MITgcm ECCOv4 Example

This Jupyter notebook demonstrates how to use xarray and xgcm to analyze data from the ECCO v4r3 ocean state estimate.

This notebook can be viewed and executed interactively viaPangeo Gallery.

First we import our standard python packages:

import xarray as xr
import numpy as np
from matplotlib import pyplot as plt
%matplotlib inline

Load the data

The ECCOv4r3 data was converted from its raw MDS (.data / .meta file) format to zarr format, using the xmitgcm package. Zarr is a powerful data storage format that can be thought of as an alternative to HDF. In contrast to HDF, zarr works very well with cloud object storage. Zarr is currently useable in python, java, C++, and julia. It is likely that zarr will form the basis of the next major version of the netCDF library.

If you’re curious, here are some resources to learn more about zarr: - https://zarr.readthedocs.io/en/stable/tutorial.html - https://speakerdeck.com/rabernat/pangeo-zarr-cloud-data-storage - https://mrocklin.github.com/blog/work/2018/02/06/hdf-in-the-cloud

The ECCO zarr data currently lives in Google Cloud Storage as part of the Pangeo Data Catalog. This means we can open the whole dataset using one line of code.

This takes a bit of time to run because the metadata must be downloaded and parsed. The type of object returned is an Xarray dataset.

import intake
cat = intake.open_catalog("https://raw.githubusercontent.com/pangeo-data/pangeo-datastore/master/intake-catalogs/ocean.yaml")
ds = cat.ECCOv4r3.to_dask()
ds

