code: # Copyright (c) 2008,2015,2016,2017,2018,2019 MetPy Developers.
  # Distributed under the terms of the BSD 3-Clause License.
  # SPDX-License-Identifier: BSD-3-Clause
  """Contains a collection of thermodynamic calculations."""
  import warnings
  
  import numpy as np
  import scipy.integrate as si
  import scipy.optimize as so
  
  from .tools import (_greater_or_close, _less_or_close, _remove_nans, find_bounding_indices,
                      find_intersections, first_derivative, get_layer)
  from .. import constants as mpconsts
  from ..cbook import broadcast_indices
  from ..interpolate.one_dimension import interpolate_1d
  from ..package_tools import Exporter
  from ..units import check_units, concatenate, units
  from ..xarray import preprocess_xarray
  
  exporter = Exporter(globals())
  
  sat_pressure_0c = 6.112 * units.millibar
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[temperature]', '[temperature]')
  def relative_humidity_from_dewpoint(temperature, dewpoint):
      r"""Calculate the relative humidity.
  
      Uses temperature and dewpoint in celsius to calculate relative
      humidity using the ratio of vapor pressure to saturation vapor pressures.
  
      Parameters
      ----------
      temperature : `pint.Quantity`
          air temperature
      dewpoint : `pint.Quantity`
          dewpoint temperature
  
      Returns
      -------
      `pint.Quantity`
          relative humidity
  
      See Also
      --------
      saturation_vapor_pressure
  
      """
      e = saturation_vapor_pressure(dewpoint)
      e_s = saturation_vapor_pressure(temperature)
      return (e / e_s)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[pressure]')
  def exner_function(pressure, reference_pressure=mpconsts.P0):
      r"""Calculate the Exner function.
  
      .. math:: \Pi = \left( \frac{p}{p_0} \right)^\kappa
  
      This can be used to calculate potential temperature from temperature (and visa-versa),
      since
  
      .. math:: \Pi = \frac{T}{\theta}
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          total atmospheric pressure
      reference_pressure : `pint.Quantity`, optional
          The reference pressure against which to calculate the Exner function, defaults to
          metpy.constants.P0
  
      Returns
      -------
      `pint.Quantity`
          The value of the Exner function at the given pressure
  
      See Also
      --------
      potential_temperature
      temperature_from_potential_temperature
  
      """
      return (pressure / reference_pressure).to('dimensionless')**mpconsts.kappa
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]')
  def potential_temperature(pressure, temperature):
      r"""Calculate the potential temperature.
  
      Uses the Poisson equation to calculation the potential temperature
      given `pressure` and `temperature`.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          total atmospheric pressure
      temperature : `pint.Quantity`
          air temperature
  
      Returns
      -------
      `pint.Quantity`
          The potential temperature corresponding to the temperature and
          pressure.
  
      See Also
      --------
      dry_lapse
  
      Notes
      -----
      Formula:
  
      .. math:: \Theta = T (P_0 / P)^\kappa
  
      Examples
      --------
      >>> from metpy.units import units
      >>> metpy.calc.potential_temperature(800. * units.mbar, 273. * units.kelvin)
      <Quantity(290.9665329591884, 'kelvin')>
  
      """
      return temperature / exner_function(pressure)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]')
  def temperature_from_potential_temperature(pressure, potential_temperature):
      r"""Calculate the temperature from a given potential temperature.
  
      Uses the inverse of the Poisson equation to calculate the temperature from a
      given potential temperature at a specific pressure level.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          total atmospheric pressure
      potential_temperature : `pint.Quantity`
          potential temperature
  
      Returns
      -------
      `pint.Quantity`
          The temperature corresponding to the potential temperature and pressure.
  
      See Also
      --------
      dry_lapse
      potential_temperature
  
      Notes
      -----
      Formula:
  
      .. math:: T = \Theta (P / P_0)^\kappa
  
      Examples
      --------
      >>> from metpy.units import units
      >>> from metpy.calc import temperature_from_potential_temperature
      >>> # potential temperature
      >>> theta = np.array([ 286.12859679, 288.22362587]) * units.kelvin
      >>> p = 850 * units.mbar
      >>> T = temperature_from_potential_temperature(p, theta)
  
      """
      return potential_temperature * exner_function(pressure)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[pressure]')
  def dry_lapse(pressure, temperature, reference_pressure=None):
      r"""Calculate the temperature at a level assuming only dry processes.
  
      This function lifts a parcel starting at `temperature`, conserving
      potential temperature. The starting pressure can be given by `reference_pressure`.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The atmospheric pressure level(s) of interest
      temperature : `pint.Quantity`
          The starting temperature
      reference_pressure : `pint.Quantity`, optional
          The reference pressure. If not given, it defaults to the first element of the
          pressure array.
  
      Returns
      -------
      `pint.Quantity`
         The resulting parcel temperature at levels given by `pressure`
  
      See Also
      --------
      moist_lapse : Calculate parcel temperature assuming liquid saturation processes
      parcel_profile : Calculate complete parcel profile
      potential_temperature
  
      """
      if reference_pressure is None:
          reference_pressure = pressure[0]
      return temperature * (pressure / reference_pressure)**mpconsts.kappa
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[pressure]')
  def moist_lapse(pressure, temperature, reference_pressure=None):
      r"""Calculate the temperature at a level assuming liquid saturation processes.
  
      This function lifts a parcel starting at `temperature`. The starting pressure can
      be given by `reference_pressure`. Essentially, this function is calculating moist
      pseudo-adiabats.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The atmospheric pressure level(s) of interest
      temperature : `pint.Quantity`
          The starting temperature
      reference_pressure : `pint.Quantity`, optional
          The reference pressure. If not given, it defaults to the first element of the
          pressure array.
  
      Returns
      -------
      `pint.Quantity`
         The temperature corresponding to the starting temperature and
         pressure levels.
  
      See Also
      --------
      dry_lapse : Calculate parcel temperature assuming dry adiabatic processes
      parcel_profile : Calculate complete parcel profile
  
      Notes
      -----
      This function is implemented by integrating the following differential
      equation:
  
      .. math:: \frac{dT}{dP} = \frac{1}{P} \frac{R_d T + L_v r_s}
                                  {C_{pd} + \frac{L_v^2 r_s \epsilon}{R_d T^2}}
  
      This equation comes from [Bakhshaii2013]_.
  
      """
      def dt(t, p):
          t = units.Quantity(t, temperature.units)
          p = units.Quantity(p, pressure.units)
          rs = saturation_mixing_ratio(p, t)
          frac = ((mpconsts.Rd * t + mpconsts.Lv * rs)
                  / (mpconsts.Cp_d + (mpconsts.Lv * mpconsts.Lv * rs * mpconsts.epsilon
                                      / (mpconsts.Rd * t * t)))).to('kelvin')
          return (frac / p).magnitude
  
      if reference_pressure is None:
          reference_pressure = pressure[0]
  
      pressure = pressure.to('mbar')
      reference_pressure = reference_pressure.to('mbar')
      temperature = np.atleast_1d(temperature)
  
      side = 'left'
  
      pres_decreasing = (pressure[0] > pressure[-1])
      if pres_decreasing:
          # Everything is easier if pressures are in increasing order
          pressure = pressure[::-1]
          side = 'right'
  
      ref_pres_idx = np.searchsorted(pressure.m, reference_pressure.m, side=side)
  
      ret_temperatures = np.empty((0, temperature.shape[0]))
  
      if reference_pressure > pressure.min():
          # Integrate downward in pressure
          pres_down = np.append(reference_pressure.m, pressure[(ref_pres_idx - 1)::-1].m)
          trace_down = si.odeint(dt, temperature.m.squeeze(), pres_down.squeeze())
          ret_temperatures = np.concatenate((ret_temperatures, trace_down[:0:-1]))
  
      if reference_pressure < pressure.max():
          # Integrate upward in pressure
          pres_up = np.append(reference_pressure.m, pressure[ref_pres_idx:].m)
          trace_up = si.odeint(dt, temperature.m.squeeze(), pres_up.squeeze())
          ret_temperatures = np.concatenate((ret_temperatures, trace_up[1:]))
  
      if pres_decreasing:
          ret_temperatures = ret_temperatures[::-1]
  
      return units.Quantity(ret_temperatures.T.squeeze(), temperature.units)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def lcl(pressure, temperature, dewpoint, max_iters=50, eps=1e-5):
      r"""Calculate the lifted condensation level (LCL) using from the starting point.
  
      The starting state for the parcel is defined by `temperature`, `dewpoint`,
      and `pressure`. If these are arrays, this function will return a LCL
      for every index. This function does work with surface grids as a result.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The starting atmospheric pressure
      temperature : `pint.Quantity`
          The starting temperature
      dewpoint : `pint.Quantity`
          The starting dewpoint
  
      Returns
      -------
      `pint.Quantity`
          The LCL pressure
      `pint.Quantity`
          The LCL temperature
  
      Other Parameters
      ----------------
      max_iters : int, optional
          The maximum number of iterations to use in calculation, defaults to 50.
      eps : float, optional
          The desired relative error in the calculated value, defaults to 1e-5.
  
      See Also
      --------
      parcel_profile
  
      Notes
      -----
      This function is implemented using an iterative approach to solve for the
      LCL. The basic algorithm is:
  
      1. Find the dewpoint from the LCL pressure and starting mixing ratio
      2. Find the LCL pressure from the starting temperature and dewpoint
      3. Iterate until convergence
  
      The function is guaranteed to finish by virtue of the `max_iters` counter.
  
      """
      def _lcl_iter(p, p0, w, t):
          td = globals()['dewpoint'](vapor_pressure(units.Quantity(p, pressure.units), w))
          return (p0 * (td / t) ** (1. / mpconsts.kappa)).m
  
      w = mixing_ratio(saturation_vapor_pressure(dewpoint), pressure)
      lcl_p = so.fixed_point(_lcl_iter, pressure.m, args=(pressure.m, w, temperature),
                             xtol=eps, maxiter=max_iters)
  
      # np.isclose needed if surface is LCL due to precision error with np.log in dewpoint.
      # Causes issues with parcel_profile_with_lcl if removed. Issue #1187
      lcl_p = np.where(np.isclose(lcl_p, pressure.m), pressure.m, lcl_p) * pressure.units
  
      return lcl_p, globals()['dewpoint'](vapor_pressure(lcl_p, w)).to(temperature.units)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]', '[temperature]')
  def lfc(pressure, temperature, dewpoint, parcel_temperature_profile=None, dewpoint_start=None,
          which='top'):
      r"""Calculate the level of free convection (LFC).
  
