  This study is concerned with how the attractor dimension of the
two-dimensional Navier--Stokes equations depends on characteristic length
scales, including the system integral length scale, the forcing length scale,
and the dissipation length scale. Upper bounds on the attractor dimension
derived by Constantin--Foias--Temam are analysed. It is shown that the optimal
attractor-dimension estimate grows linearly with the domain area (suggestive of
extensive chaos), for a sufficiently large domain, if the kinematic viscosity
and the amplitude and length scale of the forcing are held fixed. For
sufficiently small domain area, a slightly ``super-extensive'' estimate becomes
optimal. In the extensive regime, the attractor-dimension estimate is given by
the ratio of the domain area to the square of the dissipation length scale
defined, on physical grounds, in terms of the average rate of shear. This
dissipation length scale (which is not necessarily the scale at which the
energy or enstrophy dissipation takes place) can be identified with the
dimension correlation length scale, the square of which is interpreted,
according to the concept of extensive chaos, as the area of a subsystem with
one degree of freedom. Furthermore, these length scales can be identified with
a ``minimum length scale'' of the flow, which is rigorously deduced from the
concept of determining nodes.
