# Data Assimilation in Large-Prandtl Rayleigh-B\'enard Convection from Thermal Measurements

## TL;DR

This paper investigates a data assimilation method for large-Prandtl Rayleigh-Bénard convection using temperature observations, providing theoretical guarantees and numerical validation, with implications for geophysical and high-pressure gas applications.

## Contribution

It introduces a rigorous framework for data assimilation in large-Prandtl convection, identifying conditions for synchronization and validating them through simulations.

## Key findings

- Synchronization often occurs beyond theoretical conditions
- Conditions are practically relevant only at extremely large Prandtl numbers
- Numerical results confirm theoretical predictions

## Abstract

This work applies a continuous data assimilation scheme---a particular framework for reconciling sparse and potentially noisy observations to a mathematical model---to Rayleigh-B\'enard convection at infinite or large Prandtl numbers using only the temperature field as observables. These Prandtl numbers are applicable to the earth's mantle and to gases under high pressure. We rigorously identify conditions that guarantee synchronization between the observed system and the model, then confirm the applicability of these results via numerical simulations. Our numerical experiments show that the analytically derived conditions for synchronization are far from sharp; that is, synchronization often occurs even when the conditions of our theorems are not met. We also develop estimates on the convergence of an infinite Prandtl model to a large (but finite) Prandtl number generated set of observations. Numerical simulations in this hybrid setting indicate that the mathematically rigorous results are accurate, but of practical interest only for extremely large Prandtl numbers.