# Source code for metpy.calc.kinematics

# Copyright (c) 2009,2017,2018 MetPy Developers.
"""Contains calculation of kinematic parameters (e.g. divergence or vorticity)."""
from __future__ import division

import functools

import numpy as np

from . import coriolis_parameter
from .tools import first_derivative, get_layer_heights, gradient
from ..cbook import is_string_like, iterable
from ..constants import Cp_d, g, Rd
from ..package_tools import Exporter
from ..units import atleast_2d, check_units, concatenate, units
from ..xarray import preprocess_xarray

exporter = Exporter(globals())

def _stack(arrs):
return concatenate([a[np.newaxis] for a in arrs], axis=0)

def _is_x_first_dim(dim_order):
"""Determine whether x is the first dimension based on the value of dim_order."""
if dim_order is None:
dim_order = 'yx'
return dim_order == 'xy'

def _check_and_flip(arr):
"""Transpose array or list of arrays if they are 2D."""
if hasattr(arr, 'ndim'):
if arr.ndim >= 2:
return arr.T
else:
return arr
elif not is_string_like(arr) and iterable(arr):
return tuple(_check_and_flip(a) for a in arr)
else:
return arr

def ensure_yx_order(func):
"""Wrap a function to ensure all array arguments are y, x ordered, based on kwarg."""
@functools.wraps(func)
def wrapper(*args, **kwargs):
# Check what order we're given
dim_order = kwargs.pop('dim_order', None)
x_first = _is_x_first_dim(dim_order)

# If x is the first dimension, flip (transpose) every array within the function args.
if x_first:
args = tuple(_check_and_flip(arr) for arr in args)
for k, v in kwargs:
kwargs[k] = _check_and_flip(v)

ret = func(*args, **kwargs)

# If we flipped on the way in, need to flip on the way out so that output array(s)
# match the dimension order of the original input.
if x_first:
return _check_and_flip(ret)
else:
return ret

# Inject a docstring for the dim_order argument into the function's docstring.
dim_order_doc = """
dim_order : str or None, optional
The ordering of dimensions in passed in arrays. Can be one of None, 'xy',
or 'yx'. 'xy' indicates that the dimension corresponding to x is the leading
dimension, followed by y. 'yx' indicates that x is the last dimension, preceded
by y. None indicates that the default ordering should be assumed,
which is 'yx'. Can only be passed as a keyword argument, i.e.
func(..., dim_order='xy')."""

# Find the first blank line after the start of the parameters section
params = wrapper.__doc__.find('Parameters')
blank = wrapper.__doc__.find('\n\n', params)
wrapper.__doc__ = wrapper.__doc__[:blank] + dim_order_doc + wrapper.__doc__[blank:]

return wrapper

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
def vorticity(u, v, dx, dy):
r"""Calculate the vertical vorticity of the horizontal wind.

Parameters
----------
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.

Returns
-------
(M, N) ndarray
vertical vorticity

--------
divergence

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
dudy = first_derivative(u, delta=dy, axis=-2)
dvdx = first_derivative(v, delta=dx, axis=-1)
return dvdx - dudy

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
def divergence(u, v, dx, dy):
r"""Calculate the horizontal divergence of the horizontal wind.

Parameters
----------
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.

Returns
-------
(M, N) ndarray
The horizontal divergence

--------
vorticity

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
dudx = first_derivative(u, delta=dx, axis=-1)
dvdy = first_derivative(v, delta=dy, axis=-2)
return dudx + dvdy

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
def shearing_deformation(u, v, dx, dy):
r"""Calculate the shearing deformation of the horizontal wind.

Parameters
----------
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.

Returns
-------
(M, N) ndarray
Shearing Deformation

--------
stretching_deformation, total_deformation

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
dudy = first_derivative(u, delta=dy, axis=-2)
dvdx = first_derivative(v, delta=dx, axis=-1)
return dvdx + dudy

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
def stretching_deformation(u, v, dx, dy):
r"""Calculate the stretching deformation of the horizontal wind.

Parameters
----------
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.

Returns
-------
(M, N) ndarray
Stretching Deformation

--------
shearing_deformation, total_deformation

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
dudx = first_derivative(u, delta=dx, axis=-1)
dvdy = first_derivative(v, delta=dy, axis=-2)
return dudx - dvdy

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
def total_deformation(u, v, dx, dy):
r"""Calculate the horizontal total deformation of the horizontal wind.

