The equations governing the current column height, h, and the vertically integrated momentum, q=uh, with u being the depth-averaged current velocity, are described in non-dimensionalised conservative form as ∂h∂t+∂q∂x=0,∂q∂t+∂∂xq2h+φh2=0 with boundary conditions q=0atx=0,q=x˙Nhatx=xN(t),given thatx˙N=Frφ1/2atx=xN(t), where xN is the location of the front of the current, x˙N is the velocity of the front of the current, Fr is the Froude number, and φ=ψh is the vertically integrated volume fraction of sediment where ψ is the depth-averaged volume fraction of sediment within the flow. Through experimentation, the Froude number for a density current with a head height < 0.075 of the total water depth has been found to be 1.19 . The evolution of φ is described using ∂φ∂t+∂∂xqφh=-βφh, where β is a constant particle-settling parameter. Hence, the gravitational forcing term in (Eq. ) (the last term on the left-hand side) changes with time as φ is advected and settles out of the column.

This single layer model ignores the effect of the motion of the overlying fluid on the current. This approximation is valid for flows in which the maximum column height is significantly less than the depth of the ambient fluid . Viscous forces are also ignored. For high Reynolds number flows, the viscous forces will be negligible in relation to the buoyancy forces. found that this was valid while the Reynolds number was greater than O(1).

The amount of deposited sediment, η, is also recorded and is calculated using ∂η∂t=βφh.

The model is non-dimensionalised with the length, time, and velocity scales h0, (h0/g0′)1/2, and (h0g0′)1/2 respectively. Here, h0 is the dimensional depth of the initial sediment release, g0′=ψ0gρp-ρa/ρa is the initial reduced gravity of the current, ρp is the sediment particle density, ρa is the ambient fluid density, which is assumed to equal the interstitial fluid density, ψ0 is the initial volume fraction of sediment, and g is the acceleration due to gravity. Finally, the volume fraction is scaled such that ∫0xN(0)φdx=1.

Following , a coordinate transformation from (x,t) to (y,τ) is applied where y=x/xN(t) and t=τ. This is a convenient form for the equations as the front of the current is always at the right-hand boundary of a fixed computational domain, and hence the boundary condition at the front of the flow is applied at the right-hand side of the domain. The transformed derivatives are given by ∂∂t=∂∂τ-yx˙NxN∂∂y,∂∂x=1xN∂∂y.

Applying this coordinate transformation, but keeping t in place of τ for notational simplicity following , produces the system of equations ∂h∂t=1xNyx˙N∂h∂y-∂q∂y,∂q∂t=1xNyx˙N∂q∂y-∂∂yq2h+φh2,∂φ∂t=1xNyx˙N∂φ∂y-∂∂yqφh-βφh,∂η∂t=1xNyx˙N∂η∂y+βφh,∂xN∂t=x˙N with boundary conditions q=0aty=0,q=x˙Nhaty=1,η=0aty=1,given thatx˙N=Frφ1/2aty=1.

Note that Eq. () for xN has been introduced to close the system.

It is now shown that the boundary conditions (see Eqs. –) are sufficient to uniquely solve this system. Equations ()–() are a hyperbolic system of PDEs. For such a system to be well posed, there must be a boundary condition for each inwardly propagating characteristic. This system of equations has four characteristic velocities: dydt=c±:=1xNu-yx˙N±φ1/2,dydt=c:=1xNu-yx˙N,dydt=cη:=-1xNyx˙N.

These are obtained using the method of characteristics, where c± is the characteristic velocity of waves in shallow water, c is the advection velocity of sediment, and cη is the advection velocity of deposited sediment which is advected away from the current head as the domain length increases.

Due to the boundary conditions on momentum, the following is true: u=q/h=0 at y=0 and u=q/h=x˙N at y=1. Hence, c=0 at both y=0 and y=1. Therefore, there are three inwardly propagating characteristics: c+=φ1/2/xN at y=0, c-=-φ1/2/xN at y=1, and cη=-yx˙N/xN at y=1. Hence, three boundary conditions are required for the problem to be well posed such that the three boundary conditions (Eqs. –) are exactly what is required.

