of $A_{pq}$ and $B_{pqr}$ have been determined in Appendix A. Each equation in (\[eq:nonlinear-ODE\]) is the projection of the CBE on a subspace spanned by a weighting function. Therefore, the left hand side of (\[eq:nonlinear-ODE\]) is the projection of $\partial
f_1/\partial t$, the summation over first order terms is the projection of $-\left [f_1,{\cal H}_0 \right ]- \left [ f_0,{\cal
H}_1 \right ]$, and the second order terms are the projections of $-\left [f_1,{\cal H}_1 \right ]$. The second order terms of amplitude functions, characterized by $B_{pqr}$, show the interaction of modes in both the radial and azimuthal directions.

Distribution of angular momentum between different Fourier components provides useful information of the disk dynamics. I compute the rate of change of the total angular momentum ${\cal L}$ using (see Appendix B in JH) $${ d {\cal L}\over  d t}=-\frac {1}{4}
\int\int \left ( f_1+\overline{f}_1 \right )
{\partial \over \partial \theta_{\phi} }
\left ( V_1+\overline{V}_1 \right ) d \textbf{\textit{J}} d \Theta,
\label{eq:rate-change-L-one}$$ where a bar denotes complex conjugate. Substituting (\[eq:expansion-f1\]) and (\[eq:expansion-V1-vs-dmlj\]) in (\[eq:rate-change-L-one\]) and evaluating the integrals, yield $$\begin{aligned}
{ d {\cal L}\over  d t} &=& {\rm i}\pi^2 \!\!\!
\sum_{m,l=-\infty}^{\infty}\sum_{j,p=0}^{\infty}
\biggl \{  m\left [ a^{(-m)}_{j}(t)
+\overline{a^{m}_{j}}(t) \right ] \Lambda^{ml}_{jp}
d^{ml}_{p}(t) 