### Formulation

We consider a planar, inviscid, incompressible, immiscible gravity current of density *ρ*
_{c} in a deep ambient fluid of density *ρ*
_{a}, as shown in Fig. 1. The current propagates along a smooth horizontal bottom in the positive *x*
^{∗} direction in time *t*
^{∗}, and gravitational acceleration *g* acts in the negative *z*
^{∗} direction, where asterisks denote dimensional variables. The propagating current is initially stationary in a reservoir of length *x*
_{0} and height *h*
_{0}, and propagation occurs after a dam at *x*
^{∗}=*x*
_{0} is rapidly removed at *t*
^{∗}=0. The boundary at *x*
^{∗}=0 is a rigid wall. The flow front at \(x^{*}=x_{\mathrm {N}}^{*}(t^{*})\) is affected by the resistance of the ambient fluid, where N denotes the front. This problem is referred to as the “dam-break problem” (e.g., Ungarish 2009), and is a simple geophysical scenario.

We assume that the current is shallow, with *h*
_{0}/*x*
_{0}≪1, and is in hydrostatic equilibrium in the vertical direction (i.e., the shallow-water approximation). In the shallow-water approximation, we can obtain the vertically averaged conservation equations of mass and momentum for the flow interior \(x^{*}<x_{\mathrm {N}}^{*}\) (e.g., Ungarish 2007) as follows:

$$\begin{array}{@{}rcl@{}} &&\frac{\partial h^{*}}{\partial t^{*}} + \frac{\partial }{\partial x^{*}}(u^{*}h^{*}) = 0, \end{array} $$

(1)

$$\begin{array}{@{}rcl@{}} &&\frac{\partial }{\partial t^{*}}(u^{*}h^{*}) + \frac{\partial }{\partial x^{*}}\left(u^{*2}h^{*} + \frac{1}{2}\frac{\rho_{\mathrm{c}}-\rho_{\mathrm{a}}}{\rho_{\mathrm{c}}}gh^{*2} \right) = 0, \end{array} $$

(2)

where *h*(*x,t*) is the local height and *u*(*x,t*) is the local horizontal velocity.

At the flow front \(x^{*}=x_{\mathrm {N}}^{*}(t^{*})\), the kinematic condition (\(\mathrm {d}x_{\mathrm {N}}^{*}/\mathrm {d}t^{*}=u_{\mathrm {N}}^{*}\)) and the mass and momentum equations should be taken into account. In addition, to describe realistic gravity current dynamics, we must consider a quasi-steady balance between the buoyancy pressure driving the current front (\(\sim (\rho _{\mathrm {c}}-\rho _{\mathrm {a}})gh_{\mathrm {N}}^{*}\)) and the resistance pressure caused by the acceleration of the ambient fluid around the front (\(\sim \rho _{\mathrm {a}} u_{\mathrm {N}}^{*2}\)). This condition is known as the front condition, and can be written as follows (e.g., Ungarish 2007):

$$\begin{array}{@{}rcl@{}} u_{\mathrm{N}}^{*} = Fr \sqrt{\frac{\rho_{\mathrm{c}}-\rho_{\mathrm{a}}}{\rho_{\mathrm{a}}}g h_{\mathrm{N}}^{*}} \qquad \text{at} \quad x^{*}=x_{\mathrm{N}}^{*}(t^{*}), \end{array} $$

(3)

where *Fr*, which is an imposed frontal Froude number, is assumed to be a constant of order 10^{0} (e.g., \(\sqrt {2}\); Benjamin 1968).

Here, using *x*
_{0} as the length scale and *h*
_{0} as the height scale, we rewrite all dimensional variables to dimensionless variables as follows:

$$\begin{array}{@{}rcl@{}} x = x^{*} / x_{0}, \quad h = h^{*} / h_{0}, \quad u = u^{*} / U, \quad t = t^{*} / T, \end{array} $$

(4)

with

$$\begin{array}{@{}rcl@{}} U = \sqrt{\frac{\rho_{\mathrm{c}} - \rho_{\mathrm{a}}}{\rho_{\mathrm{c}}}gh_{0}}, \quad T = x_{0} / U. \end{array} $$

(5)

o Eqs. (1)–(3), we obtain

$$\begin{array}{@{}rcl@{}} &&\frac{\partial }{\partial t}\, {\boldsymbol{q}} + \frac{\partial }{\partial x}\, \boldsymbol{f} = \mathbf{0} \end{array} $$

