under a Creative Commons License. The Cryosphere Discussions

New analytical solutions describing the effects of small-amplitude perturbations in boundary data on flow in the shallow ice stream approximation are presented. These solutions are valid for a non-linear Weertman-type sliding law and for Newtonian ice rheology. Comparison is made with corresponding solutions of the shallow ice sheet approximation, and with solutions of the full Stokes equations. The shallow ice stream approximation is commonly used to describe large-scale ice stream flow over a weak bed, while the shallow ice sheet approximation forms the basis of most current large-scale ice sheet models. It is found that the shallow ice stream approximation overestimates the effects of bedrock perturbations on surface topography for wavelengths less than about 5 to 10 ice thicknesses, the exact number depending on values of surface slope and slip ratio. For high slip ratios, the shallow ice stream approximation gives a very simple description of the relationship between bed and surface topography, with the corresponding transfer amplitudes being close to unity for any given wavelength. The shallow ice stream estimates for the timescales that govern the transient response of ice streams to external perturbations are considerably more accurate than those based on the shallow ice sheet approximation. In contrast to the shallow ice sheet approximation, the shallow ice stream approximation correctly reproduces the short-wavelength limit of the kinematic phase speed. In accordance with the full system solutions, the shallow ice sheet approximation predicts surface fields to react weakly to spatial variations in basal slipperiness with wavelengths less than about 10 to 20 ice thicknesses.

EGU scalings results in ∂ x (4hη∂ x u + 2hη∂ y v) + ∂ y (hη(∂ x v + ∂ y u)) − (u/c) 1/m = ρgh∂ x s cos α − ρgh sin α, 5 which are two coupled partial differential equations for u and v. The horizontal velocity components (u and v) are constant over depth, and the vertical velocity component (w) varies linearly with depth. In these equation s is the surface, h is ice thickness, η is the effective ice viscosity, and c is the basal slipperiness. The parameter m and the basal slipperiness c are parameters in the sliding law defined by Eq. (A5) in Appendix A. 10 For a linear viscous media (n=1) and a non-linear sliding law (m arbitrary) these equations can be linearised and solved analytically using standard methods. We write f =f +∆f , where f stands for some relevant variable entering the problem, and look for a zeroth-order solution wheref is independent of x and y and time t, while the first-order field ∆f is small but can be a function of space and time.
This transfer function describes the relation between surface and bedrock geometry, where ∆s(k, l , t)=T sb (k, l , t)∆b (k, l ). Other transfer functions are defined in an analogous manner. 15 In Gudmundsson (2003) the relaxation time scale is referred to as the decay time scale and denoted by t d . As pointed out by Hindmarsh (2004), depending on the situation, the term "growth rate" is presumably more descriptive. Here the term "relaxation time scale" will be used as the time scale t r determines how long it takes for the the transient solution to "relax" toward the steady-state limit. 20 The relationship between surface velocity and bedrock is found to be given by and for the longitudinal (u) and the transverse (v) components, respectively.

Perturbations in basal slipperiness
Transfer functions describing the effects of a spatial variations in basal slipperiness 10 on surface geometry and surface velocities can be derived in a similar fashion. In Appendix B the solution procedure is outlined and expressions for the corresponding transfer functions listed.

Surface perturbations
We now determine the transient evolution of a surface undulation prescribed at t=0. 15 By writing h=h+∆s, s=s+∆s, b=b, u=ū+∆u, v=∆v, w=∆w, and c=c, inserting into (1) and (2) and solving the resulting equations together with (12) and the kinematic boundary condition at the surface EGU it is found that the surface evolution as a function of time is given by s(k, l , t) = T ss 0 s(k, l , t = 0), where The velocity components are given by where ψ = ikh cot α, and 10 T vs 0 = il (3ikη + ν cot α)τ d νξ e p t .
2.4 Non-dimensional forms of the transfer functions The transfer functions listed above are all in dimensional form. It is often much more convenient to work with the transfer functions in a non-dimensional form. For this purpose the same scalings as used in Gudmundsson (2003) will be employed. The 15 velocity is given in units of mean deformational velocity of the full-system solution. All spatial scales are in units of mean ice thickness (h) and stresses in units of driving stress (τ d ). It follows from these scalings that the non-dimensional viscosity, E , is given by E =ū d η/(hτ d )=1/2, and the mean non-dimensional basal slipperinessC is

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The mean non-dimensional slipperiness is therefore equal to the slip ratio, i.e. the ratio between mean basal sliding velocity (ū b ) and the mean forward deformational velocity (ū d ). One obtains the non-dimensional form of the transfer functions from the dimensional one using the substitutionsc →C, η → 1/2,h → 1,ū →C, k → k, l → l , γ → (mC) −1 , and ρgh sin α → 1.

