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# Copyright (c) 2008-2015 MetPy Developers. |
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# Distributed under the terms of the BSD 3-Clause License. |
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# SPDX-License-Identifier: BSD-3-Clause |
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"""Contains calculation of kinematic parameters (e.g. divergence or vorticity).""" |
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from __future__ import division |
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import functools |
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import warnings |
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import numpy as np |
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from ..cbook import is_string_like, iterable |
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from ..constants import Cp_d, g |
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from ..package_tools import Exporter |
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from ..units import atleast_2d, check_units, concatenate, units |
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exporter = Exporter(globals()) |
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def _gradient(f, *args, **kwargs): |
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"""Wrap :func:`numpy.gradient` to handle units.""" |
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if len(args) < f.ndim: |
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args = list(args) |
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args.extend([units.Quantity(1., 'dimensionless')] * (f.ndim - len(args))) |
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grad = np.gradient(f, *(a.magnitude for a in args), **kwargs) |
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if f.ndim == 1: |
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return units.Quantity(grad, f.units / args[0].units) |
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return [units.Quantity(g, f.units / dx.units) for dx, g in zip(args, grad)] |
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def _stack(arrs): |
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return concatenate([a[np.newaxis] for a in arrs], axis=0) |
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def _get_gradients(u, v, dx, dy): |
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"""Return derivatives for components to simplify calculating convergence and vorticity.""" |
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dudy, dudx = _gradient(u, dy, dx) |
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dvdy, dvdx = _gradient(v, dy, dx) |
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return dudx, dudy, dvdx, dvdy |
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def _is_x_first_dim(dim_order): |
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"""Determine whether x is the first dimension based on the value of dim_order.""" |
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if dim_order is None: |
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warnings.warn('dim_order is using the default setting (currently "xy"). This will ' |
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'change to "yx" in the next version. It is recommended that you ' |
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'specify the appropriate ordering ("xy", "yx") for your data by ' |
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'passing the `dim_order` argument to the calculation.', FutureWarning) |
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dim_order = 'xy' |
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return dim_order == 'xy' |
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def _check_and_flip(arr): |
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"""Transpose array or list of arrays if they are 2D.""" |
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if hasattr(arr, 'ndim'): |
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if arr.ndim >= 2: |
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return arr.T |
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else: |
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return arr |
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elif not is_string_like(arr) and iterable(arr): |
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return tuple(_check_and_flip(a) for a in arr) |
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else: |
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return arr |
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def ensure_yx_order(func): |
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"""Wrap a function to ensure all array arguments are y, x ordered, based on kwarg.""" |
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@functools.wraps(func) |
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def wrapper(*args, **kwargs): |
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# Check what order we're given |
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dim_order = kwargs.pop('dim_order', None) |
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x_first = _is_x_first_dim(dim_order) |
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# If x is the first dimension, flip (transpose) every array within the function args. |
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if x_first: |
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args = tuple(_check_and_flip(arr) for arr in args) |
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for k, v in kwargs: |
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kwargs[k] = _check_and_flip(v) |
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ret = func(*args, **kwargs) |
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# If we flipped on the way in, need to flip on the way out so that output array(s) |
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# match the dimension order of the original input. |
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if x_first: |
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return _check_and_flip(ret) |
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else: |
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return ret |
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# Inject a docstring for the dim_order argument into the function's docstring. |
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dim_order_doc = """ |
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dim_order : str or ``None``, optional |
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The ordering of dimensions in passed in arrays. Can be one of ``None``, ``'xy'``, |
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or ``'yx'``. ``'xy'`` indicates that the dimension corresponding to x is the leading |
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dimension, followed by y. ``'yx'`` indicates that x is the last dimension, preceded |
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by y. ``None`` indicates that the default ordering should be assumed, |
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which will change in version 0.6 from 'xy' to 'yx'. Can only be passed as a keyword |
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argument, i.e. func(..., dim_order='xy').""" |
