Riemannian Geometry of Nonlinear Deformation

The nonlinear (finite) deformation of flow is studied from the geometric point of view. First- and second-order covariant rates of deformation tensors are derived in the context of an arbitrary connection on frame bundles with torsion and Riemannian curvature. Also, the compatibility conditions of continua are extended to deformations on Einstein manifolds using a pullback Ricci curvature. We relate our compatibility conditions with existing formulations on Euclidean space.A particular interest in studying nonlinear deformation here is to develop evolution equations for the principle rates and directions of finite deformation, which are quantities that are widely used to understand the flow topology. To this end, we present a spectral decomposition of two-point tensors on Riemannian manifolds. We derive evolution equations for the spectral decomposition, namely for the eigenvalues and eigenvectors of deformation tensors on Riemannian manifolds in terms of arbitrary covariant rates and using intrinsic Lie derivatives. We demonstrate the analogy between the spectral decomposition of deformation and attitude kinematics of moving frames. Geometric numerical integration of the evolution equations is explored through Lie group actions, namely the Lie algebra of quaternions. Our formulation extends attitude kinematics of deformation with Euler angles on 2-manifolds with the Riemannian metric. Numerical and analytic examples for both Euclidean space and Hamiltonian flows on Kählerian (symplectic) manifolds are given.Accompanying our theoretical framework, a comprehensive high-performance computing platform for the Lagrangian analysis of flow is developed. The platform has been integrated into an online gateway with server-client architecture and available online as a community resource. Our platform is capable of processing up to billion-point grids and has been employed in real-time field experiments. We demonstrate practical applications of globe scale geophysical flows using comprehensive datasets