Thermodynamic and dynamic contributions to future changes in regional precipitation variance: focus on the Southeastern United States

Abstract

The frequency and severity of extreme events are tightly associated with the variance of precipitation. As climate warms, the acceleration in hydrological cycle is likely to enhance the variance of precipitation across the globe. However, due to the lack of an effective analysis method, the mechanisms responsible for the changes of precipitation variance are poorly understood, especially on regional scales. Our study fills this gap by formulating a variance partition algorithm, which explicitly quantifies the contributions of atmospheric thermodynamics (specific humidity) and dynamics (wind) to the changes in regional-scale precipitation variance. Taking Southeastern (SE) United States (US) summer precipitation as an example, the algorithm is applied to the simulations of current and future climate by phase 5 of Coupled Model Intercomparison Project (CMIP5) models. The analysis suggests that compared to observations, most CMIP5 models (~60 %) tend to underestimate the summer precipitation variance over the SE US during the 1950–1999, primarily due to the errors in the modeled dynamic processes (i.e. large-scale circulation). Among the 18 CMIP5 models analyzed in this study, six of them reasonably simulate SE US summer precipitation variance in the twentieth century and the underlying physical processes; these models are thus applied for mechanistic study of future changes in SE US summer precipitation variance. In the future, the six models collectively project an intensification of SE US summer precipitation variance, resulting from the combined effects of atmospheric thermodynamics and dynamics. Between them, the latter plays a more important role. Specifically, thermodynamics results in more frequent and intensified wet summers, but does not contribute to the projected increase in the frequency and intensity of dry summers. In contrast, atmospheric dynamics explains the projected enhancement in both wet and dry summers, indicating its importance in understanding future climate change over the SE US. The results suggest that the intensified SE US summer precipitation variance is not a purely thermodynamic response to greenhouse gases forcing, and cannot be explained without the contribution of atmospheric dynamics. Our analysis provides important insights to understand the mechanisms of SE US summer precipitation variance change. The algorithm formulated in this study can be easily applied to other regions and seasons to systematically explore the mechanisms responsible for the changes in precipitation extremes in a warming climate.

Appendices

Appendix 1: Summertime hydroclimate over the SE US

On temporal scales longer than 10 days, the atmospheric branch of the regional hydrological cycle is characterized as the moisture balance between precipitation, evapotranspiration, and net moisture transport across the domain lateral boundary (Brubaker et al. 1993). Thus, SE US summertime hydrological cycle can be quantified as:

$$\rho_{w} g\left( {\left[ {\bar{P}} \right] - \left[ {\bar{E}} \right]} \right) = - \left[ {\nabla \cdot \overline{{\int\limits_{0}^{{p_{s} }} {q\vec{V}dp} }} } \right]$$

(7)

where P is precipitation; E is evapotranspiration; and $\nabla \cdot \int_{0}^{{p_{s} }} {q\vec{V}dp}$ quantifies the net moisture transport across the domain lateral boundary (hereafter, “moisture transport (MT)” for abbreviation). The bars and [] denote the average over the JJA season and the domain average over the terrestrial SE US, respectively.

The MT in Eq. 7 can be further partitioned into: MT caused by JJA mean circulation (MTM), subseasonal-scale eddies (MTE), and surface properties (MTS), respectively (e.g. Trenberth and Guillemot 1995; Seager et al. 2010; Seager and Henderson 2013):

$$\nabla \cdot\underbrace {{\overline{{\int\limits_{0}^{{p_{s} }} {q\vec{V}dp^{{}} } }} }}_{{MT}} = \underbrace {{\int\limits_{0}^{{p_{s} }} {\nabla \cdot\left( {\bar{q}\overrightarrow {{\bar{V}}} } \right)dp} }}_{{MTM}} + \underbrace {{\int\limits_{0}^{{p_{s} }} {\nabla \cdot\overline{{\left( {\overline{{q\prime }} \overrightarrow {{V\prime }} } \right)}} dp} }}_{{MTE}} + \underbrace {{\overline{{q_{s} \vec{V}_{s} \cdot\nabla p_{s} }} }}_{{MTS}}$$

(8)

In Eq. 8, the JJA mean (denoted by bar) is calculated using the average of 6-h (3-h for NARR) data in each summer. The primes are the 6-h (3-h for NARR) deviation from the JJA mean.

Equations 7 and 8 indicate that the variance of SE US summer precipitation can be introduced by evapotranspiration, MT processes associated with JJA mean flow (MTM in Eq. 8), subseasonal-scale eddies (MTE in Eq. 8), and surface properties (MTS in Eq. 8: surface moisture flux across the surface pressure gradient), or the interaction of these processes. Thus, a comprehensive analysis of the regional hydrological cycle according to Eqs. 7 and 8 enables the diagnosis and quantification of the processes that contributes to SE US summer precipitation variance.

