Seed grids and flow maps

This notebook is about the structures and the API, not about advection.

A SeedGrid holds a grid of release positions. SeedGrid.to_parcels_pset() emits them as flat lon, lat lists. SeedGrid.pset_to_flowmap(...) ingests the advected positions and returns a FlowMap, which computes the deformation gradient, the Cauchy–Green tensor and the FTLE. No Parcels is imported below; the advection is replaced by particles that do not move.

There are two stencils, and each is a pair of classes. NeighborSeedGrid releases one particle per grid point and differences against the neighbouring grid points. AuxiliarySeedGrid releases four arms around each grid point and differences across the arms.

import numpy as np

from lcs_parcels import AuxiliarySeedGrid, NeighborSeedGrid

Grid and window

The axes set the grid. The times are not attached to anything until ingest. t1 > t0 makes the window \(T = t_1 - t_0\) positive, so the FTLE below is the forward one.

lon_axis = np.linspace(-25.0, -20.0, 6)
lat_axis = np.linspace(15.0, 20.0, 5)

t0 = np.datetime64("2020-01-01")
t1 = np.datetime64("2020-01-11")

The neighbour stencil

One particle per grid point: the release positions lon_0/lat_0 are the diagnostic grid points lon_grid/lat_grid, stored twice rather than left to be reconstructed. The outermost ring of grid points has no neighbour to difference against on one side, so the FTLE is undefined there.

seed = NeighborSeedGrid.from_axes(lon=lon_axis, lat=lat_axis)
print(seed)
seed.ds

<NeighborSeedGrid 6x5 grid, lon -25.00..-20.00, lat 15.00..20.00>

<xarray.Dataset> Size: 1kB
Dimensions:   (i: 6, j: 5)
Coordinates:
  * i         (i) int64 48B 0 1 2 3 4 5
  * j         (j) int64 40B 0 1 2 3 4
    lon_grid  (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_grid  (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
    lon_0     (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_0     (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
Data variables:
    *empty*

xarray.Dataset

• Dimensions:
  • i: 6
  • j: 5
• Coordinates: (6)
  • i
    (i)
    int64
    0 1 2 3 4 5

    long_name :

    logical grid index along i

    array([0, 1, 2, 3, 4, 5])

  • j
    (j)
    int64
    0 1 2 3 4

    long_name :

    logical grid index along j

    array([0, 1, 2, 3, 4])

  • lon_grid
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_grid
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

  • lon_0
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude of the reference release position x_0

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_0
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude of the reference release position x_0

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

• Data variables: (0)

The auxiliary stencil

Four arms (east, north, west, south) at aux_separation_m around each grid point, so lon_0/lat_0 gain a displacement dim and there are four particles per grid point. The finite-difference step is the arm separation rather than the grid spacing, and the FTLE is defined at every grid point, including the boundary.

aux_seed = AuxiliarySeedGrid.from_axes(
    lon=lon_axis, lat=lat_axis, aux_separation_m=1_000.0
)
print(aux_seed)
aux_seed.ds

<AuxiliarySeedGrid 6x5 grid, lon -25.00..-20.00, lat 15.00..20.00>

<xarray.Dataset> Size: 3kB
Dimensions:       (displacement: 4, i: 6, j: 5)
Coordinates:
  * displacement  (displacement) <U5 80B 'east' 'north' 'west' 'south'
    lon_0         (i, j, displacement) float64 960B -24.99 -25.0 ... -20.0
    lat_0         (i, j, displacement) float64 960B 15.0 15.01 ... 20.0 19.99
  * i             (i) int64 48B 0 1 2 3 4 5
  * j             (j) int64 40B 0 1 2 3 4
    lon_grid      (i, j) float64 240B -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_grid      (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
Data variables:
    *empty*

xarray.Dataset

• Dimensions:
  • displacement: 4
  • i: 6
  • j: 5
• Coordinates: (7)
  • displacement
    (displacement)
    <U5
    'east' 'north' 'west' 'south'

    long_name :

    auxiliary stencil arm

    array(['east', 'north', 'west', 'south'], dtype='<U5')

