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The centre of Tropical Storm Eta was expected to pass near or over the Florida Keys Sunday night and early Monday, the US National Hurricane Center reported.
As of 10pm Sunday, the storm had sustained winds of 65 miles per hour and was moving in a northwesterly direction at about 14 miles per hour.
Eta is expected to strengthen Monday and is forecast to become a hurricane when it moves over the southeastern Gulf of Mexico.
The NHC said tropical-storm-force winds were extend outward up to 310 miles from the centre on Sunday night.
The storm is forecast to bring up to 12 inches of rain to the central and southern Florida peninsula.
Eta passed over Cuba on Sunday, bringing heavy rains and strong winds. Local reports said the storm swelled rivers and flooded coastal zones in Cuba, where 25,000 had been evacuated.
Tropical storm conditions were expected to spread across the Florida Keys Sunday night, with hurricane conditions forecast for a portion of the Florida Keys by early Monday morning.
Tropical storm conditions are expected to continue in the northwestern Bahamas and the Florida peninsula through early Monday.
The NHC said tornadoes were also possible through Monday across southern Florida and the Keys.
Following the passage of Eta about 30 miles from Grand Cayman on Saturday, sea swells were continuing to impact the west and south coasts of the island on Sunday night. Wave heights on Sunday evening through Monday were expected to be 4-6 feet.
A small craft warning remained in effect Sunday night.
According to Cayman’s National Weather Service’s 10pm bulletin on Sunday, the Cayman Islands was expected to experience isolated showers, and moderate winds and seas, over the following 24 hours.