Parameters
----------
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.

Returns
-------
(M, N) ndarray
Total Deformation

--------
shearing_deformation, stretching_deformation

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
dudy, dudx = gradient(u, deltas=(dy, dx), axes=(-2, -1))
dvdy, dvdx = gradient(v, deltas=(dy, dx), axes=(-2, -1))
return np.sqrt((dvdx + dudy)**2 + (dudx - dvdy)**2)

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
r"""Calculate the advection of a scalar field by the wind.

The order of the dimensions of the arrays must match the order in which
the wind components are given.  For example, if the winds are given [u, v],
then the scalar and wind arrays must be indexed as x,y (which puts x as the
rows, not columns).

Parameters
----------
scalar : N-dimensional array
Array (with N-dimensions) with the quantity to be advected.
wind : sequence of arrays
Length M sequence of N-dimensional arrays.  Represents the flow,
with a component of the wind in each dimension.  For example, for
horizontal advection, this could be a list: [u, v], where u and v
are each a 2-dimensional array.
deltas : sequence of float or ndarray
A (length M) sequence containing the grid spacing(s) in each dimension. If using
arrays, in each array there should be one item less than the size of scalar along the
applicable axis.

Returns
-------
N-dimensional array
An N-dimensional array containing the advection at all grid points.

"""
# This allows passing in a list of wind components or an array.
wind = _stack(wind)

# If we have more than one component, we need to reverse the order along the first
# dimension so that the wind components line up with the
# order of the gradients from the ..., y, x ordered array.
if wind.ndim > scalar.ndim:
wind = wind[::-1]

# Gradient returns a list of derivatives along each dimension. We convert
# this to an array with dimension as the first index. Reverse the deltas to line up
# with the order of the dimensions.

# Make them be at least 2D (handling the 1D case) so that we can do the
# multiply and sum below

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
def frontogenesis(thta, u, v, dx, dy, dim_order='yx'):
r"""Calculate the 2D kinematic frontogenesis of a temperature field.

The implementation is a form of the Petterssen Frontogenesis and uses the formula
outlined in [Bluestein1993]_ pg.248-253.

.. math:: F=\frac{1}{2}\left|\nabla \theta\right|[D cos(2\beta)-\delta]

* :math:F is 2D kinematic frontogenesis
* :math:\theta is potential temperature
* :math:D is the total deformation
* :math:\beta is the angle between the axis of dilitation and the isentropes
* :math:\delta is the divergence

Parameters
----------
thta : (M, N) ndarray
Potential temperature
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.

Returns
-------
(M, N) ndarray
2D Frontogenesis in [temperature units]/m/s

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

Conversion factor to go from [temperature units]/m/s to [temperature units/100km/3h]
:math:1.08e4*1.e5

"""
# Get gradients of potential temperature in both x and y
ddy_thta = first_derivative(thta, delta=dy, axis=-2)
ddx_thta = first_derivative(thta, delta=dx, axis=-1)

# Compute the magnitude of the potential temperature gradient
mag_thta = np.sqrt(ddx_thta**2 + ddy_thta**2)

# Get the shearing, stretching, and total deformation of the wind field
shrd = shearing_deformation(u, v, dx, dy, dim_order=dim_order)
strd = stretching_deformation(u, v, dx, dy, dim_order=dim_order)
tdef = total_deformation(u, v, dx, dy, dim_order=dim_order)

# Get the divergence of the wind field
div = divergence(u, v, dx, dy, dim_order=dim_order)

# Compute the angle (beta) between the wind field and the gradient of potential temperature
psi = 0.5 * np.arctan2(shrd, strd)
beta = np.arcsin((-ddx_thta * np.cos(psi) - ddy_thta * np.sin(psi)) / mag_thta)

return 0.5 * mag_thta * (tdef * np.cos(2 * beta) - div)

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
def geostrophic_wind(heights, f, dx, dy):
r"""Calculate the geostrophic wind given from the heights or geopotential.

Parameters
----------
heights : (M, N) ndarray
The height field, with either leading dimensions of (x, y) or trailing dimensions
of (y, x), depending on the value of dim_order.
f : array_like
The coriolis parameter.  This can be a scalar to be applied
everywhere or an array of values.
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of heights along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of heights along the applicable axis.

