The ionic mass-transfer rates accompanying natural convective electrolyte flow in a CuSO4 aqueous electrolyte solution acidified with an excess amount of H2SO4 are numerically analyzed. The effects of a supporting electrolyte and an interaction behavior between both cathodic upward and anodic downward natural convections are examined. Both anodic and cathodic current density distributions along the vertical height are also calculated. A mathematical model is extended by incorporating an additional boundary condition at the limiting current. A measure of ionic migration effect epsilon, a ratio of limiting current to limiting diffusion current, involving the transference number of a discharging metallic ion is introduced for this purpose. The present calculation predicts the oscillation behaviors in transient variations in both electrode surface concentration and maximal natural convective velocity, which are deeply related to the periodic fluctuating electrolyte flow patterns distorted by secondary flow. The addition of H2SO4 maintains an epsilon value around 1 and prevents the further development to the transition or turbulent natural convection.