      This works by finding the first intersection of the ideal parcel path and
      the measured parcel temperature. If this intersection occurs below the LCL,
      the LFC is determined to be the same as the LCL, based upon the conditions
      set forth in [USAF1990]_, pg 4-14, where a parcel must be lifted dry adiabatically
      to saturation before it can freely rise.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The atmospheric pressure
      temperature : `pint.Quantity`
          The temperature at the levels given by `pressure`
      dewpoint : `pint.Quantity`
          The dewpoint at the levels given by `pressure`
      parcel_temperature_profile: `pint.Quantity`, optional
          The parcel temperature profile from which to calculate the LFC. Defaults to the
          surface parcel profile.
      dewpoint_start: `pint.Quantity`, optional
          The dewpoint of the parcel for which to calculate the LFC. Defaults to the surface
          dewpoint.
      which: str, optional
          Pick which LFC to return. Options are 'top', 'bottom', 'wide', 'most_cape', and 'all'.
          'top' returns the lowest-pressure LFC, default.
          'bottom' returns the highest-pressure LFC.
          'wide' returns the LFC whose corresponding EL is farthest away.
          'most_cape' returns the LFC that results in the most CAPE in the profile.
  
      Returns
      -------
      `pint.Quantity`
          The LFC pressure, or array of same if which='all'
      `pint.Quantity`
          The LFC temperature, or array of same if which='all'
  
      See Also
      --------
      parcel_profile
  
      """
      pressure, temperature, dewpoint = _remove_nans(pressure, temperature, dewpoint)
      # Default to surface parcel if no profile or starting pressure level is given
      if parcel_temperature_profile is None:
          new_stuff = parcel_profile_with_lcl(pressure, temperature, dewpoint)
          pressure, temperature, dewpoint, parcel_temperature_profile = new_stuff
          parcel_temperature_profile = parcel_temperature_profile.to(temperature.units)
  
      if dewpoint_start is None:
          dewpoint_start = dewpoint[0]
  
      # The parcel profile and data may have the same first data point.
      # If that is the case, ignore that point to get the real first
      # intersection for the LFC calculation. Use logarithmic interpolation.
      if np.isclose(parcel_temperature_profile[0].to(temperature.units).m, temperature[0].m):
          x, y = find_intersections(pressure[1:], parcel_temperature_profile[1:],
                                    temperature[1:], direction='increasing', log_x=True)
      else:
          x, y = find_intersections(pressure, parcel_temperature_profile,
                                    temperature, direction='increasing', log_x=True)
  
      # Compute LCL for this parcel for future comparisons
      this_lcl = lcl(pressure[0], parcel_temperature_profile[0], dewpoint_start)
  
      # The LFC could:
      # 1) Not exist
      # 2) Exist but be equal to the LCL
      # 3) Exist and be above the LCL
  
      # LFC does not exist or is LCL
      if len(x) == 0:
          # Is there any positive area above the LCL?
          mask = pressure < this_lcl[0]
          if np.all(_less_or_close(parcel_temperature_profile[mask], temperature[mask])):
              # LFC doesn't exist
              x, y = np.nan * pressure.units, np.nan * temperature.units
          else:  # LFC = LCL
              x, y = this_lcl
          return x, y
  
      # LFC exists. Make sure it is no lower than the LCL
      else:
          idx = x < this_lcl[0]
          # LFC height < LCL height, so set LFC = LCL
          if not any(idx):
              el_pres, _ = find_intersections(pressure[1:], parcel_temperature_profile[1:],
                                              temperature[1:], direction='decreasing',
                                              log_x=True)
              if np.min(el_pres) > this_lcl[0]:
                  x, y = np.nan * pressure.units, np.nan * temperature.units
              else:
                  x, y = this_lcl
              return x, y
          # Otherwise, find all LFCs that exist above the LCL
          # What is returned depends on which flag as described in the docstring
          else:
              return _multiple_el_lfc_options(x, y, idx, which, pressure,
                                              parcel_temperature_profile, temperature,
                                              dewpoint, intersect_type='LFC')
  
  
  def _multiple_el_lfc_options(intersect_pressures, intersect_temperatures, valid_x,
                               which, pressure, parcel_temperature_profile, temperature,
                               dewpoint, intersect_type):
      """Choose which ELs and LFCs to return from a sounding."""
      p_list, t_list = intersect_pressures[valid_x], intersect_temperatures[valid_x]
      if which == 'all':
          x, y = p_list, t_list
      elif which == 'bottom':
          x, y = p_list[0], t_list[0]
      elif which == 'top':
          x, y = p_list[-1], t_list[-1]
      elif which == 'wide':
          x, y = _wide_option(intersect_type, p_list, t_list, pressure,
                              parcel_temperature_profile, temperature)
      elif which == 'most_cape':
          x, y = _most_cape_option(intersect_type, p_list, t_list, pressure, temperature,
                                   dewpoint, parcel_temperature_profile)
      else:
          raise ValueError('Invalid option for "which". Valid options are "top", "bottom", '
                           '"wide", "most_cape", and "all".')
      return x, y
  
  
  def _wide_option(intersect_type, p_list, t_list, pressure, parcel_temperature_profile,
                   temperature):
      """Calculate the LFC or EL that produces the greatest distance between these points."""
      # zip the LFC and EL lists together and find greatest difference
      if intersect_type == 'LFC':
          # Find EL intersection pressure values
          lfc_p_list = p_list
          el_p_list, _ = find_intersections(pressure[1:], parcel_temperature_profile[1:],
                                            temperature[1:], direction='decreasing',
                                            log_x=True)
      else:  # intersect_type == 'EL'
          el_p_list = p_list
          # Find LFC intersection pressure values
          lfc_p_list, _ = find_intersections(pressure, parcel_temperature_profile,
                                             temperature, direction='increasing',
                                             log_x=True)
      diff = [lfc_p.m - el_p.m for lfc_p, el_p in zip(lfc_p_list, el_p_list)]
      return (p_list[np.where(diff == np.max(diff))][0],
              t_list[np.where(diff == np.max(diff))][0])
  
  
  def _most_cape_option(intersect_type, p_list, t_list, pressure, temperature, dewpoint,
                        parcel_temperature_profile):
      """Calculate the LFC or EL that produces the most CAPE in the profile."""
      # Need to loop through all possible combinations of cape, find greatest cape profile
      cape_list, pair_list = [], []
      for which_lfc in ['top', 'bottom']:
          for which_el in ['top', 'bottom']:
              cape, _ = cape_cin(pressure, temperature, dewpoint, parcel_temperature_profile,
                                 which_lfc=which_lfc, which_el=which_el)
              cape_list.append(cape.m)
              pair_list.append([which_lfc, which_el])
      (lfc_chosen, el_chosen) = pair_list[np.where(cape_list == np.max(cape_list))[0][0]]
      if intersect_type == 'LFC':
          if lfc_chosen == 'top':
              x, y = p_list[-1], t_list[-1]
          else:  # 'bottom' is returned
              x, y = p_list[0], t_list[0]
      else:  # EL is returned
          if el_chosen == 'top':
              x, y = p_list[-1], t_list[-1]
          else:
              x, y = p_list[0], t_list[0]
      return x, y
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]', '[temperature]')
  def el(pressure, temperature, dewpoint, parcel_temperature_profile=None, which='top'):
      r"""Calculate the equilibrium level.
  
      This works by finding the last intersection of the ideal parcel path and
      the measured environmental temperature. If there is one or fewer intersections, there is
      no equilibrium level.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The atmospheric pressure profile
      temperature : `pint.Quantity`
          The temperature at the levels given by `pressure`
      dewpoint : `pint.Quantity`
          The dewpoint at the levels given by `pressure`
      parcel_temperature_profile: `pint.Quantity`, optional
          The parcel temperature profile from which to calculate the EL. Defaults to the
          surface parcel profile.
      which: str, optional
          Pick which LFC to return. Options are 'top', 'bottom', 'wide', 'most_cape', and 'all'.
          'top' returns the lowest-pressure EL, default.
          'bottom' returns the highest-pressure EL.
          'wide' returns the EL whose corresponding LFC is farthest away.
          'most_cape' returns the EL that results in the most CAPE in the profile.
  
      Returns
      -------
      `pint.Quantity`
          The EL pressure, or array of same if which='all'
      `pint.Quantity`
          The EL temperature, or array of same if which='all'
  
      See Also
      --------
      parcel_profile
  
      """
      pressure, temperature, dewpoint = _remove_nans(pressure, temperature, dewpoint)
      # Default to surface parcel if no profile or starting pressure level is given
      if parcel_temperature_profile is None:
          new_stuff = parcel_profile_with_lcl(pressure, temperature, dewpoint)
          pressure, temperature, dewpoint, parcel_temperature_profile = new_stuff
          parcel_temperature_profile = parcel_temperature_profile.to(temperature.units)
  
      # If the top of the sounding parcel is warmer than the environment, there is no EL
      if parcel_temperature_profile[-1] > temperature[-1]:
          return np.nan * pressure.units, np.nan * temperature.units
  
      # Interpolate in log space to find the appropriate pressure - units have to be stripped
      # and reassigned to allow np.log() to function properly.
      x, y = find_intersections(pressure[1:], parcel_temperature_profile[1:], temperature[1:],
                                direction='decreasing', log_x=True)
      lcl_p, _ = lcl(pressure[0], temperature[0], dewpoint[0])
      idx = x < lcl_p
      if len(x) > 0 and x[-1] < lcl_p:
          return _multiple_el_lfc_options(x, y, idx, which, pressure,
                                          parcel_temperature_profile, temperature, dewpoint,
                                          intersect_type='EL')
      else:
          return np.nan * pressure.units, np.nan * temperature.units
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def parcel_profile(pressure, temperature, dewpoint):
      r"""Calculate the profile a parcel takes through the atmosphere.
  
      The parcel starts at `temperature`, and `dewpoint`, lifted up
      dry adiabatically to the LCL, and then moist adiabatically from there.
      `pressure` specifies the pressure levels for the profile.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The atmospheric pressure level(s) of interest. This array must be from
          high to low pressure.
      temperature : `pint.Quantity`
          The starting temperature
      dewpoint : `pint.Quantity`
          The starting dewpoint
  
      Returns
      -------
      `pint.Quantity`
          The parcel temperatures at the specified pressure levels.
  
      See Also
      --------
      lcl, moist_lapse, dry_lapse
  
      """
      _, _, _, t_l, _, t_u = _parcel_profile_helper(pressure, temperature, dewpoint)
      return concatenate((t_l, t_u))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def parcel_profile_with_lcl(pressure, temperature, dewpoint):
      r"""Calculate the profile a parcel takes through the atmosphere.
  
