Kinetic energy of rain and its functional relationship with intensity

Abstract

The rain kinetic energy (KE) is a widely used indicator of the potential ability of rain to detach soil. However, rain kinetic energy is not a commonly measured meteorological parameter. Therefore, empirical laws linking the rain kinetic energy to the more easily available rain intensity (I) have been proposed based on drop-size and drop-velocity measurements. The various mathematical expressions used to relate rain kinetic energy and rain intensity available from the literature are reported in this study. We focus our discussion on the two expressions of the kinetic energy used: the rain kinetic energy expended per volume of rain or volume-specific kinetic energy (KE_mm, J m⁻² mm⁻¹) and the rain kinetic energy rate or time-specific kinetic energy (KE_time, J m⁻² h⁻¹). We use statistical and micro-physical considerations to demonstrate that KE_time is the most appropriate expression to establish an empirical law between rain kinetic energy and rain intensity. Finally, considering the existing drop-size distribution models from literature, we show that the most suitable mathematical function to link KE and I is a power law. The constants of the power law are related to rain type, geographical location and measurement technique.

Introduction

Empirical and process-based soil erosion models often use rain kinetic energy (KE) as the rain erosivity index: e.g. in splash erosion modelling (e.g. Poesen, 1985) and in modelling sheet and rill erosion, such as in SLEMSA (Elwell, 1978), in EUROSEM (Morgan et al., 1998a, Morgan et al., 1998b) or in RUSLE (Renard et al., 1997).

Basically, the rain kinetic energy results from the kinetic energy of each individual raindrop that strikes the soil. The information provided by drop-size distribution (DSD) measurements combined with fall velocity measurements or empirical laws linking terminal fall velocity (V_t) and drop diameter (D), allow one to calculate the rain kinetic energy. DSD data have been obtained using various techniques (e.g. flour pellet, filter paper, oil immersion, electro-mechanical or optical devices, meteorological radar). Such measurements usually do not provide continuous data in space and time. An exception is the study from Doelling et al. (1998) who reports 7 years of DSD measurements in northern Germany. Hence, the introduction of more specific devices that allow continuous and direct rain kinetic energy measurements (e.g. Madden et al., 1998, Jayawardena and Rezaur, 2000a) will hopefully enlarge the availability of rain kinetic energy datasets. Nevertheless, rain kinetic energy is still widely calculated from DSD measurements combined with empirical V_t(D) laws (e.g. Laws, 1941, Gunn and Kinzer, 1949, Beard, 1976). Due to the sporadic availability of DSD measurements, data obtained from measurement campaigns were analysed in order to establish empirical relationships between KE and rain intensity (I). Assuming that the DSD samples used to establish the KE–I relationship were representative, KE can be calculated directly for any rainfall event from I using the KE–I relationship. Actually, rain intensity data, which are widely available, are obtained in a straightforward manner in comparison to KE.

The objective of this study is to demonstrate how the rain kinetic energy should be expressed when one wants to relate KE to I and then to find the most suitable mathematical expression linking both parameters. In Section 2, the two existing expressions of KE are discussed. A (non-exhaustive) review of the literature yields the different formulations used to relate KE and I in Section 3. Next, we discuss the statistical and micro-physical basis which needs to be considered when linking KE and I.