Convection as a creep phenomenon

Figure 1 shows the idealized creep behavior of ice deduced from constant stress and constant strain-rate creep tests for polycrystalline and single-crystal ice specimens (Reference Griggs and ColesGriggs and Coles, 1954; Reference GlenGlen, 1955; Reference Higashi, Higashi, Koinuma and MaeHigashi and others, 1964,1965,1968; Reference Tabor and WalkerTabor and Walker, 1970; Reference Hawkes and MellorHawkes and Mellor, 1972; Ramseier, unpublished). Elastic deformation occurs during Stage I, decelerating creep deformation occurs during Stage II, constant creep deformation controlled by hard glide occurs during Stage III, accelerating creep deformation occurs during Stage IV, and constant creep deformation controlled by easy glide occurs during Stage V. Hard glide is slip on non-basal crystallographic planes (probably prismatic planes, which are parallel to the open axis of an ice crystal). Easy glide is slip on basal crystallographic planes (basal planes are normal to the optic axis of an ice crystal). Decelerating, constant, and accelerating creep are sometimes called primary (transient), secondary (steady-state), and tertiary creep. Decelerating creep (Reference MottMott, 1953) and constant creep (Reference Weertman, Whalley, Whalley and GoldWeertman, 1973) can be explained by dislocation climb theories. Accelerating creep is related to re-crystallization in polycrystalline ice, during which hard glide resulting from a randomly oriented crystal fabric is replaced by easy glide resulting from a crystal fabric with a strong preferred orientation. Accelerating creep in ice single crystals is related to the upper yield stress phenomenon (Reference Weertman, Whalley, Whalley and GoldWeertman, 1973).

The classical empirical expression of the creep curve considers total strain ϵ to be the sum of an elastic strain ϵ_e and a visco-plastic strain ϵ_v:

Fig. 1.

Idealized creep curves in ice. Shown are the effects of increasing temperature T, time t, strain ϵ, strain rate and stress σ, far (a) creep in polycrystalline ice under a constant stress, (b) creep in polycrystalline ice under a constant strain-rate, (c) creep in single-crystal ice under a constant stress, and (d) creep in single-crystal ice under a constant strain-rate. Stage I elastic deformation. Stage II is transient creep deformation. Stage III is steady-state creep deformation controlled by hard glide. Stage IV is creep deformation during recrystallization and during transition from hard glide to easy glide. Stage V is steady- state creep deformation controlled by easy glide. Solid lines are creep curves. Dotted lines separate creep stages.

where

is a temporary strain-rate, is a steady-state strain-rate, t is time, and m is a constant determined by the relative contributions of easy glide and bard glide to creep. In this paper, temporary creep means time-dependent creep rates and steady-state creep means time- independent creep rates. The terms “primary”, “secondary”, and “tertiary” creep are avoided because they imply a time sequence of creep that is only observed in polycrystalline ice having an initially random fabric. Polycrystalline ice having an initial single-maximum fabric or single-crystal ice both might begin with “tertiary” creep which becomes “secondary” creep, and “primary” creep never occurs during the creep experiment (see Fig. 1c).

The total strain-rate given by Equation (3) is

where

because and are constants. If inertial effects are important (which they might be in ice falls, calving ice, or surges) then perhaps the total differential form of Equation (4) is necessary. Such effects will be ignored in this paper.

Assume that thermally activated power-law dislocation mechanisms control both temporary and steady-state creep in ice. In this case

where

is a thermally activated strain-rate, σ_o is a visco-plastic stress, and n is a visco-plastic parameter, all of which are constant at a given stress and temperature provided that recrystallization does not occur. Q is the activation energy of creep, T is absolute temperature R is the ideal gas constant, and is the creep rate appropriate to ice having a given fabric, texture, and purity. Substituting Equation (5) into Equation (4) gives:

where single prime terms refer to time-dependent creep and double prime terms refer to time-independent creep.

For decelerating creep rates, o < m < 1 and for accelerating creep rates 1 < m < ∞ where decelerating creep rates are observed in Stage II creep for polycrystalline ice having a random fabric and single-crystal ice oriented for hard glide, and accelerating creep rates are observed in Stage IV creep for polycrystalline ice during recrystallization to an oriented fabric and single-crystal ice oriented for easy glide. For perfectly viscous flow n = 1 and for perfectly plastic flow n = ∞, where η = σ_o|

is the fluid viscosity and E = σ₀|ϵ_eis the elastic modulus. Any creep curve can he represented by Equation (6) provided that appropriate values of σ_o’, σ_o,” , Q’, Q” m and n are chosen.

Thermal convection in polar ice sheets involves the creep of polycrystalline ice. Convection flow driven by the buoyancy stress probably initiates as plumes rising vertically from the base of the ice sheet, where either a random crystal fabric will prevail owing to continuous recrystallization or an oriented fabric will prevail owing to basal shear. In either case, convection flow normal to the bed will initially be controlled by hard glide (except along the sides of ice streams where shear should generate an ice fabric favoring easy glide normal to the bed). Reference GlenGlen (1955) was the first to make a comprehensive study of creep in polycrystalline ice having a random fabric. His results and subsequent work, as reviewed by Reference BuddBudd (1969), Reference Weertman, Whalley, Whalley and GoldWeertman (1973), and Reference GowGlen (in press), show that m =⅓, n = 3, and Q’ = 3Q” are appropriate for the creep rates common to glacial flow. Using these values and the results of Reference RigsbyRigsby (1958), Equation (6) reduces to:

Where T _M is the melting temperature at hydrostatic pressure P,

and K is constant given by the expression (Reference WeertmanWeertman, 1970,1973)

Stage II strain is viscous because the transient creep term B _tdominates when t ≈ 0, and Stage III strain is visco-plastic because the steady-state term B _s dominates when t ≫ 0.

