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Extratropics Land Theory

get_delta_temp_quant_theory(temp_quant_ocean, sphum_quant_ocean, temp_quant_land, sphum_quant_land, pressure_surface, const_rh=False, delta_mse_ratio=None, taylor_level='linear_rh_diff')

Computes the theoretical temperature difference between simulations of neighbouring optical depth values for each percentile, \(\delta T(x)\), according to the assumption that changes in MSE are equal to the change in mean MSE, \(\delta h(x) = \delta \overline{h}\):

\[\delta T_L(x) = \gamma^T \delta T_O(x) + \gamma^{\Delta r} \delta (r_O(x) - r_L(x))\]

This above equation is for the default settings, but a more accurate equation can be used with the delta_mse_ratio and taylor_level arguments.

If data from n_exp optical depth values provided, n_exp-1 theoretical temperature differences will be returned for each percentile.

Parameters:

Name Type Description Default
temp_quant_ocean ndarray

float [n_exp, n_quant]
temp_quant_ocean[i, j] is the percentile quant_use[j] of near surface ocean temperature of experiment i. Units: K.
Note that quant_use is not provided as not needed by this function, but is likely to be np.arange(1, 100) - leave out x=0 as doesn't really make sense to consider \(0^{th}\) percentile of a quantity.

 required
sphum_quant_ocean ndarray

float [n_exp, n_quant]
sphum_quant_ocean[i, j] is the percentile quant_use[j] of near surface ocean specific humidity of experiment i. Units: kg/kg.

 required
temp_quant_land ndarray

float [n_exp, n_quant]
temp_quant_land[i, j] is the percentile quant_use[j] of near surface land temperature of experiment i. Units: K.

 required
sphum_quant_land ndarray

float [n_exp, n_quant]
sphum_quant_land[i, j] is the percentile quant_use[j] of near surface land specific humidity of experiment i. Units: kg/kg.

 required
pressure_surface float

Near surface pressure level. Units: Pa.

 required
const_rh bool

If True, will return the constant relative humidity version of the theory, i.e. \(\gamma^T \delta \overline{T}\). Otherwise, will return the full theory.

 False
delta_mse_ratio Optional[ndarray]

float [n_exp-1, n_quant]
delta_mse_ratio[i] is the change in \(x\) percentile of MSE divided by the change in the mean MSE between experiment i and i+1: \(\delta h(x)/\delta \overline{h}\). If not given, it is assumed to be equal to 1 for all \(x\).

 None
taylor_level str

This specifies the level of approximation that goes into the taylor series for \(\delta q(x)\) and \(\delta \overline{q}\):

  • squared: Includes squared, \(\delta T^2\), nonlinear, \(\delta T \delta r\), and linear terms.
  • nonlinear: Includes nonlinear, \(\delta T \delta r\), and linear terms.
  • linear: Includes just linear terms so \(\delta T_L(x) = \gamma^T \delta T_O(x) + \gamma^{r_O} \delta r_O(x) +\gamma^{r_L} \delta r_L(x)\)
  • linear_rh_diff: Same as linear, but does another approximation to combine relative humidity contributions so: \(\delta T_L(x) = \gamma^T \delta T_O(x) + \gamma^{\Delta r} \delta (r_O(x) - r_L(x))\)
 'linear_rh_diff'

Returns:

Type Description
ndarray

float [n_exp-1, n_quant].
delta_temp_quant_theory[i, j] refers to the theoretical temperature difference between experiment i and i+1 for percentile quant_use[j].
Source code in isca_tools/thesis/extrop_land_theory.py 
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def get_delta_temp_quant_theory(temp_quant_ocean: np.ndarray, sphum_quant_ocean: np.ndarray, temp_quant_land: np.ndarray,
                                sphum_quant_land: np.ndarray, pressure_surface: float, const_rh: bool = False,
                                delta_mse_ratio: Optional[np.ndarray] = None,
                                taylor_level: str = 'linear_rh_diff') -> np.ndarray:
    """
    Computes the theoretical temperature difference between simulations of neighbouring optical depth values for each
    percentile, $\delta T(x)$, according to the assumption that changes in MSE are equal to the change in mean MSE,
    $\delta h(x) = \delta \overline{h}$:

    $$\delta T_L(x) = \gamma^T \delta T_O(x) + \gamma^{\Delta r} \delta (r_O(x) - r_L(x))$$

    This above equation is for the default settings, but a more accurate equation can be used with the `delta_mse_ratio`
    and `taylor_level` arguments.

