(CNN) The White House moved to tamp down suggestions that ties with Saudi Arabia are fraying, with administration officials saying that President Barack Obama "really cleared the air" with King Salman at a meeting Wednesday.

Yet even as White House officials stressed that the leaders made progress, a prominent member of the Saudi royal family told CNN "a recalibration" of the U.S.-Saudi relationship was needed amid regional upheaval, dropping oil prices and ongoing strains between the two longtime allies.

Obama landed in Riyadh earlier Wednesday for a summit with Gulf leaders and spent two-and-a-half hours meeting with the 80-year-old monarch on issues that have recently strained the alliance, including the conflict in Yemen, the role of Iran, Lebanon's instability and the fight against ISIS, U.S. officials said.

Statements after the meeting made clear that deep differences remain on several of these points, with the two sides agreeing to disagree and a U.S. official characterizing the encounter as the start of a discussion rather than a venue for solutions.

But the two leaders glossed over some of the thorniest matters, including a Saudi threat to dump U.S. assets if Obama signs into law a bill that could make the kingdom liable for damages stemming from the September 11 terror attacks.

There is going to have to be "a recalibration of our relationship with America," former Saudi Intelligence Chief Prince Turki Al-Faisal told CNN's Christiane Amanpour. "How far we can go with our dependence on America, how much can we rely on steadfastness from American leadership, what is it that makes for our joint benefits to come together," Turki said in a significant departure from usual Saudi rhetoric. "These are things that we have to recalibrate."

An unhappy marriage

For all the crosscurrents buffetting U.S.-Saudi relations, analysts and former officials say the two countries aren't at the end of a love affair so much as in an unhappy marriage in which both sides, for better or worse, are stuck with each other.

"Despite all these differences, Saudi Arabia and America are not getting divorced," said Bruce Riedel, director of the Intelligence Project at the Brookings Institution and a former CIA official. "We need each other."

It's tough going, though. The Saudis have little confidence in Obama's commitment to their security and fear he's shifting U.S. attentions to its rival, Iran; Obama has described the Saudis as "so-called allies" and has complained their policies fuel anti-U.S. terror and regional chaos.

In the U.S. Congress, a growing drumbeat of criticism about Saudi Arabia is finding expression in efforts to restrict arms sales to Riyadh, expose alleged Saudi involvement in the September 11 terror attacks and allow it to be sued for that day's destruction and death.

The clamor coincides with increasing domestic energy resources that lessen the U.S. need for foreign oil. Moreover, the allies are divided by a slew of issues including the approach to the wars in Syria and Yemen , the Iranian nuclear deal and the influence Tehran wields in Iraq.

These regional issues are topping Obama's agenda during his visit this week as he looks for backing for the fight against ISIS in Syria and Iraq. And they are dynamics that are set to persist and color the U.S.-Saudi relationship for the next occupant of the Oval Office as well. Turki told CNN that the changes underway will last long after Obama leaves office.

"I don't think that we should expect any new president in America to go back to, as I said, the yesteryear days when things were different," Turki said.

A chilly welcome

The prince made his unprecedented comments as Obama landed in Riyadh to a reception that social media critics termed a snub, but U.S. officials strongly disputed. The Saudi government dispatched the governor of Riyadh and Foreign Minister Adel Al-Jubair to shake Obama's hand, a departure from the scene at the airport earlier in the day when King Salman was shown on state television greeting the leaders of other Gulf nations on the tarmac.

A U.S. official said Salman's absence upon arrival was not taken as a snub and noted that Obama rarely greets foreign leaders when they land in the U.S. for meetings. Obama went immediately to the Erga Palace to meet the King shortly after landing, but the perceived slight on his arrival was seen as one more sign that a relationship long lubricated by barrels of oil is encountering friction.

Fawaz Gerges, an expert on Islamic-Western relations at the London School of Economics, called their current dynamic "an estrangement" but not a break that would end U.S. involvement in the Middle East.

He's among many analysts who say that, as unlikely as the union between a rigidly conservative Islamic monarchy with a questionable human rights record and a secular democratic republic may seem, neither will be able to cut the ties the bind them.

The two countries are connected by military links and sales, a shared fight against terrorism, the need to leverage each other's diplomatic clout and, for the U.S., the necessity of ensuring that world oil supplies flow freely.

Though the U.S. imports fewer barrels of Saudi crude and petroleum than it did on the day of Obama's first inauguration, the energy needs of its allies -- particularly in Asia -- are crucial to global and U.S. economic health.

"U.S. energy independence doesn't really change the equation that much because of the global strategic importance of the oil supplies," said Hussein Ibish, a senior resident scholar at the Arab Gulf States Institute in Washington.

Saudi Arabia also carries diplomatic weight in the region that the U.S. has used to serve its interests.

The "Saudis are such an influential actor in the Middle East and broader Muslim world that no secretary of state or president has truly wanted to go it without them," said David Weinberg, a Saudi Arabia expert at the Foundation for the Defense of Democracies.

As the Arab Spring has devolved into chaotic violence, Saudi Arabia has provided funds that have stabilized key U.S. allies, including Egypt, Bahrain and Jordan, and it has developed stronger ties with one of its longtime enemies, Israel, the closest U.S. ally in the region.

