We infer a flow field, u(x,y,) from magnetic evolution over a time interval, assuming: Ideality assumed:  t B n = -c(  x E), but E = -(v x B)/c, so perpendicular flows drive all evolution. Démoulin & Berger (2003) argue u = v h – (v z /B z ) B h, but Schuck (2006, 2008) argues u ≃ v h.

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 (u x, u y ). Thick: frame-to-frame autocorrelation, at 96 minute lag. Thin: initial-to-n th 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?

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 ΔB n 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, 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.

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 |B LOS | > 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 |B z | 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 2 n x2 n 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 B LOS, 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 |B LOS | in full-resolution pixels. As expect- ed, 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 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.

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 magneto- grams 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 u x and u y, 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 |B LOS | 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 B LOS, B z, 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 u x in subregions are shown with red +’s, u y are blue x’s, and fits to these are red and blue solid lines. The red ∆’s (green ∇ ’s) are lifetimes of u x versus subregion-averaged |B z | (|B|) from the SP data, and the red dashed (green solid) line is a fit.

Black numerals correspond to log 2 (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 u x (+) and u y (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: 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, peak lifetime scales approximately as -2.

Lifetimes of flow divergences and curls are also longer in subregions with higher field strengths. This trend is true for B LOS, B z, 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 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, cm 2 /sec.