To quantify the action of solar storms we used a modification, recently proposed by some of us, of the so-called E model of photosynthesis (Rodríguez-López et al 2021):
$$\frac{\text{P}\left(\text{z}\right)}{{\text{P}}_{\text{s}}}=\frac{1-{\text{e}}^{-{\text{E}}_{\text{P}\text{A}\text{R}}\left(\text{z}\right)/{\text{E}}_{\text{s}}}}{1+{\text{f}}_{\text{i}\text{r}}\left(\text{z}\right)+{\text{E}}_{\text{U}\text{V}}^{\text{*}}\left(\text{z}\right)}$$
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where P is the photosynthesis rate at depth z, PS is the maximum possible photosynthesis rate, EPAR(z) is the irradiance of photosynthetically active radiation (PAR) at depth z, ES is a parameter accounting how efficiently the species uses PAR, E*UV(z) is the irradiance of ultraviolet radiation (UV), convolved with a biological action spectrum measuring how much each UV wavelength inhibits photosynthesis (the reason for the asterisk), and fir(z) is the function formally introduced by some of us in (Rodríguez-López et al 2021) to represent the influence of ionizing radiation. To account for the effects of UV on photosynthesis we used a biological action spectrum typical of temperate phytoplankton (Neale 2014, personal communication).
The irradiances of PAR and UV at sea level were calculated with the radiative transfer code Tropospheric Ultraviolet and Visible, developed at the National Centre for Atmospheric Research of USA, free for download (https://www2.acom.ucar.edu/modeling/tropospheric-ultraviolet-and-visible-tuv-radiation-model). It was assumed a solar zenital angle of 45 degrees (moderate radiational regime), an ozone column of 300 Dobson units, an ocean albedo of 0,065; a cloud layer between 4 and 5 km above sea level with an optical depth of 0,00 (clear sky conditions); aerosols with an optical depth of 0,235 and a single scattering albedo of 0,990. The radiation transfer model in the atmosphere was pseudo-spherical with two streams. The radiation transfer model in the ocean included Lambert-Beer’s law of Optics:
$$\text{E}\left({\lambda },\text{z}\right)=\text{E}\left({\lambda },{0}^{-}\right){\text{e}}^{-\text{K}\left({\lambda }\right).\text{z}}$$
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where E(λ,z) are the spectral irradiances at depth z, E(λ,0−) are the spectral irradiances just below the ocean surface, and K(λ) are the (wavelength-dependent) attenuation coefficients, which were taken from Jerlov’s reference tables (Jerlov 1976) and further interpolated according to (Peñate-Alvariño et al 2010). To get a wide range of potential responses, we used ocean optical types I and III, which are the clearest and darkest in Jerlov’s classification. For the same reason, calculations were also made for coastal waters C1 and C9 of above-mentioned classification. In a further study, we intend to include freshwater ecosystems.
It was assumed that solar storms can increase at ocean surface both the muon flux and their average energy up to 10% respect to ordinary conditions. However, these increments were first treated separately, in order to weigh their relative importance, and then were considered together. As in (Rodríguez-López et al 2018), the penetration of muons in the ocean was modeled through:
$$\text{I}\left(\text{z}\right)={\text{I}}_{0}{\text{e}}^{-\left({\rho }/\text{l}\right)\text{z}}$$
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where I0 and I(z) are the particle fluxes (m− 2) at ocean surface and at depth z, respectively; ρ is the density of water and l is a parameter measuring the penetrating efficiency of the particles of ionizing radiation (the bigger l, the more penetrating the particle). In this first modeling, it was not considered the disintegration of muons in their way down the water column, and it was assumed that the penetrating power depends linearly on their average energy < E>:
$$\text{l}=\text{n}⟨\text{E}⟩$$
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The average energy < ESS> of muons from solar storms can be written:
$$⟨{\text{E}}_{\text{S}\text{S}}⟩=\text{m}⟨\text{E}⟩$$
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where m is a proportionality constant. Thus, the penetrating power lSS of ‘’solar’’ muons can be stated as:
$${\text{l}}_{\text{S}\text{S}}=\text{n}⟨{\text{E}}_{\text{S}\text{S}}⟩=\text{n}\text{m}⟨\text{E}⟩=\text{m}\text{l}$$
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Following an ansatz formally analogous to the one used in (Atri and Melott 2011; Rodríguez-López, Cárdenas-Ortiz and Rodríguez-Hoyos 2013), we propose as the function of ionizing radiation:
$${\text{f}}_{\text{i}\text{r}}=\frac{{\text{I}}_{\text{S}\text{S}}\left(\text{z}\right)}{\text{I}\left(\text{z}\right)}=\frac{{\text{I}}_{0,\text{S}\text{S}}{\text{e}}^{-\left(\frac{{\rho }}{{\text{l}}_{\text{S}\text{S}}}\right)\text{z}}}{{\text{I}}_{0}{\text{e}}^{-\left(\frac{{\rho }}{\text{l}}\right)\text{z}}}$$
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where the subscript ss means the scenario of the solar storm. Applying Eq. (6) to (7) we get:
$${\text{f}}_{\text{i}\text{r}}=\frac{{\text{I}}_{0,\text{S}\text{S}}}{{\text{I}}_{0}}{\text{e}}^{\left\{-\left[(\text{m}-1)/\text{m}\right]\left(\frac{{\rho }}{\text{l}}\right)\text{z}\right\}}$$
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For our calculations we used l = 104 kg/m2, a typical value for muons from ordinary cosmic rays. We used three particular cases of Eq. (8). If there is only an increase in muon flux and average energy remains constant, it means m = 1 in Eq. (5), which implies the following form for the function of ionizing radiation:
$${\text{f}}_{\text{i}\text{r}}=\frac{{\text{I}}_{0,\text{S}\text{S}}}{{\text{I}}_{0}}$$
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Assuming a 10% of increase of the muon flux means fir(z) = 1,1. On another hand, if muon flux is constant and average energy increases in 10%, this means m = 1,1 in Eq. (5), so Eq. (8) results:
$${\text{f}}_{\text{i}\text{r}}={\text{e}}^{\left\{-\text{0,09}\left(\frac{{\rho }}{\text{l}}\right)\text{z}\right\}}$$
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The third case is an increase of 10% in both variables, implying:
$${\text{f}}_{\text{i}\text{r}}={\text{1,1}\text{e}}^{\left\{-\text{0,09}\left(\frac{{\rho }}{\text{l}}\right)\text{z}\right\}}$$
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