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Keywords:

  • probable maximum precipitation;
  • PMP;
  • Clausius–Clapeyron scaling;
  • rainfall extremes;
  • climate change

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model set-up
  5. 3. Results
  6. 4. Summary and conclusions
  7. Acknowledgements
  8. References

The sensitivity of squall rainfall to changes in atmospheric temperature is investigated. For instantaneous rainrates and accumulations up to one hour, extreme rainfall scales with Clausius-Clapeyron (CC) for temperatures below 24°C and at up to twice CC above 24°C. For longer accumulation periods and higher temperatures the scaling breaks down due to increased propagation of the squall line. For all periods, the storm average rainfall is found to scale at approximately 1.5 times CC over the entire range of temperatures. These results have implications for design parameters for infrastructure that is vulnerable to flooding and for climate change projections. Copyright © 2012 Royal Meteorological Society


1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model set-up
  5. 3. Results
  6. 4. Summary and conclusions
  7. Acknowledgements
  8. References

The relationship between the water-holding capacity of the atmosphere and atmospheric temperature is governed by the Clausius–Clapeyron (CC) equation. This equation infers that for a 1 K increase in temperature, the moisture-holding capacity of the atmosphere will increase by approximately 7% (Trenberth et al., 2003). It would therefore seem reasonable to expect that heavy precipitation will follow a similar scaling assuming that it is constrained by the precipitable water already in the atmosphere. General Circulation Models (GCMs) provide support for scaling of heavy precipitation with temperature, albeit weaker than CC (Allen and Ingram, 2002; Pall et al., 2007; O'Gorman and Schneider, 2009). Some observational studies suggest excess of CC scaling (Lenderink and van Meijgaard, 2008, 2010; Liu et al., 2009). Haerter and Berg (2009) and Berg et al. (2009) suggest that an observed shift to twice CC scaling results from the increasing influence of convective rainfall compared to large-scale rainfall on the statistics as local temperature increases. However, Lenderink and van Meijgaard (2010) have shown through analysis of observations and simulations with two regional models that this is unlikely to be true for summer rainfall.

An inherent weakness of using daily rainfall statistics from observations and models is that they do not resolve the life cycle of individual storms, and will potentially mask the effects of more intense storms. Furthermore, the GCMs and regional climate models used in previous studies are of relatively coarse resolution and do not explicitly resolve the convective processes that result in the heavy precipitation in the summer months. Here we take the approach of investigating the effects of temperature on an individual storm using a high-resolution convection-resolving model in order to identify any scaling that exists at the time-scale of the storm itself. For the individual storm, we simulate an idealized squall line. The idealized approach is preferable so that factors other than near-surface temperature, such as changes in large-scale forcing, changes in vertical profiles of moisture and temperature, and changes in the distribution of atmospheric aerosols, are removed from the analysis. A squall line was chosen as it is a well understood convective system (Weisman et al., 1988; Houze, 2004).

2. Model set-up

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model set-up
  5. 3. Results
  6. 4. Summary and conclusions
  7. Acknowledgements
  8. References

We used version 3.0 of the Weather Research and Forecasting (WRF) model with the Advanced Research WRF (ARW) dynamical core for this study. The WRF-ARW dynamics uses a mass coordinate system and casts the governing equations in flux form. In this way the discretization of the equations exactly conserves mass and scalars (Skamarock et al., 2008). The idealized squall line was initialized with vertical profiles of temperature and water vapour typical of midlatitude convection following Weisman et al. (1988), and a vertical wind profile with a 12 m.s−1 shear layer in the lowest 2.5 km of the atmosphere. Convection was initiated by a y-oriented line thermal with 1.5° C temperature excess with x-radius 10 km and z-radius 1.5 km placed 1.5 km from the ground. It was given three-dimensional structure by adding random temperature perturbations with a range of −0.1 to +0.1° C to the thermal. The model was run with a 1 km horizontal resolution on a domain measuring 800 km in the x-direction, 160 km in the y-direction and 20 km in the z-direction using 80 vertical levels, with open x-boundaries, periodic y-boundaries and a rigid lid with a 5 km absorbing layer. This domain was sufficiently large to prevent gravity wave reflections at the open boundaries. Cloud microphysics processes were parametrized for six classes of hydrometeor (water vapour, cloud water, rain, cloud ice, snow and graupel) (Hong and Lim, 2006) and for simplicity Coriolis force, surface physics and atmospheric radiative transfer were not included. Data were output at 5-minute intervals.

To examine the temperature response, the initial temperature in the model was increased uniformly throughout the atmosphere and the relative humidity was held constant. In this way the atmosphere remains close to dynamic balance for all temperatures. Furthermore, such an approach is consistent with GCM simulations that suggest that as global temperatures increase, relative humidity will remain largely unchanged as the hydrological cycle is strengthened (Held and Soden, 2006).

