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An Alfvén wave is just one of the types of waves found in plasma that do not occur in neutral gas (plasma waves aka MHD waves): there are a variety of wave-types that incorporate electromagnetic forces from the currents and magnetic fields that can form in plasma. Among them:
Waves that incorporate ions clearly must shift considerably more mass than do those involving only electrons, which puts them in a generally-slower frequency-range.
The waves are of interest regarding the Solar atmosphere (chromosphere, corona, etc.), the ionosphere of Earth and other solar system planets, and the interplanetary medium and interstellar medium.