    If data from `n_exp` optical depth values provided, `n_exp-1` theoretical temperature differences will be returned
    for each percentile.

    Args:
        temp_quant_ocean: `float [n_exp, n_quant]`</br>
            `temp_quant_ocean[i, j]` is the percentile `quant_use[j]` of near surface ocean temperature of
            experiment `i`. Units: *K*.</br>
            Note that `quant_use` is not provided as not needed by this function, but is likely to be
            `np.arange(1, 100)` - leave out `x=0` as doesn't really make sense to consider $0^{th}$ percentile
            of a quantity.
        sphum_quant_ocean: `float [n_exp, n_quant]`</br>
            `sphum_quant_ocean[i, j]` is the percentile `quant_use[j]` of near surface ocean specific humidity of
            experiment `i`. Units: *kg/kg*.
        temp_quant_land: `float [n_exp, n_quant]`</br>
            `temp_quant_land[i, j]` is the percentile `quant_use[j]` of near surface land temperature of
            experiment `i`. Units: *K*.</br>
        sphum_quant_land: `float [n_exp, n_quant]`</br>
            `sphum_quant_land[i, j]` is the percentile `quant_use[j]` of near surface land specific humidity of
            experiment `i`. Units: *kg/kg*.
        pressure_surface: Near surface pressure level. Units: *Pa*.
        const_rh: If `True`, will return the constant relative humidity version of the theory, i.e.
            $\gamma^T \delta \overline{T}$. Otherwise, will return the full theory.
        delta_mse_ratio: `float [n_exp-1, n_quant]`</br>
            `delta_mse_ratio[i]` is the change in $x$ percentile of MSE divided by the change in the mean MSE
            between experiment `i` and `i+1`: $\delta h(x)/\delta \overline{h}$.
            If not given, it is assumed to be equal to 1 for all $x$.
        taylor_level: This specifies the level of approximation that goes into the taylor series for $\delta q(x)$
            and $\delta \overline{q}$:

            - `squared`: Includes squared, $\delta T^2$, nonlinear, $\delta T \delta r$, and linear terms.
            - `nonlinear`: Includes nonlinear, $\delta T \delta r$, and linear terms.
            - `linear`: Includes just linear terms so $\delta T_L(x) = \\gamma^T \delta T_O(x) +
                \\gamma^{r_O} \delta r_O(x) +\\gamma^{r_L} \\delta r_L(x)$
            - `linear_rh_diff`: Same as `linear`, but does another approximation to combine relative humidity
                contributions so: $\delta T_L(x) = \gamma^T \delta T_O(x) +
                \gamma^{\Delta r} \delta (r_O(x) - r_L(x))$

    Returns:
        `float [n_exp-1, n_quant]`.</br>
            `delta_temp_quant_theory[i, j]` refers to the theoretical temperature difference between experiment `i` and
            `i+1` for percentile `quant_use[j]`.
    """
    n_exp, n_quant = temp_quant_land.shape
    alpha_quant_l = clausius_clapeyron_factor(temp_quant_land, pressure_surface)
    alpha_quant_o = clausius_clapeyron_factor(temp_quant_ocean, pressure_surface)
    sphum_quant_sat_l = sphum_sat(temp_quant_land, pressure_surface)
    sphum_quant_sat_o = sphum_sat(temp_quant_ocean, pressure_surface)
    r_quant_l = sphum_quant_land / sphum_quant_sat_l
    r_quant_o = sphum_quant_ocean / sphum_quant_sat_o
    delta_temp_quant_o = np.diff(temp_quant_ocean, axis=0)

    delta_r_quant_o = np.diff(r_quant_o, axis=0)
    delta_r_quant_l = np.diff(r_quant_l, axis=0)
    if const_rh:
        # get rid of relative humidity contribution if constant rh
        delta_r_quant_o = 0 * delta_r_quant_o
        delta_r_quant_l = 0 * delta_r_quant_l

