Anisotropic turbulence in rotating magnetoconvection

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1 Anisotropic turbulence in rotating magnetoconvection André Giesecke Astrophysikalisches Institut Potsdam An der Sternwarte Potsdam MHD-Group seminar, 2006 André Giesecke (AIP) Anisotropic turbulence Seminar / 25

2 Content 1 Introduction: Anisotropy in convection driven turbulence Solar convection Convection driven dynamo in the Earth s fluid core Restrictions in numerical simulations 2 Local 3D MHD simulations of rotating magnetoconvection Structure of the convection pattern Turbulence intensity and vertical heat flux Quenching Reynold stresses and horizontal heat flux 3 Conclusions André Giesecke (AIP) Anisotropic turbulence Seminar / 25

3 Deeper solar convection zone Coriolis force is dominating: Co = 2Ωd u rms 1 Simulations of rotating convection by Rüdiger, Egorov et al. (2005): estimation of Λ-Effekt, heat flux and turbulence intensity for the sun meridional heat flux is important for the solar rotation law (interactions between Λ-effect and large scale meridional flow) westward oriented azimuthal heat-flux (unimportant in axisymmetric system) André Giesecke (AIP) Anisotropic turbulence Seminar / 25

Sunspots radial magnetic field inhibits convection cooler area sunspot horizontal oriented field in penumbra causes filament like structure unsolved issues: hot (bright)

4 Sunspots radial magnetic field inhibits convection cooler area sunspot horizontal oriented field in penumbra causes filament like structure unsolved issues: hot (bright) rings around a sunspot? Weiss et al abundance of sun spots close to the poles in fast rotating A-stars André Giesecke (AIP) Anisotropic turbulence Seminar / 25

5 Sunspots André Giesecke (AIP) Anisotropic turbulence Seminar / 25

Convection in the Earth s fluid outer core solid inner core (SIC) fluid outer core (FOC) fast rotator: Ro= u 2Ωd 1 Ek= ν 10 15 2Ωd 2 weak stratification: ρ sic /ρ foc 1.

6 Convection in the Earth s fluid outer core solid inner core (SIC) fluid outer core (FOC) fast rotator: Ro= u 2Ωd 1 Ek= ν Ωd 2 weak stratification: ρ sic /ρ foc 1.2 strong magnetic field: E m E k B 2 Λ = 2Ωρµ 0 η 1 backreaction of the magnetic field R sic =1280km R foc = 3400km André Giesecke (AIP) Anisotropic turbulence Seminar / 25

7 Simulations of convection driven turbulence restricted resolution restricted scale range André Giesecke (AIP) Anisotropic turbulence Seminar / 25

8 Simulations of convection driven turbulence restricted resolution restricted scale range (enhanced) scalar parameters resemble the turbulent values of the diffusivities anisotropy is neglected and diffusion is overestimated André Giesecke (AIP) Anisotropic turbulence Seminar / 25

9 Simulations of convection driven turbulence restricted resolution restricted scale range (enhanced) scalar parameters resemble the turbulent values of the diffusivities anisotropy is neglected and diffusion is overestimated turbulence models to include the effects of small-scale motions: André Giesecke (AIP) Anisotropic turbulence Seminar / 25

10 Simulations of convection driven turbulence restricted resolution restricted scale range (enhanced) scalar parameters resemble the turbulent values of the diffusivities anisotropy is neglected and diffusion is overestimated turbulence models to include the effects of small-scale motions: tensor modells (Phillips & Ivers 2001, 2003) André Giesecke (AIP) Anisotropic turbulence Seminar / 25

11 Simulations of convection driven turbulence restricted resolution restricted scale range (enhanced) scalar parameters resemble the turbulent values of the diffusivities anisotropy is neglected and diffusion is overestimated turbulence models to include the effects of small-scale motions: tensor modells (Phillips & Ivers 2001, 2003) large eddy simulations with subgrid-scale modelling (Buffett 2003, Matsui & Buffett 2005) André Giesecke (AIP) Anisotropic turbulence Seminar / 25

12 Simulations of convection driven turbulence restricted resolution restricted scale range (enhanced) scalar parameters resemble the turbulent values of the diffusivities anisotropy is neglected and diffusion is overestimated turbulence models to include the effects of small-scale motions: tensor modells (Phillips & Ivers 2001, 2003) large eddy simulations with subgrid-scale modelling (Buffett 2003, Matsui & Buffett 2005) properties of the turbulence in the fluid outer core are more or less unknown: André Giesecke (AIP) Anisotropic turbulence Seminar / 25

13 Simulations of convection driven turbulence restricted resolution restricted scale range (enhanced) scalar parameters resemble the turbulent values of the diffusivities anisotropy is neglected and diffusion is overestimated turbulence models to include the effects of small-scale motions: tensor modells (Phillips & Ivers 2001, 2003) large eddy simulations with subgrid-scale modelling (Buffett 2003, Matsui & Buffett 2005) properties of the turbulence in the fluid outer core are more or less unknown: typical velocities and scales André Giesecke (AIP) Anisotropic turbulence Seminar / 25

