Chaotic and Periodic Behavior of Finite-Amplitude Baroclinic Waves

Abstract

Numerical integrations of the amplitude equations governing the dynamics of a weakly unstable finite-amplitude baroclinic wave have revealed a new and unexpectedly complex dependence on the degree of dissipation. We have found for very small dissipation the simple limit cycle behavior noted earlier by Pedlosky (1971) and Smith and Reilly (1977). For slightly higher values of the dissipation further solution bifurcations reveal an ever-increasing family of periodic solutions whose periods are even multiples of the fundamental. The critical value of the dissipation parameter γ for which the nth cycle emerges, appears to satisfy the Feigenbaum (1978) relation (γn−γn−1)/(γn+1−γn)=4.669201. Above a limit point γ∞ the solutions are aperiodic and chaotic. However, at isolated “islands” within this chaotic regime we have again found periodic solutions. Although their phase plane trajectories differ from those found below γ∞, their periods are always multiples of half the fundamental. For higher γ, equili... Abstract Numerical integrations of the amplitude equations governing the dynamics of a weakly unstable finite-amplitude baroclinic wave have revealed a new and unexpectedly complex dependence on the degree of dissipation. We have found for very small dissipation the simple limit cycle behavior noted earlier by Pedlosky (1971) and Smith and Reilly (1977). For slightly higher values of the dissipation further solution bifurcations reveal an ever-increasing family of periodic solutions whose periods are even multiples of the fundamental. The critical value of the dissipation parameter γ for which the nth cycle emerges, appears to satisfy the Feigenbaum (1978) relation (γn−γn−1)/(γn+1−γn)=4.669201. Above a limit point γ∞ the solutions are aperiodic and chaotic. However, at isolated “islands” within this chaotic regime we have again found periodic solutions. Although their phase plane trajectories differ from those found below γ∞, their periods are always multiples of half the fundamental. For higher γ, equili...