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## Appendix A. Derivation Of 1-D Transversely Integrated Shelf Equations

This 1-D approximation has not received the same attention from mathematicians as its vertical equivalent (e.g. Baral and others, 2001; Schoof and Hindmarsh, 2010), but a basic demonstration of its derivation as an asymptotic approximation is given by Hindmarsh (2006). Expanded forms of the momentum equations (1) are given by

Following Van der Veen (1999), 1-D transversely integrated flows are obtained by assuming the transverse shear stress varies linearly with transverse position, *y*, giving

where the transversely meaned values of *H* and τ_{
xx
} are approximated by their centre-line values, *H*
_{c} and

To compute the centre-line velocity, *u*
_{c}, we note that for the transversely integrated stream or shelf, the *x*-direction momentum balance is

where *τ*
_{
l
} = *τ*_{xy}
(Ω). Integration of the shear relationship,

gives the expression for *u*
_{c}, the *x*-direction centre-line velocity

where *σ* is a margin-softening factor. Combining Eqns (A2) gives the standard form

The upstream boundary conditions used in the 2-D modeling are given by

with *u*
_{c} given by Eqn (A2d).

Under the assumption that shear stress varies linearly with transverse distance, we may write

where *τ*_{l}
is the shear stress at the lateral margin and *η* = *y*/Ω, which may be rewritten as

and where we have also defined

Since, by definition,

we arrive at an algebraic equation for ω as a function of η, the normalized distance from the ice-sheet centre,

The solution for ω is used in the equation for lateral shearing, again using the assumption that shear stress varies linearly across the ice shelf,

and following a transverse integration we find

and the rheological softening parameter is

This can be solved iteratively in numerical solutions of the 1-D equation set. The standard form for the centre-line velocity may be retrieved by setting *P* ≡ 1. The hybridization used here is not as complex as the L1L2 approximation of Hindmarsh (2004) and Schoof and Hindmarsh (2010), and further work is needed to investigate the scaling properties of this and other hybrid formulation

## Appendix B. Calving-Front Scaling Relationships and Relation to Schoof Grounding Line Formula

Here we show how the algebraic technique deriving flux formulae may also be used to derive the Schoof flux formula for basally resisted ice streams. Considerthe momentum and mass conservation for a basally resisted ice stream in plane flow with bed slope much smaller than thickness gradient,

where *C* is a drag coefficient. We assume the horizontal stress gradient terms are small and, eliminating *∂*
_{
x
}
*H*, we obtain

and for small *a* the Schoof flux formula follows:

## Appendix C. Conditions For 2-D Flows to be Scale-Free

Our aim here is to show that, in principle, the 2-D equations can be scaled to depend on only one parameter, *H*
^{*}. (This was done in Section 2.2 for the 1-D equations.) For algebraic simplicity we adopt a slightly more general scaling, allowing the x-direction length scale to be free. Let the scale for the thickness at the calving front be *H** and choose as scales

where *x*
^{*} is a longitudinal length scale of interest, which in practice will be the x-length scale of one of the boundary layers; we do not need to specify *x** now. We assume that *δ* is a constant independent of Ω. We then use in Eqn (9a) to obtain the dimensionless scale-invariant form for the momentum balance equation

with tildes denoting dimensionless variables. A scale- invariant form also exists for the continuity equation that motivates the choice of scales

substitution of these in the continuity equation yields a scale- invariant form when *a* ≪ *H∂*
_{
x
}
*u*
_{c}. The scale invariance arises from the fact that the shear stress scale, is defined by *H**, which is a result of the boundary layer theory through the universality of the traction number.

We now consider the 2-D flows and make the choice of scales

The relationship for *∂*
_{y}
*v** is consistent with the choice of scale for The fact that *∂*
_{y}
*v** is independent of Ω implies that *v*, like *u*, scales with Ω, which is necessary for scale invariance from a kinematical point of view. This, and the fact that must also do so. With these choices of scales, the 2-D momentumbalance equations (A1) become

which are scale invariant, holding for any combination of *H** and Ω. The choices that and ∂_{y}
*H*
^{*} scale with *H*
^{*} are therefore consistent with the 1-D scaling. It is straightforward to see that the continuity equation,

is scale invariant under these same conditions. In consequence,scaling considerations demonstrate that the calvingfrontboundary layers of 2-D flows have the same scaling relationship as the 1-D flows, leading to the same condition of flows being parameterized by the calving-front thickness. At first sight, it is slightly puzzling that *τ*
_{yy}* and *∂*
_{y}
*v*∗ donot tend to zero as the shelf becomes infinitely wide. The whole analysis, however, is based on the idea that the lateralmargins play a significant role. The longitudinal velocity,*u*, increases with width, and this results in the transversevelocity and horizontal stress not decreasing in value.

## Appendix D. Boundary Layer Scale Estimates

Here we obtain expressions for the horizontal extent of the two boundary layers. In the 1-D transversely integrated form, only one boundary layer exists, in the region where *τ*
_{xx} changes from its interior value to its frontal value, set by the boundary condition. Letting asterisked quantities represent scale magnitudes, and letting *L*∗ represent the unknown longitudinal extent of boundary layer, we write scale versions of the left-hand side of Eqn (9a)

to find

The longitudinal boundary layer has extent roughly equal to the width, *W* = 2Ω, of the stream. The quantity is close to unity, while the sensitivity to any margin-softening factor, *σ*, is small for *n* = 3.

In 2-D, there are two frontal boundary layers in narrow ice shelves. One of these arises for identical reasons to the one just derived for the transversely integrated case. The second boundary layer arises from the flow in the interior (zones A and B) being close to a simple shear, with one of the planes aligned with the direction of flow. This implies a nonzero tangential traction on a vertical plane across the ice stream. However, at the calving front, the shear stress in the plane parallel to the front must be zero, so there must be a large term, (∂*H*
_{τxy
})/∂_{
x
} near the calving front. Figure 1 confirms this suggestion.

An analysis for the *x*-momentum balance gives the boundary layer length as similar to the width of the shelf. Ignoring the drive term, γ*H∂*
_{
x
}
*H*, and the small transverse stress, the *x*-direction momentum balance equation (A1a) becomes

and replacing quantities with their asterisked scale magnitudes

so that Using

where *ν*is the viscosity, we find *L*
^{*} = 2Ω and In fact, Eqn (17a) makes a stronger statement regarding the ratio , which can be evaluated and is very close to unity.

The transverse boundary layer is more difficult to analyse, and we restrict attention to the longitudinal extent of zone C, and assume (as is demonstrated by numerical solutions) that the transverse gradients in thickness are small, implying that the transverse driving force and transverse thickness gradients are small. Then, expanding Eqn (A1b) and dropping terms involving ∂_{
y
}
*H*, we find

We can also use the normal stress condition at the calving front, *S*
_{
xx
} = γ*H*/2, to write

and use these two last expressions to eliminate *τ*
_{
yy
}, obtaining

which, when replaced by scale magnitudes, is

where *D*
^{*} is the longitudinal extent of the transverse boundary layer. Then, using

the 6 stemming ultimately from the definition of the stress tensor S.

## Appendix E. Stress Invariant At Calving Front

The stress invariant at the front, where *τ*
_{
xy
} = 0, is given by

and from the boundary condition, is the stress for an infinitely wide ice shelf, we find

with minimum value 1 at *μ* = 1. Thus, nonzero *τ*
_{
yy
} causes an increase in the stress invariant near the calving front, compared with the plane flow situation, despite the constraint imposed by the boundary condition.