Abstract
For land glaciers with simply connected free surface, the general solution of the balance conditions and the boundary conditions of vanishing boundary stresses on the free surface is expressed by three components of the stress tensor or the deviatoric stress tensor, where these three components can be taken as arbitrary functions within the framework of the general solution. For a glacier with a heavy rock on its surface, such that the free surface surrounding the rock is not simply connected, the general solution of the balance boundary and load conditions is given, where the load conditions are defined by the weight and torque exerted by the rock. For land glaciers with small strain rates, so-called “quasi-stagnant models” are introduced. These quasi-stagnant models are candidates for realistic models and can be represented with reasonable effort without aiming for a precision by too much effort, which cannot be achieved anyway due to information deficits.
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Notes
- 1.
- 2.
- 3.
See paragraph 1 and paragraph 6, Sect. 9.1.2.
- 4.
See No. 1, Sect. 3.4.1.
- 5.
See point 2, Sect. 3.4.1.
- 6.
- 7.
The functions \( {\partial}_z^{-1}\rho \), \( {\partial}_x{\partial}_z^{-2}\rho \) and \( {\partial}_y{\partial}_z^{-2}\rho \) vanish at the free surface, since ρ by definition vanishes beyond the free surface.
- 8.
The functions \( {\partial}_z^{-1}{S}_{xx} \), \( {\partial}_x{\partial}_z^{-2}{S}_{xx} \) etc. vanish at the free surface, since Sxx etc. vanish by definition beyond the free surface.
- 9.
The transformations in Sect. 18.5 are used.
- 10.
See No. 5, Sect. 3.1.
- 11.
In the following, the term “ice depth” stands for both the name and the length of the distance in the z-direction from the free surface to the point considered in the glacier.
- 12.
The integrals or double integrals are expressions of the type \( {\partial}_z^{-1}\left(\theta \cdot \ast \right) \) or \( {\partial}_z^{-2}\left(\theta \cdot \ast \right) \). See Chap. 3 on integral operators.
- 13.
- 14.
See paragraph 1 and paragraph 6, Sect. 9.1.2.
- 15.
The oriented free surface must be transverse and synchronous to the model cone K xyz, which is generated by the model cone vectors ex, ey and ez.
- 16.
See point 2, Sect. 3.4.1.
- 17.
See Sect. 10.1.1 paragraphs 1–4 and the subsequent discussion of these properties.
- 18.
- 19.
See paragraph 1, and paragraph 6, Sect. 9.1.2.
- 20.
The oriented free surface must be transverse and synchronous to the model cone \( {K}_z^{\odot } \). See para. 1, Sect. 9.1.2.
- 21.
See point 2, Sect. 3.4.1.
- 22.
See point 6, Sect. 9.1.2.
- 23.
See paragraphs 1–4, Sect. 10.1.1 and the subsequent discussion of these properties.
- 24.
The functions q1, q2 and q3 are calculated for the matrix elements of T0 from the independent stress components \( {S}_{xx}^{\prime } \),\( {S}_{yy}^{\prime } \), Sxy and for the matrix elements of Sf from the ice density ρ. The calculation procedure is given in Sect. 18.6. The independent stress components and the ice density are written as the product of the step function θ (10.46) and a smooth function.
- 25.
- 26.
The function G(r′ − r) (10.48) is non-zero only in the points r′ that lie in the dependence cone starting from the point r.
- 27.
- 28.
The total boundary surface Λ where no boundary conditions are known consists of two separate and connected parts: the contact surface Λ1 with the rock and the contact area Λ0 with the bedrock and the glacier area not under consideration.
- 29.
- 30.
The stress tensor field T∗∗ is constructed according to the procedure described in Sect. 16.2.
- 31.
- 32.
The term “stagnant stress tensor field” is a shorthand term for the stress tensor field of a stagnant glacier. Stagnant stress tensor fields have vanishing deviatoric components and are therefore scalar multiples of the unit tensor.
- 33.
In the stagnant case, all symbols are marked with the accent “˘”.
- 34.
Despite this comparison with the hydrostatic case, we do not use the seemingly obvious term “glaciostatic” here. In fact, all models in this paper are glaciostatic, since there is complete balance of all forces and all torques. The special case of stagnant glaciers is thus not characterized by glaciostatics, but by the isotropy of the stress tensor fields, which is equivalent to deviatoric stress components disappearing everywhere.
- 35.
- 36.
This can only be decided after further investigation in each specific case.
- 37.
See Sect. 8.2.
- 38.
“Simple” means that the quasi-stagnant stress tensor field can be represented mathematically in a relatively simple way.
- 39.
- 40.
In this case, the surface slope tanα in formula (10.113) disappears.
- 41.
This tangential vector \( {\mathbf{l}}_0^{\prime } \) at the bed is obtained by projecting the tangential vector l0 of the ice surface, pointing in the direction of the surface slope, vertically onto the tangential plane of the bed and then normalizing it to length 1.
References
Nye, J.F.: A comparison between the theoretical and the measured long profile of the Unteraar Glacier. J. Glaciol. 2, 103–107 (1952)
Paterson, W.S.B.: The Physics of Glaciers. Elsevier, Oxford (1994)
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Halfar, P. (2022). Land Glaciers. In: Stresses in glaciers . Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-66024-9_10
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DOI: https://doi.org/10.1007/978-3-662-66024-9_10
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