# Zonal wavenumber

In meteorological applications, a zonal wavenumber or hemispheric wavenumber is the dimensionless number of wavelengths fitting within a full circle around the globe at a given latitude.[1]

500mb geopotential height averaged between October 9–21, 2010 illustrating Rossby wave pattern with the zonal wavenumber 4. DOE AMIP reanalysis data.
${\displaystyle k={\frac {2\pi r\cos \varphi }{\lambda }}}$

where λ is the wavelength, r is the Earth's radius, and ${\displaystyle \varphi }$ is the latitude.

Zonal wavenumbers are typically counted on the upper level (say 500-millibar) geopotential maps by identifying troughs and ridges of the waves. Wavenumber 1 has one trough and one ridge, i.e. one wavelength fits ${\displaystyle 2\pi =360}$ degrees. Wavenumber 2 has two ridges and two troughs around 360 degrees.

Wavenumber 0 corresponds to zonal (symmetric) flow. Wavenumbers 1–3 are called long waves and often synonymous in meteorological literature with the mid-latitude planetary Rossby waves, while wavenumbers 4-10 are often referred to as "synoptic" waves.[2] In the Northern Hemisphere, wavenumbers 1 and 2 are important for the time-mean circulation due to topography (Tibetan Plateau and Rocky Mountains),[3][4] whereas in the Southern Hemisphere, tropical convection is responsible for the presence of mainly zonal wavenumber 3.[5]