Some theoretical considerations upon the transient topographical perturbation of a zonal current

Abstract

The author presents some linear theories on the large-scale or ultralarge-scale transient topographical perturbation of a two-dimensional zonal current in the rotating system on the Coriolis plane (β = 0, ƒ ≠ 0) with the upper free boundary condition. The following three different cases are dealt with here: (1) the basic current is constant for t > 0, (2) the basic current is changing for t ≥ 0, and (3) the lower boundary changes profile with the constant basic current (simulating cold air intrusion, sustention, and evacuation). For the first two cases the lower boundaries are fixed (simulating the fixed terrestrial topographies). Some remarkable characteristics shown by the analytical solutions obtained herewith are pointed out as follows. The synoptic solitary aggregated wave (to be called hereafter the ensemble wave) induced by the topographies forms itself successively around the barrier with a period depending upon the basic current speed. The successive solitary ensemble waves are coexistent and flow downstream, the speed changing somewhat around the value of the basic zonal current velocity both for constant or changing basic current cases when the barrier range remains within about one third of the latitudinal circumference (120° stretched). When the lower boundary changes (i.e., the cold air intrudes, sustains, and evacuates) and especially when it flattens (the cold air evacuates), the solitary ensemble wave of individual group waves flows downstream very fast, 2 or 3 times as fast as the individual group waves' phase velocity. The solitary ensemble wave of individual group waves oscillates remarkably in a crippled fashion around the barrier when the barrier range is stretched over more than one third of the latitudinal circumference, while no traveling solitary wave can be seen in such a case. Such a phenomenon is synoptically well known. The individual group waves of each wavelength behave themselves in accordance with other synoptic experiences; e.g., the ultralong group wave of wave number one retrogrades in this case, while the Rossby or Rossby-Haurwitz group waves of wave number one remain progressive, etc. Thus the synoptic solitary ensemble wave can be interpreted as, so to speak, the group wave of individual group waves. Consequently, the Fourier analyses for each wave number of the actual synoptic data, say, for the configurations of streamline at a 500-mbar surface, give only the group wave value of each wave number instead of the single wave value of each wave number, a procedure that is against the customary expectation. Thus the synoptic planetary waves may be finally interpreted as the ensemble waves composed of inertia-gravity group waves on the Coriolis plane (β = 0,ƒ ≠ 0) instead of as Rossby waves on the β plane (β ≠ 0,ƒ ≠ 0), which was the same with the steady case.