As the cell size grows throughout the simulation, it is possible to use a much larger time step at the end of the simulation than at the start of the simulation. To exploit this property, an adaptive time-stepping scheme is used in this model. A new time-dependent variable is introduced, Δt, which will vary according to a CFL criteria, C, based upon a velocity scale, x˙N, and the mesh element size such that Δt=CΔxx˙N=CxNΔyx˙N, where Δx is the mesh element size in x, and Δy is the mesh element size in the transformed coordinate system y. The time-dependent model variables are defined as a vector U=h,q,φ,η,xN,x˙N,ΔtT.

The system is discretised in time using a second-order explicit Runge–Kutta time discretisation . An implicit term is added to the semi-discrete system in order to solve for the diagnostic variables x˙N and Δt. With Un as the solution at the beginning of the time step, Un+1 as the solution at the end of the time step, and U(0), U(1), and U(2) as intermediate values, the system of equations (discretised in time) can be written as U(0)=Un,U(1)=A(U(0))+L(U(0))+K(U(1)),U(2)=A(U(1))+L(U(1))+K(U(2)),Un+1=12U(0)+12U(2), where A(U)=h,q,φ,η,xN,0,0T,L(U)=Δt1xNyx˙N∂f1(U)∂x-∂f2(U)∂x+f3(U),f1=h,q,φ,η,0,0,0T,f2=q,q2h+φh2,qφh,0,0,0,0T,f3=0,0,-βφh,βφh,0,0,0T. A(U) is non-zero where there is a time derivative term. L(U) is the explicit right-hand side term multiplied by Δt. Note that K(U) contains the implicit right-hand side terms. K(U) can only be easily described well in weak form, so this is defined later.

The spatially weak form of the semi-discrete system (Eq. ) is obtained as Ψ and integrated over the domain Ω. This gives Ω. For all Ψ in an appropriately chosen test space, ∫ΩΨ⋅U(0)dΩ=∫ΩΨ⋅UndΩ,∫ΩΨ⋅U(1)dΩ=∫ΩΨ⋅A(U(0))dΩ+∫ΩΨ⋅L(U(0))dΩ+∫ΩΨ⋅K(U(1))dΩ,∫ΩΨ⋅U(2)dΩ=∫ΩΨ⋅A(U(1))dΩ+∫ΩΨ⋅L(U(1))dΩ+∫ΩΨ⋅K(U(2))dΩ,∫ΩΨ⋅Un+1dΩ=12∫ΩΨ⋅Un+U(2)dΩ.

Piecewise linear discontinuous Galerkin (DG) elements are used to discretise the spatially varying state variables. Thus, the spatial and temporal discretisations both have second-order accuracy. DG element types are known to be particularly suitable for advection dominated problems . They are good at preserving discontinuities as they produce stable discretisations without the need for diffusive stabilisation strategies, such as streamline upwinding . These are important features in shallow water particle-laden density current models.

In order to construct a DG formulation, a regular partition Th={e} of Ω into non-overlapping subdomains Ωe∈Ω with boundaries ∂Ωe is considered. The piecewise linear DG function space is denoted DG1. For this function space, piecewise linear test functions with no global continuity requirement are considered; i.e. functions that have the potential to be double valued on ∂Ωe. xN, x˙N, and Δt are defined on a function space, R, which is constant throughout the spatial domain. Therefore, the vector of model unknowns U is defined on a mixed function space, Vh=DG14×R3. The function in the mixed function space is denoted with Ψh∈Vh and the discretised approximation of the state variable with Uh∈Vh.

Notice that L(U) contains derivatives of discontinuous functions. Its undiscretised weak form is ∫ΩΨ⋅L(U)dΩ=Δt∫ΩΨ⋅1xNyx˙N∂f1(U)∂x-∂f2(U)∂x+f3(U)dΩ.

