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II. Mechanics of Rainfall Erosion

II. Mechanics of Rainfall Erosion

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114



DWIGHT D. SMITH AND WALTER H. WISCHMEIER



servation Service started their work on raindrop velocity, size distribution, and energy in relation to intensity. This marked a turning point in

water erosion research. It was the beginning of the concept that erosion

is a work process for which the energy is supplied by the falling raindrops and the slope of the land down which the runoff flows.

A. R A ~ ~ R O

CHARA~~ERISTICS

P

Study of rainfall momentum and energy in relation to erosion requires

knowledge of the determining factors-raindrop mass, size, size distribution, shape, velocity, and direction. Direct measurement of momentum

has been attempted by use of torsion balances (Neal and Baver, 1937) as

discussed in an unpublished manuscript by N. W. Hudson,l but difficulties of wind shielding and adhering water drops have prevented successful development of the instrument. Hudson experimented with a

kinetic energy recorder using a circular fan or paddle wheel. His most

successful device utilizes a receiving diaphragm over a metal cylinder in

which the noise of the impinging drop is picked up by a sensitive microphone. The output voltage from the microphone is passed through an

amplifier, rectifier, and special recording milliammeter to give a trace

with time that is proportional to the momentum of the drops.

Drop size distribution in natural rain was first investigated with respect to erosion as a phase of rainfall simulator development (Laws and

Parsons, 1943). Laws and Parsons used the flour method of drop size

measurement (Bentley, 1904). Their data and those of others show an

increase in median drop size with increases in rain intensity.

Drop size distribution is described by a parameter, D50,commonly

called the median drop size. Median, in this case, refers to the midpoint

of the total volume. The combined volume of all drops smaller than D5,,

equals the combined volume of all drops larger than D50.

Drop size distributions measured by different investigators are frequently compared by the relationships of median drop size to rainfall intensity in the respective sets of data. Laws and Parsons combined data

by Defant and Lenard with their own measurements secured at Washington, D. C. The relationship of median drop size to intensity was described by the equation:

DjO = 2.23 ZOJS2

(1)

in which Z is intensity in inches per hour.

Another equation relating the two parameters was reported by Best

(1950):

DZo = 0.69l”’ APT

(2)

1 Senior Research Engineer, Federal Department of Conservation and Extension,

Henderson Research Station, P.B. 4, Mazoe, Southern Rhodesia.



RAINFALL EROSION



115



in which n, A, and p are empirically derived constants, and I is intensity.

Spilhaus (194813) concluded that a rain can be completely described by

the median drop diameter and rain intensity.

The increased interest in measuring drop size distribution in the late

1940’s was due to measurement of rainfall by radar and the need for

understanding the mechanics of precipitation (Atlas and Plank, 19%).

These studies related drop size distribution and liquid water volumes

per unit volume of the atmosphere (m?) to the radar reflectivity factor 2

by empirical equations of the exponential type.

Drop size measurements of nonorographic rain have all shown fair

to good agreement when compared on the basis of the relationship of

median drop diameter to intensity. Variations within groups, however,

has been high. These data were from widely scattered locations including, in addition to that of Laws and Parsons, Ottawa, Canada

(Marshall and Palmer, 1948), Hawaii (Anderson, 1948; Blanchard, 1953;

Blanchard and Spencer, 1957), Japan (Mihara, 1952), Illinois (Jones,

1956), England (Mason and Andrews, M O ) , and Southern Rhodesia

(Hudson, 1961a).

Several of the investigations recognized a different relationship of

diameter to intensity for different types of rain. Orographic rain, in

which drops are formed at low altitude and in warm cloud conditions, is

an example. In these rains, drops seldom exceed 2.0 mm. in diameter and

intensities generally do not exceed 1 inch per hour. Median drop diameters for this type rain are about half those in nonorographic rain of the

same intensity (Blanchard, 1953).

