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Aquaplanet Theory

get_delta_temp_quant_theory(temp_mean, sphum_mean, temp_quant, sphum_quant, 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, \(\delta h(x) = \delta \overline{h}\):

\[\delta T(x) = \gamma^T \delta \overline{T} + \gamma^{\Delta r} \delta (\overline{r} - r(x))\]

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_mean ndarray

float [n_exp]
Average near surface temperature of each simulation, corresponding to a different optical depth, \(\kappa\). Units: K.

 required
sphum_mean ndarray

float [n_exp]
Average near surface specific humidity of each simulation. Units: kg/kg.

 required
temp_quant ndarray

float [n_exp, n_quant]
temp_quant[i, j] is the percentile quant_use[j] of near surface 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 ndarray

float [n_exp, n_quant]
sphum_quant[i, j] is the percentile quant_use[j] of near surface 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 \(\delta T(x) = \gamma^T \delta \overline{T} + \gamma^{\bar{r}} \delta \overline{r} +\gamma^{r} \delta r(x)\)
  • linear_rh_diff: Same as linear, but does another approximation to combine relative humidity contributions so: \(\delta T(x) = \gamma^T \delta \overline{T} + \gamma^{\Delta r} \delta (\overline{r} - r(x))\)
 '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/aquaplanet_theory.py 
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def get_delta_temp_quant_theory(temp_mean: np.ndarray, sphum_mean: np.ndarray, temp_quant: np.ndarray,
                                sphum_quant: 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,
    $\delta h(x) = \delta \overline{h}$:

    $$\delta T(x) = \gamma^T \delta \overline{T} + \gamma^{\Delta r} \delta (\overline{r} - r(x))$$

    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_mean: `float [n_exp]`</br>
            Average near surface temperature of each simulation, corresponding to a different
            optical depth, $\kappa$. Units: *K*.
        sphum_mean: `float [n_exp]`</br>
            Average near surface specific humidity of each simulation. Units: *kg/kg*.
        temp_quant: `float [n_exp, n_quant]`</br>
            `temp_quant[i, j]` is the percentile `quant_use[j]` of near surface 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: `float [n_exp, n_quant]`</br>
            `sphum_quant[i, j]` is the percentile `quant_use[j]` of near surface 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 $\delta T(x) = \\gamma^T \delta \\overline{T} +
                \\gamma^{\\bar{r}} \delta \\overline{r} +\\gamma^{r} \\delta r(x)$
            - `linear_rh_diff`: Same as `linear`, but does another approximation to combine relative humidity
                contributions so: $\delta T(x) = \gamma^T \delta \overline{T} +
                \gamma^{\Delta r} \delta (\overline{r} - r(x))$

    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.shape
    alpha_quant = clausius_clapeyron_factor(temp_quant, pressure_surface)
    alpha_mean = clausius_clapeyron_factor(temp_mean, pressure_surface)
    sphum_quant_sat = sphum_sat(temp_quant, pressure_surface)
    sphum_mean_sat = sphum_sat(temp_mean, pressure_surface)
    r_quant = sphum_quant / sphum_quant_sat
    r_mean = sphum_mean / sphum_mean_sat
    delta_temp_mean = np.diff(temp_mean)

    delta_r_mean = np.diff(r_mean)
    delta_r_quant = np.diff(r_quant, axis=0)
    if const_rh:
        # get rid of relative humidity contribution if constant rh
        delta_r_mean = 0 * delta_r_mean
        delta_r_quant = 0 * delta_r_quant

