Solving quasi-geostrophic equations and predicting vorticity in a chaotic regime

Vorticity is the rotation of individual particles in a flow. Vorticities in the ocean and the atmosphere play an important role in the change of the weather. A model describing the vorticity under influence of the Coriolis force due to the rotating earth is given by the quasigeostrophic equations (QGE’s). These QGE’s are differential equations describing the change of the vorticity at a particular point in a grid. The grid consists of a finite number of elements in the plane considered. To solve these QGE’s, both the values of the vorticity and the streamfunction at all the points in the grid have to be known and are therefore stored in matrices of which the elements represent the values at the different gridpoints. With the ability to solve the QGE’s, simulations can be performed for the evolution of the vorticity and predictions can be made. If the vorticity is measured in a physical system, then the values of the variables in the system (viscosity, Coriolis parameter and gain with which the forcing is multiplied) are unknown. Using the measurement data of a restricted number of measurement points in a grid, the unknown values of the variables and the initial conditions for the prediction can be determined by minimizing a cost function, which is a value for the difference between the values of the vorticity resulting from the model and the real measured values. A potentially chaotic system is considered, for which estimating the correct values of the variables and initial conditions is crucial since in chaotic systems perturbations in initial conditions will diverge, resulting in bad estimations. For systems that behave chaotically, finding the correct values of the unknown variables and initial states is much harder than for ordinary nonlinear systems. Performing measurements at four or more points in a grid that consists of sixteen points, the obtained values of the estimated unknown variables and initial conditions are sufficient close to the real values, resulting in accurate predictions for 600 seconds. The Lyapunov exponent is a measure for the speed at which a perturbation of initial condition in a system grows, a positive value means that the perturbation in initial conditions will diverge away from the normal trajectory. In this way, the exponents can be used to calculate how long predictions are accurate