The number of drop size samples secured at intensities higher than

2 inches per hour has been limited. Maximum intensities sampled were

approximately as follows: Laws and Parsons 4.5; Jones 4.0; Mason and

Andrews 4.0; Mihara 5.0; and, Hudson 7.6 inches per hour. Hudson

found a progressive trend toward larger drop sizes up to and including

his class interval ending at 4.5 inches per hour. His median drop diameter peaked between 3.0 and 4.0 inches per hour and decreased thereafter. A similar trend had been noted by Mihara.

The shape of raindrops as they strike the surface of the earth is not

spherical owing to differential air pressure created by the falling drop.

The resultant shape approximates an ellipsoid flattened on the bottom.

This change in shape is of little consequence for drops less than 2.6 mm.

diameter (Spilhaus, 1948a). The three principal factors that control the

shape of larger water drops are surface tension, hydrostatic pressure,

and external aerodynamic pressure ( McDonald, 1954).

The change in shape of a raindrop has significance from an erosion

standpoint in that it affects the velocity (Laws, 1941) and the impaot



116



DWIGHT D. SMITH AND WALTER H. WISMMEIER



force per unit area of soil (Ekern, 1951). Laws describes the action of

the large drops as vibrating between vertical and horizontal oblateness

with a frequency depending upon size. It is generally agreed, however,

that drop shape is stable after terminal velocity has been attained. The

phenomenon of drop vibration has been studied by other meteorologists

(Blanchard, 1949, 1950; Magono, 1954; Gunn, 1949). This change in

shape of larger drops is of high importance in rain simulator work where

large drops at low fall height are used to secure high impact energy

values.

Study of the change in shape of water drops has been conducted

largely in the laboratory. Natural raindrops were measured in Illinois by

photographic methods (Jones, 1959). Of 1783 drops measured, 569 were

spherical, 496 oblate, 331 prolate, and 387 unclassified. All drops of a

given size did not have the same axis ratio, although definite trends were

apparent with change in drop diameter. When all drops were included

and averaged by class intervals, ratios were generally larger than when

only oblate drops were considered. Values taken from Jones’ hand-fitted

curves are shown in Table I.

TABLE I

Axial Ratios of Raindrops As Measured by Jonesa



D



All raindropsb



( mm. 1



( b’a)

0.99



0.92



0.85

0.78

0.71

a

b



Oblate dropsb

( b’a)

0.93

0.87

0.81

0.74

0.68



From Table 2 of Jones ( 1959 ) .

Ratio of the vertical to the horizontal drop axes = b/*.



The theoretical values of Spilhaus are in good agreement with the

oblate values of Jones except for the maximum size drops.

The fall velocity of raindrops was studied by Laws to assist in understanding the action of rain in the eroding of soil (Laws, 1941). He used

photographic equipment to measure drop velocity. His studies covered

a drop range from a little over 1 mm. to 6 mm. diameter. Laws’ values

were significantly higher than earlier velocity measurements, probably

because of less air turbulence and a more accurate method of measurement. They were essentially equal to the terminal velocities of water

drops in stagnant air that Gunn and Kinzer (1949) measured by inducing

an electric charge and producing pulses on an oscillograph record.

Terminal velocities read from their data and that of Mihara (1952) are

shown in Table 11.



117



RAINFALL EROSION



In natural rain, air turbulence can act either to increase or to decrease

drop velocity. The magnitude of air turbulence during rainfall and the

effects of drop velocity have not been studied. Winds, however, have an

appreciable effect on drop velocity. A horizontal wind increases terminal

drop velocity by the reciprocal of the cosine of the angle of inclination

of the rain with the vertical. In a heavy, driving rain with a 3-mm.

median drop size and a 30-degree angle of inclination, the velocity would

be increased 17 per cent and the kinetic energy would be increased 36

per cent. In detailed erosion studies where rain intensity, momentum or

kinetic energy are related to soil movement, this factor cannot be

neglected (Hudson, 1961a). It could not be considered in analyses of

TABLE I1

Terminal Velocity of Waterdrops and Distance Necessary to Attain 95 Per Cent

Terminal Velocity



Drop

sue



(mm. 1

0.25

0.50

1.00

2.00

3.00

4.00

5.00

6.00

5



b

c



Terminal

velocitya

(m./sec.)