      The parcel starts at `temperature`, and `dewpoint`, lifted up
      dry adiabatically to the LCL, and then moist adiabatically from there.
      `pressure` specifies the pressure levels for the profile. This function returns
      a profile that includes the LCL.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The atmospheric pressure level(s) of interest. This array must be from
          high to low pressure.
      temperature : `pint.Quantity`
          The atmospheric temperature at the levels in `pressure`. The first entry should be at
          the same level as the first `pressure` data point.
      dewpoint : `pint.Quantity`
          The atmospheric dewpoint at the levels in `pressure`. The first entry should be at
          the same level as the first `pressure` data point.
  
      Returns
      -------
      pressure : `pint.Quantity`
          The parcel profile pressures, which includes the specified levels and the LCL
      ambient_temperature : `pint.Quantity`
          The atmospheric temperature values, including the value interpolated to the LCL level
      ambient_dew_point : `pint.Quantity`
          The atmospheric dewpoint values, including the value interpolated to the LCL level
      profile_temperature : `pint.Quantity`
          The parcel profile temperatures at all of the levels in the returned pressures array,
          including the LCL.
  
      See Also
      --------
      lcl, moist_lapse, dry_lapse, parcel_profile
  
      """
      p_l, p_lcl, p_u, t_l, t_lcl, t_u = _parcel_profile_helper(pressure, temperature[0],
                                                                dewpoint[0])
      new_press = concatenate((p_l, p_lcl, p_u))
      prof_temp = concatenate((t_l, t_lcl, t_u))
      new_temp = _insert_lcl_level(pressure, temperature, p_lcl)
      new_dewp = _insert_lcl_level(pressure, dewpoint, p_lcl)
      return new_press, new_temp, new_dewp, prof_temp
  
  
  def _parcel_profile_helper(pressure, temperature, dewpoint):
      """Help calculate parcel profiles.
  
      Returns the temperature and pressure, above, below, and including the LCL. The
      other calculation functions decide what to do with the pieces.
  
      """
      # Find the LCL
      press_lcl, temp_lcl = lcl(pressure[0], temperature, dewpoint)
      press_lcl = press_lcl.to(pressure.units)
  
      # Find the dry adiabatic profile, *including* the LCL. We need >= the LCL in case the
      # LCL is included in the levels. It's slightly redundant in that case, but simplifies
      # the logic for removing it later.
      press_lower = concatenate((pressure[pressure >= press_lcl], press_lcl))
      temp_lower = dry_lapse(press_lower, temperature)
  
      # If the pressure profile doesn't make it to the lcl, we can stop here
      if _greater_or_close(np.nanmin(pressure.m), press_lcl.m):
          return (press_lower[:-1], press_lcl, units.Quantity(np.array([]), press_lower.units),
                  temp_lower[:-1], temp_lcl, units.Quantity(np.array([]), temp_lower.units))
  
      # Find moist pseudo-adiabatic profile starting at the LCL
      press_upper = concatenate((press_lcl, pressure[pressure < press_lcl]))
      temp_upper = moist_lapse(press_upper, temp_lower[-1]).to(temp_lower.units)
  
      # Return profile pieces
      return (press_lower[:-1], press_lcl, press_upper[1:],
              temp_lower[:-1], temp_lcl, temp_upper[1:])
  
  
  def _insert_lcl_level(pressure, temperature, lcl_pressure):
      """Insert the LCL pressure into the profile."""
      interp_temp = interpolate_1d(lcl_pressure, pressure, temperature)
  
      # Pressure needs to be increasing for searchsorted, so flip it and then convert
      # the index back to the original array
      loc = pressure.size - pressure[::-1].searchsorted(lcl_pressure)
      return np.insert(temperature.m, loc, interp_temp.m) * temperature.units
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[dimensionless]')
  def vapor_pressure(pressure, mixing_ratio):
      r"""Calculate water vapor (partial) pressure.
  
      Given total `pressure` and water vapor `mixing_ratio`, calculates the
      partial pressure of water vapor.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          total atmospheric pressure
      mixing_ratio : `pint.Quantity`
          dimensionless mass mixing ratio
  
      Returns
      -------
      `pint.Quantity`
          The ambient water vapor (partial) pressure in the same units as
          `pressure`.
  
      Notes
      -----
      This function is a straightforward implementation of the equation given in many places,
      such as [Hobbs1977]_ pg.71:
  
      .. math:: e = p \frac{r}{r + \epsilon}
  
      See Also
      --------
      saturation_vapor_pressure, dewpoint
  
      """
      return pressure * mixing_ratio / (mpconsts.epsilon + mixing_ratio)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[temperature]')
  def saturation_vapor_pressure(temperature):
      r"""Calculate the saturation water vapor (partial) pressure.
  
      Parameters
      ----------
      temperature : `pint.Quantity`
          air temperature
  
      Returns
      -------
      `pint.Quantity`
          The saturation water vapor (partial) pressure
  
      See Also
      --------
      vapor_pressure, dewpoint
  
      Notes
      -----
      Instead of temperature, dewpoint may be used in order to calculate
      the actual (ambient) water vapor (partial) pressure.
  
      The formula used is that from [Bolton1980]_ for T in degrees Celsius:
  
      .. math:: 6.112 e^\frac{17.67T}{T + 243.5}
  
      """
      # Converted from original in terms of C to use kelvin. Using raw absolute values of C in
      # a formula plays havoc with units support.
      return sat_pressure_0c * np.exp(17.67 * (temperature - 273.15 * units.kelvin)
                                      / (temperature - 29.65 * units.kelvin))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[temperature]', '[dimensionless]')
  def dewpoint_from_relative_humidity(temperature, relative_humidity):
      r"""Calculate the ambient dewpoint given air temperature and relative humidity.
  
      Parameters
      ----------
      temperature : `pint.Quantity`
          air temperature
      relative_humidity : `pint.Quantity`
          relative humidity expressed as a ratio in the range 0 < relative_humidity <= 1
  
      Returns
      -------
      `pint.Quantity`
          The dewpoint temperature
  
      See Also
      --------
      dewpoint, saturation_vapor_pressure
  
      """
      if np.any(relative_humidity > 1.2):
          warnings.warn('Relative humidity >120%, ensure proper units.')
      return dewpoint(relative_humidity * saturation_vapor_pressure(temperature))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]')
  def dewpoint(vapor_pressure):
      r"""Calculate the ambient dewpoint given the vapor pressure.
  
      Parameters
      ----------
      e : `pint.Quantity`
          Water vapor partial pressure
  
      Returns
      -------
      `pint.Quantity`
          dewpoint temperature
  
      See Also
      --------
      dewpoint_from_relative_humidity, saturation_vapor_pressure, vapor_pressure
  
      Notes
      -----
      This function inverts the [Bolton1980]_ formula for saturation vapor
      pressure to instead calculate the temperature. This yield the following
      formula for dewpoint in degrees Celsius:
  
      .. math:: T = \frac{243.5 log(e / 6.112)}{17.67 - log(e / 6.112)}
  
      """
      val = np.log(vapor_pressure / sat_pressure_0c)
      return 0. * units.degC + 243.5 * units.delta_degC * val / (17.67 - val)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[pressure]', '[dimensionless]')
  def mixing_ratio(partial_press, total_press, molecular_weight_ratio=mpconsts.epsilon):
      r"""Calculate the mixing ratio of a gas.
  
      This calculates mixing ratio given its partial pressure and the total pressure of
      the air. There are no required units for the input arrays, other than that
      they have the same units.
  
      Parameters
      ----------
      partial_press : `pint.Quantity`
          Partial pressure of the constituent gas
      total_press : `pint.Quantity`
          Total air pressure
      molecular_weight_ratio : `pint.Quantity` or float, optional
          The ratio of the molecular weight of the constituent gas to that assumed
          for air. Defaults to the ratio for water vapor to dry air
          (:math:`\epsilon\approx0.622`).
  
      Returns
      -------
      `pint.Quantity`
          The (mass) mixing ratio, dimensionless (e.g. Kg/Kg or g/g)
  
      Notes
      -----
      This function is a straightforward implementation of the equation given in many places,
      such as [Hobbs1977]_ pg.73:
  
      .. math:: r = \epsilon \frac{e}{p - e}
  
      See Also
      --------
      saturation_mixing_ratio, vapor_pressure
  
      """
      return (molecular_weight_ratio * partial_press
              / (total_press - partial_press)).to('dimensionless')
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]')
  def saturation_mixing_ratio(total_press, temperature):
      r"""Calculate the saturation mixing ratio of water vapor.
  
      This calculation is given total pressure and the temperature. The implementation
      uses the formula outlined in [Hobbs1977]_ pg.73.
  
      Parameters
      ----------
      total_press: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          air temperature
  
      Returns
      -------
      `pint.Quantity`
          The saturation mixing ratio, dimensionless
  
      """
      return mixing_ratio(saturation_vapor_pressure(temperature), total_press)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def equivalent_potential_temperature(pressure, temperature, dewpoint):
      r"""Calculate equivalent potential temperature.
  
      This calculation must be given an air parcel's pressure, temperature, and dewpoint.
      The implementation uses the formula outlined in [Bolton1980]_:
  
      First, the LCL temperature is calculated:
  
      .. math:: T_{L}=\frac{1}{\frac{1}{T_{D}-56}+\frac{ln(T_{K}/T_{D})}{800}}+56
  
      Which is then used to calculate the potential temperature at the LCL:
  
      .. math:: \theta_{DL}=T_{K}\left(\frac{1000}{p-e}\right)^k
                \left(\frac{T_{K}}{T_{L}}\right)^{.28r}
  
      Both of these are used to calculate the final equivalent potential temperature:
  
      .. math:: \theta_{E}=\theta_{DL}\exp\left[\left(\frac{3036.}{T_{L}}
                                                -1.78\right)*r(1+.448r)\right]
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          Temperature of parcel
      dewpoint: `pint.Quantity`
          Dewpoint of parcel
  
      Returns
      -------
      `pint.Quantity`
          The equivalent potential temperature of the parcel
  
      Notes
      -----
      [Bolton1980]_ formula for Theta-e is used, since according to
      [DaviesJones2009]_ it is the most accurate non-iterative formulation
      available.
  
      """
      t = temperature.to('kelvin').magnitude
      td = dewpoint.to('kelvin').magnitude
      p = pressure.to('hPa').magnitude
      e = saturation_vapor_pressure(dewpoint).to('hPa').magnitude
      r = saturation_mixing_ratio(pressure, dewpoint).magnitude
  
      t_l = 56 + 1. / (1. / (td - 56) + np.log(t / td) / 800.)
      th_l = t * (1000 / (p - e)) ** mpconsts.kappa * (t / t_l) ** (0.28 * r)
      th_e = th_l * np.exp((3036. / t_l - 1.78) * r * (1 + 0.448 * r))
  
      return th_e * units.kelvin
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]')
  def saturation_equivalent_potential_temperature(pressure, temperature):
      r"""Calculate saturation equivalent potential temperature.
  