<xarray.Dataset>
Dimensions:    (face: 13, i: 90, i_g: 90, j: 90, j_g: 90, k: 50, k_l: 50, k_p1: 51, k_u: 50, time: 288, time_snp: 287)
Coordinates:
    Depth      (face, j, i) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    PHrefC     (k) float32 dask.array<chunksize=(50,), meta=np.ndarray>
    PHrefF     (k_p1) float32 dask.array<chunksize=(51,), meta=np.ndarray>
    XC         (face, j, i) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    XG         (face, j_g, i_g) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    YC         (face, j, i) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    YG         (face, j_g, i_g) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    Z          (k) float32 dask.array<chunksize=(50,), meta=np.ndarray>
    Zl         (k_l) float32 dask.array<chunksize=(50,), meta=np.ndarray>
    Zp1        (k_p1) float32 dask.array<chunksize=(51,), meta=np.ndarray>
    Zu         (k_u) float32 dask.array<chunksize=(50,), meta=np.ndarray>
    drC        (k_p1) float32 dask.array<chunksize=(51,), meta=np.ndarray>
    drF        (k) float32 dask.array<chunksize=(50,), meta=np.ndarray>
    dxC        (face, j, i_g) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    dxG        (face, j_g, i) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    dyC        (face, j_g, i) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    dyG        (face, j, i_g) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
  * face       (face) int64 0 1 2 3 4 5 6 7 8 9 10 11 12
    hFacC      (k, face, j, i) float32 dask.array<chunksize=(50, 13, 90, 90), meta=np.ndarray>
    hFacS      (k, face, j_g, i) float32 dask.array<chunksize=(50, 13, 90, 90), meta=np.ndarray>
    hFacW      (k, face, j, i_g) float32 dask.array<chunksize=(50, 13, 90, 90), meta=np.ndarray>
  * i          (i) int64 0 1 2 3 4 5 6 7 8 9 ... 80 81 82 83 84 85 86 87 88 89
  * i_g        (i_g) int64 0 1 2 3 4 5 6 7 8 9 ... 80 81 82 83 84 85 86 87 88 89
    iter       (time) int64 dask.array<chunksize=(1,), meta=np.ndarray>
    iter_snp   (time_snp) int64 dask.array<chunksize=(1,), meta=np.ndarray>
  * j          (j) int64 0 1 2 3 4 5 6 7 8 9 ... 80 81 82 83 84 85 86 87 88 89
  * j_g        (j_g) int64 0 1 2 3 4 5 6 7 8 9 ... 80 81 82 83 84 85 86 87 88 89
  * k          (k) int64 0 1 2 3 4 5 6 7 8 9 ... 40 41 42 43 44 45 46 47 48 49
  * k_l        (k_l) int64 0 1 2 3 4 5 6 7 8 9 ... 40 41 42 43 44 45 46 47 48 49
  * k_p1       (k_p1) int64 0 1 2 3 4 5 6 7 8 9 ... 42 43 44 45 46 47 48 49 50
  * k_u        (k_u) int64 0 1 2 3 4 5 6 7 8 9 ... 40 41 42 43 44 45 46 47 48 49
    rA         (face, j, i) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    rAs        (face, j_g, i) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    rAw        (face, j, i_g) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
    rAz        (face, j_g, i_g) float32 dask.array<chunksize=(13, 90, 90), meta=np.ndarray>
  * time       (time) datetime64[ns] 1992-01-15 1992-02-13 ... 2015-12-14
  * time_snp   (time_snp) datetime64[ns] 1992-02-01 1992-03-01 ... 2015-12-01
Data variables:
    ADVr_SLT   (time, k_l, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    ADVr_TH    (time, k_l, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    ADVx_SLT   (time, k, face, j, i_g) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    ADVx_TH    (time, k, face, j, i_g) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    ADVy_SLT   (time, k, face, j_g, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    ADVy_TH    (time, k, face, j_g, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    DFrE_SLT   (time, k_l, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    DFrE_TH    (time, k_l, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    DFrI_SLT   (time, k_l, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    DFrI_TH    (time, k_l, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    DFxE_SLT   (time, k, face, j, i_g) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    DFxE_TH    (time, k, face, j, i_g) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    DFyE_SLT   (time, k, face, j_g, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    DFyE_TH    (time, k, face, j_g, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    ETAN       (time, face, j, i) float32 dask.array<chunksize=(1, 13, 90, 90), meta=np.ndarray>
    ETAN_snp   (time_snp, face, j, i) float32 dask.array<chunksize=(1, 13, 90, 90), meta=np.ndarray>
    GEOFLX     (face, j, i) float32 dask.array<chunksize=(7, 90, 90), meta=np.ndarray>
    MXLDEPTH   (time, face, j, i) float32 dask.array<chunksize=(1, 1, 90, 90), meta=np.ndarray>
    SALT       (time, k, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    SALT_snp   (time_snp, k, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    SFLUX      (time, face, j, i) float32 dask.array<chunksize=(1, 13, 90, 90), meta=np.ndarray>
    TFLUX      (time, face, j, i) float32 dask.array<chunksize=(1, 13, 90, 90), meta=np.ndarray>
    THETA      (time, k, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    THETA_snp  (time_snp, k, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    UVELMASS   (time, k, face, j, i_g) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    UVELSTAR   (time, k, face, j, i_g) float32 dask.array<chunksize=(1, 50, 1, 90, 90), meta=np.ndarray>
    VVELMASS   (time, k, face, j_g, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    VVELSTAR   (time, k, face, j_g, i) float32 dask.array<chunksize=(1, 50, 1, 90, 90), meta=np.ndarray>
    WVELMASS   (time, k_l, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    WVELSTAR   (time, k_l, face, j, i) float32 dask.array<chunksize=(1, 50, 1, 90, 90), meta=np.ndarray>
    oceFWflx   (time, face, j, i) float32 dask.array<chunksize=(1, 13, 90, 90), meta=np.ndarray>
    oceQsw     (time, face, j, i) float32 dask.array<chunksize=(1, 13, 90, 90), meta=np.ndarray>
    oceSPtnd   (time, k, face, j, i) float32 dask.array<chunksize=(1, 50, 13, 90, 90), meta=np.ndarray>
    oceTAUX    (time, face, j, i_g) float32 dask.array<chunksize=(1, 1, 90, 90), meta=np.ndarray>
    oceTAUY    (time, face, j_g, i) float32 dask.array<chunksize=(1, 1, 90, 90), meta=np.ndarray>

Note that no data has been actually download yet. Xarray uses the approach of lazy evaluation, in which loading of data and execution of computations is delayed as long as possible (i.e. until data is actually needed for a plot). The data are represented symbolically as dask arrays. For example:

SALT       (time, k, face, j, i) float32 dask.array<shape=(288, 50, 13, 90, 90), chunksize=(1, 50, 13, 90, 90)>

The full shape of the array is (288, 50, 13, 90, 90), quite large. But the chunksize is (1, 50, 13, 90, 90). Here the chunks correspond to the individual granuales of data (objects) in cloud storage. The chunk is the minimum amount of data we can read at one time.

