Meander formation in supraglacial streams



[1] Meandering streams on the surface of glaciers are similar in planform geometry to meanders in alluvial and bedrock rivers, despite fundamental differences in the mechanisms and timescales of incision. We develop depth-averaged conservation equations for flow in such supraglacial channels with erodible boundaries and solve the linear stability problem for harmonic perturbations to an initially straight channel. Meander formation in supraglacial streams is driven by channel curvature, which enhances heat production and heat transfer to the surrounding ice at bend apexes. This leads to enhanced melting and lateral channel migration, with near constant channel width maintained by the competition of lateral erosion and broadscale ablation of the glacier surface. We find that meandering occurs for a wide but finite range of hydraulic and thermal parameters in both subcritical and supercritical flows and that meanders usually propagate downstream. Predicted meander wavelengths are in general agreement with an empirical scaling between supraglacial channel width and meander wavelength derived from glacial environments worldwide.

1 Introduction

[2] Supraglacial streams are melt channels that form in the ablation zone of glaciers and ice sheets during the summer season when surface meltwater (sometimes augmented by rainwater) thermally erodes channels into the glacier's surface. These streams form networks that drain off the glacier, most often into supraglacial lakes, moulins, or crevasses. They thus form a primary link between surface, englacial, and subglacial hydrologic systems [Fountain and Walder, 1998]. Although there is likely to be some inheritance of large channels from year to year, supraglacial hydrology re-evolves from initial channelization to large-scale organization each year as surface melting waxes and wanes. Over the glacier as a whole, albedo changes caused by such meltwater channelization help set the fine structure of glacial surface ablation, a positive feedback between channelization and melting.

[3] Supraglacial streams are mechanistically distinct from alluvial or bedrock rivers, where mechanical erosion forms the principle mode of incision and channel slope regularization that occurs over much longer timescales [e.g., Sklar and Dietrich, 2004]. Yet despite the dominance of thermal erosion, differences in substrate strength and a lack of significant suspended sediment, supraglacial streams exhibit similar qualitative morphologic features such as meandering (including cutoff loops), anastomosing channels, and propagating knickpoints [Knighton, 1981; Marston, 1983]. Understanding the morphodynamics of these streams thus informs in a broader sense the study of sinuous channels on other terrestrial planets, where potential thermally (or chemically) eroded channels are observed but the substrate characteristics, melt composition, and volumes are often poorly constrained.

[4] This study focuses on meanders because of their ubiquity in supraglacial channels (Figure 1) and utilizes techniques developed to study meandering in alluvial settings [Seminara, 2006]. We develop a model for flow in curved channels with fixed cross-sectional geometry and then perform a linear analysis of depth-integrated mass, momentum, and energy conservation equations for harmonic perturbations to flow variables and channel boundaries. This work generalizes previous modeling of Parker [1975], following a more general approach taken in recent meandering models of alluvial rivers [e.g., Blondeaux and Seminara, 1985; Camporeale et al., 2007]. We end by comparing our model predictions to a compilation of supraglacial stream data from the literature and our own field work.

Figure 1.

(a) Supraglacial stream network on the west Greenland ice sheet, from the near-infrared band of the IKONOS satellite. Image courtesy of Jason Box, Ohio State University, donated by Geoeye to Impossible Pictures UK; (b) Single wavelength meander sequence on the Mendenhall Glacier, Alaska, August 2010, person for scale; (c) Small well-developed meander bend on the Mendenhall Glacier, Alaska, August 2009.

2 Meander Formation

[5] In alluvial settings, variations in boundary shear stresses that exceed the threshold for sediment motion drive bank erosion and deposition. In supraglacial streams, there is no similar mechanism for deposition of bank material (bank re-freezing is not effective due to long durations of daily solar radiation during the melt season) and generally little sediment (ice or rock) with which to mechanically erode the channel. Meandering supraglacial streams thus exemplify a problem long recognized in bedrock channels [Leopold and Wolman, 1960]: How does channel sinuosity evolve in the absence of bank deposition?

[6] We propose that supraglacial streams migrate through the interaction of two erosional processes: thermal erosion occurring over the entire glacial surface forced by solar radiation and thermal erosion within the channel. Channelized meltwater will attain a higher mean temperature than the surrounding ice due to its lower albedo and the heat dissipated by flow, providing a mechanism to establish long-lived channels that may persist for the duration of the melt season and beyond. A balance between stream incision and large-scale surface lowering will determine the channel geometry. Observations [e.g., Marston, 1983] suggest that in many cases an apparently steady channel cross section is attained, although some channels may continue to cut down until the ice above them creeps and isolates them from the surface [Jarosch and Gudmundsson, 2012].

