* Sudden rise in migrants risking Mediterranean crossing

* Factional conflict in Libya helps people smugglers

* Libyan coast guard depleted after years of chaos (Adds comments after Italo-German summit)

By Steve Scherer and Ahmed Elumami

ROME/TRIPOLI, Aug 31 (Reuters) - Calmer seas and Libya's lawlessness have opened the way for smugglers to ship thousands of migrants across the Mediterranean this week, in a striking reminder of how far Europe is from ending the migrant crisis.

In just four days, Italy's coastguard and European vessels pulled 13,000 migrants from packed wooden boats and rubber dinghies crossing from Libya's coast through the Strait of Sicily, one of the shortest routes from North Africa.

Images from rescue vessels showed migrants crammed into fragile boats, some in orange life jackets, others jumping into the water to swim as rescuers shouted for them to stop. Many were women and children, most of them Subsaharan Africans.

Some 1,800 migrants were saved in 12 rescue missions on Wednesday, 3,000 migrants on Tuesday, 6,600 on Monday, 1,600 on Sunday and 500 on Saturday, according to Italy's coastguard, which coordinates sea rescues between Libya and Italy.

Aid agencies and Libyan officials say calm waters after rough winds in July may have prompted smugglers to dispatch more boats for migrants who can wait weeks for a chance to sail.

"It's not like people were not coming out, but on Monday they came out by the thousands," Nicholas Papachrysostomou, coordinator for Medecins Sans Frontieres on board the Dignity 1 rescue ship, told Reuters.

"In the 10 days before this weekend, we didn't rescue anyone, or very few. It wasn't bad weather, but there were 1 metre to 1-1/2 metre waves. The rubber boats won't come out in those conditions."

In Tripoli, naval authorities say they are undermanned and ill-equipped. Years of conflict and chaos have left their naval and coastguard services in tatters.

Since the 2011 NATO-backed revolution toppled Muammar Gaddafi, Libya has descended into chaos, with rival governments and armed factions battling for control, leaving both migrant smugglers and Islamist militants space to expand.

"This is the season for illegal migrants. The calm seas have helped them take advantage," said Ayoub Qassem, a spokesman for Libya's naval forces in Tripoli. "Also our vessels are in severe need of maintenance and spare parts to operate again."

SUDDEN SPIKE, STEADY FLOW

Still, this week's rescues were on a huge scale. On Monday, when some 6,500 were rescued, the Dignity 1 responded to a distress call for a large wooden boat in the morning.

After pulling more than 600 migrants off that boat, MSF's Papachysostomou suddenly realised that many more were on their way. "These boats just appeared on the horizon. At about noon you could see rubber boats everywhere, in 360 degrees," he said.

The International Organization for Migration said around 105,000 migrants have reached Italy by boat in 2016. An estimated 2,726 men, women and children died trying to make the journey. Three more bodies were retrieved on Wednesday.

Italian Prime Minister Matteo Renzi and German Chancellor Angela Merkel discussed the immigration crisis at a meeting in northern Italy on Wednesday, saying more must be done to improve living standards in Africa.

"Italy will continue to save lives, but there are limits," Renzi told reporters. Merkel said it was important that migrants who had no right to claim asylum should be repatriated, adding that they needed incentives to stay at home.

"That's a very big challenge, but there is no reasonable alternative than to cooperate with these countries and give their citizens the outlook for a better future," she said.

This week's surge in new arrivals might also be due to the situation in Libya, with smugglers perhaps trying to avoid armed rivals or police by shipping as many people as possible, said Carlotta Sami, spokeswoman for the UN refugee agency in Italy.

"There are moments in which no one leaves from Libya, and then there are moments in which there's a surprising peak in departures...It's like a roller coaster."

A new U.N.-backed government has been in place in Tripoli since March, but it is struggling to end factional fighting and impose its authority over hardliners.

With EU naval vessels patrolling international waters, smugglers elude authorities by launching their boats from varied locations.

Remote beaches along some parts of Libya's coast are sometimes littered with clothing, bottles and shoes migrants have left behind; or the wooden bases of inflatable rafts.

"Most of our patrol vessels do not work at the moment," Tripoli naval officer Qassem said. "So we can not patrol in the western region of Tripoli where most of the boats of illegal migrants set off."

(Additional reporting by Antonella Cinelli and Crispian Balmer in Rome, Stephanie Ulmer-Nebehay in Geneva and Andrea Shalal in Berlin; writing by Patrick Markey; editing by Ralph Boulton)