CMS-Flow:Equilibrium Concentrations and Transport Rates
Equilibrium Concentrations and Transport Rates
In order to close the system of equations describing the sediment transport, bed change, and bed sorting equations, the fractional equilibrium depth-averaged total-load concentration (Ctk*) must be estimated from an empirical formula. The depth-averaged equilibrium concentration is defined as
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk*} = \frac{q_{tk*}}{Uh}} | (1) |
where qtk* is the total-load transport for the kth sediment size class estimated from an empirical formula. For convenience, Ctk* is written in general form as
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_{tk*} = p_{1k}C_{tk}^*} | (2) |
where pidis the fraction of the sediment size (k) in the first (top) bed layer, and Ctk* is the potential equilibrium total-load concentration. The potential concentration (Ctk*) can be interpreted as the equilibrium concentration for uniform sediment of size dk. The above equation is essential for the coupling of sediment transport, bed change, and bed sorting equations.
Lund-CIRP
Camenen and Larson (2005, 2007, and 2008) developed a general sediment transport formula for bed and suspended load under combined waves and currents. These are refered to as the Lund-CIRP transport formulas. The general transport formulas can be used for both symmetric and asymmetric waves but for simplicity the waves are assumed to be symmetric. The current-related bed- and suspended transport rate wave stirring s given by
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{q_{b}}{\sqrt{(s-1) g d_{50}^3}} = f_b \rho_s \sqrt{\theta_c} \theta_{cw,m}\exp{ \biggl (-4.5 \frac{\theta_{cr}}{\theta_{cw}}} \biggr )} | (3) |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{q_{s*}}{\sqrt{(s-1)gd_{50}^3}} = f_s \rho_s c_R U \frac{\epsilon}{\omega_s} \left[1 - exp \left(- \frac{\omega_s h}{\epsilon} \right)\right] } | (4) |
where:
- qb* = equilibrium bed-load transport rate [kg/m/s]
- qs* = equilibrium suspended-load transport rate [kg/m/s]
- Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Theta_c} = Shields parameters due to currents [-]
- Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Theta_{cw,m}} = mean Shields parameters due to waves and currents [-]
- Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Theta_{cw}} = maximum Shields parameters due to waves and currents [-]
- Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \Theta_{cr}</math. = critical Shields parameter [-] :<math>\epsilon} = vertical sediment diffusivity [m2/s]
- cR = reference bed concentration [kg/m3]
bf = bed-load scaling factor (default 1.0) [-] sf = suspended-load scaling factor (default 1.0) [-].
The current-related suspended load transport with wave stirring is given by
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{q_s}{\sqrt{ (s-1) g d_{50}^3 }} = U c_R \frac{\varepsilon}{\omega_s} \biggl[ 1 - \exp{ \biggl( - \frac{w_s h}{\varepsilon}} \biggr) \biggr] } | (2) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U} is the depth-averaged current velocity, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle h} is the total water depth, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega_s} is the sediment fall velocity, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \varepsilon } is the sediment diffusivity, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_R} is the reference bed concentration. The reference bed concentration is calculated from
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle c_R = A_{cR} \exp{ \biggl( - 4.5 \frac{\theta_{cr}}{\theta_{cw}}} \biggr)} | (3) |
where the coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{cR}} is given by
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{cR} = 3.5 \times 10^3 \exp{ \bigl( - 0.3 D_{*} } \bigr) } | (4) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \nu } the kinematic viscosity of water, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_{*} } the dimensionless grain size
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_{*} = d_{50} \biggl[ \frac{(s-1) g}{ \nu} \biggr] } | (5) |
The sediment fall velocity is calculated using the formula by Soulsby (1997)
|
Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \omega_s = \frac{\nu}{d} \bigg[ \big( 10.36^2 + 1.049 D_{*}^3 \big)^{1/2} -10.36 \bigg] } |
(6) |
The sediment mixing coefficient is calculated as
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \epsilon = h \biggl( \frac{k_b^3 D_b + k_c^3 D_c + k_w^3 D_w}{\rho} \biggr)^{1/3}} | (7) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_b, k_c, and k_w} are coefficients, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_b} is the wave breaking dissipation, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_c} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle D_w} are the bottom friction dissipation due to currents and waves respectively. For more details see Camenen and Larson (2008).
van Rijn
The van Rijn (1984ab) transport equations are used with the recalibrated coefficients of van Rijn (2007ab) are given by
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_b = 0.015 \rho_s U h \biggl( \frac{U_e - U_{cr} }{ \sqrt{(s-1) g d_{50}} } \biggr)^{1.5} \biggl( \frac{d_{50}}{h} \biggr)^{1.2} } | (8) |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_s = 0.012 \rho_s U d_{50} \biggl( \frac{U_e - U_{cr} }{ \sqrt{(s-1) g d_{50}}} \biggr)^{2.4} D_{*}^{-0.6} } | (9) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{cr}} is the critical depth-averaged velocity for initiation of motion, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_e} is the effective depth averaged velocity calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_e = U + 0.4 U_w} in which Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_w} is the peak orbital velocity based on the significant wave height
According to van Rijn (2007) bed load transport formula predicts transport rates with a factor of 2 for velocities higher than 0.6 m/s, but underpredicts transports by a factor of 2-3 for velocities close to initiation of motion.
