Gulf stream name when it crosses the atlantic

Gulf Stream begins it the tip of the Florida, flows up the East Coast of the US and crosses the Atlantic Ocean. It causes warmer climate in the northwestern part of the Europe. It is the part of a clockwise-rotating system of currents in the North Atlantic. After crossing the Atlantic one part of the stream flow towards the British Isles and the Norwegian Seas and form the North Atlantic Current (NAC). Another part flow south joining the Canary Current.