# CR-07-1:Chapter5

## A Uniﬁed Sediment Transport Formula for Coastal Inlet Application

### Summary of total load formula

In the following, a summary is given of the governing equations in the sediment transport model in concise form. Previous presentation of the model included many aspects of the equations that were developed and tested, which could make it difficult for the reader to extract the most suitable equations needed for computations.

Camenen and Larson (2005a, b) developed a formula for the bed-load transport based on the Meyer-Peter and Müller (1948) formula. The bed-load transport (qsb) may be expressed as follows: \begin{align} & \frac{{{q}_{sbw}}}{\sqrt{\left( s-1 \right)g\ d_{50}^{3}}}={{a}_{w}}\frac{{{\theta }_{cw,net}}}{\sqrt{\left| {{\theta }_{cw,net}} \right|}}{{\theta }_{cw,m}}\exp \left( -b\frac{{{\theta }_{cr}}}{{{\theta }_{cw}}} \right) \\ & \frac{{{q}_{sbn}}}{\sqrt{\left( s-1 \right)g\ d_{50}^{3}}}={{a}_{n}}\frac{{{\theta }_{cn}}}{\sqrt{\left| {{\theta }_{cn}} \right|}}{{\theta }_{cw,m}}\exp \left( -b\frac{{{\theta }_{cr}}}{{{\theta }_{cw}}} \right) \\ \end{align} (225)

where the subscripts w and n correspond, respectively, to the wave direction and the direction normal to the waves, s is the relative density between sediment (s) and water (), g the acceleration due to gravity, d50 the median grain size, aw, an, and b are empirical coefficients (to be discussed), cw,m the mean Shields parameter, cw the maximum Shields parameter due to wave-current interaction, $\text{ }{{\theta }_{cn}}=0.5{{f}_{c}}U_{c}^{2}\sin \varphi \left| \sin \varphi \right|/\left( \left( s-1 \right)g{{d}_{50}} \right)$, where fc is the current related friction factor, Uc the steady current velocity, and  the angle between the wave and the current direction. To simplify the calculations, the mean and maximum Shields parameter due to wave-current interaction is obtained by vector addition: ${{\theta }_{cw,m}}={{\left( \theta _{c}^{2}+\theta _{w,m}^{2}+2{{\theta }_{w,m}}{{\theta }_{c}}\cos \varphi \right)}^{1/2}}$ and ${{\theta }_{cw}}={{\left( \theta _{c}^{2}+\theta _{w}^{2}+2{{\theta }_{w}}{{\theta }_{c}}\cos \varphi \right)}^{1/2}}$, where  is the angle between the mean current direction and the wave incidence direction, and c, w,m, and w are the current, mean wave, and maximum wave Shields number, and w,m = 0.5 w for a sinusoidal wave profile. The net sediment transporting velocity in Equation 225 is given by: ${{\theta }_{cw,net}}=\left( 1-{{\alpha }_{pl,b}} \right){{\theta }_{cw,onshore}}+\left( 1+{{\alpha }_{pl,b}} \right){{\theta }_{cw,offshore}}$ (226)

where cw,onshore and cw,offshore are the mean values of the instantaneous shear stress over the two half periods Twc and Twt (Tw = Twc + Twt, in which Tw is the wave period), and pl,b a coefficient accounting for the phase lag (Camenen and Larson 2006). In the same way as for the Dibajnia and Watanabe (1992) formula, the mean values of the instantaneous shear stress over a half period are defined as follows (Figure 93): \begin{align} & {{\theta }_{cw,onshore}}=\frac{1}{{{T}_{wc}}}\int_{0}^{{{T}_{wc}}}{\frac{1/2{{f}_{cw}}{{\left[ {{u}_{w}}\left( t \right)+{{U}_{c}}\text{cos}\varphi \right]}^{2}}}{\left( s-1 \right)g{{d}_{50}}}dt} \\ & {{\theta }_{cw,offshore}}=-\frac{1}{{{T}_{wt}}}\int_{{{T}_{wc}}}^{{{T}_{w}}}{\frac{1/2{{f}_{cw}}{{\left[ {{u}_{w}}\left( t \right)+{{U}_{c}}\text{cos}\varphi \right]}^{2}}}{\left( s-1 \right)g{{d}_{50}}}dt} \\ \end{align} (227)

