic activity observed in our model can be interpreted as anatural outcome of storage and amplification processes of magnetic fields in the thin banded structure of vortexsheets. The turbulent vortical convection, which stochastically penetrates into the overshoot zone, might inducethis horizontally-banded structure of vortex sheets via an inverse cascade process of vorticity merging.Subject headings: instabilities—convection— MHD: magnetic fields — stars: sun

1. INTRODUCTION

Bx,p(z, t) =

! t0

!Sp

[!Bx(z)/!t] dSp dt!

SpdSp

when "z > 0 , (1)

Bx,n(z, t) =

! t0

!Sp

[!Bx(z)/!t] dSn dt!

SndSn

when "z < 0 , (2)

One of long-standing issues in the solar physics is the originof the Sun’s magnetic field, that is the solar dynamo process.The final goal of solar dynamo researches is to reveal forma-tion and cyclic variation mechanisms of large-scale magneticfields, which are responsible for sunspot and active regionwith 22 year variation cycle, self-consistently with observa-tions and magneto-hydrodynamics. Although there exists alot of numerical works, studying magnetic dynamos sustainedby turbulent convections in global spherical shell geometry(e.g., Gilman 1983; Glatzmaier 1985; Brun et al. 2004 ) andlocal Cartesian geometry (e.g., Nordlund et al. 1992; Bran-denburg et al. 1996; Ziegler & Rüdiger 2003; Cattaneo etal. 2003), the formation mechanism of large-scale magneticfields in the Sun still remains an open issue under the existingconditions (Ossendrijver 2003; Miesch & Toomre 2009).

Recently, a growing body of evidence is accumulating todemonstrate that large-scale magnetic fields, which is ac-countable for the sunspot field, can be organized from tur-bulent convection in spherical shell simulations. Browning etal. (2006) found, for the first time, an emergence of strongaxisymmetric toroidal magnetic fields within the convectivelystable layer in anelastic spherical shell dynamo simulations(see also Browning et al. 2007; Miesch et al. 2009) The polar-ity reversal of large-scale magnetic components is reported in

1 Department of Computational Science, Graduate School of Sys-tem Informatics, Kobe University; Nada, Kobe 657-8501:E-mail:[email protected]

2 Institute of Laser Engineering, Osaka University, 1-1, Yamadaoka,Suita, Japan: E-mail:[email protected]

the similar framework of spherical shell dynamos by Ghizaruet al. (2010) (see also Racine et al. 2011).

Not only in the global shelluler geometries, the large-scalemagnetic fields and its cyclic variations were observed alsoin local Cartesian simulations of rotating turbulent convectionby Käpylä et al. (2009a, 2011) (see Käpylä et al. 2008 for thecase with an imposed shear flow). Their parameter study sug-gested that large-scale magnetic field is generated by a turbu-lent #-effect only when the rotation is rapid enough. It wouldbe interesting that mean-field model with dynamo coefficientsobtained by test-field method gives a reasonable prediction onthe dynamo excitation in direct simulations. Turbulent trans-port coefficients that describes the evolution of large-scalemagnetic fields was studied in detail by Käpylä et al. (2009b).

Despite numerical manifestations of large-scale magneticfields in both global spherical shell and local Cartesian simu-lations, there is ongoing debate as to its reversal mechanism(e.g., Miesch et al. 2009; Ghizaru et al. 2010). The aim of ourwork is to gain an insight into the cycle variation mechanismof large-scale magnetic fields. In this letter, we report numer-ical results obtained in our local Cartesian simulation of ro-tating penetrative convection. The storage, amplification, andcyclic variation mechanisms of large-scale magnetic fields arestudied with an emphasis on physical properties and possibleroles of horizontally-banded vortex sheets, which are main-tained in bottom convection zone and might be a key structurefor magnetic activity cycle observed in our numerical model.

