They could see the outbreak coming three days in advance. It was the classic set-up. Up high, a mass of cold air, the moisture wrung out of it by the Rockies, screamed east across the Great Plains. Down low, near the surface, soupy air surged north from the Gulf of Mexico.

At 2:23 a.m. Central time Wednesday, government forecasters in Oklahoma sent out a 3-day Convective Outlook, warning that on Friday afternoon a line of storms would race across the eastern United States and could involve “supercells and accordingly enhanced tornado potential.”

That proved tragically correct. Friday’s tornado outbreak killed more than three dozen people in Indiana, Kentucky, Ohio, Alabama and Georgia, with the death toll rising steadily Saturday as rescuers searched through the rubble across a vast region east of the Mississippi.

As of Saturday afternoon, the death toll stood at 38, the Associated Press reported . Among those injured was a 2-year-old girl found alone in a field after her parents and two siblings were killed by a tornado in New Pekin, Ind., according to the Louisville Courier-Journal.

The girl was taken to Kosair Children’s Hospital in Louisville. Hospital spokesman Brian Rublein said Saturday afternoon that the toddler was in critical condition. He confirmed that her immediate family had been killed but said she had relatives at her bedside. The hospital would not release further details.

One long-track twister, packing winds that scientists said might have hit 200 mph, chewed through the Ohio River Valley at more than 50 mph and obliterated much of Henryville, Ind.

Narrow escapes were numerous: A Henryville school bus driver, attempting to return children to their homes, saw the twister coming and returned to the town’s junior-senior high school just in time for the remaining kids to take shelter as the twister ripped the roof off the gymnasium and ravaged the school. The bus impaled a restaurant across the street.

Indiana Gov. Mitch Daniels (R), who toured the Henryville devastation Saturday, said in an official statement released by his office, “We’ve learned so much and improved so much in disaster preparedness, warning systems and responder communications but still we are no match for Mother Nature at her worst.”

The tornado outbreak affected a huge area as the line of storms swept east like a harvest sickle. The tornadoes came close to, but spared, such major cities as Louisville, Cincinnati, Nashville and Atlanta.

The Friday storms weren’t as destructive as the extraordinary outbreak April 27, 2011, when 317 people were killed by 199 tornadoes in the Deep South, with Alabama the hardest hit. Last year, there were outbreaks on three other days in April — the 15th, 16th and 26th — and an outbreak on May 22 in which a monster tornado leveled Joplin, Mo.

No one knows whether this year’s storm season will match last year’s mayhem. Researchers dream of being able to make a reliable seasonable forecast rather than one that looks out only a few days or a week ahead.

The connection between climate change and tornado formation and intensity is a subject of ongoing research. A warmer atmosphere holds more moisture and can lead to greater deluges, but meteorologists have not established a causal connection between climate change and twisters. “We don’t particularly have strong expectations for changes in tornado occurrence with climate change,” Harold Brooks, research meteorologist with NOAA’s National Severe Storms Laboratory, said Saturday.

What’s certain is that the 2012 tornado season is off to an early, and roaring, start. Tornadoes earlier in the week touched down in Missouri and Illinois. Although an accounting of the number of tornadoes Friday is incomplete — as of Saturday afternoon, the official tally stood at 51 — the outbreak will likely rank as one of the biggest on record for the month of March.

“A March tornado outbreak of similar scope to yesterday occurs roughly once a decade,” Russell Schneider, director of NOAA’s Storm Prediction Center in Norman, Okla., said Saturday.

In producing the “Day 3 Severe Thunderstorm Outlook” on Wednesday morning, the Storm Prediction Center put a bullseye on central Kentucky and central Tennessee and parts of adjoining states. The next day the forecasters ramped up the possibility of severe storms in the same area, noting in their bulletin, “long-track/significant tornadoes are possible.”

Finally, on Friday morning, the center declared, “Outbreak of strong tornadoes and damaging winds likely today over a large area. . . . Storms currently over [southeastern Missouri and Illinois] should thrive as they move across a wide and destabilizing warm sector. This should favor long lived and fast moving tornadic supercells capable of extensive damage.”

So it came to pass. TV meteorologists in places such as Louisville and Cincinnati found themselves monitoring multiple incoming supercells and coaching viewers on when to take shelter.

Although experts can see a tornado day coming, they can’t yet say precisely where they will form. Tornadoes remain unpredictable in part because most supercell storms, even the most frightening ones, don’t give birth to twisters.

“We really don’t know what causes some thunderstorms to cause tornadoes and others not to,” said Steve Weiss, chief of the science support branch at the Storm Prediction Center.

Technology has enabled forecasters to improve their ability to issue a warning that a tornado is about to form. In the 1980s, three out of four tornadoes formed before any warning had been issued, Brooks said. But he said only about one in four now form prior to a warning.

Brooks said the creation in the early 1990s of a national network of doppler radars — which can detect rotational motion within a storm — gave the profession a huge boost.

“We’re able to bring all the radars together so we can see a national mosaic of radar data. You can actually watch storms moving all the way across the country,” Brooks said.

Typically, there’s still only 20 minutes between the issuance of a tornado warning in a specific location and the arrival of the tornado, he said. It’s unclear whether more lead time would be helpful, he added, because people might not take immediate shelter.

“If you tell people you’ve got 45 minutes, they may decide, ‘Wow, I really do need to run to the store and get those batteries.’ Then if your timing is off on the arrival, you may have a lot of people driving around, and that’s a bad thing.”

Weiss said that in the 1970s he and his colleagues could forecast severe weather one day in advance. Now, there’s a severe weather forecast that looks a week ahead.

He noted that social media, such as Facebook and Twitter, help spread information about severe weather. Fewer people are caught completely off guard by severe storms.

“By the time you hear the warning, if you haven’t considered what to do if a storm is going to threaten you, that may be too late,” Weiss said.