CMS-Flow:Boundary Conditions: Difference between revisions

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== Water Level ==
= Wall Boundary=


== Water Level and Current Velocity==
The wall boundary condition is a closed boundary and is applied at any cell face between wet and dry cells. Any unassigned boundary cell at the edge of the model domain is assumed to be closed and is assigned a wall boundary. A zero normal flux to the boundary is applied at closed boundaries. Two boundary conditions are available for the tangential flow:


== Tidal Constituent==
# Free-slip: no tangential shear stress (wall friction)
The CMS follows the National Oceanographic and Atmospheric Administration (NOAA) National Ocean Service (NOS) convention for astronomical water level predictions. The water level predictions are based on a harmonic equation with several arguments
# Partial-slip: tangential shear stress (wall friction) calculated based on the log law.


{{Equation|<math> \eta(t) =  \sum_{i=1}^N f_i A_i \cos (\omega_i t + V_{i}^0 + u_i - \kappa_i)  </math>|2=1}}
Assuming a log law for a rough wall, the partial-slip tangential shear stress is given by


where <math>A</math> is the constituent ''mean '' amplitude, <math>f</math> is a factor that reduces the mean amplitude and varies in time, <math>V_0+u</math> are the constituents ''equilibrium '' phase and <math>\kappa</math> is the constituent phase lag or epoch.
{{Equation|<math>\tau_{wall} = \rho c_{wall}U^2_\parallel</math>|1}}


