Stochastic Exit Problems with Applications to Geophysical and Atmospheric Problems.

• Grant number: 24/20960-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: May 01, 2025
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Applied Mathematics
• Principal Investigator: Breno Raphaldini Ferreira da Silva
• Grantee: Jhuann Piedro Alves Nogueira
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/16513-9 - Nonlinear dynamics, wave interactions and coherent structures in atmospheric and solar flows, AP.JP

Abstract

Several geophysical and atmospheric models present a bi-stable character: they remain stable for long times in a particular statefollowed by short bursts that will bring the system to a different state, then after some time a new burst will bring the system backto the previous state.Perhaps one of the most well known geophysical systems to present this type of behavior is the geomagnetic field. Thegeomagnetic field is dominated by a strong dipole component that remains closely aligned with the rotation axis at most times.Occasionally, the axial geomagnetic dipole reverses its polarity. The duration of the polarity intervals is highly variable, rangingfrom ¿ 10 a thousand years to ¿ 40 a million years with an average rate of ¿ 200 thousandyears. Several studies represent geomagnetic reversals with the dynamics of a particle in a bi-stable potential with random forcing. Recent studies indicate that observed statistics of polarity intervals departsfrom the exponential type distribution expected in a bistable potential with white noise forcing , whetherincluding memory effects or a periodic forcing due to Earth's orbital period can better represent the reversal statistics still needs further investigation.Another important example of bi-stable phenomena in geophysical sciences is that of glaciation cycles. One instance of wellstudied glacial transition is the Dansgaard-Oeschger events, which consist of rapid warming episodes that take place in theNorthern Hemisphere and that occur in cycles of approximately 1500 years. The statistics of theseevents is well represented by a bi-stable model driven by Levy alpha stable noise instead of white noise. The mechanism ofstochastic resonance was first proposed to explain strong periodicities in paleo climatic datasets. Bistablestochastic models are still extensively used in the context of glacial transitions.The idea of this project is to use this simple mathematical model as a way to explore aspects of observed geophysical time-series, such as the amplitude statistics and exit time statistics. To achieve these goals, a combination of analytic techniques, numerical and statistic methods will be learned. Including elements of stochastic differential equations, the derivation of theFokker-Planck equation, and its adjoint (Kolmogorov backward equations), the derivation of the equation for exit times as aboundary value problem, the Euler-Maruyama method for solving stochastic differential equation and finally, how to infer theparameters of the bistable model (for instance the form of the potential and the diffusion coefficient) from data. After that, thestatistics of exit times will be numerically obtained for a number of cases including periodic forcing, memory effects in the noiseterm forcing and Levy-type jumps in the noise forcing.