Spatial stabilization and intensification of moistening and drying rate patterns under future climate change

Abstract

Precipitation projections are usually presented as the change in precipitation between a fixed current baseline and a particular time in the future. However, upcoming generations will be affected in a way probably more related to the moving trend in precipitation patterns, i.e. to the rate and the persistence of regional precipitation changes from one generation to the next, than to changes relative to a fixed current baseline. In this perspective, we propose an alternative characterization of the future precipitation changes predicted by general circulation models, focusing on the precipitation difference between two subsequent 20-year periods. We show that in a business-as-usual emission pathway, the moistening and drying rates increase by 30–40 %, both over land and ocean. As we move further over the twenty-first century, more regions exhibit a significant rate of precipitation change, while the patterns become geographically stationary and the trends persistent. The stabilization of the geographical rate patterns that occurs despite the acceleration of global warming can be physically explained: it results from the increasing contribution of thermodynamic processes compared to dynamic processes in the control of precipitation change. We show that such an evolution is already noticeable over the last decades, and that it could be reversed if strong mitigation policies were quickly implemented. The combination of intensification and increasing persistence of precipitation rate patterns may affect the way human societies and natural ecosystems adapt to climate change, especially in the Mediterranean basin, in Central America, in South Asia and in the Arctic.

Appendices

Appendix 1: Additivity of precipitation rates

In order to preserve additivity of both drying and moistening rates, the computation of multi-model mean values is conducted as follows. For each model l, each realization r and each year t, $\Delta P_{20}$ is expressed as:

$$\begin{aligned} \Delta P_{20} = \varPi _+ \cdot \Delta P^+_{20} - \varPi _- \cdot \Delta P^-_{20} \end{aligned}$$

(6.1)

where $\varPi _+$ ($\varPi _-$) is the fraction of moistening (drying) regions.

To ensure that Eq. (6.1) is valid for multi-model mean values, the multi-model means of the drying and moistening rates are weighted by the fraction of drying and moistening regions:

$$\begin{aligned} \left\{ \begin{aligned} \overline{\Delta P^+_{20}}^{l,r}&= \frac{\sum _{l,r} \left( \varPi _{+,l,r} \cdot \Delta P^+_{20,l,r}\right) }{\sum _{l,r} \varPi _{+,l,r}} \\ \overline{\Delta P^-_{20}}^{l,r}&= \frac{\sum _{l,r} \left( \varPi _{-,l,r} \cdot \Delta P^-_{20,l,r}\right) }{\sum _{l,r} \varPi _{-,l,r}} \end{aligned} \right. \end{aligned}$$

(6.2)

with $\overline{\cdot \cdot \cdot }^{l,r}$ representing the average of all models and all realizations, i.e. the multi-model mean.

Appendix 2: Additive conditions of natural and inter-model variability

Indicators are computed with L models including $R_l$ realizations. Raw prediction X for each model l, realization r and year t can be expressed as:

$$\begin{aligned} X_{l,r,t} = x_{t} + x'_{l,t} + \varepsilon _{l,r,t} \end{aligned}$$

(7.1)

where $x_t$ is the multi-model mean value, $x'_{l,t}$ is the difference between $x_t$ and the multi-realization mean value of each model and $\varepsilon _{l,r,t}$ is the difference between the multi-run mean value of the considered model and $X_{l,r,t}$.

Within the scope of this work, we consider two hypotheses:

• H1 The inter-model and natural variabilities are two independent processes, which is true as a first approximation and is an assumption of this analysis.

• H2 The multi-realization mean $\overline{\varepsilon _{l,r,t}}^r$ of $\varepsilon _{l,r,t}$ for each model l and each time t equals zero, which is true by definition of the multi-realization mean.