    # Pad all delta variables so same size as temp_quant - will not use this in calculation but just makes it easier
    pad_array = ((0, 1), (0, 0))
    if delta_mse_ratio is None:
        delta_mse_ratio = np.ones_like(temp_quant_land)
    else:
        # make delta_mse_ratio the same size as all other quant variables
        delta_mse_ratio = np.pad(delta_mse_ratio, pad_width=pad_array)
    delta_temp_quant_o = np.pad(delta_temp_quant_o, pad_width=pad_array)
    delta_r_quant_o = np.pad(delta_r_quant_o, pad_width=pad_array)
    delta_r_quant_l = np.pad(delta_r_quant_l, pad_width=pad_array)

    if taylor_level == 'squared':
        # Keep squared, linear and non-linear terms in taylor expansion of delta_sphum_quant
        coef_a = 0.5 * L_v * alpha_quant_l * sphum_quant_land * (alpha_quant_l - 2 / temp_quant_land[0])
        coef_b = c_p + L_v * alpha_quant_l * (sphum_quant_land + sphum_quant_sat_l * delta_r_quant_l)
        coef_c = L_v * sphum_quant_sat_l * delta_r_quant_l - delta_mse_ratio * (
            0.5 * L_v * alpha_quant_o * sphum_quant_ocean * (alpha_quant_o - 2 / temp_quant_ocean) * delta_temp_quant_o ** 2 +
            (c_p + L_v * alpha_quant_o * (sphum_quant_ocean + sphum_quant_sat_o * delta_r_quant_o)) * delta_temp_quant_o +
            L_v * sphum_quant_sat_o * delta_r_quant_o)
        delta_temp_quant_theory = np.asarray([[np.roots([coef_a[i, j], coef_b[i, j], coef_c[i, j]])[1]
                                               for j in range(n_quant)] for i in range(n_exp - 1)])
    elif taylor_level in ['nonlinear', 'non-linear']:
        # Keep linear and non-linear terms in taylor expansion of delta_sphum_quant
        coef_b = c_p + L_v * alpha_quant_l * (sphum_quant_land + sphum_quant_sat_l * delta_r_quant_l)
        coef_c = L_v * sphum_quant_sat_l * delta_r_quant_l - delta_mse_ratio * (
            (c_p + L_v * alpha_quant_o * (sphum_quant_ocean + sphum_quant_sat_o * delta_r_quant_o)) * delta_temp_quant_o +
            L_v * sphum_quant_sat_o * delta_r_quant_o)
        delta_temp_quant_theory = -coef_c[:-1] / coef_b[:-1]
    elif taylor_level == 'linear':
        # Only keep linear terms in taylor expansion of delta_sphum_quant
        coef_b = c_p + L_v * alpha_quant_l * sphum_quant_land
        coef_c = L_v * sphum_quant_sat_l * delta_r_quant_l - delta_mse_ratio * (
            (c_p + L_v * alpha_quant_o * sphum_quant_ocean) * delta_temp_quant_o +
            L_v * sphum_quant_sat_o * delta_r_quant_o)
        delta_temp_quant_theory = -coef_c[:-1] / coef_b[:-1]
    elif taylor_level == 'linear_rh_diff':
        # combine mean and quantile RH changes with same prefactor
        # This is a further taylor expansion of sphum_quant_sat around sphum_quant_mean
        gamma_t, gamma_rdiff = get_gamma(temp_quant_ocean, sphum_quant_ocean, temp_quant_land, sphum_quant_land, pressure_surface)
        delta_temp_quant_theory = (gamma_t * delta_mse_ratio * delta_temp_quant_o +
                                   gamma_rdiff * (delta_mse_ratio * delta_r_quant_o -
                                                  delta_r_quant_l))[:-1]
    else:
        raise ValueError(f"taylor_level given is {taylor_level}. This is not valid, it must be either: 'squared', "
                         f"'nonlinear', 'linear' or 'linear_rh_diff'")
    return delta_temp_quant_theory

get_gamma(temp_quant_ocean, sphum_quant_ocean, temp_quant_land, sphum_quant_land, pressure_surface)

This function returns the sensitivity parameters in the theory. One for changes in ocean temperature, \(\delta T_O(x)\), and one for difference between land and ocean relative humidity, \(\delta (r_O(x) - r_L(x))\):

\[\gamma^T = \frac{c_p + L_v \alpha_O q_O}{c_p + L_v \alpha_L q_L};\quad \gamma^{\Delta r} = \frac{L_v q_{O, sat}}{c_p + L_v \alpha_L q_L}\]

Parameters:

