Large−scale flow in two−dimensional simulation of
solar convection
Oskar Steiner, Kiepenheuer−Institut fuer Sonnenphysik, Freiburg
Poster in A4/PDF format available under http://www.kis.uni−freiburg.de/~steiner
Cork tracers and tracks
Flow pattern on the solar surface
Data analysis
The solar surface shows a pattern of hierarchically
ordered cells in the velocity field of the plasma:
The title bar of this poster shows an example snapshot of the simulation data; in this case, isotherms.
Density, velocity, temperature, and entropy are given on a 5000 × 50 grid every 45 s. Each run
encompasses a time period of about 12h real time.
Flow pattern
characteristic characteristic
length
time
Granulation
1.5 Mm
8 min
Mesogranulation
8 Mm
5h
Supergranulation
30 Mm
1d
Giant cells
< 10◦ × 40◦
120 d
“Corks” are markers that float on the surface (near
τ 500 nm = 1), transported by the horizontal velocity
field. Starting with 5000 equidistantly distributed
corks, they are swept to the intergranular lanes
within minutes and are subsequently expelled from
divergence centers of the large-scale flow field.
Fourier decomposition of vx:
vx(x) is periodic, vx (x) = vx(x + kL), allowing for
a discrete Fourier decomposition:
Recent results from local helioseismology suggest
that supergranular cells are shallow, pancake-like
objects with a depth of less than 5 Mm, quite different from the traditional view that the depth of
the cells were related to depths of ionization zones.
This poster attempts to answer the question:
vx =
∞
X
a0
ak cos kω0 + bk sin kω0
+
2
k=1
2π
k ∈ N , ω0 =
L
ak =
Does a large scale flow pattern evolve in numerical simulation of a thin surface layer of
the solar convection zone?
2
L
Z
L
vx(x) cos kω0xdx,
bk =
0
With vx k =
is given by:
q
2
L
Z
Fig. 3: Corktracks in case B . The corks are expelled from
L
vx(x) sin kω0xdx
supergranule centers located at x ≈ 30 Mm, 128 Mm, and,
0
possibly around 80 Mm.
2
a2
k + bk , the spectral representation
After 10 min of floating the corks are transported
to the intergranular lanes but barely show any displacement due to the large scale flow field (Fig. 4).
∞
X
a
vx = 0 +
vx k sin(kω0 x + ϕk )
2
k=1
Hydrodynamic simulation
where
a
ϕk = arctan k .
bk
Numerical intergration of the hydrodynamic equations in 2D:
x-direction k to surface
z-direction ⊥ to surface
Physical processes included:
Fig. 1 shows hvx k it near the surface of τ 500 nm = 1.
With this time average each frequency and time
step is equally weighted, independent of phase.
This averaging keeps short-lived components (e.g.
granules) salient.
- Radiation transfer (LTE, one frequency point,
Rosseland mean opacity, OPAL data)
Fig. 4: Cork positions (left) and intergranular lanes (right).
Each cork was floating for 10 min.
- Hydrogen ionization
After 1 1/2 h of floating the corks are expelled
from intergranular lanes of short-lived granules and
mesogranule centers (Fig. 5). Clusters of corks
congregate at the boundaries of mesogranulation,
being separated by a mean distance of 6 Mm.
- Subgrid-scale turbulence model
Computational domain:
Horizontal extent: L = 150 Mm
Vertical extent: H = 1.5 Mm
grid spacing: ∆x = ∆y = 30 km
1 Mm comprises the surface layers of the convection zone below the τ 500 nm = 1 surface, 0.5 Mm
the photosphere up to the temperature minimum.
The horizontal width of 150 Mm allows for the
evolution of roughly 5 supergranular cells.
Fig. 1:
Boundary conditions:
Fig. 5: Cork positions after 1 1/2 h of floating time. Cork
Time average of the spectrum of the horizontal
clusters are separated by a mean distance of 6 Mm, corre-
velocity over 6 h real time. Both spectra show a maximum
The boundary condition most sensible to large scale
flow is the one applied at the lower boundary of the
computational domain.
sponding to the mesogranular scale.
at a mesogranular scale. Case B shows a global maximum
at a supergranular scale. In both cases, the granular scale
of about 3 Mm is only hardly visible, being swamped by the
Conclusion
large scale flow over time. (Spectrum A is shifted downwards
∂T
∂v
≈0;
≈0
∂z
∂z
m tot = const
periodic
by half a decade.)
“wave absorbing
layer”: T = T0
and v = 0 at top
of layer
z
a) vx(x) of a snapshot
x
pg = pg(t) ; inflow: s = sin
∂s
=0
outflow:
∂z
What drives meso and supergranulation?
b) Time average of vx(x) over 6h: case
Two runs with different velocity conditions at the
lower boundary were performed, run A and run B.
A
inflow:
vx = 0;
B
∂hvz ix
= 0;
∂z
vz = hvz ix
∂v
=0
∂z
outflow:
∂vx
= 0;
∂z
inflow:
c) Time average of vx(x) over 6h: case
∂hvz ix
= 0;
∂z
A
B
Fig. 2: a) vx(x) of a snapshot.
The sawtooth curve is
characteristic of granular flow with the zero-crossing of the
inclined part representing the granule center and the rapid
change of sign occurring at the confluence in the intergranu-
A constrains to vertical inflow.
- Note, lar lane. About 60 granules with a mean size of 2.5 Mm are
visible. b) Averaging over 6 h reveals a long-lived horizontal
B constrain to a horizontally
A and - Both, uniform vz -component of the inflow.
flow pattern with a distinctly lower wave number than in a),
which is of mesogranular scale of about 5 Mm. c) Time av-
A can be expected to show a more rigid be- haviour with respect to the horizontal flow.
eraging in case of run B clearly shows a supergranular flow
field with a scale of about 50 Mm.
phys. process
turb. mixing
vz = hvz ix
∂v
=0
∂z
outflow:
rad. cooling
The present 2D hydrodynamic simulation shows,
additional to granular motion, a horizontal velocity
field with a characteristic scale of 5 Mm and, depending on the lower boundary condition, one with
a scale of 50 Mm.
The various horizontal flow scales can be directly
seen when comparing time averages with a snapshot of vx(x).
periodic
buoyancy breaking
2nd stage
granulation
1st stage
supergran.
Mesogranulation may simply be a consequence of
granular evolution. Repeatedly fragmenting (active) granules are surrounded by smaller, short-lived
granules, the fragments. Therefore, they are separated from each other by a few granular diameters
(see also Müller et al. 2000, Sol. Phys. 203, 211).
Supergranulation, on the contrary, may be driven
by a more fundamental physical process. We propose buoyancy-breaking in two-stages taking place
in the shallow subsurface layers (Fig. 6): Ascending
entropy-rich material undergoes a first buoyancybreaking when mixing with cool, weak downflow
plumes in a shallow (a few Mm thick) turbulent
boundary layer. This first stage of buoyancy-breaking leads to the supergranulation. A second stage
of buoyancy-breaking sets in when radiative cooling takes effect at a depth of less than one pressure scale-height, leading to granulation. Given
the depths of these layers, the horizontal extent of
both, granuar and supergranular cells may be estimated with the “tin-can model” of granulation.
Fig. 6: Sketch of the two-stage buoyancy breaking. The
buoyancy breaking that leads to the supergranular flow is
caused by a turbulent mixing layer of a few Mm depth. Granulation is due to the onset of radiative losses in a thin (< Hp )
supergranular scale
meso scale
surface layer.