The numerical model Hyper (abbreviation for hyperpycnal flow) is a two-dimensional depth-averaged model. This finite volume model solves the depth-averaged equation of mass, momentum and sediment conservation of density driven flow along with the Exner Equation of bed sediment continuity. The model is based on earlier works of Bradford and Katopodes, 1999a, Bradford and Katopodes, 1999b, and Imran and Syvitski (2000). To incorporate the effect of alongshore current on the development of hyperpycnal flow, the governing equations used in the earlier work of Bradford and Katopodes, 1999a, Bradford and Katopodes, 1999b have been slightly modified through coordinate transformation. The resulting governing equations for conservation of fluid mass, momentum and sediment concentration are as follows:

Continuity

X-momentum

Y-momentum

Conservation of suspended sediment

The bed sediment conservation equation has the form

In the above equations, h represents current thickness, u and v are depth-averaged velocity in x- and y-direction, w is the alongshore current of constant magnitude. Though alongshore current may vary with the depth of water column, we consider constant magnitude of alongshore current for the entire water column due to the use of depth-averaged equations in the present model. Ci represents vertically averaged volume concentration of ith sediment and CT is the summation of all sediment fractions. The parameter Ri = (ρsi − ρ) / ρ, where ρsi is the density of ith sediment and ρ is the density of ambient water. The shear velocities are defined as

Here, CD is the bed drag coefficient which ranges from 0.002 to 0.05 depending on the flow type (Garcia, 1990). The bed slopes in x- and y-directions are represented by sx and sy, respectively. In order to close the problem, it is necessary to use some internal relationships. Fluid entrainment coefficient Ew is specified by using the relation of Parker et al. (1986).

where, Ri is the bulk Richardson number and is defined as

The expression developed by Garcia and Parker (1993) for sediment entrainment coefficient Esi of ith sediment is used for model closure, i. e.

where,

and

is the particle Reynolds number, D is characteristic grain size, ν denotes the kinematic viscosity of water. The parameters (α1, α2) take respective values (1, 0.6) for Rp > 2.36 and (0.586, 1.23) for Rp ≤ 2.36. The fall velocity vs is calculated using the empirical relationship developed by Dietrich (1982).

The near bed concentration of ith sediment cbi is calculated using the expression developed by Garcia (1994),

where Dsg denotes the geometric mean size of the suspended sediment mixture.

The governing equations of the present model are solved using a finite volume method. In order to get a second order accuracy in time, a predictor–corrector time stepping approach is adopted to solve the governing equations for conservation of fluid mass, momentum and sediment concentration. For computing the interfacial fluxes, Roe's approximate Riemann solver has been used in the present model. The bed continuity equation has been solved after solving the equations governing the flow field. In order to maintain the second order accuracy, Heun's predictor and corrector method has been used to integrate the equation. More details on the numerical scheme can be found in Bradford and Katopodes (1999a).

Due to the presence of source term in the continuity equation, some oscillations are observed near the front of the turbidity current head. These oscillations become higher when the current spreads out into the large scale domain. To dampen these oscillations, a constant coefficient artificial viscosity model given by Jameson et al. (1981) has been used. This procedure works to smooth large gradients while leaving the smooth areas relatively undisturbed. For the present model, we apply this smoothing procedure for current thickness (h) only.