Atlantic tropical cyclones water budget in observations and CNRM-CM5 model

Abstract

Water budgets in tropical cyclones (TCs) are computed in the ERA-interim (ERAI) re-analysis and the CNRM-CM5 model for the late 20th and 21st centuries. At a 6-hourly timescale and averaged over a 5$^{\circ } \times 5^{\circ }$ box around a TC center, the main contribution to rainfall is moisture convergence, with decreasing contribution of evaporation for increasing rainfall intensities. It is found that TC rainfall in ERAI and the model are underestimated when compared with the tropical rainfall measuring mission (TRMM), probably due to underestimated TC winds in ERAI vs. observed TCs. It is also found that relative increase in TC rainfall between the second half of the 20th and 21st centuries may surpass the rate of change suggested by the Clausius–Clapeyron formula. It may even reach twice this rate for reduced spatial domains corresponding to the highest cyclonic rainfall. This is in agreement with an expected positive feedback between TC rainfall intensity and dynamics.

Appendix: Technical procedures

1.1 Compositing technique

Composite databases were constructed relative to the TC center at its lifetime-maximum area averaged rainfall intensity. For the ERAI reanalysis, we chose to base the compositing on the observed IBTrACS dataset so that rainfall percentiles could be compared with those obtained for TRMM. For the CNRM-CM simulations, we used the objective TC tracking software developed at CNRM and described in Chauvin et al. (2006). This software was also used in Daloz et al. (2012) for the same simulations as the ones described here. In order to ensure independency in the variables, the choice was made to consider only one occurrence per TC. Since we are interested in extreme rainfall, the lifetime-maximum area averaged rainfall intensity (over the 5$^{\circ }\times 5^{\circ }$ box surrounding the TC) was searched for and the associated TC occurrence selected. Only TCs south of 30$^{\circ }$N are considered and previous occurrence of the TC step must be defined in order to allow computation of the column water tendency as well as the MC component. For observations, the tropical storm level was a necessary condition for selection while for the model, a threshold of 15 m$s^{-1}$ was considered. In the end, we constructed a database of TC fields for computation of the water budget as a collection of 5$^{\circ }\times 5^{\circ }$ square boxes centered on the TC centers. The numbers of selected TCs for each dataset are 310 for ERAI, 159 for ERAI-TRMM, 688 for the present simulated climate and 453 for the future.

Moisture and wind are instantaneous prognostics in the models while rainfall, evaporation and column water are accumulated over the last 6 h. Consequently, we calculated the average moisture convergence between the occurrence of maximum rainfall and the previous 6-hourly prognostic. All the water budget terms are representative of the same 6-h period preceding the maximum rainfall. For an exact water budget computation, MC should be calculated at the model timestep and then averaged over the 6 h preceding the maximum rainfall. Unfortunately, this diagnostic is often not available in model outputs and we assume that the fluctuations of wind and moisture within a 6 h period are sufficiently small to be neglected in the 5$^{\circ }\times 5^{\circ }$ area average.

1.2 Confidence intervals

Figures 1 and 3 have been calculated in the same way, based on the rainfall percentiles distribution. For each TC occurrence selected, a spatial average is performed on each component of the water budget over the 5$^{\circ }\times 5^{\circ }$ box. Area-averaged rainfall values are sorted in an ascending order and all terms of the water budget are sorted accordingly. A cubic spline is applied to the resulting series. The aim of smoothing is to consider an average of the water budget components inside a rainfall percentile (i.e. the conditional expectation of the component given rainfall). The water budget is expected to be closed as for each percentile all the water budget components are computed from the same TC occurrences and in the same way. To construct confidence intervals of the water budget terms, a bootstrap process is constructed as follows:

• 1. generate one thousand random samples of N rainfall among N with replacements, N being the number of TC occurrences,

• 2. class the area-averaged rainfall in ascending order for each sample and sort the other terms with the same ranking as rainfall,

• 3. cubic spline smoothing of all terms, except rainfall, with 100 equivalent degrees of freedom in order to smooth sample fluctuations due to the following percentile selection,

• 4. select rainfall percentiles and corresponding smoothed conditional expectations of the other terms,

• 5. calculate the mean and the 5 and 95% probabilities for each rainfall percentile or conditional expectation of the other budget terms.

The sorting is applied to rainfall, thus the smoothing is not necessary on the rainfall component since the sorted distribution is already very smooth.

For Fig. 5, a slightly different method was adopted. Rainfall percentiles were calculated first for the present and future climate simulations. Then, differences between percentiles were calculated. Indeed, it is not possible to calculate differences between the raw series since the number of selected cases are not the same in present and future simulations. Since we are only considering rainfall, no need for smoothing is required.