5
Note that since in the shallow ice stream approximationū d =O(δ 2 ) and we are ignoring all fields to this order, we haveū → C and notū → C+1 as is the case for the full-system solutions and the solutions of the shallow ice sheet approximation. In these non-dimensional units, the requirement that the slip ratio is O(δ −2 ) for the shallow ice stream approximation to be valid impliesC+1≈C orC≫1 for small surface slopes. 10 In non-dimensional form the solutions often take a considerably simpler shape. For example the non-dimensional forms of the timescales t d and t r , and the transfer function T sb are, respectively, , 15 and EGU non-linear media and a non-linear sliding law (Nye, 1960;Fowler, 1982;Jóhannesson, 1992). Small-amplitude FS solutions have so far only been derived for linear media and a linear sliding law (Reeh, 1987;Jóhannesson, 1992;Gudmundsson, 2003). The SSTREAM solutions, derived here for the first time, are valid for linear media and a non-linear sliding law. It follows that intercomparision with the correct FS solutions can 5 only be done for n=1 and m=1, and for this reason most of the discussion is limited to this case. The only exception to this is a brief description given below of the the relationship between bed and surface geometry in the SSTREAM approximation for non-linear sliding law. A full discussion of the non-linear aspects of the solutions will be done elsewhere.

Time scales
As seen from the solutions listed above and in Appendix B the transient behaviour is completely determined by two time scales: the phase time scale t p , and the relaxation time scale t r . The term "phase time scale" is used for t p because it determines how quickly the phase of the surface fields changes with time (see for example Eqs. 24 and 15 25). As mentioned above the t r time scale determines how quickly the solutions "relax" toward the steady state limit. The properties of the phase time scale can most easily be understood by looking at the phase velocity v p which is, by definition, given by in dimensional units. The angle θ is the angle between the wave vector k=(k, l ) and the x axis. This is also the angle between the x axis and a vector lying in the xy plane and normal to the crests of the sinusoidal perturbations. We have cos θ=k/j and λ=2π/j where λ is the wavelength.

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It is instructive to compare phase speeds for different approximations. Figure 1a shows the phase speed for θ=0 as a function of wavelength for the shallow ice stream approximation (dashed line), the shallow ice sheet approximation (dashed-dotted line), and for the full-system solution (solid line) for m=1 and n=1. Note that in the figure the phase speeds have been normalised by the surface velocity as given by the full-system 5 solution (equal toC+1 in non dimensional units).
In the shallow ice sheet (SSHEET) approximation the phase speed (dashed-dotted curve in Fig. 1a) is, for θ fixed, independent of the wavelength. For θ=0 the SSHEET phase speed is (n+1)u d +(m+1)u b , where u d is the deformational velocity and u b the basal sliding velocity. This is a well known (Nye, 1960) and often used expression for 10 the speed of surface waves of on glaciers. The long-wavelength limit of the SSHEET phase speed is correct, but the short wavelength SSHEET limit is incorrect. The correct FS limit for θ=0 as λ→0 is |v p |=u s which, in physical terms, is clearly also the correct expression.
As can be seen from direct inspection of Eq. (30), and is demonstrated by Fig. 1a, 15 the SSTREAM phase speed (dashed curve) is quite similar to the FS phase speed for C≫1. In particular, in contrast to the SSHEET phase speed, the SSTREAM phase speed given by Eq. (30), valid for arbitrary m and n=1, is not independent of λ. The λ→+∞ SSTREAM limit is |v p |=ū(1+m) cos θ, which is the FS limit for u d =0, and for λ → 0 the SSTREAM phase speed is to equal to mean surface speed. Hence, in both 20 the λ→0 and the λ → +∞ limits, the SSHEET phase speed is equal to the FS phase speed.
The differences between the FS and the SSTREAM phase speeds are small for parameter values typical of an active ice stream. This can be seen most easily from direct inspection of the differences between t p given by Eq. (16)  EGU only minor differences. The wavelength dependency of the phase velocity gives rise to dispersion in the FS and the SSTREAM solutions. In these theories it is therefore somewhat misleading to interpret the phase velocity as the velocity by which surface disturbances propagate. The group velocity is a better measure of this velocity, and as shown in Gudmundsson 5 (2003) the FS group velocity can be significantly different from the phase velocity and for some wavelengths even smaller than the material surface velocity.
One finds that in the SSTREAM approximation the x and the y components of the group velocity, (u g , v g ), are given by 10 and The x component of the group velocity is shown in Fig. 1b for m=1 as a function of wavelength for the same surface slope and slip ratio values as used in Fig. 1. The figure shows that the SSTREAM approximation is closer to the exact FS solution than 15 the SSHEET approximation. Note also that for the particular values of surface slope and slipperiness used in the Fig. 1b, the group velocity is smaller than the mean surface speed for wavelength between 8 to 50 mean ice thicknesses. In Fig. 3 the relaxation time scale (t r ) is plotted in dimensional units (years) as a function of wavelength forC=100 and α=0.002 and m=1. (Note that the relaxation time 20 does not depend on the angle θ.) As indicated by the figure, and direct inspection of the corresponding equations shows, the SSTREAM relaxation time scale closely approximates the FS relaxation time scale down to wavelengths of about 10 ice thicknesses. Of particular interest is the fact that in accordance with the full-system solution, but in contrast to the SSHEET approximation, there is a range of wavelengths over which the

