Mass Conservation

The continuity equation describing the water volume conservation in 1D is given by

1) \displaystyle \frac{\partial A}{\partial t} + \frac{\partial Q}{\partial x} = q_l

where

t = time [T]
Q is the flow [L3/T]
A is the cross-sectional area [L2]
q_l is the lateral inflow per unit length [L2/T]

Momentum Conservation

The momentum equation describing the conservation of momentum in 1D is written as:

2) \displaystyle \frac{\partial V}{\partial t} + V \frac{\partial V}{\partial x} + g \left ( \frac{\partial H}{\partial x} +S_f + S_h\right ) = \nu_t \frac{\partial^2 V}{\partial x^2} + \frac{\tau _{s,x}}{\rho h}

where

V = cross-sectionally averaged velocity [L/T]
z_s = water surface elevation [L]
g = gravitational acceleration [L/T2]
\nu_t = turbulent eddy viscosity [L2/T]
S_f = friction slope [-]
S_h = added force term [-]
\tau_s = wind surface stress [M/L/T2]
\rho = water density [M/L3]
h = water depth [L]

The above equation can be written for the channel and left and right floodplains as (ignoring the wind stresses):

3) \displaystyle \frac{\partial V_i}{\partial t} + V_i \frac{\partial V_i}{\partial x} + g \left ( \frac{\partial z_s}{\partial x} +S_f + S_h\right ) = \nu_{t,i} \frac{\partial^2 V_i}{\partial x^2} + \frac{\tau _{s,x}}{\rho h_i} + M_i, \: \: i \in \{ch,rob,lob\}

where ch,lob,rob indicate the channel, and left and right overbanks, respectively. M_i represents the momentum exchange with neighboring cross-sectional areas. 

Bottom Friction

Bottom friction represents the energy loss due to skin and form drag on the bed and any other sources of drag including vegetation. The friction slope in the 1D momentum equation is given by

4) S_f = \sqrt \frac{Q}{K}=\frac{n^2}{R^{4/3}}|V|V

where

n = Manning’s roughness coefficient [s/m1/3]
R = hydraulic radius [L]
V = current velocity [L/T]
 Q= discharge [L3/T]
K = conveyance [L3/T]
 A= cross-sectional area [L2]

The bottom shear stress is given by

5) \tau_b = \gamma R S_f = \rho C_D |V| V = \rho R c_f V

where

\gamma = \rho g = water unit weight [M/L2/T2]
\rho = water density [M/L3]
C_D = drag coefficient [-]
c_f = friction coefficient [1/T]
V = current velocity [L/T]

The drag coefficient, C_D, is related to the nonlinear friction coefficient, c_f, by 

6) c_f = \frac{C_D}{R} |V|

The bottom shear velocity is given by

7) u_* = \sqrt { \tau_b / \rho}

Eddy Viscosity

Turbulence is a complex phenomenon of chaotic (turbulent) fluid motion and eddies spanning a wide range of length scales. Many of the length scales are too small to be feasibly resolved by a discrete numerical model, so turbulent flow mixing is modeled as a gradient diffusion process. The eddy viscosity is computed as follows,

8) \nu_t = D u_* h

where D is the mixing coefficient, u_* is the shear velocity, and h is the water depth. 

Wind Shear Stress

The wind surface stress vector is calculated as:

9) \tau_s = \rho_a C_d |W_{10}| W_{10}

where \rho_a is the air density at sea level (~1.29 kg/m3), C_d is the wind drag coefficient, W_{10} is the 10-m height wind velocity. The wind speed is calculated using either an Eulerian or Lagrangian reference frame as:

10) W_{10} = W_{10}^E - \gamma_W V

in which W_{10}^E is the 10-m wind velocity relative to the solid earth (Eulerian wind speed), and \gamma_W is equal to zero for the Eulerian reference frame or one for the Lagrangian reference frame.