Understanding dynamo mechanisms from 3D convection simulations of the Sun

Transcription

1 Understanding dynamo mechanisms from 3D convection simulations of the Sun Jörn Warnecke Max Planck Institute for Solar System Research Axel Brandenburg, Nordita & CU Boulder Petri J. Käpylä, AIP Maarit J. Käpylä, MPS Matthias Rheinhardt, Aalto University

2 Solar Activity differential rotation turbulent convective motions alpha effect pumping diffusion no direct measurements 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 2

3 Dynamos = r (u B) r J B = B + b 0 u = U + = r (U B + u0 b 0 ) r J r (U B) =(B r)u B(r U) (U r)b Shear Advection 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 3

4 Electromotive force E = a B + b B +... E i = a ij B j + b j B k +... E = α B + γ B β ( B) δ ( B) κ ( B) (S ), 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 4

5 Test-field method Schrinner et al. 2005, 2007, 2012 B t = (u B + u B ) η B, E = α B + γ B β ( B) δ ( B) κ ( B) (S ), B t = ( u B T + u B + u B u B ) η B. b κλr, b κλϑ i B (i) see talk by Fred Gent Tr r 0 0 ϑ 0 0 B (i) Tϑ r 0 0 ϑ 0 B (i) Tϕ r 0 0 ϑ 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 5 (5)

6 The Simulation 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 6

7 Global convective dynamo simulations A t = u B + r2 A D ln = u Dt Du Dt = g 2 0 u + 1 (J B p + 2 S) T Ds Dt = 1 (K T + t T s)+2 S 2 + µ 0 J 2 cool(r), high-order finite-difference code scales up efficiently to over cores compressible MHD 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 7

8 Warnecke et al.: Influence of Käpylä et al. 2012, 2013, 2016, Mean toroidal Warnecke et al. magnetic field Bφ evolution 2014, 2016 saturated stage for kinetic Runs A1c2, Fig. 2. Time-averaged and A1, magnetica1c, α coefficients, α, α an, no malized by α = u /3, and normalized differential rotation Ω/Ω fo m row).run The I. dashed horizontal lines indicate 0 6th of December 2016 rms MPPC Meeting, Princeton, US K M 0 8

9 Results 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 9

10 Warnecke et al.: Turbulent transport coefficients of (6) nated check t tenones e that time.6 we due to mean ue as mpare from urg & d Au- α. All north lower simin the (7) ω = -scale urrent d ρ is ional udinal ose of α, in s, see Carte- et al. (2009), where the transport coefficients for convection in a pointwise parity Cartesian box have been obtained by the testfield method, is not possible with respect to αrr as it was not determined there. In the! "2 αsi j middle of the convection zone αrr is much weaker than above P(αi j ) =! "2 αsi j and below; but the values are still high compared to the other components of α. The latitudinal dependency shows strong dewhere αs,a i j (r, θ) cay from low to high latitudes. The coefficient αθθ is around 6 torially symme In the bottom p and 2 times weaker than αrr and αφφ, respectively, and shows Fig. 3. Time-averaged main-diagona expected, its va withα, α +α (d) and the n P of K K M responding to a multiple sign reversals on cylindrical contours. A region of neg surface (black ofl ative (positive) αθθ in the northern (southern) hemisphere coin- 90 θ in the northern hemisphere coincide with v r = 0.98 R (black), r = 0.84 R (red), significance. A cides with a local minimum of the local rotation rate Ω(r, θ) = viations from dashed lines in the (d) indicate αk t Ω0 + uφ /r sin θ as seen in Figure 3 of Warnecke et al. (2014) and we overplot 2 cos θ and 2 P(α φφ cosθθ2) θ= P(α with of a coefficient a maximum of negative latitudinal shear (dω/dθ < 0), see up- respectively. Values in (a)-(d) are nor with additional per left panel of Figure 5. Further, αφφ shows concentrations at nator, see Figs have P! 0.99 low and mid to high latitudes near the surface, but also inside all equatorial s the tangent cylinder, where the sign is reversed. The sign rever- ilar to α, but the sign P (U θ ) = 0.99i reversal θθ sal with depth is most pronounced in αφφ, but also visible in αθθ. more pronounced and at high lati 4.2. Magnetic The meridional profile of αφφ is overall similar to that of αk, Already inspection by eye sug To investigate t even though the strength is around 4 times smaller, see also Figto mean ormagne ant ure 13 and the related discussion in Section 4.7. As indicated in are almost fully symmetric Ω and α effects order to study the symmetry prop Figure 3, the latitudinal dependency of αφφ as well as of αk does along with the respect to the equator quantitative from the merid not follow a typical cos or cos2 distribution as found by, e.g., weaker, ity of its components ascantly et al. (2013) an Käpylä et al. (2006) for moderate rotation. However, beside the the contribution lower amplitude, αφφ follows roughly the latitudinal dependency time-aver! "2! "2been of αk. Therefore, the main reason for the mismatch of αφφ with a ea B we first con αes α i αj (a-c) together i j magnetic cycle Fig. 2. Time-averaged main-diagonal components of cosine dependency (at surface) seems to be due with to the mismatch P(α the ins = Eq.,! (8))"2(e) over! lati- "2selected α, α +α (d) and the parity P(αi j ) ) (see es different radii: ea toroidal magne of the kinetic helicity with a cos profile. tude 90 θ in the northern hemisphere and for three α j and + dashed αi j the correspondi 10 16th Januaryof 2017 SOLARNET IV MEETING, Lanzarote, Spain r = 0.98 R (black), r = 0.84 R (red), r = 0.72 R (blue). isolid normalized by α = u /3 and norfig. 1. of Components α and α We employ lines in (d): α and α + α, respectively. Values in (a)-(d) are normalmalized differential rotation Ω/Ω ; all quantities are time averaged. NuThe non-diagonal components of α have similar strengths merals at the bottom right at each panel: overall parity P, see Eq. (8). ized by α = u /3. mean magnetic K K,M 0 K M rr rms 0 0 K rms K M

