* Storm "Alexa" brings first snowfall to Syria and Lebanon
* Newly arrived refugees say aid is not enough
* Meteorologist says more rain and cold likely
By Stephen Kalin
TEL SARHAN CAMP, Lebanon, Dec 11 (Reuters) - The season's first snow settled in parts of Lebanon on Wednesday and refugee children who have fled the war in Syria took the opportunity to have a snowball fight outside their tents.
But the worst of winter is yet to come for 2.2 million refugees living outside Syria and millions more displaced inside the country.
A storm named Alexa is sweeping across Syria and Lebanon, bringing with it high winds and freezing temperatures - and marking the beginning of the third winter since the Syrian conflict began in March 2011.
The snow wreaked havoc across the region and grounded the start of a humanitarian airlift that was meant to start bringing supplies from Iraq into the northeastern Kurdish areas of Syria, where tens of thousands of people have been out of reach.
In a tented settlement a few kilometres (miles) from the border in Lebanon's Bekaa Valley, more than 1,000 people live in rudimentary shelters.
Refugee Ibrahim, 27, spoke to Reuters in his tent where the dirt floor had turned to mud and strong winds blew snow in the entrance. Children gathered in one corner around a fire in a metal crate.
"The storm will finish us. It's freezing now. I seek refugee in God," he said.
His family does not have enough money for food or to build a sturdier shelter.
Men filled bags with dirt to hold down tents and placed car tires and bricks atop the flimsy wooden structures to prevent the wind from tearing them apart.
In Lebanon, more that 835,000 refugees live in tented camps, unused buildings or with friends or family. The Lebanese government has decided not to house them in formal camps due to local sensitivities that they will stay permanently.
Aid agencies can help them with food, tents, blankets and clothes but they cannot set up formal refugee camps.
(Click here for a graphic on refugees in Lebanon.)
Ibrahim, who arrived last week, said the aid is not sufficient for the new arrivals.
"We came here in the winter but it would have been better if we had stayed in Syria. At least if you die, you die in your own house," he said.
Simon Ingram, a spokesman for the U.N. Children's Fund, said the agency has been able to mobilise winter supplies for Syrians in the country and its neighbours but "the needs will outstrip what we and our partners are able to provide."
The start of winter coincides with a polio vaccination campaign that seeks to vaccinate 750,000 children in Lebanon after an outbreak of the disease was confirmed in eastern Syria in October.
"Polio spreads through water and sewage," Ingram said. "This is one the big dangers - overflowing drains."
NO END IN SIGHT
Chief of Lebanon's Meteorological Department, Mark Whaybeh, said that in past years, snow and rainfall had increased and temperatures decreased over December and into January.
"We are still at the beginning of the season," he said. "We should have rain and cold periods during the next two months."
Whaybeh said Alexa will last until Saturday night and that temperatures could plummet to minus 7 degrees Celsius in some mountainous areas of Lebanon.
Minister for Social Affairs Wael Abu Faour said the government was "trying its best" and that the army had been called in to make refugee shelters ready for winter.
He told Reuters his main worry was that some refugees are living in a flood area near the Litani river."
Abu Faour said he was trying to rehouse refugees near the river, which flooded during storms in January but that so far there was no place available.
"It's beyond our capacity," he said. "We tried mosques, we tried schools, we have tried everything. It will be a disaster."
Snowfall in Syria has not halted the fighting between rebels and forces loyal to President Bashar al-Assad, which has killed more than 100,000 people in the past 2-1/2 years.
Shelling continued on Wednesday across the country, including in the strategic Qalamoun area between Damascus and the Lebanese border, activists said.
Footage filmed by opposition activists and posted to the Internet on Wednesday showed the corpse of a baby they said had frozen to death in the central Syrian town of Rastan. Reuters could not independently confirm the report.
The weather has also forced the U.N. to cancel an airlift announced Tuesday which aimed to deliver food and winter supplies from Iraq to northeast Syria.
"Qamishli airport (in Syria) has suspended all flights due to weather conditions, snow and poor visibility," UNHCR spokesman Dan McNorton said.
"We're not going to be able to make those flights happen until the weather improves," he said.