Meridional Overturning - Tracer Space

Calculating the AMOC in latitude-tracer coordinates¶

Description:¶

This recipe shows how to calculate the Atlantic Meridional Overturning Circulation (AMOC) stream function in potential density-coordinates using annual-mean outputs from the National Oceanography Centre Near-Present-Day global eORCA1 configuration of NEMO forced using JRA55-do from 1976-2024.

For more details on this model configuration and the available outputs, users can explore the Near-Present-Day documentation here.

Background:¶

The diapycnal overturning stream function is routinely used to characterise the strength and structure of the AMOC in density-space (e.g., $\sigma_{0}$ or $\sigma{2}$) as a function of latitude $\phi$ and can be defined at time $t$ as follows:

$$\Psi_{\sigma_{0}}(\phi, \sigma_{0}, t) = \int_{x_w}^{x_e} \int_{z(\lambda, \phi, \sigma_{0})}^{\eta} v(\lambda, \phi, z', t) \ dz' \ dx$$

where the meridional velocity $v(\lambda, \phi, z, t)$ is first accumulated vertically from the sea surface $\eta$ to a specified isopycnal depth $z(\lambda, \phi, \sigma_{0})$ (decreasing downward) before being integrated zonally between the western $x_w$ and eastern $x_e$ boundaries of the basin.

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# -- Import required packages -- #
import gsw
import numpy as np
import xarray as xr
from nemo_cookbook import NEMODataTree
# -- Import required packages -- #
import gsw
import numpy as np
import xarray as xr
from nemo_cookbook import NEMODataTree

Using Dask¶

Optional: Connect Client to Dask Local Cluster to run analysis in parallel.

Note that, although using Dask is not strictly necessary for this simple example using eORCA1, if we wanted to generalise this recipe to eORCA025 or eORCA12 outputs, using Dask would be essential to avoid unnecessary slow calculations using only a single process.

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# -- Initialise Dask Local Cluster -- #
import os
import dask
from dask.distributed import Client, LocalCluster

# Update temporary directory for Dask workers:
dask.config.set({'temporary_directory': f"{os.getcwd()}/dask_tmp",
                 'local_directory': f"{os.getcwd()}/dask_tmp"
                 })

# Create Local Cluster:
cluster = LocalCluster(n_workers=4, threads_per_worker=3, memory_limit='5GB')
client = Client(cluster)
client
# -- Initialise Dask Local Cluster -- #
import os
import dask
from dask.distributed import Client, LocalCluster

# Update temporary directory for Dask workers:
dask.config.set({'temporary_directory': f"{os.getcwd()}/dask_tmp",
                 'local_directory': f"{os.getcwd()}/dask_tmp"
                 })

# Create Local Cluster:
cluster = LocalCluster(n_workers=4, threads_per_worker=3, memory_limit='5GB')
client = Client(cluster)
client

Accessing NEMO Model Data¶

Let's begin by loading the grid variables for our eORCA1 NEMO model from the JASMIN Object Store.

Alternatively, you can replace the domain_filepath below with a file path to your domain_cfg.nc file and read this with xarray's open_dataset() function.

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# Define directory path to ancillary files:
domain_filepath = "https://noc-msm-o.s3-ext.jc.rl.ac.uk/npd-eorca1-jra55v1/domain"

# Open eORCA1 NEMO model domain_cfg:
ds_domain = xr.open_zarr(domain_filepath, consolidated=True, chunks={})

ds_domain
# Define directory path to ancillary files:
domain_filepath = "https://noc-msm-o.s3-ext.jc.rl.ac.uk/npd-eorca1-jra55v1/domain"

# Open eORCA1 NEMO model domain_cfg:
ds_domain = xr.open_zarr(domain_filepath, consolidated=True, chunks={})

ds_domain

Out[2]:

<xarray.Dataset> Size: 669MB
Dimensions:        (y: 331, x: 360, t: 1, z: 75)
Dimensions without coordinates: y, x, t, z
Data variables: (12/58)
    atlmsk         (y, x) float32 477kB dask.array<chunksize=(331, 360), meta=np.ndarray>
    bathy_metry    (t, y, x) float32 477kB dask.array<chunksize=(1, 331, 360), meta=np.ndarray>
    bottom_level   (t, y, x) int32 477kB dask.array<chunksize=(1, 331, 360), meta=np.ndarray>
    e1f            (t, y, x) float64 953kB dask.array<chunksize=(1, 331, 360), meta=np.ndarray>
    e1t            (t, y, x) float64 953kB dask.array<chunksize=(1, 331, 360), meta=np.ndarray>
    e1u            (t, y, x) float64 953kB dask.array<chunksize=(1, 331, 360), meta=np.ndarray>
    ...             ...
    nav_lev        (z) float32 300B dask.array<chunksize=(75,), meta=np.ndarray>
    nav_lon        (y, x) float32 477kB dask.array<chunksize=(331, 360), meta=np.ndarray>
    pacmsk         (y, x) float32 477kB dask.array<chunksize=(331, 360), meta=np.ndarray>
    socmsk         (y, x) float32 477kB dask.array<chunksize=(331, 360), meta=np.ndarray>
    time_counter   (t) float64 8B dask.array<chunksize=(1,), meta=np.ndarray>
    top_level      (t, y, x) int32 477kB dask.array<chunksize=(1, 331, 360), meta=np.ndarray>
Attributes:
    NCO:      netCDF Operators version 5.2.2 (Homepage = http://nco.sf.net, C...
    history:  Fri Oct  4 10:58:13 2024: ncks -O --4 --no_abc --cnk_plc=xpl --...

Next, we need to import the conservative temperature and absolute salinity stored at T-points in a single dataset.

Typically, NEMO model outputs defined on T-grid points are stored together in netCDF files. In this case, you can replace xr.merge() with a single call to xarray's open_dataset() function passing the file path to your _gridT.nc file(s).

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# Define directory path to model output files:
output_dir = "https://noc-msm-o.s3-ext.jc.rl.ac.uk/npd-eorca1-jra55v1/T1y"

# Construct NEMO model grid dataset, including vertical grid cell thicknesses (m) and meridional velocities (m/s):
ds_gridT = xr.merge([xr.open_zarr(f"{output_dir}/{var}", consolidated=True, chunks={})[var] for var in ['thetao_con', 'so_abs']], compat="override")
ds_gridT = ds_gridT.sel(time_counter=slice("1976-01", "2023-12"))

# Calculate potential density anomaly referenced to the sea surface (kg/m3):
ds_gridT['sigma0'] = gsw.density.sigma0(CT=ds_gridT['thetao_con'], SA=ds_gridT['so_abs'])
ds_gridT['sigma0'].name = 'sigma0'

ds_gridT
# Define directory path to model output files:
output_dir = "https://noc-msm-o.s3-ext.jc.rl.ac.uk/npd-eorca1-jra55v1/T1y"

# Construct NEMO model grid dataset, including vertical grid cell thicknesses (m) and meridional velocities (m/s):
ds_gridT = xr.merge([xr.open_zarr(f"{output_dir}/{var}", consolidated=True, chunks={})[var] for var in ['thetao_con', 'so_abs']], compat="override")
ds_gridT = ds_gridT.sel(time_counter=slice("1976-01", "2023-12"))

# Calculate potential density anomaly referenced to the sea surface (kg/m3):
ds_gridT['sigma0'] = gsw.density.sigma0(CT=ds_gridT['thetao_con'], SA=ds_gridT['so_abs'])
ds_gridT['sigma0'].name = 'sigma0'

ds_gridT

Out[3]:

<xarray.Dataset> Size: 7GB
Dimensions:        (deptht: 75, y: 331, x: 360, time_counter: 48)
Coordinates:
  * deptht         (deptht) float32 300B 0.5058 1.556 ... 5.698e+03 5.902e+03
    nav_lat        (y, x) float64 953kB dask.array<chunksize=(331, 360), meta=np.ndarray>
    nav_lon        (y, x) float64 953kB dask.array<chunksize=(331, 360), meta=np.ndarray>
    time_centered  (time_counter) datetime64[ns] 384B dask.array<chunksize=(1,), meta=np.ndarray>
  * time_counter   (time_counter) datetime64[ns] 384B 1976-07-02 ... 2023-07-...
Dimensions without coordinates: y, x
Data variables:
    thetao_con     (time_counter, deptht, y, x) float32 2GB dask.array<chunksize=(1, 25, 331, 360), meta=np.ndarray>
    so_abs         (time_counter, deptht, y, x) float32 2GB dask.array<chunksize=(1, 25, 331, 360), meta=np.ndarray>
    sigma0         (time_counter, deptht, y, x) float64 3GB dask.array<chunksize=(1, 25, 331, 360), meta=np.ndarray>
Attributes:
    cell_methods:        time: mean
    interval_operation:  1 yr
    interval_write:      1 yr
    long_name:           sea_water_conservative_temperature
    online_operation:    average
    standard_name:       sea_water_conservative_temperature
    units:               degC

Next, we need to import the meridional velocity and vertical grid cell thicknesses stored at V-points in a single dataset.