      This calculation must be given an air parcel's pressure and temperature.
      The implementation uses the formula outlined in [Bolton1980]_ for the
      equivalent potential temperature, and assumes a saturated process.
  
      First, because we assume a saturated process, the temperature at the LCL is
      equivalent to the current temperature. Therefore the following equation
  
      .. math:: T_{L}=\frac{1}{\frac{1}{T_{D}-56}+\frac{ln(T_{K}/T_{D})}{800}}+56
  
      reduces to
  
      .. math:: T_{L} = T_{K}
  
      Then the potential temperature at the temperature/LCL is calculated:
  
      .. math:: \theta_{DL}=T_{K}\left(\frac{1000}{p-e}\right)^k
                \left(\frac{T_{K}}{T_{L}}\right)^{.28r}
  
      However, because
  
      .. math:: T_{L} = T_{K}
  
      it follows that
  
      .. math:: \theta_{DL}=T_{K}\left(\frac{1000}{p-e}\right)^k
  
      Both of these are used to calculate the final equivalent potential temperature:
  
      .. math:: \theta_{E}=\theta_{DL}\exp\left[\left(\frac{3036.}{T_{K}}
                                                -1.78\right)*r(1+.448r)\right]
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          Temperature of parcel
  
      Returns
      -------
      `pint.Quantity`
          The saturation equivalent potential temperature of the parcel
  
      Notes
      -----
      [Bolton1980]_ formula for Theta-e is used (for saturated case), since according to
      [DaviesJones2009]_ it is the most accurate non-iterative formulation
      available.
  
      """
      t = temperature.to('kelvin').magnitude
      p = pressure.to('hPa').magnitude
      e = saturation_vapor_pressure(temperature).to('hPa').magnitude
      r = saturation_mixing_ratio(pressure, temperature).magnitude
  
      th_l = t * (1000 / (p - e)) ** mpconsts.kappa
      th_es = th_l * np.exp((3036. / t - 1.78) * r * (1 + 0.448 * r))
  
      return th_es * units.kelvin
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[temperature]', '[dimensionless]', '[dimensionless]')
  def virtual_temperature(temperature, mixing_ratio, molecular_weight_ratio=mpconsts.epsilon):
      r"""Calculate virtual temperature.
  
      This calculation must be given an air parcel's temperature and mixing ratio.
      The implementation uses the formula outlined in [Hobbs2006]_ pg.80.
  
      Parameters
      ----------
      temperature: `pint.Quantity`
          air temperature
      mixing_ratio : `pint.Quantity`
          dimensionless mass mixing ratio
      molecular_weight_ratio : `pint.Quantity` or float, optional
          The ratio of the molecular weight of the constituent gas to that assumed
          for air. Defaults to the ratio for water vapor to dry air.
          (:math:`\epsilon\approx0.622`).
  
      Returns
      -------
      `pint.Quantity`
          The corresponding virtual temperature of the parcel
  
      Notes
      -----
      .. math:: T_v = T \frac{\text{w} + \epsilon}{\epsilon\,(1 + \text{w})}
  
      """
      return temperature * ((mixing_ratio + molecular_weight_ratio)
                            / (molecular_weight_ratio * (1 + mixing_ratio)))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[dimensionless]', '[dimensionless]')
  def virtual_potential_temperature(pressure, temperature, mixing_ratio,
                                    molecular_weight_ratio=mpconsts.epsilon):
      r"""Calculate virtual potential temperature.
  
      This calculation must be given an air parcel's pressure, temperature, and mixing ratio.
      The implementation uses the formula outlined in [Markowski2010]_ pg.13.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          air temperature
      mixing_ratio : `pint.Quantity`
          dimensionless mass mixing ratio
      molecular_weight_ratio : `pint.Quantity` or float, optional
          The ratio of the molecular weight of the constituent gas to that assumed
          for air. Defaults to the ratio for water vapor to dry air.
          (:math:`\epsilon\approx0.622`).
  
      Returns
      -------
      `pint.Quantity`
          The corresponding virtual potential temperature of the parcel
  
      Notes
      -----
      .. math:: \Theta_v = \Theta \frac{\text{w} + \epsilon}{\epsilon\,(1 + \text{w})}
  
      """
      pottemp = potential_temperature(pressure, temperature)
      return virtual_temperature(pottemp, mixing_ratio, molecular_weight_ratio)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[dimensionless]', '[dimensionless]')
  def density(pressure, temperature, mixing_ratio, molecular_weight_ratio=mpconsts.epsilon):
      r"""Calculate density.
  
      This calculation must be given an air parcel's pressure, temperature, and mixing ratio.
      The implementation uses the formula outlined in [Hobbs2006]_ pg.67.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          air temperature
      mixing_ratio : `pint.Quantity`
          dimensionless mass mixing ratio
      molecular_weight_ratio : `pint.Quantity` or float, optional
          The ratio of the molecular weight of the constituent gas to that assumed
          for air. Defaults to the ratio for water vapor to dry air.
          (:math:`\epsilon\approx0.622`).
  
      Returns
      -------
      `pint.Quantity`
          The corresponding density of the parcel
  
      Notes
      -----
      .. math:: \rho = \frac{p}{R_dT_v}
  
      """
      virttemp = virtual_temperature(temperature, mixing_ratio, molecular_weight_ratio)
      return (pressure / (mpconsts.Rd * virttemp)).to(units.kilogram / units.meter ** 3)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def relative_humidity_wet_psychrometric(pressure, dry_bulb_temperature, web_bulb_temperature,
                                          **kwargs):
      r"""Calculate the relative humidity with wet bulb and dry bulb temperatures.
  
      This uses a psychrometric relationship as outlined in [WMO8-2014]_, with
      coefficients from [Fan1987]_.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      dry_bulb_temperature: `pint.Quantity`
          Dry bulb temperature
      web_bulb_temperature: `pint.Quantity`
          Wet bulb temperature
  
      Returns
      -------
      `pint.Quantity`
          Relative humidity
  
      Notes
      -----
      .. math:: relative_humidity = \frac{e}{e_s}
  
      * :math:`relative_humidity` is relative humidity as a unitless ratio
      * :math:`e` is vapor pressure from the wet psychrometric calculation
      * :math:`e_s` is the saturation vapor pressure
  
      See Also
      --------
      psychrometric_vapor_pressure_wet, saturation_vapor_pressure
  
      """
      return (psychrometric_vapor_pressure_wet(pressure, dry_bulb_temperature,
                                               web_bulb_temperature, **kwargs)
              / saturation_vapor_pressure(dry_bulb_temperature))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def psychrometric_vapor_pressure_wet(pressure, dry_bulb_temperature, wet_bulb_temperature,
                                       psychrometer_coefficient=6.21e-4 / units.kelvin):
      r"""Calculate the vapor pressure with wet bulb and dry bulb temperatures.
  
      This uses a psychrometric relationship as outlined in [WMO8-2014]_, with
      coefficients from [Fan1987]_.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      dry_bulb_temperature: `pint.Quantity`
          Dry bulb temperature
      wet_bulb_temperature: `pint.Quantity`
          Wet bulb temperature
      psychrometer_coefficient: `pint.Quantity`, optional
          Psychrometer coefficient. Defaults to 6.21e-4 K^-1.
  
      Returns
      -------
      `pint.Quantity`
          Vapor pressure
  
      Notes
      -----
      .. math:: e' = e'_w(T_w) - A p (T - T_w)
  
      * :math:`e'` is vapor pressure
      * :math:`e'_w(T_w)` is the saturation vapor pressure with respect to water at temperature
        :math:`T_w`
      * :math:`p` is the pressure of the wet bulb
      * :math:`T` is the temperature of the dry bulb
      * :math:`T_w` is the temperature of the wet bulb
      * :math:`A` is the psychrometer coefficient
  
      Psychrometer coefficient depends on the specific instrument being used and the ventilation
      of the instrument.
  
      See Also
      --------
      saturation_vapor_pressure
  
      """
      return (saturation_vapor_pressure(wet_bulb_temperature) - psychrometer_coefficient
              * pressure * (dry_bulb_temperature - wet_bulb_temperature).to('kelvin'))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[dimensionless]')
  def mixing_ratio_from_relative_humidity(pressure, temperature, relative_humidity):
      r"""Calculate the mixing ratio from relative humidity, temperature, and pressure.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          Air temperature
      relative_humidity: array_like
          The relative humidity expressed as a unitless ratio in the range [0, 1]. Can also pass
          a percentage if proper units are attached.
  
      Returns
      -------
      `pint.Quantity`
          Dimensionless mixing ratio
  
      Notes
      -----
      Formula adapted from [Hobbs1977]_ pg. 74.
  
      .. math:: w = (relative_humidity)(w_s)
  
      * :math:`w` is mixing ratio
      * :math:`relative_humidity` is relative humidity as a unitless ratio
      * :math:`w_s` is the saturation mixing ratio
  
      See Also
      --------
      relative_humidity_from_mixing_ratio, saturation_mixing_ratio
  
      """
      return (relative_humidity
              * saturation_mixing_ratio(pressure, temperature)).to('dimensionless')
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[dimensionless]')
  def relative_humidity_from_mixing_ratio(pressure, temperature, mixing_ratio):
      r"""Calculate the relative humidity from mixing ratio, temperature, and pressure.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          Air temperature
      mixing_ratio: `pint.Quantity`
          Dimensionless mass mixing ratio
  
      Returns
      -------
      `pint.Quantity`
          Relative humidity
  
      Notes
      -----
      Formula based on that from [Hobbs1977]_ pg. 74.
  
      .. math:: relative_humidity = \frac{w}{w_s}
  
      * :math:`relative_humidity` is relative humidity as a unitless ratio
      * :math:`w` is mixing ratio
      * :math:`w_s` is the saturation mixing ratio
  
      See Also
      --------
      mixing_ratio_from_relative_humidity, saturation_mixing_ratio
  
      """
      return mixing_ratio / saturation_mixing_ratio(pressure, temperature)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[dimensionless]')
  def mixing_ratio_from_specific_humidity(specific_humidity):
      r"""Calculate the mixing ratio from specific humidity.
  
      Parameters
      ----------
      specific_humidity: `pint.Quantity`
          Specific humidity of air
  
      Returns
      -------
      `pint.Quantity`
          Mixing ratio
  
      Notes
      -----
      Formula from [Salby1996]_ pg. 118.
  
