Kine%c helicity and anisotropic turbulent stresses of solar supergranula%on Boulder, August 2016 Damien Fournier, Jan Langfellner, Bjoern Loep%en, Laurent Gizon Max Planck Ins%tute for Solar System Research and University of GoNngen, Germany Manfred Kueker IAP Postdam
Outline Solar differen%al rota%on Diagnos%cs of convec%ve veloci%es near the solar surface Local correla%on tracking of small features (e.g. granules) Local helioseismology Supergranula%on (~30 Mm, Ro=1-2) and larger scales (giant cells) Reynolds stresses Important to understand the large scale dynamics Giant cells (Ro << 1) Supergranula%on scales and smaller (Ro >= 1) Kine%c helicity and vor%cal flows in supergranula%on Correla%on ver%cal vor%city and horizontal divergence vs. la%tude Resolved vor%city around the average supergranule Evolu%on of supergranula%on: Travelling- wave convec%on Main purpose of this talk is to describe some of the observa(onal techniques used to characterize turbulent convec%ve flows near the solar surface. 2
Space observa%ons for solar physics and helioseismology 1996.05.01 15 years of data SOHO/MDI 2010.04.30 2014.05.19 2011.07.06 SDO/HMI
MDI (1996-2011) HMI (2010+) Helioseismic and Magnetic Imager 4k x 4k line- of- sight Doppler velocity @45 s High resolu%on, full disk, all the %me One pixel = one seismometer Granules (1.5 Mm) are resolved
Oscilla%on power spectrum frequency à Individual mode frequencies!nlm with l m l can be measured for l < 150 and n < 25. p modes horizontal wavenumber à
Internal rota%on from helioseismology Frequencies of acoustic (p) modes are split by rotation! nlm =! nl0 + mh i nlm,where (r, ) is the angular velocity Schou et al. 1997 Tachocline Near-surface shear layer (NSSL) red is faster (25 day) blue slower (35 day)
Observa%onal correc%ons & NSSL Reanalysis of MDI data accoun%ng for all known sources of systema%c errors (Larson and Schou 2008) Barekat et al. 2014 Old processing: Corbard & Thompson (2002)
NSSL and Reynolds stresses Velocity decomposition v =(r sin ˆ + v mer )+v 0. Conservation of angular momentum: @ @t ( r2 sin 2 ) = div r sin hv 0 v 0 i + r 2 sin 2 v mer, Turbulent stresses Q,j = hv 0 vj 0 i control the global dynamics. Stress-free boundary condition: T r = Q r = 0. Parametrization of Reynolds stress (Kippenhahn 1963, Rüdiger 1989): Q r = T r sin @ @r + S T sin. B.C. implies that @ ln @ ln r = S near the surface, where anisotropy parameter is S = h v 0 2 i/hvr 02 i 3 1 S = 1 means hv 02 r i =2hv 02 i =2hv 02 i, in line with simulations by Käpylä et al. (2011). 8
Can we measure Reynolds stresses? Rota%on perturbs convec%on à anisotropic turbulent Reynolds stresses drive differen%al rota%on and meridional circula%on (cf. Kitcha%nov 2008, Ruediger s 2014 book) On the other hand, viscous forces tend to smooth out differen%al rota%on Kueker et al. 2014 Theory relies on a parameteriza%on of the effects of rota%on on convec%on (the Lambda effect). à Need measurements of sta(s(cal proper(es of rota(ng convec(on in the convec(on zone, including turbulent Reynolds stresses
Transition in dynamics between NSSL (Ro large) and deeper convection zone (Ro small). This transition may take place around the supergranulation scale (l~ 120). (cf. N. Featherstone s talk) 1.2 1.0 Coriolis number 0.8 0.6 0.4 0.2 0.0 0.90 0.92 0.94 0.96 0.98 1.00 Fractional radius Co 1 =Ro=2 c (Küker 2015) Ro = U /L (Greer et al. 2016)
Surface gravity (f) waves 2 2 frequency à! = gk, g = 274 m/s propagate horizontally in the top 2 Mm. f modes horizontal wavenumber à
f- mode %me- distance helioseismology A technique to measure the %me it takes for wave packets to travel between any two points A and B (Duvall et al. 1993) Method is based on the cross- covariance between the oscilla%on signal at A and the oscilla%on signal at B C(r A, r B,t)= Z T 0 (r A,t 0 ) (r B,t 0 + t) dt Wave travel times (r A! r B ) and (r B! r A ) are extracted from C(t >0) and C(t <0). (r A! r B ) (r B! r A ) 2 R B 1 v ds A v 2 Can infer the horizontal components of flows in top 2 Mm B. v A δc.
