  A new technique to derive delay models from systems of partial differential
equations, based on the Mori-Zwanzig formalism, is used to derive a delay
difference equation model for the Atlantic Multidecadal Oscillation. The
Mori-Zwanzig formalism gives a rewriting of the original system of equations
which contains a memory term. This memory term can be related to a delay term
in a resulting delay equation. Here the technique is applied to an idealized,
but spatially extended, model of the Atlantic Multidecadal Oscillation. The
resulting delay difference model is of a different type than the delay
differential model which has been used to describe the El Ni\~no- Southern
Oscillation. In addition to this model, which can also be obtained by
integration along characteristics, error terms for a smoothing approximation of
the model have been derived from the Mori-Zwanzig formalism. Our new method of
deriving delay models from spatially extended models has a large potential to
use delay models to study a range of climate variability phenomena.
