PARAMETERS FOR ANALYTIC SOLUTIONS

Overview

The analytic solutions presented in the prior sections require a number of parameters. These parameters were derived from a wide range of experimental data. Parameters are summarized in the following sections for:

a) aerodynamic roughness,

b) friction velocity,

c) wind barriers,

d) threshold friction velocity,

e) emission parameters,

f) abrasion parameters,

g) breakage parameters, and

h) trapping and interception parameters.

Aerodynamic Roughness

In WEPS friction velocity was selected to drive erosion, but the meteorological input parameter is wind speed. However, for any given wind speed under neutral atmospheric conditions in the surface boundary layer, friction velocity is proportional to the natural logarithm of the surface aerodynamic roughness (Panofsky and Dutton, 1984).

Hence, to determine friction velocity, the aerodynamic roughness term of the log-law wind speed profile must be determined first. For surfaces without standing biomass, the surface aerodynamic roughness is controlled by roughness of both the soil and flat biomass cover. In EROSION, tillage ridges are characterized by their height, spacing, orientation, and top bed width. For ridge heights greater than 5 mm, the normalized aerodynamic roughness is (Fig. E-11) (Hagen and Armbrust, 1992):

where

WZ0rg = aerodynamic roughness of the ridges (mm),

SZrg = ridge height (mm), and

SXPrg = ridge spacing parallel the wind direction (mm).

Figure E-11. Ratio of aerodynamic roughness to ridge height as a function of the ridge height to spacing ratio; predicted is equation E-60. (Hagen and Armbrust, 1992).

The ridge spacing parallel to the wind direction is:

q

where

SXrg = ridge spacing (mm),

AWAdir = daily wind direction (degrees), and

SArg = ridge orientation, clockwise from north and parallel to the

ridge(degrees).

For random roughness without standing biomass, the aerodynamic roughness is estimated as

where

WZOrr = aerodynamic roughness of random roughness including any flat biomass cover (mm), and

SZrr = random roughness (mm).

The final aerodynamic roughness for the surface, WZ0, is calculated as the maximum of the ridge or random aerodynamic roughness.

If standing plant biomass is present, the aerodynamic roughness length of the canopy is

calculated.. In these calculations , the standing biomass is considered composed of two parts – the current growing crop and all other decomposing biomass. First, effective leaf and stem area indexes are calculated for all standing biomass as

and

where

BRlai = biomass effective leaf area index (m2/m2),

BRsai = biomass effective stem area index (m2/m2),

BDlai = decomposing biomass leaf area index (m2/m2),

BDsai = decomposing biomass stem area index (m2/m2),

Clai = growing crop leaf area index (m2/m2),

Csai = growing crop stem area index (m2/m2),

Rcrow = reduction of effective leaf area when crop rows spaced > 5 crop heights, and

Rrg = reduction of effective leaf and stem area when crop partly sheltered in furrow.

Empirical estimates of the reduction factors are

and

Where

CZ = growing crop height (m)

CXrow = growing crop row spacing (m)

SZrg = ridge height (mm)

Finally, an effective biomass drag coefficient (BRcd )is calculated as

The effective drag coefficient of small amounts leaf area is significantly reduced, because the leaves tend to orient parallel to the wind streamlines (Armbrust and Bilbro, 1995).

For a relatively smooth underlying surface, the standing biomass aerodynamic roughness calculated for an average stem diameter of 20 mm is (Fig. E-12) (Hagen and Armbrust, 1994):

However, underlying surfaces are often rough and contribute to the total above-canopy aerodynamic roughness when Brcd is less than about 0.1. In this case, the aerodynamic roughness is estimated as (Fig. E-13):

Minimum aerodynamic roughness is that of the below-canopy surface. While aerodynamic roughness may decrease slightly with stem diameter of residue, a constant value was assumed in this model.

Figure E-12. Biomass aerodynamic roughness as a function of effective drag

coefficient of the biomass; predicted is equation E68; data are

Raupach, 1992.

Figure E-13. Normalized aerodynamic roughness for 200mm-tall biomass with

levels of soil surface aerodynamic roughness (WZO) ranging from 0.5 to

40 mm.

Friction Velocity

Friction velocity at the simulation region is calculated in two steps. First, the friction velocity at the weather station, where wind speeds are measured, is calculated for strong winds with neutral stability using the log-law profile (Panofsky and Dutton, 1984):

where

WUF = friction velocity at the weather station (m/s),

WU = wind speed at weather station (m/s),

WZ = anemometer height at the weather station (mm); (wind speeds were adjusted to 10 m height in WEPS data base), and

WZZ0 = aerodynamic roughness at weather station, assumed to be 25 mm in WEPS.