Name Type Description Default
temp_quant_ocean ndarray

float [n_exp, n_quant]
temp_quant_ocean[i, j] is the percentile quant_use[j] of near surface ocean temperature of experiment i. Units: K.
Note that quant_use is not provided as not needed by this function, but is likely to be np.arange(1, 100) - leave out x=0 as doesn't really make sense to consider \(0^{th}\) percentile of a quantity.

 required
sphum_quant_ocean ndarray

float [n_exp, n_quant]
sphum_quant_ocean[i, j] is the percentile quant_use[j] of near surface ocean specific humidity of experiment i. Units: kg/kg.

 required
temp_quant_land ndarray

float [n_exp, n_quant]
temp_quant_land[i, j] is the percentile quant_use[j] of near surface land temperature of experiment i. Units: K.

 required
sphum_quant_land ndarray

float [n_exp, n_quant]
sphum_quant_land[i, j] is the percentile quant_use[j] of near surface land specific humidity of experiment i. Units: kg/kg.

 required
pressure_surface float

Near surface pressure level. Units: Pa.

 required

Returns:

Type Description
ndarray

gamma_t: float [n_exp, n_quant]
The sensitivity to change in ocean temperature for each experiment and quantile.
ndarray

gamma_rdiff: float [n_exp, n_quant]
The sensitivity to change in relative humidity difference from land to ocean for each experiment and quantile.
Source code in isca_tools/thesis/extrop_land_theory.py 
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def get_gamma(temp_quant_ocean: np.ndarray, sphum_quant_ocean: np.ndarray, temp_quant_land: np.ndarray,
              sphum_quant_land: np.ndarray, pressure_surface: float) -> Tuple[np.ndarray, np.ndarray]:
    """
    This function returns the sensitivity parameters in the theory.
    One for changes in ocean temperature, $\\delta T_O(x)$, and  one for difference between land and ocean
    relative humidity, $\delta (r_O(x) - r_L(x))$:

    $$\gamma^T = \\frac{c_p + L_v \\alpha_O q_O}{c_p + L_v \\alpha_L q_L};\quad
    \gamma^{\Delta r} = \\frac{L_v q_{O, sat}}{c_p + L_v \\alpha_L q_L}$$

    Args:
        temp_quant_ocean: `float [n_exp, n_quant]`</br>
            `temp_quant_ocean[i, j]` is the percentile `quant_use[j]` of near surface ocean temperature of
            experiment `i`. Units: *K*.</br>
            Note that `quant_use` is not provided as not needed by this function, but is likely to be
            `np.arange(1, 100)` - leave out `x=0` as doesn't really make sense to consider $0^{th}$ percentile
            of a quantity.
        sphum_quant_ocean: `float [n_exp, n_quant]`</br>
            `sphum_quant_ocean[i, j]` is the percentile `quant_use[j]` of near surface ocean specific humidity of
            experiment `i`. Units: *kg/kg*.
        temp_quant_land: `float [n_exp, n_quant]`</br>
            `temp_quant_land[i, j]` is the percentile `quant_use[j]` of near surface land temperature of
            experiment `i`. Units: *K*.</br>
        sphum_quant_land: `float [n_exp, n_quant]`</br>
            `sphum_quant_land[i, j]` is the percentile `quant_use[j]` of near surface land specific humidity of
            experiment `i`. Units: *kg/kg*.
        pressure_surface: Near surface pressure level. Units: *Pa*.

    Returns:
        `gamma_t`: `float [n_exp, n_quant]`</br>
            The sensitivity to change in ocean temperature for each experiment and quantile.
        `gamma_rdiff`: `float [n_exp, n_quant]`</br>
            The sensitivity to change in relative humidity difference from land to ocean for each experiment and
            quantile.
    """
    alpha_quant_l = clausius_clapeyron_factor(temp_quant_land, pressure_surface)
    alpha_quant_o = clausius_clapeyron_factor(temp_quant_ocean, pressure_surface)
    sphum_quant_sat_o = sphum_sat(temp_quant_ocean, pressure_surface)
    denom = c_p + L_v * alpha_quant_l * sphum_quant_land
    gamma_t = (c_p + L_v * alpha_quant_o * sphum_quant_ocean) / denom
    gamma_rdiff = L_v / denom * sphum_quant_sat_o
    return gamma_t, gamma_rdiff