JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 92, NO. B9, PAGES 9083-9100, AUGUST 10, 1987
Glacier Surge Mechanism Based on Linked Cavity Configuration
of the Basal Water Conduit System
BARCLAY KAMB
Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena
Based on observations of the 1982-1983 surge of Variegated Glacier, Alaska, a model of the
surge mechanism is developed in terms of a transition from the normal tunnel configuration of the
basal water conduit system to a linked cavity configuration that tends to restrict the flow of water,
resulting in increased basal water pressures that cause rapid basal sliding. The linked cavity system
consists of basal cavities formed by ice-bedrock separation (cavitation), ~1 m high and ~10 m in
horizontal dimensions, widely scattered over the glacier bed, and hydraulically linked by narrow
connections where separation is minimal (separation gap •< 0.1 m). The narrow connections, called
orifices, control the water flow through the conduit system; by throttling the flow through the large
cavities, the orifices keep the water flux transmitted by the basal water system at normal levels
even though the total cavity cross-sectional area (-200 m2) is much larger than that of a tunnel
system (-10 m2). A physical model of the linked cavity system is formulated in terms of the
dimensions of the "typical" cavity and orifice and the numbers of these across the glacier width.
The model concentrates on the detailed configuration of the typical orifice and its response to basal
water pressure and basal sliding, which determines the water flux carried by the system under
given conditions. Configurations are worked out for two idealized orifice types, step orifices that
form in the lee of downglacier-facing bedrock steps, and wave orifices that form on the lee slopes
of quasisinusoidal bedrock waves and are similar to transverse "N channels." The orifice
configurations are obtained from the results of solutions of the basal-sliding-with-separation
problem for an ice mass constituting a near half-space of linear rheology, with nonlinearity
introduced by making the viscosity stress-dependent on an intuitive basis. Modification of the
orifice shapes by melting of the ice roof due to viscous heat dissipation in the flow of water
through the orifices is treated in detail under the assumption of local heat transfer, which guaran-
tees that the heating effects are not underestimated. This treatment brings to light a melting-
stability parameter E that provides a measure of the influence of viscous heating on orifice
cavitation, similar but distinct for step and wave orifices. Orifice shapes and the amounts of roof
meltback are determined by E. When E •> 1, so that the system is "viscous-heating-dominated," the
orifices are unstable against rapid growth in response to a modest increase in water pressure or in
orifice size over their steady state values. This growth instability is somewhat similar to the
j6kulhlaup-type instability of tunnels, which are likewise heating-dominated. When E <• 1, the
orifices are stable against perturbations of modest to even large size. Stabilization is promoted by
high sliding velocity v, expressed in terms of a v 4/2 and v-• dependence of E for step and wave
cavities. The relationships between basal water pressure and water flux transmitted by linked
cavity models of step and wave orifice type are calculated for an empirical relation between water
pressure and sliding velocity and for a particular, reasonable choice of system parameters. In all
cases the flux is an increasing function of the water pressure, in contrast to the inverse flux-versus-
pressure relation for tunnels. In consequence, a linked cavity system can exist stably as a system of
many interconnected conduits distributed across the glacier bed, in contrast to a tunnel system,
which must condense to one or at most a few main tunnels. The linked cavity model gives basal
water pressures much higher than the tunnel model at water fluxes •>1 m3/s if the bed roughness
features that generate the orifices have step heights or wave amplitudes less than about 0.1 m. The
calculated basal water pressure of the particular linked cavity models evaluated is about 2 to 5 bars
below ice overburden pressure for water fluxes in the range from about 2 to 20 m3/s, which
matches reasonably the observed conditions in Variegated Glacier in surge; in contrast, the
calculated water pressure for a single-tunnel model is about 14 to 17 bars below overburden over
the same flux range. The contrast in water pressures for the two types of basal conduit system
furnishes the basis for a surge mechanism involving transition from a tunnel system at low pressure
to a linked cavity system at high pressure. The parameter E is about 0.2 for the linked cavity
models evaluated, meaning that they are stable but that a modest change in system parameters
could produce instability. Unstable orifice growth results in the generation of tunnel segments,
Copyright 1987 by the American Geophysical Union.
Paper number 6B6328.
