Regarding the Boussinesq equations of motion:
I was reading a paper which stated the following:
Shouldn't the $\alpha$ before the $t$ be $\alpha^{-1}$ instead? Could be a typo or maybe I am not thinking about this properly. It seems like a straightforward operation, what I did was substitute the second form in Eq 29a.
(I can provide link to paper if necessary but it is probably irrelevant, so I will omit for now.)
The time derivatives transform as (given $t'=\alpha t$ where $t'$ is the rescaled time variable) $$ \frac{\partial}{\partial t}=\frac{\partial}{\partial t'}\frac{\partial t'}{\partial t}=\alpha\frac{\partial}{\partial t'}. $$ Spatial variables $x$ and $y$ stay unchanged, so 29b becomes, for example, $$ \alpha\frac{\partial}{\partial t'}\left(\alpha^2 \rho(x,y,t')\right)+\alpha \vec U\cdot\left(\alpha^2\nabla\rho\right)=0 $$ and you can see that both terms have a factor of $\alpha^3$, so $\rho$ and $\vec U$ still satisfy 29b.
29a will have a common factor of $\alpha^2$, and 29c will have a common factor of $\alpha$.