To obtain the original Stommel model we just ignore the advective derivative, which will be valid if $\boxed{|\zeta| \sim Z \ll \beta L}$ where $Z = U/L$ is a representative value of vorticity. This condition is equivalent to $$ \underbrace{R_\beta}_{\text{beta-Rossby number}} \equiv \frac{\underbrace{U}_{\text{velocity scale}}}{\underbrace{\beta L^2}_{\text{planetary vorticity gradient term}}} \ll 1 $$ $R_\beta$ is called the beta-Rossby number. On sufficiently large scales, $\beta \sim f/L$ and \(R_\beta\) is similar to a small Rossby number assumption