Figure 6 shows the pair-wise relationship between terms in Eqs. 7 and 8 using observed precipitation and the reanalysis ensemble. Generally, both evapotranspiration and MT linearly correlate with precipitation (Fig. 6). However, the contribution of MT to precipitation variance is far larger than that of evapotranspiration. The linear regression between MT and precipitation has an R² value of 0.64, indicating that MT can explain the majority of SE US summer precipitation variance. In contrast, the variance explained by evapotranspiration is less than 10 %, suggesting its role is secondary compared to MT processes.

Fig. 6

Pairs plot showing the SE US summertime (JJA) moisture budget in the 1948–2007. The diagonal panel plots are the empirical PDFs constructed based on 60 data samples corresponding to the six components in moisture budget (Eqs. 7–8). The off diagonal panels show the scatter plot between the paired moisture budget components. The solid red curves are the locally weighted scatterplot smoothing (LOWESS) curves fitted to the data samples. The dashed red curves represent the upper and lower bounds of the 95 % confidence interval of the fitted LOWESS curve. The dashed blue lines are the best least square fitting lines

Further, the MTM predominates the variance of MT and precipitation. The reanalysis ensemble suggests that the MTM explains more than 90 % of the MT variance, and shows a near one-to-one relationship with precipitation. Its contributions to summer precipitation outweigh that of the MTE and MTS (Fig. 6). The MTE (associated with subseasonal-scale systems, such as hurricane land fallings and frontal systems) usually contributes to extreme weather events over the SE US (e.g., Kunkel et al. 2010; Barlow 2011; Prat and Nelson 2013). However, on seasonal scales, their contributions to SE US summer precipitation variance are relatively small and nonlinear (Fig. 6).

Overall, the analysis of the atmospheric hydrological cycle suggests that the variance of SE US summer precipitation is primarily controlled by the variance of MT, while evapotranspiration is secondary. Furthermore, most MT variance (90 %) can be explained by the MTM, indicating the importance of the seasonal mean circulation in regulating the variance of SE US summer precipitation.

Appendix 2: Variance change partition without permutation

Without permutation, the MTM variance in state 1 to state 2 are expressed as $\text{var} \left( {{\mathbf{Y}}_{{\mathbf{1}}} } \right) = \text{var} \left( {\int_{0}^{{p_{s} }} {\nabla \cdot \left( {q_{1} \vec{V}_{1} } \right)dp} } \right)$, and $\text{var} \left( {{\mathbf{Y}}_{2} } \right) = \text{var} \left( {\int_{0}^{{p_{s} }} {\nabla \cdot \left( {q_{2} \vec{V}_{2} } \right)dp} } \right)$, respectively. Accordingly, the changes of thermodynamic components (specific humidity) between state 1 and state 2 are Δq = q ₂ − q ₁, and the changes of dynamic components (wind) are $\Delta \vec{V} = \vec{V}_{2} - \vec{V}_{1}$. Thus, the changes of MTM variance in state 2 can be expressed as:

$$\text{var} \left( {{\mathbf{Y}}_{2} } \right) = \text{var} \left( {\int\limits_{0}^{{p_{s} }} {\nabla \cdot \left( {q_{1} \vec{V}_{1} } \right)dp} + \int\limits_{0}^{{p_{s} }} {\nabla \cdot \left( {\Delta q\vec{V}_{1} } \right)dp} + \int\limits_{0}^{{p_{s} }} {\nabla \cdot \left( {q_{1}\Delta \vec{V}} \right)dp} + \int\limits_{0}^{{p_{s} }} {\nabla \cdot \left( {\Delta q\Delta \vec{V}} \right)dp} } \right)$$

(9)

Unlike the term $\int_{0}^{{p_{s} }} {\nabla \cdot \left( {\Delta q_{\pi } \Delta \vec{V}_{\pi } } \right)dp}$ in Eq. (1), the term $\int_{0}^{{p_{s} }} {\nabla \cdot \left( {\Delta q\Delta \vec{V}} \right)dp}$ in Eq. (9) cannot be neglected because Δq and $\Delta \vec{V}$ is not necessarily a small deviation from the original state i.e., state 1. To verify this, we use hypothetical Monte-Carlo samples as examples. Assume ${\mathbf{Y}}_{{\mathbf{1}}} = a_{1} \cdot b_{1}$, and ${\mathbf{Y}}_{{\mathbf{2}}} = a_{2} \cdot b_{2}$. We draw 1,000 samples of both a₁ and b₁ from N(10, 0.8). In state 2, we keep a₂ the same as a₁, but draw b₂ from N(12, 1.0). Thus the variance difference between Y ₂ and Y ₁ is:

$$\delta \text{var} = \text{var} \left( {{\mathbf{Y}}_{{\mathbf{2}}} } \right) - \text{var} \left( {{\mathbf{Y}}_{{\mathbf{1}}} } \right) = \text{var} \left( {a_{1} \cdot b_{1} + \Delta a \cdot b_{1} + a_{1} \cdot \Delta b + \Delta a \cdot \Delta b} \right) - \text{var} \left( {a_{1} \cdot b_{1} } \right)$$

(10)

According to the 1000-time Monte-Carlo simulations of Y₁ and Y₂, the maximum likelihood estimator (MLE) of $\delta {\text{var}}$ is 83.81.