Typically, NEMO model outputs defined on V-grid points are stored together in netCDF files. In this case, you can replace xr.merge() with a single call to xarray's open_dataset() function passing the file path to your _gridV.nc file(s).

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# Define directory path to model output files:
output_dir = "https://noc-msm-o.s3-ext.jc.rl.ac.uk/npd-eorca1-jra55v1/V1y"

# Construct NEMO model grid dataset, including vertical grid cell thicknesses (m) and meridional velocities (m/s):
ds_gridV = xr.merge([xr.open_zarr(f"{output_dir}/{var}", consolidated=True, chunks={})[var] for var in ['e3v', 'vo']], compat="override")
ds_gridV = ds_gridV.sel(time_counter=slice("1976-01", "2023-12"))

ds_gridV
# Define directory path to model output files:
output_dir = "https://noc-msm-o.s3-ext.jc.rl.ac.uk/npd-eorca1-jra55v1/V1y"

# Construct NEMO model grid dataset, including vertical grid cell thicknesses (m) and meridional velocities (m/s):
ds_gridV = xr.merge([xr.open_zarr(f"{output_dir}/{var}", consolidated=True, chunks={})[var] for var in ['e3v', 'vo']], compat="override")
ds_gridV = ds_gridV.sel(time_counter=slice("1976-01", "2023-12"))

ds_gridV

Out[4]:

<xarray.Dataset> Size: 3GB
Dimensions:        (depthv: 75, y: 331, x: 360, time_counter: 48)
Coordinates:
  * depthv         (depthv) float32 300B 0.5058 1.556 ... 5.698e+03 5.902e+03
    nav_lat        (y, x) float64 953kB dask.array<chunksize=(331, 360), meta=np.ndarray>
    nav_lon        (y, x) float64 953kB dask.array<chunksize=(331, 360), meta=np.ndarray>
    time_centered  (time_counter) datetime64[ns] 384B dask.array<chunksize=(1,), meta=np.ndarray>
  * time_counter   (time_counter) datetime64[ns] 384B 1976-07-02 ... 2023-07-...
Dimensions without coordinates: y, x
Data variables:
    e3v            (time_counter, depthv, y, x) float32 2GB dask.array<chunksize=(1, 25, 331, 360), meta=np.ndarray>
    vo             (time_counter, depthv, y, x) float32 2GB dask.array<chunksize=(1, 25, 331, 360), meta=np.ndarray>
Attributes:
    cell_methods:        time: mean (interval: 3600 s)
    interval_operation:  3600 s
    interval_write:      1 yr
    long_name:           V-cell thickness
    online_operation:    average
    standard_name:       cell_thickness
    units:               m

Creating a NEMODataTree¶

Next, let's create a NEMODataTree to store our domain and T- & V-grid variables for the eORCA1 model.

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# Define dictionary of grid datasets defining eORCA1 parent model domain with no child/grand-child nests:
# Note: domain_cfg z-dimension is expected to be named 'nav_lev'.
datasets = {"parent": {"domain": ds_domain.rename({"z": "nav_lev"}), "gridT": ds_gridT, "gridV": ds_gridV}}

# Initialise a new NEMODataTree whose parent domain is zonally periodic & north-folding on F-points:
nemo = NEMODataTree.from_datasets(datasets=datasets, iperio=True, nftype="F")

nemo
# Define dictionary of grid datasets defining eORCA1 parent model domain with no child/grand-child nests:
# Note: domain_cfg z-dimension is expected to be named 'nav_lev'.
datasets = {"parent": {"domain": ds_domain.rename({"z": "nav_lev"}), "gridT": ds_gridT, "gridV": ds_gridV}}