[7] Melting in supraglacial channels is analogous to sediment transport and mechanical erosion in alluvial settings in that it provides the mechanism by which the channel incises. Melting is, in general, a better understood process than sediment transport, and the stability of melting/solidification fronts is well known [e.g., Mullins and Sekerka, 1963]. Channel sinuosity provides a natural vehicle for streamwise variations in heat transfer as channel bends locally decrease the thickness of the boundary layers separating flowing meltwater and glacial ice, increasing lateral temperature and velocity gradients. This mechanism thus provides a positive feedback that increases frictional dissipation and heat transfer in regions of high channel curvature that leads to increased thermal erosion. Other sources of spatial variations in heat transfer are possible: for example, anisotropic shielding from solar radiation, nonuniform surface slope, small-scale variations in glacial ice structure and composition, or input/subtraction of water from the system. However, given the ubiquity of meanders in supraglacial streams, we focus on formative mechanisms that do not depend on the particulars of external forcing.

[8] Parker [1975] developed a model for supraglacial stream meanders based on a linear stability analysis of a vertically integrated heat balance, assuming potential flow in an initially straight stream channel. Harmonic perturbations to the bed of the stream were found to be unstable, with a finite range of unstable wave numbers. In particular, Parker [1975] predicts that supercritical flow (Froude number Fr>1) is a necessary condition for meander formation.

[9] Parker's approach is conceptually similar to “bar” models of alluvial meandering [Callander, 1969]: The modeled channel axis remains fixed, but it is assumed that small amplitude perturbations of the bed force eventual lateral channel migration. However, it has since been shown for alluvial rivers that flow perturbations due to curvature in the channel axis are largely responsible for the primary lateral instability, which is reenforced by bed perturbations in the form of point bars and overdeepenings [Ikeda et al., 1981; Blondeaux and Seminara, 1985; Johannesson and Parker, 1989]. We are interested in applying this more general treatment of meandering to the linear stability of supraglacial streams.

3 Model

[10] To address the planform evolution of supraglacial channels, we model both the flow within a supraglacial stream and the substrate melting that drives lateral migration of the channel bank. Because the timescale for water advection is much shorter than the timescale for melting, flow is effectively steady on bank migration timescales. Spatially variable heat transfer occurs as a result of this flow field, which then drives bank melting and migration.

[11] We do not attempt to construct a complete description of supraglacial meltwater routing and note here several important assumptions in our model. We utilize depth-averaged, linearized governing equations, studying the initial perturbations to straight channel geometry rather than finite amplitude effects. We assume constant channel width and neglect vertical structure within the flow. We neglect the time variation of solar forcing (diurnal and seasonal), and the total surface energy balance that provides a temporarily and spatially varying meltwater flux to channels. We also neglect the variations in channel slope that often occur over the length of a supraglacial stream, and hence develop a local model in which these parameters are fixed. Our work is a parameter-sensitivity test that aims to identify a physical mechanism for the onset of meander instabilities, rather than a study of broader-scale channel sinuosity development. In what follows, symbols are defined in the text, with a complete list of variables and parameter ranges in Tables 1 and 2.

Table 1. Model Variables
s,nDownstream, cross-stream coordinatesm
X,YCartesian coordinatesm
zDepth coordinatem
inline imageGlacier surface heightm
inline imageBank perturbation wave number1/m
kFlow perturbation wave number1/m
ωPerturbation frequency1/s
κStream radius of curvature1/m
NMetric coefficient
u,UDownstream velocitym/s
v,VCross-stream velocitym/s
wVertical stream velocitym/s
inline imagePerturbation amplitudes
inline imageWater speedm/s
d,DChannel depthm
h,HWater surface heightm
inline imageChannel bank, channel centerlinem
WIce thermal boundary layerm
TWater temperature°C
inline imageTurbulent diffusivity of heatm2/s
inline imageTurbulent diffusivity of momentumm2/s
aViscous dissipationW/m2
inline imageWall shear stressPa
Lateral boundary layer in streamm
ϵExpansion parameter
CfWall friction
Table 2. Model Parameters With the Range of Values Considered
inline imageChannel half width[0.1, 1]m
inline imageChannel scale depth[0.05,0.5]m
inline imageScale velocity[0.25, 2.5]m/s
inline imageScale radius of curvature1/m
τScale times
LLatent heat of fusion334kJ/kg
ρIce/water density1000kg/m3
gGravitational constant9.81m/s2
γSolar forcing constant[10,50]W/m2 K
inline imageIce thermal conductivity2W/m2 K
αExponent, boundary layer scaling0.5
μWater kinematic viscosity10−6m2/s
inline imageSpecific heat capacity4200J/kg K
inline imageTemperature difference water and air10°C
inline imageTemperature difference water and ice[0.05,0.5]°C
inline imageCoupling constant temperature model10−5,10−2s/m
inline imageCoupling constant boundary layer10−1,101W /m2K
CbProportionality between lateral and vertical melting0.25,4
ζWall roughness height0.001m
 Dimensionless parameters
C0Wall friction base state
ν0Ratio half width to radius of curvature
βRatio half width to depth
FrFroude number
EcEckert number
Π0Ratio of melting to dissipation
Π1Ratio of advection to dissipation
Π2Ratio of conduction to dissipation
Π3Ratio of solar forcing to dissipation
Π4Ratio of lateral heat transfer to dissipation

