  Tropical convective clouds evolve over a wide range of temporal and spatial
scales, and this makes them difficult to simulate numerically. Here, we propose
that their statistical properties can be derived within a simplified
time-independent co-ordinate system of cloud number $n$, saturated static
energy $h^\star$, and cloud perimeter $\lambda$. Under the constraint that
circulations around cloud edge compete for buoyant energy and air, we show that
the product of cloud number and cloud perimeter $n\lambda$ is invariant with
$\lambda$ and that cloud number follows a negative exponential with respect to
cloud-edge deviations of $h^\star$ from the mean. Overall, the summed perimeter
of all clouds scales as the square root of the atmospheric static stability,
which suggests that the complexity of cloud field structures can be viewed
statistically as an emergent property of atmospheric bulk thermodynamics.
Analytically derived conclusions are compared with a detailed tropical cloud
field simulation and found generally to agree to within $<$13%. For the sake of
developing hypotheses about cloud temporal evolution that are testable in high
resolution simulations, the shapes of tropical cloud perimeter distributions
are predicted to be invariant as climate warms, although with a modest increase
in total cloud amount.
