There's no "default" weather app. Default means that it's the app that runs when ... whatever. Like when you "run" a .pdf file, your default PDF reader runs it.
There's no weather app that will run in response to something outside it like that - about the closest you'll find is one that will pop up a notice if there's a storm headed toward you, or something like that. Go to the Play store and look for weather widget and pick the lone you like. Many of them allow you to keep a little (or not so little) icon on the home screen showing the current temperature, weather (like the sun, clouds, lightning [meaning thunderstorms], snowflakes, etc.)