Cook and Bell scored at a pedestrian pace early on, relying on the middle order to come in and smash the Kiwis out of contention.
Bell, who nurdled and nipped the ball around without really getting going edged behind on just 18 from Southee.
Cook was next to go, wafting uncharacteristically well wide of off and edging through to Ronchi.
Trott and Root tried to play themselves in but the Kiwi attack limited them to very few boundaries.
Both had looked to have battled themselves in when they got out, having taken up precious overs without the subsequent glut of runs.
Morgan and Buttler failed to fire, instead joining in with an impromptu mini collapse, before Woakes and Bresnan steadied the ship adding some much needed runs in the only decent partnership of the innings.
First Woakes fell, holing out to Guptill on the off-side boundary from Mills, before Bresnan was beaten all ends up by a beautiful slower ball from Southee.
Swann and Anderson at least looked combative in the final overs; swinging at everything, and occasionally hitting something.
Swann lobbed the last ball of the over to McClenaghan at mid-off.
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