The natural variability $\sigma ^2_{R,l}$ for each model l is defined as the temporal variance of $\varepsilon _{l,r,t}$. It is time- and realization-independent for each model. The multi-model mean $\sigma ^2_{R}$ of $\sigma ^2_{R,l}$ is computed in accordance with an equal weight amongst all models. The inter-model variability $\sigma ^2_{L,t}$ is estimated from the variance between multi-realization mean values of each model at each time step and is thus time dependent. In the following calculation, we give an equal weight to each run for simplicity, whatever the model or the realization. The final results of the calculation remain identical. The total variability of the indicator is computed using Eq. (7.1) as follows:

$$\begin{aligned} \sigma ^2_{T,t}&= {\mathrm {Var}}_{l,r} \left( X_{l,r,t} \right) \nonumber \\&= \frac{1}{L} \sum _{l} \left[ \frac{1}{R_l} \sum _r \left( X_{l,r,t} - x_t \right) ^2 \right] \nonumber \\&\mathop {=}\limits ^{(B.1)} \frac{1}{L} \sum _{l} \left[ \frac{1}{R_l} \sum _r \left( x'^2_{l,t} + 2x'_{l,t}\cdot \varepsilon _{l,r,t} + \varepsilon ^2_{l,r,t} \right) \right] \nonumber \\&= \overline{x'^2_{l,t}}^l + 2\cdot \overline{x'_{l,t}\cdot \overline{\varepsilon _{l,r,t}}^r}^l + \overline{\varepsilon ^2_{l,r,t}}^{l,r} \nonumber \\&\mathop {=}\limits ^{(H2)} \overline{x'^2_{l,t}}^l + \overline{\varepsilon ^2_{l,r,t}}^{l,r} \end{aligned}$$

(7.2)

$\sigma ^2_{L,t}$ and $\sigma ^2_{R}$ can be expressed the same way:

$$\begin{aligned} \sigma ^2_{L,t}&= {\mathrm {Var}}_l \left( x_{t} + x'_{l,t} \right) \nonumber \\&= \frac{1}{L} \sum _l \left( x_t +x'_{l,t} - x_t \right) ^2 \nonumber \\&= \overline{x'^2_{l,t}}^l \end{aligned}$$

(7.3)

$$\begin{aligned} \sigma ^2_{R}&= \frac{1}{L} \sum _{l} \left[ \frac{1}{R_l} \sum _r {\mathrm {Var}} _{r,t} \left( \varepsilon _{l,r,t} \right) \right] \nonumber \\&= \frac{1}{L} \sum _{l} \left[ \frac{1}{R_l} \sum _r \left( \varepsilon _{l,r,t} - \overline{\varepsilon _{l,r,t}}^{r,t} \right) ^2 \right] \nonumber \\&= \overline{\varepsilon ^2_{l,r,t} - 2\varepsilon _{l,r,t} \cdot \overline{\varepsilon _{l,r,t}}^{r,t} + \left( \overline{\varepsilon _{l,r,t}}^{r,t}\right) ^2}^{l,r} \nonumber \\&\mathop {=}\limits ^{(H2)} \overline{\varepsilon ^2_{l,r,t}}^{l,r} \end{aligned}$$

(7.4)

The total variability is the sum of both natural and inter-model variabilities:

$$\begin{aligned} \sigma ^2_{T,t} = \sigma ^2_{R} + \sigma ^2_{L,t} \end{aligned}$$

(7.5)

The additivity of $\sigma ^2_{R}$ and $\sigma ^2_{L,t}$ is thus the direct consequence of the way we constructed the decomposition of each source of uncertainty.