1.0

2.0

4.0

6.5

8.1

8.8

9.1

9.3



Fall to reach

95 per cent

terminal

velocityb

(m.)



2.2

5.0

7.2



Terminal

velocityc

(m./sec.)



2.0

4.1

6.3

7.6



7.8



8.5



7.6

7.2



8.8

9.0



Laws (1941) and GUM and Kinzer (1949).

Laws (1941).

Mihara ( 1952).



United States data because only recently were wind recording instruments installed at runoff plot sites.

Kinetic energy of rainfall is important in erosion studies since erosion

is a work process and much of the energy required to accomplish this

work is derived from the falling raindrops. Kinetic energy of rainfall

can be more easily computed than directly measured. The key to the

computation is the intensity-drop size relationship since intensities may

be secured from recording rain-gauge records. With terminal velocities

and drop mass known, the calculation may be easily made (Wischmeier

and Smith, 1958; Mihara, 1952; Hudson, 1961a). Drop size distribution

as published by Laws and Parsons provided the desired intensity classes

for this purpose. Energy values computed from median drop size data

alone are from 10 to 15 per cent too high.



118



DWIGHT D. SMITH A N D WALTER H. WISCHMEIER



The energy equation by Wischmeier and Smith is as follows:



El, = 916



+ 331 loglo Z



(3)



in which Ek is kinetic energy in foot-tons per acre-inch of rain, and Z is

intensity in inches per hour.

Hudson compared kinetic energies computed from rain-intensity relationships by several investigators. These are shown in Fig. 2. The difference in energy is probably accounted for by the drop size distribu-



W

I



KW



B

w

5 200-



0'



7.0

4.4



Hudron



* 300-



Miham



i



i



i



3.8



4



i



Rain intensity- Inches per Hour



-



i



-!



FIG.2. Kinetic energy of rain as determined by different investigators. (From

Hudson, 1961a.)



tion-intensity relationship used in the computations since drop velocity

measurements all agree rather closely. The increasing spread above 4

inches per hour between the curve of Hudson and that of Wischmeier

and Smith may indicate an overestimation of kinetic energy by the

Wischmeier and Smith equation for the higher intensities. Laws and

Parsons' data used in the computations contained few samples above 2

inches per hour intensity. Both Hudson and Mihara observed a change

in drop size distribution for the higher intensities with the median size

drops tending to predominate. This could result in leveling of the energy

curve for the higher intensities.



RAINFALL EROSION



119



B. RAINDROP IMPACT

AND SPLASH

The place of raindrop impact in water erosion began to emerge

during the 1930's. The concentration of soil in runoff water was found

to increase rapidly with raindrop energy (Laws, 1940). Intercepting

raindrop impact with a straw mulch reduced erosion 95 per cent in rain

simulator studies in Ohio (Borst and Woodburn, 1942). To isolate the

mechanism of the reduction, a wire platform was used to support

the straw mulch 1 inch off the bare soil so as to remove the energy of

the drops but not to retard the flow of runoff. Raising the mulch off the

ground did not decrease its effectiveness. This demonstrated that it was

the impact of the drops on the bare soil and not the runoff velocity that

detached the large quantities of soil washed from the unprotected plot.

Splash erosion has been established as the initial phase of the water

erosion process (Ellison, 1944a, b, 1947a,b, c). It is the true sheet erosion

process. Ellison's studies demonstrated that erosion can proceed without

runoff due to progressive movement of soil particles downslope by splash

action. His studies based on direct measurement of splash showed that 75

per cent of soil splash on a 10 per cent slope moves downslope and only

25 per cent moves upslope. This compares with 60 per cent downslope

and 40 per cent upslope movement of spashed sand on a slope of the

same steepness (Ekern and Muckenhim, 1947). Ekern developed the

equation that relative downslope movement by splash is approximately

equal to 50 per cent plus the per cent slope; he substantiated the equation by laboratory measurements. This differential movement by vertical

raindrops is explained by the fact that the downhill splash travels farther

before recontacting the soil surface and that the angle of impact results

in a greater downslope conponent. This is particularly important on

extremely steep slopes (Mihara, 1952). Wind effects in the field, however, may upset this pattern. Losses from oriented soil pans exposed to

natural rainfall in New York (Free, 1952) showed three times the losses

from pans facing the direction of the storm as from pans facing the opposite direction.