Equations (6) and (7) follow from Equation (3), which predicts a steady-state contribution to creep at all strains, as illustrated in Figure 2 for polycrystalline ice with m = ⅓ and n = 3. Dislocation theory predicts the

transient creep relationship (Reference MottMott, 1953) and the steady-state creep relationship (Reference Weertman, Whalley, Whalley and GoldWeertman, 1973), but at present there is no theory which predicts a steady-state creep component at all strains. According to present theories of steady-state creep, such behavior would require a nearly instantaneous generation of dislocations when the stress is applied (Reference Weertman, Whalley, Whalley and GoldWeertman, 1973). Experimentally, Figure 1 shows that steady-state creep begins sooner as stress and temperature increase, and that stress relaxation occurs sooner in easy glide orientations than in hard glide orientations. If stress is applied to polycrystalline ice having a random fabric, therefore, crystals oriented for easy glide will deform readily and soon reach steady-state creep. Crystals oriented for hard glide may never reach steady-state creep under glacial shear stresses. Hence, steady-state creep appears first in the crystal best oriented for easy glide and last in the crystal best oriented for hard glide, with a whole spectrum in between. The net effect in randomly oriented poly-crystalline ice is a steady-state creep component at all strains.

Fig. 2.

a. Components of creep in randomly oriented polycrystalline ice. Shown using an arbitrary scale are the elastic strain ϵ_e transient strain ϵ_t, and steady-state strain ϵ_s components summed to give the total strain at a given time t prior to recrystallization.

Equations (6) and (7) are not the only expressions capable of representing the creep curves in Figure 1. Reference Weertman, Whalley, Whalley and GoldWeertman (1973) suggests alternative equations which do not require a steady-state creep component at small strains. However, a significant fraction of ice crystals near the bed of a polar ice sheet will have high dislocation densities due to advection strain, so that a steady-state component will exist in these grains when convection strain begins. Therefore, Equations (6) and (7) should satisfactorily represent all strains related to convection in polar ice sheets. Figure 2 is drawn using m = ⅓ and n = 3 in Equation (3), where

= 2.7 arbitrarily taken, and shows the expected creep behavior of thermal convection flow in polar ice sheets for polycrystalline ice having a random fabric. Figure 2 can be represented by Equation (7).

Transient creep and the initiation of convection flow

Thermal convection in polar ice sheets should begin as linear viscous flow, according to Equation (7). Hence, the classical theory for the initiation of thermal convection in a horizontal fluid layer heated from below can be applied (Reference StruttStrutt, 1916). This theory assumes that heat transfer via conduction dominates so that convective heat transfer can be treated as a small perturbation of conduction heat transfer. While adequate for transient creep during the initiation of convection, this assumption breaks down when convective flow stabilizes because heat transport via convection dominates and because non-linear visco-plastic flow probably dominates during steady-state creep. Reference WeertmanWeertman (1967) overcame these difficulties by developing a block model for thermal convection in a crystalline solid heated from below. He developed his model specifically for the Earth’s mantle but it can be applied to polar ice sheets with relatively minor alterations, as shown in Figure 3.

Fig. 3.

A block model of connection in a crystalline solid. Shown are the variation of horizontal velocity u in the vertical direction z (left) and convection flow lines (right) for a connecting layer having a semi-rigid upper boundary and a rigid lower boundary (top), a semi-free upper boundary and a free lower boundary (middle), and the block model approximating semi-free upper and lower boundaries (bottom). The distance of the density inversion (dashed line) below the upper surface determines the extent to which the density inversion is a free, surface. The degree of uncoupling between the lower surface and its bed determines the extent to which this interface is a free surface. Convection flow creates regions where tension, compression, and shear dominate. These regions are designated by letters, T, C, and S, respectively, and blocks I through 6 identify the regions. Active convection flow occurs below the density inversion (sinusoidal variation of u with z), and passive convection flow occurs above the density inversion (exponential variation of u with z). This figure is modified from Reference WeertmanWeertman (1967 figs 1, 2 a and 3).

Active thermal convective flow in polar ice sheets occurs below the density inversion since the buoyancy stress exists only in this region. Passive thermal convection flow occurs above the density inversion in polar ice sheets since this region is carried by active convective flow below the density inversion. Hence, Figure 3 shows a sinusoidal variation of the horizontal component u of convection velocity with vertical distance z through the ice sheet below the density inversion and an exponential decrease of u above the density inversion. The sinusoidal variation is weighted in favor of the increasing buoyancy stress and temperature toward the bed of the ice sheet, where u is zero at a frozen ice interface and u is maximized at a thick ice-water interface. These extremes and their influence on the pattern of convection flow are illustrated in Figure 3a and b, respectively. Cellular convection generates compression at the base of ascending flow and the top of descending flow, tension at the top of ascending flow and the bottom of descending flow, and shear between ascending and descending flow. These regions arc denoted by the letters C, T, and S. respectively, in Figure 3. Note that these stresses do not distort the convecting layer in Figure 3a where the base of the ice sheet is a rigid boundary, but do distort the convecting layer in Figure 3b where the base of the ice sheet is a free boundary. Figure 3c shows conditions when the ice—rock interface is partly uncoupled by an intervening water layer, and is therefore intermediate between Figure 3a and b. In this case, basal sliding makes longitudinal advective flow important near the bed and shear advective flow important further up (Reference LliboutryLliboutry, 1966). Longitudinal flow and shear flow develop multiple-maximum and single-maximum ice fabrics, respectively. Hence, u is controlled by the warmer temperature toward the bed and by the preferred fabric toward the density inversion. Figure 3c shows these effects on u as balanced for simplicity. Figure 3c also approximates the smoothly distorted convection layer of Figure 3b with a series of blocks displaced with respect to each other.