0148-0227/87/006B-632505.00
9083
9084 KAMB: GLACIER SUROE MECHANISM
which may connect up in a cooperative fashion, leading to conversion of the linked cavity system
to a tunnel system, with large decrease in water pressure and sliding velocity. This is what
probably happens in surge termination. Glaciers for which E •< 1 can go into surge, while those for
which E •> 1 cannot. Because E varies as c•3/z (where c• is surface slope), low values of E are more
probable for glaciers of low slope, and because slope correlates inversely with glacier length in
general, the model predicts a direct correlation between glacier length and probability of surging;
such a correlation is observed (Clarke et al., 1986). Because E varies inversely with the basal shear
stress x, the increase of x that takes place in the reservoir area in the buildup between surges causes
a decrease in E there, which, by reducing E below the critical value ~1, can allow surge initiation
and the start of a new surge cycle. Transition to a linked cavity system without tunnels should
occur spontaneously at low enough water flux, in agreement with observed surge initiation in
winter.
1. INTRODUCTION than in the nonsurging state [Brugman, 1986]. After surge
termination the mean water transport speed along the length of the
The fastest known glacier flow motions occur in glacier glacier (lower half) was 0.7 m/s, typical of basal water flow
surges. There has been much theorizing as to the cause(s) of the speeds found in nonsurge-type glaciers; during surge, in contrast,
fast flow (summarized by Paterson [1981, p. 288]), but few if any the mean water transport speed was only 0.025 m/s.
firm conclusions have emerged because observations of the 5. In the dye-tracing experiment during surge the dye was
processes in action, which could guide physical reasoning, have dispersed across the width of the glacier, appearing in all outflow
been inadequate. Studies of the 1982-1983 surge of Variegated streams, whereas after surge termination the dye appeared in one
Glacier, Alaska, provide a new, enlarged body of observations stream only.
[Kamb et al., 1985]. I present here a physical model of the surge 6. The outflow stream water during the surge was extremely
mechanism developed on the basis of these observations. A brief turbid (suspended sediment content at concentration ~100 kg/m3
sketch of the model has been given by Kamb et al. [1985, p. 478]. for particle sizes _<10 [tm), much more turbid than after surge or in
While the treatment is based on a "hard bed" model of the surge normal, nonsurging glaciers (sediment content ~1-10 kg/m3)
mechanism, it seems likely, as explained in section 10, that a
number of the important results are also applicable at least
qualitatively to "soft bed" models, in which sliding is over
deformable basal till. The discussion concentrates on the
mechanism of surging in spring and summer when relatively large
amounts of water are available to the basal water conduit system.
[Brugman, 1986, p. 88].
3. FORMULATION OF A MODEL OF THE SURGE MECHANISM
The foregoing observations throw a sharp focus on what
needs to be explained by a physical model of the surge mecha-
The surge mechanism in wintertime can be considered by nism. The high sliding speeds are explained by the high basal
extending the concepts developed here to conditions of low water
flow; this will be done in a subsequent paper.
2. OBSERVATIONAL BASIS
The following observations from the surge of Variegated
Glacier [Kamb et al., 1985; Raymond, this issue] form the direct
basis of the surge model:
1. The fast flow motion during the surge is due to rapid basal
sliding.
2. During surge, the pressure of water in the basal conduit
system is high, within 2-5 bars of the ice overburden pressure, and
occasionally reaching overburden; in the nonsurging state it is
distinctly lower, generally 4-16 bars below overburden, but with
occasional peaks to higher levels. Peaks in pressure, particularly
those in which the water pressure rises to near overburden,
correspond to peaks in sliding motion, both in surge and out.
These facts are taken as indication that the direct cause of the high
sliding speed in surge is high basal water pressure.
3. Major slowdowns in surge motion, and particularly surge
termination, are accompanied by large flood peaks in the terminus
outflow streams and by a drop of the glacier surface by 0.1-0.7 m.
This, in conjunction with observations of uplift followed by drop
of the glacier surface in minisurges [Kamb and Engelhardt,
1987], is interpreted as an indication that the high sliding speeds
and high basal water pressures in surge and in minisurges are
coupled with extensive basal cavitation, as expected theoretically
[Lliboutry, 1968; Kamb, 1970, p. 720; Iken, 1981; Fowler, 1987].
4. The flow of water through the basal water conduit system,
as indicated by dye-tracing experiments, is much slower in surge
water pressures. A detailed explanation requires a detailed model
of the relation between basal water pressure and sliding speed.
Such a model can be developed within the framework of the basal
sliding mechanism discussed here (or see Fowler [1986]), but
because, as will be argued, the detailed relationship between
water pressure and sliding speed is not the essential element of the
model needed to explain surging, I will here pass over these
details and instead assume a simple empirical relation between
water pressure and sliding.