The discretised DG formulation of Eq. () is then ∑e∈Th∫ΩeΨh⋅L(Uh)dΩ=Δt∑e∈Th∫ΩeΨh⋅1xNyx˙N∂f1(Uh)∂x-∂f2(Uh)∂x+Ψhf3(Uh)dΩ.

Integrating the gradient terms by parts and slightly rearranging yields ∑e∈Th∫ΩeΨh⋅xNΔtL(Uh)dΩ=-∑e∈Th∫Ωe∂∂xΨhyx˙N⋅f1(Uh)dΩ+∑e∈Th∫∂ΩeΨh^⋅yx˙Nf1^(Uh)n^dσ+∑e∈Th∫Ωe∂Ψh∂x⋅f2(Uh)dΩ-∑e∈Th∫∂ΩeΨh^⋅f2^(Uh)n^dσ+∑e∈Th∫ΩeΨh⋅xNf3(Uh)dΩ, where  ⋅^  indicates that the function is double valued and special attention is required. The various summations can now be rewritten as integrals over the entire domain Ω, all element interfaces Σh, and the domain boundaries Γh. Note that ∑e∈Th∫∂ΩeΨh⋅Uh^dσ≡∫ΣhΨh⋅Uh^dσ+∫ΓhΨh⋅U0dσ,and∑e∈Th∫ΩeΨh⋅UhdΩ≡∫ΩΨh⋅UhdΩ.

Additionally, note that within domain boundary integrals, the ⋅^ notation is dropped as the function is single valued at this location. Uh is also replaced with U0, which is either the boundary value if a Dirichlet boundary condition is present or the function value at the boundary if it is not. Note that in the case of the boundary condition for q at y=1, U0 is still a function of Uh. Applying Eqs. () and () to () yields ∫ΩΨh⋅xNΔtL(Uh)dΩ=-∫Ω∂∂xΨhyx˙N⋅f1(Uh)dΩ+∫ΣhΨh^⋅yx˙Nf1^(Uh)n^dσ+∫ΓhΨh⋅yx˙Nf1(U0)ndσ+∫Ω∂Ψh∂x⋅f2(Uh)dΩ-∫ΣhΨh^⋅f2^(Uh)n^dσ-∫ΓhΨh⋅f2(U0)ndσ+∫ΩΨh⋅xNf3(Uh)dΩ.

A choice of flux term must be made to handle the double-valued terms. This will involve some coupling between the elements on either side of the interface. An upwind flux is used for the advection term, f1^, and based upon experience an average flux works well for f2^. This gives ∫ΩΨh⋅xNΔtL(Uh)dΩ=-∫Ω∂∂xΨhyx˙N⋅UhdΩ+∫ΓhΨh⋅y(x˙Nn)downf1(U0)dσ+∫ΣhΨh+-Ψh-⋅yf1+(Uh)(x˙Nn+)up+f1-(Uh)(x˙Nn-)updσ+∫Ω∂Ψh∂x⋅f2(Uh)dΩ-∫ΓhΨh⋅f2(U0)ndσ-∫ΣhΨh+-Ψh-⋅12f2(Uh)++f2(Uh)-n+dσ+∫ΩΨh⋅xNf3(Uh)dΩ, where (⋅)+ and (⋅)- indicate the function values on either side of an interior element boundary. (⋅)up is equal to (⋅) where x˙Nn±>0 and 0 otherwise. Conversely, (⋅)down is equal to (⋅) where x˙Nn±<0 and 0 otherwise. K(U) can be described in weak discretised form as ∫ΩΨh⋅K(Uh)dΩ=∫ΩΨh⋅KΩ(Uh)dΩ+∫∂ΩRΨh⋅Kσ(Uh)dσ,KΩ(Uh)=0,0,0,0,0,0,CxNΔyx˙N,Kσ(Uh)=0,0,0,0,0,Fr(φ)1/2,0, where ∂ΩR is the right-hand boundary at y=1 such that a solution for x˙N is obtained by solving only at the front of the current.