where <math>U_\parallel</math> is the magnitude of the wall parallel current velocity, and <math>c_{wall}</math> is the wall friction coefficient equal to
{{Equation|<math>
c_{wall} = \left[\frac{\kappa}{ln(y_P / y_0)} \right]^2
</math>|2}}
Here, <math>y_0</math> is the roughness length of the wall and is assumed to be equal to that of the bed <math>(i.e, y_0 = z_0)</math>. The distance from the wall to the cell center is <math>y_P</math>.
=Flux Boundary=
The flux boundary condition is typically applied to the upstream end of a river or stream and specified as either a constant or time series of total water volume flux ''(Q)''. In a 2DH model, the total volume flux needs to be distributed across the boundary in order to estimate the depth-averaged velocities. This is done using a conveyance approach in which the current velocity is assumed to be related to the local flow depth ''(h)'' and Manning’s ''(n)'' as <math>U \propto h^r / n</math>. Here, ''r'' is an empirical conveyance coefficient equal to approximately 2/3 for uniform flow. The smaller the ''r'' value the more uniform the current velocities are across the flux boundary. The water volume flux ''<math>(q_i )</math>'' at each boundary cell ''(i)'' is calculated as
{{Equation|<math>\overrightarrow{q}_i = h\overrightarrow{U}_i =
\frac{f_{Ramp}Q} {\bigg | \sum\limits_i (\hat{e}\cdot\hat{n})
\frac{h_i^{r+1}} {n_i}  \Delta l_i\bigg |}
\frac{h_i^{r+1}}{n_i} \hat{e}
</math>|3}}
where:
:''i'' = subscript indicating a boundary cell
:<math>\overrightarrow{q_i}</math> = volume discharge at boundary cell ''i'' per unit width [m<sup>2</sup>/s]
:<math>\hat{e}</math> = unit vector for inflow direction = <math>(sin\ \varphi, cos\ \varphi)</math>
:<math>\varphi</math> = inflow direction measured clockwise from North [deg]
:<math>\hat{n}</math> = boundary face unit vector (positive outward)
:''Q'' = specified total volume flux across the boundary [m<sup>3</sup>/s]
:''n'' = Manning’s coefficient [s/m<sup>1/3</sup>]
:''r'' = empirical constant equal to approximately 2/3
:<math>\Delta l</math> = cell width in the transverse direction normal to flow [m]
:<math>f_{Ramp}</math>= ramp function [-] (described in Chapter 3).
The total volume flux is positive into the computational domain. Since it is not always possible to orient all flux boundaries to be normal to the inflow direction, the option is given to specify an inflow direction <math>(\varphi)</math>. The angle is specified in degrees clockwise from true North. If the angle is not specified, then the inflow angle is assumed to be normal to the boundary. The total volume flux is conserved independently of the inflow direction.
=Water Level Boundary=
The general formula for the boundary water surface elevation is given by
{{Equation|<math>
\bar{n}_B = f_{Ramp}(\bar{\eta}_E + \Delta \bar{\eta} + \bar{\eta}_G + \bar{\eta}_C) + (1 - f_{Ramp})\bar{\eta}_0
</math>|4}}
where:
:<math>\bar{\eta}_B</math> = boundary water surface elevation [m]
:<math>\bar{\eta}_E</math> = specified external boundary water surface elevation [m]
:<math>\Delta \bar{\eta}</math> = water surface elevation offset [m]
:<math>\bar{\eta}_0</math> = initial boundary water surface elevation [m]
:<math>\bar{\eta}_C</math> = correction to the boundary water surface elevation based on the wind and wave forcing [m]
:<math>\bar{\eta}_G</math> = water surface elevation component derived from user specified gradients [m]
:<math>f_{Ramp}</math> = ramp function [-] (described in Chapter 3).
The external water surface elevation <math>(\bar{\eta}_E )</math> may be specified as a time series, both spatially constant and varying or calculated from tidal/harmonic constituents. When a time series is specified, the values are interpolated using piecewise Lagrangian polynomials. By default, second order interpolation is used but can be changed by the user. If tidal constituents are specified, then <math>\bar{\eta}_E</math> is calculated as
{{Equation|<math>
\bar{\eta}_E (t) =  \sum f_i A_i \cos (\omega_i t + V_{i}^0 + \hat{u}_i - \kappa_i)
</math>|5}}
where:
:''i'' = subscript indicating a tidal constituent
:<math>A_i</math> = mean amplitude [m]
:<math>f_i</math> = node (nodal) factor [-]
:<math>\omega_i</math> = frequency [deg/hr]
:''t'' = elapsed time from midnight of the starting year [hrs]
:<math>V_i^0 + \hat{u}_i</math> = equilibrium phase [deg]
:<math>\kappa_i</math> = phase lag [deg].
The mean amplitude and phase may be specified by the user or interpolated from a tidal constituent database. The nodal factor is a time-varying correc-tion to the mean amplitude. The equilibrium phase has a uniform com-ponent <math>(V_i^0)</math> and a relatively smaller periodic component. The zero-superscript of <math>V_i^0</math> indicates that the constituent phase is at time zero. Table 1 provides a list of tidal constituents currently supported in CMS. More information on US tidal constituent values can be obtained from the US National Oceanographic and Atmospheric Administration (NOAA) https://www.noaa.gov/ and National Ocean Service (NOS) https://oceanservice.noaa.gov/.
<center>'''Table 1. Tidal constituents implemented in CMS. Constituent speeds are in degrees per mean solar hour.'''
{| border="1"
! Constituent !! Speed !! Constituent !! Speed !! Constituent !! Speed !! Constituent !! Speed
|-
| SA  || 0.041067 || SSA  || 0.082137 || MM  || 0.54438 || MSF  || 1.0159
|-
| MF  || 1.098 || 2Q1*  || 12.8543 || Q1  || 13.3987 || RHO1 || 13.4715
|-
| O1  || 13.943 || M1*  || 14.4967 || P1  || 14.9589 || S1  || 15.0
|-
| K1  || 15.0411 || J1  || 15.5854 || OO1  || 16.1391 || 2N2  || 27.8954
|-
| MU2  || 27.9682 || N2  || 28.4397 || NU2  || 28.5126 || M2  || 28.9841
|-
| LDA2 || 29.4556 || L2  || 29.5285 || T2  || 29.9589 || S2  || 30
|-
| R2  || 30.0411 || K2  || 30.0821 || 2SM2 || 31.0159 || 2MK3 || 42.9271