Let T _i be the temperature at the density inversion and be the temperature at the bed. If ρ is the mean density of the convection layer, ρ-δ ρ is the density of blocks 1 and 2 where warm ascending flow dominates, ρ+δ ρ is the density of blocks 3 and 4 where cold descending flow dominates, and ρ is the density of blocks 5 and 6 where horizontal shear flow dominates. This density difference is caused by a mean temperature difference δT between ascending and descending flow, so that:

where αv is the volume coefficient of thermal expansion. Equation (9) follows from the definition of αv:

where M is mass, Vis unit volume, Lis unit length, αL is the linear coefficient of thermal expansion.

Figure 4 shows the stresses acting on blocks 1 through 6. These arc the tensile, compressive, and shear deviator stresse σT, σC, and σS, respectively; and the hydrostatic pressure P. Under equilibrium conditions, the size and shape of all blocks are constant with time. This requirement means that the forces exerted across sections y’-y’ and y”-y” above the line z are equal and opposite. The thermal stress caused by the density inversion can be estimated by considering the horizontal forces acting on blocks 1, 4, and 5. As shown in Figure 4,

where

Here h is the total ice thickness, d is the ice thickness below the density inversion, and Δd is the displacement of blocks 1 and 2 above blocks 3 and 4 due to thermal buoyancy, where

Fig. 4.

Horizontal stress variations in the vertical direction for the block model of convection in Figure 3. Deflator components are the tensile stress oT the compressive stress or. and the shear stress as- Spherical components are the hydrostatic pressures P1 through P6, including p1’ and P5’ Other symbols defined are in the text.

Several simplifying assumptions are needed to extract the thermal buoyancy stress from Equation (11). Of course and σ_T and σ_Care equal, and they must approximately equal σ_Ssince these deviator stresses are smoothly varying functions of position. However, if ice above the density inversion is passively transported by active convective flow below the density inversion, the stress σ_Sshould be small and will be neglected. This approximation is best when the density inversion is closest to the surface, because the firn air interface is a free boundary. The ratio λ|d, where λ is the distance between centers of ascending (or descending) flow in a horizontal fluid layer heated from below, depends on whether convective circulation is in the form of polygonal platform cells or elongated rolls, and on whether the top and bottom surfaces are free or rigid (Reference KnopoffKnopoff, 1964). In Figure 3c the density inversion and basal interfaces are intermediate between free and rigid, where 2 < λ/d < 2√2 is the range over these extremes. The approximate treatment given here will assume that

Using Equations (9) (13), and (14), the thermal buoyancy stress from Equation (11) is

where terms involving are ignored.(Δp)² (Δd)²

Equation (15) is identical to the equation Reference WeertmanWeertman (1967) derived for his four-block convection model which had free top and bottom surfaces with no overlying layer of passive convection flow. This encourages an application of the Weertman block convection model to polar ice sheets partly uncoupled from bedrock by a basal water layer.

Continuing with the Weertman block-convection model, thermal convection begins when the thermal buoyancy stress σ_T overcomes the visco-plastic resistance of the ice sheet. Equation (15) shows that σ_T varies with the mean temperature difference δT between ascending and descending convection currents. The equation of heat flow must be solved to obtain δT. This equation is:

where K IS the thermal dittusivity, w is the vertical velocity of convection now, z is veirtical distance measured from the base of the ice sheet, and

for steady-state convection flow. Under these conditions

where w ₁, w ₂, w ₃, w ₄, w ₅, and w ₆ are vertical velocities in blocks 1 through 6, respectively, and is the vertical strain-rate, which is always positive in Equations (17).

Setting T = T_i + ∆T at ɀ = 0 and T = T_i at ɀ = d as boundary conditions and using Equations (17) to specify the variation of w, steady-state solutions of Equation (16) for blocks 1 and 2 are:

where replacing

with gives T ₄ from Equation (18a) and T ₃ from Equation (18b)

Most heat is transported via conduction when

at the onset of convection. For vertical heat transport mainly via conduction from the bed at z = 0 to the thermal density inversion at z = d:

where T* is a temperature perturbation caused by the onset of convection. Letting

in Equations (18) gives the temperature perturbations in blocks 1 through 4 at the onset of convection:

The mean temperature difference between acending and descending currents is

Where

for T _1* using Equation (20a) and for T _3* using Equation (20b), since these are average block temperatures. Combining Equations (15) and (21) gives the thermal buoyancy strees σT* at the onset of convection:

Convenction flow

is resisted by σ₀ in Equation (5), so that

Equations (7) and (8) indicate that σ₀ → σ₀' when σ or t are small, σ₀ → σ₀”; when σ or t are large,

and η₀ is the effective viscosity defined as follows:

Letting σ = σT convection begins when σT* = σ_o Therefore the ratio

is unity at the onset of convection. Here the dimensionless Rayleigh number is

and the critical Rayleigh number for the onset of convection is

According to classical convection theory, 657 ≤ (Ra)* ≤ 1 708 for convection in a horizontal fluid layer heated from below, where the lower value is for free boundary conditions and the higher value is for rigid boundary conditions (Reference KnopoffKnopoff, 1964). The value of (Ra)* in Equation (27) is a direct consequence of taking a free boundary λ/d ratio in Equation (14c). Even so, the square-wave solution from the Weertman (1967) block model gives (Ra)* values about half those given by the sinusoidal wave solution from the Rayleigh classical model (Reference StruttStrutt, 1916). Higher (Ra)* values are expected if convection flow is not permitted to deform free boundaries, and this was a restriction imposed on the sinusoidal wave model shown in Figure 4a which was removed for the square-wave model shown in Figure 4b and c. Hence, Equation (27) probably underestimates (Ra)* but not as much as might be expected (Reference WeertmanWeertman, 1967).

