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Australia appeared to be cruising as they reached 144/2 thanks to a 107-run third-wicket partnership from captain Aaron Finch (73) and Marnus Labuschagne (48) before capitulating to 207 all out.
Barbados-born Archer was one of the standout performers in England's maiden 50-overs World Cup triumph earlier this month.
Archer took his tally for the series to 22 wickets, becoming the first England bowler to take two six-wicket hauls in an Ashes series since Ian Botham in 1981.
White said he had offered assurances to Archer that there would be extra security to ensure there was no repeat when the second Test started in Hamilton on Friday.
The NZC said it doesn't tolerate any kind of abuse and promised to identify the man and bring him to book.
However, Smith sounded comfortable with the prospect of an even heavier short ball barrage at Old Trafford.
The Australian coach's comments came after star batsman Steve Smith was felled by a bouncer in the second Ashes Test against England at Lord's.
Results 31-37
