## Abstract

The nonlinear dynamics of baroclinically unstable waves in a time-dependent zonal shear flow is considered in the framework of the two-layer Phillips model on the beta plane. In most cases considered in this study the amplitude of the shear is well below the critical value of the steady shear version of the model. Nevertheless, the time-dependent problem in which the shear oscillates periodically is unstable, and the unstable waves grow to substantial amplitudes, in some cases with strongly nonlinear and turbulent characteristics. For very small values of the shear amplitude in the presence of dissipation an analytical, asymptotic theory predicts a self-sustained wave whose amplitude undergoes a nonlinear oscillation whose period is amplitude dependent. There is a sensitive amplitude dependence of the wave on the frequency of the oscillating shear when the shear amplitude is small. This behavior is also found in a truncated model of the dynamics, and that model is used to examine larger shear amplitudes. When there is a mean value of the shear in addition to the oscillating component, but such that the total shear is still subcritical, the resulting nonlinear states exhibit a rectified horizontal buoyancy flux with a nonzero time average as a result of the instability of the oscillating shear. For higher, still subcritical, values of the shear, a symmetry breaking is detected in which a second cross-stream mode is generated through an instability of the unstable wave although this second mode would by itself be stable on the basic time-dependent current. For shear values that are substantially subcritical but of order of the critical shear, calculations with a full quasigeostrophic numerical model reveal a turbulent flow generated by the instability. If the beta effect is disregarded, the inviscid, linear problem is formally stable. However, calculations show that a small degree of nonlinearity is enough to destabilize the flow, leading to large amplitude vacillations and turbulence. When the most unstable wave is not the longest wave in the system, a cascade up scale to longer waves is observed. Indeed, this classically subcritical flow shows most of the qualitative character of a strongly supercritical flow. This result supports previous suggestions of the important role of background time dependence in maintaining the atmospheric and oceanic synoptic eddy field.

*Corresponding author address:* Joseph Pedlosky, Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, MA 02543. Email: jpedlosky@whoi.edu