Second, the friction velocity at the simulation region is calculated based on the ratio of aerodynamic roughness at the simulation region to that at the wind speed measurement station. This equation is an approximation of a procedure suggested by Letteau (Panofsky and Dutton, 1984). If there is no standing biomass, then

where

WU* = friction velocity used to drive the erosion simulation (m/s).

However, if there is standing biomass, then

where

WU*v = friction velocity above the standing biomass (m/s), and

WZOv = as defined by equation E-69.

Next, the friction velocity below the standing biomass is calculated (Fig. E-14) (Hagen and Armbrust, 1994).

where

WU* =	friction velocity below the standing biomass at the surface that is used to drive the erosion simulation (m/s).

At this point in the calculations, the influence of barriers or hills on friction velocity are still neglected.

Figure E-14. Reduction in friction velocity through biomass canopy as a function of biomass drag coefficient; predicted is equation E-73; data are Lyles and Allison, 1976;Van de Ven, Fryrear, and Spaan, 1989.

Wind Barrier Parameters

Barrier effects are simulated by calculating the fraction of open field surface friction velocity that occurs at the center of each grid cell as a result wind speed reduction by individual barriers. When two or more barriers shelter the same area, the lowest fraction calculated for an individual barrier shelter is selected by the submodel.

To obtain the fraction of open field friction velocity in sheltered areas, we used the following procedure. First, the distances along the wind direction between barrier and the center of each grid cell were calculated. (Details of the distance calculations are illustrated in Appendix ?????). Next, the distances were normalized by the barrier height. The optical porosity of the barrier (pbroptical ) was also adjusted to an 'effective' porosity (pbr) based on normalized barrier width (xw) (Figure E-15).

Figure E-15. Effective barrier porosity as a function of normalized barrier width.

Finally, the fraction of open field friction velocity, fu*, at each grid cell was calculated as

Where

pbr = effective barrier porosity (fraction), and

x = distance, in barrier heights, along wind direction to barrier.

For all other grid cells which are not in the lee shelter region, but are within 5 barrier heights from the barrier, we determine fraction of open field friction velocity at the grid cell using the same equation for fu*, but with x as a negative value. The relationship between fu* -vs-x for various levels of barrier optical porosity are illustrated in Fig. E-16.

Figure E-16. Distribution of fraction of open field friction velocity near barriers with various levels of porosity

Hills

{This subroutine is currently not compiled with WEPS}

The subroutine "sbhill" reads an input file that assigns to each grid cell a normalized wind speed reduction or speed-up factor as influenced by topography. Individual factors are assigned for each of 16 wind directions. Because WEATHER simulates a single wind direction for each day, only one set of factors is used for each day with erosion.

The velocity at which numerous aggregates begin to saltate is defined as the static threshold friction velocity. Static threshold friction velocity is calculated in each grid cell as influenced by aggregate size and density, clod/crust cover, surface roughness, flat biomass cover, and surface soil wetness at noon.

As noted earlier, the numerical values of aggregated and crusted surface sum to one.

However, to calculate the true faction of bare, immobile surface, one must correct for both small aggregates and rock fraction, if present. Hence, the fraction of bare surface with immobile cover is :

where

SFcv	=	soil fraction covered by clod, crust, and rock so it does not emit,

SFcr	=	soil fraction covered by crust, but excluding the fraction of rock-covered area,

SFlos	=	soil fraction covered with loose, erodible soil on the crusted area,

SVroc	=	soil volume rock > 2.0 mm. (m3/m3), and

SF84	=	soil fraction covered with aggregates < 0.84 mm in diameter on the non-
crusted area, but excluding the fraction of rock-covered area.

The value of SF84 is calculated from the modified lognormal aggregate size distribution (Wagner and Ding, 1994) input to EROSION from other submodels as:

where

SLagn	=	lower limit of size distribution (mm),

SLagx	=	upper limit of size distribution (mm),

SLagm	=	geometric mean of size distribution (mm), and

SOags	=	geometric standard deviation of size distribution.

To determine threshold friction velocities for bare soil surface, estimating equations were fitted to wind tunnel data (Hagen, 1991b; Chepil and Woodruff, 1963) to give (Figs. E-17 and E-18):

where

WUB*ts = static threshold friction velocity of bare surface (m/s), and

WZO = aerodynamic surface roughness (mm).

The minimum static threshold friction velocities selected to be 0.24 m s-1 for mixed sand surfaces and 0.19 m s-1 for less dense soil aggregates. The threshold velocity increases with roughness, because part of the above-ground friction velocity is converted to form drag at the surface. The dynamic threshold velocity is increased toward static threshold linearly in proportion to shelter area (SFA12).

