Rossby waves appear on planetary scales. These waves have frequencies longer than \(\Omega\) (1 per day). Their phase speed is proportional to \(\beta\), therefore they are much slower than Kelvin and gravity waves. The dispersion relation is negative and \(\beta\) is always positive, therefore these waves always move westward (in the negative x-direction). This also means that when we turn off \(\beta\), Rossby waves will not appear. You can read more about Rossby waves in chapter 13.13 in "Fluid Mechanics" (Kundu, Cohen and Dowling, 2016).

(1) Use the extended model with the \(\beta\)-plane to simulate Rossby waves and find their phase speed. What are the challenges? Does the phase speed match the expected?