3.4 Towards a quantitative physical model

Several methods have been developed historically to convert measured cosmogenic-isotope data into a solar activity index, ranging from very simple regressions to physics-based models. A new step in long-term solar-activity reconstruction has been made recently, which is the development of the proxy method in which physics-based models are used, instead of a phenomenological regression, to link SN with cosmogenic-isotope production (Usoskin et al., 2003cJump To The Next Citation Point2007Jump To The Next Citation PointSolanki et al., 2004Jump To The Next Citation PointVonmoos et al., 2006Jump To The Next Citation PointMuscheler et al., 2007Jump To The Next Citation PointSteinhilber et al., 2008Jump To The Next Citation Point). Due to recent theoretical developments, it is now possible to construct a chain of physical models to simulate the entire relationship between solar activity and cosmogenic data. UpdateJump To The Next Update Information

The physics-based reconstruction of solar activity (in terms of sunspot numbers) from cosmogenic proxy data includes several steps:

Presently, all these steps can be completed using appropriate models. Although the uncertainties of the models may be considerable, the models allow a full basic quantitative reconstruction of solar activity in the past. However, much needs to be done, both theoretically and experimentally, to obtain an improved reconstruction.

3.4.1 Regression models

Mathematical regression is the most apparent and often used (even recently) method of solar-activity reconstruction from proxy data (see, e.g., Stuiver and Quay, 1980Jump To The Next Citation PointOgurtsov, 2004). The reconstruction of solar activity is performed in two consecutive steps. First, a phenomenological regression (either linear or nonlinear) is built between proxy data set and a direct solar-activity index for the available “training” period (e.g., since 1750 for WSN or since 1610 for GSN). Then this regression is extended backwards to evaluate SN from the proxy data. The main shortcoming of the regression method is that it depends on the time resolution and choice of the “training” period. The former is illustrated by Figure 10View Image, which shows the scatter plot of the 10Be concentration vs. GSN for the annual and 11-year smoothed data.

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Figure 10: Scatter plot of smoothed group sunspot numbers vs. (2-year delayed) 10Be concentration. a) Annual (connected small dots) and 11-year averaged (big open dots) values. b) Best-fit linear regressions between the annual (dashed line) and 11-year averaged values (solid line). The dots are the same as in panel (a). (After Usoskin and Kovaltsov, 2004Jump To The Next Citation Point).

One can see that the slope of the 10Be-vs-GSN relation (about –500 g/atom) within individual cycles is significantly different from the slope of the long-term relation (about –100 g/atom, i.e., individual cycles do not lie on the line of the 11-year averaged cycles. Therefore, a formal regression built using the annual data for 1610 – 1985 yields a much stronger GSN-vs-10Be dependence than for the cycle-averaged data (see Figure 10View Imageb), leading to a potentially-erroneous evaluation of the sunspot number from the 10Be proxy data.

It is equally dangerous to evaluate other solar/heliospheric/terrestrial indices from sunspot numbers, by extrapolating an empirical relation obtained for the last few decades back in time. This is because the last few decades (after the 1950s), which are well covered by direct observations of solar, terrestrial and heliospheric parameters, correspond to a very high level of solar activity. After a steep rise in activity level between the late 19th and mid 20th centuries, the activity remained at a roughly constant high level, being totally dominated by the 11-year cycle without an indication of long-term trends. Accordingly, all empirical relations built based on data for this period are focused on the 11-year variability and can overlook possible long-term trends (Mursula et al., 2003). This may affect all regression-based reconstructions, whose results cannot be independently (directly or indirectly) tested. In particular, this may be related to solar irradiance reconstructions, which are often based on regression-like models, built and verified using data for the last three solar cycles, when there was no strong trend in solar activity.