|-
| M3  || 43.4762 || MK3  || 44.0252 || MN4  || 57.4238 || M4  || 57.9682
|-
| MS4  || 58.9841 || S4  || 60.0 || M6  || 86.9523 || S6  || 90.0
|-
| M8  || 115.9364
|}
</center>
If a harmonic boundary condition is applied, then the node factors are set to one and the equilibrium arguments are set to zero. The harmonic boundary condition is provided as an option for simulating idealized or hypothetical conditions.
The water surface elevation offset <math>(\Delta \bar{\eta})</math> is assumed spatially and temporally constant and may be used to correct the boundary water surface elevation for vertical datums and sea level rise. The component <math>\bar{\eta}_G</math> is intended to represent regional gradients in the water surface elevation, is assumed to be constant in time, and is only applicable when <math>\bar{\eta}_E</math> is spatially constant. When applying a water level boundary condition to the nearshore, local flow reversals and boundary problems may result if the wave- and wind-induced setup is not included. This problem is avoided by adding a correction <math>(\bar{\eta}_C )</math> to the local water level to account for the wind and wave setup as
{{Equation|<math>
\rho g h_B \frac{\partial}{\partial x} (\bar{\eta}_E + \Delta \bar{\eta} + \bar{\eta}_G + \bar{\eta}_C) = \tau_{sx} + \tau_{wx} - \tau_{bx}</math>|6}}
where <math>h_B</math> is the boundary total water depth, and <math>\tau_{sx}, \tau_{wx} \ and\  \tau_{bx}</math> are the wind, wave, and bottom stresses in the boundary direction ''(x)''. The wave forcing term is equal to
{{Equation|<math>
\tau_{wi} = - \frac{\partial}{\partial x_j}(S_{ij} + R_{ij} - \rho h U_{wi} U_{wj})
</math>|7}}
The correction <math>\bar{\eta}_C</math> is only applicable when <math>\bar{\eta}_E</math> is spatially constant as in the case of a single water surface elevation time series.
=Cross-shore Boundary=
In the implicit flow solver, a cross-shore boundary condition is applied in the nearshore by solving the 1D cross-shore momentum equation including wave and wind forcing (Wu et al. 2010). Along a cross-shore boundary, it is assumed that a well-developed longshore current exists (quasi-steady conditions with longshore gradients in advection, diffusion, and water levels equal to zero). These assumptions are valid for relatively long coasts with shore-parallel contours and simplify the alongshore (y-direction) momentum equation to
{{Equation|<math>
\frac{\partial}{\partial x} \left(\rho v_t h \frac{\partial V_y}{\partial x}\right) = \tau_{sy} + \tau_{Wy} - \tau_{by}
</math>|8}}
where <math>\tau_{sy}, \tau_{Wy}, \ and\  \tau_{by}</math> are the surface, wave, and bottom stresses in the longshore direction, respectively. Equation (8) is solved iteratively to determine the longshore current velocity. The cross-shore ''(x)'' component of the velocity is assigned a zero-gradient boundary condition. The longshore current velocity is applied when the flow is directed inwards. When the flow is directed outwards, a zero-gradient boundary condition is applied to the longshore current velocity.
The water level due to waves and wind at the cross-shore boundary can be determined by assuming a zero alongshore gradient of flow velocity and negligible cross-shore current velocity. For this case, the cross-shore momentum equation reduces to
{{Equation|<math>
\rho g h \frac{\partial \bar{\eta}} {\partial x}= \tau_{sx} + \tau_{Wx} - \tau_{bx}
</math>|9}}
where <math>\tau_{sx}, \tau_{Wx}, \ and\ \tau_{bx}</math>
are the surface, wave, and bottom stresses in the cross-shore direction. The water level boundary condition is applied for the case when the flow is directed outwards. When the flow is directed inwards, a zero-gradient boundary condition is applied to the water level.
= Tidal Constituent=
The water level predictions are based on a harmonic equation with several arguments
{{Equation|<math>
\eta(t) =  \sum_{i=1}^N f_i A_i \cos (\omega_i t + V_{i}^0 + u_i - \kappa_i)
</math>|1}}
where <math>A</math> is the constituent ''mean '' amplitude, <math>f</math> is a factor that reduces the mean amplitude and varies in time, <math>V_0+u</math> are the constituents ''equilibrium '' phase and <math>\kappa</math> is the constituent phase lag or epoch. Table 1 shows a list of the currently supported tidal constituents in CMS.
{| border="1"
! Constituent !! Speed !! Constituent !! Speed !! Constituent !! Speed !! Constituent !! Speed
|-
| SA  || 0.041067 || SSA  || 0.082137 || MM  || 0.54438 || MSF  || 1.0159
|-
| MF  || 1.098 || 2Q1  || 12.8543 || Q1  || 13.3987 || RHO1 || 13.4715
|-
| O1  || 13.943 || M1  || 14.4967 || P1  || 14.9589 || S1  || 15
|-
| K1  || 15.0411 || J1  || 15.5854 || OO1  || 16.1391 || 2N2  || 27.8954
|-
| MU2  || 27.9682 || N2  || 28.4397 || NU2  || 28.5126 || M2  || 28.9841
|-
| LDA2 || 29.4556 || L2  || 29.5285 || T2  || 29.9589 || S2  || 30
|-
| R2  || 30.0411 || K2  || 30.0821 || 2SM2 || 31.0159 || 2MK3 || 42.9271
|-
| M3  || 43.4762 || MK3  || 44.0252 || MN4  || 57.4238 || M4  || 57.9682
|-
| MS4  || 58.9841 || S4  || 60 || M6  || 86.9523 || S6  || 90
|-
| M8  || 115.9364
|}
== Flux ==
The water flux is specified as m^3/sec per cell along the cell string. This value is multiplied by the number of cells in the cell string to obtain the total flux.
The total flux is then redistributed along the boundary according to
{{Equation|<math>
q_i  = \frac{Q_{b}}{ \sum \Delta s_i h_i^{1+r}/n_i} \frac{ h_i^{1+r}} {n_i}
</math>|2}}
= Cross-shore =
Along a cross-shore boundary, it is assumed that a well-developed longshore current exists. Thus, the y (alongshore) momentum equation can be reduced as follows
{{Equation|<math>
0 =  \frac{\partial}{\partial x} \biggl( \nu_t h \frac{\partial V}{\partial x} \biggr) + \tau_{sy} + \tau_{wy} - \tau_{by}
</math>|3}}
 