11 Magnetic field generation A&A proofs: manuscript no. paper 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 11

12 into symmetric 3. Test-field and method anti-symmetric parts, we can rewrite Eq tion (3) as t B pol = Turbulent pumping 3.1. Theoretical background We consider the induction equation in the mean-field appro E = α B + γ B β ( B) δ ( B) κ ( B) (S ), where B α is the symmetric part of a giving rise to the so-called effect (Steenbeck = (u et al. B ), u B γ ) i = ϵ η ijk a jk /2characterizes B, t anti-symmetric part of a and describes changes of the magne field as where it were frozen into the velocity γ (also: turbulent pum ing ) (e.g. Ossendrijver et al. 2002), β is the symmetric part u B = E [...+ [...+ (γ pol + U pol) B pol ] (16) (γ tor + U tor) ] B pol the rank two tensor acting upon B, whichcharacterizes ( turbulent 1 t B tor diffusion, the = γ pol vector + U pol) δ quantifies B tor its antisymmetric p + and enables what is known as the Rädler effect (Rädler 196 Article number, page 2 of 17page.17 (17) ( B) (S ) is the symmetric part of the derivative tensor and κ arank-threetensor,whoseinterpretationisstillnotfullyund stood. 16th of Calculating January 2017 thesesolarnet transport IV MEETING, coefficients Lanzarote, Spain will lead to an und 12

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14 Magnetic quenching 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 14

15 sport coefficients shown as 2D histograms of α rr (a), γ r (b), β rr (c), α φφ zed energy density of the mean field B 2 /B 2 eq. Data taken from the entire Fig. 11. Quenching of transp averages, over the normalize median, respectively; blue contours: margins of range, in which 50% of and yellow lines: mean and m in log-log scale for zoomed-in range; dotted anddashedgreen: 1/ ( 1 + Inlays: average and median in t (= time-averaged) part and a part with alled variation (indicated by superscript 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 15 Fig. 12. Average cycle dependency of selected transport coefficients. Mean azimuthal and radial magnetic field, B (10) variations of the diagonal components, δ and the corresponding toroidal and pical cycle, the definition and computabed in Section 4.2. All coefficients show (top), together with the be divided into a constant temporal average zero, cal v), such as on short time scales, w appearance when avera To quantify the vari values, defined as α = α t + α v. In Fig. 12, we plot the va of α and β, ij alongwithγ, t, etc. radial mean field for a typi tion of which was describe clear cyclic variations wit α V ij = α v2 For all shown coefficien latitudes. Near the surf

16 Conclusions Test-field method is one way to understand dynamo simulations. Alpha deviates from helicity expression. Complicated mixture of dynamo effects. Turbulent pumping changes significantly the eff. flow. Quenching does not depends analytical on B Strong cyclic variations of coefficients 16th of January 2017 SOLARNET IV MEETING, Lanzarote, Spain 16