Introduction
Cirques and stepped longitudinal profiles of trough valleys are common features of glaciated, and only glaciated, mountain relief. Therefore, it is glacial erosion that causes these morphological forms. Davis wrote as far back as 1906 “that it savors of extreme conservatism any longer to deny the efficacy of glacial erosion”. However, extreme conservatism is extremely conservative — after decades we have to demonstrate the statement again.
Modern estimates (Reference Serebryannyy and OrlovSerebryannyy and Orlov, 1982; Reference Gros'vald and GlazovskiyGros'vald and Glazovskiy, 1983) suppose that the rate of glacial erosion is sufficiently high to create considerable morphological forms. But the next question arises: why glacial erosion is not homogeneous along the glacial bed, localizes in particular zones, and thus generates cirques and stepped longitudinal profiles within trough valleys. The main object of this paper is to demonstrate the reasons for self-organization of the longitudinal glacial trough profiles in principle. So some statements are only mentioned without detail.
According to widespread opinion, heterogeneities of glacial erosion occur because of a variety of different non-glacial and ad hoc reasons: pre-glacial bed configuration, lithological differences of bedrock, and others (see Reference ColmanColman’s (1976) critical review). An alternative approach is to connect the heterogeneities of glacial erosion to heterogeneities of glacier flow. Reference Clark and LewisClark and Lewis (1951) used the rotational flow of glaciers. Reference Nye and MartinNye and Martin (1968) proposed the connection between the heterogeneities of glacial erosion and the active and passive flow of glaciers (Reference NyeNye, 1952). However, both approaches require a particular pre-glacial relief to generate rotational or active and passive glacier flow, and therefore spatial scales of the erosional forms are prescribed in advance.
A glacier bed does not only vary under the impact of glacial erosion: glacier flow, in turn, varies according to changes in the bed. Therefore, we encounter a purely glaciological self-supported process for the generation of heterogeneities of glacier flow and bed configuration. It appears adequate to connect the dynamics of an eroding glacier and eroded bed to describe the genesis process. According to synergetic ideology (Reference HakenHaken, 1978), we should investigate the dynamics of small random perturbations against the background of a homogeneous glacier flow and a homogeneous glacier bed.
The approach proposed here has been used for transverse perturbations to explain the generation of ice-sheet ice streams and associated large-scale trough valleys (Reference MazoMazo, 1987). In this case, the heterogeneities are due to the instability of the perturbations of larger scales and the stability of the perturbations of smaller scales, or due to negative or positive dissipation.
In this paper, the same approach is applied for longitudinal perturbations to investigate the formation of cirques and stepped longitudinal profiles within glacial troughs. The analysis below shows that, in contrast to the transverse perturbations, the longitudinal ones dampen out for all spatial scales: this attenuation is small for the perturbations of larger scales and is rapid for those of moderate and smaller scales. However, for the longitudinal perturbations, there exists another mechanism that creates the heterogeneities: the longitudinal perturbations propagate with non-zero velocities up the glacier bed and these velocities are different for the perturbations of different spatial scales. In other words, velocity dispersion arises. Moreover, for a certain spatial scale stationary phase (Reference WhithamWhitham, 1974) occurs. The result is that the longitudinal perturbations of this spatial scale stand out against the others.
Glacier Flow Down a Perturbed Bed
A glacier is modelled here as a layer of a very viscous linear incompressible liquid flowing down an inclined bed by gravity. The following scales are chosen: spatial scale d, time-scale d2/v, velocity scale d2g sin θ/2ν, pressure scale pgd, where d is the average thickness of the layer, θ is the average angle of the bed slope, ρ and ν are the density and the kinematic viscosity of the liquid, and g is the acceleration of gravity. All equations are cast in terms of dimensionless variables. The X-axis is in the ice-flow direction, the r-axis is normal to the surface, positive upwards, z= h(x) and z = f(x) are the equations of the bed and free surface, h(x) = f(x) - b(x) is the thickness of the layer (see Fig. 1).
Longitudinal section of a glacier.
The ice-layer flow is described by a system of the incompressibility and equilibrium equations and boundary conditions on the bed and free surface for the pressure and x- and z-components of ice flow u and w. The latter may be, in principle, found as a function of the boundaries b and f. However, it is difficult because of the complexity of the full system and, more important, unnecessary because the short-scale perturbations dampen rapidly and, therefore, may be excluded from consideration. We will find the pressure, velocity components u and w and, in addition, the basal shear stress in long-wave approximation when the spatial scales of the perturbations are supposed to be much longer than the averaged ice thickness.
An additional boundary condition on the free surface is the kinematic equation
which in the long-wave approximation (see Appendix, Equation (A8)) is reduced to the non-linear parabolic equation of the free-surface relaxation
where h = f - b and the bed is supposed to be given. This equation is a convenient tool to describe free-surface dynamics of thin layers for given beds. In particular, the relaxation equation describes propagation of free-surface waves which are shown to tend to the steady-state wave associated with the given bed wave.
The steady-state free-surface wave satisfies the stationary equation
where l (steady-state variables are indicated by underlined italics). The steady-state equation may be used to solve the inverse problem of reconstructing the bed configuration for the given free surface. However, only the long-wave perturbations may be reconstructed: in the longwave approximation, the inverse problem for short waves is incorrect.
