[(a)] By entering into a bond for costs or to stay the execution of any proceedings to enforce a judgment, the surety submits oneself to the jurisdiction of the court, irrevocably appoints the clerk of the court as the surety's agent upon whom any papers affecting the surety's liability on the bond may be served, and agrees that the surety's liability may be enforced on motion without the necessity of an independent action.
[(b)] The papers served on the clerk as statutory agent for the surety shall be mailed by the clerk to the surety if the surety's address is known.