Washington (CNN) President Donald Trump just can't keep out of the Democratic presidential race.

Every time they have a nominating contest -- for instance, Tuesday's New Hampshire primary -- he shows up in Air Force One to loom over their party. When there's a debate, no one can stop talking about the commander-in-chief. His intervention in the race to dig dirt on former Vice President Joe Biden has already earned him the infamy of impeachment

And just as the Democratic race heats up , the President is suddenly beginning to look like a formidable opponent after three years of weak polling that suggested in might be an easy target. He skated free from his Senate trial last week with his party -- minus Mitt Romney -- in lock step marching toward battle in November. And a low turnout last week in Iowa is worrying Democratic Party leaders who had made assumptions on massive enthusiasm among liberal voters desperate to deprive the President of a second term.

Trump's headline grabbing all-time Gallup poll approval rating high of 49% has injected a note of urgency into calls for the Democratic Party to unite behind a candidate -- a seemingly distant scenario.

Most fundamentally, Trump is provoking the question defining the 2020 race: Can any Democratic candidate tame Trump's fury in November?

This is not a President riding a strong economy and a Rose Garden strategy to reelection. Trump keeps getting in the Democrats' faces, and flew to New Hampshire on Monday in a show of strength before the Granite State's primary.

During the rally, Trump suggested his supporters vote in the Democratic primary the next day.

"I hear a lot of Republicans tomorrow will vote for the weakest candidate possible of the Democrats," he said. New Hampshire law allows independent voters to pick which primary they will vote in. Voters who register as Republicans or Democrats must vote in their party's primary.

The President's comments represent more than an in-person version of the Twitter trolling to which he resorts when one Democratic candidate seems to build a bit of steam.

His event, highlighting a larger crowd than most Democrats managed in the state, underlined potency in an important swing state albeit one with four electoral votes.

It mirrored a stop in Iowa, three days before last week's caucuses, and another show of force when the President blitzed the Hawkeye State with family members and political allies.

None of this suggests the President is in trouble in the GOP primary. It's a way of stealing Democratic media coverage and inflaming his own political base in key states.

Such tactics reveal a big difference between Trump's seat of the pants effort in 2016 -- more of a lark than a traditional campaign and the sophisticated political machine he's bought with millions of fundraising dollars this time.

The President stood at 50% approval in New Hampshire in January, according to polling by CNN and the University of New Hampshire, after languishing in the mid-to-low 40s for most of 2019.

Trump's prospects suddenly look brighter than for months, since for most of the campaign, match-up polls have suggested that almost every Democrat would beat the President. But the imprecision of such surveys months from a general election matchup and Trump's formidable standing among his own base voters means that Democrats would be wise not to underestimate the President -- even if he has never yet cracked 50% in a job approval poll during his three years in office.

Trump visit opens a lane for Biden

Biden returned to the core theme of his campaign on Monday -- that he is the Trump destroyer -- after a weekend trying to revive his flagging fortunes by attacking Pete Buttigieg over his limited experience as a mayor.

"Folks, you know, you might've heard, Trump is coming to New Hampshire today," Biden said. "I can hardly wait."

"He's coming in person. You know, that's how interested he is in the Democratic primary I guess," Biden said, before lashing the President over "Hobbesian" choice economic policies that he said meant that the middle class get "clobbered."

Biden's case is however being challenged by his own halting performance so far in the Democratic race, posing the question that if he can't beat his fellow party hopefuls, how will he fare against a political street fighter like the President?

Trump couldn't resist a dig at the former vice president's underwhelming primary run last week.

"It's also very sad how he's doing -- how he's doing in the polls," Trump told reporters.

The Democratic race has included some of the most detailed debates about policy -- especially on health care -- of a Trumpian age when outrage has more often driven politics.

Yet the President's looming presence has sometimes seemed to distort the Democratic race, just as it has many other aspects of political and national life.

Daniel Guild, a Democratic activist from New Hampshire who is voting for Sen. Elizabeth Warren , said some voters were choosing a candidate based on their perception of the President.

"A lot of people have personal reactions to Donald Trump," he said, adding that voters who decry chaos might look for normality in a candidate like Biden or Minnesota Sen. Amy Klobuchar . Others might seek to flush away Trump's untruths with a candidate who shown consistency in his principles for decades, such as Vermont Sen. Bernie Sanders

"Some people are looking for intelligence. I think that's why Pete Buttigieg has done well over the last two months," Guild told CNN's Kate Bolduan. "Again, this all goes back to my own personal opinion (about) how we react differently to Donald Trump."

Each Democratic candidate has their own narrative about why they are uniquely well placed to tackle the President in November.

Sanders is pushing back on the warnings that his "democratic socialism" would be a gift for the President by arguing that he's the only candidate with the grass roots movement and low-dollar fundraising stream needed for victory.

"Donald Trump says that he would love to run against Bernie Sanders. Well, maybe not. Donald Trump lies a whole lot of the time and in fact, what I have read is some of his advisers tell him that I will be the toughest candidate for him to run against," Sanders said at a "Politics and Eggs" breakfast in New Hampshire last week.

'Unwinnable fights'

Warren says that her history of taking on difficult causes is proof that she is the best fit for an hour in which Democrats are suddenly realizing Trump might be a tough opponent.

"People are saying they're worried that we may not be able to win the fight against Donald Trump," the Massachusetts senator said on Sunday. "My view is, I've been winning unwinnable fights pretty much all my life."

Klobuchar is leaning into the argument that Sanders is too left-wing to win the support of critical center ground voters.

"Truthfully, Donald Trump's worst nightmare is a candidate that will bring people in from the middle," Klobuchar said at Friday's Democratic debate in Manchester, New Hampshire.

"The people that are tired of the noise and the nonsense. And they are tired of the tweets and the mean stuff and they are looking for someone else. And I would submit that that is me."

Buttigieg says his fresh face is the best way to win and that the choice is about "making sure we bring the perspective that will be needed into the Oval Office."

"It's not that I don't accept or don't understand the way that Washington works. But we can't accept the division and the dysfunction that is going on. And I think our best shot to change it, and by the way our best shot to beat Donald Trump, is a focus on the future," he told CNN's John Berman on "New Day."

The campaign of Mike Bloomberg , who is not competing in the first four nominating contests, seized on a new Quinnipiac University national poll showing the former New York mayor at 15% behind Sanders and Biden as proof of his electability.

"Mike is the strongest Democrat to go head-to-head against Trump," campaign spokesperson Galia Slayen said.

No doubt the President -- who has always seemed to relish publicity good or bad -- enjoys his notoriety in the Democratic race. His name was mentioned 60 times in the debate.

It's a sign that whoever the Democrats pick to carry their banner, November's election, like the turmoil of last three years will revolve around one man: Donald Trump.