Heavy on atmosphere as well as heavy-handed symbolism, “The Ballad of Jack and Rose” is an absorbing but strangely unconvincing new film by Rebecca Miller. Worst of all, it lacks a coherent narrative and strong dialogue as well, enough to sink a lesser work. To its credit, “Ballad” borrows the isolated atmosphere of Jane Campion’s beautiful “The Piano.” Yet Miller (the daughter of the late playwright Arthur Miller) fails to capture that film’s warm humanity and believability, instead falling for a series of cliched and absurdly conceived plot twists.

At its best, “Ballad” is gorgeously and sensitively filmed by acclaimed cinematographer Ellen Kuras. The film’s setting, an island off the coast of New England, lends itself to continual shots of surprising beauty. Miller expertly creates evocative moments out of graceful and fluid visuals, unanticipated but effective editing choices and a masterful use of music (Bob Dylan, Creedence Clearwater Revival and Nina Simone among others). But Miller’s obvious plot overpowers these mesmerizing attributes about half an hour into the film.

The film stars Daniel Day-Lewis, the brilliant method actor — and Miller’s husband — who acts here in only his second film in eight years. He plays Jack Slavin, an ex-hippie who’s the only remaining member of a commune founded in the ’60s. Jack, whose wife long ago abandoned him, lives in isolation with his young daughter Rose (the promising Camilla Belle), with no connection to the outside world besides occasional visits to the mainland.

Jack is a difficult character — at once hypocritical and elitist (he insults and threatens real estate developers) — and Day-Lewis nails him in all his complexities with effortless skill. Jack’s health issues (he suffered a heart attack several years earlier) provide a constantly ominous threat that complicates their life together.

It is apparent early in the film that Rose’s relationship with her father is unusual, if not outright disturbed. Rose, who was taken out of school at 11 and barely has had any human contact since then (not even television), latches onto her father with both filial and apparently sexual love. Her near-incestuous obsession leads her to repeatedly declare her intention to commit suicide when he dies, a promise that captures the film’s dire moroseness. To Belle’s credit, she is largely convincing in this ever-difficult role, though she occasionally exaggerates in the more emotionally fraught scenes.

On the other hand, Catherine Keener — who plays Jack’s lover Kathleen — is outright melodramatic, detracting painfully from the film. “Ballad” takes a terrible turn when she moves into Jack’s home, along with her teenage sons, in order to take care of the increasingly sick man. Kathleen’s boys provide Rose with her first contact with people her own age, introducing her to sex, drugs and other temptations. As expected, she lashes out against the woman who rapidly takes her place as the object of her father’s affections, and the results are nearly disastrous.

The heavy-handed themes of forbidden love, lust, temptation and innocence come to full fruition in the second half of the film, and Miller isn’t stingy with the religious imagery. Rose’s first sexual experience occurs on her bed as a snake slithers underneath. Kathleen, who is explicitly referred to as a “savior” early in the movie, obviously turns into a temptress who corrupts Jack and Rose’s relationship. Miller even uses a storm — destroying Rose’s tree house, an annoyingly overt symbol of her innocence — to signal the watershed moment when Kathleen and her children interrupt the calm of Jack and Rose’s lives.

In addition to her penchant for vulgar symbolism, Miller’s dialogue is often hackneyed, obvious and at worst pretentious. And the inconceivable plot twists at the film’s climax, involving Kathleen and her children, violently alienates the audience. A chase scene involving the snake is even unintentionally amusing, disrupting the mood of the film. Given the wonderfully reflective and atmospheric tone of the film, the amount of action Miller tries to cram into its end unnecessarily weighs it down.

Despite these gripes, there is a great deal to like about “The Ballad of Jack and Rose.” Miller is a talented director, and her pacing and visuals are perpetually superb. With the help of Day-Lewis’ quiet intensity and Belle’s star-making turn, she creates an engrossing world populated by two complex and believable characters with wonderful chemistry. If Miller had made the central narrative of her film less busy and more convincing, “Ballad” could have been a genuinely deep and profoundly affecting film.