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## Abstract

A truncated spectral model of the forced, dissipative, barotropic vorticity equation on a cyclic β-plane is examined for multiple stationary and periodic solutions. External forcing on one scale of the motion provides a barotropic analog to thermal heating.

For forcing of any (finite) magnitude at the maximum or minimum scale in the truncation, the truncated solution converges in the limit as *t* → ∞ to the known solution of the corresponding linear model. If the forcing is constant, this limit solution represents a globally attracting stationary point in phase space. These results extend the well-known spectral blocking theorem of Fjortøft (1953) to forced, dissipative flows.

The main results, however, obtain from a low-order model describing two disturbance components interacting with a constant, forced, basic-flow component of intermediate scale. The zonal dependence of either the basic flow or the disturbances is flexible and determined by the choice of component wave vectors. For low-wavenumber disturbances and β ≠ 0, the basic flow represents a unique stationary solution, which becomes unstable when the forcing exceeds a critical value. An application of the Hopf bifurcation theorem in the neighborhood of critical forcing reveals the existence of a periodic solution or limit cycle, which is then derived explicitly in phase space as a closed circular orbit whose frequency is described by a linear combination of the normal-mode Rossby-wave frequencies.

The limit cycle radius, which physically represents the ultimate enstrophy of the disturbances, can be depicted as a response surface on the control plane defined by the independent forcing and beta parameters. If the forcing is zonally dependent, the response surface may exhibit a pronounced fold, which arises from the existence of a snap-through bifurcation. The projection of this fold onto the parameter control plane defines a bimodal or hysteresis region in which multiple stable solutions exist for given parameters. The boundary of the hysteresis region represents parameter states at which the model can exhibit sudden flow regime transitions, analogous to those observed in the laboratory rotating annulus.

This study demonstrates that the degree of nonlinearity, the scale of the forcing, and the spatial dependence of the disturbances and the forcing all crucially influence both the multiplicity and temporal nature of the stable limit solutions in a low-order, forced, dissipative model. Thus, choices in this rather complex array of physical degrees of freedom must be carefully considered in any model of the long-term evolution of large-scale atmospheric flow.

## Abstract

A truncated spectral model of the forced, dissipative, barotropic vorticity equation on a cyclic β-plane is examined for multiple stationary and periodic solutions. External forcing on one scale of the motion provides a barotropic analog to thermal heating.

For forcing of any (finite) magnitude at the maximum or minimum scale in the truncation, the truncated solution converges in the limit as *t* → ∞ to the known solution of the corresponding linear model. If the forcing is constant, this limit solution represents a globally attracting stationary point in phase space. These results extend the well-known spectral blocking theorem of Fjortøft (1953) to forced, dissipative flows.

The main results, however, obtain from a low-order model describing two disturbance components interacting with a constant, forced, basic-flow component of intermediate scale. The zonal dependence of either the basic flow or the disturbances is flexible and determined by the choice of component wave vectors. For low-wavenumber disturbances and β ≠ 0, the basic flow represents a unique stationary solution, which becomes unstable when the forcing exceeds a critical value. An application of the Hopf bifurcation theorem in the neighborhood of critical forcing reveals the existence of a periodic solution or limit cycle, which is then derived explicitly in phase space as a closed circular orbit whose frequency is described by a linear combination of the normal-mode Rossby-wave frequencies.

The limit cycle radius, which physically represents the ultimate enstrophy of the disturbances, can be depicted as a response surface on the control plane defined by the independent forcing and beta parameters. If the forcing is zonally dependent, the response surface may exhibit a pronounced fold, which arises from the existence of a snap-through bifurcation. The projection of this fold onto the parameter control plane defines a bimodal or hysteresis region in which multiple stable solutions exist for given parameters. The boundary of the hysteresis region represents parameter states at which the model can exhibit sudden flow regime transitions, analogous to those observed in the laboratory rotating annulus.

This study demonstrates that the degree of nonlinearity, the scale of the forcing, and the spatial dependence of the disturbances and the forcing all crucially influence both the multiplicity and temporal nature of the stable limit solutions in a low-order, forced, dissipative model. Thus, choices in this rather complex array of physical degrees of freedom must be carefully considered in any model of the long-term evolution of large-scale atmospheric flow.