An initial zonally symmetric quasigeostrophic potential vorticity (PV) distribution q i ( y) is subjected to complete or partial mixing within some finite zone jyj , L, where y is latitude. The change in M, the total absolute angular momentum, between the initial and any later time is considered. For standard quasigeostrophic shallow-water beta-channel dynamics it is proved that, for any q i ( y) such that dq i /dy . 0 throughout jyj , L, the change in M is always negative. This theorem holds even when ''mixing'' is understood in the most general possible sense. Arbitrary stirring or advective rearrangement is included, combined to an arbitrary extent with spatially inhomogeneous diffusion. The theorem holds whether or not the PV distribution is zonally symmetric at the later time. The same theorem governs Boussinesq potential-energy changes due to buoyancy mixing in the vertical. For the standard quasigeostrophic beta-channel dynamics to be valid the Rossby deformation length L D ) L where is the Rossby number; when L D 5 ' the theorem applies not only to the beta channel but also to a single barotropic layer on the full sphere, as considered in the recent work of Dunkerton and Scott on ''PV staircases.'' It follows that the M-conserving PV reconfigurations studied by those authors must involve processes describable as PV unmixing, or antidiffusion, in the sense of time-reversed diffusion. Ordinary jet self-sharpening and jet-core acceleration do not, by contrast, require unmixing, as is shown here by detailed analysis. Mixing in the jet flanks suffices. The theorem extends to multiple layers and continuous stratification. A least upper bound and greatest lower bound for the change in M is obtained for cases in which q i is neither monotonic nor zonally symmetric. A corollary is a new nonlinear stability theorem for shear flows.

Abstract

An adaptive candidate approach for the fast orthogonal search (FOS) algorithm is presented for detecting signals the frequencies of which are much more closely spaced than can be resolved by the fast Fourier transform (FFT). The algorithm is shown to give solutions comparable to a global minimisation of the sum squared error, but with much greater speed. It can also surpass the Cramer-Rao lower bound on frequency resolution.Introduction: Fast orthogonal search (FOS) is a modelling technique that has been shown to be able to accurately estimate the frequency and magnitude of closely spaced signals with up to ten times the fast Fourier transform (FFT) resolution [1-3]. These results were found using a set of candidate functions with closely spaced frequencies and fitting the candidate functions one frequency at a time in order of significance. By fitting sinusoids in candidate sets of two or more frequencies and choosing the set with the lowest sum square error (SSE), it is possible to resolve candidate frequencies with an even smaller frequency separation. Simulated time series with varying signal-to-noise ratios (SNRs) are created and the adaptive FOS technique presented here is used to estimate the frequency, magnitude and phase of the closely spaced sinusoidal signals.

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