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# Find the first blank line after the start of the parameters section |
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params = wrapper.__doc__.find('Parameters') |
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blank = wrapper.__doc__.find('\n\n', params) |
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wrapper.__doc__ = wrapper.__doc__[:blank] + dim_order_doc + wrapper.__doc__[blank:] |
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return wrapper |
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@exporter.export |
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@ensure_yx_order |
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def v_vorticity(u, v, dx, dy): |
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r"""Calculate the vertical vorticity of the horizontal wind. |
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The grid must have a constant spacing in each direction. |
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Parameters |
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---------- |
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u : (M, N) ndarray |
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x component of the wind |
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v : (M, N) ndarray |
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y component of the wind |
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dx : float |
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The grid spacing in the x-direction |
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dy : float |
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The grid spacing in the y-direction |
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Returns |
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------- |
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(M, N) ndarray |
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vertical vorticity |
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See Also |
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-------- |
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h_convergence, convergence_vorticity |
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""" |
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_, dudy, dvdx, _ = _get_gradients(u, v, dx, dy) |
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return dvdx - dudy |
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@exporter.export |
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@ensure_yx_order |
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def h_convergence(u, v, dx, dy): |
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r"""Calculate the horizontal convergence of the horizontal wind. |
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The grid must have a constant spacing in each direction. |
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Parameters |
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---------- |
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u : (M, N) ndarray |
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x component of the wind |
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v : (M, N) ndarray |
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y component of the wind |
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dx : float |
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The grid spacing in the x-direction |
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dy : float |
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The grid spacing in the y-direction |
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Returns |
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------- |
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(M, N) ndarray |
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The horizontal convergence |
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See Also |
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-------- |
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v_vorticity, convergence_vorticity |
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""" |
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dudx, _, _, dvdy = _get_gradients(u, v, dx, dy) |
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return dudx + dvdy |
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@exporter.export |
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@ensure_yx_order |
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def convergence_vorticity(u, v, dx, dy): |
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r"""Calculate the horizontal convergence and vertical vorticity of the horizontal wind. |
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The grid must have a constant spacing in each direction. |
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Parameters |
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---------- |
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u : (M, N) ndarray |
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x component of the wind |
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v : (M, N) ndarray |
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y component of the wind |
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dx : float |
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The grid spacing in the x-direction |
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dy : float |
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The grid spacing in the y-direction |
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Returns |
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------- |
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convergence, vorticity : tuple of (M, N) ndarrays |
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The horizontal convergence and vertical vorticity, respectively |
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See Also |
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-------- |
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v_vorticity, h_convergence |
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Notes |
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----- |
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This is a convenience function that will do less work than calculating |
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the horizontal convergence and vertical vorticity separately. |
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""" |
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dudx, dudy, dvdx, dvdy = _get_gradients(u, v, dx, dy) |
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return dudx + dvdy, dvdx - dudy |
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@exporter.export |
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@ensure_yx_order |
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def shearing_deformation(u, v, dx, dy): |
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r"""Calculate the shearing deformation of the horizontal wind. |
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The grid must have a constant spacing in each direction. |
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Parameters |
