Is it correct to say for the following: $$u_{k}\sim N(0,\gamma ^{2})$$
that "$u_{k}$ follows the Gaussian distribution with a mean of zero and a covariance of $\gamma ^{2}$" ?
I am a bit sceptical regarding the term covariance. Shouldn't that be variance?
It is not correct. Covariance makes no sense here:
There are no two random variables $X,Y$ given, and covariance measures $E[XY]-E[X]E[Y]$ which depends on how one multiplies two random variables $X,Y$ (thus covariance cannot be calculated with distributions of the random variables alone).
Correct would be: $\mu_k$ follows a Gaussian distribution with mean $0$ and variance $\gamma^2$.