So for just the second time in history, the Greek alphabet will be used, the World Meteorological Organization (WMO) said.
Arthur was the first storm on the 21-name list prepared for the 2020 season, which ends on Nov. 30.
Hurricanes Paulette and Sally and Tropical Storms Teddy and Vicky are still active, and a low pressure system off West Africa could develop into a cyclone and would be christened Wilfred.
"Once we exhaust the last name on the list, which is Wilfred, we then have to switch to the Greek alphabet and you will start to see names such as Alpha, Beta, Gamma, Delta, Epsilon, Zeta etc," the WMO's Clare Nullis told reporters.
This year's season is set to be the busiest ever with the US National Oceanic and Atmospheric Administration forecasting a record 25 storms which it attributes to warm sea temperatures and low wind-shear. Climate change is thought to be increasing their intensity.
As of Monday, there were advisories for five cyclones swirling in the Atlantic for only the second time in history. However one of them, Rene, has since dissipated but it still means the 'R' name choice had gone.
No names have been prepared X, Y or Z this year because there are few appropriate names that begin with them, WMO said.
Since, 1953, tropical storms have been named for ease of communication with mariners and the public and were initially only female. Historical storms were identified by their coordinates or the things they destroyed.
Meteorologists now rotate through six, alphabetical stock lists of alternating male and female names maintained by a WMO committee. The last time it resorted to Greek names was in the busy 2005 season which included Hurricane Katrina.
Occasionally, a storm name is "retired" by the committee if it proves to be very destructive, such as Mitch which struck Honduras in 1998.
Nullis said the WMO is then bombarded with replacement requests. "Either people are outraged that a storm has their name or they say: please name it after my husband or wife."