Thermal convection is possible when (Ra) ≥ (Ra)* and comparing Equations (7) and (8) with Equations (25) through (27) shows that (Ra) ∝t ^-2/3 for t ≈ 0 and (Ra) ∝ σ² t ≫ o.for This illustrates an important distinction between convection in crystalline solids and convection in fluids. The viscous flow creep component in crystalline solids is time dependent, whereas viscous flow in fluids is time independent. Furthermore, the time dependence is such that when convection begins at t = o, the strain-rate

of vertical flow decreases from the infinite strain-rate of elastic deformation in Stage I strain, whereas strain-rate is always constant and finite for viscous fluids. In short, convection in crystalline solids begins as transient creep and convection in viscous fluids begins as steady-state creep.

Steady-state creep and stable convection flow

Thermal convection in polar ice sheets is important only if thermal conduction alone cannot satisfy the equation of heat How, as expressed by Equation (16). In the absence of convection, w = o and Equation (16) becomes

where k is the thermal conductivity, cp is the specific heat capacity at constant pressure, and setting w = o neglects accumulation and ablation of ice. In the absence of conduction, K = o and Equation (16) becomes:

where w is the mean vertical velocity in incremental distance Δz. Hence, K is obtained from the ratio of the temporal variation of temperature to the gradient of the spatial variation of temperature across distance Δz when conduction overwhelms convection, and

is obtained from the ratio of the temporal and spatial variations of temperature across distance Δz when convection overwhelms conduction. In polar ice sheets, the former would characterize Stage II creep during the initiation of transient convective flow and the latter would characterize Stage IV creep during the transition from Stage III to Stage V steady-state convective flow. During steady-state convective flow so that Equation (16) becomes:

Steady-state convective flow can therefore be discussed in terms of a characteristic distance z*, where

When convection overwhelms conduction, z* is the thickness of the thermal boundary layer of sharp temperature gradients which develops at the top of blocks 1 and 5, at the bottom of blocks 3 and 6, and between blocks 4 and 5, blocks 5 and 6, and blocks 6 and 2, as shown in Figure 5. Heat transport across this boundary layer is via conduction because

normal to these interfaces is either nil or non-existent. Convection overwhelms conduction in regions between the thermal boundary layer, where is large in blocks 1 through 4 and in blocks 5 and 6.

Heat transport via convection dominates heat transport via conduction above a critical creep rate

which exists when z* ≈ d, so that from Equation (30)

Solutions of Equations (18) in terms of

_e and at large are

in the range

In the range

Fig. 5.

Thermal regimes predicted by the block model of convection. Shown are the regime during Stage III steady-state strain (top), Stage V steady-state strain (bottom), and the zone of sharp temperature gradients (diagonal hatching). Here, T i and T₁ + ∆T are ice temperatures at the density inversion and at the bed, respectively, before convection began.

in the range and

in the range

. The average temperature difference between ascending and descending currents is now

Comparing Equations (21) and (33) shows that δT varies with

when conduction dominates and is independent of when convection dominates.

Convection in a polar ice sheet underlain by a substantial basal water layer occurs at a rate which keeps ΔT nearly constant below the density inversion because T = T _Mat the bed and T = T _i at the density inversion, where T _i is close to ambient mean annual air temperature. Therefore the position of the density inversion is such that heat supplied at the bed equals heat transported across d. The minimum amount of heat supplied from the bed is the geothermal heat flux H _G, and the maximum value of d is the total ice thickness h (disregarding the firn layer). Consequently the thickness of the actively convecting layer is controlled by heat transport across this layer.

In Stage III steady-state strain,

so heat transport via conduction dominates. Therefore, the heat H generated per unit horizontal distance (in the two-dimensional case illustrated in Figure 3) per unit time near the base of the ice sheet is given by the conduction equation

In Stage V steady-state strain,

so heat transport via convection dominates. Convective heat transported upward is the product of the temperature difference ΔT across d, the average vertical velocity of ascending flow, the heat capacity (ρc _p), and the length (A/8) at the base of block 2 in Figure 5:

However,

_z across the interfaces between blocks 5 and 6, where block 5 is warmer than block 6 when Hence, heat will be transported downward via conduction across this interface at a rate controlled by the temperature gradient ΔT across the thermal boundary layer z* shown in Figure 5. This heat flux should equal the heat supplied at the basal thermal boundary layer and lost at the density inversion thermal boundary layer. These layers have length of 3λ/8 in Figure 5. Therefore conductive heat transported downward is

Heat transported upward by convection and downward by conduction must equal the total heat generated at the bed. This heat is H(λ/2) in Figure 5, where λ/2 is the total length of the bed under blocks, 2, 3, and 6. Therefore, the heat blance equation is

Substituting from Equations (35) and (36) gives the heat transport when

This is approximately

When

heat transport via conduction dominates and the thermal buoyancy stress driving convection is obtained by combining Equations (15), (21), and (27):

For constant heat transport, ΔT is obtained from Equation (34) and Equation (39) becomes

When

heat transport via convection dominates and the thermal buoyancy strees driving convection is obtained by combining Equations (15) and (33):

For constant heat transport, ΔT is obtained from Equation (38) and Equation (41) becomes

Note that both σTand σH are proportional to

when but that σT is independent of and σHis inversely proportional to when

Figure 6 is a plot of stress versus strain-rate comparing Equations (39) through (42) with Equation (7), including both transient and steady-state creep components. Transient creep is controlled by a viscous flow stress σ_η:

Steady-state creep is controlled by a visco-plastic flow stress σn

Fig. 6.