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SSTREAM relaxation time scale is independent of λ (see Fig. 3). The only qualitative aspect of t r not captured by the SSTREAM approximation is the increases in t r with decreasing λ for λ less than about ten ice thicknesses. ForC≫1 the expression for t r given by the SSTREAM solution is a much better approximation to t r than the one given by the SSHEET solution. For the particular set of parameters used in Fig. 3 both 5 the SSTREAM and the FS solutions give, for the range 10h<λ<100h, a constant value for t r on the order of ten years. The SSHEET solution gives, for the same range values, estimates of t r ranging from hours to days. Of the two time scales t r and t p it can be argued that t r is the more important one. A surface wave will travel a distance equal to its wavelength in the time 2πt p . In 10 the same time its amplitude will decrease by the factor e 2πt p /t r (See Eq. 25). The ratio 2πt p /t r , thus, can be thought of as giving the relative importance of relaxation/diffusion to wave propagation. As pointed out by Jóhannesson (1992) this ratio is, for typical situations encountered in glaciology, usually larger than one. It follows that the time scale for local mass redistribution on glaciers and ice sheets is essentially given by 15 t r , and takes place much faster than the time scale t p would suggest. The close agreement between t r as given by the FS and the t r as calculated on basis of the SSTREAM approximation gives added confidence in the applicability of the SSTREAM approximation to situations typical of active ice streams.

Bedrock perturbations
20 Figure 4 shows the steady-state bedrock-to-surface transfer amplitude (|T sb |) as a function of wavelength. The figure shows the FS, the SSHEET, and the SSTREAM transfer amplitudes, respectively (based on Eqs. 82 and 26 in Gudmundsson, 2003, and Eq. 19). Surface slope (α=0.002) and slip ratio (C=100) values are typical for active ice streams. The three curves differ in a number of important ways.

25
The steady-state SSHEET transfer amplitude (dotted line in Fig. 4) is strictly increasing as a function of wavelength and does not show the pronounced peak in bed-to- EGU surface transfer for wavelengths from about 1 to 10 ice thicknesses seen in FS solution (solid line). The SSTREAM solution (dashed line) overestimates the transfer at short wavelengths and gives a physically wrong limit of |T sb | → 1 for λ→0. Despite the incorrect limit for λ → 0, the SSTREAM solution for T sb is, and this is generally the case for high 5 slip ratios, a considerably better approximation to the FS solution than the corresponding SSHEET solution. For the particular set of values used in Fig. 4 the SSTREAM solution agrees within a few percent to the FS solution down to wavelengths of about eight ice thicknesses. The SSHEET solution is only similarly accurate for wavelengths larger than about 100 ice thicknesses. 10 From Eq. (19) it follows that the steady state SSTREAM transfer function for l =0 is where dimensionless units have been used. Hence for θ=0, T sb → 1 asC → ∞ irrespective of the values for surface slope α, wavelength λ, and sliding law exponent m. The value ofC=100 used in Fig. 4 can hardly be considered very large for typical 15 active ice streams, and if a value ofC=1000 is used together with typical surface slopes of about 0.002 to 0.004, it follows that |T sb | is fairly close to unity for all wavelengths. The minimum in the SSTREAM transfer amplitude given by Eq. (33) is reached for the wavelength

20
where kλ x =2π, and the minimum is given by

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For bedrock undulations aligned transversely to the main flow direction the steadystate ratio between surface and bedrock amplitudes is in the SSTREAM approximation always larger than that given by Eq. (35). The wavelength given by Eq. (34) does not depend on surface slope α. As Fig. 4 suggests, the (global) minimum in SSTREAM transfer amplitude (T sb ) closely approximates the corresponding (local) minimum of the 5 FS transfer amplitude forC≫1. Examples for non-steady T sb amplitudes are shown in Fig. 5. The dashed lines in the figure are calculated using Eq. (19) and are based on the shallow ice stream approximation, using the additional small-amplitude assumption ∆b/h≪1. The solid lines are calculated using Eq. (82) in Gudmundsson (2003) and are correct for any 10 value ofC and any wavelength λ as long as ∆b/h≪1. The times are given in nondimensional units. These can be translated to dimensional units through multiplication withh/ū d , whereū d is the mean deformational velocity. As an example, for a 1000 m thick ice stream where the surface velocity is 1 m d −1 , t=0.001 corresponds to about 3.3 months. 15 The figure shows the relatively slow increase in |T sb | with time for long wavelengths (larger than about 100 mean ice thicknesses) toward the steady-state long-wavelength limit of |T sb |=1. The rate of increase toward the steady-state limit is determined by t r which, for long wavelengths, increases quadratically as a function of wavelength (see Eq. 17 and Fig. 3), hence the slow increases in |T sb | for long wavelengths. Over 20 wavelengths less than about 5 mean ice thicknesses the SSTREAM relaxation time is smaller than the FS relaxation time (see Fig. 3). Consequently, over this range of wavelengths the SSTREAM amplitudes grow faster with time than the FS amplitudes. Another noticeable aspect of Fig. 5 is the oscillating behaviour of the transfer amplitudes with wavelength. These are caused by temporal fluctuations (kinematic oscillations) 25 in |T sb | that are governed by the phase time scale t p . As the figure shows, transient amplitudes can be larger than unity, and when this happens surface amplitudes are larger than the bedrock amplitudes. As follows from inspection of Eq. (19) and Eq. (82) in Gudmundsson (2003), kinematic oscillation are particularly pronounced for 0<t<t r EGU whenever t p <t r . The consequence can be an up to twofold increase in transfer amplitudes as compared to the corresponding steady-state limit. As mentioned above, the steady-state λ→0 limit of the SSTREAM T sb amplitude is unity, and for short wavelengths (λ/h<1) the transient value can be as large as two, whereas the correct value for these wavelengths (as given by the FS solution) is always close to zero.

