We infer a flow field, u(x,y,) from magnetic
evolution over a time interval, assuming:
Ideality assumed: tBn = -c( x E), but E = -(v x B)/c,
so perpendicular flows drive all evolution.
Démoulin & Berger (2003) argue u = vh – (vz/Bz) Bh,
but Schuck (2006, 2008) argues u ≃ vh .
Motivation: What is the optimal Δt? What can we
learn from the coherence time of flows?
Welsch et al. (2009) autocorrelated active region flows in MDI
magnetograms and found flow lifetimes of ~6 hours on supergranular scales (c. 15 Mm). But what about other, smaller scales?
Caption: Autocorrelation of LOS magnetic field (black) and flow
components (ux, uy). Thick: frame-to-frame autocorrelation, at
96 minute lag. Thin: initial-to-nth frame autocorrelation.
What constrains choice of time interval, Δt?
There are two regimes for which inferred flows u do not
accurately reflect plasma velocities v:
1) Noise-dominated: If Δt is very short, then B is
~constant, so ΔBn is due to noise, not flows. But all
changes in B are interpreted as flows!
==> estimates of u are noise-dominated.
2) Displacement-dominated: If Δt is too long, then v will
evolve significantly over Δt; so u is inferred from
displacements due to the average of the velocity over Δt.
We tracked Stokes’ V/I images at 2 min. cadence,
w/0.3’’/pix, from Hinode NFI, over Dec 12-13, 2006.
Left: Initial magnetogram in the ~13 hr. sequence, at full resolution (0.16” pixels) with the
saturation level set at ±500 Mx cm−2.
Right: Red (blue) is cumulative frame-to-frame shifts in x (y) removed in image coregistration prior to tracking. Total flux (unsigned in thin black, negative of signed in
thick black) is overplotted. Note periodicty similar to Hinode 98-minute orbital freq.
We estimated flows with several choices of
tracking parameters.
• We varied the time interval ∆t between tracked
magnetograms, with
∆t ∈ {2,4,8,16,32,64,128,256} minutes.
• We varied the apodization (windowing)
parameter, σ ∈ {2,4,8,16} pixels.
• We re-binned the data into macropixels, binning
by ∆x ∈ {2,4,8,16,32,64} 0.3’’ pixels.
Histogramming the magnetograms shows a “noise
core” consistent with a noise level around 15 G.
Accordingly, we tracked all pixels with |BLOS| > 15 G.
Pixels from “the bubble” in the filter were excluded from all
subsequent quantitative analyses.
We also co-registered
a resampled vector
magnetogram from
the SP instrument,
produced by Schrijver
et al. (2008).
This enables a crude
calibration of |Bz|
and |B| for each flow,
at least at low spatial
resolution.
To baseline flow coherence times, we investigated
magnetic field coherence times.
Colored: autocorrelation coeffs. over a range of lags with 2nx2n binning.
Black: frame-to-frame autocorr. coeffs. at full res., linear = dashed, rank-order.
We autocorrelated field structure in subregions, and
fitted decorrelation as e-t/τ ; “lifetime” is τ at e-1.
Red: rank-order autocorrelations
of BLOS, in (32 x 32)-binned
subregions vs. lag time.
The the vertical range in each cell
is [-0.5, 1.0], with a dashed black
line at zero correlation.
Blue: one-parameter fits to the
decorrelation in each subregion,
assuming exponential decay, with
the decay constant as the only
free parameter.
Only subregions with median
occupancy of at least 20% of
pixels above our 15 G threshold
were fit.
Background gray contours show
50 G and 200 G levels of |BLOS| in
full-resolution pixels. As expected, field structures persist longer
in stronger-field regions.
Lifetimes of magnetic field structures are longer in
subregions with higher field strengths.
This trend is true for B_LOS,
B_z, and |B|.
This is entirely consistent
with convection
reconfiguring fields, but
strong fields inhibiting
convection.
For short ∆t, frame-to-frame flow correlations can increase
with increasing ∆t’s, and averaging prior to tracking.
Dashed lines show frameto-frame correlation coeffs
for un-averaged
magnetograms.
Solid lines show
correlations for averaged
magnetograms.
Note complete lack of
correlation at ∆t = 2 min.