  • lon_0
    (i, j, displacement)
    float64
    -24.99 -25.0 ... -20.01 -20.0

    long_name :

    longitude of the reference release position x_0

    units :

    degrees_east

    array([[[-24.99068954, -25.        , -25.00931046, -25.        ],
            [-24.99063255, -25.        , -25.00936745, -25.        ],
            [-24.99057035, -25.        , -25.00942965, -25.        ],
            [-24.99050277, -25.        , -25.00949723, -25.        ],
            [-24.99042962, -25.        , -25.00957038, -25.        ]],
    
           [[-23.99068954, -24.        , -24.00931046, -24.        ],
            [-23.99063255, -24.        , -24.00936745, -24.        ],
            [-23.99057035, -24.        , -24.00942965, -24.        ],
            [-23.99050277, -24.        , -24.00949723, -24.        ],
            [-23.99042962, -24.        , -24.00957038, -24.        ]],
    
           [[-22.99068954, -23.        , -23.00931046, -23.        ],
            [-22.99063255, -23.        , -23.00936745, -23.        ],
            [-22.99057035, -23.        , -23.00942965, -23.        ],
            [-22.99050277, -23.        , -23.00949723, -23.        ],
            [-22.99042962, -23.        , -23.00957038, -23.        ]],
    
           [[-21.99068954, -22.        , -22.00931046, -22.        ],
            [-21.99063255, -22.        , -22.00936745, -22.        ],
            [-21.99057035, -22.        , -22.00942965, -22.        ],
            [-21.99050277, -22.        , -22.00949723, -22.        ],
            [-21.99042962, -22.        , -22.00957038, -22.        ]],
    
           [[-20.99068954, -21.        , -21.00931046, -21.        ],
            [-20.99063255, -21.        , -21.00936745, -21.        ],
            [-20.99057035, -21.        , -21.00942965, -21.        ],
            [-20.99050277, -21.        , -21.00949723, -21.        ],
            [-20.99042962, -21.        , -21.00957038, -21.        ]],
    
           [[-19.99068954, -20.        , -20.00931046, -20.        ],
            [-19.99063255, -20.        , -20.00936745, -20.        ],
            [-19.99057035, -20.        , -20.00942965, -20.        ],
            [-19.99050277, -20.        , -20.00949723, -20.        ],
            [-19.99042962, -20.        , -20.00957038, -20.        ]]])

  • lat_0
    (i, j, displacement)
    float64
    15.0 15.01 15.0 ... 20.0 19.99

    long_name :

    latitude of the reference release position x_0

    units :

    degrees_north

    array([[[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]]])

  • i
    (i)
    int64
    0 1 2 3 4 5

    long_name :

    logical grid index along i

    array([0, 1, 2, 3, 4, 5])

  • j
    (j)
    int64
    0 1 2 3 4

    long_name :

    logical grid index along j

    array([0, 1, 2, 3, 4])

  • lon_grid
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_grid
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

• Data variables: (0)

Emit the particle sets

to_parcels_pset() flattens the release positions to two plain lists, one entry per particle: 30 grid points for the neighbour stencil, 4 arms each for the auxiliary one. The two lists are everything Parcels needs, and ingest reattaches the advected positions to the grid by their order.

lon_0, lat_0 = seed.to_parcels_pset()
aux_lon_0, aux_lat_0 = aux_seed.to_parcels_pset()
print("neighbour particles:", len(lon_0))
print("auxiliary particles:", len(aux_lon_0))

neighbour particles: 30
auxiliary particles: 120

Advect, without Parcels

The particles sit still: \(u = v = 0\) for ten days. The flow map is the identity, \(\nabla F = I\), so the FTLE is zero wherever it is defined. A non-zero window keeps \(1/T\) finite.

lon_1, lat_1 = lon_0, lat_0
aux_lon_1, aux_lat_1 = aux_lon_0, aux_lat_0

Ingest the advected positions

pset_to_flowmap puts the flat positions back on the grid as lon/lat beside the release positions lon_0/lat_0, and records t0 and the signed window T it derives from t0 and t1. Both stencils ingest through the same call; the datasets differ only where their release positions did.