The water level setup due to waves and winds at the cross-shore boundary can be determined by assuming a zero alongshore gradient of water level, or using the following equation reduced from the x (cross-shore) momentum equation
{{Equation|<math>
0 = \rho g \frac{\partial \eta}{\partial x} + \tau_{sx} + \tau_{wx}
</math>|4}}
----
----
{| border="1"
{| border="1"
! Symbol!! Description
! Symbol!! Description
Line 27: Line 226:
|  <math>\kappa</math>  || Constituent phase or epoch
|  <math>\kappa</math>  || Constituent phase or epoch
|}
|}
=References=
* Wu, W., A. Sánchez, and M. Zhang. 2010. An implicit 2-D depth-averaged finite-volume model of flow and sediment transport in coastal waters. In Proceedings of the International Conference on Coastal Engineering, No. 32. Paper Number: Sediment 23. Shanghai, China.


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[[CMS#Documentation_Portal | Documentation Portal]]
[[CMS#Documentation_Portal | Documentation Portal]]

Latest revision as of 19:03, 9 August 2023

Wall Boundary

The wall boundary condition is a closed boundary and is applied at any cell face between wet and dry cells. Any unassigned boundary cell at the edge of the model domain is assumed to be closed and is assigned a wall boundary. A zero normal flux to the boundary is applied at closed boundaries. Two boundary conditions are available for the tangential flow:

  1. Free-slip: no tangential shear stress (wall friction)
  2. Partial-slip: tangential shear stress (wall friction) calculated based on the log law.

