Velocity Potential (Spherical Harmonics)

Velocity potential describes the irrotational, divergent part of the horizontal wind. This example calculates it with easyclimate.spec.calc_velocity_potential and compares it with easyclimate.spec.calc_velocity_potential_rs.

import cartopy.crs as ccrs
import xarray as xr
import matplotlib.pyplot as plt
import easyclimate as ecl

Open the tutorial zonal and meridional wind components, combine them into one dataset, and select one 500 hPa time slice for the calculation.

u_data = ecl.open_tutorial_dataset("uwnd_2022_day5").uwnd
v_data = ecl.open_tutorial_dataset("vwnd_2022_day5").vwnd

uvdata = xr.Dataset()
uvdata["uwnd"] = u_data
uvdata["vwnd"] = v_data

uvdata_500_202201 = uvdata.sel(level=500).isel(time = 3)
uvdata_500_202201

<xarray.Dataset> Size: 85kB
Dimensions:  (lon: 144, lat: 73)
Coordinates:
  * lon      (lon) float32 576B 0.0 2.5 5.0 7.5 10.0 ... 350.0 352.5 355.0 357.5
  * lat      (lat) float32 292B 90.0 87.5 85.0 82.5 ... -82.5 -85.0 -87.5 -90.0
    time     datetime64[ns] 8B 2022-01-04
    level    float32 4B 500.0
Data variables:
    uwnd     (lat, lon) float32 42kB ...
    vwnd     (lat, lon) float32 42kB ...

xarray.Dataset

• Dimensions:
  • lon: 144
  • lat: 73
• Coordinates: (4)
  • lon
    (lon)
    float32
    0.0 2.5 5.0 ... 352.5 355.0 357.5

    standard_name :

    longitude

    long_name :

    Longitude

    units :

    degrees_east

    axis :

    X

    array([  0. ,   2.5,   5. ,   7.5,  10. ,  12.5,  15. ,  17.5,  20. ,  22.5,
            25. ,  27.5,  30. ,  32.5,  35. ,  37.5,  40. ,  42.5,  45. ,  47.5,
            50. ,  52.5,  55. ,  57.5,  60. ,  62.5,  65. ,  67.5,  70. ,  72.5,
            75. ,  77.5,  80. ,  82.5,  85. ,  87.5,  90. ,  92.5,  95. ,  97.5,
           100. , 102.5, 105. , 107.5, 110. , 112.5, 115. , 117.5, 120. , 122.5,
           125. , 127.5, 130. , 132.5, 135. , 137.5, 140. , 142.5, 145. , 147.5,
           150. , 152.5, 155. , 157.5, 160. , 162.5, 165. , 167.5, 170. , 172.5,
           175. , 177.5, 180. , 182.5, 185. , 187.5, 190. , 192.5, 195. , 197.5,
           200. , 202.5, 205. , 207.5, 210. , 212.5, 215. , 217.5, 220. , 222.5,
           225. , 227.5, 230. , 232.5, 235. , 237.5, 240. , 242.5, 245. , 247.5,
           250. , 252.5, 255. , 257.5, 260. , 262.5, 265. , 267.5, 270. , 272.5,
           275. , 277.5, 280. , 282.5, 285. , 287.5, 290. , 292.5, 295. , 297.5,
           300. , 302.5, 305. , 307.5, 310. , 312.5, 315. , 317.5, 320. , 322.5,
           325. , 327.5, 330. , 332.5, 335. , 337.5, 340. , 342.5, 345. , 347.5,
           350. , 352.5, 355. , 357.5], dtype=float32)

  • lat
    (lat)
    float32
    90.0 87.5 85.0 ... -87.5 -90.0

    standard_name :

    latitude

    long_name :

    Latitude

    units :

    degrees_north

    axis :

    Y

    array([ 90. ,  87.5,  85. ,  82.5,  80. ,  77.5,  75. ,  72.5,  70. ,  67.5,
            65. ,  62.5,  60. ,  57.5,  55. ,  52.5,  50. ,  47.5,  45. ,  42.5,
            40. ,  37.5,  35. ,  32.5,  30. ,  27.5,  25. ,  22.5,  20. ,  17.5,
            15. ,  12.5,  10. ,   7.5,   5. ,   2.5,   0. ,  -2.5,  -5. ,  -7.5,
           -10. , -12.5, -15. , -17.5, -20. , -22.5, -25. , -27.5, -30. , -32.5,
           -35. , -37.5, -40. , -42.5, -45. , -47.5, -50. , -52.5, -55. , -57.5,
           -60. , -62.5, -65. , -67.5, -70. , -72.5, -75. , -77.5, -80. , -82.5,
           -85. , -87.5, -90. ], dtype=float32)

