  An attempt is made to quantitatively demonstrate the difference between a
complex adaptive combat model and conventional combat models. The work shows
that complex adaptive models may give rise to "turbulent" dynamics, which
emerge once the battlefield is no longer "linear", i.e. once military
formations no longer form ordered lines or columns. This is done using a
cellular automaton model. This model exhibits a high degree of complexity,
leading to a rich variety of behaviour. Conventional statistical methods fail
to adequately capture this richness. Particular attention is paid to the
properties of the attrition function. It is found that this function is
discontinuous and possesses non-linear properties such as clustering of
casualties, scaling of the statistical moments of the data and fat-tailed
probability distributions. Fractal methods are found to be capable of
quantifying these properties. A particularly significant result is the
non-linearity of the pay-off of improved weapons performance. This contrasts
with conventional models, where the attrition rate usually depends in a linear
way on kill probability. Additionally, the model is found to possess at least
two attractive states. A transition exists between the two cases in some
situations. This transition appears to behave in an analogous way to the
transition between laminar and turbulent states in fluid dynamics.
