Bottom Friction

Bottom Friction

The mean (short-wave averaged) 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_{bi}} , 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 \tau_{bi} = \lambda_{wc}\tau_{ci}} (2-8)

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 \lambda_{wc} = } nonlinear bottom friction enhancement factor 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 \left( \lambda_{wc} \geq 1 \right)} [-]

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 = } current-related bed shear stress vector [Pa]

The current 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_{ci} = \rho c_b UU_i} (2-9)

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 \rho = } water density (~1025 kg/m³)

c_b = bed friction 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 =} current magnitude = 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 \sqrt{U_i^2 + U_j^2}} [m/s]

The bottom roughness is specified with either a Manning's roughness 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 \eta} , Nikuradse roughness height k_s , or bed friction coefficient c_b . It is important to note that the roughness value is held constant throughout the simulation and is not changed according to bed composition and bedforms. This is a common engineering approach which can be justified by the lack of data to initialize the bed composition, and the large error in estimating the bed composition evolution and bedforms. In addition using a constant bottom roughness simplifies the model calibration. In future versions of CMS, it option to automatically estimate the bed roughness from the bed composition and bedforms will be added.

The bed friction coefficient c_b is related to the Manning’s roughness 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 \eta} by (Graf and Altinakar 1998)

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_b = \left( \frac{\kappa}{ln (z_0/h) + 1 }\right)^2} (2-11)

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 \kappa =} 0.4. The roughness length is related to the Nikuradse roughness (roughness height specified in SMS interface for CMS-Flow) 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 z_0 = k_z /30.} SOMETHING IS MISSING provides mean roughness heights from a large number of field measurements over natural sea beds.

In the presence of waves, the mean (wave-averaged) bed friction is enhanced beyond the value that would result from a linear superposition of the current- and wave-related components. This enhancement of the bed friction is due a nonlinear interaction between waves and currents in the bottom boundary layer. In CMS, the bottom friction enhancement is included through an explicit factor in the quadratic bottom friction equation (Eq. 2-4) that simplifies the numerical discretization. The nonlinear bot-tom friction enhancement factor is calculated using one of the following formulations (name abbreviations are given in parenthesis):

1. Quadratic formula (QUAD)

2. Soulsby (1995) empirical two-coefficient data fit (DATA2)

3. Soulsby (1995) empirical thirteen-coefficient data fit (DATA13)

4. Fredsoe (1984) analytical wave-current boundary layer model (F84)

5. Huynh-Thanh and Temperville (1991) numerical wave-current boundary layer model (HT91)

6. Davies et al. (1988) numerical wave-current boundary layer model (DSK88)

7. Grant and Madsen (1979) analytical wave-current boundary layer model (GM79)

In the case of the quadratic formula (QUAD),

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 \lambda_{wc} = \frac{\sqrt{ U^2 + c_w u^2_w}}{U}} (2-12)

where c_w is an empirical coefficient equal to approximately 0.5 and u_w is the bottom wave orbital velocity amplitude. For random waves, u_w is calculated based on the significant wave height and peak wave period, from linear wave theory 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_{wc} = \frac{\pi H_s}{T_p sinh(kh)}} (2-13)

The DATA2, DATA13, F84, and HT91 formulations are calculated using general parameterization of Soulsby (1995),

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 \lambda_{wc} = 1 + bX^P (1 - X)^q} (2-14)

where $X=\tau _{c}/(\tau _{c}+\tau _{w})$ and b, P and q are coefficients that depend on the formulation selected. 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} 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 \tau_w} are the current- and wave-related bed shear stress magnitudes. The current 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_c = \rho c_b U^2} (2-15)

and 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 f_w u^2_w} (2-16)

where f_w is the wave friction factor. For random waves, u_w is set to an equivalent or representative bottom orbital velocity amplitude equal to 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 = \sqrt{2}u_{rms}} , where u_rms the root-mean-squared bottom orbital velocity amplitude defined here as (Soulsby 1987; 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 u_{rms} = \frac{\pi H_{rms}}{T_p \sqrt{2} sinh(kh)}} (2-18)

Wiberg and Sherwood (2008) reported that 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} } estimates 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 H_{rms}} 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 T_p} agree reasonably well with field measurements (except for 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 T_p } < 8.8 s) and produces better estimates than other combinations with 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_{rms} , H_s , T_p } and the zero-crossing wave period 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 T_z} . The zero-crossing wave period is calculated as the average period (time lapse) between consecutive upward or downward intersections of the water level time series with the zero water line. A better approach is to assume a spectral shape (e.g. JONSWAP, Pierson-Moskowitz, etc.), and obtain an explicit curve for 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}} by summing the contributions from each frequency (Soulsby 1987; Wiberg and Sherwood 2008). A simple explicit expression is provided in Equation 2-19 below based on the JONSWAP (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 \gamma} = 3.3) spectrum following the work of Soulsby (1987)

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} = 0.134 \frac {H_s}{T_n} \left[1 + tanh\left(-7.76 \frac {T_n}{T_p} +1.34 \right) \right]} (2-19)

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 T_n = \sqrt {h/g}}