5
The T sb transfer amplitudes in Figs. 4 and 5 are plotted as functions of longitudinal wavelength, that is for sinusoidal bedrock undulations aligned transversely to the main flow direction. This corresponds to the situation θ=0. Figs. 6a and b show the SSTREAM and the FS transfer amplitudes, respectively, as functions of both θ and λ. As Fig. 4 showed for θ=0, the main difference between the SSTREAM and the 10 FS T sb amplitudes is the short-wavelength limit. Irrespectively of θ the SSTREAM short-wavelength limit always equal to unity, whereas the correct limit is zero. The only exception to this is for the θ = π/2 where both transfer functions are identically zero.
In Figs. 7a and b the T ub transfer amplitudes of, respectively, the SSTREAM and FS solutions are shown in a non-dimensionalised form. A simple way of interpreting 15 the numerical contour values is to think of them as ratios between mean deformational velocity and mean ice thickness (ū d /h). In the figures, a slip ratio of 99 is used and the mean surface FS velocity is equal to 100. Where the transfer amplitudes are equal to, lets say 50, a sinusoidal bedrock perturbation with an amplitude corresponding to 10% of the mean ice thickness produces a 0.1×50=5ū d large perturbation in surface 20 velocity or a 5 % perturbation in mean surface velocity.
As the Figs. 7a and b suggest, and inspection of the corresponding analytical solutions confirms, the short wavelength limits of the SSTREAM and the FS T ub transfer amplitudes are both equal to zero. This is physically the correct limit and there is therefore no problem similar to that of the T sb transfer amplitude for λ → 0. Quantitatively the 25 T ub amplitudes tend to be somewhat underestimated by the SSTREAM approximation when the shallow-ice condition λ/h≫1 is not fulfilled.
For bedrock disturbances running along the flow (k=0) the resulting perturbation in EGU surface velocity is, irrespectively of t, given by as can readily be derived from Eq. (21). The maximum value of |T ub | is reached for k=0 in the limit λ → ∞. In the FS theory this limit isC+1 which is the physically correct limit. In the SSTREAM theory this limit is, for m=1 where these results can be 5 compared, equal toC, the difference being due to the simple fact that the SSTREAM approximation ignores the internal ice deformation. For the transverse amplitudes (|T vb |) shown in Figs. 8a and b the only qualitative difference is found for θ=π/4. For this value of θ the SSTREAM transfer amplitude |T vb | has one local maximum as function of λ but the FS solution two. Again the SSTREAM T vb amplitudes are somewhat underestimated when the shallow-ice condition λ/h≫1 is not fulfilled. For the particular set of parameters using in Figs. 7a and b, and Figs. 8a and b the SSTREAM approximation underestimates the effects of bed perturbations on surface velocities for wavelengths less than about 10 ice thicknesses.
3.2.1 Surface topography and non-linear sliding 15 The discussion given above has mostly dealt with the n=1 and m=1 case. In Fig. 9 the SSHEET and the SSTREAM ratios between surface and bedrock amplitude (|T sb |) are plotted for m=1 and m=3 for n=1. In addition the FS ratio is shown for m=1. There are no analytical FS solutions known for m = 1. The SSTREAM ratios shown (blue lines in Fig. 9) (2003), and the SSHEET ratios (red lines) from Eq. (4.4.7) in Jóhannesson (1992).
As the Jóhannesson (1992) thesis is presumably not widely available I list here his SSHEET expression for the T sb transfer function. It is particularly simple and valid for any value of n and m. It can be written in the form is in units of mean ice thickness. The long and the short wavelength limits of SSHEET transfer amplitude as given by Eq. (37) are independent of n and m, and the range of wavelengths where the transfer is, in absolute terms, significantly dependent on n 5 and m is determined by the value of λ ⋆ . For moderate values of n and high slip ratios Hence, the surface slope is the most important parameter affecting the SSHEET bedrock-to-surface transfer characteristics for high slip ratios and the transfer is only 10 moderately affected by the value of m.
Presumably the most interesting aspect of Fig. 9 is that the differences between the m=1 and the m=3 cases are only of any significance for wavelengths longer than about 100 ice thicknesses. This is so for both the SSHEET and the SSTREAM solutions. By analysing the SSTREAM transfer function given by Eq. (19) it is found that the 15 sensitivity of |T sb | to m is small at large and small wavelengths and decreases with decreasing slope. In particular, for wavelengths smaller than the wavelength for which the minimum in SSTREAM amplitude is reached (see Eq. 34), the transfer amplitudes are not significantly affected by the value of the sliding law exponent m.
It is not clear if this insensitivity of T sb to the value of the sliding law exponent m for 20 wavelength smaller than the one given by Eq. (34) is also true of the FS solutions. If so, then that would suggest surface topography on active ice streams to be a weak function of m. The only study on the effects of non-linear sliding on the bed-to-surface transfer amplitudes seems to be that of Raymond and Gudmundsson (2005). They calculated numerically the T sb amplitude for various values of m. Figure 11 in Raymond Gudmundsson (2005) suggests that for wavelengths smaller than the one for which the local minimum in transfer amplitudes of the FS solutions is reached, changing m from 1 to 3 has almost no effect on transfer amplitudes.

