We present a theoretical argument suggesting that steady, adiabatic convection probably cannot occur throughout a closed-magnetic-field-line region that extends into a long magnetotail. If plasma in a typical flux tube in the Earth's outer plasma sheet were compressed adiabatically as it convected into the near-Earth part of the plasma sheet, the plasma pressure would be absurdly large there. This excess-pressure problem is demonstrated numerically for several standard models of the magnetospheric magnetic field, for the case of isotropic pressure. We argue that the excess-pressure problem results from the general shapes of field lines in the inner and outer plasma sheet, and not from simple inaccuracies in all the magnetic-field models. We hypothesize that sunward convection must necessarily be time-dependent, and that the magnetospheric substorm may be the essential time-dependent process in which the plasma is suddenly and non-adiabatically released from plasma-sheet flux tubes.