A transform method is developed for the fast calculation of vector-coupled sums appearing in the spectrally truncated vorticity equation. The method involves a kind of finite convolution theorem for expansions in surface harmonics. The method succeeds because it turns out to be much faster to transform the spectral representation to physical space, multiply the physical-space functions, and then inverse transform back to the spectral representation, than to evaluate the vector-coupled sums directly in spectral form. Direct evaluation of the sums requires order N5 operations when the spectral representation is truncated at surface harmonies of degree N, while the transform method requires about 10N3+40N2 log2N real operations per time step. Direct evaluation of the sums also requires storage of order N5, while the transform method requires storage of roughly 3N3 real words. An explanation is also given how to specialize the transform method to the “hemispheric” models of Baer and Platzman with a s... Abstract A transform method is developed for the fast calculation of vector-coupled sums appearing in the spectrally truncated vorticity equation. The method involves a kind of finite convolution theorem for expansions in surface harmonics. The method succeeds because it turns out to be much faster to transform the spectral representation to physical space, multiply the physical-space functions, and then inverse transform back to the spectral representation, than to evaluate the vector-coupled sums directly in spectral form. Direct evaluation of the sums requires order N5 operations when the spectral representation is truncated at surface harmonies of degree N, while the transform method requires about 10N3+40N2 log2N real operations per time step. Direct evaluation of the sums also requires storage of order N5, while the transform method requires storage of roughly 3N3 real words. An explanation is also given how to specialize the transform method to the “hemispheric” models of Baer and Platzman with a s...