If random flat biomass cover is present, the increase in soil surface area protected from emission is:

where

SFCcv	=	fraction change in soil surface area protected from emission, and

BFFcv	=	biomass fraction of flat cover.

Figure E-17. Static threshold friction velocities for various levels of

aerodynamic roughness and surface cover as predicted by equations E-83, E-84, and E-85.

Prediction eq. was modified – need to recheck this graph}

Figure E-18. Predicted threshold friction velocities as a function of measured threshold friction velocities on random rough and ridged surfaces (Hagen, 1991b; Hagen and Armbrust, 1992).

The increase in static threshold friction velocity caused by flat biomass cover is (Fig. E-
19) (Hagen, 1995):

where

WUC*ts = change in static threshold friction velocity caused by flat biomass cover (m/s).

Figure E-19. Increase in static threshold friction velocity of erodible sand (0.29-0.42 mm); predicted is equation E-87, for flat biomass cover (Hagen, 1995).

If the surface is wet, threshold velocity increases as the slope of Fig. E-20 (Saleh and Fryrear, 1995).

where

WUCW*ts	=	increase in static threshold friction velocity from surface wetness (m/s),

HR0wc	=	surface soil water content (kg/kg), and

HR15wc	=	surface soil water content at 1.5 MPa (kg/kg).

Finally, static threshold velocity with wetness and flat cover is:

where

WU*ts	=	surface static threshold velocity accounting for both flat biomass cover and wetness effects (m/s).

If friction velocity exceeds the threshold, the dynamic threshold velocity, WU*t ,is used in saltation/creep transport capacity equations (Bagnold, 1943) and is estimated as

Figure E-21. Static threshold velocity change with water content relative to

1.5 Mpa water content; predicted slope is equation E-88 (Saleh and

Fryrear, 1995).

??? revise figure with new curve

Emission Parameters

As outlined in the theory section, the erosion process is modeled as the quasi-steady state conservation of mass using linked partial differential equations for three size classes of eroding soil. These are saltation/creep size (0.1 to 2.0 mm), suspension size (<0.1 mm), and PM-10 size (<0.01 mm).

When the wind transport capacity exceeds the saltation/creep discharge, the net entrainment rate of loose surface soil into the airstream is directly proportional the coefficient of emission, Cen (m-1). A fraction of the emitted soil, SFSSen, becomes part of the suspension discharge, and a fraction of the suspended soil from the emission, SF10en, is PM-10. Estimations for the preceding three parameters are presented in this section.

For the complex surfaces simulated in EROSION, auxiliary equations were developed to estimate Cen. The emission coefficient is calculated as a function of surface complexity as

where

Ceno = coefficient of emission for a bare, smooth, loose, erodible soil. A typical field value is about 0.06 (1/m).

Renv = emission ratio for flat, random vegetation, and

Renb = emission ratio for immobile cover and surface roughness.

The fractional emission factor to account for flat biomass cover is (Hagen, 1995) (Fig. E-
21),

Figure E-22. Fractional emission factor of loose soil as function of

increasing biomass flat cover; predicted is equation E-93 (Hagen, 1995).

On rough surfaces, increasing the fraction of surface sheltered from direct saltation impacts by macro-roughness also reduces emission rates. A shelter angle at a point is defined as the largest angle above horizontal to the top of any upwind point To predict the fraction of surface sheltered from saltation impacts, the random and ridge roughness can be used to estimate the sheltered area. . The shelter angle distribution is described by a two parameter Weibull distribution; the two parameters are a scale factor and a shape factor (Potter, Zobeck, and Hagen, 1990). The average shape factor measured over a range of roughness was about 0.77 (Potter and Zobeck, 1990). The scale factor was related to the random roughness as

where

SACrr	=	Weibull scale factor for shelter angle (degrees) of random roughness, (mm) and

SZrr	=	random roughness (mm)

The fraction horizontal area with shelter angles greater than 12 degrees caused by random roughness, SFA12rr, is (Fig. E-22)

Figure E-23. Fraction of horizontal area with shelter angles > 12 degrees as a function

of random roughness estimated by Eq. E-95.

Similarly, the fraction of horizontal area sheltered by ridges, SFA12rg, can be calculated as (Potter and Zobeck, 1990)

where

SACrg = Weibull scale factor for ridge shelter (mm),

Szrg = ridge height (mm),

SXPrg = ridge spacing parallel the wind direction (mm), and

SFA12rg = surface fraction of area with shelter angles > 12 degrees for ridges.