As an example let us consider an attempt (Belov et al., 2006Jump To The Next Citation Point) to reconstruct cosmic-ray intensity since 1610 from sunspot numbers using a (nonlinear) regression. The regression between the count rate of a neutron monitor and sunspot numbers, established for the last 30 years, yields an agreement at a 95% level for the period 1976 – 2003. Based on that, Belov et al. (2006Jump To The Next Citation Point) extrapolated the regression back in time to produce a reconstruction of cosmic-ray intensity (quantified in NM count rate) to 1560 (see Figure 11View Image). One can see that there is no notable long-term trend in the reconstruction, and the fact that all CR maxima essentially lie at the same level, from the Maunder minimum to modern times, is noteworthy. It would be difficult to dispute such a result if there was no direct test for CR levels in the past. Independent reconstructions based on cosmogenic isotopes or theoretical considerations (e.g., Usoskin et al., 2002bJump To The Next Citation PointScherer et al., 2004Scherer and Fichtner, 2004) provide clear evidence that cosmic-ray intensity was essentially higher during the Maunder minimum than nowadays. This example shows how easy it is to overlook an essential feature in a reconstruction based on a regression extrapolated far beyond the period it is based on. Fortunately, for this particular case we do have independent information that can prevent us from making big errors. In many other cases, however, such information does not exist (e.g., for total or spectral solar irradiance), and those who make such unverifiable reconstructions should be careful about the validity of their models beyond the range of the established relations.

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Figure 11: An unsuccessful attempt at the reconstruction of cosmic-ray intensity in the past using a regression with sunspot numbers (After Belov et al., 2006). Dots represent the observed cosmic-ray intensity since 1951. Note the absence of a long-term trend.

3.4.2 Reconstruction of heliospheric parameters

The modulation potential ϕ (see Section 3.1.1) is most directly related to cosmogenic isotope production in the atmosphere. It is a parameter describing the spectrum of galactic cosmic rays (see the definition and full description of this index in Usoskin et al., 2005aJump To The Next Citation Point) and is sometimes used as a stand-alone index of solar (or, actually, heliospheric) activity. We note that, provided the isotope production rate Q is estimated and geomagnetic changes can be properly accounted for, it is straightforward to obtain a time series of the modulation potential, using, e.g., the relation shown in Figure 7View Image. Several reconstructions of modulation potential for the last few centuries are shown in Figure 12View Image. While being quite consistent in the relative changes, they differ in the absolute level and fine details. Reconstructions of solar activity often end at this point, representing solar activity by the modulation potential, as some authors (e.g., Beer et al., 2003Jump To The Next Citation PointVonmoos et al., 2006Jump To The Next Citation PointMuscheler et al., 2007Jump To The Next Citation Point) believe that further steps (see Section 3.4.3) may introduce additional uncertainties. However, since ϕ is a heliospheric, rather than solar, index, the same uncertainties remain when using it as an index of solar activity. Moreover, the modulation potential is a model-dependent quantity (see discussion in Section 3.1.1) and therefore does not provide an unambiguous measure of heliospheric activity. In addition, the modulation potential is not a physical index but rather a formal fitting parameter to describe the GCR spectrum near Earth (Usoskin et al., 2005aJump To The Next Citation Point) and, thus, does not seem to be a universal solar-activity index.

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Figure 12: Several reconstructions of the decade-averaged modulation potential ϕ for the last few centuries: from sunspot numbers (SN(U02) – Usoskin et al., 2002bJump To The Next Citation Point), from 14C data (14C(S04), 14C(M05), 14C(M07) – Solanki et al., 2004Jump To The Next Citation PointMuscheler et al., 2005Jump To The Next Citation Point2007Jump To The Next Citation Point, respectively), from Antarctic 10Be data (10Be(U03), 10Be(MC04) – Usoskin et al., 2003cJump To The Next Citation PointMcCracken et al., 2004, respectively). The thick black NM curve is based on direct cosmic-ray measurements by neutron monitors since 1951 (Usoskin et al., 2005a) and ionization chambers since 1936 (McCracken and Beer, 2007Jump To The Next Citation Point).

