1. THEORY OF MERIDIONAL FLOW AND DIFFERENTIAL ROTATION
L. L. Kitchatinov,
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

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

3. 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

4. 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)

5. 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).

6. 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

7. 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:

8. 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)

9. 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

10. 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

11. Slow rotation
Moderate rotation
Fast rotation
The Sun

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

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

14. 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).

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

16. 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

17. 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.