Laboratory and numerical experiments are reported on dye advection processes in geostrophic turbulence. The experimental setup is the classical rotating annulus with differential heating which mimics the most essential features of midlatitude atmospheric flow. The main control parameter is the temperature contrast. Fluorescent dye is used as passive tracer, and dispersion is evaluated by digital image processing. The results are compared with tracer dispersion computations which are performed by means of global reanalysis wind fields at the pressure height of 500 hPa covering a time interval of one year. Apart from initial transient periods, the characteristic behavior for intermediate time scales is ballistic dispersion in both systems, where the zonal extent of the tracer cloud increases linearly in time (Batchelor scaling). The long-time evolution cannot be followed by the experimental technique, however, the numerical tests suggest a slower diffusive dispersion (Taylor regime) after 70-80 revolutions (days), in agreement with expectations. Richardson-Obukhov scaling (superdiffusion with an exponent value of 3/2) is neither observed in the laboratory nor in the numerical tests. Our findings confirm recent experimental results on the classic prediction by Batchelor that the initial pair separation is an essential parameter of the subsequent time evolution of tracers.