where uw(t) is the instantaneous wave orbital velocity, t the time, and fcw the friction coefficient due to wave-current interaction introduced by Madsen and Grant (1976): ${{f}_{cw}}={{X}_{v}}{{f}_{c}}+\left( 1-{{X}_{v}} \right){{f}_{w}}$ (228)

with ${{X}_{v}}=\left| {{U}_{c}} \right|/\left( \left| {{U}_{c}} \right|+{{U}_{w}} \right)$, where Uw is the average of the peak velocities during the wave cycle (the root-mean-square value is used for random waves). Based on comparison with an extensive data set (Camenen and Larson 2005b), the following relationship is proposed for the transport coeﬃcient aw: ${{a}_{w}}=6+6\ Y$ (229)

in which Y = c/(c + w). The coefficient perpendicular to the waves, where only the current transport sediment, is set to an = 12, and the coefficient in the term describing initiation of motion is b = 4.5 (Equation 225). The phase lag is introduced through the coefficient pl,b = c - t following Camenen and Larson (2006):

Figure 93. Deﬁnition of current and wave direction and velocity variation at bed in direction of wave propagation. ${{\alpha }_{j}}=\frac{{{v}^{0.25}}U_{cw,j}^{0.5}}{{{W}_{s}}\ T_{j}^{0.75}}\exp \left[ -{{\left( \frac{{{U}_{w,cr,sf}}}{{{U}_{cw,j}}} \right)}^{2}} \right]$ (230)

where Uw,cr,sf is the critical wave orbital velocity for the inception of sheet flow (Equation 57), and Ucw,j is the root-mean-square value of the velocity (wave and current) over the half period Twj, and the subscript j should be replaced either by c (crest or onshore) or t (trough or offshore) (Figure 93): \begin{align} & U_{cw,onshore}^{2}=\frac{1}{{{T}_{wc}}}\int_{0}^{{{T}_{wc}}}{{{\left[ {{u}_{w}}\left( t \right)+{{U}_{c}}\text{cos}\varphi \right]}^{2}}dt} \\ & U_{cw,offshore}^{2}=\frac{1}{{{T}_{wt}}}\int_{{{T}_{wc}}}^{{{T}_{w}}}{{{\left[ {{u}_{w}}\left( t \right)+{{U}_{c}}\text{cos}\varphi \right]}^{2}}dt} \\ \end{align} (231)

In determining the suspended load qss, following the simplified approach by Madsen (1993) and Madsen et al. (2003), the vertical variation in the horizontal velocity was neglected and an exponential-law profile assumed for the sediment concentration. Thus, the suspended sediment load may be obtained from (Camenen et al. 2005; Camenen and Larson 2007): \left\{ \begin{align} & {{q}_{ssw}}={{U}_{c,net}}\ {{c}_{R}}\frac{\varepsilon }{{{W}_{s}}}\left[ 1-\exp \left( -\frac{{{W}_{s}}h}{\varepsilon } \right) \right] \\ & {{q}_{ssn}}={{U}_{c}}\sin \varphi \ {{c}_{R}}\frac{\varepsilon }{{{W}_{s}}}\left[ 1-\exp \left( -\frac{{{W}_{s}}h}{\varepsilon } \right) \right] \\ \end{align} \right. (232)

where h is the water depth, Uc,net the net mean current over a wave period, cR the reference concentration at the bottom, Ws the sediment fall speed, and  the sediment diﬀusivity. In calculating the integral, the ratio Wsh/ may often be assumed large, implying that the exponential term is close to zero. However, such an assumption that integrating to infinity or to h produces about the same result may not be valid if a strong mixing by wave breaking is present. The bed reference concentration is obtained from: ${{c}_{R}}={{A}_{cR}}\ {{\theta }_{cw,m}}\exp \left( -4.5\frac{{{\theta }_{cr}}}{{{\theta }_{cw}}} \right)$ (233)

The coefficient AcR is written as follows: ${{A}_{cR}}=3.5\ {{10}^{-3}}\exp \left( -0.3{{d}_{*}} \right)$ (234)

where ${{d}_{*}}=\sqrt{\left( s-1 \right)g/{{v}^{2}}}{{d}_{50}}$ is the dimensionless grain size. A multiplying factor is introduced if plunging breakers occurs: ${{C}_{Rb}}=1.0+\tanh \left( 50\left( {{\xi }_{\infty }}-0.15 \right) \right)\quad \text{if}\quad {{\xi }_{\infty }}>0.15$ (235)

