θ is the angle between the z axis and ηηη(θ), φ is the angle between the x axis and ηηη(θ)

1 Benchmark problem

The physical Model















∂_τ(h) + ∂_ξ(κh) = −αρM

∂_τ(hu) + ∂_ξ(hu⊗ κ + p) = −αρuM + hS(u, θ)

∂_τx = −(θ − Θ_n)E(h, u, θ)ηηη(θ)

(1)

where h is the , u is the fluid velocity,x(ξ, t) is the position interface position, ξ is the lagrangian coordinate, Θ_n is the neutral angle, θ = (θ, φ) is the local inclination angles, ηηη(θ) is the local unit normal to the interface. θ is the angle between the z axis and ηηη(θ), φ is the angle between the x axis and ηηη(θ).

κ(θ) = u

cos θ and ηηη(θ) =



−sin θ cos θ



 (2)

The pressure is defined according to the Mohr− Coulomb behavior

p(h, θ, κ) = h²β(κ, θ)

2 , with β(κ, θ) =cos²θK(κ) where the factor K(κ) is define, for ε = ^H_Lref^ref, as

K(κ) =







K−(ε, θ_φ, θ_δ) for ∂_ξκ ≥ 0 K₊(ε, θ_φ, θ_δ) for ∂_ξκ < 0

K_±(ε, θ_φ, θ_δ) = ε







2

1±



1− ^cos_cos²2^θθ^φδ



cos²θ_φ − 1







where θ_φ and θ_δ are respectively the internal and the basal friction angles.

1.1 Model for the net driving acceleration

S(u, θ) = cos θsin θ− ε^βsgn(u)µ cos θ where µ = 0.9∗ tan(θ_f) with θ_f the friction angle.

2

1.2 Erosion/deposition

E(h, u, θ) = Fe(h, u, θ)Fh(h) Fe(h, u, θ) = α_e

(1− tanh (eα(||κ|| − κth)) )

2 H(θ − Θn) +H(Θn− θ)



Acordind to the relation H(Θn− θ) = 1 − H(θ − Θn), we can also use the simplified form

F_e(h, u, θ) = α_e



1−(1 + tanh (e_α(||κ|| − κ_th)) )

2 H(θ − Θ_n)



Fh(h) = h + α_h√

h, κ_th= α_v(θ− Θ)²

1.3 Benchmark Data

θ_φ= 34^o, Θ_n = 34^o, θ_δ = 23^o θ_f = 33^o

α_h = 0.05, α_e= 2.0, α_v = 1.0, α_ρ= 0.9, e_α = 20, ε = 1.0.

1.4 Initial condition

Computational domain [−3, 3]. Initial uniform mesh according to x and mesh sizes of : 100, 200, 400.

The initial basal surface

B(τ = 0, x) = B0exp



−32x² 9



The initial layer profile

h(τ = 0, x) cos θ₀(x) = L₀exp



−32x² 9



− B0exp



−32x² 9



L₀ = 2, T est1 T est2 T est3 B₀ = 0 0.2 0.5

3