Mr Obama has made explicit what many in the British military and political establishment think, but have not said out loud: that in the battle against Islamist extremism, the priority should be Afghanistan, not Iraq. He wants to pull US forces out of Iraq in 16 months and send two more brigades, or about 7,000 more troops, to Afghanistan, shifting the emphasis from what he calls the Bush administration's "single-minded" focus on Iraq.
After an uncomfortable year in partnership with Tony Blair's great ally, George Bush, the Prime Minister appears to be looking forward to standing shoulder to shoulder with a US president far more in sympathy with his own vision. Certainly Mr Brown is in a hurry to get out of Iraq: last autumn, before circumstances changed on the ground, he announced Britain's intention to scale down its contingent from 4,000 to 2,500 by this spring. Unlike Mr Obama, he has not announced ambitions to send a specific number of extra troops to Afghanistan, but military sources say that the next rotation of forces this autumn will see more British soldiers arriving in Afghanistan than the number going home.
Crucially, the Prime Minister's plans to cut back the British presence in Iraq will not run into opposition in Baghdad, or even in Washington. Last week Mr Bush, who has always derided talk of any schedule for US troop withdrawals as dangerous, was pushed by the Iraqi Prime Minister, Nouri al-Maliki, into setting a "general time horizon" for a pullout.
The Iraqi leader makes no secret of wanting British and American troops to leave his country as soon as possible: according to the German magazine Der Spiegel, he made this explicit in an interview in which he also indicated a belief that Mr Obama was odds-on for the presidency. "US presidential candidate Barack Obama talks about 16 months," he said. "That, we think, would be the right time frame for a withdrawal, with the possibility of slight changes." After his meeting in Baghdad yesterday with Mr Brown, Mr Maliki's office suggested a total British pullout could come as soon as next July, though that was hastily played down by Downing Street sources.
As for America's objections, which stalled the British drawdown from Iraq this spring, Paul Smyth of the Royal United Services Institute (RUSI), a London-based military thinktank, said the US perception of the threat in southern Iraq had improved. Early this year American commanders were concerned about the security of their crucial supply route from Kuwait, and complained that any British troops that left would have to be replaced. The British force was hunkered down in its base at Basra airport, and half a dozen Shia militias were battling for control of the streets.
Six months later, the militias have been driven back; the curfew has been lifted and young men and women are holding hands in public, no longer afraid of being murdered by Islamist zealots. In spring an Iraqi army force, accompanied by American MTTs, or military transition teams, cracked down on the city, and the British force came out in support. In the wake of this successful operation, some American critics bluntly expressed the view that with the Iraq adventure so unpopular at home, British commanders never had the will to seize control of Basra, but Paul Smyth believes this is not entirely fair.
"This operation could only have been carried out by Iraqi soldiers dealing with their own people," he said. "The Iraqis were able to put in four times the number of troops than the entire British force, which had helped to train many of them. In that sense Britain can claim some of the credit, but it is true that the initiative was not theirs."
With Basra much calmer, however, Britain has signalled a change in strategy to what a senior British official called "moving beyond the conflict phase". The British force will formally step back from the frontline and concentrate on training the Iraqi army, with the Royal Navy carrrying out similar duties with Iraqi sailors in the Gulf and the RAF working to hand Basra airport over to civilian control. Unless Basra slips back into anarchy, it is hard to see any of these tasks taking longer than the 16 months Mr Obama would give himself next January to get all 160,000 American soldiers out of Iraq.
That is four times the American force in Afghanistan, where many others besides the Democratic presidential candidate believe the "war on terror" will be won or lost. In May and June, for the first time since 2003, more coalition soldiers were lost in Afghanistan than in Iraq. The number of British troops killed in June, 13, was the highest for any month since the 2001 intervention to drive out the Taliban and its al-Qa'ida allies.
Last year, Britain indicated the change in its priorities when the size of its force in Afghanistan overtook its deployment in Iraq. There are now 7,800 British troops in the country, according to the MoD, nearly all in the southern province of Helmand. But as in Iraq, the British force cannot make a decisive impact without American help.
"It is again a problem of scale," Mr Smyth said. "The marines have helped the British to do more in their part of Helmand." Even though British troops are still being lost in the same relatively small area, causing some to question whether the force is big enough to hold territory it has cleared of insurgents, the analyst said it was significant that most deaths were caused by roadside bombings rather than in firefights.
Even the most senior US commanders agree with Mr Obama that more forces are needed in Afghanistan, with the chairman of the joint chiefs of staff, Admiral Michael Mullen, saying "we need more forces there", but adding that he had none to spare. The most significant comment yesterday may have come from the top US commander in Iraq, General David Petraeus.
According to Gen Petraeus, even al-Qa'ida may be seeing the main battleground as shifting from Iraq to Afghanistan. There was "some intelligence" that the network could be diverting fighters to the Afghan frontier area, he said, adding: "There are unsubstantiated rumours and reflections that perhaps some foreign fighters originally intended for Iraq may have gone to the Fata" – the lawless federally administered tribal areas of Pakistan along the border with Afghanistan, where militants have a secure staging ground to prepare attacks on the Nato forces next door. Until now, said the general, al-Qa'ida communications have made clear that Iraq is its highest priority for battle. "That could be under review."
To have your say on this or any other issue visit www.independent.co.uk/IoSblogs
Comments