Abstract
The dust aerosols floating in the atmosphere of Mars cause an attenuation of the solar radiation traversing the atmosphere that cannot be modeled through the use of classical diffusion processes. However, the definition of a type of fractional diffusion equation offers a more accurate model for this dynamic and the second order moment of this equation allows one to establish a connection between the fractional equation and the Ångstrom law that models the attenuation of the solar radiation. In this work we consider both one and three dimensional wavelength-fractional diffusion equations, and we obtain the analytical solutions and numerical methods using two different approaches of the fractional derivative.
Highlights
The scattering of solar radiation by the dust particles floating in the atmosphere is a relevant phenomenon in the study of Mars’ atmosphere [1,2]
The attenuation of the solar radiation traversing the atmosphere is modeled by the Lambert-Beer-Bouguer law, which establishes that the direct solar irradiance F (λ) at the Mars’ surface at wavelength λ is given by
The last component, τa (λ), depends directly on the medium and its value can be obtained by direct solar spectral irradiance measurements
Summary
The scattering of solar radiation by the dust particles floating in the atmosphere is a relevant phenomenon in the study of Mars’ atmosphere [1,2]. Radiative transfer analysis for Mars atmosphere [2] gives the values α = 1.2 and β = 0.3, corresponding to an aerosol optical thickness τa = 0.6 which cannot be accounted for with a model such as (17) In this context where the classical model of the process does not provide a good enough description of this kind of dynamics, the use of Fractional Calculus could provide an adequate description and simulation of the complex systems and processes observed [4,5,6]. The Fractional Calculus offers new scenarios of modeling to describe physical phenomena, like the dynamic of the Martian atmosphere In this manuscript, we continue the work started in [7] and we define both one and three dimensional wavelength-fractional diffusion equations to obtain a more accurate model of the attenuation of the solar radiation traversing the atmosphere.
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