(6)

$$\begin{array}{@{}rcl@{}} &&u_{\mathrm{N}} = Fr \sqrt{\rho_{\mathrm{c}}/\rho_{\mathrm{a}}} \sqrt{h_{\mathrm{N}}} \qquad \text{at} \quad x = x_{\mathrm{N}}(t) \end{array} $$

(7)

with

$$\begin{array}{@{}rcl@{}} \boldsymbol{q} = \left(\begin{array}{c} h \\ uh \end{array}\right); \quad {\boldsymbol{f}} = \left(\begin{array}{c} uh \\ u^{2}h + \frac{1}{2} h^{2} \end{array}\right). \end{array} $$

(8)

Note that the density ratio *ρ*
_{c}/*ρ*
_{a} is included only in the front condition (7). Hence, to capture the effects of *ρ*
_{c}/*ρ*
_{a}, it is important to calculate the front condition correctly (Ungarish 2007).

The behavior of the analytical solutions for the above equations depends on *ρ*
_{c}/*ρ*
_{a} (Fig. 2; Ungarish 2007). The analytical solutions of the dam-break problem consist of an initial “slumping” stage and a subsequent “self-similar” stage (Fig. 2
a; e.g., Hogg 2006). During the slumping stage, the front moves with a constant speed and height. During this stage, an initial backward-propagating rarefaction wave arises from the rapidly removed dam, and then a wave arises from the reflection of this rarefaction wave at the back wall *x*=0 at *t*=1. The slumping stage continues until the front is caught by this reflection wave. After the slumping stage, the solution is asymptotic to a self-similar solution as time tends to infinity (i.e., the self-similar stage). During this stage, the velocity and height of the front decrease with time. The dependence of the solution on *ρ*
_{c}/*ρ*
_{a} is clearly observed in the behavior of the flow front. When *ρ*
_{c}/*ρ*
_{a}∼10^{0}, the front height *h*
_{N} is on the order of 10^{−1} during the slumping stage and in the early self-similar stage (Fig. 2
a). On the other hand, when *ρ*
_{c}/*ρ*
_{a}∼10^{3},*h*
_{N} is much smaller than 10^{−1}, even from the beginning, the front velocity *u*
_{N} is substantially greater than *u*
_{N} for *ρ*
_{c}/*ρ*
_{a}∼10^{0} (Fig. 2
b). These differences can be interpreted as follows: the momentum lost due to the resistance of the ambient fluid at the front becomes less significant with respect to the momentum of the current as *ρ*
_{c}/*ρ*
_{a} increases. We aim to numerically reproduce these features of the analytical solution below.

### Numerical methods

In this study, we developed a numerical method for modeling gravity currents for a wide range of *ρ*
_{c}/*ρ*
_{a} by discretizing the dimensionless mass and momentum conservation equations (Eqs. (6) and (8)). As these equations are nonlinear and hyperbolic, shocks may develop in the currents. Consequently, we used a finite volume method with shock-capturing capability (e.g., LeVeque 2002; Toro 2001). The finite volume method updates a piecewise constant function \(\boldsymbol {Q}_{i}^{n}\) that approximates the average value of the solution *q* in each grid cell *i* at time step *n*, using the expression

$$\begin{array}{@{}rcl@{}} {\boldsymbol{Q}}^{n+1}_{i} = \boldsymbol{Q}_{i}^{n} - \frac{\Delta t}{\Delta x}({\boldsymbol{F}}_{i+1/2} - {\boldsymbol{F}}_{i-1/2}), \end{array} $$

(9)

where *Δ*
*x* is the constant cell length and *Δ*
*t* is the time interval. *F*
_{
i+1/2}, which is the intercell flux between cells *i* and *i*+1, is obtained by using an exact Riemann solver or an approximate Riemann solver, such as the Roe scheme (e.g., LeVeque 2002; Toro 2001). The time interval *Δ*
*t* is limited by the Courant–Friedrichs–Lewy condition (e.g., LeVeque 2002; Toro 2001).

As mentioned above, if we are to capture the effects of *ρ*
_{c}/*ρ*
_{a}, it is important to calculate the front condition (7) correctly. Previously, two types of numerical models have been proposed to calculate the front condition. In one, the front condition is calculated as a boundary condition at each time step (e.g., Ungarish 2009). We refer to this model as the Boundary Condition (BC) model (Fig. 3
a). In the other, the front condition is calculated by setting a thin artificial bed ahead of the front (e.g., Toro 2001). We refer to this as the Artificial Bed (AB) model (Fig. 3
b). In the AB model, the resistance of the ambient fluid at the flow front is modeled by the reaction of the force pushing the artificial bed at the flow front. These models will be described below.

