Air travelers in the US hoping for clear skies on Tuesday following a disastrous week of weather-related flight cancellations and delays will have to extend their patience a few more days -- particularly if they're flying with Southwest Airlines.

More than 2,845 flights within, into or out of the US have already been canceled for Tuesday as of 6:15 a.m. ET, according to flight tracking website FlightAware.

Of those canceled flights, 2,509 are operated by Southwest.

Airports most affected by the Tuesday cancellations are Denver International, followed by Chicago Midway International, Baltimore/Washington International, Harry Reid International Airport in Las Vegas, Dallas Love Field and Nashville International.

Tuesday's cancellations follow a full day of post-Christmas travel chaos, with 3,989 flights canceled on Monday -- 2,909 of those being Southwest flights.

Southwest warns that such cancellations and delays are expected to continue for several more days, with representatives saying the Dallas-based airline is planning to dial back its flight schedule in order to get operations on track.

Southwest Airlines CEO Bob Jordan told the Wall Street Journal the company plans to operate just over a third of its schedule in upcoming days to give itself the ability for crews to get into the right positions.

According to WSJ.com, Jordan added that reduced schedule could be extended.

"We had a tough day today. In all likelihood we'll have another tough day tomorrow as we work our way out of this," Jordan said in an interview Monday evening with WSJ.com.

"This is the largest scale event that I've ever seen."