      .. math:: w = \frac{q}{1-q}
  
      * :math:`w` is mixing ratio
      * :math:`q` is the specific humidity
  
      See Also
      --------
      mixing_ratio, specific_humidity_from_mixing_ratio
  
      """
      try:
          specific_humidity = specific_humidity.to('dimensionless')
      except AttributeError:
          pass
      return specific_humidity / (1 - specific_humidity)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[dimensionless]')
  def specific_humidity_from_mixing_ratio(mixing_ratio):
      r"""Calculate the specific humidity from the mixing ratio.
  
      Parameters
      ----------
      mixing_ratio: `pint.Quantity`
          mixing ratio
  
      Returns
      -------
      `pint.Quantity`
          Specific humidity
  
      Notes
      -----
      Formula from [Salby1996]_ pg. 118.
  
      .. math:: q = \frac{w}{1+w}
  
      * :math:`w` is mixing ratio
      * :math:`q` is the specific humidity
  
      See Also
      --------
      mixing_ratio, mixing_ratio_from_specific_humidity
  
      """
      try:
          mixing_ratio = mixing_ratio.to('dimensionless')
      except AttributeError:
          pass
      return mixing_ratio / (1 + mixing_ratio)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[dimensionless]')
  def relative_humidity_from_specific_humidity(pressure, temperature, specific_humidity):
      r"""Calculate the relative humidity from specific humidity, temperature, and pressure.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          Air temperature
      specific_humidity: `pint.Quantity`
          Specific humidity of air
  
      Returns
      -------
      `pint.Quantity`
          Relative humidity
  
      Notes
      -----
      Formula based on that from [Hobbs1977]_ pg. 74. and [Salby1996]_ pg. 118.
  
      .. math:: relative_humidity = \frac{q}{(1-q)w_s}
  
      * :math:`relative_humidity` is relative humidity as a unitless ratio
      * :math:`q` is specific humidity
      * :math:`w_s` is the saturation mixing ratio
  
      See Also
      --------
      relative_humidity_from_mixing_ratio
  
      """
      return (mixing_ratio_from_specific_humidity(specific_humidity)
              / saturation_mixing_ratio(pressure, temperature))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]', '[temperature]')
  def cape_cin(pressure, temperature, dewpoint, parcel_profile, which_lfc='bottom',
               which_el='top'):
      r"""Calculate CAPE and CIN.
  
      Calculate the convective available potential energy (CAPE) and convective inhibition (CIN)
      of a given upper air profile and parcel path. CIN is integrated between the surface and
      LFC, CAPE is integrated between the LFC and EL (or top of sounding). Intersection points
      of the measured temperature profile and parcel profile are logarithmically interpolated.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The atmospheric pressure level(s) of interest, in order from highest to
          lowest pressure.
      temperature : `pint.Quantity`
          The atmospheric temperature corresponding to pressure.
      dewpoint : `pint.Quantity`
          The atmospheric dewpoint corresponding to pressure.
      parcel_profile : `pint.Quantity`
          The temperature profile of the parcel.
      which_lfc : str
          Choose which LFC to integrate from. Valid options are 'top', 'bottom', 'wide',
          and 'most_cape'. Default is 'bottom'.
      which_el : str
          Choose which EL to integrate to. Valid options are 'top', 'bottom', 'wide',
          and 'most_cape'. Default is 'top'.
  
      Returns
      -------
      `pint.Quantity`
          Convective Available Potential Energy (CAPE).
      `pint.Quantity`
          Convective INhibition (CIN).
  
      Notes
      -----
      Formula adopted from [Hobbs1977]_.
  
      .. math:: \text{CAPE} = -R_d \int_{LFC}^{EL} (T_{parcel} - T_{env}) d\text{ln}(p)
  
      .. math:: \text{CIN} = -R_d \int_{SFC}^{LFC} (T_{parcel} - T_{env}) d\text{ln}(p)
  
  
      * :math:`CAPE` Convective available potential energy
      * :math:`CIN` Convective inhibition
      * :math:`LFC` Pressure of the level of free convection
      * :math:`EL` Pressure of the equilibrium level
      * :math:`SFC` Level of the surface or beginning of parcel path
      * :math:`R_d` Gas constant
      * :math:`g` Gravitational acceleration
      * :math:`T_{parcel}` Parcel temperature
      * :math:`T_{env}` Environment temperature
      * :math:`p` Atmospheric pressure
  
      See Also
      --------
      lfc, el
  
      """
      pressure, temperature, dewpoint, parcel_profile = _remove_nans(pressure, temperature,
                                                                     dewpoint, parcel_profile)
      # Calculate LFC limit of integration
      lfc_pressure, _ = lfc(pressure, temperature, dewpoint,
                            parcel_temperature_profile=parcel_profile, which=which_lfc)
  
      # If there is no LFC, no need to proceed.
      if np.isnan(lfc_pressure):
          return 0 * units('J/kg'), 0 * units('J/kg')
      else:
          lfc_pressure = lfc_pressure.magnitude
  
      # Calculate the EL limit of integration
      el_pressure, _ = el(pressure, temperature, dewpoint,
                          parcel_temperature_profile=parcel_profile, which=which_el)
  
      # No EL and we use the top reading of the sounding.
      if np.isnan(el_pressure):
          el_pressure = pressure[-1].magnitude
      else:
          el_pressure = el_pressure.magnitude
  
      # Difference between the parcel path and measured temperature profiles
      y = (parcel_profile - temperature).to(units.degK)
  
      # Estimate zero crossings
      x, y = _find_append_zero_crossings(np.copy(pressure), y)
  
      # CAPE
      # Only use data between the LFC and EL for calculation
      p_mask = _less_or_close(x.m, lfc_pressure) & _greater_or_close(x.m, el_pressure)
      x_clipped = x[p_mask].magnitude
      y_clipped = y[p_mask].magnitude
      cape = (mpconsts.Rd
              * (np.trapz(y_clipped, np.log(x_clipped)) * units.degK)).to(units('J/kg'))
  
      # CIN
      # Only use data between the surface and LFC for calculation
      p_mask = _greater_or_close(x.m, lfc_pressure)
      x_clipped = x[p_mask].magnitude
      y_clipped = y[p_mask].magnitude
      cin = (mpconsts.Rd
             * (np.trapz(y_clipped, np.log(x_clipped)) * units.degK)).to(units('J/kg'))
  
      # Set CIN to 0 if it's returned as a positive value (#1190)
      if cin > 0 * units('J/kg'):
          cin = 0 * units('J/kg')
      return cape, cin
  
  
  def _find_append_zero_crossings(x, y):
      r"""
      Find and interpolate zero crossings.
  
      Estimate the zero crossings of an x,y series and add estimated crossings to series,
      returning a sorted array with no duplicate values.
  
      Parameters
      ----------
      x : `pint.Quantity`
          x values of data
      y : `pint.Quantity`
          y values of data
  
      Returns
      -------
      x : `pint.Quantity`
          x values of data
      y : `pint.Quantity`
          y values of data
  
      """
      crossings = find_intersections(x[1:], y[1:], np.zeros_like(y[1:]) * y.units, log_x=True)
      x = concatenate((x, crossings[0]))
      y = concatenate((y, crossings[1]))
  
      # Resort so that data are in order
      sort_idx = np.argsort(x)
      x = x[sort_idx]
      y = y[sort_idx]
  
      # Remove duplicate data points if there are any
      keep_idx = np.ediff1d(x.magnitude, to_end=[1]) > 1e-6
      x = x[keep_idx]
      y = y[keep_idx]
      return x, y
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def most_unstable_parcel(pressure, temperature, dewpoint, height=None,
                           bottom=None, depth=300 * units.hPa):
      """
      Determine the most unstable parcel in a layer.
  
      Determines the most unstable parcel of air by calculating the equivalent
      potential temperature and finding its maximum in the specified layer.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Atmospheric pressure profile
      temperature: `pint.Quantity`
          Atmospheric temperature profile
      dewpoint: `pint.Quantity`
          Atmospheric dewpoint profile
      height: `pint.Quantity`, optional
          Atmospheric height profile. Standard atmosphere assumed when None (the default).
      bottom: `pint.Quantity`, optional
          Bottom of the layer to consider for the calculation in pressure or height.
          Defaults to using the bottom pressure or height.
      depth: `pint.Quantity`, optional
          Depth of the layer to consider for the calculation in pressure or height. Defaults
          to 300 hPa.
  
      Returns
      -------
      `pint.Quantity`
          Pressure, temperature, and dewpoint of most unstable parcel in the profile.
      integer
          Index of the most unstable parcel in the given profile
  
      See Also
      --------
      get_layer
  
      """
      p_layer, t_layer, td_layer = get_layer(pressure, temperature, dewpoint, bottom=bottom,
                                             depth=depth, height=height, interpolate=False)
      theta_e = equivalent_potential_temperature(p_layer, t_layer, td_layer)
      max_idx = np.argmax(theta_e)
      return p_layer[max_idx], t_layer[max_idx], td_layer[max_idx], max_idx
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[temperature]', '[pressure]', '[temperature]')
  def isentropic_interpolation(levels, pressure, temperature, *args, axis=0,
                               temperature_out=False, max_iters=50, eps=1e-6,
                               bottom_up_search=True, **kwargs):
      r"""Interpolate data in isobaric coordinates to isentropic coordinates.
  
      Parameters
      ----------
      levels : array
          One-dimensional array of desired potential temperature surfaces
      pressure : array
          One-dimensional array of pressure levels
      temperature : array
          Array of temperature
      axis : int, optional
          The axis corresponding to the vertical in the temperature array, defaults to 0.
      temperature_out : bool, optional
          If true, will calculate temperature and output as the last item in the output list.
          Defaults to False.
      max_iters : int, optional
          The maximum number of iterations to use in calculation, defaults to 50.
      eps : float, optional
          The desired absolute error in the calculated value, defaults to 1e-6.
      bottom_up_search : bool, optional
          Controls whether to search for levels bottom-up, or top-down. Defaults to
          True, which is bottom-up search.
      args : array, optional
          Any additional variables will be interpolated to each isentropic level.
  
      Returns
      -------
      list
          List with pressure at each isentropic level, followed by each additional
          argument interpolated to isentropic coordinates.
  
      Notes
      -----
      Input variable arrays must have the same number of vertical levels as the pressure levels
      array. Pressure is calculated on isentropic surfaces by assuming that temperature varies
      linearly with the natural log of pressure. Linear interpolation is then used in the
      vertical to find the pressure at each isentropic level. Interpolation method from
      [Ziv1994]_. Any additional arguments are assumed to vary linearly with temperature and will
      be linearly interpolated to the new isentropic levels.
  