Basal slipperiness perturbations
We now consider the effects of spatial variations in basal slipperiness on surface 5 fields. Basal slipperiness is here defined as the function c(x, y) in the basal sliding law (see Eq. A5). The non-dimensional slipperiness is written using an upper case letter (i.e. C(x, y)). We have C(x, y)=c(x, y)τ d /ū d , and write C(x, y)=C(1+∆C(x, y)) whereC is the spatially averaged slipperiness and ∆C(x, y) the (fractional) slipperiness perturbation introduced at t=0. The transfer functions listed in Appendix B give the relationships between surface fields and the basal slipperiness perturbation ∆C(x, y) in frequency space. We have, for example, s(k, l , t)=T sc ∆C(k, l ). Our main focus here is on the differences between the corresponding SSHEET, SSTREAM and the FS solutions and therefore the discussion is mostly limited to the m=1 and n=1 case where small-amplitude analytical solutions to all of FS, SSHEET, and the SSTREAM problems 15 are available. Figure 10 shows basal-slipperiness to surface geometry transfer amplitudes (T sc ) based on Eq. (B3) (circles) for the SSTREAM theory. For comparison the predictions of the FS (solid lines ) and the SSHEET (crosses) theories (Eqs. 83 and 27 in Gudmundsson, 2003) for the same parameter set are shown as well. The biggest difference 20 between the SSHEET and the FS solution is at long-wavelengths. Both the FS and the SSHEET long wavelength limits (λ→∞ for θ=0) are equal toC/(2(1+C)). For the SSTREAM solution this limit is 1/2 independently ofC (see Eq. B7). ForC≫1 these two different expressions, of course, give numerically quite similar answers. ForC≥19 the difference is, for example, less than 5%. Interestingly, irrespectively of how slippery the bed is, the SSHEET approximation gives an equally good or better description than the SSTREAM approximation of the relationship between surface topography and basal slipperiness.  Figure 11 shows the velocity transfer amplitudes |T uc |, |T vc |, and |T wc |. Shown are both the FS solutions (solid lines) and those based on the SSTREAM approximation (dashed lines). The SSTREAM solutions are given by Eqs. (B4), (B5) and (B6). The FS solutions can be found, or easily derived, from Gudmundsson (2003). The transfer functions are plotted in non-dimensional form. The scale for the transfer functions 5 follows from the definitions ∆u=T uc ∆C, ∆v=T vc ∆C, where ∆C is the (fractional) slipperiness perturbation. Since the velocity is scaled with the mean deformational velocitȳ u d and ∆C has no dimensions,ū d is the scale for these transfer functions.
Generally speaking there is a good agreement between the velocity transfer functions of the FS and the SSTREAM solutions (see Fig. 11). Nevertheless, there are a 10 number of noticeable differences. Like the T sc transfer amplitude, the long-wavelength limit of the T uc is incorrect. For n=1 and m=1 it isC/2 for the SSTREAM solution, but C 2 /(2(1+C)) for the FS solution. The effects of basal slipperiness perturbations are therefore somewhat overestimated for long-wavelengths, although for high slip ratios typical of active ice streams this error is small. Over wavelengths less than about 100 15 ice thicknesses the horizontal forward velocity component (u) reacts weakly to basal slipperiness perturbations. In Fig. 11, for example, the FS T uc transfer amplitudes (solid line) are less than 5% of mean surface speed for this wavelength range. The SSTREAM approximation further underestimates this weak response (see Fig. 11).
In comparison to the FS amplitude the SSTREAM T wc amplitudes are too large (see 20 Fig. 11). This difference is in most situations of no real concern, however. The most conspicuous aspect of both the FS and the SSTREAM T wc transfer amplitudes is how small they are both in absolute terms and in comparison to |T uc | and |T vc |. In fact the |T wc | amplitudes are so small that for active ice streams the vertical velocity component can be considered to be effectively insensitive to any spatial variations in basal slipper-25 iness. As an example, forC=10 and α=0.002, which are the values used in Fig. 11, the maximum of the FS T wc amplitude is about 0.025. The corresponding perturbation in the vertical surface velocity component is therefore 0.025ū d which for most active ice streams where the surface velocity is a few hundred to a few thousand times larger EGU thanū d is negligible in comparison to the mean horizontal velocity. In contrast, the maximum perturbation in longitudinal velocity (u) isC/2 forC≫1 or about 50 % of the mean surface velocity, and as can be seen from Fig. 11 or direct inspection of Eq. (B5). Equally large perturbations in transverse velocity are possibly for perturbations in basal slipperiness that are sufficiently misaligned with respect to the mean flow direction.

