The large deviation principle for the stochastic 3D primitive equations with transport noise

Abstract

We prove the small-noise large deviation principle for the three-dimensional primitive equations with transport noise and turbulent pressure. Transport noise is important for geophysical fluid dynamics applications, as it takes into account the effect of small scales on the large scale dynamics. The main mathematical challenge is that we allow for the transport noise to act on the full horizontal velocity, therefore leading to a non-trivial turbulent pressure, which requires an involved analysis to obtain the necessary energy bounds. Both Stratonovich and Itô noise are treated.

Summary

This paper provides a rigorous proof of a Large Deviation Principle (LDP) for the 3D primitive equations with transport noise, which are fundamental in modeling ocean and atmosphere dynamics. A key advancement is handling transport noise acting directly on the full horizontal velocity, a mathematically challenging and physically relevant scenario not addressed in previous works.

Key Insights

• •The authors successfully prove an LDP for the 3D primitive equations with transport noise acting on the full horizontal velocity, overcoming mathematical challenges related to the lack of coercivity that often hinders such proofs.
• •The paper leverages a recent abstract LDP result tailored for L2 settings lacking coercivity, reformulating the stochastic primitive equations into a suitable form for application.
• •The rate function, which quantifies the likelihood of rare events, is characterized in terms of a control function and the solution to a deterministic 'skeleton' equation, providing a theoretical framework for understanding the probabilities of extreme events.

Practical Implications

• •This work lays the foundation for analyzing the impact of small-scale noise on large-scale geophysical flows, potentially leading to a better understanding of phenomena like the collapse of ocean currents.
• •Future research can focus on extending the LDP to more general noise structures and exploring the implications for specific geophysical phenomena, such as quantifying the probability of extreme weather events in climate models.
• •The demonstrated proof of an LDP without relying on a standard coercivity condition broadens the scope of applicability of LDP theory to a wider class of stochastic partial differential equations (SPDEs).

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