Variational Derivation of the Weak Form for Viscous Flows Involving Free Surfaces with Implications for Geodynamic Simulations
Abstract
Geodynamic simulations involve solving the Stokes equations, where boundary conditions play a crucial role in governing internal dynamics and surface topography evolution. A common approach is to model the interface in contact with air as a free surface. This interface is stress-free and deforms in response to internal force balances, which is essential for understanding processes ranging from rifted margins to subduction. However, a free surface can introduce instabilities and oscillations in the numerical solution, commonly known as the "drunken sailor" instability, forcing the use of small time steps. Geodynamic models therefore often employ stabilization terms based on the gravitational effect of free-surface motion. To date, these terms have been derived primarily on a heuristic basis. Here, we apply variational calculus to the energy-minimization formulation of incompressible Stokes flow. Using the Gateaux derivative, we derive the complete weak form for a deforming free-surface domain and show that the stabilization terms arise naturally from the free-surface assumption. We consider the linear-viscosity case and identify two boundary-integral contributions: one associated with internal viscous deformation and the other with gravitational body force. Two benchmark cases demonstrate the consistency of the derived formulation with previously proposed stabilization approaches. Our results provide a mathematical foundation for the commonly used gravity stabilization and identify an additional viscous boundary contribution.
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BibTeX
@misc{moghanjoghi2026variational,
title = {{Variational Derivation of the Weak Form for Viscous Flows Involving Free Surfaces with Implications for Geodynamic Simulations}},
author = {Karim Norouzi Moghanjoghi and Javier Garcia Pintado and Daniel R. Shapero and Marta Perez-Gussinye},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12152},
eprint = {2610.12152},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12152},
}
RIS
TY - UNPB
TI - Variational Derivation of the Weak Form for Viscous Flows Involving Free Surfaces with Implications for Geodynamic Simulations
AU - Karim Norouzi Moghanjoghi
AU - Javier Garcia Pintado
AU - Daniel R. Shapero
AU - Marta Perez-Gussinye
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2610.12152
UR - https://arxiv.org/abs/2610.12152
AB - Geodynamic simulations involve solving the Stokes equations, where boundary conditions play a crucial role in governing internal dynamics and surface topography evolution. A common approach is to model the interface in contact with air as a free surface. This interface is stress-free and deforms in response to internal force balances, which is essential for understanding processes ranging from rifted margins to subduction. However, a free surface can introduce instabilities and oscillations in the numerical solution, commonly known as the "drunken sailor" instability, forcing the use of small time steps. Geodynamic models therefore often employ stabilization terms based on the gravitational effect of free-surface motion. To date, these terms have been derived primarily on a heuristic basis. Here, we apply variational calculus to the energy-minimization formulation of incompressible Stokes flow. Using the Gateaux derivative, we derive the complete weak form for a deforming free-surface domain and show that the stabilization terms arise naturally from the free-surface assumption. We consider the linear-viscosity case and identify two boundary-integral contributions: one associated with internal viscous deformation and the other with gravitational body force. Two benchmark cases demonstrate the consistency of the derived formulation with previously proposed stabilization approaches. Our results provide a mathematical foundation for the commonly used gravity stabilization and identify an additional viscous boundary contribution.
ER -