3.1 Flow in Meandering Coordinates

[12] We now develop a model for the flow field in supraglacial streams, using meandering coordinates to express the appropriate conservation equations in a translating coordinate system that tracks the channel centerline described by coordinates inline image (Figure 2). This coordinate system, in which sand n represent downstream and cross-stream positions (asterisks indicate dimensional variables), is mapped to a fixed cartesian basis (X,Y,Z) via

display math(1)

from which we obtain the metric coefficients

display math(2)

where κ(s)=dφ/dsis the curvature of the channel centerline. The angle of the centerline from the X axis is φ, and the slope of the channel axis is assumed to be small. More details about the derivation of this coordinate system and the resulting curvilinear differential operators may be found in the alluvial meander literature, for example, Seminara and Tubino [1992]. The radius of curvature scales with its typical length scale inline image as inline image.

Figure 2.

Model geometry and coordinate system definition.

3.2 Governing Equations for Flow

[13] In the meandering coordinate system, flow variables scale as

display math(3)

where (u,v,w) are components of the velocity vector v taking characteristic value inline image, and (h,D,z) are the water surface height, water depth, and height above a datum, respectively (Figure 2), that all scale with the characteristic depth inline image. inline image is the channel half width. The Reynolds equations for momentum and mass conservation are

display math(4)

where N=(1+ν0nκ)−1is the longitudinal metric coefficient of the coordinate system with inline image, inline image is the half width to depth ratio, inline image, is the Froude number, Cf is a dimensionless friction coefficient, while inline image is a dimensionless eddy viscosity. Hydrostatic pressure is incorporated into equations (4), while the Reynolds stresses and lateral bed stress have been neglected.

[14] After applying the depth averaging operation inline image, equations (4) become

display math(5)

Here U=〈u〉,V=〈v〉 are the depth-averaged velocities and H=h/Fr2, while σs,σn are the downstream and cross-stream shear stresses. Equations (5) are identical to those of Blondeaux and Seminara [1985], who similarly neglect the vertical structure of the flow. Such an approximate description of the flow is appropriate for channels wider than they are deep, a condition that is generally satisfied by supraglacial streams although we note that channel aspect ratios are generally smaller than their alluvial counterparts. We also make the common assumption [e.g., Smith and McLean, 1984] that depth averages may be decomposed for general variables Ψ,Θ as 〈ΨΘ〉∼〈Ψ〉〈Θ〉.

3.3 Heat Transfer

[15] Equations (5) are coupled to the Reynolds-averaged heat equation. We do not attempt a global energy balance which would set the total meltwater production for a given glacial surface lowering, thus, the slope does not enter explicitly into our formulation (slope sets the scale for flow variables). Instead, we use a local energy equation for which the dimensional viscous dissipation a, turbulent diffusivity inline image, and water temperature Tscale as

display math(6)

where ρ is water density and inline image is the bulk temperature difference between the stream water and the glacier ice.

[16] The Reynolds-averaged dimensionless heat equation in meandering coordinates may be written as

display math(7)

where inline imageis the Eckert number measuring the relative importance of kinetic energy and enthalpy and ΓT is a turbulent diffusivity defined in equation (6). We note that in meandering coordinates the divergence of a vector r=(rs,rn,rz) is

display math(8)

while the gradient operator is inline image. Equation (7) utilizes a mixing length model to relate the time average of fluctuating temperature and velocity field to the mean temperature gradient in the flow.

