# What is the correct velocity to calculate propagation speed of a Rossby Wave?

Rossby Waves have different phase- and group velocities.

For a pure zonal wave with wave number k they are

$$v_p=\overline u - \frac{\beta}{k^2}$$

$$v_g=\overline u + \frac{\beta}{k^2}$$

So group velocity is bigger than phase velocity.

I'm wondering what speed I should use to calculate the approximate speed of a wave like this:

It propagates eastward, but since it appears to consist of only a single wave number k (the wavelength is roughly the distance from the east to west coast of the US), why should I use group velocity? The group velocity is the velocity of the envelope of a wave packet tightly centered around a mean wave number. How can I know if a through or ridge (like in the picture) is a wave packet or a single harmonic wave?

• Here is how I think about it: The phase of a wave will determine where low and high pressure systems occur. So within your picture I would estimate the wavelength of the (single?) wave. Phase speed + mean flow will give an estimate on how pressure systems move zonally. Why not group? The group/envelope will determine a larger area where pressure systems can possibly form (somewhere within the envelope). Estimating the group velocity will help you identify how the wave package distributes energy since $\frac{\partial \overline{E}}{\partial t}+\nabla \cdot c_gE= 0$. Jan 8 at 18:44
• If you think about water waves hitting the shore we observe something similar: The waves break when they approach the shore and we can clearly see that the wave breaking event is related to the waves peak position - which is determined by the phase. Possibly a crude analogy for many reasons but maybe it helps a little. Jan 8 at 18:51
• But what is the "wave packet" in my example (in contrast to the single mode)? I cannot figure out what it should be. Jan 8 at 20:17
• I personally picture it as the envelope that alters the amplitude of your Rossby wave (and usually this envelope moves at a different speed - the group velocity - than individual wave phases). If you look upstream (west of the USA) the amplitude looks a lot smaller than downstream. I'm short of a formal definition, though. So plainly speaking, to obtain the envelope you just find your maxima and connect them by some interpolant, then do the same for the minima. Or use e.g. a Hilbert transform. Jan 8 at 21:48