# wind velocity derivations

1. meridional wind velocity from wind speed and direction

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(u\) | zonal wind velocity | \(m/s\) | zonal_wind_velocity {:}
   \(s\) | wind speed | \(m/s\) | wind_speed {:}
   \(\theta\) | wind direction | \(deg\) | wind_direction {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[u = s \cos((180 - \theta)\frac{\pi}{180})\]

2. zonal wind velocity from wind speed and direction

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(v\) | meridional wind velocity | \(m/s\) | meridional_wind_velocity {:}
   \(s\) | wind speed | \(m/s\) | wind_speed {:}
   \(\theta\) | wind direction | \(deg\) | wind_direction {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[v = s \sin((180 - \theta)\frac{\pi}{180})\]

3. wind direction from zonal and meridional wind velocity

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(u\) | zonal wind velocity | \(m/s\) | zonal_wind_velocity {:}
   \(v\) | meridional wind velocity | \(m/s\) | meridional_wind_velocity {:}
   \(\theta\) | wind direction | \(deg\) | wind_direction {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[\theta = 180 - \arctan2(u,v)\frac{180}{\pi}\]

4. wind speed from zonal and meridional wind velocity

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(u\) | zonal wind velocity | \(m/s\) | zonal_wind_velocity {:}
   \(v\) | meridional wind velocity | \(m/s\) | meridional_wind_velocity {:}
   \(s\) | wind speed | \(m/s\) | wind_speed {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[s = \sqrt{u^2 + v^2}\]

5. surface meridional wind velocity from surface wind speed and direction

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(u\) | zonal wind velocity | \(m/s\) | surface_zonal_wind_velocity {:}
   \(s\) | wind speed | \(m/s\) | surface_wind_speed {:}
   \(\theta\) | wind direction | \(deg\) | surface_wind_direction {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[u = s \cos((180 - \theta)\frac{\pi}{180})\]

6. surface zonal wind velocity from surface wind speed and direction

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(v\) | meridional wind velocity | \(m/s\) | surface_meridional_wind_velocity {:}
   \(s\) | wind speed | \(m/s\) | surface_wind_speed {:}
   \(\theta\) | wind direction | \(deg\) | surface_wind_direction {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[v = s \sin((180 - \theta)\frac{\pi}{180})\]

7. surface wind direction from surface zonal and meridional wind velocity

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(u\) | zonal wind velocity | \(m/s\) | surface_zonal_wind_velocity {:}
   \(v\) | meridional wind velocity | \(m/s\) | surface_meridional_wind_velocity {:}
   \(\theta\) | wind direction | \(deg\) | surface_wind_direction {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[\theta = 180 - \arctan2(u,v)\frac{180}{\pi}\]

8. surface wind speed from surface zonal and meridional wind velocity

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(u\) | zonal wind velocity | \(m/s\) | surface_zonal_wind_velocity {:}
   \(v\) | meridional wind velocity | \(m/s\) | surface_meridional_wind_velocity {:}
   \(s\) | wind speed | \(m/s\) | surface_wind_speed {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[s = \sqrt{u^2 + v^2}\]