[17] We make the further assumption that mean temperature gradients track velocity gradients via the Reynolds analogy inline image where the flow speed is inline image. The empirical and dimensional constant inline imagemeasures the strength of coupling between velocity and temperature gradients. For flow past a flat plate with Prandtl number Pr equal to unity, inline image [Schlichting, 1960], but we leave it here as an adjustable parameter.

[18] After applying continuity and modeling inline image, the depth average of equation (7) is

display math(9)

where q|z=hD is the (dimensional) vertical heat transport in the water at the water-ice interface and q|z=h is the vertical heat transport in the water at the water-air interface. Boundary terms associated with turbulent advection of heat are ignored, and we have simplified the gradient operator in the mixing length model by neglecting terms associated with the meandering coordinate system. These terms drop out in the linearization of equation (9) performed in section 4, so are unnecessary for our study. Equation (9) balances the rate of heat generation by viscous dissipation at the smallest length scales in the stream with heat transport at the top and bottom boundaries, as well as advection by the mean flow.

[19] At the air-water interface, transport of both sensible and latent occurs by a combination of radiation (solar forcing) and advection by wind. We assume that

display math(10)

with inline image the temperature difference between water and air and γthe coefficient of heat transfer at the water-air interface.

[20] Heat transfer at the bottom boundary is normal to the ice-water interface and follows a kinematic Stefan condition for boundary melting

display math(11)

where ρis the density of ice (assumed equal to the density of water) and L is the latent heat of fusion.

[21] The heat flux through the glacier inline imageis modeled by

display math(12)

where inline image are the mean air and bulk glacier ice temperatures, Wis a conductive boundary layer in the ice that scales with the flow depth inline image, and inline image is thermal conductivity.

[22] Because supraglacial stream water is strongly buffered toward 0°C [Isenko et al., 2005], the water temperature is a good approximation to the ice melting temperature in equation (12). We thus neglect diurnal fluctuations in stream water temperature. In supraglacial streams, inline image will be a function of latitude and altitude, close to zero in temperate glaciers and larger in polar settings.

[23] The characteristic time for channel lowering is a melting timescale, inline image, which is far greater than the timescale for advection inline image. Thus, the problem of flow in supraglacial streams is largely decoupled from bank and basal melting. However, this balance sets the equilibrium depth of the channel and thus determines the flow field.

[24] Dissipation 〈a〉 in the stream is modeled following Parker [1975]:

display math(13)

with the near-bed shear stress terms (σs,σn) represented as

display math(14)

[25] The friction factor Cf is given by an empirical approximation for rough-walled pipe flow [Colebrook, 1939; Parker, 1975]:

display math(15)

The Reynolds number is inline image, with μ the water kinematic viscosity and ζa characteristic roughness height of the channel walls.

[26] Inserting models (11)(15) into equation (9) gives the final energy equation, balancing the divergence of heat from the channel cross-section boundary with internal frictional dissipation:

display math(16)

[27] The dimensionless numbers

display math(17)

compare dissipation with melting over time (Π0), advection of mean temperature (Π1), conduction in the ice (Π2), and solar forcing (Π3). We use the melting timescale inline imageso that Π0=1 in what follows. For a range of physical parameter values (Table 2), all the terms that appear on the right-hand side of equation (16) are of similar magnitude.

3.4 Lateral Channel Migration and Boundary Conditions

[28] A kinematic (Stefan) condition governs the lateral migration of the channel bank inline image on the melting timescale, normal to the lateral channel boundaries:

display math(18)

where inline image and inline image are the lateral heat fluxes at the ice-water interface through water and ice.

[29] Heat transfer at the bank scales with boundary layer thickness , which depends on flow Reynolds number to some power αas ∼Reα.

display math(19)

with inline image a dimensional scaling factor.

[30] Heat lost to ice is given by equation (12), but with the opposite sign because of how channel migration is defined. This implies that the thermal boundary layer in glacier ice is similar on the sides and the bottom of the stream. In dimensionless form, the lateral migration condition is

display math(20)


display math(21)

is a dimensionless ratio of melting and boundary layer heat transfer at the channel walls. We use α=0.5 for turbulent flow [Schlichting, 1960].

