Fickian Diffusion and Newtonian Cooling: A Concept for Noise Induced Climate Variability With Long-Term Memory?

Observed near surface air and soil temperature time series reveal a long-term memory, which is associated with a power-law scaling of the frequency spectra, S(ω) ~ ω - β with β ~ 0.6, lying between white and flicker noise, 0 < β < 1. As this power law scaling is not consistent with the Brownian motion concept of climate variability, Fickian diffusion is added to a Newtonian cooling relaxation to provide a more suitable analog of climatic fluctuations: (i) Diffusive plus random heat fluxes parametrise the turbulent mixing by synoptic scale eddy life cycles, affect tropospheric and near surface temperatures and excite a long-term memory regime with a β ~ 0.5 scaling. (ii) Newtonian cooling describes the near surface temperatures relaxing towards a global mean deep soil temperature and stabilises the system to a white noise response at very low frequencies. The long-term memory regime emerges from the high frequency scaling (β ~ 1.5), once temperatures become correlated in space due to diffusion, so that spatially averaged fluctuations correlate for times beyond the diffusion time scale. The long-term memory regime disappears into a white noise plateau (β ~ 0), when low frequencies exceed the damping time scale of Newtonian cooling. This system may be interpreted as a diffusive system relaxing towards the deep soil restoration temperature with an almost infinitely large time scale.