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.  

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Space  observa%ons  for  solar  physics  and  helioseismology  

1996.05.01

SOHO/MDI 2011.07.06

2010.04.30 2014.05.19

SDO/HMI 15 years of data

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    fre

quen

cy à

horizontal  wavenumber  à    

p modes

Individual mode frequencies !nlm with �l m lcan be measured for l < 150 and n < 25.

red  is  faster  (25  day)    blue  slower  (35  day)  

Schou  et  al.  1997  

Internal  rota%on  from  helioseismology    

Near-surface shear layer (NSSL)

Tachocline

Frequencies of acoustic (p) modes are split by rotation

!nlm = !nl0 +mh⌦inlm, where ⌦(r, ✓) is the angular velocity

Reanalysis  of  MDI  data  accoun%ng  for  all  known  sources  of  systema%c  errors  (Larson  and  Schou  2008)  

Old processing: Corbard & Thompson (2002)

Observa%onal  correc%ons  &  NSSL  

Barekat et al. 2014

NSSL  and  Reynolds  stresses  

8  

Velocity decomposition v = (r sin ✓⌦�+ vmer) + v0.

Conservation of angular momentum:

@@t (⇢r

2sin

2 ✓ ⌦) = �div

⇣⇢r sin ✓hv0�v0i+ ⇢r2 sin2 ✓ ⌦vmer

⌘,

Turbulent stresses Q�,j = hv0�v0ji control the global dynamics.

Stress-free boundary condition: T�r = �⇢Q�r = 0.

Parametrization of Reynolds stress (Kippenhahn 1963, Rudiger 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|v0|2i/hv02r i � 3 � �1

S = �1 means hv02r i = 2hv02� i = 2hv02✓ i,in line with simulations by Kapyla et al. (2011).

Note: Q�✓ controls the latitudinal di↵erential rotation.

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    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  

Kueker  et  al.  2014  

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Coriolis number

0.90 0.92 0.94 0.96 0.98 1.00 Fractional radius

0.00.2

0.4

0.6

0.8

1.01.2

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)

Co

�1= Ro = 2⌧c⌦ (Kuker 2015) Ro = U⌦/L (Greer et al. 2016)

Surface  gravity  (f)  waves  fre

quen

cy à

horizontal  wavenumber  à    

f modes

!2= gk, g = 274 m/s

2

propagate horizontally in the top 2 Mm.

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             .

.

δc

v

C(rA, rB , t) =

Z T

0 (rA, t

0) (rB , t0 + t) dt

B

A

Wave travel times ⌧(rA ! rB) and ⌧(rB ! rA)are extracted from C(t > 0) and C(t < 0).

⌧(rA ! rB)� ⌧(rB ! rA) ⇠ �2

R BA

1v2�v · ds

Can infer the horizontal components of flows in top 2 Mm

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,

Horizontal divergence

divh v

Outflow is ~ 300 m/s for SG

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(top 200 km)

v✓, v�

Local  correla%on  tracking  (LCT)  

Test and calibration of LCT velocity amplitudes

15  

Observed SDO Doppler line-of-sight velocity

LCT horizonal velocities projected onto the line of sight

Power spectra for horizontal divergence: For kR < 300, general consistency between f-modes (top 1 Mm) and granulation tracking (surface)

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Horizontal Reynolds stress: Giant cells from LCT on the solar surface (Hathaway, Upton, & Colegrove, Science 2014)

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.

Q✓�

Horizontal  Reynolds  stresses  at  supergranula%on  scales  

latitude-60° -40° -20° 0° 20° 40° 60°

R θφ (m

2 /s2 )

-400

-200

0

200

400Giant Cells: Hathaway,Upton, Colegrove (2013)Supergranulation: Fournier,Gizon, Langfellner (2015)

−40° −20° 0° 20° 40°−600

−400

−200

0

200

400

600

latitude

Qeq

(m2 /s

2 )

SupergranulationGiant cells

LCT and TD agree!

latitude-60° -40° -20° 0° 20° 40° 60°

R θφ (m

2 /s2 )

-400

-200

0

200

400Giant Cells: Hathaway,Upton, Colegrove (2013)Supergranulation: Fournier,Gizon, Langfellner (2015)

0.75 0.80 0.85 0.90 0.95-400

-200

0

200

400

At 30 deg North

radius Kueker et al. (theory): Near the surface, the viscous term dominates over the Lambda effect

Horizontal  Reynolds  stresses  at  supergranula%on  scales  

Q✓�(✓) = QD✓�(✓) +Q⇤

✓�(✓)

0.75 0.80 0.85 0.90 0.95-400

-200

0

200

400

At 30 deg North

radius

Horizontal  Reynolds  stresses  at  supergranula%on  scales  

Q✓�(✓) = QD✓�(✓) +Q⇤

✓�(✓)

Note: theory is sensitive to⌫T ⇠ 1

3 luc =13↵MLTHpuc

Observations provide a valuableconstraint at the surface.

Reynolds  stress                    small  at  surface  (nearly  stress-­‐free  B.C.?)  

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Qr�

vr ⇠ �(d ln ⇢/dr)�1divhv

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  

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hv · curlvi = hvzcurlzi+ hvh · curlvi' H⇢hdivhcurlzi+ hvh · curlvi

(1)

Coriolis number Co = 2⌧c⌦ ' 0.5 for ⌧c = 1 day (SG).

Mean field theory predicts that the e↵ect of

the Coriolis force on turbulent flows gives:

hdivhcurlzi ' g ·⌦/⌧c ⇠ �10

�11⌦(�) sin�/⌦eq s

�2

where h·i is a horizontal average (Ruediger et al. 1999).

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Travel times of surface-gravity waves (top 2 Mm) (top 200 km)

v✓, v�divh v hcurlz vi =RC v · ds

by Stokes theorem

Flow components that can be measured near the surface:

v�, v✓, horizontal divergence, vertical vorticity,vr = �(d ln ⇢/dr)�1

divhv

Helioseismology LCT

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Horizontal divergence Vertical vorticity

Signal?

Vor%city  correlates  with  divergence:  Effect  of  Coriolis  force  on  convec%on  

 

(J.  Langfellner,  L.  Gizon,  A.C.  Birch,  2014)  

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)

26  

Can we spatially resolve the excess vorticity due to the Coriolis force?

The  average  supergranule  

Identify the position of ~10,000 supergranular outflows in divergence maps, then shift and average

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

Vertical vorticity

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. 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.

MOVIE  

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Evolu%on  of  of  the  average  supergranule:  traveling-­‐wave  convec%on  at  high  Ra  

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)  

   •  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.  

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!/2⇡ = 1.65(kR/100)0.45µHz

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.  

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