# a trick to make things work a bit faster
coords = ds.coords.to_dataset().reset_coords()
ds = ds.reset_coords(drop=True)

Visualizing Data

A Direct Plot

Let’s try to visualize something simple: the Depth variable. Here is how the data are stored:

Depth      (face, j, i) float32 dask.array<shape=(13, 90, 90), chunksize=(13, 90, 90)>

Although depth is a 2D field, there is an extra, dimension (face) corresponding to the LLC face number. Let’s use xarray’s built in plotting functions to plot each face individually.

coords.Depth.plot(col='face', col_wrap=5)

<xarray.plot.facetgrid.FacetGrid at 0x7f9c9dfd44f0>

This view is not the most useful. It reflects how the data is arranged logically, rather than geographically.

A Pretty Map

To make plotting easier, we can define a quick function to plot the data in a more geographically friendly way. Eventually these plotting functions may be provided by the gcmplots package: xecco/gcmplots. For now, it is easy enough to roll our own.

from matplotlib import pyplot as plt
import cartopy as cart
import pyresample

class LLCMapper:

    def __init__(self, ds, dx=0.25, dy=0.25):
        # Extract LLC 2D coordinates
        lons_1d = ds.XC.values.ravel()
        lats_1d = ds.YC.values.ravel()

        # Define original grid
        self.orig_grid = pyresample.geometry.SwathDefinition(lons=lons_1d, lats=lats_1d)

        # Longitudes latitudes to which we will we interpolate
        lon_tmp = np.arange(-180, 180, dx) + dx/2
        lat_tmp = np.arange(-90, 90, dy) + dy/2

        # Define the lat lon points of the two parts.
        self.new_grid_lon, self.new_grid_lat = np.meshgrid(lon_tmp, lat_tmp)
        self.new_grid  = pyresample.geometry.GridDefinition(lons=self.new_grid_lon,
                                                            lats=self.new_grid_lat)

    def __call__(self, da, ax=None, projection=cart.crs.Robinson(), lon_0=-60, **plt_kwargs):

        assert set(da.dims) == set(['face', 'j', 'i']), "da must have dimensions ['face', 'j', 'i']"

        if ax is None:
            fig, ax = plt.subplots(figsize=(12, 6), subplot_kw={'projection': projection})
        else:
            m = plt.axes(projection=projection)

        field = pyresample.kd_tree.resample_nearest(self.orig_grid, da.values,
                                                    self.new_grid,
                                                    radius_of_influence=100000,
                                                    fill_value=None)

        vmax = plt_kwargs.pop('vmax', field.max())
        vmin = plt_kwargs.pop('vmin', field.min())


        x,y = self.new_grid_lon, self.new_grid_lat

        # Find index where data is splitted for mapping
        split_lon_idx = round(x.shape[1]/(360/(lon_0 if lon_0>0 else lon_0+360)))


        p = ax.pcolormesh(x[:,:split_lon_idx], y[:,:split_lon_idx], field[:,:split_lon_idx],
                         vmax=vmax, vmin=vmin, transform=cart.crs.PlateCarree(), zorder=1, **plt_kwargs)
        p = ax.pcolormesh(x[:,split_lon_idx:], y[:,split_lon_idx:], field[:,split_lon_idx:],
                         vmax=vmax, vmin=vmin, transform=cart.crs.PlateCarree(), zorder=2, **plt_kwargs)

        ax.add_feature(cart.feature.LAND, facecolor='0.5', zorder=3)
        label = ''
        if da.name is not None:
            label = da.name
        if 'units' in da.attrs:
            label += ' [%s]' % da.attrs['units']
        cb = plt.colorbar(p, shrink=0.4, label=label)
        return ax

mapper = LLCMapper(coords)
mapper(coords.Depth);

We can use this with any 2D cell-centered LLC variable.