Sand particles apparently are more readily moved in splash than are

finer soil particles (Woodburn, 1948; Woodburn and Kozachyn, 1956).

Ellison reported that splash samples contained higher percentages of

sand and gravel than did the original soil, but that the size of particles

moved decreased with a decrease in drop size and velocity. The quantity

splashed increased with drop size, drop velocity, and rain intensity. Sand

particles 1 to 2 mm. in size were readily splashed by the larger drops,

and there was an apparent downslope movement of particles as large as

8 mm. (Ellison, 1944a).



120



DWIGHT D. ShlITH AND WALTER H. WISCHMEIER



Sand transport by splash from a 5.8-mm. drop showed a net increase

with increased height of fall (Ekern, 1951). However, a series of rises

and falls in amount of splash occurred with changes in height of fall that

were less than the height required for terminal velocity. This resulted

from drop oscillation between oblate to prolate shapes which affected

the impact force per unit area of drop impact.

The height and distance of splash depends upon the soil surface condition and the fall velocity of the drop (Mihara, 1952). The maximum

splash distance reported by Mihara was 37 inches and the maximum

height 12 inches when drops impacted on cultivated soil. On compacted

soil, the splash from 6-mm. drops reached a distance of 59 inches. Studies

in England showed that fragments of drops falling on wet paving stones

rebounded to maximum heights of 12 to 24 inches (Mason and Andrews,

1960). Ellison reported that the maximum distance of splash resulting

from a 5.9-mm. drop falling at 18 feet per second was 60 inches and the

height about 15 inches. Some stone fragments of 4-mm. size were

splashed 8 inches, and 2-mm. particles were splashed 16 inches.

With drop size and velocity constant the important aspects affecting

the magnitude of splash from a smooth soil surface without clods are

the resistance of the soil to deformation by the drop and the depth of

water film. The condition represented by the flour pan used in measuring

drop size distribution illustrates the extreme ease of penetration and resulting absence of splash. The other extreme, which produces maximum

splash or shattering of the drops, is illustrated by the paving stones with

a thin film of water (Mason and Andrews, 1960).

Ellison reported maximum splash shortly after the surface was wetted.

Thereafter splash decreased with increased time of water application,

possibly from development of a deeper water film or the removal of

readily detached soil particles. Investigations in Germany (Kuron and

Steinmetz, 1958) showed first an increase in splash with increasing depth

of water film, followed by a decrease in splash with continued increase

in depth. However, total soil transport was increased owing to increased

turbulence of the runoff. Similar results were reported from Japan

(Mihara, 1952). In this study the measured angle of splash was about

32 degrees when the water level in the sand was at the sand surface, but

as the depth increased the angle increased rapidly until at about 2 mm.

depth it was 75 degrees and at 10 mm. depth and above it was essentially

90 degrees.

Mihara studied sand deformation under raindrop impact. He found

that the diameter of the drop crater was always greater than the drop

diameter and that it increased with increasing velocity. The depth of

penetration varied with the compactness of the sand and the velocity of



121



RAINFALL EROSION



the drop. Figure 3 shows the steps in drop crater formation (Mihara,

1952).

The different investigators are not agreed on the relation of splash

to waterdrop parameters. Bisal ( 1960) shows detachment proportional

to the 1.4 power of drop velocity. Ekern (1951) shows splash proportional to kinetic energy when amount of applied water is constant. Rose

(1960) reports that soil detachment is more closely related to momentum

per unit area and time of rain than to kinetic energy. Free (196Ob)

found that splash losses from sand varied as the 0.9 power of drop

energy, but from a silt loam soil as the 1.5 power of the energy.

