Sunspot anchoring and longevity: what subsurface depth and formation history sustain week-long sunspot lifetimes?

Realistic radiative magnetohydrodynamic simulations of sunspots initiated by monolithic magnetic insertion into pre-existing convection (Rempel 2011, domains up to 16.4 Mm deep, up to 2 days evolution) reproduce penumbrae, Evershed flows, and moat flows, but decay on day time scales and leave open what keeps observed sunspots alive for weeks. The source paper extrapolated that week-long lifetimes require anchoring at roughly 30-50 Mm depth or formation through self-consistent magnetic flux emergence, both beyond its computational reach. Follow-up work has resolved several of the paper's secondary open issues - most notably the origin of the moat flow, shown to be independent of the Evershed flow - and has produced emergence-formed spots at 15-20 Mm domain depths and a whole-convection-zone flux-tube-rise calculation at reduced surface resolution. No published study, however, combines resolution fine enough to resolve penumbral filaments with domains of order 30 Mm or deeper, multi-week durations, or fully self-consistent emergence through such depths. The surviving open core is to determine the subsurface depth scale, magnetic structure, and formation history that anchor sunspots - deciding between deep anchoring and the dynamically disconnected shallow-spot picture - and to reproduce the observed emergence-stability-decay behavior of long-lived sunspots.

Background

A sunspot is a kilo-gauss-strength magnetic concentration in the solar photosphere, with a dark core (umbra) of about 4000 K surrounded by a filamentary, slightly brighter penumbra; typical spots carry magnetic flux of $10^{21}$-$10^{22}$ Mx (1 Mx is $10^{-8}$ Wb) and can persist for weeks. Two large-scale flows are associated with mature spots: the Evershed flow, a near-horizontal outflow of a few km/s along penumbral filaments at photospheric levels, and the moat flow, a slower outflow of a few hundred m/s extending to about two spot radii into the surrounding quiet Sun. Modern sunspot models solve the compressible magnetohydrodynamic equations coupled to a realistic equation of state (including partial ionization) and radiative transfer, so that the simulations develop a photosphere where optical depth $\tau$ (the integrated opacity; $\tau=1$ defines the visible surface) equals unity; depths are quoted in Mm (1 Mm = 1000 km). Before 2011 such radiative MHD simulations (e.g. with the MURaM code, a widely used solver for these equations) used domains shallower than about 10 Mm, where the bottom boundary condition strongly affected spot stability. Rempel (2011) extended these to 16.4 Mm deep domains covering 2 days, starting from an imposed monolithic magnetic structure inserted into relaxed convection. The simulations showed the subsurface field pushed into downflow lanes with substantial fragmentation, photospheric light bridges (bright lanes crossing the umbra) and pores (small flux concentrations without penumbrae) accompanying flux separation, a deep-reaching outflow (reaching about 50% of the convective rms velocity) around spots both with and without penumbrae, and slow spot decay on time scales of days. Because the effective life time of subsurface magnetic structures was found to be roughly 5-10 convective turnover times ($H_p/v_{\rm rms}$, the pressure scale height divided by the convective velocity), the longest convective time scales in a 16 Mm domain (about 30-60 hours) cannot sustain the week-long life times of observed spots. This is the anchoring problem: two classical resolutions are the cluster model, in which converging subsurface flows oppose spot decay, and deep anchoring close to the base of the convection zone, where convective turnover times reach 1-2 weeks or more. Rising magnetic flux tubes, however, suffer a tension-driven separation of the two polarities that is inconsistent with observed photospheric motions, which led Schüssler and Rempel (2005) to propose dynamical disconnection: during emergence the tube's field strength drops below the equipartition field strength (where magnetic and kinetic energy densities balance) over some depth range, so that the deep magnetic roots decouple from the photospheric flux concentration. Rempel (2011) left two questions about this scenario explicitly open: whether a flux tube can disconnect without destroying the coherence of the photospheric spot, and whether the resulting shallow (less than about 10 Mm deep) spot is sufficiently stable against convective erosion. The paper conjectured, from extrapolating decay rates with domain depth, that life times of about a week require depths of about 30-50 Mm, achievable only in deeper domains or with self-consistently emerging initial states. Subsequent work changed this landscape in parts: Rempel (2015) demonstrated in 18 Mm deep, 100-hour simulations that the moat flow is the photospheric component of a radially overturning flow driven by the reduced downflow filling factor (the smaller fraction of horizontal area occupied by cool downdrafts) around a spot - a mechanism independent of the penumbra and Evershed flow - and that suppressing submergence of horizontal magnetic field stabilizes the spot (near-stationary flux over about 4 days with penumbra, versus decay at about $10^{21}$ Mx/day for a penumbra-free spot). Emergence-based formation was achieved at moderate depth by Rempel and Cheung (2014, upper 15.5 Mm of the convection zone; spots assemble by coalescence and then decay on the domain's longest convective time scales) and by Chen, Rempel and Fan (2017), whose dynamo-driven emergence produces a mature spot anchored in a persistent downdraft lane with inflow below about 15 Mm and outflow above; Hotta and Iijima (2020) reached the entire convection zone but with 96 km surface grid spacing, too coarse to resolve penumbral filaments and Evershed flows, and only days of photospheric evolution. Observationally, Norton et al. (2017) measured emergence and decay rates of preceder and follower (leading and trailing bipolar-region) spots from HMI (Helioseismic and Magnetic Imager) data, providing the quantitative decay benchmark, and helioseismic inversions (helioseismology infers subsurface flows from the travel times of solar oscillations) have mapped only the shallow few-Mm moat-flow system (Švanda et al. 2014). What no study has established is the combination that the source problem demands: realistic near-surface physics (resolved penumbrae), sufficient depth (of order 30-50 Mm or a self-consistent emergence through such depths), and durations long enough to demonstrate or refute week-long spot coherence.

Problem Statement

Determine the subsurface depth scale, magnetic structure, and formation history required to keep a sunspot with realistic photospheric properties (umbra, penumbra, Evershed flow, moat flow) coherent for the week-long lifetimes observed on the Sun. A resolution must establish: (i) whether stability requires anchoring in layers of order 30-50 Mm depth or deeper - where convective turnover times approach the spot lifetime - or whether a shallow spot, less than about 10 Mm deep and dynamically disconnected from its magnetic roots in the sense of Schüssler and Rempel (2005), can remain coherent; (ii) the physical mechanism that stabilizes it against convective erosion (deep anchoring, converging subsurface flows, or suppression of magnetic-field submergence through a reduced downflow filling factor) and, correspondingly, the mechanism that ultimately erodes it; and (iii) flux decay rates consistent with those observed for sunspots of comparable magnetic flux. The primary expected route is realistic radiative MHD simulation with near-surface resolution sufficient to resolve penumbral filaments, evolved for multi-week durations in domains reaching at least the 30-50 Mm depth range or formed by self-consistent flux emergence through such depths; calibrated observational constraints on subsurface flows and fields beneath long-lived spots (e.g. from helioseismology) are an admissible complementary answer.

References

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