North Yorkshire County Council asked residents to consider whether they need to travel and be prepared for "strong winds, rain and possible snow".
Residents should secure garden furniture and other loose items that might be susceptible to strong winds, the council added.
At the coast, people are advised to avoid seafronts in case of overtopping by waves.
The council said: "Storm Barra is forecast to bring strong winds and heavy rain eastwards into the afternoon before clearing to the north-east in the evening.
"The rain may turn to snow, especially over higher ground. Gusts of 45 to 50mph are forecast, possibly increasing to between 55 and 65mph in the afternoon and early evening along the coast. The combination of strong winds and snow may cause drifting and poor visibility, particularly on the highest routes.
"The Met Office has issued a Yellow Snow Warning for parts of North, West and South Yorkshire. The forecast indicates that snow is most likely in the Harrogate and Leyburn areas.
"Our gritters will be out and you can find more information at https://www.northyorks.gov.uk/road-gritting"