flowmap = seed.pset_to_flowmap(lon=lon_1, lat=lat_1, t0=t0, t1=t1)
print(flowmap)
flowmap.ds

<NeighborFlowMap 6x5 grid, lon -25.00..-20.00, lat 15.00..20.00,
                 t0 2020-01-01T00:00:00, T +10.0 days>

<xarray.Dataset> Size: 2kB
Dimensions:   (i: 6, j: 5)
Coordinates:
  * i         (i) int64 48B 0 1 2 3 4 5
  * j         (j) int64 40B 0 1 2 3 4
    lon_grid  (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_grid  (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
    lon_0     (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_0     (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
    t0        datetime64[s] 8B 2020-01-01
    T         timedelta64[s] 8B 10 days
Data variables:
    lon       (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat       (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0

xarray.Dataset

• Dimensions:
  • i: 6
  • j: 5
• Coordinates: (8)
  • i
    (i)
    int64
    0 1 2 3 4 5

    long_name :

    logical grid index along i

    array([0, 1, 2, 3, 4, 5])

  • j
    (j)
    int64
    0 1 2 3 4

    long_name :

    logical grid index along j

    array([0, 1, 2, 3, 4])

  • lon_grid
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_grid
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

  • lon_0
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude of the reference release position x_0

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_0
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude of the reference release position x_0

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

  • t0
    ()
    datetime64[s]
    2020-01-01

    long_name :

    release time t0

    array('2020-01-01T00:00:00', dtype='datetime64[s]')

  • T
    ()
    timedelta64[s]
    10 days

    long_name :

    signed integration window T = t1 - t0

    array(864000, dtype='timedelta64[s]')

• Data variables: (2)
  • lon
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude of the advected position F(x_0)

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude of the advected position F(x_0)

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

aux_flowmap = aux_seed.pset_to_flowmap(lon=aux_lon_1, lat=aux_lat_1, t0=t0, t1=t1)
print(aux_flowmap)
aux_flowmap.ds

<AuxiliaryFlowMap 6x5 grid, lon -25.00..-20.00, lat 15.00..20.00,
                  t0 2020-01-01T00:00:00, T +10.0 days>

<xarray.Dataset> Size: 5kB
Dimensions:       (displacement: 4, i: 6, j: 5)
Coordinates:
  * displacement  (displacement) <U5 80B 'east' 'north' 'west' 'south'
    lon_0         (i, j, displacement) float64 960B -24.99 -25.0 ... -20.0
    lat_0         (i, j, displacement) float64 960B 15.0 15.01 ... 20.0 19.99
  * i             (i) int64 48B 0 1 2 3 4 5
  * j             (j) int64 40B 0 1 2 3 4
    lon_grid      (i, j) float64 240B -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_grid      (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
    t0            datetime64[s] 8B 2020-01-01
    T             timedelta64[s] 8B 10 days
Data variables:
    lon           (i, j, displacement) float64 960B -24.99 -25.0 ... -20.0
    lat           (i, j, displacement) float64 960B 15.0 15.01 ... 20.0 19.99

xarray.Dataset

• Dimensions:
  • displacement: 4
  • i: 6
  • j: 5
• Coordinates: (9)
  • displacement
    (displacement)
    <U5
    'east' 'north' 'west' 'south'

    long_name :

    auxiliary stencil arm

    array(['east', 'north', 'west', 'south'], dtype='<U5')

  • lon_0
    (i, j, displacement)
    float64
    -24.99 -25.0 ... -20.01 -20.0

    long_name :

    longitude of the reference release position x_0

    units :

    degrees_east

    array([[[-24.99068954, -25.        , -25.00931046, -25.        ],
            [-24.99063255, -25.        , -25.00936745, -25.        ],
            [-24.99057035, -25.        , -25.00942965, -25.        ],
            [-24.99050277, -25.        , -25.00949723, -25.        ],
            [-24.99042962, -25.        , -25.00957038, -25.        ]],
    