The essential ingredient of the surge model is what causes the
high basal water pressures in surge. How is the high basal water
pressure maintained and indeed enhanced in spring and early
summer, when, according to the standard model of the basal water
conduit system [Rtthlisberger, 1972], the pressure should drop as
an increasing flux of water is carried by the system? Since high
basal water pressure and high basal sliding promote basal
cavitation, opening up holes (cavities) through which water could
move at the base of the glacier, and thus increasing the hydraulic
conductivity of the basal water conduit system, why does the
water not drain out from under the glacier more rapidly than in
nonsurge and thereby reduce the water pressure to subnormal
values? Why, on the contrary, does the glacier in surge show an
unusually high "retentivity" for water, as shown by the abnor-
mally low water transport speed revealed by dye tracing? These
questions go to the heart of what is in my view the essential
physical difference between the surging and nonsurging states of
the glacier. The surge model concentrates on explaining this
difference and is therefore in the first instance a model of the
basal water conduit system in surge.
The conduit system that dominates the transport of water in
KAMB: GLACIER SURGE MECHANISM 9085
the nonsurging state is a basal tunnel system of the kind discussed
in theoretical terms by R(Sthlisberger [1972], Weertman [1972],
Nye [1976], Spring and Hutter [1981, 1982], and Lliboutry
[1983]. It consists of one or two main tunnels, of order 1 or a few
meters in diameter, running along the length of the glacier at the
bed, usually near the center or deepest part, and probably fed by
smaller side tunnels heading in glacier moulins.
The water conduit system in the surging state is very
different. This is shown by the dye tracer experiments in terms of
the slow transport speed of water through the system and the high
dispersion of the injected dye pulse [Brugman, 1986]. The
system cannot consist of the normal tunnels of the nonsurging
state with addition of conduits formed by basal cavitation under
the high basal water pressure and rapid basal sliding because in
this case the water transport speed would remain high and the
total water transport (flux) would be increased so that, as noted
running transverse to the ice flow would develop a long,
connected cavity, but the transverse direction of hydraulic
communication in it would not aid longitudinal transport of
water.) When the basal water pressure becomes high enough and
the sliding velocity rapid enough, cavitation in the bed areas
intervening between the large cavities develops sufficiently to
provide hydraulic connections between the cavities, but the
connections are small features, much smaller than the cavities
they connect. It is this system of hydraulically linked cavities that
I here consider in a model of the surge mechanism. From the
evidence previously discussed, it appears that most of the pressure
drop or potential drop in the water flow through the linked cavity
system occurs in the narrow connections, or orifices, as I will call
them, and as a consequence, these orifices throttle and control the
flow. On the other hand, for a parcel of water traveling through
the system, most of its time is spent moving slowly through the
above, the high water pressure could not be maintained, at least large cavities, so that the overall transit time is tied to the 200 m2
without an abnormally large throughput of water, which is not
observed. It follows that the normal tunnel system must not be
present.
In order to transport about 5 m3/s of water at an average
longitudinal speed of 0.025 m/s, the conduit system must have a
total cross-sectional area of about 200 m2 in the transverse plane
of the glacier. (An estimated 5 m¾s is the average flux at the time
of the tracer experiment, taking into account the distributed input
of meltwater from the tracer injection point to the terminus, where
the discharge was 7 m3/s.) If the conduits are basal cavities, their
average height, areally averaged across the 1-km width of the
glacier including areas of ice-bed contact (where the height is 0),
is 0.2 m. This is compatible with the average height 0.1 m
inferred from the drop in elevation of the ice surface on surge
termination and also with the excess amounts of water released
from the glacier at surge termination and during the 4 days
thereafter, which correspond to a drop in average cavity height of
0.1 and 0.3 m, respectively. (These are upper limits because some
of the surface drop may have been due to ice strain and some of
the released water may have been stored in intraglacial porosity;
see Kamb et al. [1985, p. 477].) Basal cavities of height ~1-2 m,
distributed widely over the glacier bed and occupying ~ 10-20% of
the bed area, would provide passageways for water flow of the
dimensions needed and would involve the widespread contact
between basal water and the ice-bed interface that could account
for the extremely high turbidity of the Outflow water in surge
(section 2, observation 6). The pattern of basal cavitation
visualized seems reasonable in relation to cavities that have been
actually observed under glaciers [Carol, 1947; Kamb and La
Chapelle, 1964; Vivian and Bocquet, 1973] or that can be inferred
from detailed observations of the abrasion markings on former
glacier beds.
