A scientific advisor, sometimes capitalised as Scientific Advisor, was an in-house scientific consultant to an organisation whose focus was not primarily scientific in nature. The title notably existed within the hierarchy of UNIT, and, starting in the second half of the 20th century, was held by the Doctor across various incarnations.
The Intrusion Counter-Measures Group, a predecessor to UNIT, had a "scientific advisor" role. In 1963, Rachel Jensen filled it. (TV: Remembrance of the Daleks) The platoon of soldiers whom a young Alistair Gordon Lethbridge-Stewart led against the Great Intelligence's infiltration of the London Underground, Edward Travers and his daughter Anne Travers tagged along, as well as the Second Doctor; (TV: The Web of Fear) subsequently, after Lethbridge-Stewart became Brigadier of the British branch of a new, special force dedicated to combatting alien threats, he officially named Anne Travers as his first Scientific Advisor. (AUDIO: Memories of Tomorrow)
However, with the British Army requisitioning Anne Travers so that she could work at the Vault, (PROSE: The Forgotten Son) Lethbridge-Stewart was forced to find a different UNIT Scientific Advisor. He interviewed Liz Shaw, of Cambridge, about the possibility of her taking the position. In the end, the role went to the Third Doctor, with Liz staying on as his assistant. The Doctor served as UNIT's scientific advisor throughout his exile on Earth. (TV: Spearhead from Space)
Although the Doctor ceased to stay on Earth on a regular basis after he regenerated into his fourth incarnation, his affiliation with UNIT remained, with others such as Colonel Mace noting that the Doctor had never officially resigned his position. (TV: Robot, The Sontaran Stratagem) Even when not actively working with UNIT, the Doctor wasn't above using his UNIT credentials to establish authority to investigate such matters as disappearing Concordes. (TV: Time-Flight) Even in 2001, the Doctor retained a legendary reputation among UNIT, with the Brigadier musing that a three-second silence after learning that the Doctor was present during a crisis was an impressive incident. (PROSE: The Shadow in the Glass)
However, while the Doctor retained official Scientific Advisor status, in practice the position was left vacant by the Fourth Doctor's departure, leading to the promotion of Elizabeth Klein as Scientific Advisor for the UK branch. (AUDIO: Dominion)
While investigating the strange abilities of apparent medium James Lees for Queen Victoria, the Fifth Doctor was officially appointed Queen Victoria's scientific advisor, being offered to remain in the role even after his departure. (PROSE: Empire of Death)