           [[-23.99068954, -24.        , -24.00931046, -24.        ],
            [-23.99063255, -24.        , -24.00936745, -24.        ],
            [-23.99057035, -24.        , -24.00942965, -24.        ],
            [-23.99050277, -24.        , -24.00949723, -24.        ],
            [-23.99042962, -24.        , -24.00957038, -24.        ]],
    
           [[-22.99068954, -23.        , -23.00931046, -23.        ],
            [-22.99063255, -23.        , -23.00936745, -23.        ],
            [-22.99057035, -23.        , -23.00942965, -23.        ],
            [-22.99050277, -23.        , -23.00949723, -23.        ],
            [-22.99042962, -23.        , -23.00957038, -23.        ]],
    
           [[-21.99068954, -22.        , -22.00931046, -22.        ],
            [-21.99063255, -22.        , -22.00936745, -22.        ],
            [-21.99057035, -22.        , -22.00942965, -22.        ],
            [-21.99050277, -22.        , -22.00949723, -22.        ],
            [-21.99042962, -22.        , -22.00957038, -22.        ]],
    
           [[-20.99068954, -21.        , -21.00931046, -21.        ],
            [-20.99063255, -21.        , -21.00936745, -21.        ],
            [-20.99057035, -21.        , -21.00942965, -21.        ],
            [-20.99050277, -21.        , -21.00949723, -21.        ],
            [-20.99042962, -21.        , -21.00957038, -21.        ]],
    
           [[-19.99068954, -20.        , -20.00931046, -20.        ],
            [-19.99063255, -20.        , -20.00936745, -20.        ],
            [-19.99057035, -20.        , -20.00942965, -20.        ],
            [-19.99050277, -20.        , -20.00949723, -20.        ],
            [-19.99042962, -20.        , -20.00957038, -20.        ]]])

  • lat_0
    (i, j, displacement)
    float64
    15.0 15.01 15.0 ... 20.0 19.99

    long_name :

    latitude of the reference release position x_0

    units :

    degrees_north

    array([[[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]]])

  • i
    (i)
    int64
    0 1 2 3 4 5

    long_name :

    logical grid index along i

    array([0, 1, 2, 3, 4, 5])

  • j
    (j)
    int64
    0 1 2 3 4

    long_name :

    logical grid index along j

    array([0, 1, 2, 3, 4])

  • lon_grid
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_grid
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

  • t0
    ()
    datetime64[s]
    2020-01-01

    long_name :

    release time t0

    array('2020-01-01T00:00:00', dtype='datetime64[s]')

  • T
    ()
    timedelta64[s]
    10 days

    long_name :

    signed integration window T = t1 - t0

    array(864000, dtype='timedelta64[s]')

• Data variables: (2)
  • lon
    (i, j, displacement)
    float64
    -24.99 -25.0 ... -20.01 -20.0

    long_name :

    longitude of the advected position F(x_0)

    units :

    degrees_east

    array([[[-24.99068954, -25.        , -25.00931046, -25.        ],
            [-24.99063255, -25.        , -25.00936745, -25.        ],
            [-24.99057035, -25.        , -25.00942965, -25.        ],
            [-24.99050277, -25.        , -25.00949723, -25.        ],
            [-24.99042962, -25.        , -25.00957038, -25.        ]],
    
           [[-23.99068954, -24.        , -24.00931046, -24.        ],
            [-23.99063255, -24.        , -24.00936745, -24.        ],
            [-23.99057035, -24.        , -24.00942965, -24.        ],
            [-23.99050277, -24.        , -24.00949723, -24.        ],
            [-23.99042962, -24.        , -24.00957038, -24.        ]],
    
           [[-22.99068954, -23.        , -23.00931046, -23.        ],
            [-22.99063255, -23.        , -23.00936745, -23.        ],
            [-22.99057035, -23.        , -23.00942965, -23.        ],
            [-22.99050277, -23.        , -23.00949723, -23.        ],
            [-22.99042962, -23.        , -23.00957038, -23.        ]],
    
           [[-21.99068954, -22.        , -22.00931046, -22.        ],
            [-21.99063255, -22.        , -22.00936745, -22.        ],
            [-21.99057035, -22.        , -22.00942965, -22.        ],
            [-21.99050277, -22.        , -22.00949723, -22.        ],
            [-21.99042962, -22.        , -22.00957038, -22.        ]],
    
