Summary
Solar Meridional Circulation and Differential Rotation as Viewed
Through Numerical Simulation
Abstract
The large-scale, concerted motions established by global scales of convection in the solar convection zone almost
certainly play a crucial role in the nature and timing of the solar dynamo. The latitudinal differential rotation
established by such motions, along with radial shear associated with the thin tachocline at the base of the
convection zone, are likely to be the dominant generators of mean toroidal magnetic fields in the Sun. The length
of the solar cycle is in turn thought to depend intimately on the transport of magnetic fields by the meridional
circulation associated with this differential rotation. Whereas the differential rotation is well measured throughout
the bulk of the solar convection zone, meridional circulations are well measured throughout only its uppermost
layers. Some insight into the nature of the deep meridional circulation may be gained, however, through numerical
modeling of solar convection. We examine the interaction of meridional circulations and differential rotation
driven by convective motion in the Sun. Using 3-dimensional hydrodynamic simulations carried out with the ASH
code, we present a class of solar convection models that demonstrate the existence of two distinct regimes of
meridional circulation. These two regimes depend predominantly on the vigor of the convective driving and
posses, in one instance, a single monolithic cell of circulation in each hemisphere, and in the other instance, a
single cell at high latitudes with multiple cells at low latitudes. We explore the long-term evolution of these two
regimes and demonstrate that the action of a single cell is to speed up the poles relative to the equator. The
compatibility of such a circulation with a solar-like differential rotation profile then relies somewhat delicately on
the ability of convective Reynolds stresses to balance these effects, or on the ability of convection to establish
sufficiently strong latitudinal entropy gradients (warm pole, cool equator) which work to suppress the strength of
this cell.
Differential Rotation & Meridional Circulation:
What we (don’t) Know
Nicholas A. Featherstone & Mark S. Miesch
High Altitude Observatory, NCAR
Modeling Approach
Vary Diffusivities:
Resolution (Nr x Nθ x Nφ):
200 x 256 x 512 &
200 x 512 x 1024
500 nHz
Ω
Ω - Ωframe (nhz)
+
-
Vary Rotation Rate:
0.75Ω ≤ Ω ≤ 2Ω
ν0 ≤ ν ≤ 2ν0
ν0 = 4 x 1012 cm2 s-1
Schou et al. 2002
± 50
± 50
Flux Balance
1
~20 m s-1
poleward
Radiative
Heating
Meridional
Circulation
• Can we gain some insight into the deep meridional circulation
through numerical modeling of the convection zone?
Entropy Profile
CCW
CW
Enthalpy Flux
L/L
• Deep meridional circulations must also be an important role in the
distribution of angular momentum throughout the convection zone
Conduction
dSdr = 0
Single-Celled
Multi-Celled
Meridional
Circulation
KE Flux
0
0.72
S=0
0.97
r/R
0.72
r/R
• Two distinct regimes of differential rotation result
from different levels of rotational influence.
0.97
(0.95 R )
Generation of the Reynolds Stresses
Polar Spin-up
Differential
Rotation
1
High Latitudes (weak Coriolis)
Equatorward transport by Reynolds
stresses (fluctuating flows)
± 45
± 90 (nhz)
Fast
Equator
± 45
± 90 (nhz)
L /L0
2κ0
2κ0
• Solar-like regimes tend to exhibit multi-celled
circulations within each hemisphere.
The Reynolds Stress Transition
Multi-Cellular Transition with Increased Convective Driving
± 50
-20
+ 200
-20
+ 160
Model Heat Transport & Convective Driving
Entropy
Howe et al. 2000;
• The deep meridional circulation is thought to mediate the timing
of the solar dynamo through flux transport processes.
-
0.75 Ω
360 nHz
• Only the shallow meridional circulation is measured throughout
the solar near-surface shear layer.
Vr
0.80 Ω
0.84 Ω
EQ
± 50
• Convection zone only
• 3.5 density scale heights
• non-magnetic
κ0 ≤ κ ≤ 2κ0
κ0 = 1.6 x 1013 cm2 s-1
1Ω
Differential
Rotation
3-D anelastic equations
Rotating deep spherical shell
Solar-like stratification (1-D stellar structure model)
Pseudo-spectral Spherical Harmonics
4th order finite-differences in radial direction
Efficient parallelization (scalable to 34,000 cores)
Model Parameters
• The latitudinal and radial shear are thought to be crucial
components of the solar dynamo.
2Ω
We model the solar convection zone using the Anelastic Spherical Harmonic (ASH) code.
•
•
•
•
•
•
Retrograde Equator
Prograde Equator
Numerical Method
Differential
Rotation
• The solar differential rotation is well-measured throughout the
convection zone.
