2.1.1. Overland Flow Routing

In WRF-Hydro, the surface runoff is generated through infiltration-excess and saturation-excess of a supersaturated soil column. It can be routed either two-way or one-way (along the largest gradient of slope), depending on the routing method specified in the model name list. As the physics and algorithms of both routing options are identical, only the equations used for two-way routing are presented here. The overland flow is assumed to be fully unsteady and non-uniform. The diffusive wave formulation, which is a simplification of Saint-Venant equations for a shallow water wave, is applied:

Continuity Equation:

$$\frac{\partial \mathbf{h}}{\partial \mathbf{t}} + \frac{\partial q\_x}{\partial \mathbf{x}} + \frac{\partial q\_y}{\partial y} = \mathbf{i}\_e \tag{1}$$

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Momentum Equation:

$$\begin{cases} S\_{fx} = S\_{\alpha x} - \frac{\partial h}{\partial y} \\ S\_{fy} = S\_{\alpha y} - \frac{\partial h}{\partial y} \end{cases} \tag{2}$$

where *h* is the surface flow depth (L), *qx*, *qy* are the unit discharge in *x* and *y* direction, respectively (L2 T<sup>−</sup>1) (in 1-way routing only discharge in the steepest direction is calculated), and *ie* is the infiltration excess (L T<sup>−</sup>1). *Sf x*, *Sf y* are the friction slope in x and y directions, *Sox*, *Soy* are the bed slope in *x* and *y* directions, and <sup>∂</sup>*<sup>h</sup>* <sup>∂</sup>*<sup>x</sup>* , <sup>∂</sup>*<sup>h</sup>* <sup>∂</sup>*<sup>y</sup>* are the gradient of surface flow depth in *x* and *y* directions.

Manning's equation is used to calculate *qx* and *qy* in order to solve Equation (1),

$$q = ah^{\beta}, \; \beta = 5/3$$

$$a = \frac{s\_f^{1/2}}{n} \tag{3}$$

where *q* is the unit discharge in *x* or *y* direction (L<sup>2</sup> T<sup>−</sup>1), *h* is the surface flow depth (L), *Sf* is the friction slope in *x* or *y* direction, *n* is the Manning roughness coefficient of land surface.

Since WRF-Hydro's performance is steady with the one-way routing in a parallel mode, we use the one-way overland routing method to simulate overland flow and provide the hydraulic parameters needed to drive the sediment model.

#### 2.1.2. Channel Flow Routing

Once overland flow gets into the channel network, the water will be routed as channel flow. Currently, WRF-Hydro provides three channel routing options: Muskingum, Muskingum-Cunge, and DiffusiveWave Routing. As the first two options are usually applied for vector-based reaches, we use the third option for gridded channel routing. An explicit, one-dimensional, variable time-stepping diffusive wave formulation is used as follows:

Continuity Equation:

$$\frac{\partial A}{\partial t} + \frac{\partial Q}{\partial \mathbf{x}} = q\_{\text{lat}} \tag{4}$$

Momentum Equation:

$$\frac{\partial \mathcal{Z}}{\partial \mathbf{x}} = -\mathcal{S}\_f \tag{5}$$

where *A* is the cross-section area (L2), t is the time (T), *Q* is the flow discharge, which is the product of cross-section area and mean flow velocity perpendicular to cross-section area (L3 T−1), *qlat* is the unit discharge of the lateral flow (L<sup>2</sup> T−1), *Z* is the water surface elevation, which is the sum of bed elevation and water depth (L), *Sf* is the friction slope.

The friction slope *Sf* is solved as follows:

$$S\_f = \left(\frac{Q}{K}\right)^2\tag{6}$$

$$K = \frac{\mathbb{C}\_m}{n} A R^{\frac{2}{3}} \tag{7}$$

where *K* is the flow conveyance coefficient (L<sup>3</sup> T<sup>−</sup>1), *Cm* is the dimensional constant (1.0 for SI units), *n* is the Manning roughness coefficient, *R* is the hydraulic radius (L).