Selecting data

The entire ECCOv4e3 dataset is contained in a single Xarray.Dataset object. How do we find a view specific pieces of data? This is handled by Xarray’s indexing and selecting functions. To get the SST from January 2000, we do this:

sst = ds.THETA.sel(time='2000-01-15', k=0)
sst

<xarray.DataArray 'THETA' (face: 13, j: 90, i: 90)>
dask.array<getitem, shape=(13, 90, 90), dtype=float32, chunksize=(13, 90, 90), chunktype=numpy.ndarray>
Coordinates:
  * face     (face) int64 0 1 2 3 4 5 6 7 8 9 10 11 12
  * i        (i) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
  * j        (j) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
    k        int64 0
    time     datetime64[ns] 2000-01-15
Attributes:
    long_name:      Potential Temperature
    standard_name:  THETA
    units:          degC

Still no data has been actually downloaded. That doesn’t happen until we call .load() explicitly or try to make a plot.

mapper(sst, cmap='RdBu_r');

/srv/conda/envs/notebook/lib/python3.8/site-packages/matplotlib/colors.py:576: RuntimeWarning: overflow encountered in multiply
  xa *= self.N

Do some Calculations

Now let’s start doing something besides just plotting the existing data. For example, let’s calculate the time-mean SST.

mean_sst = ds.THETA.sel(k=0).mean(dim='time')
mean_sst

<xarray.DataArray 'THETA' (face: 13, j: 90, i: 90)>
dask.array<mean_agg-aggregate, shape=(13, 90, 90), dtype=float32, chunksize=(13, 90, 90), chunktype=numpy.ndarray>
Coordinates:
  * face     (face) int64 0 1 2 3 4 5 6 7 8 9 10 11 12
  * i        (i) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
  * j        (j) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
    k        int64 0

As usual, no data was loaded. Instead, mean_sst is a symbolic representation of the data that needs to be pulled and the computations that need to be executed to produce the desired result. In this case, the 288 original chunks all need to be read from cloud storage. Dask coordinates this automatically for us. But it does take some time.

%time mean_sst.load()

CPU times: user 18.9 s, sys: 1.76 s, total: 20.6 s
Wall time: 42.7 s

<xarray.DataArray 'THETA' (face: 13, j: 90, i: 90)>
array([[[ 0.        ,  0.        ,  0.        , ...,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , ...,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , ...,  0.        ,
          0.        ,  0.        ],
        ...,
        [ 0.22906627,  0.20126666,  0.18094039, ...,  0.04199786,
          0.06655131,  0.09593879],
        [ 0.43254805,  0.42036152,  0.4126501 , ...,  0.20539196,
          0.24391538,  0.29138586],
        [ 0.6535146 ,  0.65845466,  0.65769684, ...,  0.37179062,
          0.42764866,  0.49818808]],

       [[ 0.87210137,  0.89154845,  0.88329524, ...,  0.52988845,
          0.60233873,  0.6977836 ],
        [ 1.0961676 ,  1.1175991 ,  1.0779033 , ...,  0.68887544,
          0.774091  ,  0.89128804],
        [ 1.3072966 ,  1.3048965 ,  1.2159245 , ...,  0.85709995,
          0.9489381 ,  1.0793763 ],
...
        [27.479395  , 27.666166  , 27.793968  , ...,  1.4822977 ,
          1.3396592 ,  1.190825  ],
        [27.444382  , 27.641308  , 27.776764  , ...,  1.3742981 ,
          1.2040414 ,  1.031747  ],
        [27.411293  , 27.615599  , 27.76121   , ...,  1.314467  ,
          1.1131614 ,  0.91215134]],

       [[ 4.6964245 ,  4.2194605 ,  3.719968  , ...,  0.        ,
          0.        ,  0.        ],
        [ 4.747999  ,  4.2700696 ,  3.7787225 , ...,  0.        ,
          0.        ,  0.        ],
        [ 4.754464  ,  4.278542  ,  3.7964838 , ...,  0.        ,
          0.        ,  0.        ],
        ...,
        [ 1.0251521 ,  0.82211953,  0.58778673, ...,  0.        ,
          0.        ,  0.        ],
        [ 0.85189605,  0.6512579 ,  0.43876594, ...,  0.        ,
          0.        ,  0.        ],
        [ 0.71174276,  0.5037161 ,  0.30439517, ...,  0.        ,
          0.        ,  0.        ]]], dtype=float32)
Coordinates:
  * face     (face) int64 0 1 2 3 4 5 6 7 8 9 10 11 12
  * i        (i) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
  * j        (j) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
    k        int64 0

mapper(mean_sst, cmap='RdBu_r');