3. Results

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model set-up
  5. 3. Results
  6. 4. Summary and conclusions
  7. Acknowledgements
  8. References

3.1. Rainfall response

The idealized squall line was modelled for initial 2 m temperatures ranging from 19°C to 29°C at intervals of 1°C. Cooler initial temperatures did not result in organized convection in the model. Figure 1 shows Hovmüller diagrams of the mean rainfall intensity in the along-line direction, derived from 5-minute rainfall accumulations. For all temperatures, precipitation commenced approximately 30 minutes after the model initialization, with the system initially stationary. The rainfall pattern shows that, for all temperatures, the modelled squall line exhibits the typical structure of heavy rainfall due to strong updraughts at the leading edge with a large trailing region and a smaller leading region of lighter stratiform precipitation (Houze, 2004). For temperatures less than 24°C the system begins to propagate to the right, the direction of the initial wind shear, approximately 3 hours after initialization. For temperatures greater than 24°C, the system begins to propagate approximately 1 hour after initialization. As the temperature increases, the squall line propagates more quickly to the right. For all temperatures, the most intense precipitation occurs soon after initialization and continues up to about 4 hours.

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Figure 1. Hovmüller diagrams of the mean rainfall intensity (mm.h−1) in the along squall line direction for different initial temperatures (2 m).

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We investigate the relationship of precipitation intensity with temperature for the instantaneous rain rate (inferred from 5-minute accumulations), 1-hour, 4-hour and 8-hour accumulations. For each output time we have accumulations for these periods for every model grid square. From these data we find the 99.9th, 90th and 75th percentiles in order to examine the effects on extreme, heavy and moderate rainfall respectively within the system. The percentiles are computed directly from the data—we investigated fitting the data to generalized Pareto, generalized extreme value and gamma distributions, and found differences in the percentiles derived from the different methods to be negligible. Figure 2 shows the percentiles of rainfall for the instantaneous intensity (Figure 2(a)) and the three aforementioned accumulation periods (Figure 2(b)–(d)). For clarity of comparison between precipitation totals for different percentiles, we present precipitation totals using an index notation whereby the precipitation total at each temperature for each percentile of each accumulation time-scale is expressed relative to the precipitation total at the lowest temperature for the same percentile and accumulation time-scale. The index is defined by scaling the precipitation totals for the lowest temperature for each percentile and accumulation time-scale to 100 and using the same scale factor for precipitation totals at the other temperatures. The process is defined in Eq. (1) for the precipitation total, R, at temperature Ti relative to the precipitation total at temperature T0:

  • equation image(1)
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Figure 2. (a) Instantaneous rainfall intensity in a grid square index as a function of initial 2 m temperature, (b) 1-hour, (c) 4-hour, and (d) 8-hour accumulations in a grid square index as a function of initial 2 m temperature. The thick black lines show the 99.9th percentile, the medium lines show the 90th percentile and the thin lines show the 75th percentile. Rainfall data are expressed as an index relative to a value of 100 for the coolest 2 m temperature of each percentile. The dotted lines show Clausius–Clapeyron scaling and the dashed lines show twice Clausius–Clapeyron scaling.

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For the instantaneous rainfall intensity (Figure 2(a)), the 99.9th percentile follows CC scaling for temperatures below 24°C, but for higher temperatures the scaling increases to approximately twice CC. The 90th percentile (Figure 2(a)) exhibits CC scaling for the lower temperatures, but for higher temperatures the scaling appears to break down; a similar trend is apparent for the 75th percentile. In other words, the extreme values of rainfall intensity in the system scale more strongly with temperature at the expense of the more moderate rainfall. For 1-hour accumulations (Figure 2(b)), the 99.9th percentile follows CC scaling for the entire range of initial temperatures, with the 90th percentile exhibiting a similar trend for all but the highest temperatures. The 75th percentile does not appear to be affected by temperature. For the longer 4-hour (Figure 2(c)) and 8-hour (Figure 2(d)) accumulation periods the 99.9th percentile increases with temperature up to 24°C before dropping off again. The 90th percentile increases with CC scaling up to an initial temperature of 24°C and at higher temperatures the scaling breaks down. For the 4-hour accumulation, the 75th percentile exhibits twice CC scaling up to a temperature of 24°C (Figure 2(c)) and for the 8-hour accumulation a slightly weaker scaling is seen (Figure 2(d)) before breaking down.

