Why is the Gulf Stream a fast, narrow river on the western side of the ocean?

Because the Earth rotates on a sphere, and the Coriolis force is not the same at every latitude. That is the short answer, and it was Henry Stommel, in a 1948 paper, who showed it.

Here is the puzzle. The trade winds blow westward in the tropics, and the westerlies blow eastward at mid-latitudes. That wind stress has a curl, and it causes a slow flow toward the tropics across the ocean interior. The water cannot pile up, so something must carry it back toward the pole. Because of conservation of mass and of potential vorticity, it is balanced by a narrow, intense poleward current along the western coast.

The result is lopsided. Subtropical western boundary currents, such as the Gulf Stream, the Agulhas and the Kuroshio, are warm, deep, narrow, fast-flowing currents on the west side of ocean basins. Eastern boundary currents are relatively shallow, broad and slow-flowing. The Gulf Stream is typically 100 km wide and 800 to 1,200 m deep, with a maximum speed of about 2.5 m/s. It carries 30 sverdrups (30 million cubic metres per second) through the Florida Straits, while all the rivers emptying into the Atlantic together carry 0.6.

Stommel's insight was the "beta effect": the north-south change in the Coriolis force concentrates the return flow of the slow interior gyre into fast western boundary currents. His simple model had only wind stress, bottom friction, a variable surface height and the Coriolis effect; with a Coriolis force that varies with latitude, its streamlines cluster along the western coast, and with a constant one they do not. It also gave an unlikely hero: Stommel became a full professor without an earned doctorate.

These currents matter because they carry the excess heat the Earth receives in the tropics toward the poles.