  • time
    ()
    datetime64[ns]
    2022-01-04

    standard_name :

    time

    long_name :

    Time

    axis :

    T

    array('2022-01-04T00:00:00.000000000', dtype='datetime64[ns]')

  • level
    ()
    float32
    500.0

    standard_name :

    air_pressure

    long_name :

    Level

    units :

    millibar

    positive :

    down

    axis :

    Z

    actual_range :

    [1000. 10.]

    GRIB_id :

    100

    GRIB_name :

    hPa

    array(500., dtype=float32)

• Data variables: (2)
  • uwnd
    (lat, lon)
    float32
    ...

    long_name :

    mean Daily U-wind

    units :

    m/s

    precision :

    2

    GRIB_id :

    33

    GRIB_name :

    UGRD

    var_desc :

    u-wind

    level_desc :

    Pressure Levels

    statistic :

    Mean

    parent_stat :

    Individual Obs

    dataset :

    NCEP Reanalysis Daily Averages

    actual_range :

    [-83.05 116.85]

    [10512 values with dtype=float32]

  • vwnd
    (lat, lon)
    float32
    ...

    long_name :

    mean Daily V wind

    units :

    m/s

    precision :

    2

    GRIB_id :

    34

    GRIB_name :

    VGRD

    var_desc :

    v-wind

    level_desc :

    Pressure Levels

    statistic :

    Mean

    parent_stat :

    Individual Obs

    dataset :

    NCEP Reanalysis Daily Averages

    actual_range :

    [-85.925 88.125]

    [10512 values with dtype=float32]

Prepare a scientific-notation formatter for the colorbars. Many spectral wind diagnostics have small physical units, so this keeps the labels readable.

import matplotlib.ticker as ticker
formatter = ticker.ScalarFormatter(useMathText=True, useOffset=True)
formatter.set_scientific(True)
formatter.set_powerlimits((0, 0))

The returned velocity potential keeps the input coordinates and can be plotted directly with xarray.

vp_fp = ecl.spec.calc_velocity_potential(
    u_data=uvdata_500_202201["uwnd"],
    v_data=uvdata_500_202201["vwnd"],
)

vp_rs = ecl.spec.calc_velocity_potential_rs(
    u_data=uvdata_500_202201["uwnd"],
    v_data=uvdata_500_202201["vwnd"],
)

The first plot shows the velocity potential from the Fortran-backed calculation on a map projection.

fig, ax = plt.subplots(
    figsize = (10, 5),
    subplot_kw={"projection": ccrs.Mercator(central_longitude=180)}
)
ax.coastlines()
ax.gridlines(crs=ccrs.PlateCarree(), draw_labels=["bottom", "left"], alpha = 0)

vp_fp.sortby("lat").sel(lat=slice(20, 80)).plot.contourf(
    levels=21,
    cbar_kwargs = {'location': 'bottom', 'format': formatter, 'pad': 0.1},
    transform = ccrs.PlateCarree(),
)

<cartopy.mpl.contour.GeoContourSet object at 0x7af7d7847560>

The final panel compares Fortran, Rust, and their difference for the velocity potential.

fig, ax = plt.subplots(1, 3, figsize = (15, 5))

vp_fp.sortby("lat").sel(lat=slice(20, 80)).plot.contourf(
    levels=21,
    ax = ax[0],
    cbar_kwargs = {'location': 'bottom'},
)
ax[0].set_title("Fortran")

vp_rs.sortby("lat").sel(lat=slice(20, 80)).plot.contourf(
    levels=21,
    ax = ax[1],
    cbar_kwargs = {'location': 'bottom'},
)
ax[1].set_title("Rust")

(vp_fp - vp_rs).sortby("lat").sel(lat=slice(20, 80)).plot(
    ax = ax[2],
    cbar_kwargs = {'location': 'bottom'},
)
ax[2].set_title("Diff: Fortran - Rust")

Text(0.5, 1.0, 'Diff: Fortran - Rust')