Lapse Theory

get_bulk_lapse_rate(temp1, temp2, p1, p2)

Compute the bulk environmental lapse rate, \(\Gamma\), between pressure p1 at environmental temperature temp1 and p2 at environmental temperature temp2:

\[\Gamma = \frac{g}{R}\ln\left(\frac{T_1}{T_2}\right)/\ln\left(\frac{p_1}{p_2}\right)\]

This equation assumes hydrostatic equilibrium, ideal gas equation of state and that \(\Gamma\) is constant between p1 and p2.

Parameters:

Name	Type	Description	Default
temp1	DataArray	Temperature at pressure p1. Units: K.	required
temp2	DataArray	Temperature at pressure p2. Units: K.	required
p1	Union[DataArray, float]	Pressure at environmental temperature temp1. Units: Pa.	required
p2	Union[DataArray, float]	Pressure at environmental temperature temp2. Units: Pa.	required

Returns:

Type	Description
	Bulk environmental lapse rate, positive if temp2<temp1 and p2<p1. Units are K/m.

Source code in isca_tools/thesis/lapse_theory.py

def get_bulk_lapse_rate(temp1: xr.DataArray, temp2: xr.DataArray, p1: Union[xr.DataArray, float],
                        p2: Union[xr.DataArray, float]):
    """
    Compute the bulk environmental lapse rate, $\Gamma$, between pressure `p1` at environmental temperature `temp1`
    and `p2` at environmental temperature `temp2`:

    $$\Gamma = \\frac{g}{R}\ln\\left(\\frac{T_1}{T_2}\\right)/\ln\\left(\\frac{p_1}{p_2}\\right)$$

    This equation assumes hydrostatic equilibrium, ideal gas equation of state and that $\Gamma$ is constant
    between `p1` and `p2`.

    Args:
        temp1: Temperature at pressure `p1`. Units: *K*.
        temp2: Temperature at pressure `p2`. Units: *K*.
        p1: Pressure at environmental temperature `temp1`. Units: *Pa*.
        p2: Pressure at environmental temperature `temp2`. Units: *Pa*.

    Returns:
        Bulk environmental lapse rate, positive if `temp2<temp1` and `p2<p1`. Units are *K/m*.
    """
    return g/R * np.log(temp1/temp2) / np.log(p1/p2)

interp_var_at_pressure(var, p_desired, p_surf, hyam, hybm, p0, plev_step=1000, extrapolate=False)

Function to get the value of variable var at the pressure p_desired, where p_desired is expected to be a different value at each lat and lon.

Parameters:

Name	Type	Description	Default
var	Union[DataArray, Dataset]	Variable to do interpolation of. Should have lev dimension as well as lat, lon and possibly time. If give Dataset, will do interpolation on all variables.	required
p_desired	DataArray	Desired pressure to find var at. Should have same dimension as var but no lev. Units: Pa.	required
p_surf	DataArray	Surface pressure. Should have same dimension as var but no lev. Units: Pa.	required
hyam	DataArray	Hybrid a coefficients. Should have dimension of lev only.	required
hybm	DataArray	Hybrid b coefficients. Should have dimension of lev only.	required
p0	float	Reference pressure. Units: Pa.	required
plev_step	float	Will find var at value closest to p_desired on pressure grid with this spacing, so sets accuracy of interpolation.	1000
extrapolate	bool	If True, below ground extrapolation for variable will be done, otherwise will return nan.	False

Returns:

Type	Description
Dataset	Dataset with plev indicating approximate value of p_desired used, as well as var interpolated to that pressure level.

Source code in isca_tools/thesis/lapse_theory.py

def interp_var_at_pressure(var: Union[xr.DataArray, xr.Dataset], p_desired: xr.DataArray, p_surf: xr.DataArray,
                           hyam: xr.DataArray, hybm: xr.DataArray, p0: float,
                           plev_step: float = 1000, extrapolate: bool = False) -> xr.Dataset:
    """
    Function to get the value of variable `var` at the pressure `p_desired`, where `p_desired` is expected to
    be a different value at each lat and lon.

    Args:
        var: Variable to do interpolation of. Should have `lev` dimension as well as lat, lon and possibly time.
            If give Dataset, will do interpolation on all variables.
        p_desired: Desired pressure to find `var` at.
            Should have same dimension as `var` but no `lev`. Units: *Pa*.
        p_surf: Surface pressure.
            Should have same dimension as `var` but no `lev`. Units: *Pa*.
        hyam: Hybrid a coefficients. Should have dimension of `lev` only.
        hybm: Hybrid b coefficients. Should have dimension of `lev` only.
        p0: Reference pressure. Units: *Pa*.
        plev_step: Will find var at value closest to `p_desired` on pressure grid with this spacing,
            so sets accuracy of interpolation.
        extrapolate: If True, below ground extrapolation for variable will be done, otherwise will return nan.

