A numerical shallow-water model for gravity currents for a wide range of density differences

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.

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

where Fr, which is an imposed frontal Froude number, is assumed to be a constant of order 10⁰ (e.g., \(\sqrt {2}\); Benjamin 1968).

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∼10⁰, 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∼10³,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∼10⁰ (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

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 (dx _N/dt=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).

Comparison of analytical and numerical results

Figure 6 shows the numerical results from the BC model along with the analytical results for the cases of ρ _c/ρ _a=1.01 (a) and 1000 (b). The numerical results for ρ _c/ρ _a=1.01 agree well with the analytical results from the early slumping stage to the late self-similar stage. The numerical results for ρ _c/ρ _a=1000 also appear to agree with the analytical results, but the speed of the front position, \(\dot {x}_{\mathrm {N}}\), shows a numerical oscillation that is not observed in the analytical result (Fig. 7 a). In particular, in the initial stage (t≲0.0002 in Fig. 7 a), \(\dot {x}_{\mathrm {N}}\) tends to be overestimated. These oscillation and overestimation are caused by the assumption that the values of h _FC and u _FC are uniform across the width of the front cell Δ x _FC in the present numerical method at first-order accuracy. For a large ρ _c/ρ _a, because h _N has a small value, the value of Δ x _FC tends to be overestimated when a constant h _FC is assumed (Fig. 7 b). We suggest, therefore, that the BC model is favorable for simulating gravity currents with relatively low ρ _c/ρ _a.

Figure 8 shows the numerical results from the AB model along with the analytical results. In these calculations, the values of ε for given values of ρ _c/ρ _a are set based on the relationship of Eq. (19) (see Fig. 5 b). In Fig. 8 b, the numerical results for ρ _c/ρ _a=1000 (ε=4.58×10⁻⁷) agree well with the analytical results. The numerical oscillations observed in the BC model do not occur with the AB model (Fig. 7 a). In Fig. 8 a, on the other hand, the numerical results for ρ _c/ρ _a=1.01(ε=6.58×10⁻²) agree well with the analytical results only during the slumping stage (t≲4.5), but deviate from the analytical results during the self-similar stage (\(t\gtrsim 4.5\)). This agreement during the slumping stage and deviation during the self-similar stage occurs because ε is set using the analytical relationship (Eq. (19)) for the slumping stage of the dam-break problem. During the slumping stage, h _N and u _N are constant so that ε based on Eq. (19) provides the correct front condition. During the self-similar stage, on the other hand, the driving pressure, and hence h _N and u _N, decrease with time; therefore, the assumed value of ε is no longer consistent with the front condition Eq. (7).

The good agreement in the results of the AB model for ρ _c/ρ _a=1000 reflects the fact that the dynamics of the gravity current becomes insensitive to the front condition for large values of ρ _c/ρ _a. In Fig. 5 b, ε approaches 0 as ρ _c/ρ _a increases. In the limit as ρ _c/ρ _a→∞ and ε→0,u _N asymptotically approaches its maximum value, 2, and h _N asymptotically approaches 0. For sufficiently small ε, the solution converges to that in the limit as u _N→2 and h _N→0, and it becomes insensitive to the value of ε (see Fig. 5 a). Indeed, as shown in Fig. 8 b, we can confirm that the result of the AB model with a very small ε(ε=1.0×10⁻¹⁰) is indistinguishable from that for ρ _c/ρ _a=1000(ε=4.58×10⁻⁷). According to Fig. 5, the results of the AB model for the dam-break problem are insensitive to ε when ε≲10⁻⁵, which corresponds to \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\) (Fig. 5 b). Consequently, we suggest that the AB model is favorable for simulating gravity currents with high ρ _c/ρ _a for which the dynamics of the current is insensitive to the assumed value of ε.

Applicability of the BC and AB models

Our results indicate that the BC and AB models each have their own advantages and disadvantages. The results obtained from the BC model agree well with the analytical results when ρ _c/ρ _a≲10² (Fig. 6 a), whereas they show a numerical oscillation at the flow front and tend to overestimate the front speed when \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\) (Fig. 7). No such numerical oscillation nor overestimation is observed in the results from the AB model. For currents with \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\), the AB model provides good approximations of the analytical results, given a sufficiently small ε (Figs. 5 and 8 b). For currents with ρ _c/ρ _a≲10², however, the AB model may fail to reproduce the analytical results for currents where the height and speed of the front change with time (Fig. 8 a). Accordingly, we propose that the BC model should be used for currents with ρ _c/ρ _a≲10² and the AB model is applicable only to currents with \(\rho _{\mathrm {c}}/\rho _{\mathrm {a}}\gtrsim 10^{2}\).

Because of its simple coding and numerical stability, the AB model with an arbitrarily small ε is commonly used for simulations of gravity currents in many geophysical situations (e.g., Denlinger and Iverson 2004; Doyle et al. 2007, 2008, 2011; Larrieu et al. 2006). This model would be applicable in simulating gravity currents with high values of ρ _c/ρ _a, such as debris flows (e.g., Denlinger and Iverson 2004). However, our results suggest that it may provide inaccurate results for gravity currents with ρ _c/ρ _a≲10², such as turbidity currents and dilute pyroclastic density currents. Numerical results for ρ _c/ρ _a=10 show that the problem arises mainly from the behavior of the flow front (Fig. 9). Generally, a gravity current with a relatively low value of ρ _c/ρ _a is characterized by the formation of a large front height, which is caused by the resistance of the ambient fluid. This large front height is successfully reproduced by the BC model (Fig. 9 a), while the AB model fails to capture it. The results from the AB model with ε=10⁻¹⁰ (Fig. 9 b) show that the resistance at the front is too small to develop a large front height; consequently, the flow speed is substantially overestimated.