    # 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)
    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_mean = np.pad(delta_temp_mean, pad_width=pad_array[0])
    delta_r_mean = np.pad(delta_r_mean, pad_width=pad_array[0])
    delta_r_quant = np.pad(delta_r_quant, 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 * sphum_quant * (alpha_quant - 2 / temp_quant[0])
        coef_b = c_p + L_v * alpha_quant * (sphum_quant + sphum_quant_sat * delta_r_quant)
        coef_c = L_v * sphum_quant_sat * delta_r_quant - delta_mse_ratio * np.expand_dims(
                0.5 * L_v * alpha_mean * sphum_mean * (alpha_mean - 2 / temp_mean) * delta_temp_mean ** 2 +
                (c_p + L_v * alpha_mean * (sphum_mean + sphum_mean_sat * delta_r_mean)) * delta_temp_mean +
                L_v * sphum_mean_sat * delta_r_mean, axis=-1)
        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 * (sphum_quant + sphum_quant_sat * delta_r_quant)
        coef_c = L_v * sphum_quant_sat * delta_r_quant - delta_mse_ratio * np.expand_dims(
                (c_p + L_v * alpha_mean * (sphum_mean + sphum_mean_sat * delta_r_mean)) * delta_temp_mean +
                L_v * sphum_mean_sat * delta_r_mean, axis=-1)
        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 * sphum_quant
        coef_c = L_v * sphum_quant_sat * delta_r_quant - delta_mse_ratio * np.expand_dims(
                (c_p + L_v * alpha_mean * sphum_mean) * delta_temp_mean + L_v * sphum_mean_sat * delta_r_mean, axis=-1)
        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_mean, sphum_mean, temp_quant, sphum_quant, pressure_surface)
        delta_temp_quant_theory = (gamma_t * delta_mse_ratio * np.expand_dims(delta_temp_mean, axis=-1) +
                                   gamma_rdiff * (delta_mse_ratio * np.expand_dims(delta_r_mean, axis=-1) -
                                                  delta_r_quant))[:-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_mean, sphum_mean, temp_quant, sphum_quant, pressure_surface)

This function returns the sensitivity parameters in the theory. One for changes in mean temperature, \(\delta \overline{T}\), and one for difference to the mean relative humidity, \(\delta (\overline{r} - r(x))\):

\[\gamma^T = \frac{c_p + L_v \bar{\alpha} \bar{q}}{c_p + L_v \alpha q};\quad \gamma^{\Delta r} = \frac{L_v \overline{q_{sat}}}{c_p + L_v \alpha q}\]

Parameters:

Name Type Description Default
temp_mean ndarray

float [n_exp]
Average near surface temperature of each simulation, corresponding to a different optical depth, \(\kappa\). Units: K.

 required
sphum_mean ndarray

float [n_exp]
Average near surface specific humidity of each simulation. Units: kg/kg.

 required
temp_quant ndarray

float [n_exp, n_quant]
temp_quant[i, j] is the percentile quant_use[j] of near surface 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 ndarray

float [n_exp, n_quant]
sphum_quant[i, j] is the percentile quant_use[j] of near surface 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 mean temperature for each experiment and quantile.
ndarray

gamma_rdiff: float [n_exp, n_quant]
The sensitivity to change in relative humidity difference from the mean for each experiment and quantile.
Source code in isca_tools/thesis/aquaplanet_theory.py 
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def get_gamma(temp_mean: np.ndarray, sphum_mean: np.ndarray, temp_quant: np.ndarray,
              sphum_quant: np.ndarray, pressure_surface: float) -> Tuple[np.ndarray, np.ndarray]:
    """
    This function returns the sensitivity parameters in the theory.
    One for changes in mean temperature, $\\delta \\overline{T}$, and  one for difference to the mean relative humidity,
    $\delta (\overline{r} - r(x))$:

    $$\gamma^T = \\frac{c_p + L_v \\bar{\\alpha} \\bar{q}}{c_p + L_v \\alpha q};\quad
    \gamma^{\Delta r} = \\frac{L_v \\overline{q_{sat}}}{c_p + L_v \\alpha q}$$

    Args:
        temp_mean: `float [n_exp]`</br>
            Average near surface temperature of each simulation, corresponding to a different
            optical depth, $\kappa$. Units: *K*.
        sphum_mean: `float [n_exp]`</br>
            Average near surface specific humidity of each simulation. Units: *kg/kg*.
        temp_quant: `float [n_exp, n_quant]`</br>
            `temp_quant[i, j]` is the percentile `quant_use[j]` of near surface 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: `float [n_exp, n_quant]`</br>
            `sphum_quant[i, j]` is the percentile `quant_use[j]` of near surface 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 mean temperature for each experiment and quantile.
        `gamma_rdiff`: `float [n_exp, n_quant]`</br>
            The sensitivity to change in relative humidity difference from the mean for each experiment and quantile.
    """
    alpha_quant = clausius_clapeyron_factor(temp_quant, pressure_surface)
    alpha_mean = clausius_clapeyron_factor(temp_mean, pressure_surface)
    sphum_mean_sat = sphum_sat(temp_mean, pressure_surface)
    denom = c_p + L_v * alpha_quant * sphum_quant
    gamma_t = np.expand_dims(c_p + L_v * alpha_mean * sphum_mean, axis=-1) / denom
    gamma_rdiff = L_v / denom * np.expand_dims(sphum_mean_sat, axis=-1)
    return gamma_t, gamma_rdiff

get_lambda_2_theory(temp_ft_quant, temp_ft_mean, z_quant, z_mean, pressure_ft)

