Sen. Robert Menendez (D-N.J.) arrives at court last week in Newark with daughter Alicia Menendez and son Robert Menendez Jr. (Seth Wenig/AP)

Ukrainian-born model Svitlana Buchyk was a less-than-ideal witness for federal prosecutors this week in the bribery trial of Sen. Robert Menendez.

Though she was called by the prosecution, Buchyk’s testimony undercut the government’s case on some points. She said the New Jersey Democrat and his co-defendant, Florida eye doctor Salomon Melgen, were very close — undermining the notion that the doctor had bribed the senator in exchange for government favors.

Buchyk was asked at one point whether she knew why she was in the Newark federal courtroom.

“No. I don’t know why I’m here. He has just been forcing me to be here,” she said, referring to the lead prosecutor, Peter Koski. “That’s why I’m here.”

Prosecutors say that Buchyk was a former girlfriend of Melgen’s and that when she wanted a visa to come to the United States, Menendez helped her get one. That help, according to the Justice Department, was part of a years-long pattern of corruption in which the wealthy doctor gave the senator private flights, a hotel stay and indirect campaign donations in exchange for using his influence as a senator to help Melgen with personal and professional issues.

But in her testimony Tuesday, Buchyk made clear she was not on the side of the government.

“Just very frustrating,’’ she said at one point, an apparent reference to being called as a witness. She also drew laughter in the courtroom when she complained about her time with the prosecutors, saying, “It just seems like very long when I’m around them, but it’s very, very long.”

When U.S. District Judge William Walls told her she could leave the witness stand, the model exclaimed "Wow!" and pumped her arms in the air. She smiled at Melgen as she left the courtroom.

After the first full week of testimony, the legal strategy for both sides in the Menendez trial is crystal clear. Through flight records, emails and witness testimony, prosecutors with the Justice Department’s public-integrity section are trying to show the jury that time and again, Menendez sought to use his role as a senator to help Melgen, even though Melgen was not a New Jersey constituent.

The defense strategy, drawn out of testimony from Buchyk and others, is equally plain: Convince the jury the senator had no intention of taking a bribe or acting corruptly. Instead, they insist, he was a devoted friend to Melgen — the two men so close they called each other “hermano,” the Spanish word for brother. The two weren’t trying to hide their relationship in any way, and that shows they are innocent, defense lawyers have argued.

The Menendez trial is not a fight over facts but meaning. When Melgen gave Menendez gifts, such as a nearly $5,000 stay in a luxury Paris hotel, was that a bribe or a gift from a friend? When Menendez did things for Melgen, such as intercede with federal health officials when the doctor got into an $8.9 million billing dispute with the government, was that an effort to clear up some bureaucratic confusion or a corrupt attempt to get a dishonest friend out of trouble?

Emails are a central pillar of the case, offering what prosecutors say are telling clues of the senator’s intentions.

A key bit of evidence offered at trial Thursday was a series of emails, starting with one from a Washington Post reporter in November 2012 to a Menendez staffer, asking about the senator’s travels on Melgen’s plane.

The Post reporter asked whether Melgen had taken a trip to the Dominican Republic in 2010 on Melgen’s private jet and whether he had gotten a letter from the Senate Ethics Committee approving such activity.

Three weeks after that query, Menendez got a quote from Melgen for the cost of the trips that had occurred two years earlier. The senator reimbursed the doctor for two of those flights, at a cost of $58,500.

In a case in which the jurors will be asked to infer what Menendez thought he was doing when he took things from Melgen, and when he did things for him, the fact that the payment came only after a reporter asked about it could prove significant. The payment also reflects how Menendez was in some ways already boxed in by Senate ethics rules by the time the question was raised.

Elliot Berke, a lawyer specializing in congressional ethics issues, said the Senate’s gift rule “allows senators to pay market value to prevent a gift from being given, or to avoid an impermissible gift. So it appears that’s why Senator Menendez decided to pay for the flights.’’

The trial is expected to last two months, though Walls has repeatedly voiced frustration that the lawyers may drag it out through extended questioning. The judge has had a short fuse at the trial, chastising the lawyers multiple times a day when he thinks they are straying from the rules of court procedure.

Though he has repeatedly warned the prosecutors not to try to create a “tabloid” atmosphere, it’s the judge who has shown the most emotion so far.

On Thursday, he exploded at both sides for not heeding his prior warnings.

“Some of you are so arrogant in your activities that you think I’m ignoring your disregard of what I told you,” he said. “I have no dog in this fight. I don’t care who wins or loses. But you have to follow the rules of engagement.”

Like a teacher calling out misbehaving students, Walls then went down the long line of attorneys for both sides and asked them individually to agree to abide by the rules.

At other points, though, Walls has tried to lower the temperature in the courtroom. When a former Menendez staffer finished testifying Tuesday, the judge jokingly urged him to “run.”

The witness, Mark Lopes, replied: “I will. Thank you.”

Barrett reported from Washington.