  A linear response function (LRF) relates the mean-response of a nonlinear
system to weak external forcings and vice versa. Even for simple models of the
general circulation, such as the dry dynamical core, the LRF cannot be
calculated from first principles due to the lack of a complete theory for
eddy-mean flow feedbacks. According to the Fluctuation-Dissipation Theorem
(FDT), the LRF can be calculated using only the covariance and lag-covariance
matrices of the unforced system. However, efforts in calculating the LRFs for
GCMs using FDT have produced mixed results, and the reason(s) behind the poor
performance of the FDT remains unclear. In Part 1 of this study, the LRF of an
idealized GCM, the dry dynamical core with Held-Suarez physics, is accurately
calculated using Green's functions. In this paper (Part 2), the LRF of the same
model is computed using FDT, which is found to perform poorly for some of the
test cases. The accurate LRF of Part 1 is used with a linear stochastic
equation to show that dimension-reduction by projecting the data onto leading
EOFs, which is commonly used for FDT, can alone be a significant source of
error. Simplified equations and examples of 2 x 2 matrices are then used to
demonstrate that this error arises because of the non-normality of the
operator. These results suggest that errors caused by dimension-reduction are a
major, if not the main, contributor to the poor performance of the LRF
calculated using FDT, and that further investigations of dimension-reduction
strategies with a focus on non-normality are needed.