5
The transfer amplitude T uc describing the response of the horizontal forward velocity component (u) to a basal slipperiness perturbation is shown as a function of wavelength (λ) and orientation (θ) in Fig. 12. This Figure should be compared to Fig. 7 giving the response to a bedrock perturbation for the same set of parameters. The most significant difference between T ub the T wc amplitudes is the comparatively weak response 10 of u to perturbation in basal slipperiness that are aligned approximately transversely to the mean flow (θ<40 degrees) for wavelengths less than about 100 ice thicknesses. As follows from Eq. (B4) and demonstrated in Fig. 12, the strongest response is |T uc |=C+1 for basal slipperiness perturbations aligned with the flow (θ=π/2) in the limit λ→∞.
Summarising, it is found that the SSTREAM transfer functions describing the re-15 sponse of surface fields to spatial perturbations in slipperiness compare favourably to the accurate FS solutions when the conditions λ/h≫1 andC≫1 are fulfilled. This is of course to be expected. Possibly somewhat unexpectedly, the SSHEET T sc transfer functions are almost equally accurate for this parameter range. Whenever these conditions are not fulfilled, the resulting errors do not, as far as can be seen, result in any 20 clearly non-physical description by the SSTREAM theory.

ISMIP-HOM Experiment F
As an illustration of differences between the full-system and the shallow ice stream solutions Figs. 13a

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Of the mean surface velocity, half is due to internal ice deformation and the other half to basal sliding (C=1). The flow is down an uniformly inclined plane with a mean slope of 3 degrees. The bedrock perturbation is a Gaussian shaped peak situated at (x, y)=(0, 0). The peak has a width of 10h and amplitude of 0.1h. Periodic boundary conditions are used with a periodicity of 400h in both x and y directions.
If either the slip ratio is not large compared to unity, or the horizontal scale of interest is not large compared to mean ice thickness, significant deviations between the SSTREAM and the FS solutions can be expected. In the experiment the slip ratio does not fulfil the conditionC≫1 and it comes as no surprise that there are some differences between the upper (SSTREAM solution) and the lower (FS solution) halves of Figs. 13a and b. However, if anything, the performance of the SSTREAM solutions seems surprisingly good. There are, for example, only fairly small differences seen in the perturbed surface topography (Fig. 13a). The amplitude of the FS surface topography perturbation (lower half of Fig. 13a) is a bit larger than that of the SSTREAM one (upper half of Fig. 13a) but otherwise the surface shapes are in qualitatively terms the 15 same.
The FS velocity perturbations are generally larger than the those of the SSTREAM approximation (see Fig. 13b). This is partly due to the simple fact that the mean SSTREAM velocity is equal to 1 while the mean FS velocity is twice as large. One could argue that the mean slip ratio in the SSTREAM theory should be redefined to 20 give the same mean surface velocity as the FS solution. In the experiment this would imply usingC=2 when calculating the SSTREAM solutions. When this is done, the differences between the upper and the lower halves in Fig. 13b become considerably smaller and the overall magnitude more similar. Irrespectively of which value ofC is used, the FS velocity solution has a more detailed short-scale structure. This aspect 25 of the solution can be understood by considering the corresponding transfer functions directly. Comparison of Fig. 7a with b, and Fig. 8a  EGU than those of the SSTREAM theory. If the width of the Gaussian peak is decreased from 10h to, lets say, 1h, and the slip ratio increased to, again lets say 100, the differences in the calculated FS and the SSTREAM surface response become more pronounced. The amplitude of the SSTREAM topography perturbation is then much larger than that of the FS solution, 5 and in fact quite similar to the shape of the bedrock perturbation itself. This aspect of the SSTREAM solution was discussed above and is caused by the fact that the SSTREAM T sb amplitude is close to unity for small wavelengths (see also Figs. 6a and b). 10 As expected the comparison between the analytical FS and SSTREAM solutions shows that the SSTREAM approximation is highly accurate for long wavelengths (λ/h≫1) and high slip ratios (C≫1). The SSTREAM approximation is in these circumstances a much better approximation to the FS solutions than the SSHEET approximation. However, somewhat disappointingly, when these conditions are not fulfilled 15 the SSTREAM approximation is not just inaccurate but gives rise to some physically unrealistic results. In particular, the ratio between surface and bedrock topography is vastly overestimated. Whereas the correct ratio is close to zero for λ≪h the SSTREAM gives a ratio close to unity. The SSHEET theory, which also is inaccurate for λ≪h, fails in this respect in a more physically realistic manner by underestimating the ratio and 20 giving the correct limit of zero as λ→0. A related somewhat less than satisfying aspect of the |T sb | SSTREAM transfer amplitudes is the absence of the local maxima in the T sb amplitudes as function of wavelengths seen in the FS solution.

