THEORY OF MERIDIONAL FLOW AND DIFFERENTIAL ROTATION L. L. Kitchatinov, kit@iszf.irk.ru Meridional flow Origin Structure Temporal variations Differential rotation: origin Importance of differential temperature Differential rotation models The Sun Individual stars Mass and rotation rate dependence Differential rotation & dynamos NORDITA – 2013

Origin of the meridional flow Centrifugal driving Baroclinic driving Meridional flow vorticity - viscous drag on the meridional flow - Taylor number - Grashof number

Meridional flow decreases with depth beneath the photosphere Meridional flow attains its largest velocities near the boundaries of the convection zone where the thermal wind balance is violated Meridional flow Latitude 45 Deviation from the thermal wind balance Zhao & Kosovichev (2004) Numerical model

Solar meridional flow varies considerably from year to year Small variations of the differential rotation produce (relatively) large fluctuations in the meridional flow Latitude 45 Gonzalez Hernandez et al. (2006) Numerical model (Rempel 2005)

Internal solar rotation from helioseismology (GONG) NSF’s Solar observatory Gilman & Howe (2003) Coriolis forces disturb convection. The back reaction distorts rotation to make it not uniform (Lebedinsky 1941).

Angular momentum transport by convection Angular momentum flux: is the Coriolis number is the convective turnover time The Sun and absolute majority of cool stars are rapid rotators with

The L-effect The angular momentum equation: Turbulence in rotating stratified fluids is capable of transporting angular momentum even in the case of rigid background rotation The analytical findings generally agree with 3D simulations of the angular momentum transport (Chan 2001; Kaepylae & Brandenburg 2008; Kaepylae et al. 2011) The angular momentum equation:

The differential temperature The convective heat flux, is controlled by the eddy conductivity tensor, The rotationally-induced anisotropy results not only in the dependence of radial flux on latitude (Weiss 1965), but also in a deviation of the heat flux from radial direction towards poles, so that the meridional flux is finite: Polar regions of the Sun are warmer than the equator by about 2.5 K (Rust et al. 2008)

Anisotropic heat transport, differential temperature and differential rotation from direct numerical simulations (Warnecke et al. 2013) Cross-component crq of the thermal eddy diffusivity Meridional entropy gradient Differential rotation

Model of the solar differential rotation Tachocline is due to an internal magnetic field of the radiation core with the amplitude of Bp = 10-2 G. Tachocline thickness

Slow rotation Moderate rotation Fast rotation The Sun

Meridional flow in slow and fast rotators The Sun Brandenburg et al. (1998)

Stellar mass/temperature dependence (for young stars) Observations of ZAMS stars (Barnes et al. 2005) Simulations G0 dwarf HD 171488 (V889 Hercules) has differential rotation of DW = 0.52 rad day-1 (Jeffers & Donati 2008). Prot = 1 day.

Rotation rate and stellar mass dependence On using gyrochronology (Barnes 2003, 2007, 2010), the surface differential rotation of single main-sequence stars is predicted as a function of surface temperature and rotation period, DW(T,P_rot), or B-V color and rotation period, DW(B-V,P_rot).

Prediction for the surface differential rotation of main-sequence dwarfs in terms of Prot and surface temperature Teff B-V colour

Differential rotation and dynamos DW is the amplitude of differential rotation is the eddy magnetic diffusivity H is the convection zone thickness The Large-Scale Axisymmetric Magnetic Topology of a Very-Low-Mass Fully Convective Star (Donati et al. 2006): “Our results, which demonstrate that fully convective stars are able to trigger axisymmetric large-scale poloidal fields without differential rotation, challenge existing theoretical models of field generation in cool stars.” Prot = 1 day

Conclusions Meridional flow results from slight deviations from the thermal wind balance. The meridional velocity attains its largest values on the boundaries of the convection zone. Small fluctuations of differential rotation produce relatively large fluctuations in the meridional flow. Stellar differential rotation varies slightly with rotation rate but increases sharply with stellar mass/surface temperature. Small differential rotation of M- and K-dwarfs is more efficient in winding toroidal magnetic fields than much larger differential rotation of F-stars.