#### Boundary Condition (BC) model

In the BC model, three quantities at the flow front (*x*
_{N}, *h*
_{N}, and *u*
_{N}) are calculated as boundary conditions of the current from the three equations (mass and momentum conservation equations and front condition) at each time step. In the present numerical method, because we apply a fixed spatial coordinate with constant *Δ*
*x*, the front position *x*=*x*
_{N}(*t*) generally does not coincide with the margins of the grid cells. We therefore define the cell that includes the front as the front cell (*i*=*FC*(*t*), where *FC*(*t*) is an integer), and the width of the region that the current occupies in the front cell as *Δ*
*x*
_{
FC
}(*t*)(0≤*Δ*
*x*
_{
FC
}(*t*)<*Δ*
*x*; see Fig. 4). Using *FC*(*t*) and *Δ*
*x*
_{
FC
}(*t*), we can write the front position as

$$\begin{array}{@{}rcl@{}} x_{\mathrm{N}}(t) = (FC(t)-1) \Delta x + \Delta x_{FC}(t). \end{array} $$

(10)

The values of *h*
_{N} and *u*
_{N} are approximated by the values of *h* and *u* at the front cell (i.e., *h*
_{
FC
} and *u*
_{
FC
}).

When the kinematic condition (d*x*
_{N}/d*t*=*u*
_{N}) is taken into account, the discretized equations for mass and momentum conservation at the flow front are given by

$$\begin{array}{@{}rcl@{}} \Delta x_{FC}^{n+1} h_{FC}^{n+1} = \Delta x_{FC}^{n} h_{FC}^{n} + \Delta t f_{1} \end{array} $$

(11)

and

$$ \begin{aligned} {}\Delta x_{FC}^{n+1} (uh)_{FC}^{n+1} = \Delta x_{FC}^{n} (uh)_{FC}^{n} + \Delta t \left(f_{2} - \frac{1}{2}\left(h_{FC}^{n+1}\right)^{2} \right), \qquad \end{aligned} $$

(12)

respectively, where (*f*
_{1},*f*
_{2})^{T} represents the intercell flux *F*
_{
FC−1/2}. From the front condition (i.e., Eq. (7)), we obtain

$$\begin{array}{@{}rcl@{}} \frac{(uh)_{FC}^{n+1}}{h_{FC}^{n+1}} = Fr\sqrt{\rho_{\mathrm{c}}/\rho_{\mathrm{a}}} \sqrt{h_{FC}^{n+1}}. \end{array} $$

(13)

Solving these three equations analytically (e.g., using Ferrari’s method for the solution of the quartic equation) or numerically (e.g., using the Newton–Raphson iteration method), we obtain \(h_{FC}^{n+1}\), \(u_{FC}^{n+1}\), and \(\Delta x_{FC}^{n+1}\), and hence, *h*
_{N},*u*
_{N}, and *x*
_{N} at each time step.

#### Artificial Bed (AB) model

In the AB model, the conservation equations (Eqs. (6) and (8)) are numerically solved using a shock-capturing method for not only the interior, but also the outside of the current by a priori setting a thin artificial bed ahead of the front. Through this numerical procedure, the flow front is generated as the flow following a shock formed ahead of the front without any additional calculation (see Fig. 3
b). In this model, the thickness of the artificial bed (*ε* in Fig. 3
b) is the parameter that controls the front condition (i.e., the values of *h*
_{N} and *u*
_{N} for different values of *ρ*
_{c}/*ρ*
_{a}; see section 10.8 in Toro 2001).

Here, we analytically determined the relationship between *ε* and *ρ*
_{c}/*ρ*
_{a}, as well as that between *u*
_{N} and *ε*, on the basis of the analytical solution for the slumping stage of the dam-break problem (e.g., LeVeque 2002; Toro 2001; Ungarish 2009). The initial conditions are *h*=1 and *u*=0 in the domain 0≤*x*≤1, and *h*=*ε* and *u*=0 in the domain *x*>1, at *t*=0. Let us consider the time evolution of the current before the rarefaction wave reaches the back wall *x*=0 (i.e., 0<*t*≤1).