0



sec



I

I400

I

700



1

400



I

I50

1

70



FIG.3. Steps in drop crater formation. (From Mihara, 1952.)



The dispersing action of raindrop impact on bare soil, which results

in formation of a soil crust that reduces infiltration and increases surface

runoff, has been shown by many investigators, including Lowdermilk

(1930), Hendrickson (1934), Duley and Kelly (1939), Laws (1940),

Borst and Woodburn (1942), Ellison (1947c), Levine (1952), and McIntyre (1958). It is most pronounced on fine-textured and weakly aggregated soils. Coarse-textured soils show the greater amount of splash,

but with fine-textured soils the particles tend to compact and seal during

the process.

The relative importance of splash and runoff erosion have not been

clearly established (Ellison, 1947d; Bennett et al., 1951). Splash erosion

denudes the top of the slope owing to differential downslope splash

movement; runoff erosion tends to increase as it creates rills in ever-



122



DWIGHT D. SMITH AND WALTER H. WISCHMEIER



increasing size toward the bottom of the slope. For the Shelby and

Marshall soils of the Midwest (Smith et nZ., 1945) depth of surface soil

decreased down from the crest of the slope as steepness and length increased, but when the land slope flattened the depth increased.



C. SHEETAND MICROCHANNEL

FLOW

Runoff as sheet and microchannel flow is the second phase of the

water erosion process. The raindrop impact-splash process has been

shown to have high detachment but low transport capacity. Sheet and

microchannel flow has low detachment capacity and high transport

capacity. IYhen both act together on an unprotected soil, erosion soon

reaches serious proportions.

Little (1940) considered that the erosivity of flow was a function of

turbulence and, therefore, a function of velocity squared per unit of flow

depth. He developed equations for flow in terms of rainfall, hydraulic

roughness, runoff coefficients, and ground slope profiles. He recognized

the extreme complexity of the problem and suggested that compromise

with theory, use of empirical constants, and judgment would enter into

practical applications to field problems.

In a study of runoff from a paved surface under simulated rainfall,

Izzard and Augustine (1943) found that surface detention required to

maintain a specific rate of flow under raindrop impact was appreciably

greater than the detention required after rain ceased. The impact force

of the raindrop nearly normal to the direction of flow acted to retard the

downslope flow and created the type of turbulence that is very effective

in detachment of soil particles on a bare field slope.

In a study of the results of several investigations, Ekern (1954) concluded that erosivity was proportional to the additive kinetic energy of

the raindrop at impact and of the shallow flow of water.

The effect of shallow sheet flows without water drop impact on movement of noncohesive soil particles ranging in size from fine sand to fine

gravel was studied by Lutz and Hargrove (1944) on a laboratory plot.

They concluded that particle movement was expressed by the general

relationship:

(4)

Ep = K S”Q”’

where Ep is particle movement, S is slope, and Q is flow discharge. K , n,

and m were found to vary for the different-sized particles. Their Q

values were much higher than those that would occur from rain falling

on a short slope length equal to that of their plot but may be considered

as representing quantities of flow (without silt) experienced with severe

storms on segments of field slopes of around 100 to 200 feet in length.



RAINFALL EROSION



123



The importance of slope on particle movement decreased as particle

size increased, but the importance of flow increased. The greatest loss

occurred when the point of maximum velocity of flow was at a depth between the radius and diameter of the particle.

Woodruff (1947) used the laboratory plot of Neal (1938) to compare soil loss when water was applied as sheet flow at the upper end of

the plot and as simulated rain. His results showed that for sheet flow

without drop impact the soil per cubic foot of runoff was only about 10

per cent of that with the water applied as simulated rain for slopes of

8 per cent and less. Doubling the slope to 16 per cent resulted in much

greater detachment by the flowing water. The soil carried by the sheet

flow in this latter case was nearly 60 per cent of that carried with the

water applied as simulated rain. This indicates, for the condition of this

study, that for the flatter slopes drop impact was responsible for the bulk

of erosion but for the 16 per cent slope the runoff water with its higher

energy contributed an important part of soil detachment and total loss

from the plot.

