Tundra cities can only grow if you build them on the coast and rush a harbour (since there's hardly any shields within these cities, unless you plant forests).
Another way a city can thrive in tundra is when wild game can be found around it. Set up a port city in close proximity to wild game, and you can have a thriving city. Simply plant forests in the tundra squares (best possible action with tundra squares). If there are furs there, then that's another bonus as furs increase shield production for some reason.
Once you build a harbour, watch the town grow (slowly but surely) into a city. Once such a city has an offshore platform, there'll be plenty of production within this city as well (unfortunately. offshore platforms come too late to be anything deciding).