The sediment diffusivity is related to the energy dissipation: $\varepsilon ={{\left( \frac{D}{\rho } \right)}^{1/3}}h$ (236)

in which D is the total effective dissipation: $D=k_{b}^{3}\ {{D}_{b}}+k_{c}^{3}\ {{D}_{c}}+k_{w}^{3}\ {{D}_{w}}$ (237)

where the energy dissipation from wave breaking (Db) and from bottom friction due to current (Dc) and waves (Dw) were simply added, and kb, kc, and kw are coefficients (Equations 175, 166, and 167, respectively). The coefficient kb corresponds to an efficiency coefficient, whereas kc and kw are related to the Schmidt number. Assuming a parabolic profile for the vertical sediment diffusivity, its mean value over the depth (for a steady current or waves, respectively) may be written as follows: ${{\varepsilon }_{c/w}}={{\left( \frac{{{D}_{c/w}}}{\rho } \right)}^{1/3}}h=\frac{{{\sigma }_{c/w}}}{6{{C}_{w}}}\kappa \ {{u}_{*c/w}}h$ (238)

where σc/w is the Schmidt number or ratio between the vertical eddy diffusivity of the particles v and the vertical eddy viscosity v, and u*c/w is the shear velocity due to a current or to waves only, respectively, and Cw is an integration constant that is 1 for the case of a steady current and /2 for a sinusoidal wave. The following expression for the Schmidt number was proposed: {{\sigma }_{c/w}}=\left\{ \begin{align} & {{A}_{1}}+{{A}_{2}}{{\sin }^{2}}\left( \frac{\pi }{2}\frac{{{W}_{s}}}{{{u}_{*c/w}}} \right)\quad \quad \quad \quad \quad \ \text{if}\quad \frac{{{W}_{s}}}{{{u}_{*c/w}}}\le 1 \\ & 1+\left( {{A}_{1}}+{{A}_{2}}-1 \right){{\sin }^{2}}\left( \frac{\pi }{2}\frac{{{u}_{*c/w}}}{{{W}_{s}}} \right)\quad \text{if}\quad \frac{{{W}_{s}}}{{{u}_{*c/w}}}>1 \\ \end{align} \right. (239)

where A1 = 0.4 and A2 = 3.5 in case of a steady current alone and A1 = 0.15 and A2 = 1.5 in case of waves only. For wave and current interaction, a weighted value is employed for the Schmidt number: ${{\sigma }_{cw}}=Y{{\sigma }_{c}}+\left( 1-Y \right){{\sigma }_{w}}$ (240)

The net mean current is deﬁned in a similar way to the net Shields parameter for the bed load in order to take into account sediment transport due to asymmetric waves, as well as possible phase lag between the suspended concentration and the velocity: ${{U}_{c,net}}=\left( 1-{{\alpha }_{pl,s}} \right){{U}_{cw,onshore}}+\left( 1+{{\alpha }_{pl,s}} \right){{U}_{cw,offshore}}$ (241)

where αpl,s is a coefficient for the phase lag on the suspended load (Equation 221). For a steady current, Uc,net = Uc.

#### Bottom slope

The bottom slope may influence the sediment transport, especially if it is close to the critical value given by the internal friction angle at saturated conditions of the sediment. To take into account the local slope, the transport rate (qs) may be multiplied with a function containing the local slope and a coefficient, $q_{s}^{*}={{q}_{s}}\left( 1-\beta \frac{\partial {{z}_{b}}}{{{\partial }_{s}}} \right)$ (242)

where β is a coefficient for the bottom slope (0.5 < β < 2), and zb/s is the local slope. Following Bailard (1981), the coefficient β depends on the sediment transport mode: \begin{align} & {{\beta }_{b}}=\frac{1}{\tan \phi }\quad \approx 1.5\quad \text{(bed load)} \\ & {{\beta }_{s}}=\frac{{{\varepsilon }_{s}}<\left| {\vec{u}} \right|>}{{{W}_{s}}}\quad \quad \text{ }\ \text{(suspended load)} \\ \end{align} (243)

where s = 0.02 is the suspended-load eﬃciency as given by Bailard (1981), and $\vec{u}$ is the instantaneous velocity vector (wave and current combined).