Using Eqs. (), (), and () and applying Eq. (), the full weak, discontinuous form of Eq. () can be obtained.Find Uh(0),Uh(1),Uh(2),Uh(n+1)∈Vh such that ∀Ψh∈Vh ∫ΩΨh⋅Uh(0)dΩ=∫ΩΨh⋅UhndΩ,∫ΩΨh⋅Uh(1)dΩ=∫ΩΨh⋅A(Uh(0))dΩ+∫ΩΨh⋅L(Uh(0))dΩ+∫ΩΨh⋅K(Uh(1))dΩ,∫ΩΨh⋅Uh(2)dΩ=∫ΩΨh⋅A(Uh(1))dΩ+∫ΩΨh⋅L(Uh(1))dΩ+∫ΩΨh⋅K(Uh(2))dΩ,∫ΩΨh⋅Uhn+1dΩ=12∫ΩΨh⋅Uh(0)+Uh(2)dΩ.

This set of equations is solved for each time step of the simulation as a nonlinear variational problem using Newton's method with an LU decomposition solver for the linear problems.

Discontinuous Galerkin discretisations for convection dominated problems can suffer from over- and undershoots at discontinuities that can cause instability problems . Slope limiting can be applied to solve this problem, but this typically involves discontinuous operations, which are problematic in a gradient-based optimisation framework. Therefore, we do not use slope limiting here and limit ourselves to the assumption of smooth initial conditions where slope limiting is not necessary. It would be possible to formulate a continuous slope-limiting function to overcome this limitation if it was required.

The shallow water sediment-laden density current model described above was built using the FEniCS framework , an open-source software project that provides features for the automated, efficient solution of differential equations. Using a high-level interface, the model partial differential equations are described in variational form using UFL (Unified Form Language) . This can be achieved in Python or C++ code in a way that is remarkably similar to how one would describe the equations on paper. At runtime, this model description is compiled into efficient C++ kernels that handle the assembly of the required matrices to generate the systems of equations that are then solved using PETSc .

Many laboratory experiments and computer models are based around the classical lock-release static initial condition (Fig. a). Following the release of the lock-gate, the current accelerates forwards. This is known as the dam-break stage (Fig. b). As the lock-gate is released, a shock forms which travels in the opposite direction to the front of the current. This shock carries information that sets the fluid in motion. Once this shock reaches the rear wall, all of the fluid behind the lock-gate is in motion. This marks the point of transfer from the dam-break to the slumping phase (Fig. c). For a non-depositional current (i.e. β=0) with initial h and xN=1, the slumping phase begins at t=1. The current front height and velocity remain approximately constant during this phase of motion. The rear propagating shock is reflected off the no-flow boundary and travels faster than the front of the current. A short while later, it reaches the front of the current, marking the end of the slumping phase. The current is now able to “forget” the initial condition and begins adjusting to self-similar propagation . For a non-depositional current (i.e. β=0), the reflected shock reaches the front of the current at t=3 .

showed that a similarity solution, which is a solution that looks the same at all times or at all length scales, could be obtained for a single layer shallow water density current model during the self-similar phase of propagation. This is described as xN=κt2/3,u=23κt-1/3us,h=κ-1t-2/3hs,ψ=1, where y=x/κt2/3,κ=27Fr212-2Fr21/3,us=y,hs=49κ3y24-14+1Fr2.

The domain is unit length, as in all cases for this model. This solution is valid for the model described in this paper so long as the settling velocity of particles, β, is equal to 0 (i.e. no particle settling). This analytical solution is useful in verifying the implementation of the governing equations and boundary conditions for this model. The solution to the model PDEs should converge on this analytical solution as the mesh resolution is refined at the correct rate for the temporal and spatial discretisation. The use of piecewise discontinuous linear elements and a second-order time-stepping regime means that the convergence order should be quadratic in both space and time.

For the convergence test, the analytical solution is projected onto the model function space forming the initial condition at t=3. At t=10, the L2 norm of the difference between the model variables and the analytical solution is obtained and used to measure convergence. The analysis shows that all variables converge on the analytical solution at the correct order (Fig. ). Note that the time step is adaptive and will therefore decrease along with the element size such that this test checks both spatial and temporal convergence. This verifies that the model equations are implemented correctly . A qualitative comparison shows that the solution matches the analytical solution very well (Fig. ).