/srv/conda/envs/notebook/lib/python3.8/site-packages/matplotlib/colors.py:576: RuntimeWarning: overflow encountered in multiply
  xa *= self.N

Speeding things up with a Dask Cluster

How can we speed things up? In general, the main bottleneck for this type of data analysis is the speed with which we can read the data. With cloud storage, the access is highly parallelizeable.

From a Pangeo environment, we can create a Dask cluster to spread the work out amongst many compute nodes. This works on both HPC and cloud. In the cloud, the compute nodes are provisioned on the fly and can be shut down as soon as we are done with our analysis.

The code below will create a cluster with up to five compute nodes. These will be ramped up and down based on demand. It can take a few minutes to provision our nodes.

from dask_gateway import Gateway
from dask.distributed import Client

gateway = Gateway()
cluster = gateway.new_cluster()
cluster.adapt(minimum=1, maximum=5)
cluster

# from dask_gateway import GatewayCluster
# from dask.distributed import Client
# cluster = GatewayCluster()
# cluster.scale(5)
# client = Client(cluster)
# cluster

Now we re-run the mean calculation. Note how the dashboard helps us visualize what the cluster is doing.

%time ds.THETA.isel(k=0).mean(dim='time').load()

CPU times: user 19.3 s, sys: 1.71 s, total: 21 s
Wall time: 37.5 s

<xarray.DataArray 'THETA' (face: 13, j: 90, i: 90)>
array([[[ 0.        ,  0.        ,  0.        , ...,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , ...,  0.        ,
          0.        ,  0.        ],
        [ 0.        ,  0.        ,  0.        , ...,  0.        ,
          0.        ,  0.        ],
        ...,
        [ 0.22906627,  0.20126666,  0.18094039, ...,  0.04199786,
          0.06655131,  0.09593879],
        [ 0.43254805,  0.42036152,  0.4126501 , ...,  0.20539196,
          0.24391538,  0.29138586],
        [ 0.6535146 ,  0.65845466,  0.65769684, ...,  0.37179062,
          0.42764866,  0.49818808]],

       [[ 0.87210137,  0.89154845,  0.88329524, ...,  0.52988845,
          0.60233873,  0.6977836 ],
        [ 1.0961676 ,  1.1175991 ,  1.0779033 , ...,  0.68887544,
          0.774091  ,  0.89128804],
        [ 1.3072966 ,  1.3048965 ,  1.2159245 , ...,  0.85709995,
          0.9489381 ,  1.0793763 ],
...
        [27.479395  , 27.666166  , 27.793968  , ...,  1.4822977 ,
          1.3396592 ,  1.190825  ],
        [27.444382  , 27.641308  , 27.776764  , ...,  1.3742981 ,
          1.2040414 ,  1.031747  ],
        [27.411293  , 27.615599  , 27.76121   , ...,  1.314467  ,
          1.1131614 ,  0.91215134]],

       [[ 4.6964245 ,  4.2194605 ,  3.719968  , ...,  0.        ,
          0.        ,  0.        ],
        [ 4.747999  ,  4.2700696 ,  3.7787225 , ...,  0.        ,
          0.        ,  0.        ],
        [ 4.754464  ,  4.278542  ,  3.7964838 , ...,  0.        ,
          0.        ,  0.        ],
        ...,
        [ 1.0251521 ,  0.82211953,  0.58778673, ...,  0.        ,
          0.        ,  0.        ],
        [ 0.85189605,  0.6512579 ,  0.43876594, ...,  0.        ,
          0.        ,  0.        ],
        [ 0.71174276,  0.5037161 ,  0.30439517, ...,  0.        ,
          0.        ,  0.        ]]], dtype=float32)
Coordinates:
  * face     (face) int64 0 1 2 3 4 5 6 7 8 9 10 11 12
  * i        (i) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
  * j        (j) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
    k        int64 0

Spatially-Integrated Heat Content Anomaly

Now let’s do something harder. We will calculate the horizontally integrated heat content anomaly for the full 3D model domain.