Appendix 3: Detailed calculation of the (thermo)dynamic components of $\Delta P_{20}$

The vertically integrated water budget can be diagnosed regionally as follows for each GCM, each run and each yearly mean value (Neelin 2007; Bony et al. 2013):

$$\begin{aligned} P = E - \left[ q {\varvec{\nabla }} \cdot \mathbf{V} \right] - \left[ \mathbf{V} \cdot {\varvec{\nabla }}q \right] \end{aligned}$$

(8.1)

where P is the precipitation, E the surface evapotranspiration, $\mathbf{V}$ the field of horizontal wind velocity and q the vertical profile of specific humidity. The vertically integrated horizontal moisture advection term $- \left[ \mathbf{V} \cdot {\varvec{\nabla }}q \right]$ is hereafter denoted as $H_q$. Mass continuity can be expressed as:

$$\begin{aligned} {\varvec{\nabla }} \cdot \mathbf{V} + \frac{\partial \omega }{\partial \tilde{p}} = 0 \end{aligned}$$

(8.2)

where $\omega$ is the pressure vertical velocity and $\tilde{p}$ the atmospheric pressure. Considering that $\omega = 0$ at the Earth surface and at the top of the atmosphere, a vertical integration by parts leads to:

$$\begin{aligned} \frac{{\mathrm {d}}(\omega q)}{{\mathrm {d}}\tilde{p}} = \omega \frac{\partial q}{\partial \tilde{p}} + q \frac{\partial \omega }{\partial \tilde{p}} = \omega q \arrowvert ^{\tilde{p}_{toa}}_{\tilde{p}_{surf}} = 0 \end{aligned}$$

(8.3)

The combination of (8.2) and (8.3) implies that a variation of (8.1) is expressed as:

$$\begin{aligned} \Delta P = \Delta E - \Delta \left[ \omega \frac{\partial q}{\partial \tilde{p}}\right] + \Delta H_q \end{aligned}$$

(8.4)

As the dynamic contribution in precipitation changes is only caused by global circulation changes (i.e. variation of $\omega$), (8.4) can be formulated differently:

$$\begin{aligned} \Delta P = \underbrace{\Delta E - \omega \Delta \left[ \frac{\partial q}{\partial \tilde{p}}\right] + \Delta H_q}_{\Delta P_{th}} + \underbrace{\left[ \frac{\partial q}{\partial \tilde{p}}\right] \Delta \omega }_{\Delta P_{dyn}} \end{aligned}$$

(8.5)

On the other hand, the Clausius–Clapeyron expression for vapor saturation is (e.g. Held and Soden 2006):

$$\begin{aligned} \frac{{\mathrm {d}}\ln q}{{\mathrm {d}}T} = \frac{L}{RT^2}=\alpha (T) \end{aligned}$$

(8.6)

where T is the surface air temperature, L the latent heat of vaporization and R the gas constant. At temperatures corresponding to those of the lower troposphere, $\alpha = 0.07K^{-1}$ above oceans. Then, (8.6) becomes:

$$\begin{aligned} {\mathrm {d}}q = 0.07q \cdot {\mathrm {d}}T \end{aligned}$$

(8.7)

Using the vertically integrated water budget (8.1) and the Clausius–Clapeyron relation (8.7) in (8.5), we obtain:

$$\begin{aligned} \Delta P = \Delta E + 0.07(P-E)\Delta T + \Delta P_{dyn} \end{aligned}$$

(8.8)

This expression is a valid approximation for absolute change of precipitation above oceans, but also for a running difference with the only condition that $P=\left\langle P \right\rangle _{t,t-20}$ and $E=\left\langle E \right\rangle _{t,t-20}$, the mean of annual precipitation (evapotranspiration) during the last twenty years. The rate of precipitation change $\Delta P_{20}$ as well as the moistening (drying) rates $\Delta P_{20}^+$ ($\Delta P_{20}^-$) can be split in two distinctive parts coming from thermodynamic and dynamic modifications of the atmosphere:

$$\begin{aligned} \left\{ \begin{aligned}&\Delta P_{20}^{th} = \Delta E_{20} + 0.07 \left( \left\langle P\right\rangle - \left\langle E\right\rangle \right) \Delta T_{20}\\&\Delta P_{20}^{dyn} = \Delta P_{20} - \Delta P_{20}^{th}. \end{aligned} \right. \end{aligned}$$

(8.9)