Heat and mass transport characteristics predicted by applying the visco-plastic flow law of ice to the block model of convection. Shown are stress a and strain rate £ variations with temperature T that relate viscous creep (on curve) and visco-plastic creep (on" curve) to a constant temperature difference (σT curves) and a constant heat tranpsort rate (σH curve) through the connecting layer. Details are discussed in the text, where σ = σ_z This figure is modified from Reference WeertmanWeertman (1967,fig 5).

Stage I strain is instantaneous clastic deformation (t = o, ϵ_z ≈ o, and

_z = ∞). Stage II strain dominates the range o < _z < _s)111 and is transient creep deformation. Stage III strain dominates the range and is steady-state creep before recrystallization, Stage IV strain dominates the region and occurs during recrystallization. Stage V strain dominates the range and is steady-state creep after recrystallization.

Stage II strain is marginally stable when

II because the ρ̄ curve is nearly tangential to the curve. Here creep is a viscous-flow phenomenon because if decreases, the σ_T curve remains above the σH curve but falls below the σn curve. Hence the thermal stress needed to maintain constant H is high enough to permit viscous flow but not visco-plastic flow. Stage III strain increasingly replaces Stage II strain when II. Here, creep becomes increasingly visco-plastic, and increases because the curve increasingly rises above both the and curves. Stage III steady-state strain is actually a transition zone where transient creep is modified by recrystallization, during which easy glide progressively replaces hard glide as the rate-controlling process of convective flow. The rapid increase of during recrystallization accelerates heat transport so that ΔT is reduced across the layer of active convection flow. Hence levels off and then decreases. Recrystallization results in Stage IV strain, which occurs in the region where Stage V strain progressively replaces Stage IV strain over the range The creep rate remains constant when because the σT σH and σ_η + σ _n curves coincide. An increase in could not be maintained by either σT or σH because both these curves would then fall below the σ_η + σ _n curve. On the other hand, a decrease in would be prevented because both the σT and σH curves would then lie above the σ_η + σ _n curve. Whatever oriented crystal fabric exists at is therefore the stable convection ice fabric.

Equation (38) was derived on the assumption that blocks 1 through 6 in Figure 3 were comparable in size and shape. This is true when

However, when constant heat transport requires constant mass transport and this is possible only if the width of vertical convection currents decreases as increases. Hence, blocks 1 through 4 narrow while blocks 5 and 6 widen. Narrowing decreases the efficiency of vertical heat transport via vertical mass transport. In Stage V steady-state strain, conductive heat transport across z* equals convective heat transport across (d—z*). Conductive heat transport across ( z*) is obtained from Equation (30):

Convective heat transport across (d-z*) ≈ d is obtained from Equation (38) modified by changing widths L ₁ L ₅, respectively, of blocks 1 and 5:

Equations (45) and (46) can be combined to show how the ratio (L ₁\L ₅ varies with

:

The thermal buoyancy strees at

v is obtained by comparing the balance of forces on blocks 1, 4, and 5 in Figure 4 with the blance of forces on block 5 (Reference WeertmanWeertman, 1967):

which is Equation (15) multiplied by the ratio (L ₁/L ₅Eliminating L ₁/L ₅and ΔT from Equation (48) by substituting from Equation (46) and Equation (45), respectively, gives Equation (42).

Temporary creep and turbulent convection flow

It is interesting that the relationship H ∝ (ΔT)^4/3 observed in turbulent fluid convection can be obtained by combining Equations (46) and (48) to eliminate L ₁/L ₅, combining the resulting equation with Equations (43) and (44) by setting σ_T = σ_η + σ _n and combining this resulting equation with Equations (8) and (45) to relate H with ΔT via

:

As shown by Equations (8), the terms B _t and B _s contain the visco-plastic yield stress σ₀. Constant strain-rate creep tests produce flow curves in σ₀ which varies with according to the relationship (Reference Weertman, Whalley, Whalley and GoldWeertman, 1973):

where C and c are constants. For randomly oriented polycrystalline ice, σ₀ = σ₀’and c = 3. For ice single crystals oriented for easy glide, σ₀ = σ₀” and c Hence, 15 ≤ c ≤ 3 is the range of polycrystalline ice with fabrics ranging from single-maximum aligned for easy glide to random. When the term having B _s in Equation (49) is disregarded, then B _t is obtained from Equation (8a), where

and σ₀ = σ₀’, and c = 3 in Equation (50), so that H = H _t depends on t and When the term having B _t in Equation (49) is disregarded, then B _s is obtained from Equation (8b), where, and c = 1.5 in Equation (50), so that H= H _s is independent of t and . The equations resulting from these two extremes are

Note that H ∝ (ΔT)^4/3 in both cases. However, since fluids and randomly oriented poly-crystalline ice are both isotropic, a comparison of turbulent convection in fluids and polar ice sheets is better for Equation (51) because Equation (52) applies to stage V steady-state strain after recrystallization.

