The general atmospheric circulation dictates that the winds in the Equatorial Pacific blow from east to west, which are the easterly trade winds. This tends to push the warmer Equatorial waters to the west, towards Indonesia, and leaves colder water along the coast of Peru. However, when the easterly trade winds either significantly weaken, or even at times shift to the west, that warmer water no longer is forced to the west. It can either stay in the Central Equatorial Pacific (a west-based El Nino), or if there is enough westerly forcing to completely dominate the easterly trade winds, the warmest waters can actually push eastward, all the way towards the coast of Peru. This often happens during the strongest El Nino events, and has major implications on the atmospheric circulation.
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The three most notable examples of the Modoki Nino are 2002-03, 2004-05, and 1994-95, along with 1986-87. El Nino Modoki can essentially be translated into "west based nino" as it featured pronounced warming of the central tropical pacific waters rather than the eastern pacific. What happens in terms of the physics of the event is tropical westerly trade winds intensify, pushing surface warmth to the east; however, the westerlies are not strong enough to force the warmest water to pile up near the west south american coastline (the strength of the trades are ultimately determined by an oscillating high/low atmospheric pressure regime, called the SOI, which I frequently referred to recently in support of the upcoming warmth). It's not a surprise that we've never seen a west based (Modoki) strong el nino, as strong ninos are always associated w/ extremely powerful westerly trades and thus the warmest SST's relative to normal in the eastern trop pacific. Many weak ninos as well as a handful of mod ninos have been west based and thus Modoki classification.