Compute the approximation for \(\lambda_2 = \delta h^*_{FT}(x)/\delta \overline{h^*_{FT}}\) used for the extratropical part of the theory between two simulations (n_exp should be 2):

\[\lambda_2 \approx (1+\frac{\overline{T}\delta \overline{\kappa}}{\delta \overline{z}}) \frac{c_p + L_v\alpha(x) q^*(x)}{c_p + L_v\overline{\alpha} \overline{q^*}} \frac{\delta z(x)}{\delta \overline{z}} - \frac{c_p + L_v\alpha(x) q^*(x)}{c_p + L_v\overline{\alpha} \overline{q^*}} \frac{\overline{T} \delta \kappa(x)}{\delta \overline{z}}\]

where \(\kappa=z/T\) and \(z\) and \(T\) are the free-troposphere geopotential height and temperature.

Parameters:

Name Type Description Default
temp_ft_quant ndarray

float [n_exp, n_lat, n_quant]
Free troposphere temperature for each experiment, latitude and quantile.

 required
temp_ft_mean ndarray

float [n_exp, n_lat]
Mean free troposphere temperature for each experiment and latitude.

 required
z_quant ndarray

float [n_exp, n_lat, n_quant]
Free troposphere geopotential height for each experiment, latitude and quantile.

 required
z_mean ndarray

float [n_exp, n_lat]
Mean free troposphere geopotential height for each experiment and latitude.

 required
pressure_ft float

Free troposphere pressure level. Units: Pa.

 required

Returns:

Type Description
ndarray

\(\lambda_2\) Approximation: float [n_lat, n_quant]
Approximation of \(\lambda_2\) at each latitude and quantile.
dict

prefactors: Dictionary containing the prefactors that go into the approximation. All variables are evaluated at the colder simulation.

  • z1: float [n_exp, n_lat, 1]
    \(1+\frac{\overline{T}\delta \overline{\kappa}}{\delta \overline{z}}\)
  • z2: float [n_exp, n_lat, n_quant]
    \(\frac{c_p + L_v\alpha(x) q^*(x)}{c_p + L_v\overline{\alpha} \overline{q^*}}\)
  • kappa: float [n_exp, n_lat, n_quant]
    \(-\frac{c_p + L_v\alpha(x) q^*(x)}{c_p + L_v\overline{\alpha} \overline{q^*}} \times \overline{T}\)
dict

delta_var: Dictionary containing the following changes between simulations.

  • z_quant: float [n_exp, n_lat, n_quant]
    \(\delta z(x)\)
  • z_mean: float [n_exp, n_lat, 1]
    \(\delta \overline{z}\)
  • kappa_quant: float [n_exp, n_lat, n_quant]
    \(\delta \kappa(x)\)
  • kappa_mean: float [n_exp, n_lat, 1]
    \(\delta \overline{\kappa}\)
Source code in isca_tools/thesis/aquaplanet_theory.py 
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def get_lambda_2_theory(temp_ft_quant: np.ndarray, temp_ft_mean: np.ndarray, z_quant: np.ndarray, z_mean: np.ndarray,
                        pressure_ft: float) -> Tuple[np.ndarray, dict, dict]:
    """
    Compute the approximation for $\lambda_2 = \delta h^*_{FT}(x)/\delta \overline{h^*_{FT}}$ used
    for the extratropical part of the theory between two simulations (`n_exp` should be 2):

    $$\\lambda_2 \\approx (1+\\frac{\\overline{T}\\delta \\overline{\\kappa}}{\\delta \\overline{z}})
    \\frac{c_p + L_v\\alpha(x) q^*(x)}{c_p + L_v\\overline{\\alpha} \\overline{q^*}}
    \\frac{\\delta z(x)}{\\delta \\overline{z}} -
    \\frac{c_p + L_v\\alpha(x) q^*(x)}{c_p + L_v\\overline{\\alpha} \\overline{q^*}}
    \\frac{\\overline{T} \\delta \\kappa(x)}{\\delta \\overline{z}}$$

    where $\kappa=z/T$ and $z$ and $T$ are the free-troposphere geopotential height and temperature.