The time dependence of Equation (51) is a unique feature of turbulent convection in polar ice sheets. Its significance may lie in the fact that creep in ice is initially not purely viscous, as in fluids, but is temporarily viscous according to Equations (7) and (8). Turbulent convection in polar ice sheets can be expected if the buoyancy stresses caused by the density inversion are insufficient to overcome the visco-plastic resistance of the ice even though thermal conduction cannot transport enough heat upward to eliminate the superadiabatic temperature gradient that develops when (Ra) > (Ra)*. Hence, as (Ra) for the ice sheet increases above the (Ra) * values appropriate for convection in fluids, oT builds until it. is relieved catastrophically at time

after (Ra) = (Ra)* for fluid convection. Equation (53) is obtained by setting B _tσ= B _sσ³and in Equation (7) and solving for t using Equations (8). It represents the time since the superadiabatic temperature gradient developed at t = o until the time when convection begins as catastrophic recrystallization during stage IV strain. Hence, turbulent convection in polar ice sheets is expected where (Ra) for the ice sheet greatly exceeds (Ra)* for fluids. This condition exists wherever ice is over 4 km thick in the Antarctic ice sheet.

Turbulent convection via catastrophic recrystallization in a polar ice sheet is a temporary event because the thermal buoyancy stress is suddenly relieved, The turbulent convection episode may consist of a local collapse of the relatively rigid cold ice ceiling above the relatively soft hot ice basement, causing a downward flood of cold heavy ice which pushes the hot light ice aside. Or, perhaps more likely, the turbulent convection episode may consist of a local upthrusting of the relatively soft hot ice. basement, caused by a uniform slow en masse sinking of the relatively rigid cold ice ceiling. The first process would create a local down-warping of the cold ice strata, and the second process might inject hot ice sills into the cold ice strata. Ice has a high Prandtl number (Pr) = η\ ρk and turbulent convection in fluids having high Prandtl numbers consists of unstable convection cells which constantly change in size and shape while appearing and disappearing (Reference Somerscales and DropkinSomerscales and Dropkin, 1966; Reference Somerscales and GazdaSomerscales and Gazda, 1969). By analogy, turbulent convection in polar ice sheets should be a temporary creep phenomenon.

Dike-sill convection near the centers of polar ice sheets

The turbulent solid-state convection regime of Stage V steady-state creep suggested by Equation (52) consists of narrow zones of hot rising ice and cold sinking ice separated by wide zones of stagnant ice, as described by Reference Schubert, Schubert, Turcotte and OxburghSchubert and others (1969) for fluids with a strongly temperature-dependent Newtonian viscosity. The narrow sinking zone may not exist in a convecting polar ice sheet, however, owing to the anisotropic effective viscosity of ice. In this case Equation (47) becomes:

where Δλ replaces L ₁ and λ = 2 (L₁ + L₄ + L₅ +) ~ 2L ₅ + in Figure 5. This is dike- sill convection of the type commonly observed when magma intrudes horizontal beds of sedimentary rocks in the Earth’s crust. By analogy, the strata of cold ice sinks en masse into the hot temperate ice layer at the bed, forcing the basal ice upward as dikes which inject sills between the most weakly coupled layers of the slowly sinking cold strata. In order to conserve mass flux, the ratio of dike width to distance between dikes is inversely proportional to the ratio of ice velocities rising in dikes and sinking between dikes.

Figure 7 illustrates the possible development of dike—sill convection according to plasticity theory, for which n = ∞ in Equation (5). The main difference for ice, which is visco-plastic so that n < ∞, would be a rounding of sharp corners and a widening of the shear zones bordering dikes and sills in Figure 7. One practical consequence of this is to reduce the possibility that convection dikes could be detected by radio-echo sounding, since the junctions between dikes and sills would be less able to behave like corner reflectors for radar waves (personal communication from S. Evans in 1974) Figure 7, convection begins as an up- warp of the boundary between temperate basal ice and cold overlying ice (an upward bulge of the T _M isotherm). The initial up-warps could be nucleated where variations exist in the basal temperature gradient, the strength of ice-rock coupling at the bed, and the stress regime around bedrock topography, all of which arc interrelated and can locally decrease the effective viscosity of cold ice. These up-warps collapse laterally and arc thrust upward during the five stages of strain to become dikes which inject sills into the sinking strata of cold ice. Convection dikes are unstable and form randomly during Stage III steady-state strain but may become stable to form an orderly array if Stage V steady-state strain is attained. The stable array is active only so long as the basal temperate ice layer exists. The array stagnates if convection flow redistributes the temperate basal ice in dikes and sills faster than it is formed by geothermal and frictional heat generated at the bed. Convective flow stops when an array stagnates, so advective flow will recrystallize the vertical easy glide ice fabric developed in dikes but will preserve the horizontal easy glide ice fabric developed in sills.

Fig. 7.

The initiation and growth of dike-sill convection in a polar ice sheet according to plasticity theory. Arrows show ice flow directions and orthogonal cycloid segments show the ice slip-line field in dikes and sills. In the top view, the slip,-line field is shown for a basal temperate ice layer having an effective viscosity an order of magnitude lower than the overlying cold ice [see Fig. 9) so that the cold ice and the bed (shaded zone) behave as rigid plates cornpressing the temperate ice (Reference HillHill 1950, fig. 64). The compressive pressure is relieved where irregularities in basal conditions allow doming of the basal temperate ice layer so that basal ice flows toward these domes, generating the slip-line field shown. In the middle view lateral spreading of the cold ice overlying the domes causes the domes to contract laterally and expand vertically into the cold ice becoming ascending dikes of recrystallized ice in the process. The slip-line field in the dikes is that for a plastic material injected between rigid parallel plates and forcing them apart. In the bottom view, ascending dikes have injected sills between weakly coupled layers (dashed horizontal lines) in the strata of slowly sinking cold ice between dikes, and frictional heat in the basal ice feeding the dikes has created a basal water layer (black horizontal band) which has uncoupled the ice sheet from the bed. Consequently, sills have the slip-line field of plastic material forced between rigid parallel plates and the basal temperate ice layer has the same slip line field as when one of the plates (the water layer) is a frictionless surface.