Geophysical application to pyroclastic density currents

Pyroclastic density currents (PDCs) are characterized by strong density stratification due to particle settling (e.g., Branney and Kokelaar 2002), whereby a dilute gravity current (particle suspension flow) with ρ _c/ρ _a=10⁰–10¹ overrides the dense basal gravity current (fluidized granular flow) with ρ _c/ρ _a=10²–10³. The dynamics of PDCs is complex because the dilute and dense currents are influenced by a number of physical processes such as particle settling (e.g., Bonnecaze et al. 1993), entrainment of ambient air (e.g., Johnson and Hogg 2013; Sher and Woods 2015), and basal resistance (e.g., Roche et al. 2008). In addition to the effects of these processes, our results suggest that the application of the correct numerical model to the flow front is important if we are to understand the dynamics and sedimentation of PDCs. The BC model should be applied to the overlying dilute current, while the AB model is applicable to the underlying dense current. Here, we discuss how the resistance at the flow front of the dilute part influences the dynamics of PDCs as a whole.

Before discussing the complex dynamics of PDCs with strong density stratification, we briefly assess the effects of some important physical processes on the dilute and dense currents. Figure 10 shows the results of simulations using the BC model for a dilute gravity current generated by an instantaneous release (i.e., the dam-break problem) of an initially homogeneous particle suspension with ρ _c/ρ _a=8.495. This model accounts for the effects of particle settling and entrainment of the ambient fluid following Bonnecaze et al. (1993) and Johnson and Hogg (2013), respectively. As with a homogeneous gravity current with a low density ratio (Fig. 6 a), a thick front develops in the dilute gravity current of the particle suspension, suggesting that the resistance of the ambient fluid at the front plays a significant role in the dynamics of the dilute part of PDCs regardless of the presence or absence of the effects of particle settling or entrainment.

Figure 11 compares our numerical results obtained using the AB model with experimental results for a granular flow with ρ _c/ρ _a=10²–10³ generated by an instantaneous release of an initially fluidized bed (Roche et al. 2008). Because the initial acceleration regime of the experimental setup (i.e., just after the release; (i) in Fig. 11 b) is beyond the applicable range of the shallow-water model, we focus on the subsequent regimes here. The numerical results obtained using the AB model show the formation of a wedge-shaped flow front (Fig. 11 a), which is in agreement with the experimental results of Roche et al. (2008). The results also indicate that the dynamics of the dense fluidized granular flow can be quantitatively simulated by the AB model when a suitable rheological model is applied for the basal resistance (Fig. 11 b). The shear drag model explains the behavior during the constant velocity regime ((ii) in Fig. 11 b; see Roche et al. 2008), whereas the Coulomb friction model better reproduces the features of the stopping regime ((iii) in Fig. 11 b; see Roche et al. 2008).

The interplay between the dilute and dense parts may be crudely simulated by a two-layer model comprising a dilute layer and a dense layer (Doyle et al. 2008, 2011). The previous two-layer models for PDCs apply the AB model with a small ε to both layers. Here, we follow Doyle et al. (2011) for the basic formulation of the two-layer system, but apply the BC model to the dilute layer and the AB model to the dense layer. We also consider the effects of basal shear drag in the dense layer and entrainment of ambient air into the dilute layer (see Appendix for details). Figure 12 shows a representative result of our two-layer model for a PDC generated by the instantaneous release of an initially homogeneous dilute particle suspension with ρ _c/ρ _a=8.495. When the BC model is applied, a thick front head develops in the dilute gravity current because of the resistance of the ambient fluid at the front (see Fig. 10). On the other hand, a dense gravity current with ρ _c/ρ _a=600.6 is generated by particle settling from the overlying dilute gravity current. Because the rate at which particles are supplied to the dense layer is controlled by the conditions of the overlying dilute layer (e.g., thickness and particle concentration), the evolution and dynamics of the dense layer are critically dependent on those of the dilute layer. This suggests that the behavior of the flow front of the dilute layer controls not only the dynamics of the dilute layer but also the dynamics of the stratified PDC as a whole.

The results in Fig. 12 are preliminary, and a comprehensive understanding of the dynamics of PDCs should consider many other effects, such as the expansion of entrained air due to heating by pyroclasts, density stratification inside the overlying dilute layer, diffusion of the pore pressure and entrainment of air in the underlying dense layer, and the transport of particles from the underlying dense layer to the overlying dilute layer (e.g., Andrews 2014; Breard and Lube 2017; Bursik and Woods 1996; Dufek and Bergantz 2007; Esposti Ongaro et al. 2016; Ishimine 2005; Roche et al. 2008; Wilson and Walker 1982). Nevertheless, preliminary results (not shown here) have already indicated the diversity of the interplay between the dilute and dense layers, which depends on the initial particle concentration and grain size. The interaction also influences the sedimentation process from the PDCs (Fujii and Nakada 1999). A systematic parametric study of the two-layer PDC model using the BC model is in progress, with the aim of accounting for the diversity of PDC deposits.