[31] The governing equations (5) and (16) contain four unknown flow variables and are closed with four boundary conditions. We therefore supplement equation (20) with Dirichlet conditions stating that the channel walls are impermeable to flow:

display math(22)

[32] Implicit in our model development is the assumption that a steady channel shape is maintained, which requires that large-scale lowering of the glacier surface keeps pace with lateral migration to maintain constant channel width. This assumption is grounded in observations of nearly constant width channels (Figure 2) and is similar to early alluvial meandering studies [Blondeaux and Seminara, 1985; Seminara, 2006] that separate the problem of meander formation from that of channel geometry. In our case this assumption implies that lowering of the channel relative to the glacier surface inline image[Parker, 1975] (Figure 2) is proportional to the lateral migration of channel bank inline image:

display math(23)

where Cb is a parameter of order unity. Our problem is thus completely specified, with control parameters Π01234, β, and Fr.

4 Linear Stability Analysis

[33] We now perform a linear stability analysis on governing equations (5) and (16), in a similar manner to Blondeaux and Seminara [1985], where further details may be found. We examine a channel whose centerline exhibits small-amplitude perturbations, a straight base state, and investigate the conditions for these perturbations to grow in time. We make the harmonic perturbation (noting that here and elsewhere the complex conjugate of the perturbations must be added to form a real quantity)

display math(24)

where λm is a nondimensional meander wave number scaled by the half width inline image and ω is a complex frequency that dictates the growth rate as well as propagation direction and speed of harmonic disturbances.

[34] The channel centerline is perturbed as

display math(25)

where yc, ϵ, k, and x are quantities normalized by the half width inline image and ϵ≪1 is a small parameter for the Taylor expansions involved in the linearization.

[35] For this assumption to hold, we must also have

display math(26)

We follow Blondeaux and Seminara [1985] in assuming that the flow is fully developed in the s direction, with no secondary flow. We perturb the flow variables (U,V,h,D) as an expansion in ϵ:

display math(27)

where inline image, inline image, inline image, inline image are all functions of n alone and h0 is the nondimensional unperturbed depth.

[36] The thermal boundary layers in ice surrounding the stream W are assumed to track perturbations in the flow depth D0 and channel lateral boundaries, respectively, with base states that scale with the flow depth. These are modeled by

display math(28)

Expanding Cf as a function of inline image and D in powers of ϵ, we have

display math(29)

This gives

display math(30)


display math(31)

The zeroth-order friction parameter C0 as well as the derivatives inline imageand Cf/D may be evaluated in closed form using the principal solution for w in z=wew.

4.1 Solution of the Linearized Problem

[37] Substituting all the expansions into the governing differential system (5) and (16), at inline image), we find the uniform flow solution and energy balance within the stream:

display math(32)
display math(33)

In equation (33), a nonzero right-hand side sets the average lowering rate of the stream relative to glacier surface lowering [Parker, 1975]. Background lowering of the glacier surface enters through Π2 and Π3, with Π3 measuring net solar radiation. We neglect relative motion between stream lowering and ice surface lowering.

[38] At inline image, we have

display math(34)

with χ0=βC0, and the coefficients ai(i=1,2,…,10) defined by

display math(35)

By manipulating this system, we obtain a simpler fourth-order equation in inline image

display math(36)


display math(37)
display math(38)
display math(39)
display math(40)

This is coupled to equations in the other variables. Equation (36) has the form of a forced linear oscillator, which is also characteristic of alluvial meander models [Blondeaux and Seminara, 1985].

[39] With some manipulation, the boundary conditions equations (20) and (22) at inline image become

display math(41)
display math(42)


display math(43)

We solve the differential equation (36) with boundary conditions (41) and (42) giving

display math(44)


display math(45)


display math(46)
display math(47)

Perturbation amplitudes inline image, inline image, and inline image follow from substituting equation (44) into (34).

[40] We accept only those perturbations that do not change the discharge of the stream and the overall surface slope. These are expressed at linear order through the following integral conditions:

display math(48)
display math(49)

which state that perturbations to the flow field do not affect the overall discharge and surface slope, respectively. inline image, and inline image are all odd functions of n and thus satisfy these conditions.

[41] Perturbations to the flow field will lead to harmonic meandering instabilities on the timescale of melting. The linear contribution to this balance governs the amplification of perturbations to the channel centerline, while the zeroth order contribution is lateral migration in the unperturbed state. The channel width is assumed to be fixed

display math(50)

and we also assume that the zeroth-order lateral migration is balanced by glacial surface lowering so that inline image. At linear order, however, equation (23) becomes (evaluated at the outside of the meandering bend)

display math(51)

Equation (51) provides an implicit dispersion relation between ω and λm for acceptable growth rate amplitudes (the real part of iω) and propagation velocities (the imaginary part of iω) of harmonic channel perturbations.