111.



Basic Factors Affecting Field Soil Loss



Relationships of the basic factors to soil erosion under numerous field

conditions have been empirically evaluated from plot studies throughout

the United States and in several other countries. Recent assembly of most

of these data in the Runoff and Soil Loss Data Laboratory ( Wischmeier,

1955) facilitated analyses to identify the factors and interaction effects

responsible for the frequently wide differences in results reported from

the localized studies. The basic factors affecting field soil loss are discussed under five major classifications.



A. RAINFALL

Climatic effects on field soil loss include both rainfall and temperature. Length of growing season influences the nature and quality of vegetative cover available to protect the soil surface. Thaw and melting

snow may cause serious rill and microchannel erosion from unprotected

sloping fields, especially when an impervious layer of frozen soil beneath

a shallow-thawed surface impedes infiltration. However, the climatic

feature most significantly related to field soil loss is rainfall.

Data from plots tilled similarly to corn or cotton but kept free of

vegetation were analyzed to identify and evaluate rainstorm characteristics most closely correlated with field soil loss in the absence of protective cover. The results, summarized by Wischmeier et d. (1958),

show that, even for specific storms, soil loss was poorly correlated with

rain amount. The correlation of soil loss with maximum 5-, 1 5 , or 30-



124



DWIGHT D. SMITH Ah7) WALTER H. WISCHMEIER



minute intensities was also generally poor. Good correlation with maximum 30-minute intensity was found only on steep slopes or sandy loam.

At every location for which fallow-plot data were available, both runoff

and soil loss were more highly correlated with rainfall energy than with

rain amount or any short-period maximum intensity. Momentum rated

second, but was well below energy as a predictor of soil loss from fallow.

In further regression analyses of the data, Wischmeier (1959) found

that the rainstorm parameter most highly correlated with soil loss from

fallow was a product term, kinetic energy of the storm times maximum

30-minute intensity. He called this product the “rainfall-erosion index.”

Maximum 30-minute intensity was defined as twice the greatest amount

of rain falling in any 30-minute period. A break between storms was defined as a period of 6 consecutive hours with less than 0.05 inches of rainfall. This index was selected as the most appropriate rainfall parameter

for use in the soil loss prediction equation.

The rainfall-erosion index thus defined explained from 72 to 97 per

cent of the variation in individual-storm erosion from tilled continuous

fallow on each of six widely scattered soils. The percentage of the soilloss variance explained by the index was greater than that explained by

any other of 42 factors investigated and greater than that explained by

rain amount and maximum 5, 15, and 30-minute intensities, all combined in a multiple regression equation.

The erosion index evaluates the interacting effect of total storm

energy and maximum sustained intensity. Thus it is an approximation of

the combined effects of impact energy and rate and turbulence of runoff.

Rainfall energy is a function of the specific combination of drop velocities

and rain amount. The maximum 30-minute intensity is an indication of

the excessive rainfall available for runoff.

The product terms-rain amount times 30-minute intensity, and momentum times 30-minute intensity-were also more precise estimators of

soil loss than was energy alone, although less accurate than the energyintensity product. This supports the conclusion that the erosive potential

of a rainstorm is primarily a function of the interacting effects of drop

velocity, rain amount and maximum sustained intensity. In assembled

plot data, maximum 30-minute intensity was more effective than maximum 15- or 60-minute intensity as the second element of the interaction

term.

The relationship of soil loss to the storm energy-intensity products is

linear. Therefore, the location erosion-index value for a year can be

computed by summing the storm energy-intensity products. The assembled

plot data showed that when all factors other than rainfall were constant,

specific-year soil losses from cultivated areas were directly proportional



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