The initial conditions are based upon the analytical solution for a non-depositional flow at a non-dimensional time, t=ts=3, after a column collapse as described by Eq. (). Some assumptions are therefore made as to the initial shape, sediment concentration profile, and velocity profile of the flow.

The non-dimensional particle-settling velocity, β, is calculated using the standard Stokes settling law for a particle in suspension non-dimensionalised by (h0g0′)1/2 to give β=g′D218ν(h0g0′)1/2=g′1/2D218νh0ψ01/2, where D is the average sediment diameter. The sediment-reduced gravity, g′=g0′/ψ0=(ρp-ρa)g/ρa=16, is based upon the reduced gravity of silica in water. Using these initial conditions, there are three unknowns: h0, the dimensional length scale of the current; ψ0, the initial sediment concentration throughout the current; and D, the mean sediment diameter. These become the set of input parameters that will be optimised with m=h0,ψ0,DT.

The beginning of the basin is defined as being at the front of the current at t=ts such that the current is not in the basin prior to the start of the simulation. The current enters the domain as soon as the simulation starts. The end of the simulation, tf, is defined as the time at which the total suspended sediment is less than 1 % of the starting quantity.

The aim here is to reduce the difference between the deposit profile generated by this model and the target deposit profile from field measurements. To do this, we need to map the non-dimensional, transformed results from the model back to the observation space. We also only measure the variation over the length of the measured deposit. Therefore, the functional that we will aim to minimise, J, has the form J(U(m),m)=∫0x^max⁡η̃-ηT2dx^, where ηT is the dimensional target deposit profile, η̃=ψ0h0η is the dimensionalised modelled deposit, x^max⁡= 82 000 is the extent of the measured data, and x^=x̃-x̃N(ts) is a coordinate transformation such that x^=0 at the front of the current at t=ts, x̃=yx̃N(tf) is the dimensionalised reverse of the coordinate transformation outlined in Sect. ; x̃N(t)=h0xN(t) is the dimensional length of the current.

To calculate this functional, ηT must be a function of x^. The deposit is approximated using a fourth-order polynomial ηT=∑i=04cix^i, where ci is the ith coefficient. The coefficients are obtained using the least squares method. The fourth-order approximation fits the measured data points well (Fig. ).

It is important to note that at the end of the simulation, t=tf, the length of the current does not necessarily match the length of the deposit, or x^N≠x^max⁡, where x^N(t)=x̃N(t)-x̃N(ts) is the dimensional length of the modelled deposit within the basin. This complicates the calculation of the above integral.

The calculation of J is split into two components, an integral over the lesser of the length of the modelled current (or the length of the measured data, J0) and an integral over any remaining length of measured data, J1, such that J=J0+J1. The first integral takes the form J0=∫0min⁡(x^N,x^max⁡)ηT(x^)-η̃(x^)2dx^.

This can be approximately transformed into the model coordinate system as J0=∫Ωγ0ηT(x^(y))-γ0η̃(x^(y))2dy, where γ0 is a scaled filter. This filter is 0 in the regions x^<0 and x^>x^max⁡. Elsewhere, the filter value is a constant such that the integral of the filter over the domain is equal to the length of the dimensional integral, min⁡(x^N,x^max⁡). The filter defines the region of the domain over which the integral is evaluated and scales the resultant value appropriately. The filter is defined as γ0(x^)=min⁡(x^N,x^max⁡)exp⁡min⁡(x^-x^max⁡,x^,0)sγ0,sγ0=∫Ωexp⁡min⁡(x^-x^max⁡,x^,0)dx^. It is important that the functional is differentiable. Therefore, the min⁡ and max⁡ functions are replaced by smooth approximations fmin⁡ and fmax⁡ defined as fmin⁡(a,b)=ln⁡(exp⁡(10a)+exp⁡(10b))/10,fmax⁡(a,b)=fmin⁡(-a,-b). The second integral, J1, takes the form J1=∫min⁡(x^N,x^max⁡)x^max⁡ηT(x^)2dx^, such that J1 integrates the target deposit volume beyond the extent of the modelled deposit. If the modelled deposit length exceeds the length of the measured data, this integral will be 0. Again, this can be approximately transformed into the model coordinate system as J1=∫Ωγ1ηT(x̃(y))2dy,x̃(y)=min⁡(x^N+y(x^max⁡-x^N),x^N), where γ1 is a scaled filter similar to γ0 defined as γ1(x^)=x^max⁡-min⁡(x^N,x^max⁡)⋅exp⁡min⁡(x^-x^max⁡,0)sγ0,sγ0=∫Ωexp⁡min⁡(x^-x^max⁡,0)dx^. Again, min⁡ and max⁡ are replaced by smooth differentiable alternatives.

