  Variation of voltage profiles with different time scales leads to the
redistribution of deposited energy as well as electro-hydrodynamic forces,
while the mechanism and scaling law is still unknown. On the basis of
theoretical and numerical analysis, we show that a secondary surface ionization
wave forms during the voltage rising slope when electron density decreases to a
critical level while the voltage is still rising. A characteristic map of
energy and electro-hydrodynamics force in time scales between 1~ns and 0.1~s at
atmospheric pressure is proposed, opening the door towards the target--directed
design of surface discharges.