Advective flow is a general feature that is characteristic of the entire ice sheet, whereas convective flow would be a local feature confined to narrow dikes and sills in the lower part of the ice sheet. In general, therefore, the convective flow regime should conform to the advective flow regime rather than vice versa. Figure 8 shows the slip-line field which plasticity theory predicts for advective flow from a central ice dome. Ice flow lines can be drawn as lines radiating from the ice dome and therefore are 45° diagonals to the slip-line field. In the absence of advection, the most efficient convection occurs when convection dikes intersect at 1200 angles to form a hexagonal array, since this minimizes the length of dikes penetrating cold sinking ice. In Stage V convection, therefore, hexagonal prisms of cold ice λ in diameter and separated by dikes of width Δλ should sink slowly into a temperate basal ice layer under the domes of polar ice sheets where advection is small. However, advective flow increases with increasing distance from the domes, and the dike array must intersect at 90° angles if it is to enclose prisms of cold sinking ice which correspond to the slip-line field of orthogonal logarithmic spirals for advective flow radiating from the dome, as shown in Figure 8. Furthermore, as distance from the dome increases, the prism diameter paralleling advective flow increases with respect to the prism diameter normal to advective flow. Consequently, hexagonal convection “cells” under the dome where advection is least are transformed into elongated convection “rolls” aligned with advective flow toward the margin of the ice sheet. This transformation has been observed and studied in detail for convection-advection interactions in fluid flow (Low, 1925; Reference JeffreysJeffreys, 1928; Reference Deardroff Deardorff, 1965; Reference Gallagher and MercerGallagher and Mercer, 1965; Reference Davies-JonesDavies-Jones, 1971). However, the dike-sill nature of crystalline convection proposed for ice sheets does not exist in fluid convection.

The vertical ice velocity entering dikes can be estimated from the tensor form of the flow law of ice (Reference NyeNye, 1957):

where i and j are orthogonal coordinates, u _i and u _j are velocity components in respective directions x _i and x _j,

are components of the strain-rate tensor, σij’ are components of the deviator stress tensor, T is the effective strees, and using the notation of Equation (5),

Since a convection dike is aligned in the direction x of advective flow, set i = y, j = z, xi = y, and x _j = z so that the effective strees

since σij = σij’. Advection is nil under central ice domes of polar ice sheets so the only deviator stress components used in Equation (57) are where σ_z' = σ_y' ≃ σ_yz', where σ_y' = σ_T = σ_c and in Equation (14a) and Figure 4. For Stage V steady-state strain, n = 3 and d → h in the temperate ice layer at the base of the dike. In the dike itself, the effects of developing an easy-glide fabric tend to offset the effects of decreasing temperature so that A ^-n should not change greatly for ice moving up the dike. Solving Equation (55) for i =j = z and using Equation (57) to evaluate t gives

Fig. 8.

An idealization of the interaction between convective flow and advective flow in a polar ice sheet drained by ice streams and fringed by ice shelves. This figure assumes that convection dikes form a stable polygonal array that becomes elongated in the direction of advective flow and converges on ice streams so that the entire ice stream behaves like a single dike (alternatively, the convection dikes may form randomly. be transient, and be largely independent of advective flow). Convection begins under domes (D) and saddles (S) along the ice divide where the ice sheet is thickest and advection is minimal so that dikes (thick lines) can form a hexagonal array. As ice spreads from the domes, interaction with the plastic slip-line field of radially spreading advective flow (dotted orthogonal logarithmic spirals) causes the six-sided convection polygons to become five-sided, then four-sided. and finally elongated rolls with dikes paralleling advection flow lines (thin broken lines). These flow lines converge to form ice streams, for which the plastic slip-line field is typical of extrusion flow (Reference HillHill, 1950, fig. 44) at the upper end where flow lines converge on the ice stream, compressive flow (Reference HillHill, 1950, fig. 64) in the middle where flow lines are parallel in the ice stream, and indenting flow (Reference HillHill, 1950, fig. 70) at the lower end where flow lines diverge onto the ice shelf Indenting flow (cross-section A) also characterizes ice-stream convection before ice thinning enables the shear zones alongside ice streams to penetrate the surface (cross-section B).

where σ_z’ is obtained from Equations (2) and (g). Snow accumulation and ice advection are generally least under central ice domes of polar ice sheets. Hence, the density inversion should occur not far below the firn layer so

is a reasonable approximation. Setting

If dike-sill convection in the Earth’s mantle controls sea-floor spreading and continental drift, as has been suggested (Reference HughesHughes, 1973[a] Reference Hughes[d]), then λ ≃ 2d = 6000 km for mantle-wide convection. Setting Δλ/λ = 0.032 predicts that Δλ = 200 km is the dike width in the mantle. These values of λ and Δλ are the same order as the average distance between mid-ocean ridges and the average ridge width. Hence, the spacing and width of dikes predicted for polar ice sheets arc in approximately the same ratio as those observed for crustal features that might result from convection dikes in the Earth’s mantle. This is encouraging, because the Earth’s mantle is the only other part of our planet where convection in crystalline solids is postulated.

