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# Improvements of a numerical continuation code for ocean circulation problems

Levine, R.C. (2001) Improvements of a numerical continuation code for ocean circulation problems. Master's Thesis / Essay, Mathematics.

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## Abstract

This report considers the numerical efficiency of part of a model for three-dimensional ocean circulation problems, which was developed by the oceanography group of Utrecht University. The main objective of oceanography research is to predict and explain climate change. The recent change in climate, global warming, is mainly accounted to the increased concentration of CO2 gases in the atmosphere, which are caused by human emissions. In order to lay a direct link between global warming and the increased CO2 concentration, natural causes must also be examined. The climate system consists of the atmosphere, the hydrosphere (oceans) and the cryosphere (permanent ice). So natural variations in the state of these three elements are examined. The state of the ocean can change significantly due to perturbations in the external forcings of the climate system. Examples of these forcings are the radiation received from the sun, wind stress and fresh water fluxes. The atmosphere responds relatively fast to perturbations. However the response times in the hydrosphere differ greatly. Oceans contain two basic circulation systems. The first is the wind-driven surface circulation. The second is the density-driven circulation, or thermohaline circulation, which is mainly controlled by differences in temperature and salt content. The wind-driven circulation responds faster to perturbations in the external forcings than the thermohaline circulation. Therefore response times in oceans can vary from several months to thousands of years, so the oceans play a major role in climate change occurring on longer time scales. The current research is focussed on the thermohaline circulation, the model that is used is described in [1]. The model describes the ocean circulations by a time-dependent coupled system of partial differential equations: three momentum equations, a continuity equation and equations for heat and salt transport. The state variables are the velocities in three dimensions (u, v, w), pressure p, temperature T and salinity S. To study the behaviour of the flow patterns, u(t) = (u, v, w,p, T, 5), steady states of the system are computed. Steady states are states where the system is at rest, they are defined by solutions = 0. This condition leads to solving a nonlinear system of equations. Once a steady state has been computed, its stability can be determined by solving an eigenvalue problem.The response of the ocean circulation to perturbations in the external forcings is now examined by a dynamical systems theory approach. This is done by varying the ocean model parameters, which leads to qualitative changes to the state of the system, and monitoring the steady states of the system. This dependence of the steady states on the model parameters is studied using a continuation code that is part of the ocean model. The continuation code eventually leads to solving large systems of linear equations. This is done using the MRILU solver, a linear systems solver for unstructured grids that was developed by the numerical mathematics group of Groningen University. This solver makes results for sufficiently high spatial resolution possible. The MRILU solver consists of an incomplete LU factorization as preconditioner, combined with an iterative solver. The continuation code is described in more detail in chapter 2, and the MRILU solver is described shortly in chapter 3.The objective of this report is to improve the continuation code. First the cost is reduced of the method for solving the nonlinear system of equations in the continuation code. This is done in two ways: (i) by tuning the accuracy with which the solution is computed, (ii) by implementing a simple, cheaper method for solving the nonlinear system. Finally part of the continuation code is implemented in parallel on the TERAS, the new Dutch national supercomputer. The aim is to obtain a speedup by parallelizing the most expensive components of the MRILU solver: matrix multiplications.

Item Type: Thesis (Master's Thesis / Essay) Mathematics Master's Thesis / Essay English 15 Feb 2018 07:29 15 Feb 2018 07:29 https://fse.studenttheses.ub.rug.nl/id/eprint/8867

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