If the water conduit system in surge consisted of an openly
interconnected network of basal cavities of the dimensions
suggested, the water transport speed through it would be ~ 1 m/s,
as it is through normal tunnel systems, the lateral dimensions of
tunnels being of the same order. This is in strong contradiction
with the observed transport speed, 0.025 m/s. It follows that the
water flow through the conduits of 200 m2._ cross-sectional area
must be throttled in some way. There is a natural reason why this
should happen. In the sliding of ice over an irregularly un-
dulatory bed, the distribution of normal stress across the
ice-bedrock interface, which controls ice-bed separation, is such
that the large cavities that form tend to be isolated from one
another, so that the water in them tends not to communicate
hydraulically. (A long steplike or wavelike roughness feature
total cross-sectional area of the cavities. The water flow through
the orifices is probably fast and turbulent, as will be seen later,
and the water from each orifice probably emerges as a jet into the
cavity downstream, which helps to explain how the water is able
to pick up and carry a large amount of fine sediment in
suspension.
Once the basic topology of the water conduit system in surge
is ascertained, it becomes possible to formulate a physical model
of it in sufficient detail to permit its hydraulic properties to be
determined, analogously to what has been done for the normal
tunnel system by Riithlisberger [1972]. The results provide a
basis for deciding whether the underlying picture of the surge
mechanism is satisfactory and for identifying the physical
conditions that distinguish the surging and nonsurging states of
glacier motion, from which one can reason about what causes a
glacier to be in the nonsurging or surging state.
4. THE LINKED CAVITY MODEL
The pattern of basal cavitation visualized in the linked cavity
model is shown schematically in Figures 1 and 2. Figure 2 shows
schematically the longitudinal cross-sectional shapes of the
leeside ice-bed separation cavities whose plan view shapes are
shown in Figure 1. The hachured lines in Figure 1 represent
upstream cavity boundaries where the ice separates from bedrock,
often at the edge of a topographic step or sharp break in slope of
the bedrock surface, as shown in Figure 2; the nonhachured lines
in Figure 1 are where the ice recontacts the bed downstream. The
cavitation pattern in Figure 1 is rather similar to natural examples
of basal cavitation mapped by Walder and Hallet [ 1979, Figure 7]
and by Hallet and Anderson [1980, p. 174], except for the
presence of solution-etched flow channels in the limestone
bedrock of these natural examples. Under high sliding velocity,
the cavities probably elongate greatly downglacier, but the basic
topology of the hydraulic linkage of cavities remains similar.
In Figure 1, the larger areas of the ice-bedrock separation
represent the cavities of the linked cavity model, and the narrow
connections between these are the separation-gap orifices. In
cross section these features appear as shown in Figure 2:
Figure 2a is a section through two separation cavities in
succession along the flow line, while Figure 2b is a section
through two orifices, somewhat exaggerated in size for clarity.
The pattern of water flow through the linked cavity system is
shown by the small arrows in Figure 1. Because of the geometry
of cavitation, the flow water tends to be in the lateral direction,
particularly in the orifices. This lateral flow is reflected in the
9086 KAMB: GLACIER SURGE MECHANISM
/ce
wa/er
flow
CAVITY
.........
B'
'i:i:i:!:!:i
. .......
......
i:
.....
.. .
(O) actual I•nked-cawty pattern (schemahc)
i:i ORIFICE ii::'
(b) •deal•zed hnked-cawt¾ model
:.:-:-: iiiiiiiiiili
::::::::::::
he,cjht of .... ty ce,hng = gc i!iiiiiiiiii
:::::2::::
::::::::::::
Fig. 3. Local detail and dimensions in the linked cavity system as seen in
map view, with conventions of Figure 1. (a) A schematic representation
of a realistic linkage pattern between two cavities. (b) The idealized geo-
metry of the linkage assumed in the linked cavity model. The length,
breadth, and height parameters L, l, and g are discussed in the text.
l is the average lateral dimension (breadth) of the orifice in
the direction transverse to water flow. This is always in the
/ .:i•• direction of basal sliding and represents the length of the
" ' • 10 m separation gap between ice and bedrock, which forms the orifice.
to is the average dimension (breadth) of the cavity in the
Fig. 1. Conception of the linked cavity basal water conduit system, in direction transverse to water flow. In the cavitation pattern of
map view, portraying schematically a small area of the glacier bed, of Figure 1 this dimension is generally parallel to basal sliding, but
lateral dimensions ~20 m. Areas of ice contact with the bed are shaded, in downglacier-elongated cavities it tends to be transverse to
areas of ice-bed separation (cavitation) are blank. Vertical cross sections sliding.