14 Simulations of convection driven turbulence restricted resolution restricted scale range (enhanced) scalar parameters resemble the turbulent values of the diffusivities anisotropy is neglected and diffusion is overestimated turbulence models to include the effects of small-scale motions: tensor modells (Phillips & Ivers 2001, 2003) large eddy simulations with subgrid-scale modelling (Buffett 2003, Matsui & Buffett 2005) properties of the turbulence in the fluid outer core are more or less unknown: typical velocities and scales structure of convection pattern André Giesecke (AIP) Anisotropic turbulence Seminar / 25

15 Simulations of convection driven turbulence restricted resolution restricted scale range (enhanced) scalar parameters resemble the turbulent values of the diffusivities anisotropy is neglected and diffusion is overestimated turbulence models to include the effects of small-scale motions: tensor modells (Phillips & Ivers 2001, 2003) large eddy simulations with subgrid-scale modelling (Buffett 2003, Matsui & Buffett 2005) properties of the turbulence in the fluid outer core are more or less unknown: typical velocities and scales structure of convection pattern averaged effect of small-scale fluctuations on large-scale fields André Giesecke (AIP) Anisotropic turbulence Seminar / 25

16 Local model solve the 3D-MHDequations local in a cartesian box = examine small-scale behavior of fluid motions and magnetic field AR : 8 : 8 : 1 Res. : cartesian box placed tangentially on a sphere Ω Θ x y z x points in meridional y in azimuthal and z in radial direction periodic boundary conditions in x and y closed box in z André Giesecke (AIP) Anisotropic turbulence Seminar / 25

17 Basic equations t ρ = (ρu) t (ρu) = (ρuu) p+ 1 µ 0 ( B) B+ σ+ρg 2ρΩ u t e t B = (eu) p u+σ u+ η µ 0 B 2 + (κ c T) = (u B η B) ρ(r, t) mass density g gravitation field u(r, t) fluid velocityfield Ω angular velocity p(r, t) thermal pressure µ magnetic permeability B(r, t) magnetic field ν kinematic viscosity T(r, t) temperature κ c thermal conductivity e(r, t) thermal energy density η magnetic diffusivity σ viscous stress tensor σ ij = νρ( i u j + j u i 2 3 uδ ij) André Giesecke (AIP) Anisotropic turbulence Seminar / 25

18 Parameters and boundary conditions ( ) Ra = gd4 dt Tκν dz g c p = 10 6 Pr = ν/κ = 0.5 Pm = ν/η = 0.5 Ta = 4Ω 2 d 4 ν 2 = Λ = B 2 y0 2ρµ 0 ηω = Stratification: ρ bot ρ top = 1.1 at the boundaries ρ and T are fixed to their initial values z u x = 0 z u y = 0 u z = 0 stress free z B x = 0 z B y = 0 B z = 0 perfect conductor André Giesecke (AIP) Anisotropic turbulence Seminar / 25

19 Flow pattern I slow rotation: Ta = 10 6 Λ = 1 Λ = 4 André Giesecke (AIP) Anisotropic turbulence Seminar / 25

20 Flow pattern II fast rotation:ta = 10 7 Λ = 1 Λ = 4 André Giesecke (AIP) Anisotropic turbulence Seminar / 25

21 Two-Point-Correlations Q zz (δx) = u z(x)u z(x + δx) u z (x) 2 Q zz(δx) = u z(x)u z(x + δx) u z (x) 2 high correlations only occur within a cell were the motions are oriented in the same direction Q zz serves as a measure for the correlation length λ corr. André Giesecke (AIP) Anisotropic turbulence Seminar / 25

22 Cell size ( B y ) is independent of the imposed field strength λ corr x ( B y ) resembles the increasing extension of the convection cells in direction of the imposed field λ corr y for higher rotation rate the transition from an isotropic to an elongated cell occurs at weaker field strength André Giesecke (AIP) Anisotropic turbulence Seminar / 25

23 Horizontal and vertical anisotropy A H = u 2 y u 2 x u 2 rms A V = u 2 x + u 2 y 2 u 2 z u 2 rms suppression of motions B y is more effective than B y at the pole the turbulence is dominated by the vertical component independent from the field strength and rotation rate André Giesecke (AIP) Anisotropic turbulence Seminar / 25

24 Turbulence intensity, Reynolds stresses and heat flux Quasi-linear approximation: F conv i = ρc p ui T ( T = χ ij g ) j x j c p vertical temperature gradient only case j = z can be discussed André Giesecke (AIP) Anisotropic turbulence Seminar / 25

