Diapositive 1

1
A Simple Model for the Solar Isorotation
Countours
Steven A. Balbus Ecole Normale
Supérieure Physics Department Paris, France
2

• Aoccdrnig to rsceeacrh at Cmabirgde Uinrevtisy,
  it deosnt
• mtater in waht oredr the ltteers in a wrod
  are, the olny
• iprmoatnt tihng is taht the frist and lsat
  ltteer be at the rghit
• plcae. Tihs is bcuseae the huamn mnid deos
  not raed ervey
• ltteer by istlef, but the wrod as a wlohe.

3
SOLAR DIFFERENTIAL ROTATION

• One of the most beautiful astronomical results
  of the
• last half century was the precision
  determination
• of the interior solar rotation.
• Splitting of p-mode frequencies allows an
  accurate
• determination of the angular velocity ?(r, ?),
  using
• sophisticated inversion techniques applied to
  the
• excited mode spectrum.

4
THE FINDINGS

• The only place where there is significant
  differential rotation in the sun is in the
  convective zone (CZ).
• This is thought to be the only place where
  there is a significant level of turbulence. (So
  much for enhanced viscosity models.)
• The rotation is approximately constant on cones
  of constant ? at mid latitudes, cylindrical near
  the equator, spherical (apparently) near the
  poles.

5
Howe et al. 2000
Tachocline
Surface shear
6
THE PROBLEM

• The CZ is very nearly adiabatic, PP(?),
  barotropic.
• Convective motions, except near the surface,
  are small typically 30 m s-1.
• A barotropic fluid in hydrostatic equilibrium
  must rotate on cylinders, ? ( R ). (Taylor
  columns.) The solar rotation profile is
  decidedly not constant on cylinders.
• But large scale numerical simulations generally
  do produce cylindrical contours.

7
Brun Toomre 2002 ASH Code
Miesch, 2007
8
THE ORTHODOX VIEW

• Despite the simple regularity of the rotation
  pattern, the flow is an extremely complex
  interplay between convective turbulence and
  rotation. Some handles exist, however.
• Departures from barotropic structure because
  Coriolis forces affect convection.
• Convection along the axis of rotation is more
  efficient than convection in planes of constant
  Z. Hot poles, cool equator.
• Thermal wind equation R ??2/?z e? (?P
  ???)/ ?2

9
THERMAL WIND EQUATION

• R ??2/?z e? (?P ???)/ ?2 (R, ?, z) or
  (r, ?, ?)
• R ?2 ??2/?z (??/r??) (?P/?r) - (??/?r) (?P/r??)
  
• Let S k/(?-1) ln P?- ? , CP ?k/(?-1) ,
• R ? CP ??2/?z (?P/r??) (?S/?r) - (?P/?r)
  (?S/r??)
• For SCZ RCP ??2/?z g (?S/r??) , ?g -
  (?P/?r).
• Shows relationship between large scale
  latitudinal entropy gradients due to Coriolis,
  and departures from cylindrical isotachs.
  Trend moving polewards, ? dec., S inc.

10
GETTING THE LAY OF THE LAND

• N2 g/? ? (ln P?- ?) /?r 3.8 X 10-13
  s-2,
• by requiring the solar luminosity to be carried
  by convection
• (Schwarzschild 1958).
• But ? gradient of S is estimated by different TWE
  physics
• g/? ? (ln P?- ?) /r?? R??2/?z 2 X 10-12
  s-2,
• The ? gradient of S exceeds the r gradient by
  factor of 5if thermal wind balance is valid.