along lines AA' and BB' are shown in Figure 2. The large arrow indicates
the direction of basal sliding. Ice separates from the bed along the Lo is the length of the orifice in the direction parallel to water
hachured lines and recontacts the bed along the plain lines. One large flow. Because of the curving, sinuous pattern of the ice-bedrock
cavity and two small "orifices" in the linked cavity pattern are identified. separation lines and recontacting lines (Figure 1), the dimension
Directions of water flow through the system are shown with small arrows. Lo is not sharply defined, as Figure 3a suggests, but in the model
Approximate scale is indicated by the 10-m bar.
wide lateral dispersion of dye in the tracer experiment during
surge (section 2, observation 5).
In developing a quantitative model of the linked cavity system
I will consider the system in terms of a typical cavity and a typical
linking orifice, illustrated in plan view in Figure 3. The dimen-
sions associated with the cavity and orifice, marked in Figure 3,
are as follows:
:::::::::::::::::::::::: C ::::::::::::::::::::::::::::::::: ..... . ........ '"""/C :::::::::::::::::::::::::::
~ lorn
Fig. 2. Vertical cross sections through the schematic linked cavity system
of Figure 1, along lines AA' and BB'. Heavy shading indicates bedrock,
light shading ice. The arrows show the direction of basal sliding. The
blank areas are water-filled volumes formed by ice-bed separation
(cavitation) in the sliding process. Section AA' shows two large
separation cavities, while BB' shows two separation gap orifices, whose
gap height is exaggerated for visibility in the drawing. Approximate scale
is indicated by the 10-m bar.
the distinction between cavity and orifice is sharpened by
idealizing their geometry in the way shown in Figure 3b.
Lo is the dimension of the cavity in the direction parallel to
water flow, or, more precisely, the distance between successive
orifices along the water-flow path.
A = Lo/Lo is the "head gradient concentration factor"
(section 5).
go is the average height of the cavity.
g is the local height (measured perpendicular to the bedrock
surface) of the separation-gap orifice. It is a function of position
across the orifice, from the point of ice-bedrock separation to the
point of recontact.
No is the number of independent orifices in a transverse
section across the glacier. The average lateral spacing between
independent orifices is W/No, where W is the glacier width. By
independent orifices I mean orifices that are not in succession
along water flow paths, so that their water fluxes add to make the
total water flux carried by the system.
An actual linked cavity system is an ensemble of cavities and
orifices of different shapes and sizes, resulting from cavitation by
basal sliding over the diverse and sundry roughness features of the
bed, under a spatial distribution of basal shear stress, water
pressure, and ice overburden pressure. The complexity of the
actual system is suppressed in the model developed here by
replacing the spectrum of cavity and orifice dimensions with a
single set of "typical" dimensions as indicated above and by
considering how these dimensions are controlled by a single set of
values of basal shear stress '•, water pressure Pw, and ice
overburden pressure PI, when basal sliding takes place over
KAMB: GLACIER SURGE MECHANISM 9087
roughness features of a single type, with prescribed dimensions. equivalent to the statement that all drop in hydraulic head is taken
This idealization, which is as great a simplification of the system across the orifices; hence the head gradient in the orifices is the
as can be made without losing what I regard as its essential average gradient multiplied by LdLo = A (section 4). The orifice
physical characteristics, is chosen here as the first approximation gaps are assumed to be thin compared to their breadth (g << l), as
to the behavior of the complicated natural system. It is made in seems appropriate for narrow ice-bedrock separation gaps that are
the same spirit as the analysis of basal sliding over a bed with a only marginally open. For simplicity, the typical orifice is taken
single type of roughness feature [e.g., Weertman, 1957]). to have a cross-sectional shape that is constant along its length Lo.
Because the flow of water through the linked cavity system is The flow of water through the orifice is then determined locally
controlled by the orifices, as discussed in section 3, and because by the local gap height g(x), which depends on a spatial coordi-
the orifices are the ice-bed separation gaps that are most sensitive nate x across the breadth l of the orifice. Since the flow is
to the conditions controlling separation, the hydraulic behavior of turbulent (as will be shown), it can be obtained from the Manning
the system is much more sensitive to the detailed geometry of the formula [RSthlisberger, 1972, equation (9)] by putting the
orifices than it is to the cavities. The cavities doubtless vary hydraulic radius equal to g(x)/2:
somewhat as physical conditions change, but the variation of the
orifices has a much larger effect on the behavior of the system. (_•)z (__•) •
Consequently, in analyzing the linked cavity model I will simply u-w (x)- M -1 • • (1)
assign reasonable values to lG, L•, and g• and will concentrate the
effort on how the orifice dimensions l and g are determined by the Here •w is the mean water flow velocity (averaged across the gap
sliding process and by water pressure and flow. This is another height g), M is the Manning roughness, and etA/co is the local
simplification that can reasonably be made in a first treatment of hydraulic gradient in the orifice as discussed above. The total
the system.
