Harrowes, who steps down in October, has opted to spend more time with her family. She was promoted to head marketing communications last November after Damien Whitmore left to become communications chief at Museum of Modern Art (MoMA) in New York.
The vacated role of marketing and communications director oversees the organisation's entire advertising and PR activity and heads a team of more than 50 people.
The Tate uses Wolff Olins for its advertising creative and Naked for media strategy. Most of its marketing activity revolves around promoting temporary paid-for collections such as the recent Warhol exhibition at the Tate Modern.
A spokeswoman said Harrowes' successor would be expected to renew the "reputation for experimentation of the core Tate brand, which in London competes for attention against dozens of branded visitor attractions.
The Tate operates galleries in Liverpool and St Ives, Cornwall as well as London's Tate Britain and the hugely successful Tate Modern, whose attendance figures have just topped ten million. The latter threw its doors open overnight for the first time last Saturday to cope with demand for its blockbuster Matisse-Picasso exhibition.
Tate was judged the fifth coolest brand in the UK by the Superbrands Cool Council, which explored 18- to 30-year-olds' perceptions of cool.