  Tropical cyclones are known to expand to an equilibrium size on the
$f$-plane, but the expansion process is not understood. In this study, an
analytical model for tropical cyclone size expansion on the $f$-plane is
proposed. Conceptually, the storm expands because the imbalance between latent
heating and radiative cooling drives a lateral inflow that imports absolute
vorticity. Volume-integrated latent heating increases more slowly with size
than radiative cooling, and hence the storm expands towards an equilibrium
size. The predicted expansion rate is given by the ratio of the difference in
size from its equilibrium value ($r_{t,eq}$) to an environmentally-determined
time scale $\tau_{rt}$ of $10\sim15$ days. The model is fully predictive if
given a constant $r_{t,eq}$, which can also be estimated environmentally. The
model successfully captures the first-order size evolution across a range of
numerical simulation experiments in which the potential intensity and $f$ are
varied. The model predictions of the dependencies of lateral inflow velocity
and expansion rate on latent heating rate also compare well with numerical
simulations. This model provides a useful foundation for understanding storm
size dynamics in nature.