      See Also
      --------
      potential_temperature
  
      """
      # iteration function to be used later
      # Calculates theta from linearly interpolated temperature and solves for pressure
      def _isen_iter(iter_log_p, isentlevs_nd, ka, a, b, pok):
          exner = pok * np.exp(-ka * iter_log_p)
          t = a * iter_log_p + b
          # Newton-Raphson iteration
          f = isentlevs_nd - t * exner
          fp = exner * (ka * t - a)
          return iter_log_p - (f / fp)
  
      # Get dimensions in temperature
      ndim = temperature.ndim
  
      # Convert units
      pres = pressure.to('hPa')
      temperature = temperature.to('kelvin')
  
      slices = [np.newaxis] * ndim
      slices[axis] = slice(None)
      slices = tuple(slices)
      pres = np.broadcast_to(pres[slices].magnitude, temperature.shape) * pres.units
  
      # Sort input data
      sort_pres = np.argsort(pres.m, axis=axis)
      sort_pres = np.swapaxes(np.swapaxes(sort_pres, 0, axis)[::-1], 0, axis)
      sorter = broadcast_indices(pres, sort_pres, ndim, axis)
      levs = pres[sorter]
      tmpk = temperature[sorter]
  
      levels = np.asarray(levels.m_as('kelvin')).reshape(-1)
      isentlevels = levels[np.argsort(levels)]
  
      # Make the desired isentropic levels the same shape as temperature
      shape = list(temperature.shape)
      shape[axis] = isentlevels.size
      isentlevs_nd = np.broadcast_to(isentlevels[slices], shape)
  
      # exponent to Poisson's Equation, which is imported above
      ka = mpconsts.kappa.m_as('dimensionless')
  
      # calculate theta for each point
      pres_theta = potential_temperature(levs, tmpk)
  
      # Raise error if input theta level is larger than pres_theta max
      if np.max(pres_theta.m) < np.max(levels):
          raise ValueError('Input theta level out of data bounds')
  
      # Find log of pressure to implement assumption of linear temperature dependence on
      # ln(p)
      log_p = np.log(levs.m)
  
      # Calculations for interpolation routine
      pok = mpconsts.P0 ** ka
  
      # index values for each point for the pressure level nearest to the desired theta level
      above, below, good = find_bounding_indices(pres_theta.m, levels, axis,
                                                 from_below=bottom_up_search)
  
      # calculate constants for the interpolation
      a = (tmpk.m[above] - tmpk.m[below]) / (log_p[above] - log_p[below])
      b = tmpk.m[above] - a * log_p[above]
  
      # calculate first guess for interpolation
      isentprs = 0.5 * (log_p[above] + log_p[below])
  
      # Make sure we ignore any nans in the data for solving; checking a is enough since it
      # combines log_p and tmpk.
      good &= ~np.isnan(a)
  
      # iterative interpolation using scipy.optimize.fixed_point and _isen_iter defined above
      log_p_solved = so.fixed_point(_isen_iter, isentprs[good],
                                    args=(isentlevs_nd[good], ka, a[good], b[good], pok.m),
                                    xtol=eps, maxiter=max_iters)
  
      # get back pressure from log p
      isentprs[good] = np.exp(log_p_solved)
  
      # Mask out points we know are bad as well as points that are beyond the max pressure
      isentprs[~(good & _less_or_close(isentprs, np.max(pres.m)))] = np.nan
  
      # create list for storing output data
      ret = [isentprs * units.hPa]
  
      # if temperature_out = true, calculate temperature and output as last item in list
      if temperature_out:
          ret.append((isentlevs_nd / ((mpconsts.P0.m / isentprs) ** ka)) * units.kelvin)
  
      # do an interpolation for each additional argument
      if args:
          others = interpolate_1d(isentlevels, pres_theta.m, *(arr[sorter] for arr in args),
                                  axis=axis, return_list_always=True)
          ret.extend(others)
  
      return ret
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def surface_based_cape_cin(pressure, temperature, dewpoint):
      r"""Calculate surface-based CAPE and CIN.
  
      Calculate the convective available potential energy (CAPE) and convective inhibition (CIN)
      of a given upper air profile for a surface-based parcel. CIN is integrated
      between the surface and LFC, CAPE is integrated between the LFC and EL (or top of
      sounding). Intersection points of the measured temperature profile and parcel profile are
      logarithmically interpolated.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          Atmospheric pressure profile. The first entry should be the starting
          (surface) observation, with the array going from high to low pressure.
      temperature : `pint.Quantity`
          Temperature profile corresponding to the `pressure` profile.
      dewpoint : `pint.Quantity`
          Dewpoint profile corresponding to the `pressure` profile.
  
      Returns
      -------
      `pint.Quantity`
          Surface based Convective Available Potential Energy (CAPE).
      `pint.Quantity`
          Surface based Convective INhibition (CIN).
  
      See Also
      --------
      cape_cin, parcel_profile
  
      """
      pressure, temperature, dewpoint = _remove_nans(pressure, temperature, dewpoint)
      p, t, td, profile = parcel_profile_with_lcl(pressure, temperature, dewpoint)
      return cape_cin(p, t, td, profile)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def most_unstable_cape_cin(pressure, temperature, dewpoint, **kwargs):
      r"""Calculate most unstable CAPE/CIN.
  
      Calculate the convective available potential energy (CAPE) and convective inhibition (CIN)
      of a given upper air profile and most unstable parcel path. CIN is integrated between the
      surface and LFC, CAPE is integrated between the LFC and EL (or top of sounding).
      Intersection points of the measured temperature profile and parcel profile are
      logarithmically interpolated.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          Pressure profile
      temperature : `pint.Quantity`
          Temperature profile
      dewpoint : `pint.Quantity`
          Dew point profile
      kwargs
          Additional keyword arguments to pass to `most_unstable_parcel`
  
      Returns
      -------
      `pint.Quantity`
          Most unstable Convective Available Potential Energy (CAPE).
      `pint.Quantity`
          Most unstable Convective INhibition (CIN).
  
      See Also
      --------
      cape_cin, most_unstable_parcel, parcel_profile
  
      """
      pressure, temperature, dewpoint = _remove_nans(pressure, temperature, dewpoint)
      _, _, _, parcel_idx = most_unstable_parcel(pressure, temperature, dewpoint, **kwargs)
      p, t, td, mu_profile = parcel_profile_with_lcl(pressure[parcel_idx:],
                                                     temperature[parcel_idx:],
                                                     dewpoint[parcel_idx:])
      return cape_cin(p, t, td, mu_profile)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def mixed_layer_cape_cin(pressure, temperature, dewpoint, **kwargs):
      r"""Calculate mixed-layer CAPE and CIN.
  
      Calculate the convective available potential energy (CAPE) and convective inhibition (CIN)
      of a given upper air profile and mixed-layer parcel path. CIN is integrated between the
      surface and LFC, CAPE is integrated between the LFC and EL (or top of sounding).
      Intersection points of the measured temperature profile and parcel profile are
      logarithmically interpolated. Kwargs for `mixed_parcel` can be provided, such as `depth`.
      Default mixed-layer depth is 100 hPa.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          Pressure profile
      temperature : `pint.Quantity`
          Temperature profile
      dewpoint : `pint.Quantity`
          Dewpoint profile
      kwargs
          Additional keyword arguments to pass to `mixed_parcel`
  
      Returns
      -------
      `pint.Quantity`
          Mixed-layer Convective Available Potential Energy (CAPE).
      `pint.Quantity`
          Mixed-layer Convective INhibition (CIN).
  
      See Also
      --------
      cape_cin, mixed_parcel, parcel_profile
      """
      depth = kwargs.get('depth', 100 * units.hPa)
      parcel_pressure, parcel_temp, parcel_dewpoint = mixed_parcel(pressure, temperature,
                                                                   dewpoint, **kwargs)
  
      # Remove values below top of mixed layer and add in the mixed layer values
      pressure_prof = pressure[pressure < (pressure[0] - depth)]
      temp_prof = temperature[pressure < (pressure[0] - depth)]
      dew_prof = dewpoint[pressure < (pressure[0] - depth)]
      pressure_prof = concatenate([parcel_pressure, pressure_prof])
      temp_prof = concatenate([parcel_temp, temp_prof])
      dew_prof = concatenate([parcel_dewpoint, dew_prof])
  
      p, t, td, ml_profile = parcel_profile_with_lcl(pressure_prof, temp_prof, dew_prof)
      return cape_cin(p, t, td, ml_profile)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def mixed_parcel(pressure, temperature, dewpoint, parcel_start_pressure=None,
                   height=None, bottom=None, depth=100 * units.hPa, interpolate=True):
      r"""Calculate the properties of a parcel mixed from a layer.
  
      Determines the properties of an air parcel that is the result of complete mixing of a
      given atmospheric layer.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          Atmospheric pressure profile
      temperature : `pint.Quantity`
          Atmospheric temperature profile
      dewpoint : `pint.Quantity`
          Atmospheric dewpoint profile
      parcel_start_pressure : `pint.Quantity`, optional
          Pressure at which the mixed parcel should begin (default None)
      height: `pint.Quantity`, optional
          Atmospheric heights corresponding to the given pressures (default None)
      bottom : `pint.Quantity`, optional
          The bottom of the layer as a pressure or height above the surface pressure
          (default None)
      depth : `pint.Quantity`, optional
          The thickness of the layer as a pressure or height above the bottom of the layer
          (default 100 hPa)
      interpolate : bool, optional
          Interpolate the top and bottom points if they are not in the given data
  
      Returns
      -------
      `pint.Quantity`
          The pressure of the mixed parcel
      `pint.Quantity`
          The temperature of the mixed parcel
      `pint.Quantity`
          The dewpoint of the mixed parcel
  
      """
      # If a parcel starting pressure is not provided, use the surface
      if not parcel_start_pressure:
          parcel_start_pressure = pressure[0]
  
      # Calculate the potential temperature and mixing ratio over the layer
      theta = potential_temperature(pressure, temperature)
      mixing_ratio = saturation_mixing_ratio(pressure, dewpoint)
  
      # Mix the variables over the layer
      mean_theta, mean_mixing_ratio = mixed_layer(pressure, theta, mixing_ratio, bottom=bottom,
                                                  height=height, depth=depth,
                                                  interpolate=interpolate)
  
      # Convert back to temperature
      mean_temperature = mean_theta * exner_function(parcel_start_pressure)
  
      # Convert back to dewpoint
      mean_vapor_pressure = vapor_pressure(parcel_start_pressure, mean_mixing_ratio)
  
      # Using globals() here allows us to keep the dewpoint parameter but still call the
      # function of the same name.
      mean_dewpoint = globals()['dewpoint'](mean_vapor_pressure)
  
      return (parcel_start_pressure, mean_temperature.to(temperature.units),
              mean_dewpoint.to(dewpoint.units))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]')
  def mixed_layer(pressure, *args, height=None, bottom=None, depth=100 * units.hPa,
                  interpolate=True):
      r"""Mix variable(s) over a layer, yielding a mass-weighted average.
  
