  This paper discusses some basic concepts that arise in the study of the
tropical cyclone frictional boundary layer. Part I discusses the concepts of
asymptotic triangular waves and asymptotic N-waves in the context of the
nonlinear advection equation and Burgers' equation. Connections are made
between triangular waves and single eyewalls, and between N-waves and double
eyewalls. In Part II, analytical solutions of a line-symmetric, $f$-plane, slab
model of the atmospheric boundary layer are presented. The boundary layer flow
is forced by a specified pressure field and initialized with $u$ and $v$ fields
that differ from the steady-state Ekman solution. With certain smooth initial
conditions, discontinuities in $u$ and $v$ can be produced during the transient
adjustment to the steady-state Ekman solution. Associated with these
discontinuities in the horizontal wind components are singularities in the
boundary layer pumping and the boundary layer vorticity, which can be either
divergence-preferred or vorticity-preferred. These models serve as a prototype
for understanding the role of the atmospheric boundary layer in the dynamics of
primary and secondary eyewalls in tropical cyclones