# the monthly climatology
theta_clim = ds.THETA.groupby('time.month').mean(dim='time')
# the anomaly
theta_anom = ds.THETA.groupby('time.month') - theta_clim
rho0 = 1029
cp = 3994
ohc = rho0 * cp * (theta_anom *
                   coords.rA *
                   coords.hFacC).sum(dim=['face', 'j', 'i'])
ohc

<xarray.DataArray (time: 288, k: 50)>
dask.array<mul, shape=(288, 50), dtype=float64, chunksize=(1, 50), chunktype=numpy.ndarray>
Coordinates:
  * k        (k) int64 0 1 2 3 4 5 6 7 8 9 10 ... 40 41 42 43 44 45 46 47 48 49
  * time     (time) datetime64[ns] 1992-01-15 1992-02-13 ... 2015-12-14
    month    (time) int64 1 2 3 4 5 6 7 8 9 10 11 ... 2 3 4 5 6 7 8 9 10 11 12

# actually load the data
ohc.load()
# put the depth coordinate back for plotting purposes
ohc.coords['Z'] = coords.Z
ohc.swap_dims({'k': 'Z'}).transpose().plot(vmax=1e20)

<matplotlib.collections.QuadMesh at 0x7f9c2d032610>

Spatial Derivatives: Heat Budget

As our final exercise, we will do something much more complicated. We will compute the time-mean convergence of vertically-integrated heat fluxes. This is hard for several reasons.

The first reason it is hard is because it involves variables located at different grid points. Following MITgcm conventions, xmitgcm (which produced this dataset) labels the center point with the coordinates j, i, the u-velocity point as j, i_g, and the v-velocity point as j_g, i. The horizontal advective heat flux variables are

ADVx_TH    (time, k, face, j, i_g) float32 dask.array<shape=(288, 50, 13, 90, 90), chunksize=(1, 50, 13, 90, 90)>
ADVy_TH    (time, k, face, j_g, i) float32 dask.array<shape=(288, 50, 13, 90, 90), chunksize=(1, 50, 13, 90, 90)>

Xarray won’t allow us to add or multiply variables that have different dimensions, and xarray by itself doesn’t understand how to transform from one grid position to another.

That’s whyxgcmwas created.

Xgcm allows us to create a Grid object, which understands how to interpolate and take differences in a way that is compatible with finite volume models such at MITgcm. Xgcm also works with many other models, including ROMS, POP, MOM5/6, NEMO, etc.

A second reason this is hard is because of the complex topology connecting the different MITgcm faces. Fortunately xgcm also supports this.

import xgcm

# define the connectivity between faces
face_connections = {'face':
                    {0: {'X':  ((12, 'Y', False), (3, 'X', False)),
                         'Y':  (None,             (1, 'Y', False))},
                     1: {'X':  ((11, 'Y', False), (4, 'X', False)),
                         'Y':  ((0, 'Y', False),  (2, 'Y', False))},
                     2: {'X':  ((10, 'Y', False), (5, 'X', False)),
                         'Y':  ((1, 'Y', False),  (6, 'X', False))},
                     3: {'X':  ((0, 'X', False),  (9, 'Y', False)),
                         'Y':  (None,             (4, 'Y', False))},
                     4: {'X':  ((1, 'X', False),  (8, 'Y', False)),
                         'Y':  ((3, 'Y', False),  (5, 'Y', False))},
                     5: {'X':  ((2, 'X', False),  (7, 'Y', False)),
                         'Y':  ((4, 'Y', False),  (6, 'Y', False))},
                     6: {'X':  ((2, 'Y', False),  (7, 'X', False)),
                         'Y':  ((5, 'Y', False),  (10, 'X', False))},
                     7: {'X':  ((6, 'X', False),  (8, 'X', False)),
                         'Y':  ((5, 'X', False),  (10, 'Y', False))},
                     8: {'X':  ((7, 'X', False),  (9, 'X', False)),
                         'Y':  ((4, 'X', False),  (11, 'Y', False))},
                     9: {'X':  ((8, 'X', False),  None),
                         'Y':  ((3, 'X', False),  (12, 'Y', False))},
                     10: {'X': ((6, 'Y', False),  (11, 'X', False)),
                          'Y': ((7, 'Y', False),  (2, 'X', False))},
                     11: {'X': ((10, 'X', False), (12, 'X', False)),
                          'Y': ((8, 'Y', False),  (1, 'X', False))},
                     12: {'X': ((11, 'X', False), None),
                          'Y': ((9, 'Y', False),  (0, 'X', False))}}}