      This function will integrate a data variable with respect to pressure and determine the
      average value using the mean value theorem.
  
      Parameters
      ----------
      pressure : array-like
          Atmospheric pressure profile
      datavar : array-like
          Atmospheric variable measured at the given pressures
      height: array-like, optional
          Atmospheric heights corresponding to the given pressures (default None)
      bottom : `pint.Quantity`, optional
          The bottom of the layer as a pressure or height above the surface pressure
          (default None)
      depth : `pint.Quantity`, optional
          The thickness of the layer as a pressure or height above the bottom of the layer
          (default 100 hPa)
      interpolate : bool, optional
          Interpolate the top and bottom points if they are not in the given data (default True)
  
      Returns
      -------
      `pint.Quantity`
          The mixed value of the data variable.
  
      """
      layer = get_layer(pressure, *args, height=height, bottom=bottom,
                        depth=depth, interpolate=interpolate)
      p_layer = layer[0]
      datavars_layer = layer[1:]
  
      ret = []
      for datavar_layer in datavars_layer:
          actual_depth = abs(p_layer[0] - p_layer[-1])
          ret.append((-1. / actual_depth.m) * np.trapz(datavar_layer.m, p_layer.m)
                     * datavar_layer.units)
      return ret
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[length]', '[temperature]')
  def dry_static_energy(height, temperature):
      r"""Calculate the dry static energy of parcels.
  
      This function will calculate the dry static energy following the first two terms of
      equation 3.72 in [Hobbs2006]_.
  
      Notes
      -----
      .. math::\text{dry static energy} = c_{pd} * T + gz
  
      * :math:`T` is temperature
      * :math:`z` is height
  
      Parameters
      ----------
      height : `pint.Quantity`
          Atmospheric height
      temperature : `pint.Quantity`
          Air temperature
  
      Returns
      -------
      `pint.Quantity`
          The dry static energy
  
      """
      return (mpconsts.g * height + mpconsts.Cp_d * temperature).to('kJ/kg')
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[length]', '[temperature]', '[dimensionless]')
  def moist_static_energy(height, temperature, specific_humidity):
      r"""Calculate the moist static energy of parcels.
  
      This function will calculate the moist static energy following
      equation 3.72 in [Hobbs2006]_.
  
      Notes
      -----
      .. math::\text{moist static energy} = c_{pd} * T + gz + L_v q
  
      * :math:`T` is temperature
      * :math:`z` is height
      * :math:`q` is specific humidity
  
      Parameters
      ----------
      height : `pint.Quantity`
          Atmospheric height
      temperature : `pint.Quantity`
          Air temperature
      specific_humidity : `pint.Quantity`
          Atmospheric specific humidity
  
      Returns
      -------
      `pint.Quantity`
          The moist static energy
  
      """
      return (dry_static_energy(height, temperature)
              + mpconsts.Lv * specific_humidity.to('dimensionless')).to('kJ/kg')
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]')
  def thickness_hydrostatic(pressure, temperature, mixing_ratio=None,
                            molecular_weight_ratio=mpconsts.epsilon, bottom=None, depth=None):
      r"""Calculate the thickness of a layer via the hypsometric equation.
  
      This thickness calculation uses the pressure and temperature profiles (and optionally
      mixing ratio) via the hypsometric equation with virtual temperature adjustment
  
      .. math:: Z_2 - Z_1 = -\frac{R_d}{g} \int_{p_1}^{p_2} T_v d\ln p,
  
      which is based off of Equation 3.24 in [Hobbs2006]_.
  
      This assumes a hydrostatic atmosphere.
  
      Layer bottom and depth specified in pressure.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          Atmospheric pressure profile
      temperature : `pint.Quantity`
          Atmospheric temperature profile
      mixing_ratio : `pint.Quantity`, optional
          Profile of dimensionless mass mixing ratio. If none is given, virtual temperature
          is simply set to be the given temperature.
      molecular_weight_ratio : `pint.Quantity` or float, optional
          The ratio of the molecular weight of the constituent gas to that assumed
          for air. Defaults to the ratio for water vapor to dry air.
          (:math:`\epsilon\approx0.622`).
      bottom : `pint.Quantity`, optional
          The bottom of the layer in pressure. Defaults to the first observation.
      depth : `pint.Quantity`, optional
          The depth of the layer in hPa. Defaults to the full profile if bottom is not given,
          and 100 hPa if bottom is given.
  
      Returns
      -------
      `pint.Quantity`
          The thickness of the layer in meters.
  
      See Also
      --------
      thickness_hydrostatic_from_relative_humidity, pressure_to_height_std, virtual_temperature
  
      """
      # Get the data for the layer, conditional upon bottom/depth being specified and mixing
      # ratio being given
      if bottom is None and depth is None:
          if mixing_ratio is None:
              layer_p, layer_virttemp = pressure, temperature
          else:
              layer_p = pressure
              layer_virttemp = virtual_temperature(temperature, mixing_ratio,
                                                   molecular_weight_ratio)
      else:
          if mixing_ratio is None:
              layer_p, layer_virttemp = get_layer(pressure, temperature, bottom=bottom,
                                                  depth=depth)
          else:
              layer_p, layer_temp, layer_w = get_layer(pressure, temperature, mixing_ratio,
                                                       bottom=bottom, depth=depth)
              layer_virttemp = virtual_temperature(layer_temp, layer_w, molecular_weight_ratio)
  
      # Take the integral (with unit handling) and return the result in meters
      return (- mpconsts.Rd / mpconsts.g * np.trapz(
          layer_virttemp.m_as('K'), x=np.log(layer_p.m_as('hPa'))) * units.K).to('m')
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]')
  def thickness_hydrostatic_from_relative_humidity(pressure, temperature, relative_humidity,
                                                   bottom=None, depth=None):
      r"""Calculate the thickness of a layer given pressure, temperature and relative humidity.
  
      Similar to ``thickness_hydrostatic``, this thickness calculation uses the pressure,
      temperature, and relative humidity profiles via the hypsometric equation with virtual
      temperature adjustment.
  
      .. math:: Z_2 - Z_1 = -\frac{R_d}{g} \int_{p_1}^{p_2} T_v d\ln p,
  
      which is based off of Equation 3.24 in [Hobbs2006]_. Virtual temperature is calculated
      from the profiles of temperature and relative humidity.
  
      This assumes a hydrostatic atmosphere.
  
      Layer bottom and depth specified in pressure.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          Atmospheric pressure profile
      temperature : `pint.Quantity`
          Atmospheric temperature profile
      relative_humidity : `pint.Quantity`
          Atmospheric relative humidity profile. The relative humidity is expressed as a
          unitless ratio in the range [0, 1]. Can also pass a percentage if proper units are
          attached.
      bottom : `pint.Quantity`, optional
          The bottom of the layer in pressure. Defaults to the first observation.
      depth : `pint.Quantity`, optional
          The depth of the layer in hPa. Defaults to the full profile if bottom is not given,
          and 100 hPa if bottom is given.
  
      Returns
      -------
      `pint.Quantity`
          The thickness of the layer in meters.
  
      See Also
      --------
      thickness_hydrostatic, pressure_to_height_std, virtual_temperature,
      mixing_ratio_from_relative_humidity
  
      """
      mixing = mixing_ratio_from_relative_humidity(pressure, temperature, relative_humidity)
  
      return thickness_hydrostatic(pressure, temperature, mixing_ratio=mixing, bottom=bottom,
                                   depth=depth)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[length]', '[temperature]')
  def brunt_vaisala_frequency_squared(height, potential_temperature, axis=0):
      r"""Calculate the square of the Brunt-Vaisala frequency.
  
      Brunt-Vaisala frequency squared (a measure of atmospheric stability) is given by the
      formula:
  
      .. math:: N^2 = \frac{g}{\theta} \frac{d\theta}{dz}
  
      This formula is based off of Equations 3.75 and 3.77 in [Hobbs2006]_.
  
      Parameters
      ----------
      height : `pint.Quantity`
          One-dimensional profile of atmospheric height
      potential_temperature : `pint.Quantity`
          Atmospheric potential temperature
      axis : int, optional
          The axis corresponding to vertical in the potential temperature array, defaults to 0.
  
      Returns
      -------
      `pint.Quantity`
          The square of the Brunt-Vaisala frequency.
  
      See Also
      --------
      brunt_vaisala_frequency, brunt_vaisala_period, potential_temperature
  
      """
      # Ensure validity of temperature units
      potential_temperature = potential_temperature.to('K')
  
      # Calculate and return the square of Brunt-Vaisala frequency
      return mpconsts.g / potential_temperature * first_derivative(potential_temperature,
                                                                   x=height, axis=axis)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[length]', '[temperature]')
  def brunt_vaisala_frequency(height, potential_temperature, axis=0):
      r"""Calculate the Brunt-Vaisala frequency.
  
      This function will calculate the Brunt-Vaisala frequency as follows:
  
      .. math:: N = \left( \frac{g}{\theta} \frac{d\theta}{dz} \right)^\frac{1}{2}
  
      This formula based off of Equations 3.75 and 3.77 in [Hobbs2006]_.
  
      This function is a wrapper for `brunt_vaisala_frequency_squared` that filters out negative
      (unstable) quantities and takes the square root.
  
      Parameters
      ----------
      height : `pint.Quantity`
          One-dimensional profile of atmospheric height
      potential_temperature : `pint.Quantity`
          Atmospheric potential temperature
      axis : int, optional
          The axis corresponding to vertical in the potential temperature array, defaults to 0.
  
      Returns
      -------
      `pint.Quantity`
          Brunt-Vaisala frequency.
  
      See Also
      --------
      brunt_vaisala_frequency_squared, brunt_vaisala_period, potential_temperature
  
      """
      bv_freq_squared = brunt_vaisala_frequency_squared(height, potential_temperature,
                                                        axis=axis)
      bv_freq_squared[bv_freq_squared.magnitude < 0] = np.nan
  
      return np.sqrt(bv_freq_squared)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[length]', '[temperature]')
  def brunt_vaisala_period(height, potential_temperature, axis=0):
      r"""Calculate the Brunt-Vaisala period.
  
      This function is a helper function for `brunt_vaisala_frequency` that calculates the
      period of oscillation as in Exercise 3.13 of [Hobbs2006]_:
  
      .. math:: \tau = \frac{2\pi}{N}
  
      Returns `NaN` when :math:`N^2 > 0`.
  