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---------- |
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u : (M, N) ndarray |
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x component of the wind |
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v : (M, N) ndarray |
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y component of the wind |
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dx : float |
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The grid spacing in the x-direction |
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dy : float |
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The grid spacing in the y-direction |
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Returns |
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------- |
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(M, N) ndarray |
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Shearing Deformation |
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See Also |
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-------- |
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stretching_convergence, shearing_stretching_deformation |
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""" |
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_, dudy, dvdx, _ = _get_gradients(u, v, dx, dy) |
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return dvdx + dudy |
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@exporter.export |
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@ensure_yx_order |
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def stretching_deformation(u, v, dx, dy): |
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r"""Calculate the stretching deformation of the horizontal wind. |
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The grid must have a constant spacing in each direction. |
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Parameters |
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---------- |
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u : (M, N) ndarray |
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x component of the wind |
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v : (M, N) ndarray |
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y component of the wind |
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dx : float |
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The grid spacing in the x-direction |
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dy : float |
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The grid spacing in the y-direction |
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Returns |
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------- |
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(M, N) ndarray |
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Stretching Deformation |
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See Also |
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-------- |
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shearing_deformation, shearing_stretching_deformation |
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""" |
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dudx, _, _, dvdy = _get_gradients(u, v, dx, dy) |
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return dudx - dvdy |
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@exporter.export |
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@ensure_yx_order |
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def shearing_stretching_deformation(u, v, dx, dy): |
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r"""Calculate the horizontal shearing and stretching deformation of the horizontal wind. |
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Parameters |
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---------- |
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u : (M, N) ndarray |
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x component of the wind |
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v : (M, N) ndarray |
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y component of the wind |
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dx : float |
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The grid spacing in the x-direction |
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dy : float |
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The grid spacing in the y-direction |
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Returns |
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------- |
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shearing, strectching : tuple of (M, N) ndarrays |
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The horizontal shearing and stretching deformation, respectively |
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See Also |
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-------- |
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shearing_deformation, stretching_deformation |
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Notes |
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----- |
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This is a convenience function that will do less work than calculating |
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the shearing and streching deformation terms separately. |
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""" |
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dudx, dudy, dvdx, dvdy = _get_gradients(u, v, dx, dy) |
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return dvdx + dudy, dudx - dvdy |
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@exporter.export |
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@ensure_yx_order |
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def total_deformation(u, v, dx, dy): |
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r"""Calculate the horizontal total deformation of the horizontal wind. |
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The grid must have a constant spacing in each direction. |
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Parameters |
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---------- |
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u : (M, N) ndarray |
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x component of the wind |
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v : (M, N) ndarray |
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y component of the wind |
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dx : float |
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The grid spacing in the x-direction |
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dy : float |
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The grid spacing in the y-direction |
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Returns |
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------- |
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(M, N) ndarray |
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Total Deformation |
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See Also |
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-------- |
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334
