$C_j, D_j$ are given in the Appendix, where it is shown that $C_2(k,k') = C_3(k,k-k')$, so that the $C_3$ integral is equal to the $C_2$ integral. Equations and are equivalent to Eq. , but they directly show how $\beta_1$ and $\beta_2$ interact.

An analogy can be made here to the use of Els[ä]{}sser variables in incompressible, homogeneous MHD turbulence, which are a familiar example of an eigenmode decomposition that makes explicit the nonlinear interaction of the linear eigenmodes. The linearized equations have as solutions counterpropagating, noninteracting waves of the form $\mathbf{z}^{\pm} = \mathbf{v} \pm \mathbf{b}/(4\pi\rho_0)^{1/2}$. Expressing the nonlinear equations in terms of $\mathbf{z}^{\pm}$, the nonlinearity in the equation for $\partial \mathbf{z}^{\pm}/\partial t$ is $\mathbf{z}^{\mp}\cdot\nabla\mathbf{z}^{\pm}$, which describes the nonlinear interactions between linearly noninteracting modes. In the present calculation, the linearly noninteracting $\phi_1, \phi_2$ are comparable to $\mathbf{z}^{\pm}$, and the terms proportional to $\beta_1(k')\beta_2(k'')$ and $\beta_1(k'')\beta_2(k')$ are comparable to $\mathbf{z}^{\mp}\cdot\nabla\mathbf{z}^{\pm}$. However, unlike the $\mathbf{z}^{\pm}$ equations, the $\dot{\beta}_j$ equations include other nonlinear terms that are proportional to $\beta_1(k')\beta_1(k'')$ and $\beta_2(k')\beta_2(k'')$. If all of the nonlinearities are zero except for the $C_1$ term, then the evolution of $\beta_1(k)$ is just a combination of its linear drive $i\omega_1(k)$ and three-wave interactions with $\beta_1(k')$ and $\beta_1(k-k')$, allowing $\phi_1$ to saturate through a Kolmogorov-like cascade to smaller scales. This is effectively the assumption of standard quasilinear calculations of momentum transport, where only $\phi_1,\omega_1$ are considered. Figure \[CDplot\] shows some of the nonlinear coupling coefficients plotted over a range of wavenumbers. Since $D_1, C_2$, and $C_3$ are not identically zero, there is some interaction between the eigenmodes. Systems