War on Terror: Offering Arab cover

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On the most kinetic level, the two countries are linked by counterterrorism efforts that will go on for years.

It was reported in 2013 that the U.S. operates an unacknowledged drone base out of Saudi Arabia and is relying on the country to fight al Qaeda in the Arabian Peninsula, the Yemen-based group that the Obama administration has called the most serious threat to the American homeland.

Separately, the U.S. "needs Saudi Arabia to provide Arab cover for the American-led coalition fighting the Islamic State in Iraq and Syria," said David Ottaway, a Wilson Center expert on the kingdom. "The overall U.S. war on terrorism in the Middle East cannot be won without Saudi help."

The militaries of both countries are linked in nuts-and-bolts ways, too. Saudi Arabia "is by far the world's largest purchaser of U.S. weaponry," said Nawaf Obaid, a visiting fellow at Harvard University who estimated that it has is over $100 billion worth of equipment on order from U.S. defense contractors.

Not only does that translate into jobs for U.S. workers, but the equipment also comes with support from the Pentagon in the form of training that creates deep ties between the two militaries.

The Saudis, for all their frustration with the U.S., "just don't have alternatives," said Ibish. "They can talk about Europe and China and Russia all they like, but in the end, its military is structured around the United States and only the United States can provide the leadership they're looking for."

Uncertain times for the House of Saud

The tensions are compounding an already uncertain time for the House of Saud. A transition in 2015 introduced a new and largely untested group of leaders at a time when the falling price of oil has saddled Saudi Arabia with its first budget deficits.

And it has embarked on a protracted war in Yemen that is earning it U.S. condemnation even though the Saudis see themselves as shouldering their region's security challenges just as the White House has asked.

Saudi Arabia, feeling threatened by the overthrow of a friendly government on its border, has intervened against the Iran-backed Houthi rebels while also trying to strike al Qaeda there.

"Obama was encouraging them to take responsibility for their own security and that's what the Saudis are doing," Ottaway said. "But when your allies decide they're going to act on their own, they don't necessarily do what you want them to do."

The administration has quietly criticized the humanitarian cost of the Yemen conflict, for which it is providing intelligence, weapons and ammunition as the region's strongest al Qaeda affiliate exploits the conflict.

"Al Qaeda's most dangerous branch is seeing this Saudi-led war as a godsend," Weinberg said. "That's something the Saudis haven't treated as a priority."

Saudis recoil from U.S. criticism

The criticism is another sore point for Saudi Arabia, which has long had fraught relations with Obama.

Even before he became President, Obama referred to the Saudis as "our so-called allies" at a 2002 rally. After he took office, Riyadh saw his decision to support the 2011 ouster of long-time Egyptian leader and U.S. ally Hosni Mubarak as a betrayal of the established order.

That was compounded when the U.S. announced it had been secretly meeting with Saudi Arabia's regional arch-rival Iran for talks that led to the 2015 nuclear agreement.

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Saudi fears that Obama's so-called pivot to Asia meant he was pulling away from the Middle East deepened in 2012, when the President set a red line for taking military action against the Syrian regime -- a government backed by Iran and opposed by Riyadh -- and then changed his mind once Damascus crossed it.

When Obama told The Atlantic Magazine this year that he was aggravated by "free riders" -- "people who push us to act but then showing an unwillingness to put any skin in the game" -- he earned a rare public rebuttal from the Saudis.

Though Obama hadn't named any country, Turki asked him in an open letter whether it was "because you have pivoted to Iran so much" that he had forgotten "the Kingdom's 80 years of constant friendship."

The 28 pages

After the years of increasing distance , many in Saudi Arabia have adopted a mantra that Weinberg of the FDD said boils down to "wait until the next administration."

But that may be a mistake, he warned, as there is increasingly vocal anti-Saudi sentiment in Congress and the growing gaps between the two nations are dependent on circumstance rather than individual leaders.

A bipartisan group of current and former lawmakers are pushing for the release of 28 classified pages from the 9/11 Commission that reportedly include evidence that Saudi officials in the U.S. at the time lent support to some of the 15 Saudis who were among the 19 terrorists who conducted that attack.

A large bipartisan group of Senate and House lawmakers has backed a bill that would allow families of the 9/11 victims to sue foreign states if they helped fund or support a terrorist attack on U.S. soil. While the White House is lobbying against the bill, the Saudis have reacted by warning they would sell off $750 billion in U.S. assets if it becomes law.

The veto threat led families of 9/11 victims to write Obama on Tuesday to say they "view the disregard and dismissal of our loved ones' deaths implicit in both these threats and their acceptance as disrespectful and improper."

The bill has drawn the backing of Democratic presidential candidates Hillary Clinton and Bernie Sanders and is co-sponsored by GOP candidate Ted Cruz. And Republican front-runner Donald Trump has said that Saudi Arabia should pay for more of its own defense.

Meanwhile, another group of Republican and Democratic senators banded together last week to propose a bill to limit arms sales to Saudi Arabia to protest the way it's conducting the war in Yemen.

The dynamic points to continued tension after Obama leaves office.

There are areas where the U.S. and Saudi Arabia are in conflict, and political elites across the board "reflect those feelings," according to Weinberg.

"This is not something that can be personalized down to the views of the commander in chief," he said.