Returns
-------
A 2-item tuple of arrays
A tuple of the u-component and v-component of the geostrophic wind.

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
if heights.dimensionality['[length]'] == 2.0:
norm_factor = 1. / f
else:
norm_factor = g / f

dhdy = first_derivative(heights, delta=dy, axis=-2)
dhdx = first_derivative(heights, delta=dx, axis=-1)
return -norm_factor * dhdy, norm_factor * dhdx

[docs]@exporter.export
@preprocess_xarray
@ensure_yx_order
def ageostrophic_wind(heights, f, dx, dy, u, v, dim_order='yx'):
r"""Calculate the ageostrophic wind given from the heights or geopotential.

Parameters
----------
heights : (M, N) ndarray
The height field.
f : array_like
The coriolis parameter.  This can be a scalar to be applied
everywhere or an array of values.
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of heights along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of heights along the applicable axis.
u : (M, N) ndarray
The u wind field.
v : (M, N) ndarray
The u wind field.

Returns
-------
A 2-item tuple of arrays
A tuple of the u-component and v-component of the ageostrophic wind.

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
u_geostrophic, v_geostrophic = geostrophic_wind(heights, f, dx, dy, dim_order=dim_order)
return u - u_geostrophic, v - v_geostrophic

[docs]@exporter.export
@preprocess_xarray
@check_units('[length]', '[temperature]')
def montgomery_streamfunction(height, temperature):
r"""Compute the Montgomery Streamfunction on isentropic surfaces.

The Montgomery Streamfunction is the streamfunction of the geostrophic wind on an
isentropic surface. This quantity is proportional to the geostrophic wind in isentropic
coordinates, and its gradient can be interpreted similarly to the pressure gradient in
isobaric coordinates.

Parameters
----------
height : pint.Quantity
Array of geopotential height of isentropic surfaces
temperature : pint.Quantity
Array of temperature on isentropic surfaces

Returns
-------
stream_func : pint.Quantity

Notes
-----
The formula used is that from [Lackmann2011]_ p. 69.

.. math:: \Psi = gZ + C_pT

* :math:\Psi is Montgomery Streamfunction
* :math:g is avg. gravitational acceleration on Earth
* :math:Z is geopotential height of the isentropic surface
* :math:C_p is specific heat at constant pressure for dry air
* :math:T is temperature of the isentropic surface

--------
get_isentropic_pressure

"""
return (g * height) + (Cp_d * temperature)

[docs]@exporter.export
@preprocess_xarray
@check_units('[speed]', '[speed]', '[length]', '[length]', '[length]',
'[speed]', '[speed]')
def storm_relative_helicity(u, v, heights, depth, bottom=0 * units.m,
storm_u=0 * units('m/s'), storm_v=0 * units('m/s')):
# Partially adapted from similar SharpPy code
r"""Calculate storm relative helicity.

Calculates storm relatively helicity following [Markowski2010] 230-231.

.. math:: \int\limits_0^d (\bar v - c) \cdot \bar\omega_{h} \,dz

This is applied to the data from a hodograph with the following summation:

.. math:: \sum_{n = 1}^{N-1} [(u_{n+1} - c_{x})(v_{n} - c_{y}) -
(u_{n} - c_{x})(v_{n+1} - c_{y})]

Parameters
----------
u : array-like
u component winds
v : array-like
v component winds
heights : array-like
atmospheric heights, will be converted to AGL
depth : number
depth of the layer
bottom : number
height of layer bottom AGL (default is surface)
storm_u : number
u component of storm motion (default is 0 m/s)
storm_v : number
v component of storm motion (default is 0 m/s)

Returns
-------
pint.Quantity, pint.Quantity, pint.Quantity
positive, negative, total storm-relative helicity

"""
_, u, v = get_layer_heights(heights, depth, u, v, with_agl=True, bottom=bottom)

storm_relative_u = u - storm_u
storm_relative_v = v - storm_v

int_layers = (storm_relative_u[1:] * storm_relative_v[:-1] -
storm_relative_u[:-1] * storm_relative_v[1:])