11
?S, ?? COUNTER ALIGNED ?
Clearly, e? ? ?? also much exceeds er ? ?? .
er
??
12
?S, ?? COUNTER ALIGNED ?
Clearly, e? ? ?? also much exceeds er ? ??
. What if ?? and ?S are more closely related
than just a trend? What if SS(?2) ?
?S
er
??
13
TWE would then define the isorotational surfaces.
14
Thermal Wind Equation with SS(?2)
where S is dS/d?2. Solution is ?2 is constant
along the characteristic
Since ? is constant along this characteristic, so
is S. To solve, set y sin ?. Find
, a first order linear equation.
15
With g GM?/r2, the solution is
where A is an integration constant, and B is
B/r?3 order unity or less.
16
With g GM?/r2, the solution is
where A is an integration constant, and B is
B/r?3 order unity or less.
For solving for ?, assume a fit at rr?,
?(cos2?o), where ?o is ? at rr?, the starting
point of the characteristic
17
Batman contour is typical. Spheres at small R,
cylinders at larger R, sharp upturn in between.
18
0.2
0.1
0.4
0.3
19
0.5
0.6
0.7
0.8
20
Thermal Wind Equation for SS(L2)
Solution is angular momentum is const. along
characteristic
Solution is similar to angular velocity
characteristics. Find
21
0.1
0.2
0.4
0.3
22
(No Transcript)
23
(No Transcript)
24
This is often the way it is in physics---our
mistake is not that we take our theories too
seriously, but that we do not take them
seriously enough. ---Steven
Weinberg, in The First Three Minutes
25
HOW IS IT THAT S AND ? CARE ABOUT EACH
OTHER SO MUCH?
To answer this, we need to understand something
about the stability of rotating, stratified,
magnetized plasmas.
We need to take rotation, stratification and
magnetism seriously.
26
THE PUNCHLINE
Counter alignment of the entropy and
angular velocity gradients is a rigorous
condition for marginal stability in a rotating,
convective, magnetized gas.
27
THE PUNCHLINE
The solar rotation profile can be understood as
a consequence of maintaining a state of marginal
(in)stability to the most rapidly growing axi-
and nonaxisymmetric dynamical modes.
28
THE PUNCHLINE
A magnetic field is essential to this picture.
29
Fundamental linear response of a magnetized
medium
(Boussinesq degenerate Alfvén slow modes.)
Addition of rotation introduces two new terms,
one of which is epicyclic, ?2d?2/dlnR
4?2, the other of which is tethering, and
gives rise to the MRI.
30
Schematic MRI
?2
angular momentum
?1
To rotation center
31
Schematic MRI
angular momentum
To rotation center
32
Compact form of equation
General wave numbers
Allow ?(R, z)
33
Allow S(R,z) as well
Most general, barotropic, axisymmetric
response. Stability from ??? limit
34
More clear written in terms of displacement
vector, ?n
Then,
Marginal modes exist when rotation and entropy
surfaces coincide. Explicitly (Papaloizou
Szuszkiewicz 1992, Balbus 1995)
- -
N2 d?2/dln R gt0 also required, but amply
satisfied.
35
Did we miss something? What happened to good
old-fashioned convection?
Marginalization of BV oscillations picked
out by nonaxisymmetric modes. Without a
magnetic field, these purely hydrodynamic modes
dominate the question of stability. With even a
weak magnetic field, the axisymmetric modes
become major players.
36
Z
?S
?S
?S
stable
stable
unstable
?
R
??
??
??
37
(kvA)2
unstable zone
?R/?z
(kvA)2 versus ?R/?z under conditions of
marginal instability
38
Global Simulations of the MRI, Hawley 2000
Equatorial Plane
Meridional Plane
39
SUMMARY SYNTHESIS

• A dominant balance of the vorticity equation
• corresponding to the thermal wind equation
  seems
• to hold in much of the SCZ.
• 2. Implies ?S/?? gtgt ?S/? ln r , just as seen in ?
  contours.
• TWE equation may be solved exactly with SS(? ).
• Produces isorotation contours in broad
  agreement with
• helioseismology.
• As it happens, SS(? ) corresponds precisely to
  marginal
• stability of axisymmetric, baroclinic,
  magnetized modes
• in rotating gas. Coincidence?

40
SUMMARY SYNTHESIS

• 5. Nonaxisymmetric modes couple to N2, but
  insensitive to
• magnetic couplings. Axisymmetric modes couple
  strongly to
• rotation, very sensitive to magnetic field.
  Hydro stability criteria
• very different, not near criticality.
• The gross dynamical (Batman isotachs) and
  thermal
• (adiabatic) features of the SCZ are a
  consequence of
• marginalizing the dominant magnetobaroclinic
• linear unstable modes of the system.

41
SUMMARY SYNTHESIS

• Need to resolve (kvA)2 ??2 /? ln R wavelengths
  , nominally
• difficult, not impossible. Can surely fudge
  parameters to
• bring into computational domain.
  Calibration with linear
• dispersion relation is essential.
• Ideas are generic, simple. For the future,
  hope is that
• they will prove to be useful for problems
  they were not
• designed to solve directly, e.g. latitude
  dependence of
• dynamo cycle ? N2(r, ?). (M. McIntyre).