On the behaviour of baroclinic waves undergoing horizontal deformation. II: Error-bound amplification and Rossby wave diagnostics

An expression for the maximum possible amplification of a baroclinic wave with zero potential vorticity, undergoing horizontal deformation, is obtained. This amplification provides an estimate of the growth of hypothetical analysis errors for such a system. For the geophysically relevant flows considered in detail in this paper, strain rates of ±1 × 10−5 s−1 cause a 28% reduction in the logarithmic amplification over a 48-hour period. If the effects of phase changes of water are simulated by halving the brunt-Vaisala frequency, all growth rates are larger, but strain rated of ±1 × 10−5s−1 now cause a 34% reduction logarithmic amplification. Notably, this corresponds to an 81% reduction in actual amplification. For all strain rates the increase in non-modal growth associated with lowering the Richardson number is found to be less than for norma-mode-type baroclinic waves. To understand why particular initial zonal wavelengths and vertical phase shifts optimize growth in this time-dependent basic state, new diagnostics are developed. The diagnostics are based on a mathematical model of Bretherton's qualitative description of baroclinic instability in terms of two counter-propagating Rossy waves. They provide a viewpoint from which growth results form the product of a scale term, which is proportional to the wind strength induced at opposing boundaries, and a phase term, which is maximized when the upper wave is 90° up shear of the lower Rossby wave. The perspective leads to a simple explanation of why strain affects amplification and how this effect depends on the static stability. Furthermore, it provides simple methods, based on diagnosed amplitudes of neglected 2nd and 3rd order nonlinear terms, for forecasting when linear prediction become misleading. To assess how dynamic constraints associated with the time dependence of a basis flow affect its stability, it proves useful to define a new quantity, the ‘potential maximal wave amplification’, which is the amplification that would arise if at each instant the flow was adiabatically and mass preservingly rearranged so that the Rossby waves defining the boundary buoyancy distribution were always optimally configured for growth. The results presented here also show that time-dependent systems which do not support normal modes can amplify certain wavelengths more than other, and hence be likely to impart the corresponding horizontal structures onto arbitrarily shaped initial disturbances. Such scale selection is linked to the relationship between the horizontal morphology of a Rossby wave and the strength of the wind it can induce at vertically distant points. The effect of strain on sets of flows is also found to depend on the mean meridional wave number of such flows. Notably, if this mean meridional wave number is increased to values near the short-wave cut-off for instability, the presence of strain can actually enhance growth. In narrow baroclinic jets, such as those found off ice edges in polar regions, barotropic modes often grow faster than baroclinic modes in unstrained flow. A comparison with the effects of strain on barotropic instability shows that frontogenetic strain damps barotropic growth much more than baroclinic growth. The result may be important to theories of polar-low genesis.