#### Velocity proﬁles for varying slope

Variations in the velocity profile can influence both bed load and suspended load transport because the characteristic velocity in the lower part of the water column, where the concentration is larger, may be significantly reduced. Coles (1956) showed that velocity proﬁles in a non-uniform flow can be described by a linear combination of logarithmic proﬁles representing the law of the wall and a perturbation proﬁle representing the inﬂuence of pressure gradients: $u\left( z \right)={{A}_{1}}\ {{u}_{h}}\ \ln \left( \frac{z}{{{z}_{0}}} \right)+{{A}_{2}}\ {{u}_{h}}\ F\left( \frac{z}{h} \right)$ (244)

in which uh is flow velocity at the water surface (z = h), z0 the zero-velocity level (z0 = 0.03 ks from Van Rijn and Tan 1985), and A1 and A2 are dimensionless variables. Van Rijn and Tan (1985) proposed the following perturbation proﬁle: $F\left( \frac{z}{h} \right)=2{{\left( \frac{z-{{z}_{0}}}{h-{{z}_{0}}} \right)}^{n}}-{{\left( \frac{z-{{z}_{0}}}{h-{{z}_{0}}} \right)}^{2n}}$ (245)
 Applying the boundary condition u(z) = uh for z = h, introducing the discharge integrated over the width $Q=B\ h\ {{U}_{c}}=B\ \int_{{{z}_{0}}}^{h}{udz}$ , where B is the width of the flow, and assuming that the middepth velocity is approximatively the same as for a uniform flow, a relationship for n can be numerically obtained: $0.16{{n}^{2}}-0.29n+1.02\approx \frac{\ln \left( h/{{z}_{0}} \right)-1}{\ln \left( 0.5h/{{z}_{0}} \right)}$ (246)

Figure 94 shows that this method is capable of representing a wide range of velocity proﬁles including those with flow reversal.

Figure 94. Velocity proﬁles according to Equations 244, 245, and 246 ('''Uc '= 0.4 m/sec, '''h '= 0.2 m, and '''zo '= 0.001 m). Van Rijn and Tan (1985) proposed a first-order differential equation to solve for the spatial variation of the water surface velocity uh, which yields an exponential adjustment of the surface velocity with respect to the equilibrium surface velocity uh,e, as follows: $\frac{d{{u}_{h}}}{dh}={{\alpha }_{1}}\frac{{{u}_{h,e}}}{h}-{{\alpha }_{2}}\frac{{{u}_{h}}}{h}-{{\alpha }_{3}}\frac{{{u}_{h}}}{B}$ (247)

The coefficients α1 and α2 have been found to depend on the local bottom slope with values determined by comparison to data from several experiments, and α3 takes into account lateral variations: \begin{align} & {{\alpha }_{1}}=0.28+0.11\tanh \left[ 6\left( dh/dx \right)-0.15 \right] \\ & {{\alpha }_{2}}=0.235+0.065\tanh \left[ 17\left( dh/dx-0.035 \right) \right] \\ & {{\alpha }_{3}}=0.1\tanh \left[ 10\left( dB/dx \right) \right] \\ \end{align} (248)

Equation 247 can be solved numerically by a Runge-Kutta method, with the surface velocity uh,0 as the boundary condition.

### Application to coastal inlet studies

Longshore sediment transport forms the main input required for many coastal engineering projects such as dredging of inlet navigation channels, assessment of beach evolution in the vicinity of jetties and groins, and the evolution and stability of inlets, breaches, and estuaries. For coastal inlets, complex interactions occur among the longshore current, tidal current in the inlet, and waves, which may induce many types of phenomena related to sediment transport (Figure 2).

