发表机构
University of Bremen; University of Washington（不来梅大学; 华盛顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过变分法推导含自由表面粘性流动的弱形式，明确了重力稳定的数学基础及额外粘性边界贡献，经基准案例验证其与现有稳定方法一致，可支撑地球动力学模拟。
AI 中文摘要
地球动力学模拟需要求解斯托克斯方程，边界条件对控制内部动力学和地表地形演化至关重要。常用方法是将与空气接触的界面建模为自由表面，该界面无应力且会因内部力平衡发生变形，这对理解从裂谷边缘到俯冲带的各类过程至关重要。然而，自由表面会在数值解中引入不稳定性和振荡，通常称为“醉汉水手”不稳定性，迫使使用小时间步长。因此，地球动力学模型常采用基于自由表面运动引力效应的稳定项。迄今为止，这些项主要基于启发式方法推导。本文将变分法应用于不可压缩斯托克斯流的能量最小化公式，利用Gateaux导数推导了变形自由表面域的完整弱形式，并表明稳定项自然产生于自由表面假设。本文考虑线性粘度情况，确定了两个边界积分贡献：一个与内部粘性变形相关，另一个与引力体力相关。两个基准案例证明了推导的公式与先前提出的稳定方法的一致性。本文结果为常用的重力稳定提供了数学基础，并确定了额外的粘性边界贡献。
英文摘要
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.