f-mode travel time between a point and an annulus (average over a few hr) inward minus outward travel-time difference sensitive to the horizontal divergence of the flow, div h v Outflow is ~ 300 m/s for SG Horizontal divergence
Local correla%on tracking (LCT) (top 200 km) v, v 14
Test and calibration of LCT velocity amplitudes Observed SDO Doppler line-of-sight velocity LCT horizonal velocities projected onto the line of sight 15
Power spectra for horizontal divergence: For kr < 300, general consistency between f-modes (top 1 Mm) and granulation tracking (surface)
Horizontal Reynolds stress: Giant cells from LCT on the solar surface (Hathaway, Upton, & Colegrove, Science 2014) Q is positive in the north (would drive equatorial acceleration) Note 1: we have known the existence of these scales for a while (Gizon 2003; Hindman, Gizon et al. 2004; though we did not give them a name). Note 2: Hathaway s result confirmed by granulation LCT by Loeptien. 17
Horizontal Reynolds stresses at supergranula%on scales 400 200 Giant Cells: Hathaway, Upton, Colegrove (2013) Supergranulation: Fournier, Gizon, Langfellner (2015) Q (m 2 /s 2 ) 600 400 200 0 200 LCT and TD agree! Supergranulation Giant cells R θφ (m 2 /s 2 ) 0 400 600 40 20 0 20 40 latitude -200-400 -60-40 -20 0 20 40 60 latitude
Horizontal Reynolds stresses at supergranula%on scales Q ( ) =Q D ( )+Q ( ) 400 200 Giant Cells: Hathaway, Upton, Colegrove (2013) Supergranulation: Fournier, Gizon, Langfellner (2015) 400 200 R θφ (m 2 /s 2 ) 0 0-200 -400-60 -40-20 0 20 40 60 latitude -200-400 At 30 deg North 0.75 0.80 0.85 0.90 0.95 radius Kueker et al. (theory): Near the surface, the viscous term dominates over the Lambda effect
Horizontal Reynolds stresses at supergranula%on scales Q ( ) =Q D ( )+Q ( ) 400 200 Note: theory is sensitive to T 1 3 lu c = 1 3 MLTH p u c Observations provide a valuable constraint at the surface. 0-200 -400 At 30 deg North 0.75 0.80 0.85 0.90 0.95 radius
Reynolds stress Q r small at surface (nearly stress- free B.C.?) v r (d ln /dr) 1 div h v 21
We have seen that supergranula%on is not strongly constrained by rota%on, however Ro is close to 1 and thus we expect rota%on to be perturbed by rota%on. An obvious target is the Kine(c helicity hv curlvi = hv z curl z i + hv h curlvi ' H hdiv h curl z i + hv h curlvi Coriolis number Co = 2 c ' 0.5 for c =1day(SG). Mean field theory predicts that the e ect of the Coriolis force on turbulent flows gives: hdiv h curl z i'ĝ / c 10 11 ( )sin / eq s 2 where h i is a horizontal average (Ruediger et al. 1999). 22
Flow components that can be measured near the surface: v, v, horizontal divergence, vertical vorticity, 1 Helioseismology Travel times of surface-gravity waves (top 2 Mm) LCT (top 200 km) div h v hcurl z vi = R C v ds by Stokes theorem v, v 23
Signal? Horizontal divergence Vertical vorticity 24
Vor%city correlates with divergence: Effect of Coriolis force on convec%on Signs and func%onal form are as expected. However, measured correla%on is ten %mes bigger (SDO) than mean- field predic%on Tangential velocities 10 m/s (5% effect) Effect of Coriolis force on convec%on (J. Langfellner, L. Gizon, A.C. Birch, 2014)
Can we spatially resolve the excess vorticity due to the Coriolis force? 26
The average supergranule Identify the position of ~10,000 supergranular outflows in divergence maps, then shift and average
Vertical vorticity 40 deg è Good agreement between f-mode seismology and granulation tracking Hemispheric dependence of net vorticity in outflows. à Consistent with effect of Coriolis force on flows
Evolu%on of of the average supergranule: traveling- wave convec%on at high Ra The average supergranule can be studied in the past and the future, with respect to the time of maximum outflow. Following movie shows the evolution of the horizontal divergence and of the intensity contrast. MOVIE Similar to a cross-correlation analysis. Divergence signal oscillates with period ~ 6 days. dt ~ (di/i) T/4 ~ 1.5 K Consistent with an earlier study by Rast et al. 29
Evolu%on of of the average supergranule: traveling- wave convec%on at high Ra Analysis of average SG in Fourier space confirms dispersion rela%on (G et al. 2003)!/2 =1.65(kR/100) 0.45 µhz The dispersion rela%on does not depend on la%tude Excess power in the prograde direc%on Unfortunately theory is missing. Even linear stability analysis is missing. 30
Conclusion Reynolds stresses and influence of rota%on on convec%on can be measured at the solar surface at various horizontal spa%al scales. General consistency with mean- field theory from Ruediger and colleagues, used to explain the global dynamics. Helioseismology has the poten%al to characterize convec%ve veloci%es in the convec%on zone using the deeper p modes. In doing so, we will have to resolve the ques%on of the convec%ve amplitudes along the way Order amist turbulence: supergranula%on supports traveling- wave modes - - s%ll not understood. May be important to understand mean flows as well. 31