  The use of proper orthogonal decomposition (POD) to explore the complex fluid
flows that are common in engineering applications is increasing and has yielded
new physical insights. However, for most engineering systems the dimension of
the dynamics is expected to be very large yet the flow field data is available
only for a finite time. In this context, it is important to establish the
convergence of the POD in order to accurately estimate such quantities as the
Karhunen-Lo\`{e}ve dimension. Using direct numerical simulations of
Rayleigh-B\'{e}nard convection in a finite cylindrical geometry we explore a
regime exhibiting extensive chaos and demonstrate the consequences of
performing a POD with a finite amount of data. In particular, we show that the
convergence in time of the eigenvalue spectrum, the eigenfunctions, and the
dimension are very slow in comparison with the time scale of the convection
rolls and that the errors incurred by not using the asymptotic values can be
significant. We compute the dimension using two approaches, the method of
snapshots and a Fourier method that exploits the azimuthal symmetry. We find
that the convergence rate of the Fourier method is vastly improved over the
method of snapshots. The dimension is found to be extensive as the system size
is increased and for a dimension measurement that captures 90% of the variance
in the data the Karhunen-Lo\`{e}ve dimension is about 20 times larger than the
Lyapunov dimension.
