The Met Office has upgraded its weather warning for Richmondshire tomorrow.
The yellow warning previously in place is now an amber warning over most of the district, meaning disruption is more likely.
Snow showers are forecast for much of the morning, followed by a break in the afternoon.
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Further snow is then forecast into the evening.
A Met Office spokesman said: “Snow showers or longer periods of snow are expected on Tuesday, accompanied by very low temperatures.
“The heaviest, most persistent snow looks likely to cross the area during the morning.
“There is the likelihood of travel delays on roads, with some stranded vehicles and passengers, as well as delays or cancellations to rail and air travel. Some rural communities could become cut off.
“Power cuts may also occur and other services, such as mobile phones, may be affected.”
The yellow weather warning remains in place for Thursday when further snow is forecast.
Temperatures are set to fall to -1C tomorrow, but could fall as low as -5C on Wednesday.
North Yorkshire Count Council says its gritters are on stand-by.
In addition to its fleet of 86 gritters, the county council can call on 111 farming contractors and a number of road and footway snowblowers to support the work to keep highways open.
The gritting crews are on call 24 hours a day and typically start gritting at 5am.
The council says priority routes on the county’s roads have already been well salted.