Assuming a log law for a rough wall, the partial-slip tangential 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_{wall} = \rho c_{wall}U^2_\parallel} (1)

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_\parallel}  is the magnitude of the wall parallel current 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_{wall}} is the wall friction coefficient 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 c_{wall} = \left[\frac{\kappa}{ln(y_P / y_0)} \right]^2 } (2)

Here, 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 y_0} is the roughness length of the wall and is assumed to be equal to that of the bed 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 (i.e, y_0 = z_0)} . The distance from the wall to the cell center 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 y_P} .

Flux Boundary

The flux boundary condition is typically applied to the upstream end of a river or stream and specified as either a constant or time series of total water volume flux (Q). In a 2DH model, the total volume flux needs to be distributed across the boundary in order to estimate the depth-averaged velocities. This is done using a conveyance approach in which the current velocity is assumed to be related to the local flow depth (h) and Manning’s (n) 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 \propto h^r / n} . Here, r is an empirical conveyance coefficient equal to approximately 2/3 for uniform flow. The smaller the r value the more uniform the current velocities are across the flux boundary. The water volume flux 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_i )} at each boundary cell (i) 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 \overrightarrow{q}_i = h\overrightarrow{U}_i = \frac{f_{Ramp}Q} {\bigg | \sum\limits_i (\hat{e}\cdot\hat{n}) \frac{h_i^{r+1}} {n_i} \Delta l_i\bigg |} \frac{h_i^{r+1}}{n_i} \hat{e} } (3)

where:

i = subscript indicating a boundary cell
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 \overrightarrow{q_i}} = volume discharge at boundary cell i per unit width [m2/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 \hat{e}} = unit vector for inflow direction = 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 (sin\ \varphi, cos\ \varphi)}
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 \varphi} = inflow direction measured clockwise from North [deg]
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 \hat{n}} = boundary face unit vector (positive outward)
Q = specified total volume flux across the boundary [m3/s]
n = Manning’s coefficient [s/m1/3]
r = empirical constant equal to approximately 2/3
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 \Delta l} = cell width in the transverse direction normal to flow [m]
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_{Ramp}} = ramp function [-] (described in Chapter 3).

The total volume flux is positive into the computational domain. Since it is not always possible to orient all flux boundaries to be normal to the inflow direction, the option is given to specify an inflow direction 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 (\varphi)} . The angle is specified in degrees clockwise from true North. If the angle is not specified, then the inflow angle is assumed to be normal to the boundary. The total volume flux is conserved independently of the inflow direction.

Water Level Boundary

The general formula for the boundary water surface elevation 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 \bar{n}_B = f_{Ramp}(\bar{\eta}_E + \Delta \bar{\eta} + \bar{\eta}_G + \bar{\eta}_C) + (1 - f_{Ramp})\bar{\eta}_0 } (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 \bar{\eta}_B} = boundary water surface elevation [m]
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 \bar{\eta}_E} = specified external boundary water surface elevation [m]
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 \Delta \bar{\eta}} = water surface elevation offset [m]
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 \bar{\eta}_0} = initial boundary water surface elevation [m]
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 \bar{\eta}_C} = correction to the boundary water surface elevation based on the wind and wave forcing [m]
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 \bar{\eta}_G} = water surface elevation component derived from user specified gradients [m]
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_{Ramp}} = ramp function [-] (described in Chapter 3).

The external water surface elevation 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 (\bar{\eta}_E )} may be specified as a time series, both spatially constant and varying or calculated from tidal/harmonic constituents. When a time series is specified, the values are interpolated using piecewise Lagrangian polynomials. By default, second order interpolation is used but can be changed by the user. If tidal constituents are specified, then 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 \bar{\eta}_E} 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 \bar{\eta}_E (t) = \sum f_i A_i \cos (\omega_i t + V_{i}^0 + \hat{u}_i - \kappa_i) } (5)

where:

i = subscript indicating a tidal constituent
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_i} = mean amplitude [m]
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_i} = node (nodal) 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 \omega_i} = frequency [deg/hr]
t = elapsed time from midnight of the starting year [hrs]
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 V_i^0 + \hat{u}_i} = equilibrium phase [deg]
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_i} = phase lag [deg].

