I realized scaler was not a very good name (I am submitting some changes).
Anyway, let’s assume that you want to parametrize your model using the non-centered version, for example to improve sampling and eliminate divergences (see here). Then, you can use this trick: \mathcal{N}(0, \sigma_z) = \mathcal{N}(0, 1) \cdot \sigma_z. In this equation, \sigma_z is the scaler (because it scales samples drawn from the standard normal distribution \mathcal{N}(0, 1)). In a linear mixed effects model, \sigma_z (that can be a deterministic or random variable) may represent the standard deviation of the random intercepts (or slopes). That is, the intercepts for the individual subjects are drawn from the distribution \mathcal{N}(0, \sigma_z).
In case you need it, this also holds true: \mathcal{N}(\mu_z, \sigma_z) = \mu_z + \mathcal{N}(0, 1) \cdot \sigma_z.
BTW, nice notebook, fun to read 