We are grateful to Dudley Chelton for numerous helpful and stimulating discussions during the course of this research. We thank Mike McPhaden, Billy Kessler, and an anonymous reviewer for helpful comments on the manuscript. We thank Michael Schlax for helpful critical comments on an early version of the manuscript. We also benefitted from discussions with Carl Wunsch, Jay McCreary, Ken Brink, Breck Owens, Alexey Kaplan, Deepak Cherian, and Jake Gebbie. We thank the TAO Project Office of NOAA’s Pacific Marine Environmental Laboratory for providing the dynamic height and wind data, and we thank the TAO Project and TRITON Project for their sustained efforts to collect the data. This research was funded by NASA Grant NNX10AO93G and is a contribution of the Ocean Vector Winds Science Team.
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These waves are “unit amplitude” in the sense each mode is normalized so that
As the zonal wavenumber goes to infinity, the inertia–gravity wave modes increasingly resemble pure, zonally propagating gravity waves, and the amplitude of V(m) becomes small compared to that of U(m). In this limit, the ratios
These conditions are that the measurement noise be white and the best-guess a priori estimate of the Fourier coefficients be one that assigns them all equal value, with a sum of squares equal to the expected dynamic height variance. The latter would correspond to a white a priori spectrum, but with a very particular phase arrangement. The dynamic height spectrum is not expected to be white, so our estimate is not optimal, but we have chosen the tapered least squares approach in part out of a spirit of agnosticism, to avoid giving the impression that our expectations have influenced the result.
Parseval’s theorem states that, for uniformly spaced samples, the integrated spectral power equals the variance of the data. This result relies on the orthogonality of sinusoids with the usual Fourier frequencies on a uniform sampling grid, but sinusoids on an irregular grid are not in general orthogonal, so Parseval’s theorem does not generally hold for a spectrum estimated from gappy data. In addition, a key premise of the tapered least squares method is that there is noise in the data and that a solution that represents all of the variance in the data is thus undesirable. Nonetheless, given our reasonably good SNR, we view an acceptable solution as one that has about the same amount of variance in the spectrum as in the raw data.
The confidence interval displayed in Fig. 7 is for judging whether there is a statistically significant difference in symmetric power and antisymmetric power at a particular wavenumber and frequency, which is done by using the confidence interval to determine whether the value plotted is bounded away from zero (i.e., whether the ratio of the symmetric spectrum to the antisymmetric spectrum is significantly different from 100 = 1). The upper and lower confidence intervals are each about 2.5 contour intervals, so any values in the figure that are three contour intervals away from zero represent a statistically significant symmetry preference.