Mathematical Study of Certain Geophysical Problems: Data Assimilation Algorithms and Mathematical Analysis of the Primitive Equations
Abstract
Abstract: The viscous three-dimensional primitive equations of oceanic and atmospheric dynamics are known to be globally well-posed. However, in the absence of rotation, it has been recently established that certain solutions of the inviscid primitive equations develop singularity in finite time. The first part of this project focuses on studying the effect of fast rotation on the life-span of solutions of the two- and three-dimensional inviscid primitive equations. It is expected, based on our experience with the fast rotating 2D Burgers equations and the 3D Euler equations, that fast rotation will stabilize the solutions of the inviscid primitive equations and will prolong their life-span. Moreover, it is also proposed to study the anisotropic viscous primitive equations with moisture, and to give a rigorous justification for the derivation of the primitive equations for the relevant temporal and spatial scales. The second part of this project aims at capitalizing on, and extending, a recently established mathematical framework that facilitate for the recovery of fine-scale information for time histories of spatial coarse-scale data, and for systematic, rigorous, and efficient downscaling and assimilation of coarse-scale observations. The framework is based on suitable determining projections and interpolation operators (that represent the observed data), and on the implementation of these operators in mathematical algorithms designed for recovering the fine-scales associated with these observations. Preliminary results implementing this framework have been demonstrated for the two-dimensional Navier-Stokes equations, both concerning the validity, efficiency, and optimal use of the coarse-data. This project specifically aims at (1) generalizing the application of this algorithm to various systems including buoyancy-driven flow, the 3D viscous planetary geostrophic models of oceanic circulation, and the primitive equations of oceanic and atmospheric sciences (with or without moisture); (2) improving these algorithm by reducing the the number of observed/ measured physical quantities, in particular to design data assimilation algorithms for recovering the solutions (the temperature and the velocity field) of the viscous 3D planetary geostrophic models from coarse-spatial measurements of the temperature alone; (3) conducting detailed computational studies to assess the validity and performance of the algorithms, in particular in the presence of stochastic errors in the measurements. The third part of this project is concerning the development of numerical schemes for computing statistical stationary solutions (probability invariant measures) of the Navier-Stokes equations, and various oceanic and atmospheric dynamics models. This will lead to a novel idea for computing reliable statistics of turbulent flows. In particular, we propose to employ ideas from the data assimilation algorithms, described above, to enhance the practicality of these numerical schemes for computing statistical solutions and statistical properties of turbulent flows and geophysical models. i
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- Aug 12, 2016
- Source ID
- N000141512333
Entities
People
- Edris Titi
Organizations
- Office of Naval Research
- Texas A&M University
- United States Navy