A shocking photograph posted on social media shows damage to the line at Bramhope.
Services have been cancelled between Leeds and Doncaster by LNER due to an obstruction on the line.
Leeds to Harrogate and Leeds to Bradford lines have also been fully blocked due to the weather.
Northern warned passengers not to travel.
Customers in possession of Advance fares routed Northern only & dated for Sunday 9 February can travel on any Northern service for the journey stated on their ticket.
In a statement, Northern said: We are sorry for the inconvenience this will cause our customers.
"Please check immediately before you travel.
"Anyone whose journey is delayed by 15 minutes or more can apply for compensation via Delay Repay."
The Met Office said Storm Ciara will bring a spell of very strong winds on Sunday.
Flying debris could lead to Injuries or danger to life, the forecast said.
Some damage to buildings, such as tiles blown from roofs is expected.
Commuters should expect longer journey times and cancellations, as road, rail, air and ferry services are affected.
Some roads and bridges will be closed - with the possibility of the Humber Bridge being closed for the first time ever.
Power cuts with the potential to affect other services will also hit mobile phone coverage.
Injuries and danger to life from large waves and beach material being thrown onto sea fronts, coastal roads and properties are all issues to be aware of, The Met Office stated.
Disruption to travel is also likely during Sunday.