Rapid decorrelation of
noise-dominated flows
can be seen by
comparing overlain
flow vectors from
successive flow maps
with short ∆t.
Note: these
magnetograms were
not averaged prior to
tracking.
Flows are much more
consistent from one
frame to the next for
longer ∆t, and
averaging magnetograms prior to tracking.
Flow lifetimes can be estimated via autocorrelation of
flow maps.
Flows decorrelate on longer timescales when a
larger apodization window, σ, is used.
Rebinning the data mimics use of a
larger σ --- compare solid black with
dashed blue.
Rebinning speeds tracking; using a
larger σ slows it.
The product of macropixel size ∆x and
σ defines a spatial scale of the flow,
(∆x *σ) .
The decorrelation time is longer for
longer ∆t, but saturates at ∆t = 128 min.
We also fitted exponentials to autocorrelations of
flows to determine flow lifetimes in subregions.
Red & Blue: rank-order
autocorrelations of ux and
uy, in (32 x 32)-binned
subregions vs. lag time.
The the vertical range in
each cell is [-0.5, 1.0], with
a dashed black line at zero
correlation.
Only subregions with
median occupancy of at
least 20% of pixels above
our 15 G threshold were fit.
Background gray contours
show 50 G and 200 G levels
of |BLOS| in full-resolution
pixels. As expected, field
structures persist longer in
stronger-field regions.
Lifetimes of flows are also longer in subregions with
higher field strengths.
This trend is true for BLOS, Bz, and |B|.
This is consistent with the influence of
Lorentz forces active region flows.
Lorentz forces such as buoyancy (e.g.,
Parker 1957) and torques (e.g.,
Longcope & Welsch 2000) can be
longer-lived than convective effects.
Lifetimes for ux in subregions are
shown with red +’s, uy are blue x’s,
and fits to these are red and blue
solid lines.
The red ∆’s (green ∇’s) are lifetimes
of ux versus subregion-averaged
|Bz| (|B|) from the SP data, and
the red dashed (green solid) line is
a fit.
Black numerals
correspond to
log2(binsize), and
show that average
speed decreases
with increasing ∆t
and spatial scale (∆x
*σ).
Red and blue
numerals correspond
to curl and
divergences, which
behave similarly.
Flow lifetimes for ux
(+) and uy (x) as a
function of ∆t and
spatial scale (∆x*σ).
Power-laws were fit
over a limited range
of spatial scales for
each ∆t.
Averaged over all spatial
scales (∆x*σ) at a given ∆t,
speeds decrease with ∆t.
Fitted slope: -0.34
Error bars are standard
deviations in ∆t over all
spatial scales (∆x*σ).
Lifetimes of faster flows tend
to be shorter, for all spatial
scales.
For a given average speed
<s>, peak lifetime scales
approximately as <s>-2 .
Lifetimes of flow divergences and curls are also
longer in subregions with higher field strengths.
This trend is true for BLOS, Bz, and |B|, and the correlation is independent of the
number of tracked pixels in each subregion (the “occupancy”).
This is entirely consistent with Lorentz forces driving curls and divergences in
magnetized regions.
We also fitted power
laws to the lifetimes
of curls and
divergences as
function of spatial
scale for each value
of ∆t we used.
Lifetimes scale less
than linearly with
spatial scale.
Practical Conclusions, for tracking:
• Flow estimates with a given choice of tracking
parameters (∆t, ∆x, σ) are sensitive to flows on
particular length and time scales.
• Long-lived magnetic structures imply ∆t is less
constrained in tracking magnetograms than
intensities.
• It’s unwise to track with a ∆t that’s either too short
(noise dominated) or too long (displacement
dominated).
• Average speeds are lower for longer ∆t.
Scientific Conclusions:
• Flows operate over a range of length and time scales; the term
“the flow” is imprecise.
• Magnetic structures, flows, and curls/divergences are longerlived in stronger-field regions. This is consistent with both:
– magnetic fields inhibiting convection, and
– Lorentz forces driving photospheric flows.
• Flows with faster average speeds typically exhibit shorter peak
lifetimes. The product of mean speed squared and peak
lifetime is approximately constant, with units of a diffusion
coefficient, cm2/sec.