5 Results

[42] A typical solution to the dispersion relation equation (49) is given in Figure 3, in which we see that the plotted most unstable wave number also exhibits downstream propagation (imaginary part of iω is positive) with growth rates similar to the propagation speed (real and imaginary parts of iω are similar magnitude). Both downstream and upstream meander propagation occur in our model, although the parameter space of realistic flow variables exhibits downstream propagation. We explore the behavior of the dispersion relation by varying flow variables, channel aspect ratio, and adjustable constants in the governing equations.

Figure 3.

Solution to the dispersion relation equation (51) as a function of wave number λm for inline image m/s, inline image m, inline image m, inline image s/m, inline imageC, inline image K, Cb=0.25. Solid curve (real part of iω) shows that growth rate of perturbations has a single maximum wave number, while dashed curve (imaginary part of iω) shows downstream meander propagation.

[43] We find that for a range of realistic parameters (Table 2), channel bank perturbations are unstable with a maximum wave number, indicating a fastest-growing mode. The parameter dependencies of this most unstable wave number are shown in Figure 4, plotting the dimensional wavelength (inline image) as a function of control parameters in the problem. The channel aspect ratio β generally exerts greatest control on meander wavelength, and Figure 4 compares the effect of control parameters Fr, Π1 and Π4 with β. Parameter Π2 is generally small compared to other thermal parameters in the parameter range of interest, and Π3 does not enter the inline image solution. In Figure 4, we take inline imageC (making the role of conductive heat losses small—this is chosen to match field data, see section 6), inline image s/m, inline image W/m2 K, channel half width B=0.5 m, and all other parameters as in Table 2.

Figure 4.

Regime diagram for the parameter ranges in Table 1 with inline image m, contouring dimensional wavelength (inline image, numbered) as a function of (a) Froude number Fr versus aspect ratio β, with gray bar illustrating the range of wavelengths that are consistent with natural meanders. (b) Π14 versus aspect ratio β. Dashed lines correspond to Π4 while solid lines correspond to Π1 in the same range. The dotted lines in both panels signify a transition to stable perturbations (no meander formation). Downstream meander propagation occurs for all parameter choices in this figure.

[44] We see that meander wavelength varies by about a factor of 10 for the range of parameters considered, including a range of wavelengths (inline image) that is consistent with observed meander wavelengths (gray shading, see section 6). Increasing βgenerally decreases meander wavelength, while increasing the Froude number decreases the wavelength (Figure 4a). At low Froude number, perturbations are stable (real part of iω<0) indicating no meandering. The transition from stable to unstable perturbations has a weak dependence on aspect ratio β. The transition to meandering occurs around Fr∼0.2 for β∼1 and around Fr∼0.4 for β∼5. Π1 (increasing dissipation relative to temperature advection toward boundaries) and Π4 (coupling to lateral boundary migration) have generally weak influence on meander wavelength, but in both cases a transition to no meandering occurs as these parameters increase (Figure 4b).

[45] We also explore the effect of varying adjustable constants inline image and water temperature difference inline image (Figure 5). Meander wavelength generally increases with increasing inline image, although the dependence diminishes for large temperature differences and the hydraulic parameters considered (inline image m/s, inline image m, inline image m, inline image s/m, inline image W/m2 K in Figure 5a). At low velocities, increasing inline image can make perturbations stable and shut off meandering. Meander wavelength increases with inline image (Figure 5b, fixing inline imageC). Coupling constant inline image and ratio of lateral to vertical heat transfer Cb do not affect predicted meander wavelengths more than 5% for the range of parameters explored here (Table 2).

Figure 5.

Effect on dimensional meander wavelength inline image of varying (a) bulk temperature difference inline image and (b) coupling constant inline image for temperature field advection. Curves vary the characteristic velocity scale inline image. Dotted lines signify the transition to stable perturbations for inline image m/s in Figure 5a and for inline image m/s in Figure 5b. Parameters not listed are the same as in Figure 3.

6 Discussion

[46] In our model, the origin of meandering in supraglacial channels is an instability driven by channel curvature that enhances heat transfer (hence melting) along the outside of bends. The meander wavelength and amplitude are modulated by the glacier's surface slope which sets the total potential energy available for flow, a result suggested by some field studies [e.g., Ferguson, 1973]. Flow variables in general are not independent: A surface energy balance connects solar radiation to total available meltwater flux and thus flow speed, water temperature, and channel aspect ratio. Channel geometry does not remain static in supraglacial streams, and coevolves with variable solar forcing. We do not address these complexities but rather identify a range of flow conditions for which meandering can occur.

