Fig. 1: Sketches of Rossby waves’ fundamental principles.
From: Rossby wave second harmonic generation observed in the middle atmosphere
a, b The restoring force. c–e The waveform’s velocity. In (a), an air parcel follows along latitude φ0 at an eastward velocity vE with a meridional acceleration aN = 0 when the pressure gradient force balances the Coriolis force. In b, when the parcel encounters a small displacement δφ in latitude, the Coriolis force’s gradient imposes a meridional acceleration \({a}_{N}=\delta \varphi {{{{{\rm{d}}}}}}{a}_{C}/{{{{{\rm{d}}}}}}\varphi=-\delta \varphi {v}_{E}2{{\Omega }}{{{{{{{\rm{Cos}}}}}}}}{\varphi }_{0}\) that always points against δφ when vE > 0. Here, Ω denotes the Earth’s angular frequency and \({a}_{C}=-{\nu }_{E}2{{\Omega }}{{{{{{{\rm{Sin}}}}}}}}\varphi\) is the northward Coriolis acceleration. While the parcel meanders along the blue arrowed line l in (b), its waveform travels westward as sketched in c. The absolute vorticity composes the planetary vorticity \(f=2{{\Omega }}{{{{{{{\rm{Sin}}}}}}}}\varphi\) and the relative vorticity ζ, reflecting the Earth’s rotation and the parcel’s rotation with respect to the Earth, respectively. The conservation of absolute vorticity D(ζ + f)/Dt = 0 determines a southward gradient of ζ, as denoted by the red shadow in (c). The gradient’s projection along the flow path l is typically not zero and would cause a tangential velocity vt. As an example, the path l in c is zoomed in at two green crosses, displayed in (d, e). These two crosses are associated with positive and negative gradients of ζ along l, respectively, as denoted by the red and pink arrows in (d, e). The black arrows vt denote the vector sums of the red and pink arrows bordering the crosses, both of which project zonally westward. The parcels at these crosses drift toward the green points in (c) and, visually, the path l drifts westward toward the dotted line.