    Args:
        temp_ft_quant: `float [n_exp, n_lat, n_quant]`</br>
            Free troposphere temperature for each experiment, latitude and quantile.
        temp_ft_mean: `float [n_exp, n_lat]`</br>
            Mean free troposphere temperature for each experiment and latitude.
        z_quant: `float [n_exp, n_lat, n_quant]`</br>
            Free troposphere geopotential height for each experiment, latitude and quantile.
        z_mean: `float [n_exp, n_lat]`</br>
            Mean free troposphere geopotential height for each experiment and latitude.
        pressure_ft: Free troposphere pressure level. Units: *Pa*.

    Returns:
        $\lambda_2$ Approximation: `float [n_lat, n_quant]`</br>
            Approximation of $\lambda_2$ at each latitude and quantile.
        `prefactors`: Dictionary containing the prefactors that go into the approximation. All variables are
            evaluated at the colder simulation.

            - `z1`: `float [n_exp, n_lat, 1]`</br>
                $1+\\frac{\\overline{T}\\delta \\overline{\\kappa}}{\\delta \\overline{z}}$
            - `z2`: `float [n_exp, n_lat, n_quant]`</br>
                $\\frac{c_p + L_v\\alpha(x) q^*(x)}{c_p + L_v\\overline{\\alpha} \\overline{q^*}}$
            - `kappa`: `float [n_exp, n_lat, n_quant]`</br>
                $-\\frac{c_p + L_v\\alpha(x) q^*(x)}{c_p + L_v\\overline{\\alpha} \\overline{q^*}} \\times \\overline{T}$
        `delta_var`: Dictionary containing the following changes between simulations.

            - `z_quant`: `float [n_exp, n_lat, n_quant]`</br>
                    $\delta z(x)$</br>
            - `z_mean`: `float [n_exp, n_lat, 1]`</br>
                    $\delta \overline{z}$</br>
            - `kappa_quant`: `float [n_exp, n_lat, n_quant]`</br>
                    $\delta \kappa(x)$</br>
            - `kappa_mean`: `float [n_exp, n_lat, 1]`</br>
                    $\delta \overline{\kappa}$
    """
    kappa_quant = z_quant / temp_ft_quant
    kappa_mean = z_mean / temp_ft_mean
    # Need to expand dims of mean variables so can multiply quant arrays
    temp_ft_mean = np.expand_dims(temp_ft_mean, axis=-1)

    delta_var = {'z_quant': z_quant[1] - z_quant[0], 'z_mean': np.expand_dims(z_mean[1] - z_mean[0], axis=-1),
                 'kappa_quant': kappa_quant[1] - kappa_quant[0],
                 'kappa_mean': np.expand_dims(kappa_mean[1] - kappa_mean[0], axis=-1)}

    alpha_quant = clausius_clapeyron_factor(temp_ft_quant[0], pressure_ft)
    alpha_mean = clausius_clapeyron_factor(temp_ft_mean[0], pressure_ft)
    q_quant = sphum_sat(temp_ft_quant[0], pressure_ft)
    q_mean = sphum_sat(temp_ft_mean[0], pressure_ft)

    prefactors = {'z1': 1+temp_ft_mean[0] * delta_var['kappa_mean'] / delta_var['z_mean'],
                  'z2': (c_p + L_v * alpha_quant * q_quant) / (c_p + L_v * alpha_mean * q_mean)}
    prefactors['kappa'] = -prefactors['z2'] * temp_ft_mean[0]

    lambda_2_approx = prefactors['z1'] * prefactors['z2'] * delta_var['z_quant'] / delta_var['z_mean'] + \
        prefactors['kappa'] * delta_var['kappa_quant'] / delta_var['z_mean']
    return lambda_2_approx, prefactors, delta_var