# Initialise a new NEMODataTree whose parent domain is zonally periodic & north-folding on F-points:
nemo = NEMODataTree.from_datasets(datasets=datasets, iperio=True, nftype="F")

nemo

Out[5]:

<xarray.DatasetView> Size: 768B
Dimensions:        (time_counter: 48)
Coordinates:
    time_centered  (time_counter) datetime64[ns] 384B dask.array<chunksize=(1,), meta=np.ndarray>
  * time_counter   (time_counter) datetime64[ns] 384B 1976-07-02 ... 2023-07-...
Data variables:
    *empty*
Attributes:
    nftype:   F
    iperio:   True

Calculating the AMOC diapycnal overturning stream function¶

Now we have constructed our NEMODataTree, let's calculate the diapycnal overturning stream function.

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# Define Atlantic Ocean basin mask:
atlmask = ds_domain['atlmsk'].rename({"x":"i", "y":"j"}).astype(bool)

# Add meridional volume transport [m3/s] to NEMO V-grid - meridional velocity [m/s] * area of meridional grid cell face [m2]:
nemo['/gridV']['volume_transport'] = (nemo['/gridV']['vo'] * nemo.cell_area(grid='/gridV', dim='j'))

# Transform potential density from NEMO T-grid to V-grid using linear interpolation:
nemo['/gridV']['sigma0'] = nemo.transform_scalar_to(grid='/gridT', var='sigma0', to='V')
# Define Atlantic Ocean basin mask:
atlmask = ds_domain['atlmsk'].rename({"x":"i", "y":"j"}).astype(bool)

# Add meridional volume transport [m3/s] to NEMO V-grid - meridional velocity [m/s] * area of meridional grid cell face [m2]:
nemo['/gridV']['volume_transport'] = (nemo['/gridV']['vo'] * nemo.cell_area(grid='/gridV', dim='j'))

# Transform potential density from NEMO T-grid to V-grid using linear interpolation:
nemo['/gridV']['sigma0'] = nemo.transform_scalar_to(grid='/gridT', var='sigma0', to='V')

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# Define potential density bins [kg /m3]:
sigma0_bins = np.arange(22, 29, 0.01)

# Compute meridional volume transport in latitude-potential density coords
vt_sigma0_atl = nemo.binned_statistic(grid="/gridV",
                                      vars=["sigma0"],
                                      values="volume_transport",
                                      keep_dims=["time_counter", "j"],
                                      bins=[sigma0_bins],
                                      statistic="nansum",
                                      mask=atlmask
                                      )

vt_sigma0_atl
# Define potential density bins [kg /m3]:
sigma0_bins = np.arange(22, 29, 0.01)

# Compute meridional volume transport in latitude-potential density coords
vt_sigma0_atl = nemo.binned_statistic(grid="/gridV",
                                      vars=["sigma0"],
                                      values="volume_transport",
                                      keep_dims=["time_counter", "j"],
                                      bins=[sigma0_bins],
                                      statistic="nansum",
                                      mask=atlmask
                                      )

vt_sigma0_atl

Out[7]:

<xarray.DataArray 'volume_transport' (time_counter: 48, j: 331, sigma0_bins: 699)> Size: 89MB
dask.array<reshape, shape=(48, 331, 699), dtype=float64, chunksize=(48, 331, 699), chunktype=numpy.ndarray>
Coordinates:
  * time_counter  (time_counter) datetime64[ns] 384B 1976-07-02 ... 2023-07-0...
  * j             (j) float64 3kB 1.5 2.5 3.5 4.5 ... 328.5 329.5 330.5 331.5
  * sigma0_bins   (sigma0_bins) float64 6kB 22.01 22.02 22.03 ... 28.98 28.99

Notice that the resulting DataArray includes a dask array, so we haven't actually computed the diapycnal overturning yet. To do this, we need to call the .compute() method:

Visualising the time-mean AMOC diapycnal overturning stream function¶

Finally, let's visualise the results by plotting the time-mean Atlantic Meridional Overturning stream function in potential density-coordinates:

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moc_sigma0_atl.mean(dim='time_counter').plot(y='sigma0_bins', yincrease=False)
moc_sigma0_atl.mean(dim='time_counter').plot(y='sigma0_bins', yincrease=False)

Out[9]:

<matplotlib.collections.QuadMesh at 0x12ed33cb0>

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