The mean amplitude and phase may be specified by the user or interpolated from a tidal constituent database. The nodal factor is a time-varying correc-tion to the mean amplitude. The equilibrium phase has a uniform com-ponent 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 (V_i^0)} and a relatively smaller periodic component. The zero-superscript of 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 V_i^0} indicates that the constituent phase is at time zero. Table 1 provides a list of tidal constituents currently supported in CMS. More information on US tidal constituent values can be obtained from the US National Oceanographic and Atmospheric Administration (NOAA) https://www.noaa.gov/ and National Ocean Service (NOS) https://oceanservice.noaa.gov/.

Table 1. Tidal constituents implemented in CMS. Constituent speeds are in degrees per mean solar hour.
Constituent Speed Constituent Speed Constituent Speed Constituent Speed
SA 0.041067 SSA 0.082137 MM 0.54438 MSF 1.0159
MF 1.098 2Q1* 12.8543 Q1 13.3987 RHO1 13.4715
O1 13.943 M1* 14.4967 P1 14.9589 S1 15.0
K1 15.0411 J1 15.5854 OO1 16.1391 2N2 27.8954
MU2 27.9682 N2 28.4397 NU2 28.5126 M2 28.9841
LDA2 29.4556 L2 29.5285 T2 29.9589 S2 30
R2 30.0411 K2 30.0821 2SM2 31.0159 2MK3 42.9271
M3 43.4762 MK3 44.0252 MN4 57.4238 M4 57.9682
MS4 58.9841 S4 60.0 M6 86.9523 S6 90.0
M8 115.9364

If a harmonic boundary condition is applied, then the node factors are set to one and the equilibrium arguments are set to zero. The harmonic boundary condition is provided as an option for simulating idealized or hypothetical conditions.

The water surface elevation offset 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 (\Delta \bar{\eta})} is assumed spatially and temporally constant and may be used to correct the boundary water surface elevation for vertical datums and sea level rise. The component 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 \bar{\eta}_G} is intended to represent regional gradients in the water surface elevation, is assumed to be constant in time, and is only applicable when 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 \bar{\eta}_E} is spatially constant. When applying a water level boundary condition to the nearshore, local flow reversals and boundary problems may result if the wave- and wind-induced setup is not included. This problem is avoided by adding a correction 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 (\bar{\eta}_C )} to the local water level to account for the wind and wave setup 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 \rho g h_B \frac{\partial}{\partial x} (\bar{\eta}_E + \Delta \bar{\eta} + \bar{\eta}_G + \bar{\eta}_C) = \tau_{sx} + \tau_{wx} - \tau_{bx}} (6)

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 h_B} is the boundary total water depth, 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_{sx}, \tau_{wx} \ and\ \tau_{bx}} are the wind, wave, and bottom stresses in the boundary direction (x). The wave forcing term is 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 \tau_{wi} = - \frac{\partial}{\partial x_j}(S_{ij} + R_{ij} - \rho h U_{wi} U_{wj}) } (7)

The correction 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 \bar{\eta}_C} is only applicable when 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 \bar{\eta}_E} is spatially constant as in the case of a single water surface elevation time series.

Cross-shore Boundary

In the implicit flow solver, a cross-shore boundary condition is applied in the nearshore by solving the 1D cross-shore momentum equation including wave and wind forcing (Wu et al. 2010). Along a cross-shore boundary, it is assumed that a well-developed longshore current exists (quasi-steady conditions with longshore gradients in advection, diffusion, and water levels equal to zero). These assumptions are valid for relatively long coasts with shore-parallel contours and simplify the alongshore (y-direction) momentum equation 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 \frac{\partial}{\partial x} \left(\rho v_t h \frac{\partial V_y}{\partial x}\right) = \tau_{sy} + \tau_{Wy} - \tau_{by} } (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 \tau_{sy}, \tau_{Wy}, \ and\ \tau_{by}} are the surface, wave, and bottom stresses in the longshore direction, respectively. Equation (8) is solved iteratively to determine the longshore current velocity. The cross-shore (x) component of the velocity is assigned a zero-gradient boundary condition. The longshore current velocity is applied when the flow is directed inwards. When the flow is directed outwards, a zero-gradient boundary condition is applied to the longshore current velocity.