[47] However, even in the context of this parameter study, it is interesting to compare model results with field data. Detailed hydraulic parameters are poorly constrained in general for supraglacial streams, but there are a number of studies that provide meander wavelengths and channel widths [Leopold and Wolman, 1960; Parker, 1975; Knighton, 1981; Marston, 1983]. We have compiled these meander wavelength and channel width data and collected additional data from the Juneau Icefield and from IKONOS satellite photos (July 2007) of the West Greenland Icesheet. These data are plotted in Figure 6 and show a power law relationship between channel width and meander wavelength for supraglacial streams over three orders of magnitude in channel width (Figure 6). Such behavior is consistent with the power law channel width versus meander wavelength relationship also found for alluvial and bedrock meanders. Although the coefficients of these empirical relations appear to differ by up to an order of magnitude as the composition of the erodible substrate varies [Leopold et al., 1964], the power does not differ and remains similar to unity.

Figure 6.

Compilation of supraglacial stream channel width versus meander wavelength derived from field studies, using the Levenberg-Marquardt algorithm for the power law fit. Standard error in the intercept is 0.517 and for the exponent is 0.043. Data sources: a. Authors' measurements; b. Marston [1983]; c. Knighton [1981]; d. Leopold and Wolman [1960]; e. Parker [1975]; f. pixel counting IKONOS 1 m resolution satellite images, taken July 2007.

[48] The paucity of available data on supraglacial streams makes it premature to evaluate the significance of the data in Figure 6 in the context of other meandering systems. A direct comparison between our model and the observed supraglacial channel widths and wavelengths seen in the field is also not attempted here, as the covariance of flow parameters (the hydraulic geometry and slope variations) for supraglacial streams is not yet well constrained. However, we can find a range of parameters (Figure 4a, gray bar) that are consistent with conditions often found in natural channels and that reproduce the empirical scaling between channel width and supraglacial meander wavelength in Figure 6. We find that it is necessary to choose a low temperature difference between stream water and boundary ice to generate meandering wavelengths that match field data (inline image less than 0.05°C, Figure 5a). Because water temperature is strongly buffered toward zero in these systems, such small temperature differences may be reasonable. However, it is also possible that higher order effects on the curvature-induced redistribution of heat within the stream (e.g., secondary current not modeled here [Seminara, 2006]) may be important for the details of realistic meander wavelength prediction. We expect that a more complete treatment of variable glacier surface energy balance and of flow within the stream may better constrain the temporal evolution of meanders. However, the present model appears satisfactory in predicting the onset of lateral channel instabilities.

[49] Agreement between modeled and observed meander wavelength/width scaling allows us to constrain the mechanisms for meander formation in supraglacial streams (relative magnitude of forcing terms for model solutions that overlap with observations). The gray bar in Figure 4 shows that meanders form when flow is slightly subcritical to supercritical (Fr>∼0.4) with channel aspect ratios β∼2.5−5. Comparing the relative magnitudes of the thermal control parameters, we see that the advective terms Π14 likely dominate conductive losses through ice (Π2) when balanced against dissipation. Thus, the enhanced heat production through dissipation and increased lateral heat transfer induced by curvature are the primary thermal drivers of meandering.

[50] The present model differs from previous modeling by Parker [1975] in two important aspects. First, we find that meanders migrate downstream, consistent with our field observations and those of others [Marston, 1983]. Second, we find that meanders also occur for subcritical flow (Fr<1). Neither of these conditions were predicted by the linear stability analysis of Parker [1975], which did not solve explicitly for the stream velocity field or account for the effects of channel curvature on flow. Thus, as in alluvial meanders [Seminara, 2006], consideration of the coupling between evolving channel geometry and the stream flow field appears to be an important aspect of modeling supraglacial stream morphodynamics.

[51] The rate for meander development may be estimated from our model through the growth rate of linear perturbations. For the range of parameters listed in Table 1, we find maximum growth rates in range of 1–10 mm/h. Despite the implicit neglect of time-dependent forcing in this estimate, this is roughly consistent with measured maximum supraglacial stream incision rates of several centimeters per day [Marston, 1983] in warm temperate setting where total yearly lowering of the ablation zone may exceed 10 m (E. Hood, personal communication, 2013). Such rapid incision raises the possibility that the drainage network as well as the planform geometry of supraglacial streams may evolve significantly over the course of one melt season. While there are limited data suggesting that this does occur on multiyear timescales [Leopold and Wolman, 1960], intraseason reorganization of the supraglacial network could also occur. In the context of high volume systems such as on the Greenland ice sheet, this dynamic reorganization interacts with englacial and subglacial meltwater routing through the drainage of supraglacial lakes [Box and Ski, 2007], which may affect the large-scale dynamics of ice sheet movement [der Wal et al., 2008].

