The authors would like to gratefully acknowledge useful discussion with Steven Jayne, Jonathan Lilly, David Marshall, and James Maddison. SW acknowledges the support of the U.K. Natural Environmental Research Council (NERC) (Grant NE/G001510/1), as well the Australian Research Council (ARC) (Grant RM10240) and the ARC Centre of Excellence for Climate System Science (Grant CE110001028).
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Here ζa is the absolute vorticity, β is the meridional planetary vorticity gradient, and
It is worthwhile to note here that, although the discussion of eddy shape and orientation and the drawing of velocity variance ellipses evoke a mind's eye image of coherent “closed loop” vortex motion, these diagnostics can equally be applied to velocity covariances from other “types” of eddy motion, for example, jet meandering. Also, because properties are diagnosed at a point, they describe local properties which may vary across the scale of a coherent eddy structure. The eddy feedback on the mean flow as defined here depends only on correlation terms at a point/local values of eddy shape and orientation regardless of eddy size or coherence.
Specifically x* is defined by the zonal location of maximum time-mean recirculation transport found to coincide approximately with the zonal location where the time-mean meridional absolute vorticity gradient first ceases to change sign across the jet, that is, where the time-mean jet is first necessarily stabilized to its barotropic instability. See Waterman and Jayne (2011) for further details.
The ray-tracing calculation requires initial values for the neutral wavenumbers as a function of x, and for the purpose of our thought exercise we wish to consider “typical” values. We obtain these from the gradients of the time-mean streamfunction and velocity variances as specified, which indicate the horizontal scales of dominant motions, but note several caveats associated with this choice. First, strictly time-mean values do not define the relevant initial conditions that instead are instantaneous values set by the growing instability modes, nonlinear interactions, and time-dependent Rossby wave dynamics. Second, the wavenumbers obtained in this way may not correspond to the wavenumbers of the neutral wave in the upstream region where barotropic instability dominates. For these reasons, these initial conditions should be interpreted as conceptual and not exact. They are viewed as appropriate for our purposes here as qualitative results are not critically dependent on the wavenumber magnitudes. Further, we neglect the imaginary component of l, expected to be associated with the growing modes of the unstable jet (see, for example, Talley 1983) and in the subsequent calculation consider only the evolution of neutral waves and not growing unstable modes to which the ray-tracing theory does not apply. Because the imaginary part of the propagation speed tends to be small relative to the relative real-valued propagation speed, we expect neutral wave arguments to be useful in predicting many properties of the growing waves, however, in the upstream unstable jet region, strictly we apply the theory outside its domain of applicability (as well as neglect nonlinear interactions that likely are important here) and as such results in the upstream region should be treated with caution. Finally, we take the sign of l to be positive, but recognize that the neutral waves radiated from the jet will have both positive and negative values of l to the north and south of the jet axis, respectively, as is required for energy propagation away from their energy source (the meandering jet). Assuming a positive l implies that we consider waves to the south of the jet, but we expect waves with negative l to the north of the jet to behave equivalently.