Modeling Non-Rotational Ocean Circulation and Heat Distribution in Icy Moons

Research Period
Sept 2024 – May 2025

Research Guidance
Guidance under Christopher Hill, Massachusetts Institute of Technology (MIT)

Hypothesis
Boundary conditions strongly influence the equilibrium temperature profile in a non rotational one dimensional icy moon heat transport model, leading to different predicted ice shell thicknesses.

Motivation
I am fascinated by how icy moons can sustain liquid water oceans in environments where very little sunlight reaches the surface. These oceans are kept warm by internal heating, but their long term habitability depends on how that heat is transported through the ocean and ice. I wanted to understand the physical processes that control ice shell thickness and thermal structure. This project also gave me the opportunity to work with a numerical modeling approach that I had not previously used in my research.

Research Breakdown
The research problem was deconstructed into several manageable tasks:

1. Model Formulation: I began by developing a one dimensional vertical heat transport model to represent an icy moon ocean beneath an ice shell. The ocean column was divided into discrete layers, each with its own temperature, allowing heat flow between adjacent layers to be tracked over time.
2. Heat Flux and Temperature Evolution: Heat transfer between layers was calculated using Fourier’s law, with temperature updates determined by the difference in flux entering and leaving each layer. The model accounts for layer thickness, density, and specific heat capacity using an explicit time stepping scheme.
3. Boundary Condition Testing: I explored multiple top and bottom boundary conditions to represent internal tidal or geothermal heating at the ocean floor and cooling into an overlying ice shell and space. Comparing these scenarios allowed me to examine how boundary assumptions influence the equilibrium temperature profile and predicted ice shell thickness.
4. Numerical Stability and Validation: Simplified analytical test cases were used to verify correct model behavior. I identified numerical instabilities at large time steps and constrained model parameters using a stability condition to ensure physically realistic temperature evolution.
5. Ice Layer Representation: The model was extended to include temperature dependent material properties, allowing thermal conductivity to change between liquid water and ice. This enabled tracking of the ice water interface and estimation of ice thickness as the system evolved toward equilibrium.
6. AI Assisted Model Verification: To independently verify model behavior, my advisor implemented a parallel version of the model using an AI assisted coding workflow. The AI was guided step by step to reproduce the governing equations, boundary conditions, and parameters defined in this project. Comparing results between implementations helped identify numerical issues and build confidence in the physical interpretation of the results.
   

Quantifiable Outcomes
1. Thermal Model Implementation: Developed a one dimensional numerical heat transport model that computes temperature evolution across a layered icy moon ocean column using physically motivated parameters, including thermal conductivity, density, and specific heat capacity.
2. Boundary Condition Sensitivity: Tested multiple top and bottom boundary condition configurations representing internal heating and surface cooling, demonstrating that predicted temperature profiles and ice shell thickness are highly sensitive to boundary assumptions.
3. Ice Thickness Estimation: Under one representative set of boundary conditions, the model predicted an ice fraction of approximately 40 percent by depth. This value is higher than commonly inferred for Europa or Enceladus, highlighting the tendency of simplified vertical models to overestimate ice thickness in the absence of lateral heat transport.
4. Numerical Stability Constraints: Identified numerical instabilities at large time steps and established stability limits based on model parameters. Applying these constraints ensured physically realistic temperature evolution and consistent convergence behavior.
5. Time Dependent Temperature Profiles: Generated time series of temperature evolution for each layer, enabling analysis of how the system approaches equilibrium and how thermal gradients evolve near the ice water boundary.

Skills Acquired

1. Numerical Heat Transport Modeling: Developed and analyzed a one dimensional diffusion based thermal model, gaining experience translating physical heat flow equations into stable numerical implementations.
2. Boundary Condition Analysis: Learned how different top and bottom boundary assumptions affect equilibrium temperature profiles and ice shell thickness in simplified planetary interior models.
3. Thermal Physics Interpretation: Built intuition for how material properties such as thermal conductivity, density, and specific heat influence temperature gradients across ice and liquid water.
4. Numerical Stability and Debugging: Identified and corrected numerical instabilities related to time step size and parameter choices, applying stability constraints to ensure physically meaningful results.
5. Model Verification and Comparison: Gained experience validating model behavior through independent implementations and comparing outputs to assess robustness and sensitivity.
   

Key Learnings

1. How to design a model based on the science question: Framing the model around a specific question, how boundary conditions influence heat flow and equilibrium in a non rotational icy moon ocean, guided choices about dimensionality, boundary conditions, and parameterization.
2. Analytical test cases are essential, even when outcomes are known: Setting all parameters to simple values and testing linear temperature profiles helped verify that the numerical scheme behaved correctly before exploring physically complex scenarios.
3. Stability constraints must be enforced explicitly: The model became unstable when the Courant–Friedrichs–Lewy condition was violated, reinforcing the importance of enforcing time step limits based on physical and numerical parameters.
4. Boundary conditions determine physical realism: Uniform or linear temperature profiles do not evolve meaningfully over time, motivating the use of nonlinear or custom boundary conditions to study heat flow and equilibrium in icy moon interiors.
5. Simple physical sanity checks catch failures early: Monitoring for non physical temperature ordering, such as interior layers becoming colder than the surface, proved critical for identifying numerical instabilities or incorrect equation implementation.
   
6. Initial and final temperature profiles from the icy-moon ocean model. The simulation begins from a simple linear temperature gradient and evolves under imposed boundary conditions and internal heat fluxes, producing a non-linear final profile. The dashed line indicates the depth-averaged temperature. The kink in the final state reflects the model’s response to boundary forcing rather than a fully equilibrated steady state.