# Need to manually check for masked value because sum() on masked array with non-default
# mask will return a masked value rather than 0. See numpy/numpy#11736
positive_srh = int_layers[int_layers.magnitude > 0.].sum()
positive_srh = 0.0 * units('meter**2 / second**2')
negative_srh = int_layers[int_layers.magnitude < 0.].sum()
negative_srh = 0.0 * units('meter**2 / second**2')

return (positive_srh.to('meter ** 2 / second ** 2'),
negative_srh.to('meter ** 2 / second ** 2'),
(positive_srh + negative_srh).to('meter ** 2 / second ** 2'))

[docs]@exporter.export
@preprocess_xarray
@check_units('[speed]', '[speed]', '[length]', '[length]')
def absolute_vorticity(u, v, dx, dy, lats, dim_order='yx'):
"""Calculate the absolute vorticity of the horizontal wind.

Parameters
----------
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.
lats : (M, N) ndarray
latitudes of the wind data

Returns
-------
(M, N) ndarray
absolute vorticity

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
f = coriolis_parameter(lats)
relative_vorticity = vorticity(u, v, dx, dy, dim_order=dim_order)
return relative_vorticity + f

[docs]@exporter.export
@preprocess_xarray
@check_units('[temperature]', '[pressure]', '[speed]', '[speed]',
'[length]', '[length]', '[dimensionless]')
def potential_vorticity_baroclinic(potential_temperature, pressure, u, v, dx, dy, lats,
axis=0, dim_order='yx'):
r"""Calculate the baroclinic potential vorticity.

.. math:: PV = -g \frac{\partial \theta}{\partial z}(\zeta + f)

This formula is based on equation 7.31a [Hobbs2006]_.

Parameters
----------
potential_temperature : (M, N, P) ndarray
potential temperature
pressure : (M, N, P) ndarray
vertical pressures
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.
lats : (M, N) ndarray
latitudes of the wind data
axis : int, optional
The axis corresponding to the vertical dimension in the potential temperature
and pressure arrays, defaults to 0, the first dimension.

Returns
-------
(M, N) ndarray
baroclinic potential vorticity

Notes
-----
The same formula is used for isobaric and isentropic PV analysis. Provide winds
for vorticity calculations on the desired isobaric or isentropic surface. Three layers
of pressure/potential temperature are required in order to calculate the vertical
derivative (one above and below the desired surface).

"""
if np.shape(potential_temperature)[axis] != 3:
raise ValueError('Length of potential temperature along axis '
'{} must be 3.'.format(axis))
if np.shape(pressure)[axis] != 3:
raise ValueError('Length of pressure along axis '
'{} must be 3.'.format(axis))
avor = absolute_vorticity(u, v, dx, dy, lats, dim_order=dim_order)
stability = first_derivative(potential_temperature, x=pressure, axis=axis)
# Get the middle layer stability derivative (index 1)
slices = [slice(None)] * stability.ndim
slices[axis] = 1
return (-1 * avor * g * stability[slices]).to(units.kelvin * units.meter**2 /
(units.second * units.kilogram))

[docs]@exporter.export
@preprocess_xarray
@check_units('[length]', '[speed]', '[speed]', '[length]', '[length]', '[dimensionless]')
def potential_vorticity_barotropic(heights, u, v, dx, dy, lats, dim_order='yx'):
r"""Calculate the barotropic (Rossby) potential vorticity.

.. math:: PV = \frac{f + \zeta}{H}

This formula is based on equation 7.27 [Hobbs2006]_.

Parameters
----------
heights : (M, N) ndarray
atmospheric heights
u : (M, N) ndarray
x component of the wind
v : (M, N) ndarray
y component of the wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.
lats : (M, N) ndarray
latitudes of the wind data

Returns
-------
(M, N) ndarray
barotropic potential vorticity

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
avor = absolute_vorticity(u, v, dx, dy, lats, dim_order=dim_order)
return (avor / heights).to('meter**-1 * second**-1')

[docs]@exporter.export
@preprocess_xarray
def inertial_advective_wind(u, v, u_geostrophic, v_geostrophic, dx, dy, lats):

.. math:: \frac{\hat k}{f} \times (\vec V \cdot \nabla)\hat V_g

.. math:: \frac{\hat k}{f} \times \left[ \left( u \frac{\partial u_g}{\partial x} + v
\frac{\partial u_g}{\partial y} \right) \hat i + \left( u \frac{\partial v_g}
{\partial x} + v \frac{\partial v_g}{\partial y} \right) \hat j \right]

.. math:: \left[ -\frac{1}{f}\left(u \frac{\partial v_g}{\partial x} + v
\frac{\partial v_g}{\partial y} \right) \right] \hat i + \left[ \frac{1}{f}
\left( u \frac{\partial u_g}{\partial x} + v \frac{\partial u_g}{\partial y}
\right) \right] \hat j

This formula is based on equation 27 of [Rochette2006]_.

