A self-consistent theory of time-dependent convection in the earth's magnetotail is developed. In contrast to earlier convection models our approach takes into account that the plasma sheet particles are effectively trapped on closed geomagnetic field lines. The results demonstrate that steady state convection, although possible in principle, is unlikely to occur in the earth's magnetotail. In particular, if particle or energy losses are sufficiently small and if the outer lobe field lines have convex shape, a steady state is impossible in the framework of the present polytropic model. Quantitative models for time-dependent convection are constructed. The time dependence introduces important consequences for energy storage, stability, and spatial dependence of the convection electric field. The results are consistent with a quasi-periodic evolution of the tail, where periods of quasi-static compressional convection are followed by phases of dynamic evolution.