.. _grids: Simple Grids ------------ General Concepts ~~~~~~~~~~~~~~~~ Most finite volume ocean models use `Arakawa Grids`_, in which different variables are offset from one another and situated at different locations with respect to the cell center and edge points. As an example, we will consider *C-grid geometry*. As illustrated in the figure below, C-grids place scalars (such as temperature) at the cell center and vector components (such as velocity) at the cell faces. This type of grid is widely used because of its favorable conservation properties. .. figure:: images/grid2d_hv.svg :scale: 100 :alt: C-grid Geometry Layout of variables with respect to cell centers and edges in a C-grid ocean model. Image from the `pycomodo project `_. These grids present a dilemma for the `xarray`_ data model. The ``u`` and ``t`` points in the example above are located at different points along the x-axis, meaning they can't be represented using a single coordinate. But they are clearly related and can be transformed via well defined interpolation and difference operators. One goal of xgcm is to provide these interpolation and difference operators. We use `MITgcm notation`_ to denote the basic operators that transform between grid points. The difference operator is defined as .. math:: \delta_i \Phi = \Phi_{i+1/2} - \Phi_{i-1/2} where :math:`\Phi` is any variable and ``i`` represents the grid index. The other basic operator is interpolation, defined as .. math:: \overline{\Phi} = (\Phi_{i+1/2} + \Phi_{i-1/2})/2 Both operators return a variable that is shifted by half a gridpoint with respect to the input variable. With these two operators, the entire suite of finite volume vector calculus operations can be represented. An important consideration for both interpolation and difference operators is boundary conditions. xgcm currently supports periodic, `Dirichlet `_, and `Neumann `_ boundary conditions, although the latter two are limited to simple cases, see :ref:`Boundary conditions`. The inverse of differentiation is integration. For finite volume grids, the inverse of the difference operator is a discrete cumulative sum. xgcm also provides a grid-aware version of the ``cumsum`` operator. .. _axis-positions: Axes and Positions ~~~~~~~~~~~~~~~~~~ A fundamental concept in xgcm is the notion of an "axis". An axis is a group of coordinates that all lie along the same *physical dimension* but describe different positions relative to a grid cell. There are currently five possible positions supported by xgcm. ``center`` The variable values are located at the cell center. ``left`` The variable values are located at the left (i.e. lower) face of the cell. ``right`` The variable values are located at the right (i.e. upper) face of the cell. ``inner`` The variable values are located on the cell faces, excluding both outer boundaries. ``outer`` The variable values are located on the cell faces, including both outer boundaries. The first three (``center``, ``left``, and ``right``) all have the same length along the axis dimension, while ``inner`` has one fewer point and ``outer`` has one extra point. These positions are visualized in the figure below. .. figure:: images/axis_positions.svg :alt: axis positions The different possible positions of a variable ``f`` along an axis. xgcm represents an axis internally using the :py:class:`xgcm.Axis` class. Although it is technically possible to create an ``Axis`` directly, the recommended way to to use xgcm is by creating a single :py:class:`xgcm.Grid` object, containing multiple axes for each physical dimension. Creating ``Grid`` Objects ~~~~~~~~~~~~~~~~~~~~~~~~~ The core object in xgcm is an :class:`xgcm.Grid`. A ``Grid`` object should be constructed once and then used whenever grid-aware operations are required during the course of a data analysis routine. Xgcm operates on :py:class:`xarray.Dataset` and :py:class:`xarray.DataArray` objects. A basic understanding of :ref:`xarray data structures ` is therefore needed to understand xgcm. When constructing an :class:`xgcm.Grid` object, we need to pass an :py:class:`xarray.Dataset` object containing all of the necessary coordinates for the different axes we wish to use. We also have to tell xgcm how those coordinates are related to each other, i.e. which positions they occupy along the axis. We can provide this information in two ways: manually or via dataset attributes. .. note:: In most real use cases, the input dataset to create a ``Grid`` will be a come from a netCDF file generated by a GCM simulation. In this documentation, we