WASHINGTON — The stormy aftermath of Mississippi’s Republican Senate runoff has sent Tea Party conservatives around the country to the ramparts, raising the prospect of a prolonged battle that holds the potential to depress conservative turnout in November in Mississippi — and possibly beyond.

The campaign of State Senator Chris McDaniel has formally filed a legal notice that it intends to contest the results of the June 24 runoff against Senator Thad Cochran, a six-term incumbent. At least two lawsuits from McDaniel allies are in the works, one demanding the release of election records, the other accusing Mississippi authorities of culpability in the suicide of a Mississippi Tea Party leader on June 27.

The turmoil has given Mr. Cochran’s Democratic challenger, former Representative Travis Childers, an opening to exploit the divide in what is otherwise seen as a race in which he trails badly. “Senator Cochran does not have the confidence of his state, let alone his own party,” Mr. Childers said.

Perhaps more worrisome to Washington Republicans is that if conservative outrage in Mississippi spreads, some Tea Party voters in other states where the Republican establishment’s candidate has prevailed could stay home in November. That could bolster Democrats like Kay Hagan in North Carolina, Mark Pryor in Arkansas, Mark Begich in Alaska or Michelle Nunn in Georgia.