The scaling for the 99.9th percentile of instantaneous rainfall intensity, which is twice CC for temperatures greater than 24°C (Figure 2(a)), is consistent with the observations of Lenderink and van Meijgaard (2008, 2010) for hourly precipitation, who found the increase in scaling to twice CC to occur at maximum daily temperature of around 23°C. However, the scaling for 99.9th percentile hourly precipitation (Figure 2(b)) is consistent with CC scaling for all temperatures in our case, and for longer accumulation periods no scaling with temperature is seen (Figure 2(c) and (d)). This suggests that either the rainfall intensities responsible for twice CC scaling do not last long enough to be seen in the hourly and longer accumulations, or that they do not remain over a grid square for long enough for accumulations in a grid square to represent the rainfall intensity. Figure 3 shows the propagation velocity of the squall line estimated from the gradients of the leading edge of the squall line in the Hovmüller diagrams in Figure 1. The propagation velocity appears to follow a twice CC scaling with a considerable departure from the trend between 22°C and 24°C. Twice CC scaling is seen in the 99.9th percentile of precipitation intensity at temperatures greater than 24°C. At 25°C, the propagation velocity is 4.5 m.s−1 which means that a single point in the squall line will pass over 16 grid squares in one hour, which may explain why the twice CC scaling seen in the instantaneous rainfall intensity is not reflected in the hourly rainfall, and no scaling at all for the 99.9th percentile is seen for the longer accumulation periods. Furthermore, the drop-off in propagation velocity at temperatures between 22°C and 24°C appears to be reflected in the rainfall accumulations for all three percentiles with peaks at 24°C (Figure 2(c) and (d)).

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Figure 3. Propagation velocity of the squall line in the direction perpendicular to its orientation (solid line). The dotted line shows Clausius–Clapeyron scaling and the dashed line shows twice Clausius–Clapeyron scaling.

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The point precipitation totals are heavily influenced by the propagation of the storm—as the temperature rises, increases in precipitation rate are offset by increases in the propagation velocity of the storm. By considering the maximum storm average precipitation accumulations, the effect of increased temperature on total rainfall from the storm can be quantified. We do this for the 1-hour, 4-hour and 8-hour accumulation periods, using the area of the footprint of the squall line during these periods as a proxy for storm size. Such an approach integrates all of the percentiles of the distribution and as such, the storm average accumulation for the idealized squall line will unambiguously show if storm total precipitation is constrained by the moisture-holding capacity of the atmosphere. Figure 4 shows the maximum storm average rainfall during the lifetime of the squall line for these accumulation periods as a function of initial 2 m temperature. For all three accumulation periods the scaling is approximately 1.5 times CC with no apparent increase in the scaling behaviour at higher temperatures as seen in the 99.9th percentile of the instantaneous precipitation intensity (Figure 2(a)). This provides support for our supposition that increases in the higher percentiles of rainfall intensity are offset by the breakdown in scaling with temperature for the lower percentiles. Furthermore, the scaling in excess of CC for the system as a whole suggests that the moisture-holding capacity of the atmosphere does not act to constrain rainfall intensity, but as the temperature increases a larger amount of moisture is entrained into the system from the surrounding areas.

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Figure 4. Maximum domain average rainfall accumulation as a function of initial 2 m temperature. The bottom solid line shows the 1-hour accumulation, the middle solid line shows 4-hour accumulation and the top solid line shows 8-hour accumulation. The dotted black lines show Clausius–Clapeyron scaling and the dashed black lines show 1.5 times Clausius–Clapeyron scaling.

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3.2. Vertical velocity response

For a simple single-column model, rainfall intensity, R, can be described by the product of the precipitation efficiency, κ, and the mass-moisture flux into the cloud (e.g. Wilson and Toumi, 2005) following Eq. (2):

  • equation image(2)

where q is the specific humidity, ρ is the density, w is the vertical velocity vector and zm is the height of the top of the moist layer. The overbar represents the temporal average. In our experiments the precipitation efficiency is 100% when the rainfall rate is the maximum, q is constrained by the CC equation and ρ is a constant. Therefore for scaling in excess of CC, changes in the vertical velocity vector must be responsible for increasing the rainfall intensity.

Figure 5(a) shows the maximum 5-minute mean vertical velocity at the cloud base as a function of temperature. We choose the 5-minute mean vertical velocity since the rainfall intensity is inferred from 5-minute accumulations, and we define the cloud base as the first eta level above the surface with cloud water mixing ratio greater than zero. For temperatures below 24°C there is no relationship between the vertical velocity and temperature, but at higher temperatures, vertical velocity scales with temperature at a rate approximately equal to CC scaling. To confirm that this increase in vertical velocity is responsible for the twice CC scaling seen in the 99.9th percentile of instantaneous precipitation intensity, we plot the maximum mass-moisture flux ( w) at the cloud base as a function of temperature (Figure 5(b)). As with the 99.9th percentile of instantaneous precipitation intensity, the scaling follows that of CC up to a temperature of 24°C and at higher temperatures the scaling becomes twice CC.

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Figure 5. (a) Maximum 5-minute mean vertical velocity at cloud base as a function of initial 2 m temperature. The dotted line shows Clausius–Clapeyron scaling. (b) Maximum 5-minute mass-moisture flux at cloud base as a function of initial 2 m temperature. The dotted line shows Clausius–Clapeyron scaling and the dashed line shows twice Clausius–Clapeyron scaling.