There are three distinct physical processes that, acting
together, determine the hydraulic behavior of the linked cavity
model: (1) for given roughness characteristics of the bed, given
Pw and Pi and given sliding velocity v, there is a particular orifice
geometry determined by the cavitation process in basal sliding;
(2) for given orifice and cavity geometry and given hydraulic
gradient, there will be a certain flow of water through the linked
cavity system; and (3) the water flow will result in generation of
heat by viscous dissipation, which will cause enlargement of the
flux of water Qw carried by the linked cavity system is obtained
by multiplying (1) by the local gap width g(x), integrating over
the breadth l of the orifice, and summing the contributions from
the No independent orifices (section 4):
Qw 22/3 M (•
5
[g(x)] 5 d• (2)
The local rate of heat generation by viscous heating is the
orifices by melting of the ice roof, resulting in a modification of local water flux •wg times the potential gradient pwgrctA/co
the results of process 1 and a consequent increase in the flow (where gr is gravity and Pw is the density of water). Expressed in
given by process 2. Since the effect of heat dissipation is crucial
in the functioning of a tunnel system and since for a given total
water flux down a given hydraulic gradient the dissipation of heat
will be the same in a tunnel system and a linked cavity system, it
is essential to take it into account in the linked cavity model.
Although there is a considerable parallelism between the
treatment of the linked cavity model here and the treatment of the
tunnel model by R6thlisberger [1972], the two models differ
terms of an equivalent volume rate of melting of ice by dividing
by piH, where H is the latent heat of melting and Pi is the ice
density, the local heat generation rate, per unit area of the orifice
roof, is thus
(etA/co) 3/2 5/3
th - 22/3 DM g (3)
substantially because the water flow and ice flow geometries are where D = piH/pwg• is a constant with dimensions of length (D =
very different and because the tunnel model lacks a component 31 km).
process of type 1 above. The linked cavity conduits have an The rate of melting caused by the heat generation in (3) is
essential feature of the "N channels" introduced by Nye [1973] governed by heat transfer from the water to the ice roofs of the
and considered by Weertman [1972, p. 306], namely, that they are orifices and cavities. A fully detailed treatment based on the
tied to specific topographic features of the bed. They differ in this principles of heat transfer, analogous to that carried out for tunnel
respect fundamentally from tunnels ("R channels" of Weertman
systems by Spring and Hutter [1981, 1982], is rather complicated
[1972]), which, in the model concept of R•thlisberger [1972], are and will here be avoided by the simple assumption that the heat
not tied to any topographic features of the bed. The "N channels" generated is transferred locally to the ice roof, as was assumed in
were specifically assumed to follow bedrock channels incised by the original treatments of tunnel systems [R•thlisberger, 1972;
erosion into the bed, whereas the linked cavity conduits consid- Nye, 1976]. This assumption guarantees that the effects of heat
ered here are tied to bedrock roughness features of diverse kinds. generation on the linked cavity model will not be underestimated,
because a heat transfer distributed over the cavity roofs as well as
5. WATER FLOW AND VISCOUS HEATING the orifices would reduce the melting in the orifices and thus
reduce its effect on the hydraulic behavior of the linked cavity
Given the geometry of the linked cavity model as specified in
system. Under this assumption and neglecting the effect of the
section 4, the flow of water through the system can be calculated
pressure dependence of the melting point [R6thlisberger, 1972,
as follows. Call etw the overall longitudinal hydraulic gradient p. 179], which averages to zero if etw = et, equation (3) gives the
(slope of the hydraulic grade line) along the length of the glacier. meltback rate of the orifice roof.
In the evaluation here, etw is taken to be equal to et, the surface
slope of the glacier, but this is not a necessary assumption. If the 6. BASAL CAVITATION
linked cavity system has an average tortuosity co, the average
hydraulic gradient along the water flow paths is then ct/co. The The sizes and shapes of basal cavities and separation gap
model condition that the orifices control or throttle the flow is orifices of the linked cavity model (Figure 2) are determined by