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The use of proper orthogonal decomposition (POD) to explore the complex fluidflows that are common in engineering applications is increasing and has yieldednew physical insights. However, for most engineering systems the dimension ofthe dynamics is expected to be very large yet the flow field data is availableonly for a finite time. In this context, it is important to establish theconvergence of the POD in order to accurately estimate such quantities as theKarhunen-Lo\`{e}ve dimension. Using direct numerical simulations ofRayleigh-B\'{e}nard convection in a finite cylindrical geometry we explore aregime exhibiting extensive chaos and demonstrate the consequences ofperforming a POD with a finite amount of data. In particular, we show that theconvergence in time of the eigenvalue spectrum, the eigenfunctions, and thedimension are very slow in comparison with the time scale of the convectionrolls and that the errors incurred by not using the asymptotic values can besignificant. We compute the dimension using two approaches, the method ofsnapshots and a Fourier method that exploits the azimuthal symmetry. We findthat the convergence rate of the Fourier method is vastly improved over themethod of snapshots. The dimension is found to be extensive as the system sizeis increased and for a dimension measurement that captures 90% of the variancein the data the Karhunen-Lo\`{e}ve dimension is about 20 times larger than theLyapunov dimension.

The consequences of finite-time proper orthogonal decomposition for an extensively chaotic flow field