The critical velocity is estimated as
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{cr} = \beta U_{crc} + (1-\beta) U_{crw} } | (10) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{crc}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{crw}} are the critical velocity for currents and waves respectively. As in van Rijn (2007), the critical velocity for currents and waves are calculated based on Komar and Miller (1975).
Watanabe
The equilibrium total load sediment transport rate of Watanabe (1987) is given by
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{t} = A_w \biggl[ \frac{(\tau_{b,max} - \tau_{cr}) U }{\rho g } \biggr] } | (13) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{b,max} } is the maximum shear stress, Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{cr} } is the critical shear stress of incipient motion, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A } is an empirical coefficient typically ranging from 0.1 to 2.
The critical shear stress is determined using
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{cr} = (\rho_s - \rho) g d \phi_{cr} } | (14) |
In the case of currents only the bed shear stress is determined as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{c} = \frac{1}{8}\rho g f_c U_c^2 } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_c } is the current friction factor. The friction factor is calculated as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_c = 0.24log^{-2}(12h/k_{sd}) } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_{sd} } is the Nikuradse equivalent sand roughness obtained from Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle k_{sd} = 2.5d_{50} } .
If waves are present, the maximum bed shear stress Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{b,max} } is calculated based on Soulsby (1997)
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{max} = \sqrt{(\tau_m + \tau_w \cos{\phi})^2 + (\tau_w \sin{\phi})^2 }} | (15) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_m } is the mean shear stress by waves and current over a wave cycle, math> \tau_w </math> is the mean wave bed shear stress, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \phi } is the angle between the waves and the current. The mean wave and current bed shear stress is
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{m} = \tau_c \biggl[ 1 + 1.2 \biggl( \frac{\tau_w}{\tau_c + \tau_c} \biggr)^{3.2} \biggr] } | (16) |
The wave bed shear stress is given by Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \tau_{w} = \frac{1}{2}\rho g f_w U_w^2 } where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_w } is the wave friction factor, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_w } is the wave orbital velocity amplitude based on the significant wave height.
The wave friction factor is calculated as (Nielsen 1992) Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f_w = \exp{5.5R^{-0.2}-6.3}} where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R } is the relative roughness defined as Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle R = A_w/k_{sd} } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_w } is semi-orbital excursion Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_w = U_w T / (2 \pi) } .
Soulsby-van Rijn
Soulsby (1997) proposed the following equation for the total load sediment transport rate under currents and waves
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_t = A_s U \biggl[ \biggl( U^2 + 0.018 \frac{U_{rms}^2}{C_d} \biggr)^{0.5} - U_{cr} \biggr]^{2.4} } | (20) |
where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_{rms}} is the root-mean-squared wave orbital velocity, and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_d} is the drag coefficient due to currents alone and the coefficient Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{s} = A_{sb} + A_{ss} } . The coefficients Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{sb}} and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{ss}} are related to the bed and suspended transport loads respectively and are given by
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{sb} = \frac{ 0.005 h (d_{50}/h)^{1.2} }{ [(s-1)g d_{50} ]^{1.2} } } | (21) |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle A_{s} = \frac{ 0.012 d_{50} D_{*}^{-0.6} }{ [(s-1)g d_{50} ]^{1.2} } } | (22) |
The current drag coefficient is calcualted as
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle C_d = \biggl[ \frac{0.4}{\ln{(h/z_0)}-1 } \biggr]^2 } | (23) |
with a constant bed roughness length Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle z_0 } set to 0.006 m.
| Symbol | Description | Units |
|---|---|---|
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle q_{bc} } | Bed load transport rate | m3/s |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle s } | Relative density | - |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta_{c} } | Shields parameter due to currents | - |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta_{cw} } | Shields parameter due to waves and currents | - |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \theta_{cw}} | Critical shields parameter | - |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle a_c } | Empirical coefficient | - |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle b_c } | Empirical coefficient | - |
| Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle U_c } | Current magnitude | m/s |
References
- Camenen, B., and Larson, M. (2005). "A bed load sediment transport formula for the nearshore," Estuarine, Coastal and Shelf Science, 63, 249-260.
- Camenen, B., and Larson, M. (2007). "A unified sediment transport formulation for coastal inlet applications", ERDC/CHL-TR-06-7, US Army Engineer Research and Development Center, Coastal and Hydraulics Laboratory, Vicksburg, MS.
- Camenen, B., and Larson, M., (2008). "A General Formula for Non-Cohesive Suspended Sediment Transport," Journal of Coastal Research, 24(3), 615-627.
- Soulsby, D.H. (1997). "Dynamics of marine sands. A manual for practical applications," Thomas Telford Publications, London, England, 249 p.
- van Rijn, L. C. (1984a). "Sediment transport. Part I: Bed load transport", Journal of Hydraulic Engineering, 110(10), 1431–1456.
- van Rijn, L. C. (1984b). "Sediment transport. Part II: Suspended loadtransport", Journal of Hydraulic Engineering, 110(11), 1613–1641.
- van Rijn, L.C., (2007a). "Unified View of Sediment Transport by Currents and Waves. I: Initiation of Motion, Bed Roughness, and Bed-load Transport", Journal of Hydraulic Engineering, 133(6), 649-667.
- van Rijn, L.C., (2007b). "Unified View of Sediment Transport by Currents and Waves. II: Suspended Transport", Journal of Hydraulic Engineering, 133(6), 668-689.
- Watanabe, A. (1987). "3-dimensional numerical model of beach evolution," Proceedings Coastal Sediments '87, ASCE, 802-817.