Ridges and random roughness interact in complex ways to shelter the surface from impacts. In EROSION, the fraction area with shelter angles >12 degrees, SFA12, was estimated as

On horizontal surfaces, immobile individual aggregates, rocks or pieces of crust tend to protrude slightly above the mobile surface and protect about 3 times their own horizontal area from emissions. However, these small protrusions are usually part of the surface micro-roughness and not easily measured by conventional (pin) roughness meters. For comparison, sparse, flat, random biomass protects about 4 times its horizontal cover area from emissions as illustrated in Fig. E-21. Similar to flat biomass, the areas protected by micro-roughness will begin to overlap as the cover of immobile aggregates, rock, and crust increases. Thus, the area not sheltered by either the micro-roughness or by the macro-roughness, as measured by roughness meters, can be used to estimate the emission ratio for bare soil as

The predicted effect of the macro-roughness measured by pin meters and various levels of immobile surface cover on the bare soil emission ratio, Renb, is illustrated in Fig. E-23.

Figure E-24. Bare soil emission ratios predicted by Eq. E-99 for ranges of immobile soil cover and random roughness.

{prediction eq. was modified - - recheck this graph}

Figure E-25. Predicted reduction in emission of loose soil compared to values

measured in the wind tunnel; predicted is equation E-99 (Hagen, 1991b).

The soil fraction of suspensions size in the emitted soil is estimated from the aggregate size distribution

where

SF10	=	soil fraction less than 0.10 mm diameter, and

SF200	=	soil fraction less than 2.0 mm diameter.

Fraction of PM-10 in the emitted suspension-size aggregates is estimated as (Hagen, unpublished data)

where

SFsil = suface soil fraction silt,

SFcla = surface soil fraction clay, and

WZypt = average annual precipitation (mm).

Abrasion Parameters

The abrasion of immobile soil clods and crust by saltation/creep impacts creates additional erodible aggregates. The rate of creation of additional erodible aggregates depends upon the saltation/creep discharge, q (kg m-1 s-1), the fractions of saltation/creep discharge impacting the clods and crust, Fani , and the abrasion coefficients of the target clods and crust, Cani (m-1 ). A fraction, Sfssan , of the erodible aggregates created by abrasion contribute to the suspension discharge, and a fraction, SF10an , of the suspension-size aggregates created by abrasion are PM-10. Estimating equations for these parameters - Fani, Cani, Sfssan , and SF10an - are presented in this section.

The fraction, Fan , of saltation/creep impact on both clods and crust that are not sheltered by either flat biomass or rock cover is estimated as

where

BFFcv = biomass fraction flat cover, and

SVroc = soil volume of rock in surface layer (m3 m-3).

The preceding equation assumes that both rock and flat biomass cover protrude slightly above the surrounding surface soil.

The fraction of saltation/creep impacting on immobile aggregates is estimated as

where

SF84 = soil mass fraction of aggregates < 0.84 mm diameter on aggregated surface, and

Sfcr = soil fraction of crust cover.

The fraction of saltation/creep impacting on unprotected surface crust, Fancr , is estimated as

where

SFcr = surface fraction crust cover, and

SFlos = fraction of crust covered with loose, erodible soil.

The abrasion coefficients for clods, Canag , and crust, Cancr , are a function of their dry stabilities, and these were related in wind tunnel abrasion experiments (Fig. E-25)( Hagen, Skidmore, and Saleh, 1992).

where

SEags = dry stability of immobile aggregates [ln(J kg-1)], and

SEcrs = dry stability of crust [ln(J kg-1)].

An approximate estimate for crust abrasion coefficient used in EROSION is (Zobeck, 1991)

Figure E-26. Abrasion coefficients as a function of crushing energy for soil

aggregates and crusts; predicted is equation 105 (Hagen, Skidmore, and Saleh,

1992).

The soil fraction abraded from clods and crust that is of suspension size was determined experimentally (Mirzamostafa and Hagen, 1995) and estimated as (Fig. E-??)

where

SFcla = surface soil fraction clay.

Figure E-27. Estimated soil fraction suspension-size in abraded soil as a function of clay content.