create datasets from scratch in order to make the examples self-contained and portable. Manually Specifying Axes ^^^^^^^^^^^^^^^^^^^^^^^^ To begin, let's create a simple example :py:class:`xarray.Dataset` with a single physical axis. This dataset will contain two coordinates: - ``x_c``, which represents the cell center - ``x_g``, which represents the left cell edge We create it as follows. .. ipython:: python import xarray as xr import numpy as np ds = xr.Dataset( coords={ "x_c": ( ["x_c"], np.arange(1, 10), ), "x_g": ( ["x_g"], np.arange(0.5, 9), ), } ) ds .. note:: The choice of these coordinate names (``x_c`` and ``x_g``) is totally arbitrary. xgcm never requires datasets to have specific variable names. Rather, the axis geometry is specified by the user or inferred through the attributes. At this point, xarray has no idea that ``x_c`` and ``x_g`` are related to each other; they are subject to standard :ref:`xarray broadcasting rules `. When we create an :class:`xgcm.Grid`, we need to specify that they are part of the same axis. We do this using the ``coords`` keyword argument, as follows: .. ipython:: python from xgcm import Grid grid = Grid( ds, coords={"X": {"center": "x_c", "left": "x_g"}}, autoparse_metadata=False ) grid The printed information about the grid indicates that xgcm has successfully undestood the relative location of the different coordinates along the x axis. Because we did not specify the ``periodic`` keyword argument, xgcm assumed that the data is periodic along all axes. The arrows after each coordinate indicate the default shift positions for interpolation and difference operations: operating on the center coordinate (``x_c``) shifts to the left coordinate (``x_g``), and vice versa. Detecting Axes from Dataset Attributes ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ It is possible to avoid manually specifying the axis if the dataset contains specific metadata that can tell xgcm about the relationship between different coordinates. If the ``autoparse_metadata`` kwarg is set to ``True`` (the default), xgcm looks for this metadata in the coordinate attributes. Wherever possible, we try to follow established metadata conventions, rather than defining new metadata conventions. The main relevant conventions are the `CF Conventions`_, which apply broadly to Climate and Forecast datasets that follow the netCDF data model, and the `COMODO conventions`_ and `SGRID conventions`_, both of which define specific attributes relevant to Arakawa grids. While the COMODO conventions were designed with C-grids in mind, we find they are general enough to support all the different Arakawa grids. Detection and extraction of grid information from datasets is performed by a series of metadata parsing functions that take an xarray dataset and return a (potentially modified) dataset and dictionary of extracted Grid kwargs. When used as part of the autoparsing functionality of the ``Grid`` class there is a default hierarchy imposed. For more control a user can manually use a specific autoparsing function to extract the ``Grid`` kwargs and then pass they to the ``Grid`` constructor (after any changes/additions) with ``autoparse_metadata=False``. For example: .. ipython:: python grid = xgcm.Grid(ds) will return a ``Grid`` object constructed from xgcm's best attempts to autoparse any metadata in the dataset according to internal hierarchies, whilst .. ipython:: python ds = xr.Dataset( { "grid": ( (), np.array(1, dtype="int32"), { "cf_role": "grid_topology", "topology_dimension": 1, "node_dimensions": "x_g", "face_dimensions": "x_c: x_g (padding: high)", }, ), }, attrs={"Conventions": "SGRID-0.3"}, coords={ "x_c": ( ["x_c"], np.arange(1, 10), ), "x_g": ( ["x_g"], np.arange(0.5, 9), ), }, ) ds_sgrid, grid_kwargs_sgrid = xgcm.metadata_parsers.parse_sgrid(ds) grid = xgcm.Grid(ds, coords=grid_kwargs_sgrid["coords"], autoparse_metadata=False) explicitly extracts SGRID metadata which is then used to construct a ``Grid`` object without autoparsing. SGRID data """""""""" The identifier xgcm looks for is 'SGRID' in the ``conventions`` attribute. Grid data is then contained within the ``variable`` with the ``cf_role`` of 'grid_topology'. A set of grid axes in the order ``'X', 'Y', 'Z'`` are assigned based on the dimensionality of the data. Note that SGRID treatment of 3D grids and 2D grids with a vertical coordinate is subtly different. Both cases are handled by the autoparsing functionality to form a 3D ``Grid`` object. SGRID 'node_dimensions' are extracted and correspond to xgcm's cell edges. SGRID 'face' or 