We thank three anonymous referees for their useful comments that helped improve the readability of this paper. NJM acknowledges funding from a UK NERC studentship and JGE acknowledges support from NERC Grant NE/G003122/1. This paper is dedicated to the memory of Chris Bell, who contributed many helpful suggestions during the early stages of this work.
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The present study casts doubt on the accuracy of quasi-linear wave-mean models such as that used by Smith, as it is shown explicitly in section 4d that the interaction of weakly nonlinear vortex Rossby waves with their second harmonic is equally important as their interaction with the zonal mean flow.
Had we retained the dependence on the fast time variable t here, the projection condition would appear as the “condition for the removal of secular terms” at higher orders in the expansion.