Mowlam said his son indicated that passengers are trying to make the best of a bad situation.
"So far people have been pretty much taking it in stride," Mowlam said his son told him.
Rob Mowlam told his father the ship's crew had started giving free alcohol to passengers.
"He was concerned about what that was going to lead to when people start drinking too much," Mowlam said.
Other passengers have described more dire conditions, including overflowing toilets and limited access to food.
Jay Herring, a former senior officer for Carnival Cruise Lines, said one of the biggest concerns crew members will have until the ship docks is the potential for disease outbreak, particularly norovirus, which causes vomiting and diarrhea.
"Housekeeping, others are probably working double shifts to keep the mess clean and wipe down and sanitize all the common areas," said Herring, who worked for Carnival from 2002 to 2004 and spent four months on the Triumph.
Carnival hasn't determined what caused the fire, said Oliva, the company spokeswoman.
The National Transportation Safety Board announced Tuesday it has opened an investigation into the cause of the fire. The NTSB said the Bahamas Maritime Agency will lead the investigation because the ship carries a Bahamian flag.
The ship was originally going to be towed to a port in Progreso, Mexico, but after currents pushed it northward, the company decided to take it to Alabama, saying it would make it easier for passengers without passports to get home.
Cahill said Carnival has reserved more than 1,500 hotel rooms in Mobile and New Orleans for Thursday. The company plans to return passengers back to Houston on Friday using charter flights.
A similar situation occurred on a Carnival cruise ship in November 2010. That vessel, named Splendor, was stranded with 4,500 people aboard after a fire in the engine room. When the passengers disembarked in San Diego, they described a nightmarish three days in the Pacific with limited food, power and bathroom access.
Cahill said the Spendor's fire was different because it involved a "catastrophic explosion" in a diesel generator, and the Triumph's fire had "some other cause." He could not say what the economic impact will be due to the fire aboard the Triumph. The impact from the Splendor was $40 million, he said.
Carnival canceled the Triumph's next two voyages, scheduled to depart Monday and Saturday. Passengers aboard the stranded ship will also receive a full refund.