One and one half Mars years, from 8 June 2002 to 2 April 2005, of atmospheric argon (Ar) measurements are described and studied in the context of understanding how Ar, a minor constituent of the Martian atmosphere that does not condense at temperatures found on Mars, can be used to study Martian circulation and dynamics. The Ar data are from the gamma subsystem of the gamma ray spectrometer currently operating on the Mars Odyssey spacecraft in orbit around Mars. A comprehensive data analysis including γ ray production and attenuation by the atmosphere is presented. Four discoveries are discussed. (1) There is a factor of 6 enhancement of Ar measured over south polar latitudes (75¿S to 90¿S) for 2 consecutive Mars years occurring near the onset of southern winter. (2) There is no similar strong enhancement of Ar over north polar regions during northern winter. Part of this difference is explained by the global topographic dichotomy and the fact that the duration of northern autumn and winter is only ~80% as long as that of southern autumn and winter. (3) Rapid seasonal fluctuations in Ar abundance from 60¿S to 90¿S may be evidence for wave activity at the perimeter of the southern seasonal polar cap. (4) The magnitude of the Ar enhancement over the south polar region lacks spatial coincidence with either the relatively dark "cryptic terrain" or the relatively bright albedo regions; however, a divergence in Ar abundance near Ls = 267.5¿ may be an expression of later CO2 sublimation in the bright terrain rather than in the cryptic region.