Adv Comput Math DOI 10.1007/s10444-015-9417-0
A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation Omer San1 · Traian Iliescu2
Received: 15 January 2014 / Accepted: 13 April 2015 © Springer Science+Business Media New York 2015
Abstract In this paper, a stabilized proper orthogonal decomposition (POD) reduced-order model (ROM) framework is developed for the barotropic vorticity equation. Two different closure ideas are utilized in order to model truncated modes in the ROMs. We apply the POD-ROMs to mid-latitude simplified oceanic basins, which are standard prototypes of more realistic large-scale ocean dynamics. Two closure schemes are used to model the effects of the discarded POD modes: a mode dependent eddy viscosity closure model and a Smagorinsky-type model. A sensitivity analysis with respect to the free eddy viscosity stabilization parameter is performed for various POD-ROMs with different numbers of POD modes. The POD-ROM results are validated against the Munk layer resolving direct numerical simulations using a fully conservative fourth-order Arakawa scheme. A comparison with the standard Galerkin POD-ROM without any stabilization or closure is also included in our investigation. For a four-gyre ocean circulation problem, the new POD-ROM closure models show significant improvements in accuracy over the standard Galerkin model. This first step in the numerical assessment of the POD-ROMs shows that they could represent a computationally efficient tool for large scale oceanic simulations over long time intervals.
Communicated by: Editors of Special Issue on MoRePas Omer San
[email protected] 1
Interdisciplinary Center for Applied Mathematics, Virginia Tech Blacksburg, VA 24061, USA
2
Department of Mathematics, Virginia Tech Blacksburg, VA 24061, USA
O. San, T. Iliescu
Keywords Proper orthogonal decomposition · Reduced-order modeling · Stabilization · Eddy viscosity closure · Barotropic vorticity equations · Quasigeostrophic ocean model · Double-gyre wind forcing · Four-gyre ocean circulation Mathematic Subject Classifications (2010) 37N10 · 76M25 · 76F20 · 76D99
1 Introduction Proper orthogonal decomposition (POD) is one of the most successful successful reduced-order modeling techniques of complex systems. POD has been used to generate reduced-order models (ROMs) for the optimal control and analysis of many forced-dissipative nonlinear systems in science and engineering applications [11, 30, 35, 36, 43, 52]. POD extracts the most energetic modes, which are expected to contain the dominant characteristics of these systems. The POD and its variants are also known as Karhunen-Lo`eve expansions in signal processing and feature selection [26], principal component analysis in statistics [34, 72], and empirical orthogonal functions in atmospheric science [45]. The development of accurate and reliable low dimensional models is crucial in many complex systems, such as data assimilation in weather and climate modeling [12, 22, 23]. Reduced-order modeling of such problems, usually governed by a system of coupled nonlinear partial differential equations, typically consists of a basis selection strategy to build representative modes and then a projection step to build the lowdimensional model (e.g., see [6, 13, 18, 39]). The globally supported POD modes are often constructed empirically from a database obtained from a high fidelity numerical simulation of the governing equations and are problem dependent. These bases are then used to reduce the partial differential equations to a truncated system of amplitude equations using Galerkin projection [33]. It is possible to obtain good approximations with a few POD modes in which fine scale details are embedded. The resulting systems are low dimensional but dense and provide an efficient framework for real time analysis and control applications [8, 30]. Although the POD-ROM Galerkin method provides an efficient way to generate the reduced-order system (especially for fairly smooth systems in which the energetics can be characterized by the first few modes), its applicability to complex systems is limited mainly due to errors associated with the truncation of POD modes. To model the effects of the discarded modes, several closure modeling strategies have been proposed [7, 9, 38, 54, 70, 71]. Besides closure models, several successful interpolating procedures have been developed to make the ROMs robust with respect to parameter changes (e.g., see [1, 2]). Derivative-extended POD models have also been developed by incorporating sensitivity information to improve the POD basis functions [14, 60]. Novel algorithms have been developed to handle nonlinearities (e.g., see [16, 73] for discrete empirical interpolation method (DEIM) to address the reduction of the nonlinear components). Convergence and stability characteristics of the Galerkin reduced-order models have been discussed in [37] for compressible flows applications (see also [3]). As carefully explained in [69], for
