Minimal models of the Solar cycle

Transcription

1 Minimal models of the Solar cycle Friedrich H Busse 1,3 and Radostin D Simitev 2,3 1 Institute of Physics, University of Bayreuth, Bayreuth D-954, Germany 2 School of Mathematics & Statistics, University of Glasgow, Glasgow G12 8QW, UK 3 NORDITA, AlbaNova University Center, Stockholm SE-1691, Sweden Busse@uni-bayreuth.de, Radostin.Simitev@glasgow.ac.uk Abstract. We discuss the extent to which minimal self-consistent models of convection-driven dynamos may reproduce the Solar cycle. PACS numbers:?????? (specify here) 1. Figures dated - 3 Jan 11

2 Minimal models of the Solar cycle 2 Figure 18: (color online) A dynamo oscillation in the case τ =, R = 15, P = 1, P m = 4.5, β =, stress-free at r o and no-slip at R i. The first column shows meridional lines of constant B ϕ on the left and poloidal field lines, r sin θ h/ θ on the right. The second column shows lines of constant g/ θ at r =.9 corresponding to -.9, -.8, -.7,.7,.8,.9 of the maximum absolute value of g/ θ, and the third column shows lines of constant B r at r = r o. The last column shows Re( g m=2 / θ) on the left and Im( g m=2 / θ) on the right. The five rows are separated equidistantly in by t =.28. e65p1t2r15m1p4.5mvbcfd

3 Minimal models of the Solar cycle 3 Figure 19: (color online) A dynamo oscillation in the case τ =, R = 11, P = 1, Pm = 4.5, β =, stress-free at ro and no-slip at R i. The first column shows meridional lines of constant Bϕ on the left and poloidal field lines, r sin θ h/ θ on the right. The second column shows lines of constant g/ θ at r =.9 corresponding to -.9, -.8, -.7,.7,.8,.9 of the maximum absolute value of g/ θ, and the third column shows lines of constant Br at r = ro. The last column shows Re( g m=2 / θ) on the left and Im( g m=2 / θ) on the right. The five rows are separated equidistantly in by t =.168. e65p1t2r11m1p4.5mvbcfd

4 Minimal models of the Solar cycle 4 Figure : (color online) A dynamo oscillation in the case τ =, R = 1, P = 1, Pm = 6, β =, stress-free at ro and no-slip at R i. The first column shows meridional lines of constant Bϕ on the left and poloidal field lines, r sin θ h/ θ on the right. The second column shows lines of constant g/ θ at r =.9 corresponding to -.9, -.8, -.7,.7,.8,.9 of the maximum absolute value of g/ θ, and the third column shows lines of constant Br at r = ro. The last column shows Re( g m=2 / θ) on the left and Im( g m=2 / θ) on the right. The five rows are separated equidistantly in by t =.28. e65p1t2r1m1p6mvbcb

5 Minimal models of the Solar cycle Figure 21: Butterfly diagrams: Bϕ m= + Bϕ m=1 sgn(bϕ m= ), Br m= + Br m=1 sgn(br m= ), Bϕ m= + Bϕ m=2 sgn(bϕ m= ), Br m= + Br m=2 sgn(br m= ), (from top to bottom) are plotted as functions of and in the case P = 1, τ =, R = 15, P m = 4.5.

6 Minimal models of the Solar cycle Figure 22: Butterfly diagrams: Bϕ m= + Bϕ m=1 sgn(bϕ m= ), Br m= + Br m=1 sgn(br m= ), Bϕ m= + Bϕ m=2 sgn(bϕ m= ), Br m= + Br m=2 sgn(br m= ), (from top to bottom) are plotted as functions of and in the case P = 1, τ =, R = 11, P m = 4.5.

7 Minimal models of the Solar cycle Figure 23: Butterfly diagrams: Bϕ m= + Bϕ m=1 sgn(bϕ m= ), Br m= + Br m=1 sgn(br m= ), Bϕ m= + Bϕ m=2 sgn(bϕ m= ), Br m= + Br m=2 sgn(br m= ), (from top to bottom) are plotted as functions of and in the case P = 1, τ =, R = 1, P m = 6.

8 Minimal models of the Solar cycle 8 R 1 5 clip clip P m P Figure 24: Convection-driven dynamos as a function of R, P and P m for τ = and η =.6,.65. Decaying dynamos are indicated by black dots, MD dynamos are indicated by blue diamonds, FD dynamos are indicated by red stars, quadrupolar dynamos are indicated by green triangles, and coexisting MD and FD dynamos are indicated by pink squares.

9 Minimal models of the Solar cycle 9 Figure 25: Half a period of oscillation in the case η =.65, P = 1.2, τ =, R e = 1, P m = 4.5, β = with stress-free velocity boundary condition at r = r o and no-slip at r = r i. The first column shows meridional lines of constant B ϕ to the left and of r sin θ θ h to the right. Lines of constant B horiz at r =.9, of B r at r = 1 and are shown in the second and third, respectively. The last column shows Re( g m=1 / θ) on the left and Im( g m=1 / θ) on the right. The interval between the snapshots is.224. e65p1.2t2r1m1p4.5mvbcfd