#### Validation of longshore sediment transport

To validate the present formula for the case of longshore sediment transport, two data sets were employed. Bayram et al. (2001) discussed the Sandy Duck experiments carried out at the U.S. Army Corps of Engineers Field Research Facility in Duck, NC (for a summary of the field experiments, see Miller 1998, 1999). During these experiments, the cross-shore distribution of the time-averaged longshore current and sediment concentration were measured, from which the transport rate could be estimated for six cases from 1996 to 1998. Wang et al. (2002) performed four sets of experiments in a large wave basin (Large-scale Sediment Transport Facility - LSTF) at the Coastal and Hydraulics Laboratory, Vicksburg, MS. The longshore sediment transport rate was recorded on a sandy beach exposed to random waves breaking at an incident wave angle, in one experiment as plunging breakers and in the other as spilling breakers. In the LSTF experiments, the hydrodynamics and concentration profiles were recorded at many locations across the profile. The beach profiles for the spilling breaker case were similar to the plunging case apart from the shape of the offshore bar, which was less pronounced for the spilling breaker case, implying less intensive wave breaking (and energy dissipation) in this region. Table 41 presents the main hydrodynamic conditions together with the median grain size for five of the experimental cases from Sandy Duck and the LSTF employed here for validating the sediment transport model. The main difference between the cases in the LSTF experiment was the breaker type, whereas the initial bathymetry was the same. For the Sandy Duck experiments, the bathymetry varied between the cases, where a clear bar was observed during the early cases and a terrace-shaped beach for later cases.

Table 41. Experiment conditions for studied cases on longshore sediment transport.

 Case Characteristics (-) d50(mm) Hw'(m)' Tw'(s)' 'w'(deg) LSTF Case 1 Spilling breakers 0.30 0.2 0.25 1.5 30 LSTF Case 6 Plunging breakers 0.75 0.2 0.19 3.0 30 Sandy Duck 12/03/96 Barred beach 0.20 0.20 3.1 13 5 Sandy Duck 31/03/97 Barred beach 0.15 0.18 1.4 8 5 Sandy Duck 04/02/98 Terrace beach 0.20 0.18 2.3 13 5

Figure 95. Cross-shore variations in hydrodynamic parameters and beach profile for an LSTF experimental case (Test 1 - spilling breakers) together with (a) measured longshore suspended sediment transport, and (b) calculated transport using six studied formulas.

Figure 96. Cross-shore variations in hydrodynamic parameters and beach profile for an LSTF experimental case (Test 6 - plunging breakers) together with (a) measured longshore suspended sediment transport, and (b) calculated transport using six studied formulas.

Figure 97. Cross-shore variations in hydrodynamic parameters and beach profile for Sandy Duck experiment (31 March 1997) together with (a) measured longshore suspended sediment transport, and (b) calculated transport using six studied formulas.

Figure 98. Cross-shore variations in hydrodynamic parameters and beach profile for Sandy Duck experiment (4 February 1998) together with (a) measured longshore suspended sediment transport, and (b) calculated transport using six studied formulas.

Table 42. Predictive capability of different transport formulas for longshore suspended load transport for LSTF and Sandy Duck experiments.

 Author(s) Px'''2 (%) Px'''5 (%) Mean ('''f'''('''qss)) Std ('''f'''('''qss)) LSTF Data Bijker (1968) 48 78 1.0 1.5 Bailard (1981) 60 77 0.35 1.7 Van Rijn (1989) 08 37 -1.4 1.9 Watanabe and Isobe (1992) 11 60 1.4 1.3 Dibajnia and Watanabe (1992) 35 75 -0.45 1.5 Present formula 55 75 0.10 1.7 Sandy Duck Data Bijker (1968) 33 85 0.8 0.5 Bailard (1981) 30 68 -1.3 0.8 Van Rijn (1989) 20 56 -1.3 1.1 Watanabe and Isobe (1992) 61 91 0.1 0.9 Dibajnia and Watanabe (1992) 17 64 -1.4 0.6 Present formula 41 86 0.4 0.9

Figure 99 plots the predictive results of the longshore sediment transport rate across the beach profile for both experiments using the present formula. It conﬁrms the underestimation in the swash zone observed for the LSTF data as well as a slight underestimation in the zone of incipient breaking. Because of a larger uncertainty in the measurements and calculations for the Sandy Duck data, a larger discrepancy is observed for these data. In general, the formula overestimates the transport rates, which may be due to an overestimation of the sediment diffusivity for waves at a large water depth, as previously pointed out.

#### Validation of cross-shore sediment transport

The cross-shore sediment transport rate could also be estimated for the Sandy Duck data. However, the sensitivity in the predictions by the formulas is much greater compared to the longshore transport rate. Figure 100 presents results obtained for one experimental case carried out on 12 March 1996. The Bijker (1968), Van Rijn (1989), and Watanabe and Isobe (1992) formulas induce a sediment transport that is in the same direction as the undertow, which means in the offshore direction. In contrast, the Bailard (1981), Dibajnia and Watanabe (1992) formulas, as well as the present formula, allows for transport in the opposite direction to the mean current if asymmetric waves are present. Thus, onshore sediment transport is often observed seaward of the surf zone. Because the transport rates derived from the measurements do not take into account the wave-induced sediment transport, it differs from the calculation results for the three latter formulas (for the other formulas, the estimated sediment transport rate is always in the direction of the mean current). The present formula appears to be sensitive to the balance between the undertow and wave asymmetry at the seaward end of the surf zone (compare results when Uc is used instead of Ucw,net for the suspended load; see Figure 100).

