Empiric evidence and theoretical argument indicate that magnetospheric convection is globally sub-adiabatic. Reconciliation of the sub-adiabaticity with the underlying adiabatic particle drifts is a problem that has not been conclusively demonstrated. In this paper, through an ensemble average, we show that the Rice Convection Model contains a heat flux due to thermal inequilibrium between two adjacent flux tubes of equal volume. The averaged theory is based on field variables obeying a set of partial differential equations (Eulerian formulation), instead of conservative variables advecting with fluid elements (Langrangian formulation). We apply the theory to investigate a one-dimensional case of subadiabatic variation of thermodynamic properties in the plasma sheet.