The gradient computation was verified using the Taylor remainder convergence test. Let J^(m)≡J(U(m),m), a pure function of m. The zero-order Taylor expansion states that J^(m+δm)-J^(m)=Oδm, while the first-order Taylor expansion states that J^(m+δm)-J^(m)-dJ^(m)dmδm=Oδm2.

Even small errors in the derivative destroy the second-order convergence in Eq. (). Therefore, testing the convergence of these expansions with the gradient calculated from the adjoint yields a strong indicator of whether the adjoint gradient computation is correct.

The above convergence was succesfully carried out for the implemented adjoint model with a number of different controls and functionals. As an example, Table  shows the results with m as the initial deposit length. The convergence of the remainder term is second order with respect to varying magnitudes of δm, providing strong evidence that the adjoint model and gradient computation are implemented correctly.

With confidence that the forward and backward models are working, optimisation of the input parameters, m=h0,ψ0,D, to minimise the objective functional J can now be performed as minm=J(U(m),m), with the the following bounding constraints on the input parameters: 10m≤h0≤10km,0.001%≤ψ0≤50%,1μm≤D≤1mm.

These bounding constraints are chosen based upon very loose limits of expected values that each parameter may possibly take. The principal purpose of these bounds is to avoid invalid negative values being generated for any of the parameters.

The nonlinear optimisation library, IPOPT , is used to solve this problem. This library implements a primal-dual interior point algorithm which has good global and local convergence properties . The interface to this library is supplied by dolfin-adjoint . The initial input parameters are set to m=h0=2.3kmψ0=0.07%D=200.0µm.

The aim is to recreate the sand deposit by modelling only the sand in the flow using a single average grain size, D. The value of ψ0 is based upon a combined initial volumetric concentration for the sand and mud mixture of 0.5 %, with 86 % of the mixture being mud. The starting value for h0 is based upon the area of the two-dimensional deposit profile and the value of ψ0. The average sediment grain size is a reasonable estimate of the average grain size based upon the information provided by . The input parameters provided to the optimisation algorithm, m‾, are normalised such that they are all equal to 1. Thus, mi=m‾im0, where m0 indicates the initial parameter values in Eq. (), and mi indicates the value of m after optimisation iteration i. This scaling helps the optimisation algorithm work effectively .

The criteria for finishing the parameter optimisation is based upon the relative change in J between iterations such that Ji-Ji-1Ji<1.0×10-5, where Ji is the value of J after the ith iteration. The optimisation is completed in 21 iterations with a final functional value of J=1.75 (Fig. ). The optimised deposit profile, η, compares relatively well with ηT (Fig. ). Most notably, there is a significant variation in the thickness towards the end of the deposit. This will be addressed later. The final optimised values are m=h0=2.56kmψ0=0.0494%D=103µm.

These optimised values are not completely acceptable. The value for h0 represents the initial height of the current starting from a static lock-release initial condition. This translates to an initial current height of 993.3 m at the start of this simulation and as the current enters the basin plain. This value appears to be quite large for a relatively small turbidity current. Additionally, the average sediment diameter of 103 µm is lower than expected. defines the sandstone interval as dominated by sediment grains estimated to be larger than ≈ 125 µm.