           [[-20.99068954, -21.        , -21.00931046, -21.        ],
            [-20.99063255, -21.        , -21.00936745, -21.        ],
            [-20.99057035, -21.        , -21.00942965, -21.        ],
            [-20.99050277, -21.        , -21.00949723, -21.        ],
            [-20.99042962, -21.        , -21.00957038, -21.        ]],
    
           [[-19.99068954, -20.        , -20.00931046, -20.        ],
            [-19.99063255, -20.        , -20.00936745, -20.        ],
            [-19.99057035, -20.        , -20.00942965, -20.        ],
            [-19.99050277, -20.        , -20.00949723, -20.        ],
            [-19.99042962, -20.        , -20.00957038, -20.        ]]])

  • lat
    (i, j, displacement)
    float64
    15.0 15.01 15.0 ... 20.0 19.99

    long_name :

    latitude of the advected position F(x_0)

    units :

    degrees_north

    array([[[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]],
    
           [[15.        , 15.00899322, 15.        , 14.99100678],
            [16.25      , 16.25899322, 16.25      , 16.24100678],
            [17.5       , 17.50899322, 17.5       , 17.49100678],
            [18.75      , 18.75899322, 18.75      , 18.74100678],
            [20.        , 20.00899322, 20.        , 19.99100678]]])

grid_image is the one place the two datasets are reconciled: the advected position of the grid point, on (i, j). The neighbour flow map passes its single advected position through; the auxiliary one takes the centroid of its four advected arms.

aux_flowmap.grid_image

<xarray.Dataset> Size: 1kB
Dimensions:   (i: 6, j: 5)
Coordinates:
  * i         (i) int64 48B 0 1 2 3 4 5
  * j         (j) int64 40B 0 1 2 3 4
    lon_grid  (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_grid  (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
    t0        datetime64[s] 8B 2020-01-01
    T         timedelta64[s] 8B 10 days
Data variables:
    lon       (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat       (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0

xarray.Dataset

• Dimensions:
  • i: 6
  • j: 5
• Coordinates: (6)
  • i
    (i)
    int64
    0 1 2 3 4 5

    long_name :

    logical grid index along i

    array([0, 1, 2, 3, 4, 5])

  • j
    (j)
    int64
    0 1 2 3 4

    long_name :

    logical grid index along j

    array([0, 1, 2, 3, 4])

  • lon_grid
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_grid
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

  • t0
    ()
    datetime64[s]
    2020-01-01

    long_name :

    release time t0

    array('2020-01-01T00:00:00', dtype='datetime64[s]')

  • T
    ()
    timedelta64[s]
    10 days

    long_name :

    signed integration window T = t1 - t0

    array(864000, dtype='timedelta64[s]')

• Data variables: (2)
  • lon
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude of the advected position F(x_0)

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude of the advected position F(x_0)

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

FTLE from both stencils

ftle() returns 1/s on the diagnostic grid. Both stencils give zero; they differ in where the FTLE is defined.

ftle = flowmap.ftle()
ftle

<xarray.DataArray 'ftle' (i: 6, j: 5)> Size: 240B
array([[nan, nan, nan, nan, nan],
       [nan,  0.,  0.,  0., nan],
       [nan,  0.,  0.,  0., nan],
       [nan,  0.,  0.,  0., nan],
       [nan,  0.,  0.,  0., nan],
       [nan, nan, nan, nan, nan]])
Coordinates:
  * i         (i) int64 48B 0 1 2 3 4 5
  * j         (j) int64 40B 0 1 2 3 4
    lon_grid  (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_grid  (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
    t0        datetime64[s] 8B 2020-01-01
    T         timedelta64[s] 8B 10 days
Attributes:
    long_name:  finite-time Lyapunov exponent
    units:      1/s

xarray.DataArray

'ftle'

• i: 6
• j: 5

• nan nan nan nan nan nan 0.0 0.0 ... 0.0 0.0 nan nan nan nan nan nan

  array([[nan, nan, nan, nan, nan],
         [nan,  0.,  0.,  0., nan],
         [nan,  0.,  0.,  0., nan],
         [nan,  0.,  0.,  0., nan],
         [nan,  0.,  0.,  0., nan],
         [nan, nan, nan, nan, nan]])