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shearing_deformation, stretching_deformation, shearing_stretching_deformation |
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335
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336
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Notes |
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337
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----- |
|
338
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This is a convenience function that will do less work than calculating |
|
339
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the shearing and streching deformation terms separately and calculating the |
|
340
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total deformation "by hand". |
|
341
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342
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""" |
|
343
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dudx, dudy, dvdx, dvdy = _get_gradients(u, v, dx, dy) |
|
344
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return np.sqrt((dvdx + dudy)**2 + (dudx - dvdy)**2) |
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345
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346
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347
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@exporter.export |
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348
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@ensure_yx_order |
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349
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def advection(scalar, wind, deltas): |
|
350
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r"""Calculate the advection of a scalar field by the wind. |
|
351
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|
352
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The order of the dimensions of the arrays must match the order in which |
|
353
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the wind components are given. For example, if the winds are given [u, v], |
|
354
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then the scalar and wind arrays must be indexed as x,y (which puts x as the |
|
355
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rows, not columns). |
|
356
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|
357
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Parameters |
|
358
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---------- |
|
359
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scalar : N-dimensional array |
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360
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Array (with N-dimensions) with the quantity to be advected. |
|
361
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wind : sequence of arrays |
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362
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Length N sequence of N-dimensional arrays. Represents the flow, |
|
363
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with a component of the wind in each dimension. For example, for |
|
364
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horizontal advection, this could be a list: [u, v], where u and v |
|
365
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are each a 2-dimensional array. |
|
366
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deltas : sequence |
|
367
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A (length N) sequence containing the grid spacing in each dimension. |
|
368
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|
|
369
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|
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Returns |
|
370
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|
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------- |
|
371
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|
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N-dimensional array |
|
372
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|
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An N-dimensional array containing the advection at all grid points. |
|
373
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|
374
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""" |
|
375
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|
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# This allows passing in a list of wind components or an array. |
|
376
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|
|
wind = _stack(wind) |
|
377
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|
378
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# If we have more than one component, we need to reverse the order along the first |
|
379
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# dimension so that the wind components line up with the |
|
380
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|
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# order of the gradients from the ..., y, x ordered array. |
|
381
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|
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if wind.ndim > scalar.ndim: |
|
382
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|
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wind = wind[::-1] |
|
383
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|
384
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|
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# Gradient returns a list of derivatives along each dimension. We convert |
|
385
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# this to an array with dimension as the first index. Reverse the deltas to line up |
|
386
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|
|
# with the order of the dimensions. |
|
387
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|
grad = _stack(_gradient(scalar, *deltas[::-1])) |
|
388
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|
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|
389
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# Make them be at least 2D (handling the 1D case) so that we can do the |
|
390
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# multiply and sum below |
|
391
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|
|
grad, wind = atleast_2d(grad, wind) |
|
392
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|
393
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|
|
return (-grad * wind).sum(axis=0) |
|
394
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|
395
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|
396
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@exporter.export |
|
397
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@ensure_yx_order |
|
398
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|
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def frontogenesis(thta, u, v, dx, dy, dim_order='yx'): |
|
399
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|
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r"""Calculate the 2D kinematic frontogenesis of a temperature field. |
|
400
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|
401
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|
The implementation is a form of the Petterssen Frontogenesis and uses the formula |
|
402
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|
|