'volume' dimensions are then extracted with their associated 'padding' identifier. This corresponds to xgcm's cell centers. Once the padding type has been extracted the correct xgcm 'position' can be assigned to the associated cell edge coordinate as set out in the following table: +---------------+----------+ | SGRID padding | position | +===============+==========+ | low | right | +---------------+----------+ | high | left | +---------------+----------+ | both | inner | +---------------+----------+ | none | outer | +---------------+----------+ COMODO Data """"""""""" The key attribute xgcm looks for is ``axis``. When creating a new grid, xgcm will search through the dataset dimensions looking for dimensions with the ``axis`` attribute defined. All coordinates with the same value of ``axis`` are presumed to belong to the same physical axis. To determine the positions of the different coordinates, xgcm considers both the length of the coordinate variable and the ``c_grid_axis_shift`` attribute, which determines the position of the coordinate with respect to the cell center. The only acceptable values of ``c_grid_axis_shift`` are ``-0.5`` and ``0.5``. If the ``c_grid_axis_shift`` attribute attribute is absent, the coordinate is assumed to describe a cell center. The cell center coordinate is identified first; the length of other coordinates relative to the cell center coordinate is used in conjunction with ``c_grid_axis_shift`` to infer the coordinate positions, as summarized by the table below. +--------+--------------------------+----------+ | length | ``c_grid_axis_shift`` | position | +========+==========================+==========+ | n | *None* | center | +--------+--------------------------+----------+ | n | -0.5 | left | +--------+--------------------------+----------+ | n | 0.5 | right | +--------+--------------------------+----------+ | n-1 | 0.5 or -0.5 | inner | +--------+--------------------------+----------+ | n+1 | 0.5 or -0.5 | outer | +--------+--------------------------+----------+ We create an :py:class:`xarray.Dataset` with such attributes as follows: .. ipython:: python ds = xr.Dataset( coords={ "x_c": ( ["x_c"], np.arange(1, 10), {"axis": "X"}, ), "x_g": ( ["x_g"], np.arange(0.5, 9), {"axis": "X", "c_grid_axis_shift": -0.5}, ), } ) ds (This is the same as the first example, just with additional attributes.) We can now create a ``Grid`` object from this dataset without manually specifying ``coords``: .. ipython:: python grid = Grid(ds) grid We see that the resulting ``Grid`` object is the same as in the manual example. Core Grid Operations: ``diff``, ``interp``, and ``cumsum`` ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Regardless of how our ``Grid`` object was created, we can now use it to interpolate or take differences along the axis. First we create some test data: .. ipython:: python import matplotlib.pyplot as plt da = np.sin(ds.x_c * 2 * np.pi / 9) print(da) @savefig grid_test_data.png da.plot() plt.close() We interpolate as follows: .. ipython:: python da_interp = grid.interp(da, axis="X") da_interp We see that the output is on the ``x_g`` points rather than the original ``x_c`` points. .. warning:: xgcm does not perform input validation to verify that ``da`` is compatible with ``grid``. The same position shift happens with a difference operation: .. ipython:: python da_diff = grid.diff(da, axis="X") da_diff We can reverse the difference operation by taking a cumsum: .. ipython:: python grid.cumsum(da_diff, "X") Which is approximately equal to the original ``da``, modulo the numerical errors accrued due to the discretization of the data. By default, these grid operations will drop any coordinate that are not dimensions. The keep_coords argument allow to preserve compatible coordinates. For example: .. ipython:: python da2 = da + xr.Dataset(coords={"y": np.arange(1, 3)})["y"] da2 = da2.assign_coords(h=da2.y**2) print(da2) grid.interp(da2, "X", keep_coords=True) So far we have just discussed simple grids (i.e. regular grids with a single face). Xgcm can also deal with complex topologies such as cubed-sphere and lat-lon-cap. This is described in the :ref:`grid_topology` page. .. _Arakawa Grids: https://en.wikipedia.org/wiki/Arakawa_grids .. _xarray: http://xarray.pydata.org .. _MITgcm notation: http://mitgcm.org/public/r2_manual/latest/online_documents/node31.html .. _CF Conventions: http://cfconventions.org/ .. _COMODO Conventions: https://web.archive.org/web/20160417032300/http://pycomodo.forge.imag.fr/norm.html .. _SGRID Conventions: https://sgrid.github.io/sgrid/