With the exception of the sediment diameter, the optimised values are fairly similar to those chosen as input values. This confirms that the input parameters chosen were sensible predictions of the starting conditions for the gravity current. To test this hypothesis, we ran the same situation starting from a number of alternative initial conditions. We found that there are indeed a number of local minima. An optimisation with initial conditions

m=h0=3.0kmψ0=0.02%D=100.0µm was optimised to m=h0=3.95kmψ0=0.02%D=154.0µm.

Figure  shows a comparison of the two generated deposit profiles. The two profiles are very comparable, even though the alternative profile is created by a much larger, but much less dense, initial current.

The existence of alternative minima must always be considered when running optimisations of this type. It is important to have a good understanding of the problem to choose sensible initial starting conditions and also to assess the resultant optimised values. A regularisation approach would avoid this problem but assumes prior knowledge about the target profile.

A clear omission from the model is the presence of mud in the suspension. The presence of mud will significantly alter the energy budget of the flow. A mud sediment class can easily be included so that the model produces more realistic optimised values. This is detailed below.

Investigating the effect of including mud in the sediment mixture can be achieved relatively simply by including an additional transport equation with a form identical to Eq. () , ∂φm∂t=1xNyx˙N∂φm∂y-∂∂yqφmh-βmφmh, where φm=ψmh is the vertically integrated volume fraction of mud in the suspension. ψm is the depth-averaged volume fraction of mud within the flow, and βm is the settling velocity of the mud particles. Using a single tracer equation, we approximate the distribution of mud particle sizes using a single mud diameter, the same way the distribution of sand is modelled in the flow. We neglect the flocculation of mud particles. Assuming that the density of both sediment classes is the same, (Eq. ) is modified to include this new sediment class in the gravity term: ∂q∂t=1xNyx˙N∂q∂y-∂∂yq2h+(φ+φm)h2.

Finally, φ and φm are scaled such that at the start of the simulation φ+φm=1, where previously φ=1. The aim is still to recreate the deposit of sand, and hence the equation for η stays the same. We term the sand deposit generated by this modified model η2. The discretisation for Eq. () is consistent with the rest of the model, as presented in Sect. .

The initial condition needs to be altered to include the new sediment class. The initial vertically averaged volume fraction of sand is changed to ψ=fs, and a new initial condition for the vertically averaged volume fraction of mud is introduced as ψm=1-fs. The sand fraction, fs, is estimated by to be 0.14 and is kept fixed.

βm must also be calculated. This is done in the same way as for β, except that a different sediment diameter parameter is used and optimised: Dm, the mean diameter of mud particles in the flow. The equation for βm is therefore βm=g′1/2Dm218νh0ψ01/2. Note that ψ0 is now the combined initial volume fraction of sand and mud in the flow.

The set of optimised input parameters is redefined as m=h0,ψ0,D,DmT. An additional bounding constraint is added for Dm such that the new bounding constraints for m are 10m≤h0≤10.0km,0.001%≤ψ0≤50%,1.0µm≤D≤1.0mm,1.0µm≤Dm≤100.0µm. The initial input parameters are set to m=h0=2.1kmψ0=5%D=200.0µmDm=20.0µm.

The input parameters are normalised as detailed in Sect.  before being passed to the optimisation algorithm. The criteria for finishing the optimisation are consistent with the previous optimisation (see Eq. ).

The optimisation of the model with two sediment classes is completed in 17 iterations with a final functional value of J=2.13 (see Fig. ). Therefore, the fit is quantitatively slightly worse when mud is included in the model. This is a surprising result, as the model now more closely matches reality. Qualitatively, it is very hard to determine which model fits the data better. The resultant deposit is very similar in shape to that obtained when only modelling sand in the flow (see Fig. ). The fit appears to be worse at the start of the deposit. The runout length is slightly longer when mud is included such that the fit towards the end of the deposit is slightly improved.