      Parameters
      ----------
      height : `pint.Quantity`
          One-dimensional profile of atmospheric height
      potential_temperature : pint.Quantity`
          Atmospheric potential temperature
      axis : int, optional
          The axis corresponding to vertical in the potential temperature array, defaults to 0.
  
      Returns
      -------
      `pint.Quantity`
          Brunt-Vaisala period.
  
      See Also
      --------
      brunt_vaisala_frequency, brunt_vaisala_frequency_squared, potential_temperature
  
      """
      bv_freq_squared = brunt_vaisala_frequency_squared(height, potential_temperature,
                                                        axis=axis)
      bv_freq_squared[bv_freq_squared.magnitude <= 0] = np.nan
  
      return 2 * np.pi / np.sqrt(bv_freq_squared)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def wet_bulb_temperature(pressure, temperature, dewpoint):
      """Calculate the wet-bulb temperature using Normand's rule.
  
      This function calculates the wet-bulb temperature using the Normand method. The LCL is
      computed, and that parcel brought down to the starting pressure along a moist adiabat.
      The Normand method (and others) are described and compared by [Knox2017]_.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          Initial atmospheric pressure
      temperature : `pint.Quantity`
          Initial atmospheric temperature
      dewpoint : `pint.Quantity`
          Initial atmospheric dewpoint
  
      Returns
      -------
      `pint.Quantity`
          Wet-bulb temperature
  
      See Also
      --------
      lcl, moist_lapse
  
      """
      if not hasattr(pressure, 'shape'):
          pressure = np.atleast_1d(pressure)
          temperature = np.atleast_1d(temperature)
          dewpoint = np.atleast_1d(dewpoint)
  
      it = np.nditer([pressure, temperature, dewpoint, None],
                     op_dtypes=['float', 'float', 'float', 'float'],
                     flags=['buffered'])
  
      for press, temp, dewp, ret in it:
          press = press * pressure.units
          temp = temp * temperature.units
          dewp = dewp * dewpoint.units
          lcl_pressure, lcl_temperature = lcl(press, temp, dewp)
          moist_adiabat_temperatures = moist_lapse(concatenate([lcl_pressure, press]),
                                                   lcl_temperature)
          ret[...] = moist_adiabat_temperatures[-1].magnitude
  
      # If we started with a scalar, return a scalar
      if it.operands[3].size == 1:
          return it.operands[3][0] * moist_adiabat_temperatures.units
      return it.operands[3] * moist_adiabat_temperatures.units
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]')
  def static_stability(pressure, temperature, axis=0):
      r"""Calculate the static stability within a vertical profile.
  
      .. math:: \sigma = -\frac{RT}{p} \frac{\partial \ln \theta}{\partial p}
  
      This formula is based on equation 4.3.6 in [Bluestein1992]_.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          Profile of atmospheric pressure
      temperature : `pint.Quantity`
          Profile of temperature
      axis : int, optional
          The axis corresponding to vertical in the pressure and temperature arrays, defaults
          to 0.
  
      Returns
      -------
      `pint.Quantity`
          The profile of static stability.
  
      """
      theta = potential_temperature(pressure, temperature)
  
      return - mpconsts.Rd * temperature / pressure * first_derivative(np.log(theta.m_as('K')),
                                                                       x=pressure, axis=axis)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[dimensionless]')
  def dewpoint_from_specific_humidity(pressure, temperature, specific_humidity):
      r"""Calculate the dewpoint from specific humidity, temperature, and pressure.
  
      Parameters
      ----------
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          Air temperature
      specific_humidity: `pint.Quantity`
          Specific humidity of air
  
      Returns
      -------
      `pint.Quantity`
          Dew point temperature
  
      See Also
      --------
      relative_humidity_from_mixing_ratio, dewpoint_from_relative_humidity
  
      """
      return dewpoint_from_relative_humidity(temperature,
                                             relative_humidity_from_specific_humidity(
                                                 pressure, temperature, specific_humidity))
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[length]/[time]', '[pressure]', '[temperature]')
  def vertical_velocity_pressure(w, pressure, temperature, mixing_ratio=0):
      r"""Calculate omega from w assuming hydrostatic conditions.
  
      This function converts vertical velocity with respect to height
      :math:`\left(w = \frac{Dz}{Dt}\right)` to that
      with respect to pressure :math:`\left(\omega = \frac{Dp}{Dt}\right)`
      assuming hydrostatic conditions on the synoptic scale.
      By Equation 7.33 in [Hobbs2006]_,
  
      .. math:: \omega \simeq -\rho g w
  
      Density (:math:`\rho`) is calculated using the :func:`density` function,
      from the given pressure and temperature. If `mixing_ratio` is given, the virtual
      temperature correction is used, otherwise, dry air is assumed.
  
      Parameters
      ----------
      w: `pint.Quantity`
          Vertical velocity in terms of height
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          Air temperature
      mixing_ratio: `pint.Quantity`, optional
          Mixing_ratio ratio of air
  
      Returns
      -------
      `pint.Quantity`
          Vertical velocity in terms of pressure (in Pascals / second)
  
      See Also
      --------
      density, vertical_velocity
  
      """
      rho = density(pressure, temperature, mixing_ratio)
      return (-mpconsts.g * rho * w).to('Pa/s')
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]/[time]', '[pressure]', '[temperature]')
  def vertical_velocity(omega, pressure, temperature, mixing_ratio=0):
      r"""Calculate w from omega assuming hydrostatic conditions.
  
      This function converts vertical velocity with respect to pressure
      :math:`\left(\omega = \frac{Dp}{Dt}\right)` to that with respect to height
      :math:`\left(w = \frac{Dz}{Dt}\right)` assuming hydrostatic conditions on
      the synoptic scale. By Equation 7.33 in [Hobbs2006]_,
  
      .. math:: \omega \simeq -\rho g w
  
      so that
  
      .. math:: w \simeq \frac{- \omega}{\rho g}
  
      Density (:math:`\rho`) is calculated using the :func:`density` function,
      from the given pressure and temperature. If `mixing_ratio` is given, the virtual
      temperature correction is used, otherwise, dry air is assumed.
  
      Parameters
      ----------
      omega: `pint.Quantity`
          Vertical velocity in terms of pressure
      pressure: `pint.Quantity`
          Total atmospheric pressure
      temperature: `pint.Quantity`
          Air temperature
      mixing_ratio: `pint.Quantity`, optional
          Mixing ratio of air
  
      Returns
      -------
      `pint.Quantity`
          Vertical velocity in terms of height (in meters / second)
  
      See Also
      --------
      density, vertical_velocity_pressure
  
      """
      rho = density(pressure, temperature, mixing_ratio)
      return (omega / (- mpconsts.g * rho)).to('m/s')
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]')
  def specific_humidity_from_dewpoint(pressure, dewpoint):
      r"""Calculate the specific humidity from the dewpoint temperature and pressure.
  
      Parameters
      ----------
      dewpoint: `pint.Quantity`
          dewpoint temperature
  
      pressure: `pint.Quantity`
          pressure
  
      Returns
      -------
      `pint.Quantity`
          Specific humidity
  
      See Also
      --------
      mixing_ratio, saturation_mixing_ratio
  
      """
      mixing_ratio = saturation_mixing_ratio(pressure, dewpoint)
      return specific_humidity_from_mixing_ratio(mixing_ratio)
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[pressure]', '[temperature]', '[temperature]')
  def lifted_index(pressure, temperature, parcel_profile):
      """Calculate Lifted Index from the pressure temperature and parcel profile.
  
      Lifted index formula derived from [Galway1956]_ and referenced by [Doswell-Schultz2006]_:
      LI = T500 - Tp500
      where:
      T500 is the measured temperature at 500 hPa.
      Tp500 is the temperature of the lifted parcel at 500 hPa.
  
      Calculation of the lifted index is defined as the temperature difference between the
      observed 500 hPa temperature and the temperature of a parcel lifted from the
      surface to 500 hPa.
  
      Parameters
      ----------
      pressure : `pint.Quantity`
          The atmospheric pressure level(s) of interest, in order from highest to
          lowest pressure.
      temperature : `pint.Quantity`
          The atmospheric temperature corresponding to pressure.
      parcel_profile : `pint.Quantity`
          The temperature profile of the parcel.
  
      Returns
      -------
      `pint.Quantity`
          Lifted Index.
  
      """
      # find the index for the 500 hPa pressure level.
      idx = np.where(pressure == 500 * units.hPa)
      # find the measured temperature at 500 hPa.
      T500 = temperature[idx]
      # find the parcel profile temperature at 500 hPa.
      Tp500 = parcel_profile[idx]
      # calculate the lifted index.
      lifted_index = T500 - Tp500.to(units.degC)
      return lifted_index
  
  
  @exporter.export
  @preprocess_xarray
  @check_units('[length]', '[temperature]', '[speed]', '[speed]')
  def gradient_richardson_number(height, potential_temperature, u, v, axis=0):
      r"""Calculate the gradient (or flux) Richardson number.
  
      .. math::   Ri = (g/\theta) * \frac{\left(\partial \theta/\partial z\)}
               {[\left(\partial u / \partial z\right)^2 + \left(\partial v / \partial z\right)^2}
  
      See [Holton2004]_ pg. 121-122. As noted by [Holton2004]_, flux Richardson
      number values below 0.25 indicate turbulence.
  
      Parameters
      ----------
      height : `pint.Quantity`
          Atmospheric height
      potential_temperature : `pint.Quantity`
          Atmospheric potential temperature
      u : `pint.Quantity`
          x component of the wind
      v : `pint.Quantity`
          y component of the wind
      axis : int, optional
          The axis corresponding to vertical, defaults to 0.
  
      Returns
      -------
      `pint.Quantity`
          Gradient Richardson number
      """
      dthetadz = first_derivative(potential_temperature, x=height, axis=axis)
      dudz = first_derivative(u, x=height, axis=axis)
      dvdz = first_derivative(v, x=height, axis=axis)
  
      return (mpconsts.g / potential_temperature) * (dthetadz / (dudz ** 2 + dvdz ** 2))
  
apis: numpy.any scipy.optimize.fixed_point numpy.argsort numpy.insert numpy.copy numpy.isclose numpy.log numpy.trapz numpy.ediff1d numpy.append numpy.isnan numpy.where numpy.searchsorted numpy.argmax numpy.max numpy.min numpy.broadcast_to numpy.array numpy.zeros_like numpy.empty numpy.swapaxes numpy.exp numpy.atleast_1d numpy.nditer numpy.nanmin numpy.sqrt numpy.concatenate