Temporal memory of the solar dynamo: one-cycle precursor law versus persistent multi-cycle memory under Babcock–Leighton stochasticity

The solar activity cycle is believed to be driven by a flux-transport dynamo in which the poloidal magnetic field is regenerated near the surface by the Babcock–Leighton mechanism — the decay of tilted bipolar sunspot groups — whose stochastic element (random scatter in sunspot tilt angles and emergence statistics) may or may not erase the dynamo's memory of cycles before the last one. Jiang, Chatterjee & Choudhuri (2007) formulated this as the question of whether the precursor law relating the next cycle's maximum sunspot number to the polar field (dipole moment) at the preceding minimum, $(R_{max})_n = k(DM)_{n-1}$, is complete, or whether a residual multi-cycle memory requires a functional dependence on two or more preceding minima, $(R_{max})_n = f[(DM)_{n-1},(DM)_{n-2}]$. Later work gave a model-conditional answer — memory lasts about one cycle in diffusion-dominated dynamos but up to three cycles in advection-dominated ones — so the solar question reduces to establishing the real Sun's memory structure, its physical carrier (turbulent-diffusivity transport regime, meridional-circulation fluctuations, or additional deep poloidal sources), and an observational discrimination. This remains open: the deep turbulent diffusivity is unmeasured, direct polar-field records span only a few cycles, polar-field proxies remain partially inconsistent, and the observed long-term clustering of grand minima — extended, multi-decade episodes of strongly suppressed activity such as the Maunder minimum — stands in unresolved tension with purely one-cycle memory.

Background

The roughly 11-year sunspot cycle is the visible signature of the solar magnetic cycle. In the currently dominant flux-transport (Babcock–Leighton) dynamo picture, the toroidal (azimuthal) magnetic field is amplified by differential rotation in the tachocline — the shear layer at the base of the convection zone — and rises to the surface as tilted bipolar sunspot regions. The decay of these tilted regions regenerates the poloidal (north–south) field near the surface through the Babcock–Leighton mechanism; a large-scale meridional circulation (a poleward flow at the surface with a return flow at depth) transports this poloidal field poleward and then downward, closing the loop. A key parameter is the turbulent magnetic diffusivity inside the convection zone: if it is high (of order $10^{12}\ \mathrm{cm^2\ s^{-1}}$), poloidal field diffuses directly to the tachocline within a cycle; if it is low (of order $10^{10}\ \mathrm{cm^2\ s^{-1}}$), the field is advected by the circulation over about two cycles instead. Two competing calibrated models embodied these two regimes and produced opposite dynamo-based forecasts of cycle 24. Observations show that the Sun's polar field at a sunspot minimum — operationally, the dipole moment $DM$ — correlates strongly with the strength of the following cycle's maximum, while the polar field produced at a cycle's end does not correlate with that cycle's own strength. This asymmetry implies that the Babcock–Leighton step injects randomness (chiefly the observed scatter in sunspot group tilt angles), while the rest of the dynamo loop is comparatively ordered. Within a deterministic model, a single disturbance to the poloidal field persists for at least two cycles (Charbonneau & Dikpati 2000); whether the recurring random kicks erase this memory so that only the last minimum matters — the one-cycle precursor law $(R_{max})_n = k(DM)_{n-1}$, where $(R_{max})_n$ is the maximum sunspot number of cycle $n$, $(DM)_{n-1}$ the polar-field dipole moment at the minimum ending cycle $n-1$, and $k$ a proportionality constant, as proposed in polar-field precursor studies — or whether a residual dependence on earlier minima, $(R_{max})_n = f[(DM)_{n-1},(DM)_{n-2}]$, survives, is the memory question posed by Jiang, Chatterjee & Choudhuri (2007), who noted that the limited and noisy observational record then available could not decide between the two. The question is inseparable from two further limitations they identified: the value of the deep turbulent diffusivity (which sets the transport regime and hence the memory length) lacked direct observational validation, and the historical proxies of the polar field before routine direct measurements (polar faculae counts, filament-derived large-scale field measures, and different magnetograph data sets) disagreed with one another.

Problem Statement

Determine the temporal memory structure of the solar dynamo under realistic stochastic Babcock–Leighton forcing. Specifically: (a) establish whether the dependence of the next cycle's maximum amplitude on the preceding polar field is completely described by the one-cycle precursor law $(R_{max})_n = k(DM)_{n-1}$, or whether a statistically significant residual memory of earlier cycles exists, i.e. a functional dependence $(R_{max})_n = f[(DM)_{n-1},(DM)_{n-2},\ldots]$ — here $(R_{max})_n$ is the maximum sunspot number of cycle $n$, $(DM)_{n-1}$ the polar-field dipole moment at the minimum ending cycle $n-1$, and $k$ a proportionality constant; (b) identify the physical carrier and consequent memory length — in particular whether the deep-convection-zone turbulent diffusivity places the Sun in the diffusion-dominated regime (memory of roughly one cycle) or the advection-dominated regime (memory of up to about three cycles), and what role is played by fluctuations of the meridional circulation and by additional deep poloidal sources such as a tachocline $\alpha$-effect (regeneration of poloidal field from toroidal field by helical turbulence in the shear layer at the base of the convection zone); and (c) provide an observational discrimination between these alternatives, requiring internally calibrated polar-field records (direct measurements reconciled with historical proxies such as polar faculae counts and filament-derived large-scale field measures) robust enough to test one-cycle against multi-cycle dependence over the available cycle sequence. The single scientific objective is to fix the physical basis and the intrinsic prediction horizon of solar cycle forecasting: a complete answer must address all three parts at the strength stated above — (a) how far back the dynamo remembers, (b) through which mechanism, and (c) with what observational evidence — the last establishing, from internally calibrated polar-field records, how far one-cycle dependence can be discriminated from multi-cycle dependence.

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