[52] Predicted depth perturbations are antisymmetric with respect to the channel centerline, deepening on the outside edges of bends and shallowing on the insides. It is the addition of background lowering of the channel from solar forcing that sets the balance between lateral and vertical incision. Channel depth below the glacier surface in our model is set by the difference between surface lowering rate and the zeroth-order stream lowering rate inline imagearising from equation (23). This difference may generally be a small negative quantity (net lowering), due to dissipation of heat in the stream and albedo differences between water and ice. Although we anticipate that in some cases (for example, warm rainwater input), significant deepening of channels below the surface may occur [Jarosch and Gudmundsson, 2012], in the present study we neglect the relative incision of channels with respect to the surface. For γΔTa in the range of ∼100–500 W/m2 as the solar forcing for stream incision, perturbed lateral migration rates are proportional to vertical incision perturbations through equation (23), and equation (33) predicts lowering at rates similar to migration speeds. This is consistent with the observation of approximately constant channel width in supraglacial streams. However, we do not rule out that variable absorbed solar radiation and a time-varying energy balance are reflected in downstream variations that do exist in channel width.

[53] Supraglacial streams are distinct from other meandering fluvial channels in several important ways. First, the instability driving lateral channel migration is thermal rather than mechanical, such that hydrodynamics in the stream are coupled to bank erosion through forced spatial variations in heat flux at the channel wall. This is fundamentally different than the coupling of hydrodynamics to erosion through sediment transport that occurs in alluvial settings, although it may be more similar to dissolution meanders in Karst settings [Ford and Williams, 2007].

[54] A distinctive feature of supraglacial streams is that the discharge is strongly diurnal, especially for streams in which rain or other stored meltwater does not constitute a large fraction of the total water mass. A supraglacial stream will experience a smoothly varying hydrograph each day, with the same period but slightly out of phase with solar forcing [Marston, 1983]. To attain a regular sinusoidal meandering planform, then, would imply that this pattern is either set early on or is set by the average daily maximum discharge. We believe that the latter case is more likely, because the timescale over which meanders adjust their planform geometry is rapid [Marston, 1983]. We also note that meander wavelengths are not generally constant over the full length of a reach (e.g., Figure 1a). The parameter study here predicts that as increasing slope increases average streamwise velocities, dominant meander wavelengths will get smaller. However, the regime diagram of Figure 4a suggests that all else constant, the velocity dependence of meander wavelength becomes negligible for high velocity flows.

[55] Supraglacial channels represent a mechanistic end member in the spectrum of natural channelization features that exhibit meandering (only thermal erosion), which also includes lava tubes and flows, alluvial rivers, and dissolution channels. A metric relating supraglacial channel morphology to the rates and mechanisms of incision would be a useful remote means of monitoring seasonal meltwater routing on ice sheets, but may be applicable in a much broader planetary context as well. On Earth, it has been suggested that sinuosity of bedrock channels varies as a function of underlying geology and may contain a climate signature [Stark et al., 2010]. Some sinuous channels on Mars record meandering lava flows, as they do on the Moon [Hauber et al., 2009], but others may record fluvial activity from a wetter period of Martian history [Malin and Edgett, 2003]. On Titan, present-day cycling of liquid methane is likely responsible for meandering channels on the icy surface [Lunine and Atreya, 2008], but the rates and mechanisms of incision are currently unknown. The work here proposes mechanistic controls on meandering for thermally eroded supraglacial streams. If it is possible to separate the dominant scales of thermally eroded meanders from those formed through mechanical erosion (for example, through the large intercept of the relationship between width and meander wavelength in alluvial rivers [Leopold and Wolman, 1960]), this can be used to better understand landscape evolution on terrestrial planets and moons in our solar system that exhibit sinuous channels.


[56] William E. Dietrich and Timothy Creyts are acknowledged for helpful discussions, while Eran Hood, NorthStar Trekking, and the Juneau Icefield Research Program provided logistical support for field work. We thank Jason Box and Geoeye to Impossible Pictures UK for the kind donation of IKONOS imagery of the west Greenland ice sheet. We thank two anonymous reviews and an Associate Editor for formal reviews, and Alice Thompson for checking the math. This research was funded by a graduate student grant from the Geological Society of America, the UC Berkeley Behrman fund, and NASA grant NNX09AN18G.