# create the grid object
grid = xgcm.Grid(ds, periodic=False, face_connections=face_connections)
grid

<xgcm.Grid>
X Axis (not periodic, boundary=None):
  * center   i --> left
  * left     i_g --> center
Y Axis (not periodic, boundary=None):
  * center   j --> left
  * left     j_g --> center
T Axis (not periodic, boundary=None):
  * center   time --> inner
  * inner    time_snp --> center
Z Axis (not periodic, boundary=None):
  * center   k --> left
  * left     k_l --> center
  * outer    k_p1 --> center
  * right    k_u --> center

Now we can use the grid object we created to take the divergence of a 2D vector

# vertical integral and time mean of horizontal diffusive heat flux
advx_th_vint = ds.ADVx_TH.sum(dim='k').mean(dim='time')
advy_th_vint = ds.ADVy_TH.sum(dim='k').mean(dim='time')

# difference in the x and y directions
diff_ADV_th = grid.diff_2d_vector({'X': advx_th_vint, 'Y': advy_th_vint}, boundary='fill')
# convergence
conv_ADV_th = -diff_ADV_th['X'] - diff_ADV_th['Y']
conv_ADV_th

<xarray.DataArray (face: 13, j: 90, i: 90)>
dask.array<sub, shape=(13, 90, 90), dtype=float32, chunksize=(1, 89, 89), chunktype=numpy.ndarray>
Coordinates:
  * face     (face) int64 0 1 2 3 4 5 6 7 8 9 10 11 12
  * j        (j) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
  * i        (i) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89

# vertical integral and time mean of horizontal diffusive heat flux
difx_th_vint = ds.DFxE_TH.sum(dim='k').mean(dim='time')
dify_th_vint = ds.DFyE_TH.sum(dim='k').mean(dim='time')

# difference in the x and y directions
diff_DIF_th = grid.diff_2d_vector({'X': difx_th_vint, 'Y': dify_th_vint}, boundary='fill')
# convergence
conv_DIF_th = -diff_DIF_th['X'] - diff_DIF_th['Y']
conv_DIF_th

<xarray.DataArray (face: 13, j: 90, i: 90)>
dask.array<sub, shape=(13, 90, 90), dtype=float32, chunksize=(1, 89, 89), chunktype=numpy.ndarray>
Coordinates:
  * face     (face) int64 0 1 2 3 4 5 6 7 8 9 10 11 12
  * j        (j) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89
  * i        (i) int64 0 1 2 3 4 5 6 7 8 9 10 ... 80 81 82 83 84 85 86 87 88 89

# convert to Watts / m^2 and load
mean_adv_conv = rho0 * cp * (conv_ADV_th/coords.rA).fillna(0.).load()
mean_dif_conv = rho0 * cp * (conv_DIF_th/coords.rA).fillna(0.).load()

/srv/conda/envs/notebook/lib/python3.8/site-packages/dask/core.py:121: RuntimeWarning: invalid value encountered in true_divide
  return func(*(_execute_task(a, cache) for a in args))

ax = mapper(mean_adv_conv, cmap='RdBu_r', vmax=300, vmin=-300);
ax.set_title(r'Convergence of Advective Flux (W/m$^2$)');

ax = mapper(mean_dif_conv, cmap='RdBu_r', vmax=300, vmin=-300)
ax.set_title(r'Convergence of Diffusive Flux (W/m$^2$)');

ax = mapper(mean_dif_conv + mean_adv_conv, cmap='RdBu_r', vmax=300, vmin=-300)
ax.set_title(r'Convergence of Net Horizontal Flux (W/m$^2$)');

ax = mapper(ds.TFLUX.mean(dim='time').load(), cmap='RdBu_r', vmax=300, vmin=-300);
ax.set_title(r'Surface Heat Flux (W/m$^2$)');