In-depth Investigation of Heat Transfer Mechanisms in Enceladus’s Subglacial Ocean

Academic Background

Enceladus, one of Saturn’s icy moons, has tiger stripe fractures at its south pole that are identified as sources of both thermal emission and water vapor plumes. The existence of these heat anomalies suggests active hydrothermal processes beneath Enceladus’s ice shell. However, the mechanisms responsible for these heat anomalies remain largely unknown. About 60 years ago, geoscientist George Veronis proposed a model for cold water oceans and suggested a classical ¹⁄₃ scaling relationship between vertical heat transfer and the Rayleigh number (Ra), a dimensionless number representing the strength of buoyancy driving convection within the fluid body. This study delves into Veronis’ model of steady coherent rolls to validate this classical scaling and explore its application to Enceladus’s subglacial ocean.

Paper Source

This paper was co-authored by Zhen Ouyang, Qi Wang, Kai Li, Baole Wen, and Zijing Ding from the School of Energy Science and Engineering at Harbin Institute of Technology, the Department of Earth and Space Sciences at Southern University of Science and Technology, the Institute of Mechanics at the Chinese Academy of Sciences, and the Department of Mathematics at New York Institute of Technology. It was published in the Proceedings of the National Academy of Sciences (PNAS) on February 7, 2025.

Research Process and Results

Research Process

1. Model Construction
   The study is based on Veronis’ penetrative convection model, using the Boussinesq approximation where fluid density depends quadratically on temperature in the buoyancy term. The governing equations for velocity, pressure, and temperature were formulated through non-dimensionalization of a fluid layer of height h.

2. Numerical Simulations
   Two-dimensional (2D) and three-dimensional (3D) direct numerical simulations (DNS) were conducted to simulate penetrative convection under different Rayleigh numbers (Ra). Particular attention was paid to the formation and heat transfer characteristics of steady coherent rolls. Fourier spectral methods and finite difference methods on Chebyshev points were used for discretization, while pseudo-arclength continuation techniques tracked the solutions of steady coherent rolls.

3. Data Analysis
   The analysis focused on heat transfer properties under varying levels of stratification, examining how Nusselt number (Nu) and Reynolds number (Re) change with Ra. Flow fields and temperature distributions of steady coherent rolls were also visualized.

Main Results

1. Validation of Classical Scaling
   The study confirmed that the classical ¹⁄₃ scaling relationship holds as Ra approaches infinity, supporting Veronis’ hypothesis and validating it for the first time through numerical simulations.

2. Diversity of Steady Coherent Rolls
   For low stratification levels (<0.36), steady coherent rolls with fixed aspect ratios achieved the classical scaling. For high stratification levels (≥0.36), adjusting the roll aspect ratio to maximize heat transport was necessary to achieve the classical scaling.

3. Correlation Between Heat Transfer and Roll Structure
   A strong correlation was found between steady coherent rolls and turbulent structures. Singular value decomposition (SVD) analysis of DNS results showed that steady coherent rolls closely resemble optimal structures in turbulence, suggesting they can serve as an approximate model for turbulent heat transfer.

4. Heat Flux Prediction
   Based on the findings, the study predicted the heat flux and lateral roll sizes in Enceladus’s subglacial ocean. These predictions align well with measurements obtained by the Cassini spacecraft, further validating the applicability of the ¹⁄₃ scaling law at high Ra.

Conclusion and Significance

This research validates Veronis’ proposed ¹⁄₃ classical scaling for the first time through numerical simulations and highlights the critical role of steady coherent rolls in heat transfer. The results deepen our understanding of penetrative convection mechanisms and provide new theoretical support for explaining the heat anomalies in Enceladus’s south polar tiger stripes. Additionally, steady coherent rolls are shown to be an effective tool for studying turbulent heat transfer, offering new perspectives for future research.

Key Highlights

1. First Validation of Classical Scaling
   The study provides the first numerical validation of Veronis’ ¹⁄₃ scaling law, filling a theoretical gap in this field.

2. Diversity of Steady Coherent Rolls
   It reveals the diversity of steady coherent rolls under different stratification levels and proposes methods to maximize heat transfer by adjusting aspect ratios.

3. Consistency Between Predicted Heat Flux and Observations
   The predicted heat flux and roll sizes match well with data from the Cassini mission, confirming the reliability of the theoretical model.

4. High Correlation Between Steady Coherent Rolls and Turbulent Structures
   The study finds a high correlation between steady coherent rolls and turbulent structures, offering new insights into turbulent heat transfer.

Additional Valuable Information

The study also explored how the flow intensity (represented by Re) changes with Ra under different stratification levels, revealing distinct scaling relationships. This finding offers new clues for investigating the flow characteristics of penetrative convection. Moreover, upcoming missions like ESA’s JUICE and NASA’s Europa Clipper will provide crucial on-site data, enhancing our understanding of the dynamics beneath these icy shells.