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Appendix 1: Integration of the stress equation
The normal stresses τss
are related by the flow law (Equation (3)) to the corresponding deviatoric strain rates ess
, given by Equations (13). By assumption, the relative rate of ice densification in Equations (13) does not depend on the coordinate s and hence does not noticeably affect the first term and is not present in the second term on the right-hand side of Equation (14). Thus, by means of Equations (3) and (13), after integration of Equation (14) with respect to z from z to l for a small surface elevation slope (Kl
→ 0), we find the shear stress τzs
To evaluate the two integrals in Equation (A1), we use an appropriate approximation (Expression (17)) for the ice viscosity, µ, in the cold upper part of the glacier where µ is comparatively large and significant. Furthermore, as a firstorder iteration, we substitute the vertically averaged longitudinal velocity, given by Equations (15), for u in Equation (32). That is, we put u ≈ A/Δ. This leads to the shear-stress profile
with Ψ defined in Equation (19). The second term in Equation (A2) for τzs
represents an interpolation between the boundary values at ζ = 0 and 1. The fact that τss
is small near the bottom is also taken into account.
Substituting Equations (13) and (A2) for ezs
into the flow law (Equation (3)) and integrating with respect to ζ, we obtain
0 is the sliding velocity.
Owing to the effective change in Q near the melting point (Hooke, 1981; Budd and Jacka, 1989), the ice viscosity µ, given by Equation (4), rapidly decreases with increasing temperature as the glacier bottom is approached. Only the basal values of the integrands are important in the integrals in the above relation for u. In this case, Lliboutry’s (1981) approximation (Expression (18)) becomes especially useful. Consequently, we come to a simplified presentation of the longitudinal velocity:
b is the basal porosity.
If the sliding velocity is introduced as
then Expressions (15), (A3) and (A4) straightforwardly yield Equation (19) for the surface elevation profile.
A sliding law links u
0 to the basal shear stress τzs
(at ζ = 0). At high load pressures the meltwater is supposed to penetrate into the underlying fractured volcanic rocks, and, in the absence of cavity formation (ice–rock separation), by means of Equation (A2) we write, after Fowler (1981):
0 is a basal friction factor which depends inversely on the bedrock roughness and is proportional to the longitudinal scale of the substratum corrugations. For the specific dimensions of lava and tephra blocks in volcanic craters, K
0 is expected to be relatively small.
Equation (19) can be used now to eliminate the surface gradient from Equations (A3) and (A5). This results in Expression (20) for the longitudinal velocity profile and relates parameter σ to K
Let us note that parameter σ is close to unity for small values of the ratio K
b + 3)/Δ.
Appendix 2: Vertical Ice-Mass Transfer
If the density ρ (porosity c) of the snow–firn–ice deposits depends only on depth, then, following Salamatin (1991) and calculating partial derivatives of ζ, defined by Equations (16), we can reduce Equation (22) for small K
to a relation between
At the same time, integration of Equation (12) with respect to z gives (Salamatin, 1991):
where s and ζ should be considered as independent variables. The latter two equations straightforwardly lead to the final result
The integral in the latter equation can be evaluated easily after substitution of the u-velocity profile (Expression (20)) and finally Equation (23) is obtained.
Appendix 3: Comments on Implications of a Non-Linear Flow Law
Glen’s flow law, a power-law relationship between effective deviatoric strain rates
is normally used to describe the non-Newtonian behavior of pure polycrystalline ice. Here the exponent α is the creep index, α ≥1, and the viscosity factor μ is defined by Equation (4).
We expect that Equations (13–15) are also valid in this case. If, as before, the ice densification effects are not important in the normal strain rates in Equations (13), the line of considerations of Appendix 1 can be extended to the nonlinear ice rheology.
For the cold upper (firn and bubbly-ice) stratum of the glacier, where the normal stresses prevail, in accordance with Equation (A6), we write (Salamatin and Duval, 1997)
The substitution of these expressions into Equation (14) results in a generalized form of Equation (A1), and, for u ≈ A/Δ, the corresponding analogue of Equation (A2), based on Expression (17), transforms into
For the basal shear layer, Equation (A6) gives (Salamatin and Duval, 1997):
and instead of Equation (A3) for large β
b in Equation (18) determining μ, we obtain
This approximate relation for the longitudinal velocity, together with Equations (15) and (A4), straightforwardly leads to the corresponding generalization of Equation (19). Expressions (20) and (23) remain valid with β
b+ 1 replaced with β
b + α. This also means that all applications considered in section 4 do not depend directly on ice rheology.
Appendix 4: Reduced Forms of the Heat-Transfer Model
In the framework of the above scale analysis (see section 2), the thermal conductivity in lateral directions has a small effect, of the order O(K
2), on the temperature field. The low dynamic activity of the glacier in combination with intense volcanic heating makes internal dissipation of mechanical energy also negligible. Consequently, the general heat-transfer equation in the flowline coordinate system s, ζ, in accordance with Equation (21), can be written in the following simplified form (Salamatin and others, 1995):
The principal boundary conditions on the bed and ice surface are
The basal temperature is supposed to be close to the melting temperature T
f. Hence, Equation (A8a) determines the melt rate, w
0 ≥ 0. In a changing climate the annual surface temperature, T
s, is a function of time.
Correspondingly, as explained in section 3, Equation (A7) can be reduced to a spatially one-dimensional form relevant to crater glaciers, and, for example, in the quasi-stationary state can be written as (Salamatin and Murav’yev, 1991)
The latter boundary-value problem presented by Equations (A8) and (A9) leads to Equations (25) and (26).
Next, let us note that the most significant changes in c and λ (i.e. A) take place within the upper 30–50 m, in snow and firn. Below this level, which also bounds the depth of seasonal temperature fluctuations, the variations in c and λ do not exceed 10%. With this in mind, it is relevant, at least in the deeper central area of the crater, to simulate the temperature profile, based on the assumption that the thermophysical properties of ice in Equation (A9) are constant, i.e. c ≈ 0, Λ ≈ 1. The enhanced thermal resistance of the near surface snow-and-firn stratum in this case might be taken into account by a special boundary condition (Salamatin and others, 1995) imposed on the glacier surface instead of Equation (A8b):
The apparent heat-transfer coefficient χ is defined here so that it is constant. Using Expressions (24) and (27) it can be evaluated explicitly as χ = [ac
s – (a + 1) ln(1 – c
The above correction conserves the total heat balance of the glacier and represents the temperature field as ζ → 1as an extrapolation of the temperature–depth relation from the deeper part of the glacier, neglecting rapid changes in physical properties ρ and λ within the upper snow–firn stratum. Both the temperature profile and the heat flux in the basal layer remain unchanged. This is important for simulating bottom melting processes and predicting their impact on glacier motion. Such simplified formulation of the heat-transfer equations can be especially useful in completing the model of the dynamics of a crater glacier.