2. NUMERICAL SETTINGS

Our numerical model is almost same as that adopted inBrandenburg et al. (1996) (see also, Käpylä et al. 2004). Arectangular portion of stratified spherical shell is modeled bya local Cartesian box situated at latitude $, where x representslatitude, y longitude (azimuth), and z points in the directionof gravity g. The computational domain has three layers: topcooling layer of depth 0.15d in the range z0 < z < z1, middleconvection layer of depth d in the range z1 < z < z2 and bot-tom convectively stable overshoot layer of depth 0.85d in therange z2 < z < z3. The horizontal extent of the box is 4d in

SUBMITTED TO APJ LETTER, PREPRINT TYPESET USING LATEXSTYLEPreprint typeset using LATEX style emulateapj v. 10/09/06

CYCLIC LARGE-SCALE MAGNETIC FIELDS AND BANDED VORTEX SHEETSBY TURBULENT PENETRATIVE CONVECTION

YOUHEI MASADA1, AND TAKAYOSHI SANO2

Submitted to ApJ Letter, Preprint typeset using LATEXstyle

ABSTRACTBy means of local Cartesian simulation of rotating penetrative convection, we study storage, amplification

and cyclic variation mechanisms of large-scale magnetic fields. Our numerical model delivers a well-regulatedcyclic magnetic activity with relatively short cycle period. The strong mean horizontal fields, reaching a halfof equipartition strength when maxima, are stored preferentially in bottom convection zone, and show time-reversal of their polarities. The field reversal interface then propagates toward upper convection zone. Thebroken symmetry between x- and y-directions yields different time evolutions of Bx and By field components.It is characteristic in our model that statistically stationary, horizontally-banded, vortex sheets with alternatingvortities are developed and maintained in a thin layer covering bottom convection and top overshoot zoneswhere magnetic cycles are initiated. The magnetic activity observed in our model can be interpreted as anatural outcome of storage and amplification processes of magnetic fields in the thin banded structure of vortexsheets. The turbulent vortical convection, which stochastically penetrates into the overshoot zone, might inducethis horizontally-banded structure of vortex sheets via an inverse cascade process of vorticity merging.Subject headings: instabilities—convection— MHD: magnetic fields — stars: sun

1. INTRODUCTION

Bx,p(z, t) =

! t0

!Sp

[!Bx(z)/!t] dSp dt!

SpdSp

when "z > 0 , (1)

Bx,n(z, t) =

! t0

!Sp

[!Bx(z)/!t] dSn dt!

SndSn

when "z < 0 , (2)

One of long-standing issues in the solar physics is the originof the Sun’s magnetic field, that is the solar dynamo process.The final goal of solar dynamo researches is to reveal forma-tion and cyclic variation mechanisms of large-scale magneticfields, which are responsible for sunspot and active regionwith 22 year variation cycle, self-consistently with observa-tions and magneto-hydrodynamics. Although there exists alot of numerical works, studying magnetic dynamos sustainedby turbulent convections in global spherical shell geometry(e.g., Gilman 1983; Glatzmaier 1985; Brun et al. 2004 ) andlocal Cartesian geometry (e.g., Nordlund et al. 1992; Bran-denburg et al. 1996; Ziegler & Rüdiger 2003; Cattaneo etal. 2003), the formation mechanism of large-scale magneticfields in the Sun still remains an open issue under the existingconditions (Ossendrijver 2003; Miesch & Toomre 2009).

Recently, a growing body of evidence is accumulating todemonstrate that large-scale magnetic fields, which is ac-countable for the sunspot field, can be organized from tur-bulent convection in spherical shell simulations. Browning etal. (2006) found, for the first time, an emergence of strongaxisymmetric toroidal magnetic fields within the convectivelystable layer in anelastic spherical shell dynamo simulations(see also Browning et al. 2007; Miesch et al. 2009) The polar-ity reversal of large-scale magnetic components is reported in

1 Department of Computational Science, Graduate School of Sys-tem Informatics, Kobe University; Nada, Kobe 657-8501:E-mail:[email protected]

2 Institute of Laser Engineering, Osaka University, 1-1, Yamadaoka,Suita, Japan: E-mail:[email protected]

the similar framework of spherical shell dynamos by Ghizaruet al. (2010) (see also Racine et al. 2011).

Not only in the global shelluler geometries, the large-scalemagnetic fields and its cyclic variations were observed alsoin local Cartesian simulations of rotating turbulent convectionby Käpylä et al. (2009a, 2011) (see Käpylä et al. 2008 for thecase with an imposed shear flow). Their parameter study sug-gested that large-scale magnetic field is generated by a turbu-lent #-effect only when the rotation is rapid enough. It wouldbe interesting that mean-field model with dynamo coefficientsobtained by test-field method gives a reasonable prediction onthe dynamo excitation in direct simulations. Turbulent trans-port coefficients that describes the evolution of large-scalemagnetic fields was studied in detail by Käpylä et al. (2009b).