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4. Summary and conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model set-up
  5. 3. Results
  6. 4. Summary and conclusions
  7. Acknowledgements
  8. References

Idealized modelling of weather enables conclusions to be drawn that are not constrained by local geography or seasonal variability. The response of an idealized squall line to a particular atmospheric forcing may be considered typical of any convective system in that all of the ingredients for convection play an important role in its development, namely, low-level moisture, vertical motion and atmospheric instability. The idealized squall-line simulations show that for extreme (99.9th percentile) rainfall the scaling with temperature can be as much as twice CC adding the first convection-resolving model support, at sub-daily time-scales, to an increasing body of observational evidence that suggests midlatitude extreme rainfall totals can and do increase by more than the moisture-holding capacity of the atmosphere when there is an increase in temperature (Trenberth et al., 2003; Lenderink and van Meijgaard, 2008, 2010; Liu et al., 2009). This scaling in excess of CC only seems to be apparent for very short duration rainfall at the sub-hourly time-scale. At longer time-scales the scaling breaks down at higher temperatures suggesting consistency with the analysis of Berg et al. (2009) for summer rainfall in Europe when convective systems have a greater influence on the statistics. Our analysis suggests that rather than a physical breakdown in the scaling, it is a faster propagation velocity of the system that masks the increased rainfall intensity. This hypothesis is supported by the finding that the storm average rainfall exhibits 1.5 times CC scaling for all temperatures.

The scaling in excess of CC at temperatures higher than 24°C is caused by an increase in the maximum updraught velocity in the system which leads to an increase in the vertical flux of moisture into the storm that is larger than the increase in water vapour in the atmosphere. An analysis of the energy budget (not shown) confirms the increase in updraught velocity as vertical flux divergence in the lower levels of the troposphere leads to an export of kinetic energy from these levels that is imported to the upper levels through vertical flux convergence. There is no overall strengthening of the horizontal flux convergence in the lower levels with increasing temperature, but the horizontal flux divergence in the upper levels appears to play some role in driving the increasing velocities of the updraughts.

It is important to note that the idealized modelling experiments undertaken here represent only a starting point for further research into the response of precipitation events to higher temperatures. The idealized nature of the experiments means that effects including, amongst others, synoptic-scale forcing, rotation of the Earth, topography and surface fluxes were not taken into account. In reality, all of these forcing mechanisms can have a strong influence on the initiation and evolution of convective storms. Indeed, one of the main features of extreme rainfall that results in flooding is stationarity, which was not a feature of our experiments. Stationarity may occur due to topographic forcing, as a result of synoptic features such as quasi-stationary upper-air potential vorticity filaments, or more often from a combination of external forcing mechanisms. To provide further insight into the response of convective storms to higher temperatures in otherwise similar atmospheric environments, these experiments should be repeated for a range of case-studies where extreme rainfall is influenced by forcing mechanisms such as those just described.

However, the results of the idealized experiments, alongside the observational studies of Lenderink and van Meijgaard (2008, 2010) and Liu et al. (2009), suggest that the potential for extreme rainfall scaling in excess of CC needs to be an area of concentrated research. Such scaling of rainfall in excess of CC has implications for climate change predictions, with GCMs failing to capture the super-CC scaling of precipitation extremes in the midlatitudes (Allen and Ingram, 2002; Pall et al., 2007; O'Gorman and Schneider, 2009). Regional climate studies using cloud-resolving models may be the only way to assess the true impact of global warming on extreme precipitation. There are also implications for Probable Maximum Precipitation (PMP), which is frequently used as a design parameter for critical infrastructure. The derivation of PMP is typically based on a simple CC scaling assumption although more sophisticated methods, such as the storm model approach of Collier and Hardaker (1996), have been proposed. However, for the sub-12-hour time-scales under consideration here, the storm model approach was shown to estimate similar PMP values to those of the simple CC scaling method. Our work strongly suggests that current methodologies are likely to lead to underestimates of PMP as was first suggested by a lower-resolution model study (Abbs, 1999). A conservative revision of PMP estimates assuming up to twice CC scaling for sub-daily rainfall point and catchment totals should be considered.

Acknowledgements

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model set-up
  5. 3. Results
  6. 4. Summary and conclusions
  7. Acknowledgements
  8. References

This work was funded by the Natural Environment Research Council programme: Flood Risk from Extreme Events, reference number NE/F011822/1. We would also like to thank BP plc (Upstream Environmental Technology Program) for support, and two anonymous reviewers for their comments that resulted in improvements to the manuscript.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Model set-up
  5. 3. Results
  6. 4. Summary and conclusions
  7. Acknowledgements
  8. References