Figure 99. Predictive results for longshore sediment transport rate across beach profile using present formula for (a) LSTF data, and (b) Sandy Duck data.

Table 43 presents the statistical results for all the formulas compared regarding the Sandy Duck experiments on cross-shore transport. It appears for these cases that the predictive results are poorer than for the longshore sediment transport rate. The Watanabe and Isobe (1992) formula presents the best results, which is surprising because it was calibrated for longshore transport. However, as discussed previously, the measured suspended load includes transport by the mean current only, and other mechanisms are not included.

Figure 100. Cross-shore variations in hydrodynamic parameters and beach profile for (a) Sandy Duck experimental case (12 March 1996) together with measured cross-shore suspended sediment transport, and (b) calculated transport using six studied formulas.

Figure 101 plots the prediction of the cross-shore sediment transport rate across the beach profile for the Sandy Duck experiments using the present formula with only the current-related suspended load included (Uc is used instead of Ucw,net). It conﬁrms the general overestimation and dispersion previously observed for these data. If the wave-related sediment transport is included, the direction of the sediment transport is incorrectly estimated for several data points in the offshore. However, it may be a result of the formula including wave-related sediment transport, whereas the transport rates estimated from the measurements do not.

Table 43. Predictive capability of different transport formulas regarding suspended load transport in cross-shore direction for Sandy Duck experiments.

 Author(s) Px'''2 (%) Px'''5 (%) Mean ('''f'''('''qss)) Std ('''f'''('''qss)) Bijker (1968) 14 36 2.3 1.2 Bailard (1981) 23 50 0.05 1.6 Van Rijn (1989) 24 53 0.15 1.7 Watanabe and Isobe (1992) 47 68 0.8 1.0 Dibajnia and Watanabe (1992) 35 61 0.02 1.2 Present formula 33 65 1.1 1.1

An interesting data set was provided by Dohmen-Janssen and Hanes (2002). They measured bed load and suspended load transport in a large wave ﬂume for sheet ﬂow, and obtained results for the four cases presented in Table 44. Although a small undertow was present (opposite to the wave direction), the net sediment transport was directed onshore because of the asymmetric waves. The three formulas that assume the direction of the current to be the direction of the sediment transport (the Bijker (1968), Van Rijn (1989), and Watanabe and Isobe (1992) formulas) predict the wrong direction for the net total load. The Bailard (1981) formula predicts a correct direction for the sediment transport, but tends to overestimate the total load, especially the suspended load portion. Dohmen-Janssen and Hanes (2002) observed that, in case of sheet ﬂow and nonbreaking waves, bed load was always dominant and only 10 percent of the total load was carried by the suspended load. The Bailard (1981) formula (as well as the Bijker (1968) formula) predicts that suspended load is dominant (qsb/qss = 0.11). The Dibajnia and Watanabe (1992) formula, as it was calibrated for sheet-ﬂow conditions, yields good results. Finally, the present formula also yields good results, although it tends to overestimate the suspended load.

Figure 101. Comparison of cross-shore suspended load across profile line with present formula and Sandy Duck data.

Table 44. Predictive capability of total load sediment transport in cross-shore direction for sheet-ﬂow experiments by Dohmen-Janssen and Hanes (2002).

 Author(s) Px'''2 (%) Px'''5 (%) Mean ('''f'''('''qss)) Std ('''f'''('''qss)) qsb/'''qss Bijker (1968) 0* 0* -0.2 0.5 0.03 Bailard (1981) 0 0 2.4 0.07 0.11 Van Rijn (1989) 0 0* -0.3 0.2 1.3 Watanabe and Isobe (1992) 0* 0* -0.4 0.4 - Dibajnia and Watanabe (1992) 100 100 -0.15 0.3 - Present Work 75 100 0.5 0.3 0.4 Opposite transport direction predicted.