The Effect of Rotation Rate
± 60 (nhz)
Equatorward
angular
momentum
transport across
60° latitude
Viscous
Stresses
0
Low Latitudes
(strong Coriolis)
Radial plumes nearly parallel to Ω
Tilted Cylindrical Rolls
(banana cells)
High angular momentum transported
inward
Total
Low angular momentum transported
outward
Poleward transport by meridional
circulation (mean flows)
High angular momentum
transported outward
Low angular momentum
transported inward
-1
Ω
± 70
± 60
± 30
• Reynolds Stresses work to accelerate the equator, while meridional
circulations tend to spin up the poles.
poles.
EQ
-
κ0
± 70
± 75
m s -1
+
ν0
κ0
ν0
2ν0
Meridional
Circulation
2κ0
Ω - Ωframe
+
nhz
Reynolds
Stresses
2κ0
Increasing Angular
Momentum
• To understand how such strong circulations are generated, we must look
at how the Reynolds stresses are formed.
formed.
2ν0
φ
Radially inward
• The transition to polar vortex regimes can be understood in terms of the
meridional circulations overwhelming the Reynolds Stresses (e
(e..g. Gilman
1976;; Glatzmaier & Gilman 1982
1976
1982)). Viscous stresses ultimately stabilize the
numerical system
system..
Polar
Vortices
± 80
Net Transport
Cylindrically
outward
(Accelerates
the Equator)
Equatorial Plane
Angular momentum (L) conservation → meridional circulations must balance Reynolds stresses
Gyroscopic Pumping
∇L
− ρ v m ⋅ ∇ L ≡ F = ∇ ⋅ ( ρ r sin θ v φ' v 'm )
≈ Cylindrically Outward
SingleCelled
MultiCelled
Positive F:
MC inward
+
Positive F:
MC inward
+
MC
MC
F
F
Multi--Celled
Multi
-
Negative F:
MC outward
-
Negative F:
MC outward
Single--Celled
Single
κ0
ν0
CW
CCW
2ν0
• As convective driving increases, convection tends to establish
prograde poles, while more laminar solutions exhibit a prograde
equator.
• Cases with polar vortices possess single-celled circulation profiles
much as cases as lower rotation rates did.
Acknowledgements
This work was supported by NASA grants NNX10AB81G (Heliophysics SR&T), NNH09AK14I
(Heliophysics SR&T), & NNX08AI57G (Heliophysics Theory). Additional support was provided
by the HAO Postdoctoral Fellowship Program. HAO is a division of the National Center for
Atmospheric Research, sponsored by the National Science Foundation.
κ0
ν0
+
∇ ⋅ ( ρ r sin θ v φ v )
'
'
m
• High latitudes behave as in singlecelled case.
• Convection removes angular
momentum from deep layers.
• No banana cells
• Low latitudes differ.
• Deposits angular momentum in
outer layers.
• Deposits angular momentum in deep layers.
• Convection removes angular momentum from outer layers.
2ν0
• Polar vortices become visible in convective patterns as level of
turbulence increases
• The convective Reynolds stresses exhibit noticeably different
patterns in these two regimes.
References
Summary and Conclusions
• We find two distinct regimes of differential rotation in these simulations: a
regime of strong rotational influence characterized by a prograde equator, and
a regime of weak rotational influence is characterized by prograde poles
• Prograde equatorial regimes tend to be accompanied by a multi-cellular
patterns of meridional circulation in the equatorial regions
Gilman, P.A., 1976, in IAU Symp., 71, 207
Glatzmaier, G.A. & Gilman, P.A., 1982, ApJ, 256, 316
Howe, R. et al., 2000, Science, 287, 2456
Schou, J. et al., 2002, ApJ, 567, 1234
• These double-celled patterns can be understood as a response to the
Reynolds stresses necessarily present in these systems that act to transport
angular momentum to the equatorial regions at low latitudes, and radially
inward at high latitudes.
• Our results raise the interesting possibility that the Sun may possess multicellular structures of a similar nature in the equatorial regions. Such
multiple-layered structures may have interesting consequences for flux
transport dynamo models, but we note that these convection simulations
alone cannot tell us what the Sun is really doing. It is quite possible that the
Sun exists near the transition.
• The clear trend toward double-celled circulations in regimes of strong
rotational influence suggests that stars rotating more rapidly than the Sun
may very well be expected to possess multiple circulatory cells rather than
single monolithic circulation cell. These stars may be expected to exhibit very
different cyclic properties than their more slowly rotating counterparts owing
to their different flux-transport properties.