Rossby radius of deformation

For a barotropic ocean, the Rossby radius is ${\displaystyle L_{R}\equiv {\frac {($gD)^{1/2}}{f}}} , where ${\displaystyle \,g}$ is the gravitational acceleration, ${\displaystyle \,D}$ is the water depth, and ${\displaystyle \,f}$ is the Coriolis parameter.^[2]

For f = 1×10⁻⁴s⁻¹ appropriate to 45° latitude, g = 9.81 m/s² and D = 4 km, L_R ≈ 2000 km; using the same latitude and gravity but changing D to 40 m; L_R ≈ 200 km.

The nthbaroclinic Rossby radius is:

${\displaystyle L_{R,n}\equiv {\frac {NH}{n\pi f_{0}}}}$, where ${\displaystyle \,N}$ is the Brunt–Väisälä frequency, ${\displaystyle \,H}$ is the scale height, and n = 1, 2, ....

In Earth's atmosphere, the ratio N/f₀ is typically of order 100, so the Rossby radius is about 100 times the vertical scale height, H. For a vertical scale associated with the height of the tropopause, L_{R, 1} ≈ 1000 km, which is the predominant scale seen on weather charts for cyclones and anticyclones. This is commonly called the synoptic scale.

In the ocean, the Rossby radius varies dramatically with latitude. Near the equator it is larger than 200 km, while in the high latitude regions it is less than 10 km. ^{[3] [4]} The size of ocean eddies varies similarly; in low latitude regions, near the equator, eddies are much larger than in high latitude regions.

The associated dimensionless parameter is the Rossby number. Both are named in honor of Carl-Gustav Rossby.

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