Police Scotland were made aware of the avalanche at around 3.35pm on December 30 with the Lochaber Mountain Rescue Team and a helicopter were dispatched to help two male climbers who were in the area at the time.
Mr Bessell was pronounced dead at the scene, while a 40-year-old was taken to Belford Hospital in Fort William for treatment after they were found in Ben Nevis's number two gully following a rescue operation that took eight hours.
Richard Uffendell, headteacher of Ashton Park School, paid tribute to Mr Bessell as a "well-respected and long-standing member of our staff" sending a letter to parents and praising Mr Bessell’s “positive impact" on the lives of many students.
"Mark will be deeply missed by us all as he has had such a positive impact on the lives of so many students, staff, families and the wider communities, but he will never be forgotten," Mr Uffendell said.
"We will be in touch in due course with details of how our school and community will pay tribute to Mark's life and continue the outstanding work he led within outdoor education."
Mr Bessell, a veteran of the Gloucestershire Regiment, is survived by his wife Charlie - who also works at the school - and a son from a previous relationship.
He ran the school's adventure club and organised the Duke of Edinburgh and Ten Tors challenges.
Students were being offered grief support through the schools educational psychology team.