This is technically incorrect. The Gulf Stream is not the same as the AMOC, which is not the same as the Thermohaline circulation. Consult NOAA for the correct definitions of each. (e.g. https://www.gfdl.noaa.gov/bibliography/related_files/td0802.pdf)

In the aftermath of the idiotic movie "The Day After Tomorrow" and associated discussion in Nature journal, leading oceanographer Carl Wunsch of MIT was moved to write the following letter to Nature journal:

"Sir —Your News story "Gulf Stream probed for early warnings of system failure" (Nature 427, 769; 2004) discusses what the climate in the south of England would be like "without the Gulf Stream". Sadly, this phrase has been seen far too often, usually in newspapers concerned with the unlikely possibility of a new ice age in Britain triggered by the loss of the Gulf Stream.

European readers should be reassured that the Gulf Stream's existence is a consequence of the large-scale wind system over the North Atlantic Ocean, and of the nature of fluid motion on a rotating planet. The only way to produce an ocean circulation without a Gulf Stream is either to turn off the wind system, or to stop the Earth's rotation, or both.

Real questions exist about conceivable changes in the ocean circulation and its climate consequences. However, such discussions are not helped by hyperbole and alarmism. The occurrence of a climate state without the Gulf Stream any time soon — within tens of millions of years has a probability of little more than zero."

NOAA definitions: The terms Atlantic Meridional Overturning Circulation (AMOC) and Thermohaline Circulation (THC) are often used interchangeably but have distinctly different meanings. The AMOC is defined as the total (basin-wide) circulation in the latitude depth plane, as typically quantified by a meridional transport streamfunction. Thus, at any given latitude, the maximum value of this streamfunction, and the depth at which this occurs, specifies the total amount of water moving meridionally above this depth (and below it, in the reverse direction). The AMOC, by itself, does not include any information on what drives the circulation.
In contrast, the term “THC” implies a specific driving mechanism related to creation and destruction of buoyancy. Rahmstorf (2002) defines this as “currents driven by fluxes of heat and fresh water across the sea surface and subsequent interior mixing of heat and salt.” The total AMOC at any specific location may include contributions from the THC, as well as contributions from wind-driven overturning cells. It is difficult to cleanly separate overturning circulations into a “wind-driven” and “buoyancy-driven” contribution. Therefore, nearly all modern investigations of the overturning circulation have focused on the strictly quantifiable definition of the AMOC as given above.