In the case of small amplitudes (much smaller than the average ice thickness) of the bed and, hence, of free-surface perturbations, the relaxation and stationary equations are reduced. Let us represent the perturbed variables f and b as sums of mean variables and perturbations φ and ß. The relaxation equation for small perturbations is then the linear parabolic equation
The small free-surface perturbation, φ, relaxes to the small steady-state perturbation j> satisfying the stationary equation
where δ = 3 tan Θ. There exist two solutions for Equation (2) which dampen out for large x, positive or negative
Let us suppose that the bed perturbation ß is localized within a section of the x-axis, and select a solution which decreases for large x, positive and negative. The first solution decreases for large negative x while, generally, it increases for large positive x, and hence cannot be used. The second solution decreases in both the positive and negative directions and therefore can be used. To be more exact, the second solution is only non-zero up-stream of the local section where the solution decreases exponentially as x → ∞, while it is zero everywhere down-stream of the section: bed perturbations result in free-surface perturbations that are only up-stream of the bed perturbations.
Let us proceed from the perturbations jj> and β to their Fourier transforms j£' and ß'. We find from Equation (2)
where к is the wave number. Therefore, the ratio of the amplitudes of the stationary free-surface to the bed perturbations decreases as
when the wave number к increases: the longer the bed perturbations are, the less dampened will be the free-surface perturbations. More complex calculations show that the full system (not in the long-wave approximation) gives higher rates of the dampening of the short-wave perturbations than the system in the long-wave approximation: the short-wave perturbations indeed dampen out. Also, there is a phase lag between the perturbation (tangent of the phase-lag angle is k/δ): long perturbations have a smaller lag.
Eroded Bed Dynamics
In order to describe completely the dynamics of the joint system comprising both the eroding glacier and the eroded bed, an equation controlling the eroded bed changes must be added. It is the kinematic equation
where ρ and r are x- and r-components of the bed-erosion rate (see Fig. 1). The simplest assumption connecting the rate of bed erosion to the parameters of ice flow is as follows. The direction of the erosion-rate vector is supposed to be normal to the bed
where e is the magnitude of erosion rate. The latter is supposed to be proportional to the basal shear stress τb:e = Ɛτb, where Ε is a positive constant. Then the kinematic equation for erosion is
In the long-wave approximation, the equation reduces to
Erosional changes of bed configuration occur on a time-scale s which is much longer than the time-scale l for the adjustment of the ice flow. Thus, at any given instant s the bed configuration may be taken as the boundary condition for ice flow and the previous results for stationary ice flow may be used though the bed itself is being slowly eroded. It is the parameter ε that characterizes the ratio of these time-scales: s = Ɛt. So, we may suppose that Ɛ << 1.
Using the long-wave formula for the basal shear stress τb (see Appendix, Equation (A9)), and proceeding from the shorter time-scale t to the longer one s, we find the final form of the erosional kinematic equation
In the case of small amplitudes of the bed and, hence free-surface perturbations, the erosional kinematic equation is
where μ = 2sin Θ.
Propagation Of The Longitudinal Bed Waves
Two Equations (1) and (2), or (3) and (4) (the latter in case of small-perturbation amplitudes), describe jointly the dynamics of the eroded bed and adjusted ice flow. Equations (3) and (4) are reduced to the single hyperbolic equation
for which Equations (3) and (4) are characteristic. Let us proceed from the perturbation β to its Fourier transform β'. The latter varies exponentially with time: ß' = constant exp(−a(k)s), where a(k) is a wave-number-dependent complex frequency (a(k) = ω(k) + iᵧ(k), ω(k) is a proper frequency, and ᵧ(k) is a time constant. The time constant ᵧ(k) determines the growth or dampening of the perturbation, while the phase velocity c(k) = ω(k)/k determines the propagation velocity of the perturbation. When c(k) varies with k, velocity dispersion occurs. If, in addition to a velocity dispersion, the group velocity g(k) = dw(k)/dk is zero for certain wave numbers, then stationary phase occurs and the perturbations with these wave numbers stand out against the others.
From Fquation (5) rewritten in terms of Fourier transforms, we find
so the phase velocity c(k) is negative. Therefore, all longitudinal perturbations propagate up-stream. This circumstance is explained by the lag between the bed and free-surface perturbations (see previous discussion). Also, the phase velocity c(k) is different for different wave numbers k: thus the waves are dispersive. Moreover, the group velocity
is zero for wave numbers
Thus, by the end of an initial transient period, the longitudinal bed wave with a wavelength which is
Additional calculations show that glacier-sliding conditions and/or non-linearity of ice rheology and the erosion law included in the model change the quantitative estimations slightly but do not change qualitative features of bed-wave propagation.
Acknowledgements
I thank G.K.C. Clarke and K. Hutter for their suggestions for improving both the presentation and my English.
Appendix In the model considered, glacier flow is described by the equations of incompressibility and equilibrium
for the x- and z-components of ice velocity ν and w, and the pressure p, where
is the Laplacian operator. The boundary conditions are: the no-slip conditions on the bed
and the zero-stress (normal and shear) conditions on the free surface
The additional boundary condition on the free surface is the kinematic equation
Furthermore, the basal shear stress τb is
The equation of incompressibility implies that the velocity components u and w may be deduced from the stream function
and the kinematic Equation (Al) may be rewritten as the mass-conservation law
where
The problem is to find the stream function
In the case of the small-amplitude perturbations, calculation shows that short-scale perturbations dampen rapidly and may be excluded from consideration. So we will find the stream function
with the boundary conditions
The kinematic Equation (A3) remains the same. Equation (A4) for the basal shear stress τb is reduced to
Solving the equilibrium Equation (A5), together with the boundary conditions (A6), we find
, and hence we have for the kinematic Equation (A3)
For the basal shear-stress Equation (A7), we find