Neglecting Δa·Δb on the right-hand side of Eq. (10), then the estimated variance change would be $\delta \text{var}_{est} = \text{var} \left( {a_{1} \cdot b_{1} + \Delta a \cdot b_{1} + a_{1} \cdot \Delta b} \right) - \text{var} \left( {a_{1} \cdot b_{1} } \right)$. Without permutation, the absolute error of $\delta \text{var}_{est}$ in comparison with $\delta \text{var}$ is 34.40 on average, which is equivalent to 41 % of the total variance change (Fig. 7). It indicates that Δa·Δb makes substantial contributions to variance change, and thus cannot be neglected in Eq. (10) without permutation.

Fig. 7

Absolute errors of variance change estimation without considering the Δa·Δb term in Eq. (10). The bars denote the maximum likelihood estimator of absolute errors according to 1,000-time Monte-Carlo simulations; the lower and upper limits of the error bars denote the 99 % uncertainty range

Keeping Δa·Δb in Eq. (10), $\delta \text{var}$ is expressed as:

$$\begin{aligned} \delta \text{var} &= \text{var} \left( {a_{1} \cdot b_{1} +\Delta a \cdot b_{1} + a_{1} \cdot\Delta b +\Delta a \cdot\Delta b} \right) - \text{var} \left( {a_{1} \cdot b_{1} } \right) \\ & = \text{var} \left( {\Delta a \cdot b_{1} } \right) + \text{var} \left( {a_{1} \cdot\Delta b} \right) + \text{var} \left( {\Delta a \cdot\Delta b} \right) \\ & \quad + 2\text{cov} \left( {a_{1} \cdot b_{1} ,\Delta a \cdot b_{1} } \right) + 2\text{cov} \left( {a_{1} \cdot b_{1} ,a_{1} \cdot\Delta b} \right) + 2\text{cov} \left( {a_{1} \cdot b_{1} ,\Delta a \cdot\Delta b} \right) \\ & \quad + 2\text{cov} \left( {\Delta a \cdot b_{1} ,a_{1} \cdot\Delta b} \right) + 2\text{cov} \left( {\Delta a \cdot b_{1} ,\Delta a \cdot\Delta b} \right) + 2\text{cov} \left( {\Delta a \cdot b_{1} ,a_{1} \cdot\Delta b} \right) \\ \end{aligned}$$

(11)

In Eq. (11), 4 additional terms are added compared to Eq. (3); that are $\text{var} \left( {\Delta a \cdot\Delta b} \right)$, $2\text{cov} \left( {a_{1} \cdot b_{1} ,\Delta a \cdot\Delta b} \right)$, $2\text{cov} \left( {\Delta a \cdot b_{1} ,\Delta a \cdot \Delta b} \right)$, and $2\text{cov} \left( {a_{1} \cdot \Delta b,\Delta a \cdot \Delta b} \right)$. These four terms involve the interaction between Δa and Δb, which make their contributions to $\delta \text{var}$ inseparable.

In contrast, after permutation, ${\mathbf{Y}}_{{{\mathbf{1\pi }}}} = a_{1\pi } \cdot b_{1\pi }$, and ${\mathbf{Y}}_{{{\mathbf{2\pi }}}} = a_{2\pi } \cdot b_{2\pi }$. Δa _π  = a _2π  − a _1π and Δb _π  = b _2π  − b _1π are small deviation from a _1π and b _1π , respectively. Thus, they can be neglected in Eq. (10) without introducing significant errors. Neglecting Δa _π ·Δb _π , the variance change can be approximated as $\delta \text{var}_{est\_\pi } = \text{var} \left( {a_{1\pi } \cdot b_{1\pi } + \Delta a_{\pi } \cdot b_{1\pi } + a_{1\pi } \cdot \,\Delta b_{\pi } } \right) - \text{var} \left( {a_{1\pi } \cdot b_{1\pi } } \right)$.

According to the Monte-Carlo simulations of hypothetical time series, the errors introduced by neglecting the Δa _π ·Δb _π term in the permutation case is only 2.02 (~2.4 % relative error), which is substantially smaller than that without permutation (Fig. 7). It indicates that after permutation, the variance partition can be simplified by neglecting the Δa _π ·Δb _π term. Applying the permutation, as described in Sect. 3, to MTM time series, the contributions of thermodynamic and dynamics to MTM variance change can be quantified as in Eq. (4).