======================
Basic cube mathematics
======================


The section :doc:`navigating_a_cube` highlighted that 
every cube has a data attribute; 
this attribute can then be manipulated directly::

   cube.data -= 273.15

The problem with manipulating the data directly is that other metadata may
become inconsistent; in this case the units of the cube are no longer what was
intended. This example could be rectified by changing the units attribute::

   cube.units = 'celsius'

.. note::

    :meth:`iris.cube.Cube.convert_units` can be used to automatically convert a
    cube's data and update its units attribute.
    So, the two steps above can be achieved by::

        cube.convert_units('celsius')

In order to reduce the amount of metadata which becomes inconsistent,
fundamental arithmetic operations such as addition, subtraction, division
and multiplication can be applied directly to any cube.

Calculating the difference between two cubes
--------------------------------------------

Let's load some air temperature which runs from 1860 to 2100::

    filename = iris.sample_data_path('E1_north_america.nc')
    air_temp = iris.load_cube(filename, 'air_temperature')

We can now get the first and last time slices using indexing 
(see :doc:`reducing_a_cube` for a reminder)::

    t_first = air_temp[0, :, :]
    t_last = air_temp[-1, :, :]

.. testsetup::

    filename = iris.sample_data_path('E1_north_america.nc')
    cube = iris.load_cube(filename, 'air_temperature')
    t_first = cube[0, :, :]
    t_last = cube[-1, :, :]

And finally we can subtract the two. 
The result is a cube of the same size as the original two time slices, 
but with the data representing their difference:

    >>> print t_last - t_first
    unknown / (K)                       (latitude: 37; longitude: 49)
         Dimension coordinates:
              latitude                           x              -
              longitude                          -              x
         Scalar coordinates:
              forecast_reference_time: 1859-09-01 06:00:00
              height: 1.5 m


.. note::

    Notice that the coordinates "time" and "forecast_period" have been removed 
    from the resultant cube; 
    this is because these coordinates differed between the two input cubes. 
    For more control on whether or not coordinates should be automatically 
    ignored :func:`iris.analysis.maths.subtract` can be used instead.


Combining multiple phenomena to form a new one
----------------------------------------------

Combining cubes of potential-temperature and pressure we can calculate 
the associated temperature using the equation:

.. math::
   
    T = \theta (\frac{p}{p_0}) ^ {(287.05 / 1005)}

Where :math:`p` is pressure, :math:`\theta` is potential temperature, 
:math:`p_0` is the potential temperature reference pressure 
and :math:`T` is temperature.

First, let's load pressure and potential temperature cubes::

    filename = iris.sample_data_path('colpex.pp')
    phenomenon_names = ['air_potential_temperature', 'air_pressure']
    pot_temperature, pressure = iris.load_cubes(filename, phenomenon_names)

In order to calculate :math:`\frac{p}{p_0}` we can define a coordinate which 
represents the standard reference pressure of 1000 hPa::

    import iris.coords
    p0 = iris.coords.AuxCoord(1000.0,
                              long_name='reference_pressure',
                              units='hPa')

We must ensure that the units of ``pressure`` and ``p0`` are the same,
so convert the newly created coordinate using
the :meth:`iris.coords.Coord.convert_units` method::

    p0.convert_units(pressure.units)

Now we can combine all of this information to calculate the air temperature 
using the equation above::

    temperature = pot_temperature * ( (pressure / p0) ** (287.05 / 1005) )

Finally, the cube we have created needs to be given a suitable name::

    temperature.rename('air_temperature')

The result could now be plotted using the guidance provided in the 
:doc:`plotting_a_cube` section.

.. htmlonly::

    A very similar example to this can be found in 
    :doc:`/examples/graphics/deriving_phenomena`.

.. latexonly::

    A very similar example to this can be found in the examples section, 
    with the title "Deriving Exner Pressure and Air Temperature".