The fit with the measured data is still poor towards the end of the deposit. noted that the distal section of small deposits in the Marnoso-arenacea Formation show evidence of transport in a tractional boundary layer. This simulation does not model bedload transport or erosion, which is the likely reason for the difference in the results. The velocity of the head of the turbidity current in this simulation varies between 10 and 2.4 m s-1 over the period during which sand is deposited (Fig. ). At these head velocities, erosion is very likely to occur. Models for erosion and bedload transport exist . These could be added in future work.

The final optimised values are m=h0=1.92µkmψ0=5.94×10-3D=125µmDm=28.1µm.

A comparison of these results to those obtained without a mud sediment class shows that the value of h0 has reduced by 25 % and translates to an initial current height of 745.0 m as the current enters the basin plain. The average sediment diameter has also increased by 21 % to 125 µm, bringing the average diameter in line with the estimates from the field measurements by . Arguably, the sand and mud classes should be subdivided further. described how polydisperse density currents will have longer runout distances than equivalent currents with uniform sediment at the mean value of the poydisperse current.

It is also interesting to assess the sensitivity of the model to variations in the input parameters by analysing the final gradient of the objective functional, dJdm‾=dJ/dh‾0=5.1×10-3dJ/dψ‾0=-1.8×10-3dJ/dD‾=-2.4×10-3dJ/dD‾m=1.5×10-6, where ⋅‾ indicates a parameter value normalised by its value on the initial optimisation iteration. The sensitivity of the functional to changes in the mud diameter is several orders of magnitude smaller than the sensitivity to changes in the other variables.

It is indeed found that changing this value has very little effect on the obtained deposit. The same simulation is run with the mud diameter decreased by 2 orders of magnitude such that the input parameter values are m=h0=1.92µkmψ0=5.94×10-3D=125µmDm=0.281µm.

The resulting functional value is J=2.13, which is identical to that obtained for the optimised simulation. There is no discernible difference in the resulting deposit, η3 (Fig. ). The head height and velocity only vary a small amount over the period during which sand is deposited (Fig. a and b). The current properties vary significantly after the sand has been deposited and mud is still in suspension, but this does not have any effect on the sand deposit.

Although the sandstone deposits generated by the single and two sediment class models are very similar, the properties of the turbidity currents that produced them are very different (Fig. ). The turbidity current with mud in suspension travels approximately twice as quickly due to the increased gravitational forces produced by the sand and mud mixture (Fig. b). Sand also drops out of the suspension much more rapidly (Fig. d). All of the sand is deposited within approximately 6 h. The model without mud deposits sand over a period of more than 20 h. This is due to the reduced height of the current at the start of the simulation and the faster decrease in the height of the current as a result of the higher head velocity (Fig. a and d). Clearly, the presence of mud in the suspension has a significant impact on the resultant flow and must be included in the model.

The simulated turbidity currents that produced η2 and η3 deposited sand over a similar time period. After all of the sand had fallen out of suspension, less than 25 % of the mud settled from the flow for both of these currents; the current head is > 50 m tall, and the head is moving at > 1.0 m s-1 (Fig. ). Hence, there is still a significant amount of energy in the flow. The remaining mud suspension will reach the end of the basin (x̃N≈ 130 km) and will still have a significant amount of energy left when it does so. It is very hard to predict what will happen after this point. The current may be partly reflected, and ponding of the suspended mud is likely to occur. This result is in agreement with the explanations of .

The height of the current in the optimised simulation with both sand and mud sediment classes is ≈ 750 m as it enters the basin, although this decreases very quickly as the current propagates. It is possible that including processes such as fluid entrainment, erosion, and bedload transport may reduce the necessity for such a large initial current height in producing this deposit. More complex initial and boundary conditions may also have a significant impact on this value. It is unclear what effect an inflow boundary condition with time-varying height, sediment concentration, and velocity would have on the results. This would be an interesting addition to the models capabilities.

The model also neglects variations in the bed profile. The gradient of the sea floor in the basin where the Marnoso-arenacea Formation was created was substantially less than 1 degree . Variations in gradient of this magnitude will have a negligible impact on the head velocity . However, small variations will have an impact on the velocity of the body of the current. Future work will address this.