Despite numerical manifestations of large-scale magneticfields in both global spherical shell and local Cartesian simu-lations, there is ongoing debate as to its reversal mechanism(e.g., Miesch et al. 2009; Ghizaru et al. 2010). The aim of ourwork is to gain an insight into the cycle variation mechanismof large-scale magnetic fields. In this letter, we report numer-ical results obtained in our local Cartesian simulation of ro-tating penetrative convection. The storage, amplification, andcyclic variation mechanisms of large-scale magnetic fields arestudied with an emphasis on physical properties and possibleroles of horizontally-banded vortex sheets, which are main-tained in bottom convection zone and might be a key structurefor magnetic activity cycle observed in our numerical model.

2. NUMERICAL SETTINGS

Our numerical model is almost same as that adopted inBrandenburg et al. (1996) (see also, Käpylä et al. 2004). Arectangular portion of stratified spherical shell is modeled bya local Cartesian box situated at latitude $, where x representslatitude, y longitude (azimuth), and z points in the directionof gravity g. The computational domain has three layers: topcooling layer of depth 0.15d in the range z0 < z < z1, middleconvection layer of depth d in the range z1 < z < z2 and bot-tom convectively stable overshoot layer of depth 0.85d in therange z2 < z < z3. The horizontal extent of the box is 4d in

■ 渦度が負の領域（下降流）での磁束密度の時間進化（Snはωz < 0の領域の面積） ：

ΩΩ上昇流 = diverging 下降流 = converging

δT > 0δP > 0

δT < 0δP < 0

ambient

ambient

ωz > 0 ωz < 0

Coriolis force induces C.W. motion

Coriolis force induces C.C.W. motion

Side View

Top View

上昇流と下降流での磁束密度の時間進化：Bx,p (t,z) ,Bx,n (t,z)

Bx,p

Bx,n

1.5×10-3

-1.5×10-3

1.0×10-3

-1.0×10-3

0.0-0.5×10-3

0.5×10-3

〈Bx〉

Bxの体積平均の時間進化

対流層底部(bottom C.Z.)でのBpとBnの時間進化

0.025

0.020

0.015

0.010

0.005

0.0

-0.005

-0.010

-0.015

-0.020

-0.025000 100 200

t/tcv

Bx,

p (t

,z),

Bx,

n (t

,z)

〈Bx〉

■ 大局的磁場の進化はBp（ωz > 0の領域での磁束密度）の進化と連動.■ Bn（ωz < 0の領域での磁束密度）とBpは逆極性.■ 下降流（ωz < 0）は反極性磁場を下向きに輸送 → 反転の原因？

300

対流安定層が果たす役割：対流層底部への磁場のストレージ

C.Z.

A Test Model with the same control parameters, but without overshoot zone.

(b) 対流層のみ

〈Bx 〉h

za

1.0×10-2

-1.0×10-2

0.0-0.5×10-2

0.5×10-2

〈Bx 〉h

za

1.0×10-2

0.0

-0.5×10-2

0.5×10-2

-1.0×10-2

■対流層無しの計算: 周期的な変動の兆候は示すがcoherencyは弱い. → 安定層の存在が対流層底部への磁場のストレージに寄与していることを示唆.

(downward pumping)

Overshoot Zone

C.Z.

(a) 対流層＋オーバーシュート層

20 200000 40 60 10080 120 140 180160

20 200000 40 60 10080 120 140 180160t/tcv

■ 対流ダイナモ、特に大局的磁場形成の緯度依存性を調査 (Ro = 0.03).

■ コヒーレントで大局的な磁場が赤道を除く全領域で形成.

■ 磁場反転は対流のup-down asymmetryと関係がありそうである.

今後の計画：

■ より詳しい解析で大局的磁場の生成・輸送・反転の物理を解明. ■ 太陽で実現しているRo (Rossby数) ~ 0.1-1でダイナモは起こるか？

Summary: 周期的ダイナモは局所計算でも起こる.

This work adopts control parameters Pr = 1.4, Pm = 4.0, Ra = 3.9×106, Ω＝0.4

■ 大局的磁場が出現する場所は対流層の底（赤道域以外の全モデル共通）.

■ インヤン格子を使った全球太陽ダイナモシミュレーションとの比較.

■ 緯度90°以外の場所のモデルの長時間計算.

■ 磁場反転の周期は低緯度ほど長くなる傾向（要長時間計算）.

■ 対流安定なovershoot層は磁場のストレージに寄与している.

■ “太陽の” ダイナモ機構とどう関係するかは全くわからない.