For the ratio of MC and evaporation we no longer consider conditional expectations. When considering sub-domains of the 5$^{\circ }\times 5^{\circ }$ square box, rainfall and the other components of the water budget are not always co-located. This is true not only for evaporation, the spatial structure of which is different from rainfall but also for MC, although its spatial structure is expected to be very similar to the rainfall pattern. The reason for such a shift comes from the way in which we calculated the MC component. Indeed, as we mentioned above, the MC component is computed as the average between its values at t and t−6h timesteps. Thus, the aggregation operates on fields centered on different locations while the rainfall is accumulated over the box centered on the TC at time t. Such shifts in the MC and rainfall components lead to the inability to compute the conditional expectations on sub-domains of the TC. Consequently, the distributions of the relative changes were computed from each water budget component. For rainfall, this is completely coherent with previous computations but it is different for evaporation and MC. The figure aims nevertheless to show that MC behaves like rainfall in a warming climate whilst evaporation does not. Given that MC is the main contribution to rainfall, it can be concluded that rainfall changes follow those of MC. It should be noticed that for ratios, no smoothing was applied as each variable was ranked separately, leading to smooth distributions.

1.3 Degrees of freedom for the cubic-splines

Conditional computation of the water budget to rainfall results in highly variable terms for a given rainfall percentile, due to sampling effects. To reduce the noise associated with this variability a cubic-spline is applied to the raw data before calculating the conditional expectation for each rainfall percentile. This implies the determination as objectively as possible of the degree of smoothness that we wish to apply to data. The choice of the number of degrees of freedom has a strong influence on the width of the confidence intervals shown in the figures related to the water budget. The sorted area averaged rainfall is, by definition, a smooth curve. Thus, it was not smoothed in figures showing the raw budget for present climate. In the other figures related to anomalies we applied the same smoothing as for the other terms. Since we are interested by the mean behavior of the other components as rainfall increases, we wish to consider curves as smooth as possible for these terms, without losing the link with the rainfall distribution. To illustrate the way in which we choose the degrees of freedom, Fig. 7 shows one of the bootstrap samples for the MC term as well as cubic-splines with different degrees of freedom that were applied to the raw data. It is clear from Fig. 7 that a high number of degrees of freedom do not filter the noise enough. With numbers between 10 and 20, the overall shape of the raw data is conserved and noise is substantially filtered. The slope of the curve is better represented at the boundaries of the dataset at 10 than at 5 degrees of freedom. We made the choice in this study to consider 8 degrees of freedom except in Fig. 4 where we used 5.

Fig. 7

An example of the smoothing procedure. Grey dots are the raw data conditional expectations of the MC component for one selection of TC rainfall (among the 1000 used for bootstrap) in mm day$^{-1}$. Curves correspond to different values of degrees of freedom

1.4 Rate of change of water vapor and dynamical convergence

Since water vapor and dynamical convergence are highly dependent one cannot expect that their respective contributions will be additive. Nevertheless, by making some simple assumptions, it may be nearly the case. Indeed, the MC term may be decomposed as follows (Wu et al. 2013):

$$\begin{aligned} div(VQ) = -\frac{1}{g} \nabla .\left( \int _{p}qVdp\right) = -\frac{1}{g} \int _{p}(q.\nabla V) dp -\frac{1}{g} \int _{p}(V.\nabla q) dp \end{aligned}$$

(2)

where the $\nabla$ operator represents the divergence of the wind vector V or the gradient of the scalar specific humidity q and g is the gravity acceleration (9.81 ms$^{-{2}}$).

In their study, Wu et al. (2013) refer to the first term on the right side as the wind convergence part (WCP) and to the second as the moisture advection part (MAP). The second term is of second order compared to the first one and if neglected, one can consider that the relative rate of change of MC is the sum of the relative rates of change of water vapor and dynamical convergence. But, since the terms are vertically integrated, one must be cautious to integrate the dynamical convergence with vertical weights proportional to the lapse rate of change of water vapor. If this is not done, upper level TC divergence will inhibit lower level convergence in the integral. For the computation of the dynamical convergence, we calculated the water vapor lapse rate averaged over space and time. Thus a constant set of vertical weights was applied to the present and future TC composites.

Fig. 8

Relative change of a water vapor and b dynamical convergence for the 6 classes of rainfall spatial aggregation and normalized by the mean change in SST. Dashed bold lines correspond to rates of changes of respectively one and two Clausius–Clapeyron rates of change. Units are percent by Kelvin

Figure 8a, b show that the relative rate of change of water vapor does not strictly follow the Clausius–Clapeyron formula as would be expected if TCs only experienced an increase in surface temperature. In the scenario, however, advective terms are at play which enhance the water vapour increase (this probably contradicts the assumption of their negligible role in the water budget). In contrast, the dynamical convergence increases at a rate lower than CC (maybe in coherency with fairly little increase in winds). The conclusion of this supplementary analysis is that rainfall increase at rates higher than CC may be explained by changes in dynamics as suggested by (Trenberth et al. 2007) but the respective contributions of each component are not easy to calculate since there is a highly non-linear relationship between thermal and dynamical changes.