Baroclinic instability of time-dependent currents

Abstract

The baroclinic instability of a zonal current on the beta-plane is studied in the context of the two-layer model when the shear of the basic current is a periodic function of time. The basic shear is contained in a zonal channel and is independent of the meridional direction. The instability properties are studied in the neighbourhood of the classical steady-shear threshold for marginal stability. It is shown that the linear problem shares common features with the behaviour of the well-known Mathieu equation. That is, the oscillatory nature of the shear tends to stabilize an otherwise unstable current while, on the contrary, the oscillation is able to destabilize a current whose time-averaged shear is stable. Indeed, this parametric instability can destabilize a flow that at every instant possesses a shear that is subcritical with respect to the standard stability threshold. This is a new source of growing disturbances. The nonlinear problem is studied in the same near neighbourhood of the marginal curve. When the time-averaged flow is unstable, the presence of the oscillation in the shear produces both periodic finite-amplitude motions and aperiodic behaviour. Generally speaking, the aperiodic behaviour appears when the amplitude of the oscillating shear exceeds a critical value depending on frequency and dissipation. When the time-averaged flow is stable, i.e. subcritical, finite-amplitude aperiodic motion occurs when the amplitude of the oscillating part of the shear is large enough to lift the flow into the unstable domain for at least part of the cycle of oscillation. A particularly interesting phenomenon occurs when the time-averaged flow is stable and the oscillating part is too small to ever render the flow unstable according to the standard criteria. Nevertheless, in this regime parametric instability occurs for ranges of frequency that expand as the amplitude of the oscillating shear increases. The amplitude of the resulting unstable wave is a function of frequency and the magnitude of the oscillating shear. For some ranges of shear amplitude and oscillation frequency there exist multiple solutions. It is suggested that the nature of the response of the finite-amplitude behaviour of the baroclinic waves in the presence of the oscillating mean flow may be indicative of the role of seasonal variability in shaping eddy activity in both the atmosphere and the ocean.