outlined in [Bluestein1993]_ pg.248-253. |
|
403
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|
404
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.. math:: F=\frac{1}{2}\left|\nabla \theta\right|[D cos(2\beta)-\delta] |
|
405
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|
|
406
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* :math:`F` is 2D kinematic frontogenesis |
|
407
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* :math:`\theta` is potential temperature |
|
408
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|
|
* :math:`D` is the total deformation |
|
409
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|
* :math:`\beta` is the angle between the axis of dilitation and the isentropes |
|
410
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|
|
* :math:`\delta` is the divergence |
|
411
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|
|
412
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|
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Notes: |
|
413
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|
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----- |
|
414
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|
|
Assumes dim_order='yx', unless otherwise specified. |
|
415
|
|
|
|
|
416
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|
|
Parameters |
|
417
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|
|
---------- |
|
418
|
|
|
thta : (M, N) ndarray |
|
419
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|
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Potential temperature |
|
420
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|
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u : (M, N) ndarray |
|
421
|
|
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x component of the wind |
|
422
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|
|
v : (M, N) ndarray |
|
423
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|
|
y component of the wind |
|
424
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|
|
dx : float |
|
425
|
|
|
The grid spacing in the x-direction |
|
426
|
|
|
dy : float |
|
427
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|
|
The grid spacing in the y-direction |
|
428
|
|
|
|
|
429
|
|
|
Returns |
|
430
|
|
|
------- |
|
431
|
|
|
(M, N) ndarray |
|
432
|
|
|
2D Frotogenesis in [temperature units]/m/s |
|
433
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|
434
|
|
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|
|
435
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|
|
Conversion factor to go from [temperature units]/m/s to [tempature units/100km/3h] |
|
436
|
|
|
:math:`1.08e4*1.e5` |
|
437
|
|
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|
|
438
|
|
|
""" |
|
439
|
|
|
# Get gradients of potential temperature in both x and y |
|
440
|
|
|
grad = _gradient(thta, dy, dx) |
|
441
|
|
|
ddy_thta, ddx_thta = grad[-2:] # Throw away unused gradient components |
|
442
|
|
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|
|
443
|
|
|
# Compute the magnitude of the potential temperature gradient |
|
444
|
|
|
mag_thta = np.sqrt(ddx_thta**2 + ddy_thta**2) |
|
445
|
|
|
|
|
446
|
|
|
# Get the shearing, stretching, and total deformation of the wind field |
|
447
|
|
|
shrd, strd = shearing_stretching_deformation(u, v, dx, dy, dim_order=dim_order) |
|
448
|
|
|
tdef = total_deformation(u, v, dx, dy, dim_order=dim_order) |
|
449
|
|
|
|
|
450
|
|
|
# Get the divergence of the wind field |
|
451
|
|
|
div = h_convergence(u, v, dx, dy, dim_order=dim_order) |
|
452
|
|
|
|
|
453
|
|
|
# Compute the angle (beta) between the wind field and the gradient of potential temperature |
|
454
|
|
|
psi = 0.5 * np.arctan2(shrd, strd) |
|
455
|
|
|
beta = np.arcsin((-ddx_thta * np.cos(psi) - ddy_thta * np.sin(psi)) / mag_thta) |
|
456
|
|
|
|
|
457
|
|
|
return 0.5 * mag_thta * (tdef * np.cos(2 * beta) - div) |
|
458
|
|
|
|
|
459
|
|
|
|
|
460
|
|
|
@exporter.export |
|
461
|
|
|
@ensure_yx_order |
|
462
|
|
|
def geostrophic_wind(heights, f, dx, dy): |
|
463
|
|
|
r"""Calculate the geostrophic wind given from the heights or geopotential. |
|
464
|
|
|
|
|
465
|
|
|
Parameters |
|
466
|
|
|
---------- |
|
467
|
|
|
heights : (M, N) ndarray |
|
468
|
|
|
The height field, with either leading dimensions of (x, y) or trailing dimensions |
|
469
|
|
|
of (y, x), depending on the value of ``dim_order``. |
|
470
|
|
|
f : array_like |
|
471
|
|
|
The coriolis parameter. This can be a scalar to be applied |
|
472
|
|
|
everywhere or an array of values. |
|
473
|
|
|
dx : scalar |
|
474
|
|
|
The grid spacing in the x-direction |
|
475
|
|
|
dy : scalar |
|
476
|
|
|
The grid spacing in the y-direction |
|
477
|
|
|
|
|
478
|
|
|
Returns |
|
479
|
|
|
------- |
|
480
|
|
|
A 2-item tuple of arrays |
|
481
|
|
|
A tuple of the u-component and v-component of the geostrophic wind. |
|
482
|
|
|
|
|
483
|
|
|
""" |
|
484
|
|
|
if heights.dimensionality['[length]'] == 2.0: |
|
485
|
|
|
norm_factor = 1. / f |
|
486
|
|
|
else: |
|
487
|
|
|
norm_factor = g / f |
|
488
|
|
|
|
|
489
|
|
|
# If heights has more than 2 dimensions, we need to pass in some dummy |
|
490
|
|
|
# grid deltas so that we can still use np.gradient. It may be better to |
|
491
|
|
|
# to loop in this case, but that remains to be done. |
|
492
|
|
|
deltas = [dy, dx] |
|
493
|
|
|
if heights.ndim > 2: |
|
494
|
|
|
deltas = [units.Quantity(1., units.m)] * (heights.ndim - 2) + deltas |
|
495
|
|
|
|
|
496
|
|
|
grad = _gradient(heights, *deltas) |
|
497
|
|
|
dy, dx = grad[-2:] # Throw away unused gradient components |
|
498
|
|
|
return -norm_factor * dy, norm_factor * dx |
|
499
|
|
|
|
|
500
|
|
|
|
|
501
|
|
|
@exporter.export |
|
502
|
|
|
@check_units('[length]', '[temperature]') |
|
503
|
|
|
def montgomery_streamfunction(height, temperature): |
|
504
|
|
|
r"""Compute the Montgomery Streamfunction on isentropic surfaces. |
|
505
|
|
|
|
|
506
|
|
|
The Montgomery Streamfunction is the streamfunction of the geostrophic wind on an |
|
507
|
|
|
isentropic surface. This quantity is proportional to the geostrophic wind in isentropic |
|
508
|
|
|
coordinates, and its gradient can be interpreted similarly to the pressure gradient in |
|
509
|
|
|
isobaric coordinates. |
|
510
|
|
|
|
|
511
|
|
|
Parameters |
|
512
|
|
|
---------- |
|
513
|
|
|
height : `pint.Quantity` |
|
514
|
|
|
Array of geopotential height of isentropic surfaces |
|
515
|
|
|
temperature : `pint.Quantity` |
|
516
|
|
|
Array of temperature on isentropic surfaces |
|
517
|
|
|
|
|
518
|
|
|
Returns |
|
519
|
|
|
------- |
|
520
|
|
|
stream_func : `pint.Quantity` |
|
521
|
|
|
|
|
522
|
|
|
Notes |
|
523
|
|
|
----- |
|
524
|
|
|
The formula used is that from [Lackmann2011]_ p. 69 for T in Kelvin and height in meters: |
|
525
|
|
|
.. math:: sf = gZ + C_pT |
|
526
|
|
|
* :math:`sf` is Montgomery Streamfunction |
|
527
|
|
|
* :math:`g` is avg. gravitational acceleration on Earth |
|
528
|
|
|
* :math:`Z` is geopotential height of the isentropic surface |
|
529
|
|
|
* :math:`C_p` is specific heat at constant pressure for dry air |
|
530
|
|
|
* :math:`T` is temperature of the isentropic surface |
|
531
|
|
|
|
|
532
|
|
|
See Also |
|
533
|
|
|
-------- |
|
534
|
|
|
get_isentropic_pressure |
|
535
|
|
|
|
|
536
|
|
|
""" |
|
537
|
|
|
return (g * height) + (Cp_d * temperature) |
|
538
|
|
|
|
Unidata /
MetPy