Parameters
----------
u : (M, N) ndarray
x component of the advecting wind
v : (M, N) ndarray
y component of the advecting wind
u_geostrophic : (M, N) ndarray
x component of the geostrophic (advected) wind
v_geostrophic : (M, N) ndarray
y component of the geostrophic (advected) wind
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.
lats : (M, N) ndarray
latitudes of the wind data

Returns
-------
(M, N) ndarray
x component of inertial advective wind
(M, N) ndarray
y component of inertial advective wind

Notes
-----
wind to both be the geostrophic wind. To do so, pass the x and y components
of the geostrophic with for u and u_geostrophic/v and v_geostrophic.

If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
f = coriolis_parameter(lats)

dugdy, dugdx = gradient(u_geostrophic, deltas=(dy, dx), axes=(-2, -1))
dvgdy, dvgdx = gradient(v_geostrophic, deltas=(dy, dx), axes=(-2, -1))

u_component = -(u * dvgdx + v * dvgdy) / f
v_component = (u * dugdx + v * dugdy) / f

return u_component, v_component

[docs]@exporter.export
@preprocess_xarray
@check_units('[speed]', '[speed]', '[temperature]', '[pressure]', '[length]', '[length]')
def q_vector(u, v, temperature, pressure, dx, dy, static_stability=1):
r"""Calculate Q-vector at a given pressure level using the u, v winds and temperature.

.. math:: \vec{Q} = (Q_1, Q_2)
=  - \frac{R}{\sigma p}\left(
\frac{\partial \vec{v}_g}{\partial x} \cdot \nabla_p T,
\frac{\partial \vec{v}_g}{\partial y} \cdot \nabla_p T
\right)

This formula follows equation 5.7.55 from [Bluestein1992]_, and can be used with the
the below form of the quasigeostrophic omega equation to assess vertical motion
([Bluestein1992]_ equation 5.7.54):

.. math:: \left( \nabla_p^2 + \frac{f_0^2}{\sigma} \frac{\partial^2}{\partial p^2}
\right) \omega =
- 2 \nabla_p \cdot \vec{Q} -
\frac{R}{\sigma p} \beta \frac{\partial T}{\partial x}.

Parameters
----------
u : (M, N) ndarray
x component of the wind (geostrophic in QG-theory)
v : (M, N) ndarray
y component of the wind (geostrophic in QG-theory)
temperature : (M, N) ndarray
Array of temperature at pressure level
pressure : pint.Quantity
Pressure at level
dx : float or ndarray
The grid spacing(s) in the x-direction. If an array, there should be one item less than
the size of u along the applicable axis.
dy : float or ndarray
The grid spacing(s) in the y-direction. If an array, there should be one item less than
the size of u along the applicable axis.
static_stability : pint.Quantity, optional
The static stability at the pressure level. Defaults to 1 if not given to calculate
the Q-vector without factoring in static stability.

Returns
-------
tuple of (M, N) ndarrays
The components of the Q-vector in the u- and v-directions respectively

--------
static_stability

Notes
-----
If inputs have more than two dimensions, they are assumed to have either leading dimensions
of (x, y) or trailing dimensions of (y, x), depending on the value of dim_order.

"""
dudy, dudx = gradient(u, deltas=(dy, dx), axes=(-2, -1))
dvdy, dvdx = gradient(v, deltas=(dy, dx), axes=(-2, -1))
dtempdy, dtempdx = gradient(temperature, deltas=(dy, dx), axes=(-2, -1))

q1 = -Rd / (pressure * static_stability) * (dudx * dtempdx + dvdx * dtempdy)
q2 = -Rd / (pressure * static_stability) * (dudy * dtempdx + dvdy * dtempdy)

return q1.to_base_units(), q2.to_base_units()