• Coordinates: (6)
  • i
    (i)
    int64
    0 1 2 3 4 5

    long_name :

    logical grid index along i

    array([0, 1, 2, 3, 4, 5])

  • j
    (j)
    int64
    0 1 2 3 4

    long_name :

    logical grid index along j

    array([0, 1, 2, 3, 4])

  • lon_grid
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_grid
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

  • t0
    ()
    datetime64[s]
    2020-01-01

    long_name :

    release time t0

    array('2020-01-01T00:00:00', dtype='datetime64[s]')

  • T
    ()
    timedelta64[s]
    10 days

    long_name :

    signed integration window T = t1 - t0

    array(864000, dtype='timedelta64[s]')

• Attributes: (2)

  long_name :

  finite-time Lyapunov exponent

  units :

  1/s

aux_ftle = aux_flowmap.ftle()
aux_ftle

<xarray.DataArray 'ftle' (i: 6, j: 5)> Size: 240B
array([[0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0.],
       [0., 0., 0., 0., 0.]])
Coordinates:
  * i         (i) int64 48B 0 1 2 3 4 5
  * j         (j) int64 40B 0 1 2 3 4
    lon_grid  (i, j) float64 240B -25.0 -25.0 -25.0 -25.0 ... -20.0 -20.0 -20.0
    lat_grid  (i, j) float64 240B 15.0 16.25 17.5 18.75 ... 17.5 18.75 20.0
    t0        datetime64[s] 8B 2020-01-01
    T         timedelta64[s] 8B 10 days
Attributes:
    long_name:  finite-time Lyapunov exponent
    units:      1/s

xarray.DataArray

'ftle'

• i: 6
• j: 5

• 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 ... 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

  array([[0., 0., 0., 0., 0.],
         [0., 0., 0., 0., 0.],
         [0., 0., 0., 0., 0.],
         [0., 0., 0., 0., 0.],
         [0., 0., 0., 0., 0.],
         [0., 0., 0., 0., 0.]])

• Coordinates: (6)
  • i
    (i)
    int64
    0 1 2 3 4 5

    long_name :

    logical grid index along i

    array([0, 1, 2, 3, 4, 5])

  • j
    (j)
    int64
    0 1 2 3 4

    long_name :

    logical grid index along j

    array([0, 1, 2, 3, 4])

  • lon_grid
    (i, j)
    float64
    -25.0 -25.0 -25.0 ... -20.0 -20.0

    long_name :

    longitude

    units :

    degrees_east

    array([[-25., -25., -25., -25., -25.],
           [-24., -24., -24., -24., -24.],
           [-23., -23., -23., -23., -23.],
           [-22., -22., -22., -22., -22.],
           [-21., -21., -21., -21., -21.],
           [-20., -20., -20., -20., -20.]])

  • lat_grid
    (i, j)
    float64
    15.0 16.25 17.5 ... 17.5 18.75 20.0

    long_name :

    latitude

    units :

    degrees_north

    array([[15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ],
           [15.  , 16.25, 17.5 , 18.75, 20.  ]])

  • t0
    ()
    datetime64[s]
    2020-01-01

    long_name :

    release time t0

    array('2020-01-01T00:00:00', dtype='datetime64[s]')

  • T
    ()
    timedelta64[s]
    10 days

    long_name :

    signed integration window T = t1 - t0

    array(864000, dtype='timedelta64[s]')

• Attributes: (2)

  long_name :

  finite-time Lyapunov exponent

  units :

  1/s

print("neighbour: max |ftle| =", float(abs(ftle).max()))
print("auxiliary: max |ftle| =", float(abs(aux_ftle).max()))
print("neighbour NaN points:", int(ftle.isnull().sum()))
print("auxiliary NaN points:", int(aux_ftle.isnull().sum()))

neighbour: max |ftle| = 0.0
auxiliary: max |ftle| = 0.0
neighbour NaN points: 18
auxiliary NaN points: 0