    Returns:
        Dataset with `plev` indicating approximate value of `p_desired` used, as well as `var` interpolated
            to that pressure level.
    """
    plevs = np.arange(round_any(float(p_desired.min()), plev_step, 'floor'),
                      round_any(float(p_desired.max()), plev_step, 'ceil')+plev_step/2, plev_step)
    plevs_expand = xr.DataArray(plevs, dims=["plev"], coords={"plev": np.arange(len(plevs))})
    # Expand to match dimensions in p_surf, preserving order
    for dim in p_surf.dims:
        plevs_expand = plevs_expand.expand_dims({dim: p_surf.coords[dim]})

    idx_lcl_closest = np.abs(plevs_expand - p_desired).argmin(dim='plev')
    var_out = {'plev': plevs_expand.isel(plev=idx_lcl_closest)}     # approx pressure of p_desired used

    # Note that with extrapolate, will obtain values lower than surface
    if isinstance(var, xr.DataArray):
        var_out[var.name] = interp_hybrid_to_pressure(data=var, ps=p_surf, hyam=hyam, hybm=hybm, p0=p0,
                                                      new_levels=plevs, extrapolate=extrapolate,
                                                      variable='other' if extrapolate else None).isel(plev=idx_lcl_closest)
    elif isinstance(var, xr.Dataset):
        for key in var:
            var_out[key] = interp_hybrid_to_pressure(data=var[key], ps=p_surf, hyam=hyam, hybm=hybm, p0=p0,
                                                     new_levels=plevs, extrapolate=extrapolate,
                                                     variable='other' if extrapolate else None).isel(plev=idx_lcl_closest)
    else:
        raise ValueError('Unrecognized var. Needs to be a xr.DataArray or xr.Dataset.')
    for key in var_out:
        # Drop dimension of plev in all variables
        var_out[key] = var_out[key].drop_vars('plev')
    return xr.Dataset(var_out)

reconstruct_temp(temp3, p1, p2, p3, lapse_12, lapse_23)

The temperature, \(T_1\), at \(p_1\) can be reconstructed from the lapse rate, \(\Gamma_{12}\), between \(p_1\) and \(p_2\); the lapse rate \(\Gamma_{23}\), between \(p_2\) and \(p_3\); and the temperature at \(p_3\), \(T_3\):

\[ T_1 = T_{3}\left((\frac{p_2}{p_1})^{\Gamma_{23}-\Gamma_{12}}(\frac{p_3}{p_1})^{-\Gamma_{23}}\right)^{R/g} \]

Parameters:

Name	Type	Description	Default
temp3	DataArray	Temperature at pressure p3. Units: K.	required
p1	Union[DataArray, float]	Pressure at level to reconstruct temp1. Units: Pa.	required
p2	Union[DataArray, float]	Pressure at environmental temperature temp2. Units: Pa.	required
p3	Union[DataArray, float]	Pressure at environmental temperature temp3. Units: Pa.	required
lapse_12	DataArray	Bulk environmental lapse rate between p1 and p2. Units are K/m.	required
lapse_23	DataArray	Bulk environmental lapse rate between p2 and p3. Units are K/m.	required

Returns:

Name	Type	Description
temp1		Temperature at pressure p1. Units: K.

Source code in isca_tools/thesis/lapse_theory.py

def reconstruct_temp(temp3: xr.DataArray, p1: Union[xr.DataArray, float], p2: Union[xr.DataArray, float],
                     p3: Union[xr.DataArray, float],
                     lapse_12: xr.DataArray, lapse_23: xr.DataArray):
    """
    The temperature, $T_1$, at $p_1$ can be reconstructed from the lapse rate, $\Gamma_{12}$, between $p_1$ and $p_2$;
    the lapse rate $\Gamma_{23}$, between $p_2$ and $p_3$; and the temperature at $p_3$, $T_3$:

    $$
    T_1 = T_{3}\\left((\\frac{p_2}{p_1})^{\Gamma_{23}-\Gamma_{12}}(\\frac{p_3}{p_1})^{-\Gamma_{23}}\\right)^{R/g}
    $$

    Args:
        temp3: Temperature at pressure `p3`. Units: *K*.
        p1: Pressure at level to reconstruct `temp1`. Units: *Pa*.
        p2: Pressure at environmental temperature `temp2`. Units: *Pa*.
        p3: Pressure at environmental temperature `temp3`. Units: *Pa*.
        lapse_12: Bulk environmental lapse rate between `p1` and `p2`. Units are *K/m*.
        lapse_23: Bulk environmental lapse rate between `p2` and `p3`. Units are *K/m*.

    Returns:
        temp1: Temperature at pressure `p1`. Units: *K*.
    """
    sigma_12 = p2 / p1     # if p1 is surface, this should be <1
    sigma_13 = p3 / p1
    return temp3 * (sigma_12**(lapse_23-lapse_12) * sigma_13**(-lapse_23))**(R/g)