Summary and conclusions
The physically wrong limit of the SSTREAM T sb amplitude for λ→0 is of some concern. It has, among other things, implications for surface-to-bed inversion. Because Introduction EGU might be underestimated. There is also some danger of the spatial resolving power of an SSTREAM surface-to-bed inversion to be overestimated. Further research into these issues is needed before firm quantitative statements can be made. The only previous work dealing with the effects of small-amplitude perturbations in boundary data on solutions of the SSTREAM theory is the numerical study of Hind-5 marsh (2004). He calculates T sb transfer amplitudes and both the t r and the t p timescales for linear and non-linear media. The L1L1 approximation used by Hindmarsh (2004) is an improved version of the SSTREAM theory discussed here which includes the contribution of internal ice deformation to the velocity. For high slip ratios the L1L1 approximation is effectively equal to the SSTREAM approximation. Fig. 5 in 10 Hindmarsh (2004) calculated for n = 3 shows the same general features of transfer the amplitude T sb and the times scales t p and t r for non-linear rheology as found in the analytical solutions given here valid for n=1. In particular the |T sb |→1 limit for λ→0 is also found by Hindmarsh (2004) for non-linear media (Fig. 5c). The relative insensitivity of the t r time scale to wavelengths for high slip ratios, and the chance in phase 15 speed fromū b to (m+1)ū b with increasing wavelength is also seen (Fig. 5a and 5b in Hindmarsh, 2004).
The SSTREAM solutions are much better approximations to the FS solutions than the SSHEET solutions wheneverC≫1 and λ/h≫1. For slip ratios typical of active ice streams the SSHEET solution underestimates the relaxation time scale by several 20 order of magnitudes. The SSTREAM solution for t r is, on the other hand, almost equal to the corresponding FS solution down to wavelengths of about 10h. The SSTREAM solution gives a finite number for t r in the limit λ→0 (in dimensional units the limit is 4cη tan α/u) whereas the same limit for SSHEET theory is zero. The drawback of this is that once short wavelengths are present in an SSTREAM model they do not 25 automatically decay away as they do in an SSHEET model. If, for some reason, in a numerical model based on the SSTREAM approximation an area is being resolved with a spatial resolution not fulfilling the shallow-ice condition λ/h≫1, it is presumably advisable to eliminate all wavelengths less than about five ice thicknesses through Comparing the analytical solutions of the SSTREAM and the SSHEET approximations with the FS solutions presented here and in Gudmundsson (2003) provides a quick and an easy way of assessing the applicability of these two commonly used approximations to a particular situation. Results presented by  son (2005) on small-amplitude response for non-linear media in the FS theory, and by Hindmarsh (2004) on numerical approximations to the Stokes equation for both linear and non-linear media, suggest that this approach remains useful even when the rheology of the media is non-linear. In particular, in cases where the analytical solutions reveal significant differences between the SSTREAM or the SSHEET approximations 10 as compared to the FS solutions, it is improbable that inclusion of non-linear rheology will reduce these differences. 15 We If we furthermore require the kinematic boundary condition at the surface , which seems reasonable as we can expect the vertical velocity to scale with accumulation rate for small surface slopes. We also find using the same invariant requirement of the kine-5 matic boundary condition of the surface that the time must be scaled as

