The last named storm to make landfall in Cuba during November was Paloma in 2008.
It could strengthen into a hurricane before reaching Florida late Sunday night. It will likely travel from Florida's East Coast to the West Coast near Tampa Bay.
Much of Central Cuba was already under a tropical storm warning prior to making landfall, as was the Northwestern Bahamas and the Southern part of the USA state of Florida. A Tropical Storm Warning means that tropical storm conditions are expected in the warning area within 36 hours. Some areas received more than two feet of rain and residences were damaged.
Tropical storm conditions are expected this weekend in portions of the Cayman Islands and Cuba.
Eta had maximum sustained winds of 65 mph (100 kph) at midday Sunday and it was centred north of Cuba, about 145 miles (235 kilometres) southeast of Marathon, Florida, and about 170 miles (275 kilometres) south-southeast of Miami.
Rains over the past few days have triggered landslides, including 13 that blocked roads, and the overflow of five rivers, the Chiapas civil protection agency said.
Rainfall continues to be the greatest risk with Eta, however, with locally over 2 feet of rain possible in parts of Cuba.
Unsafe storm surges near Cuba will raise water levels as much as four feet above normal tide levels. A tropical storm warning also was in effect, including for Lake Okeechobee. Sporadic power outages from downed trees or power lines should be planned for in these areas when the strongest winds arrive.
The U.S. National Hurricane Center in Miami said the storm could become a hurricane before reaching the Florida Keys.
Hurricane watches are in effect for the Florida coastline from Deerfield Beach to Bonita Beach and the Florida Keys from Ocean Reef to Dry Tortugas. Most models aren't bringing it any farther north.
It is expected to cause a unsafe storm surge and flash floods in the country.
The Florida Keys' Mayor Heather Carruthers' orders include mandatory evacuations for campgrounds and R/V parks by no later than 9 a.m. Sunday and evacuations of trailer parks, boat liveaboards, and people residing in low-elevation areas.
"The situation is somewhat favorable for tornadoes", the weather service said.
'The deepest water will occur along the immediate coast in areas of onshore winds, where the surge will be accompanied by large and risky waves, ' the report said. The Weather Service predicts gusts of 50 to 55 miles per hour in the Miami to Homestead corridor - with the strongest occurring from Sunday night into Monday - with a few 60 miles per hour gusts mixed in on Florida's southernmost coast.
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