Modulation of GCR in the heliosphere (see Section 3.1.1) is mostly defined by the turbulent heliospheric magnetic field (HMF), which ultimately originates from the sun and is thus related to solar activity. It has been shown, using a theoretical model of the heliospheric transport of cosmic rays (e.g., Usoskin et al., 2002b), that on the long-term scale (beyond the 11-year solar cycle) the modulation potential ϕ is closely related to the open solar magnetic flux Fo, which is a physical quantity describing the solar magnetic variability (e.g., Solanki et al., 2000Jump To The Next Citation PointKrivova et al., 2007Jump To The Next Citation Point).

Sometimes, instead of the open magnetic flux, the mean HMF intensity at Earth orbit, B, is used as a heliospheric index (Caballero-Lopez and Moraal, 2004Jump To The Next Citation PointMcCracken, 2007Jump To The Next Citation Point). Note that B is linearly related to Fo assuming constant solar-wind speed, which is valid on long-term scales. An example of HMF reconstruction for the last 600 years is shown in Figure 13View Image. In addition, the count rate of a “pseudo” neutron monitor (i.e., a count rate of a neutron monitor if it was operated in the past) is considered as a solar/heliospheric index (e.g., Beer, 2000Jump To The Next Citation PointMcCracken and Beer, 2007Jump To The Next Citation Point).

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Figure 13: An example of reconstruction of the heliospheric magnetic field at Earth orbit for the last 600 years from 10Be data (McCracken, 2007Jump To The Next Citation Point).

3.4.3 A link to sunspot numbers

The open solar magnetic flux Fo described above is related to the solar surface magnetic phenomena such as sunspots or faculae. Modern physics-based models allow one to calculate the open solar magnetic flux from data of solar observation, in particular sunspots (Solanki et al., 2000Jump To The Next Citation Point2002Krivova et al., 2007Jump To The Next Citation Point). Besides the solar active regions, the model includes ephemeral regions. Although this model is based on physical principals, it contains one adjustable parameter, the decay time of the open flux, which cannot be measured or theoretically calculated and has to be found by means of fitting the model to data. This free parameter has been determined by requiring the model output to reproduce the best available data sets for the last 30 years with the help of a genetic algorithm. Inversion of the model, i.e., the computation of sunspot numbers for given Fo values is formally a straightforward solution of a system of linear differential equations, however, the presence of noise in the real data makes it only possible in a numerical-statistical way (see, e.g., Usoskin et al., 2004Jump To The Next Citation Point2007Jump To The Next Citation Point). By inverting this model one can compute the sunspot-number series corresponding to the reconstructed open flux, thus forging the final link in a chain quantitatively connecting solar activity to the measured cosmogenic isotope abundance. A sunspot-number series reconstructed for the Holocene using 14C isotope data is shown in Figure 14View Image. While the definition of the grand minima (Section 4.3) is virtually insensitive to the uncertainties of paleomagnetic data, the definition of grand maxima depends on the paleomagnetic model used (Usoskin et al., 2007Jump To The Next Citation Point). Since the Y00 paleomagnetic model forms an upper bound for the true geomagnetic strength (Section 3.1.2), the corresponding solar-activity reconstructions may underestimate the solar-activity level. Accordingly, the grand maxima defined using the Y00 model are robust and can be regarded as “maximum maximorum” (see Section 4.4).

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Figure 14: Long-term sunspot-number reconstruction from 14C data (after Usoskin et al., 2007Jump To The Next Citation Point). All data are decade averages. Solid (denoted as ‘Y00’) and grey (‘K05’) curves are based on the paleo-geomagnetic reconstructions of Yang et al. (2000Jump To The Next Citation Point) and Korte and Constable (2005Jump To The Next Citation Point), respectively. Observed group sunspot numbers (Hoyt and Schatten, 1998) are shown after 1610.

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