Shallow ice stream scalings
As a scale for the stresses we use which is motivated by the expectationǫ xx ∼ a/H and Glen's flow law whereǫ is the effective strain rate defined throughǫ : =(ǫ i jǫi j /2) 1/2 and τ i j are the deviatoric stress components. We are considering a situation where the vertical shear components are small compared to all other stress components. A set of scalings which reflects this situation is 15 (σ xx , σ yy , σ zz , σ xy , σ xz , σ yz ) Same scale is used for the pressure, that is p= The analysis is done in a coordinate system which is tilted forward in x direction by the angle α. The scaled momentum equations are EGU whereτ : =[z]ρg sin α. We obtain a consistent set of equation for α=O(δ). The slope α is then no longer just an arbitrary tilt angel andτ can be interpreted as a scale for the basal shear stress. From the scalings (A1) we then have [τ]=δ [σ], and it follows that the two non-zero terms on the right hand side of the system of Eq. (A4) are of order unity. Note that only terms of order δ 2 are being dropped from the momentum 5 equations.
We write the basal sliding law on the form where T b is the basal stress vector given by T b =σn−(n T · σn)n, withn being a unit normal vector to the bed pointing into the ice. The function c(x, y) is referred to as the 10 basal slipperiness. We find that components of the scaled basal stress vector (T * b ) are given by kinematic boundary condition demonstrates that to this order the sliding law is fully consistent with the basal kinematic boundary condition. Usingǫ xz =Aτ n−1 σ xz one finds that ∂ z u=O(δ 2 ). Thus, to second order the horizontal velocity components u and v are independent of z, from which it follows thatǫ zz is independent of depth as well and that w varies linearly with depth. A further conse-25 quence is that since the sliding velocity is of order unity and the deformational velocity 51 Introduction EGU u d =O(δ 2 ) the slip ratioū b /ū d =O(δ −2 ). Note that in both the field equations and all boundary conditions all first order terms are identically equal to zero. The theory is therefore correct to second order in δ.
Only collecting zeroth-order terms followed by vertical integration over depth and some simple manipulation leads to the following two coupled differential equations for 5 the horizontal velocity components u and v 10 Note that we have now gone back to the dimensional variables. The quantity η is the effective viscosity defined through τ i j =2ηǫ i j . For Glen's flow law the effective strain rate is given bẏ an expression that is correct to second order.

EGU
In some limiting cases these expressions obtain much simpler forms. In dimensionless units the steady state T uc transfer functions for l =0 is and for k=0 one finds  Fig. 2. (b) The shallow-ice-stream phase speed as a function of wavelength λ and orientation θ. As in Fig. 2a the mean surface slope is α=0.002 and the slip ratio isC=100 and the plot has been normalised with the non-dimensional surface velocityū=C+1=101 of the full-system solution. 2,2008 Shallow ice stream approximation       Fig. 6. (a) The SSTREAM amplitude ratio (|T sb |) between surface and bedrock topography (Eq. 19). Surface slope is 0.002 and the slip ratioC=99. λ is the wavelength of the sinusoidal bedrock perturbation and θ is the angle with respect to the x axis, with θ=0 and θ=90 corresponding to transverse and longitudinal bedrock undulations, respectively.  Fig. 6. (b) The FS amplitude ratio between surface and bedrock topography (|T sb |) from Eq. (75) in Gudmundsson (2003). The shape of the same transfer function for the same set of parameters based on the SSTREAM approximation is shown in Fig. 6a.  Fig. 7. (b) The steady-state amplitude ratio (|T ub |) between longitudinal surface velocity (∆u) and bedrock topography (∆b) (Eq. 75 in Gudmundsson, 2003). The shape of the same transfer function for the same set of parameters, but based on the shallow ice stream approximation, is shown in Fig. 7a.  Fig. 8. (a) The steady-state amplitude ratio (|T vb |) between transverse velocity (∆v) and bedrock topography (∆b) in the shallow ice stream approximation (Eq. 22). Surface slope is 0.002 and the slip ratio is 99.  Fig. 8. (b) The steady-state amplitude ratio (|T vb | between transverse velocity (∆v) and bedrock (∆b) ) from Eq. (75) in Gudmundsson (2003). The shape of the same transfer function for the same set of parameters, but based on the shallow ice stream approximation, is shown in Fig. 8a Fig. 11. Steady-state response of surface longitudinal (u), transverse (v), and vertical (w) velocity components to a basal slipperiness perturbation. The surface slope is 0.002 and the slip ratioC=10. The T uc and T wc amplitudes are calculated for slipperiness perturbations aligned transversely to the flow direction (θ=0). For T vc , θ=45 degrees. Of the two y axis the scale to the left is for the horizontal velocity components (T uc and T wc ), and the one to the right is the scale for T uc . The mean flow direction is from left to right. Surface slope is 3 degrees and mean basal velocity equal to mean deformational velocity (C=1). The spatial unit is one mean ice thickness (h). The Gaussian-shaped bedrock disturbance has a width of 10h and it's amplitude is 0.1h. The problem definition is symmetrical about the x axis (y=0) and any deviations in the figure from this symmetry are due to differences in the FS and the SSTREAM solutions.  Fig. 13. (b) Response in surface velocity to a Gaussian-shaped bedrock perturbation. All parameters are equal to those in Fig. 13a. The contour lines give horizontal speed and the vectors the horizontal velocities. The velocity unit is mean-deformational velocity (ū d ). The slip ratio is equal to one, and the mean surface velocity is 2ū d . The upper half of the figure is the SSTREAM solution and the lower half the corresponding FS solution.