The water level due to waves and wind at the cross-shore boundary can be determined by assuming a zero alongshore gradient of flow velocity and negligible cross-shore current velocity. For this case, the cross-shore momentum equation reduces 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 \rho g h \frac{\partial \bar{\eta}} {\partial x}= \tau_{sx} + \tau_{Wx} - \tau_{bx} } (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 \tau_{sx}, \tau_{Wx}, \ and\ \tau_{bx}} are the surface, wave, and bottom stresses in the cross-shore direction. The water level boundary condition is applied for the case when the flow is directed outwards. When the flow is directed inwards, a zero-gradient boundary condition is applied to the water level.

Tidal Constituent

The water level predictions are based on a harmonic equation with several arguments

  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(t) = \sum_{i=1}^N f_i A_i \cos (\omega_i t + V_{i}^0 + u_i - \kappa_i) } (1)

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 A} is the constituent mean amplitude, 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} is a factor that reduces the mean amplitude and varies in time, 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 V_0+u} are the constituents equilibrium phase 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 \kappa} is the constituent phase lag or epoch. Table 1 shows a list of the currently supported tidal constituents in CMS.

Constituent Speed Constituent Speed Constituent Speed Constituent Speed
SA 0.041067 SSA 0.082137 MM 0.54438 MSF 1.0159
MF 1.098 2Q1 12.8543 Q1 13.3987 RHO1 13.4715
O1 13.943 M1 14.4967 P1 14.9589 S1 15
K1 15.0411 J1 15.5854 OO1 16.1391 2N2 27.8954
MU2 27.9682 N2 28.4397 NU2 28.5126 M2 28.9841
LDA2 29.4556 L2 29.5285 T2 29.9589 S2 30
R2 30.0411 K2 30.0821 2SM2 31.0159 2MK3 42.9271
M3 43.4762 MK3 44.0252 MN4 57.4238 M4 57.9682
MS4 58.9841 S4 60 M6 86.9523 S6 90
M8 115.9364

Flux

The water flux is specified as m^3/sec per cell along the cell string. This value is multiplied by the number of cells in the cell string to obtain the total flux. The total flux is then redistributed along the boundary according 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 q_i = \frac{Q_{b}}{ \sum \Delta s_i h_i^{1+r}/n_i} \frac{ h_i^{1+r}} {n_i} } (2)

Cross-shore

Along a cross-shore boundary, it is assumed that a well-developed longshore current exists. Thus, the y (alongshore) momentum equation can be reduced as follows

  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 0 = \frac{\partial}{\partial x} \biggl( \nu_t h \frac{\partial V}{\partial x} \biggr) + \tau_{sy} + \tau_{wy} - \tau_{by} } (3)

The water level setup due to waves and winds at the cross-shore boundary can be determined by assuming a zero alongshore gradient of water level, or using the following equation reduced from the x (cross-shore) momentum equation

  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 0 = \rho g \frac{\partial \eta}{\partial x} + \tau_{sx} + \tau_{wx} } (4)
Symbol Description
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} Time
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} Constituent mean amplitude
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} Constituent nodal 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 V^0} Constituent equilibrium argument
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} Constituent equilibrium argument
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} Constituent phase or epoch

References

  • Wu, W., A. Sánchez, and M. Zhang. 2010. An implicit 2-D depth-averaged finite-volume model of flow and sediment transport in coastal waters. In Proceedings of the International Conference on Coastal Engineering, No. 32. Paper Number: Sediment 23. Shanghai, China.

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