Description

Vorticity can be decomposed locally into shear and curvature terms as follows (e.g., Holton 2004);

where V denotes scalar wind speed, n the direction perpendicular to the flow, and Rs the radius of curvature. The first and second terms of the RHS in the above equation represent shear vorticity and curvature vorticity, respectively, which can be calculated as

where u and v denote local zonal and meridional wind velocities, respectively, and subscripts x and y partial derivatives in the zonal and meridional directions, respectively. 

The curvature, defined as

can be derived from the definition of curvature of a two-dimensional curve implicitly represented by 

The curvature or curvature vorticity enables us to circumvent the difficulties in determining areas of cyclonic and anticyclonic circulations, because it is free from shear vorticity and thus extracts vortex circulation with a certain radius.

The curvature-based methodology enables us to evaluate contributions from cyclonic and anticyclonic eddies separately to Eulerian statistics, by accumulating instantaneous contributions only at grid points where cyclonic or anticyclonic curvature is observed.

Examples:

• Eulerian eddy statistics (such as v't'850, v'v'300) and eddy feedback forcing of westerly wind acceleration: Okajima, S., H. Nakamura, Y. Kaspi: Cyclonic and anticyclonic contributions to atmospheric energetics, Scientific Reports, 11, 13202, 2021.

• Air-sea turbulent heat flux, precipitation, and associated hydrological cycle: Okajima, S., Nakamura, H., Spengler, T. (2024). Midlatitude oceanic fronts strengthen the hydrological cycle between cyclones and anticyclones. Geophysical Research Letters, 51, e2023GL106187.

Here is a press release for the curvature-based methodology

Is it possible to calculate the curvature directly from height fields? 

Yes, it is possible to calculate the curvature from height fields, by assuming geostrophic equilibrium. Under the assumption of geostrophic equilibrium, the stream function is proportional to the geopotential; ψ = gz/f. Using this relationship, you can calculate the curvature from the geopotential height field.

(The equation of the curvature from the stream function appears to be the same as the definition of curvature of an implicitly represented two-dimensional curve in the above equation.)

Did you filter winds while calculating the curvature from horizontal winds? 

No, we did not filter winds. This is because we wanted to obtain cyclonic & anticyclonic domains that correspond directly to those that appear on weather charts. (Those systems are identified based on unfiltered SLP, for example)
If you apply filtering to wind fields, we will obtain cyclonic/anticyclonic regions that are artificially too symmetric because of filtering. This will make it unable to investigate cyclone-anticyclone asymmetry.

I am still not sure which curvature threshold to adopt.

Your curvature threshold will depend on what you are focusing on. Our methodology evaluates the local curvature (or, equivalently, the curvature radius) of the flow, which represents the flow topology in a physically straightforward manner.

If you want to partition all grid points into cyclonic and anticyclonic regions, a threshold of zero curvature would work well. If you are to obtain "transition zones" between cyclonic and anticyclonic regions, a non-zero curvature threshold would be effective. It will be according to the features you want to focus on. For transient eddies, a curvature radius of 2,500km or 3,000km would work, as in our paper. Ultimately, it will correspond to what you regard as "cyclonic and anticyclonic regions", which will depend on purposes and contexts. 

You can find more detailed information about the "materialization" in the article in Tenki (in Japanese).

岡島悟 (2022): ３次元格子データのmaterialization―3Dプリンタによる”物質化”―，「天気」Vol.69, No.8.PDF