The fraction of PM-10 in the abraded suspended soil is estimated as (Hagen, unpublished data)

Breakage Parameters

As saltation/creep-size aggregates are transported, they undergo multiple impacts and partial breakdown into suspension-size aggregates. Continuous movement of a single aggregate causes it to breakdown into primary particles, which may be a sand grain at its core. However, individual moving saltation/creep aggregates are frequently interchanged with similar aggregates on the surface, so the rate of breakdown can be estimated by a constant coefficient, Cbk (m-1), as (Mirzamostafa, 1996)

The transport of the non-breakable portion of the saltation discharge, qs , is estimated as

where

SFsan = soil fraction sand, and

q = saltation/creep discharge (kg m-1 s-1)

A fraction, SF10bk , of the soil broken to suspension-size is PM-10 and is estimated as (Hagen, 2004)

where

100 < WZypt < 800 mm annual precipitation, and

0.017 < SFcla < 0.42 surface soil fraction clay.

Trapping and Interception Parameters

There are three different cases of trapping and two of interception simulated in EROSION. Examples of the first two cases can occur when a soil discharge approaches a wind barrier.

In the first case, the grid cell transport capacity, qen , is zero, but the incoming saltation/creep, qi , is greater than zero. In this case, saltation/creep leaving the grid cell is set equal to zero and qi is the deposition rate for saltation/creep in the grid cell. In addition, the deposition of both the suspension and PM-10 are simulated by simple exponential decay equations as

where

i = subscript denoting values at upwind grid cell boundary,

qss = suspension discharge (kg m-1 s-1),

q10 = PM-10 discharge (kg m-1 s-1),

Cdp = coefficient of deposition for suspension,

estimated as 0.02 m-1 , and

C10dp = coefficient of deposition for PM-10,

estimated as 0.001 m-1 .

In the second case, the incoming saltation/creep discharge, qi , is greater than the transport

capacity, qen , in a grid cell. In this case, the saltation/creep discharge for the cell is set to qen , while suspension and PM-10 discharge are calculated using their usual estimating equations. The difference between qi and qen is assigned as the deposition rate for saltation/creep in the grid cell.

In the third case, non-uniform surfaces, such as highly erodible tillage ridges, often emit more erodible saltation/creep than the wind can transport, and the excess emission is trapped in downwind furrows. Trapping is simulated when the surface cover is > 60 percent mobile aggregates, and the surface is rough, i.e., ridge roughness > 50 mm or random roughness > 10 mm. The rate of trapping of the saltation/creep discharge depends both the surface conditions and a trapping coefficient, Ct (m-1) . The trapping leads to the commonly-observed flattening of ridges and filling of furrows during erosion events.

An estimate for Ct , can be derived as follows: At sufficient distance downwind the saltation/creep discharge should approach the surface transport capacity. In this condition the emission flux, Gen , must equal the trapping flux, Gtp . Expanding and equating these values gives

Letting q approach qcp and solving for Ct gives

However, when qen >> qcp , the transport capacity will be between them. Hence, a coefficient was substituted in the prior equation to give

where

Ctf = empirical coefficient, with value about 1.2.

Plant interception by standing biomass also decreases the discharge of saltation/creep. The deflection of saltation/creep by stems reduces their saltation jump along the wind direction and surface impact kinetic energy. These effects are estimated by an empirical coefficient of interception, Ci (m-1) , that is a function of biomass stem area, row orientation, and biomass height. The coefficient of interception is estimated as

where

SAI = stem silhouette area index,

BZ = biomass height

SXrgs = ridge spacing (mm), and

SXPrg = ridge spacing parallel the wind direction.

.

Figure E-27. Plot of results from equation for estimating plant interception coefficient

Standing biomass leaves and stems also intercept a portion of the suspension-size aggregates. In this case, we assumed the suspended soil from upwind was moving above the biomass so only part of the suspension-size soil moving from below to above the biomass in a grid cell was intercepted.. The fraction of the vertical suspension flux intercepted (Cssi ) is estimated as

Cssi = 0.1*SAI

.

OUTPUT FROM EROSION TO FILES

Two subroutines output files of information from the EROSION submodel (Fig. E-22). The first subroutine, "sbout", is a specialized subroutine that is located within the daily erosion cycle. It is to be modified by those developing, verifying, and validating the EROSION submodel.

The second subroutine, "sbrpt", has several outputs:

1.	The Julian day,

2.	the total soil loss/deposition at each grid point since last output,

3.	the total suspension at each grid point since last output, and

4.	the total PM-10 production at each grid point since last output.

The outputs to a file from calls to subroutine "sbrpt" are controlled by a flag from the MAIN subroutine.

The grid points that may be included in each user-selected accounting region must form a rectangle, but there are no other restrictions. However, all the information for standard reports for all potential accounting regions is located in the report file generated by subroutine "sbrpt". When MAIN calls the subroutine "sbprnt" for a printed report, the data in the report file are analyzed for each of the specified accounting regions and printed.

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REFERENCES

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