25 Turbulence intensity, Reynolds stresses and heat flux Quasi-linear approximation: F conv i = ρc p ui T ( T = χ ij g ) j x j c p vertical temperature gradient only case j = z can be discussed ρ, T, g are constant define normalized heat flux z F conv i = u i T = F i conv ρc p André Giesecke (AIP) Anisotropic turbulence Seminar / 25

26 Turbulence intensity, Reynolds stresses and heat flux Quasi-linear approximation: F conv i = ρc p ui T ( T = χ ij g ) j x j c p vertical temperature gradient only case j = z can be discussed ρ, T, g are constant define normalized heat flux z F conv i = u i T = F i conv ρc p Express temperature fluctuations by velocity fluctuations: u i T χ ij = 1 2 τ corrq ij = 1 2 τ corr u i (x, t)u j (x, t) André Giesecke (AIP) Anisotropic turbulence Seminar / 25

27 vertical components, B y imposed F Coriolis F Lorentz Ra crit is minimal maximum around Λ 1 (but only at the pole) turbulence suppressed for Λ > 4 but heat flux remains at a high level presence of a magnetic field changes the latitudinal dependence of turbulence and heat flux André Giesecke (AIP) Anisotropic turbulence Seminar / 25

28 vertical components, B z imposed field has no influence on latitudinal behavior significant suppression of turbulence/transport of heat only for B > B Earth no catastrophic quenching! André Giesecke (AIP) Anisotropic turbulence Seminar / 25

29 Quenching comparision with non-rotating magnetoconvection non-rotating magnetoconvection: u 2 z 1 ( Bz 1 + Rm B eq rotating magnetoconvection: u 2 z 1 ( Bz 1 + B eq ) ) 3 André Giesecke (AIP) Anisotropic turbulence Seminar / 25

30 Reynold stresses and horizontal heat flux meridional and azimuthal heat flux are negative and increase with magnetic field strength Heat is transported to the poles and westwards quasi-linear relation is only confirmed for the azimuthal flux André Giesecke (AIP) Anisotropic turbulence Seminar / 25

31 Angular dependence: meridional heat flux meridional heat flux is always negative and increase with magnetic field strength Heat is transported to the poles André Giesecke (AIP) Anisotropic turbulence Seminar / 25

32 Angular dependence: azimuthal heat flux azimuthal heat flux is always negative and increases towards the equator Heat is transported westwards André Giesecke (AIP) Anisotropic turbulence Seminar / 25

33 Angular dependence: radial heat flux for Λ < 1 a minimum occurs at 30 (also obtained by Käpylä et al. 2004) André Giesecke (AIP) Anisotropic turbulence Seminar / 25

34 Conclusions I André Giesecke (AIP) Anisotropic turbulence Seminar / 25

35 Conclusions I significant change in convection pattern occurs at lower field strength for faster rotating systems André Giesecke (AIP) Anisotropic turbulence Seminar / 25

36 Conclusions I significant change in convection pattern occurs at lower field strength for faster rotating systems no catastrophic quenching for turbulence in rotating magnetoconvection André Giesecke (AIP) Anisotropic turbulence Seminar / 25

37 Conclusions I significant change in convection pattern occurs at lower field strength for faster rotating systems no catastrophic quenching for turbulence in rotating magnetoconvection turbulent thermal diffusivity exhibits strong dependence of field strength and direction André Giesecke (AIP) Anisotropic turbulence Seminar / 25

38 Conclusions II André Giesecke (AIP) Anisotropic turbulence Seminar / 25

39 Conclusions II for sufficient fast rotation a horizontal magnetic field enhances the vertical transport of heat (cooling is facilitated) André Giesecke (AIP) Anisotropic turbulence Seminar / 25

40 Conclusions II for sufficient fast rotation a horizontal magnetic field enhances the vertical transport of heat (cooling is facilitated) rotation induced horizontal transport of heat occurs westwards and towards the poles André Giesecke (AIP) Anisotropic turbulence Seminar / 25

41 Conclusions II for sufficient fast rotation a horizontal magnetic field enhances the vertical transport of heat (cooling is facilitated) rotation induced horizontal transport of heat occurs westwards and towards the poles warmer poles might act as a source for a large scale meridional flow André Giesecke (AIP) Anisotropic turbulence Seminar / 25

42 Conclusions II for sufficient fast rotation a horizontal magnetic field enhances the vertical transport of heat (cooling is facilitated) rotation induced horizontal transport of heat occurs westwards and towards the poles warmer poles might act as a source for a large scale meridional flow interaction with non-axisymmetric boundary conditions that prescribe the heat flow at the core mantle boundary André Giesecke (AIP) Anisotropic turbulence Seminar / 25

Meridional Flow, Differential Rotation, and the Solar Dynamo

Meridional Flow, Differential Rotation, and the Solar Dynamo Manfred Küker 1 1 Leibniz Institut für Astrophysik Potsdam, An der Sternwarte 16, 14482 Potsdam, Germany Abstract. Mean field models of rotating