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TACKLING
TURBULENCE
DIFFERENTIAL
ROTATION
 he
Sun regularly reminds us that we are at its mercy. Recurring sunspots,
erupting solar flares, and coronal mass ejections spew hot plasma
into space, often flooding the Earth with high-energy particles.
When the Sun is magnetically active, its outbursts can disrupt
satellite communications and knock out power grids. “One
of the great puzzles in nature has been why the Sun has 22-year
cycles of magnetic activity,” said Juri Toomre, a professor
in JILA (formerly called the Joint Institute for Laboratory Astrophysics)
and the Department of Astrophysical and Planetary Science at the
University of Colorado, Boulder. Trying to sort out the Sun’s
magnetic mechanism requires charting the turbulent flows at various
depths and latitudes, using both theoretical models and deductions
based on helioseismic data.
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Figure 1:
Modeling Solar Convection
(Nic
Brummell, Juri Toomre)
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| Figure 2: Layers
of Turbulence |
Just beneath the surface
of our closest star lies a deep shell of raging turbulence. Hot
gases erupt upward and plummet downward while being contorted by
rotation and shear. Such complex flows within this shell, which
occupies the outer 30 percent of the Sun, lead to a striking differential
rotation—at the solar surface, equatorial regions rotate once
in about 25 days and the poles in about 33 days. Helioseismology,
the study of acoustic waves of the deep interior that are detectable
at the surface, has revealed that such differential rotation cuts
down through the layer, terminating at the base of the convection
zone in a shear layer called the tachocline.
The differential rotation is thought to have a crucial role in the
Sun’s ability to build magnetic fields by dynamo processes.
Toomre and his colleagues at the University of Colorado have been
generating 3-D models of the solar convection zone to understand
the physics that link turbulent convection, rotation, and magnetism.
“The solar differential rotation and magnetism raise difficult
questions, the resolution of which requires focused and sustained
simulation efforts,” said Toomre. In particular, he and his
group are interested in the convection zone of intense turbulence,
and in the tachocline of shear that separates it from the deeper
radiative interior. The tachocline is the most likely location of
the “solar dynamo,” essentially an enormous factory that
builds and regulates the magnetic fields of the Sun. Recent helioseismic
studies have shown that there are repetitive speedups and slowdowns
of rotation rate above and below the tachocline with a period of
about 1.4 years. This discovery of a solar “heartbeat”
most likely arises from the dynamo.
TACKLING
TURBULENCE
Toomre and his colleagues
create computational models that can be compared to the results
of seismic probing of the solar interior. They have developed
an approach for simulating turbulent convection that involves
two codes, an anelastic spherical harmonic (ASH) code and a hybrid
pseudo-spectral (HPS) code. The ASH code generates a global picture
of the dynamics of convection, rotation, and magnetism within
a full spherical shell, while HPS simulates a higher-resolution,
localized planar domain within that shell. Recently, the group
made significant gains in optimizing the two codes—both performing
efficiently on scalable parallel architectures, including NPACI’s
Blue Horizon supercomputer at SDSC.
“The two approaches provide information that adds up to a
more complete picture,” said Nic Brummell, assistant professor
at the University of Colorado. “HPS extracts a small piece
of the sphere, allowing for a much higher degree of turbulence.
ASH models a full shell with the correct geometry.”
Brummell and Toomre have used HPS to compute what happens to an
unstable layer of fluid when a stable layer sits below it, mimicking
the conditions at the base of a stellar convective envelope. The
simulation allows the researchers to peer beneath the smooth surface
to the boiling turbulence below (Figure 1).
In a paper accepted for publication by the Astrophysical Journal,
Brummell and Toomre report that turbulent plumes originating from
the convection zone bore into the stable radiative zone beneath
it. This leads to transport of material and magnetic fields from
one layer into another. Adding rotation to the system decreased
the penetration by deflecting some of the plumes’ energy
into horizontal mixing. Further, the interaction studied here
between the highly turbulent interior and the rotation has provided
insight into the disparity of helioseismic and early numerical
results for the differential rotation.
“We now have access to sufficient resolution to study convective
turbulence as opposed to laminar or mildly turbulent flows,”
Brummell said. HPS ran on the Cray T3E and now runs on Blue Horizon.
“These simulations have led to an understanding of mechanisms
that are essential to the solar global dynamo. For example, the
overshooting turbulent plumes can remove magnetic fields from
the convection zone and deposit them in the tachocline.”
DIFFERENTIAL
ROTATION
Toomre and Sacha Brun,
a postdoctoral researcher at JILA, follow a second approach to
investigating solar convection, using the ASH code to generate
a global model of how rotation influences turbulent convection
within a full spherical shell. This provides information that
cannot be detected in the localized domains studied with HPS,
such as global properties of differential rotation. Using ASH,
they have created a series of 3-D simulations, each incorporating
more turbulence than the last. With every increase, the patterns
of convection changed, with larger banana-shaped areas of convection
giving way to more frequent and stronger vortices. In addition,
they examined the turbulence at different depths in the models,
peeling back layers of the Sun to examine how structures related
to turbulence spread through the convection zone. They found that
networks of relatively persistent downflows sink deep into the
unstable layer, providing a mechanism for the redistribution of
angular momentum (Figure 2).
Brun said that “using our global ASH code on teraflops computers
like Blue Horizon allows us to perform numerical experiments that
are beginning to capture the richness of turbulent dynamics and
magnetism within our nearest star.”
In another paper accepted for publication by the Astrophysical
Journal, Toomre and Brun report that their simulations of turbulent
convection under the influence of rotation closely match helioseismic
pictures of the differential rotation within the bulk of the convection
zone. They investigated factors such as the Reynolds stresses
in transporting angular momentum toward the Sun’s equator
to create the differential rotation. In studies that are just
now becoming computationally feasible, Toomre and his colleagues
hope to use ASH to investigate problems such as the role of shear
at the edges of the convection zone.
“The solar turbulent convection zone has striking dynamical
properties that continue to challenge basic theory,” said
Toomre. Despite those challenges, such dual computational approaches
for investigating solar convection is providing evidence that
matches and explains solar observations.
“A novel aspect of our research is that we are able to test
and evaluate the result of the simulations by turning to ongoing
helioseismic observations and their deductions about dynamics
deep within the Sun for both guidance and inspiration,” said
Toomre. —CF 
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PRINCIPAL INVESTIGATOR
Juri Toomre
JILA, University of Colorado, Boulder
COLLABORATORS
Nic Brummell, Sacha Brun, Marc DeRosa, Kelly Cline
University of Colorado, Boulder
Tom Clune
SGI
Laura Carrington,
Stuart Johnson
SDSC
Mark Miesch
High Altitude Observatory, NCAR
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