We infer a flow field, u ( x,y ,) from magnetic evolution over a time interval, assuming:
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Transcript of We infer a flow field, u ( x,y ,) from magnetic evolution over a time interval, assuming:
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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 .
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Motivation: What is the optimal Δt? What can we learn from the coherence time of flows?
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.
Welsch et al. (2009) autocorrelated active region flows in MDI magnetograms and found flow lifetimes of ~6 hours on super-granular scales (c. 15 Mm). But what about other, smaller scales?
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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.
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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 co-registration 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.
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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.
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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.
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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.
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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.
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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 expect-ed, field structures persist longer in stronger-field regions.
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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.
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For short ∆t, frame-to-frame flow correlations can increase with increasing ∆t’s, and averaging prior to tracking.
Dashed lines show frame-to-frame correlation coeffs for un-averaged magnetograms.
Solid lines show correlations for averaged magnetograms.
Note complete lack of correlation at ∆t = 2 min.
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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.
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Flows are much more consistent from one frame to the next for longer ∆t, and averaging magneto-grams prior to tracking.
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Flow lifetimes can be estimated via autocorrelation of flow maps.
Flows decorrelate on longer timescales when a larger apodization window, σ, is used.
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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.
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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.
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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.
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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.
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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.
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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 .
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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.
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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.
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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.
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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 longer-lived 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.