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J. Gavin Esler

Abstract

The zonal modulation of baroclinic disturbances is studied in a quasigeostrophic two-layer periodic channel. The system is relaxed toward an unstable state with a uniform flow in each layer. For small criticality, two weakly nonlinear systems are then developed, which differ in the choice of boundary condition used for the correction to the basic flow. Each system is described by an amplitude equation that determines the evolution of the wave envelope over “long” time- and space scales. For the first system the amplitude equation allows wave packet formation. Depending upon the ratio of the length scale of the packets to the channel length, either a steady wave train, stable solitonlike wave packets, or chaotically evolving wave packets are observed. The mechanism that leads to wave packet formation is then discussed with reference to the instability criterion of the amplitude equation. For the second system the amplitude equation is found to allow convergence to a steady, uniform wave train only.

A numerical model is then used to investigate the finite criticality extension of the second weakly nonlinear system. At low criticality, the assumptions that underpin the weakly nonlinear theory are tested by analyzing the convergence to a uniform wave train. As the criticality is increased, the effects of full nonlinearity cause the weakly nonlinear theory to become invalid. Initially, resonant triads of waves that have fixed amplitudes become excited owing to the dissipative nature of the system. As the criticality is increased further, other waves are excited and the system approaches full baroclinic chaos. Wave packet–like structures are then observed that evolve rapidly, growing, decaying, merging, and dividing.

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Lorenzo M. Polvani, J. Gavin Esler, and R. Alan Plumb

Abstract

Using a global, one-layer shallow water model, the response of a westerly flow to a localized mountain is investigated. A steady, linear response at small mountain heights successively gives way first to a steady flow in which nonlinearities are important and then to unsteady, but periodic, flow at larger mountain heights. At first the unsteady behavior consists of a low-frequency oscillation of the entire Northern Hemisphere zonal flow. As the mountain height is increased further, however, the oscillatory behavior becomes localized in the diffluent jet exit region downstream of the mountain. The oscillation then takes the form of a relatively rapid vortex shedding event, followed by a gradual readjustment of the split jet structure in the diffluent region. Although relatively simple, the model exhibits a surprisingly high sensitivity to slight parameter changes. A linear stability analysis of the time-averaged flow is able to capture the transition from steady to time-dependent behavior, but fails to capture the transition between the two distinct regimes of time-dependent response. Moreover, the most unstable modes of the time-averaged flow are found to be stationary and fail to capture the salient features of the EOFs of the full time-dependent flow. These results therefore suggest that, even in the simplest cases, such as the one studied here, a linear analysis of the time-averaged flow can be highly inadequate in describing the full nonlinear behavior.

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J. Gavin Esler, Lorenzo M. Polvani, and R. Alan Plumb

Abstract

The effect of a simple representation of the Hadley circulation on the propagation and nonlinear reflection of planetary-scale Rossby waves in the winter hemisphere is investigated numerically in a single-layer shallow-water model.

In the first instance, waves are forced by a zonal wavenumber three topography centered in the extratropics. In the linear limit the location of the low-latitude critical line at which the waves are absorbed is displaced poleward by the Hadley circulation. At finite forcing amplitude the critical layer regions where the waves break are found to be displaced poleward by a similar distance. The Hadley circulation is also found to inhibit the onset of nonlinear reflection by increasing the dissipation of wave activity in the critical layer.

Second, for waves generated by an isolated mountain, the presence of the Hadley circulation further inhibits nonlinear reflection by generating a strong westerly flux of wave activity within the critical layer. This westerly flux is shown to be largely advective and is explained by the poleward displacement of the critical line into the region of westerly flow. A simple expression is derived for the minimum zonal wind strength allowing propagation in the case of a quasigeostrophic β-plane flow when the mean meridional wind υ > 0.

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