Drivers should avoid flooded roads, as it can take as little as 12 inches of rain to float a vehicle.
The U.S. National Hurricane Center said Rosa should move over Baja California and Sonora, weakening and bringing 3 to 6 inches (7.5 to 15 centimeters) of rain, with its remnants expected hitting the southwestern United States on Tuesday.
The Utah National Guard was activated Monday to assist in flood mitigation efforts in Utah County.
Elsewhere in the state, flash flood watches in Phoenix and other areas were set to expire later Wednesday as the storm left the region and headed for Colorado and Utah.
"These rainfall amounts may produce life-threatening flash flooding", the National Hurricane Center said.
As Rosa trudges northeast, the storm will dump 2 to 4 inches of rain on much of Arizona and up to 6 inches in the mountains.
Phoenix meteorologist Jaret Rogers said more showers were expected across central Arizona later in the day before a weakened Rosa arrives in the area Tuesday morning. At least six roads in Tucson were closed because of flash flooding from washes that overflowed.
The National Weather Service says Phoenix Sky Harbor International Airport has received more than 2 inches (5 centimeters) of ran so far Tuesday.
Rosa, which was a hurricane until last Sunday, has already brought heavy rains to northwestern Mexico and the Baja California Peninsula.
The wet weather was a factor in numerous Phoenix-area freeway wrecks but no serious crashes were reported, said Trooper Kameron Lee of the Arizona Department of Public Safety.
"Rain is also pushing up into Arizona just south of Phoenix and will most likely move into metro Phoenix in the next few hours", Guy said. But some areas could see between 2 to 3 inches.
Gary Herbert has declared a state of emergency in advance of potential flooding from the remnants of Tropical Storm Rosa. "It doesn't mean something's imminent".
The storm has maximum sustained winds of 50 mph, with tropical-force winds extending about 150 miles from its center.
A tropical storm warning was in effect for the west coast of the Baja California Peninsula from Punta Abreojos to Cabo San Quintin and a tropical storm watch was in effect for the east coast of the Baja California Peninsula from Bahia de los Angeles to San Felipe.
Meanwhile, a separate storm, Hurricane Sergio, grew to major hurricane status in the Pacific on Tuesday, though it posed no immediate threat to land.
Another storm, Tropical Storm Sergio, is about 695 miles southwest of the Manzanillo, Mexico 805 miles south of the southern tip of Baja California.
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