For hyperbolic equations such as those used in the present system (i.e., Eqs. (6) and (8)), the relationships between the variables (i.e., *h* and *u*) on the characteristics \(c_{\pm }=u\pm \sqrt {h}\) are represented as follows:

$$\begin{array}{@{}rcl@{}} \Gamma_{\pm} = u \pm 2\sqrt{h} = \text{const} \qquad \text{on} \quad \frac{\mathrm{d}x}{\mathrm{d}t} = c_{\pm}, \end{array} $$

(14)

where *Γ*
_{±} are the “Riemann Invariants”. Considering that *c*
_{+} characteristics from the domain with one initial condition (*h*=1,*u*=0) enter the front domain (*h*=*h*
_{N},*u*=*u*
_{N}), we can obtain

$$\begin{array}{@{}rcl@{}} u_{\mathrm{N}} = 2\left(1-\sqrt{h_{\mathrm{N}}}\right) \end{array} $$

(15)

from Eq. (14). The equation provides the relationship between *h*=*h*
_{N} and *u*=*u*
_{N} inside the current.

On the other hand, when an artificial bed with *h*=*ε* and *u*=0 is set, a shock wave traveling with speed *S* occurs ahead of the front. Across this shock wave, the Rankine–Hugoniot condition,

$$\begin{array}{@{}rcl@{}} \boldsymbol{f}(\boldsymbol{q}_{\mathrm{R}}) - \boldsymbol{f}(\boldsymbol{q}_{\mathrm{L}}) = S (\boldsymbol{q}_{\mathrm{R}} - \boldsymbol{q}_{\mathrm{L}}) \end{array} $$

(16)

should hold. Here, the subscript R denotes the state on the right side of the shock and L denotes the state on the left side. From Eq. (16), we obtain the state of the front domain behind the shock (i.e., the relationship between *h*=*h*
_{N} and *u*=*u*
_{N}) as

$$\begin{array}{@{}rcl@{}} u_{\mathrm{N}} = (h_{\mathrm{N}} - \varepsilon) \sqrt{\frac{1}{2} \left(\frac{h_{\mathrm{N}} + \varepsilon}{h_{\mathrm{N}} \varepsilon} \right) }, \end{array} $$

(17)

and the shock speed as

$$\begin{array}{@{}rcl@{}} S = \sqrt{\frac{1}{2} \frac{(h_{\mathrm{N}} + \varepsilon)h_{\mathrm{N}}}{\varepsilon}}. \end{array} $$

(18)

Eliminating *h*
_{N} from Eqs. (15), (17), and (18), we obtain *u*
_{N} and *S* as a function of *ε* (Fig. 5
a). Using the front condition (7) as well as these equations, we also obtain the relationship between the artificial bed thickness *ε* and the density ratio *ρ*
_{c}/*ρ*
_{a} (Fig. 5
b) as

$$ {{\begin{aligned} &{} \left(1 - \frac{2}{Fr\sqrt{\rho_{\mathrm{c}}/\rho_{\mathrm{a}}}+2} \right) \frac{4\sqrt{2 \varepsilon}}{Fr\sqrt{\rho_{\mathrm{c}}/\rho_{\mathrm{a}}}+2}\\ &{} \qquad \qquad = \left\{ \left(\frac{2}{Fr\sqrt{\rho_{\mathrm{c}}/\rho_{\mathrm{a}}}+2} \right)^{2} - \varepsilon \right\} \sqrt{\left(\frac{2}{Fr\sqrt{\rho_{\mathrm{c}}/\rho_{\mathrm{a}}}+2} \right)^{2} + \varepsilon }. \qquad \end{aligned}}} $$

(19)

Note that because we use Eq. (15) here, these relationships (Fig. 5) are in the slumping stage.

In Fig. 5
a, *S* is larger than the front velocity, *u*
_{N}, because of the accumulation of the artificial bed at the flow front (see Fig. 3
b). This deviation of *S* from *u*
_{N} is substantial for \(\varepsilon \gtrsim 10^{-3}\). This implies that the position of the shock does not always approximate the flow front. If we are to extract the correct position of the flow front, we must calculate an advection equation for a passive tracer concentration, *ϕ* (*ϕ*=1 for 0≤*x*≤1, and *ϕ*=0 for *x*>1, at *t*=0):

$$\begin{array}{@{}rcl@{}} \frac{\partial \phi}{\partial t}+u\frac{\partial \phi}{\partial x}=0 \end{array} $$

(20)

after solving the equations of fluid motion (see section 13.12 in LeVeque 2002 for details).