Vertical convection velocity w _e is a maximum at the base of a dike and decreases in steps to zero at the top of the dike, with one step for each sill intersected and fed by the dike. Solving Equation (58) for the above values of

km/a. This compares with surge velocities, as might be expected if dike-sill convection in crystalline solids is analogous to turbulent convection in fluids. The sinking velocity of cold ice between dikes is (Δλ/λ)w _e = 67 m/a at the top of the basal temperate ice layer. Hence, this is the annual layer of ice which must be heated to the pressure melting point if dike-sill convection is to remain in a stable, steady-state condition near the bed. Otherwise, the basal temperate ice layer will thin to zero thickness and dike-sill convection will stop. The temperature gradient above the basal temperate ice layer is therefore important.

Ice-stream convection near the margins of polar ice sheets

Ice streams form near the margins of polar ice sheets when advective flow concentrates in channels of the subglacial topography. The increased shear deformation in these regions of concentrated flow generates frictional heat which melts the basal ice, permitting basal sliding, which erodes the channels and thereby further increases the ice discharged through them. Once formed, therefore, ice streams are self-perpetuating. Shear zones border the sides of ice streams, and create characteristic crevasse patterns when ice-sheet thinning near the margin is sufficient to allow the shear zones to penetrate to the surface. The ice fabric generated in the shear zones favors vertical easy glide, and if shear deformation at the base and along the sides of the ice stream generates sufficient Frictional heat to make ice streams warmer than ice between ice streams, a buoyancy force arises which may lift the ice stream slab en masse from its bed. This would be ice-stream convection.

Figure 8 illustrates the possible development of ice-stream convection before and after the flanking ice-stream shear zones penetrate the surface of the ice sheet, according to plasticity theory. Before penetration, the warm ice stream presses against the cold overlying ice as if it were a rigid prism indenting a plastic plate. If d and A are the respective heights and widths of the ice stream, where d < h and the ice stream temperature is uniformly δT hotter than the surrounding ice, the buoyancy force it experiences arises from a thermal stress σ_T = ρgdαvδT Applying the Reference OrowanOrowan (1965) analysis to ice streams, resisting this force are the plastic properties of the surrounding colder ice. This includes a uniaxial stress

exerted on the overlying cold ice and shear stresses exerted on the flanking cold ice, where is the uniaxial yield stress of the cold ice and σ _zy acts on six vertical surfaces: the two sides of the ice stream, the two sides of flanking ice, and the sides of the two adjacent ice streams. Summing vertical forces:

Solving for δT:

When the ice stream penetrates the surface of the ice sheet, σ_z vanishes, d = h and Equation (60) becomes

Note that δT is independent of d in this case.

Antarctic ice streams are generally visible as heavily crevassed slabs on the ice sheet surface when h ≤ 1 km, and they are typically A ≈ 25 km in width. The plastic yield stress of ice is commonly taken as σ₀= 1 bar. Using these values and the values of p, g, and αv previously cited gives =δT = 178 deg in Equation (60) and δT = 8.7 deg in Equation (61). Hence ice-stream convection is unlikely when d < .h, but may occur when d = .h, provided that ice streams average several degrees warmer than surrounding ice. However, since σ_o ≪ 1 bar for visco-plastic flow, ice-stream convection is still possible when d < h if Ʌ → d Major ice streams often form by the confluence of smaller ice streams for whichɅ → d, and small ice streams may originate from converging convection dikes. The displacement of a convecting ice stream from the bed is

For = 1 km,

, and δT = 8.7 deg, Equation (62) gives Δd = 1.3 m. This displacement would substantially uncouple the ice stream from the bed and radio-echo sounding has revealed subglacial bodies of water at least this thick under Antarctic ice streams, leading Reference Robin.de, Robin.de, Swithinbank and SmithRobin and others (1970) to call such ice streams “pseudo ice shelves”. Basal uncoupling of this order should cause an ice stream to surge, and since ice streams drain ninety per cent of the West Antarctic ice sheet, surges via ice-stream convection should result in disintegration of the ice sheet (Reference HughesHughes, 1970,1972[b],1973[b],[c]).

Figure 8 shows the slip-line field and the resulting flow lines which plasticity theory predicts for an ice stream draining an ice sheet onto an ice shelf. In ice-stream convection, the ice stream is buoyed up en masse as a hot slab as shown in Figure 8, This slab acts like a rigid prism which plastically indents the overlying cold ice before the ice stream breaks through the surface of the ice sheet.

Rayleigh numbers for initiating convection

Convection begins under the central domes of polar ice sheets where ice thickness and the temperature difference between surface and bed are greatest and where snow accumulation and ice advection are least, since (Ra) ∝ d ³ ΔT in Equation (26) and d → h under these conditions. The temperature profile under polar ice domes is given by the equation (Reference Robin.de, Robin.de, Swithinbank and SmithRobin, 1955):

where z is measured from the base,

is the basal temperature gradient, and is the ice accumulation-ablation rate. For a frozen bed and no advection, For the temperature range of polar ice sheets will be used where Θ is the Celsius temperature (Reference PounderPounder, 1965). In the lower part of a polar ice sheet, where convection would be most active, Equation (63) is adequately approximated by the expression (Reference HughesHughes, 1972[c]):

where T _b is the basal ice temperature. The effective viscosity of ice no in Equation (26) is obtained from Equation (55) and can be written in terms of the basal effective viscosity of ice η_b using z/h and N in Equation (64) as follows (Reference HughesHughes, 1972[c]:

where K = 25.3 in Equation (7) and Figure 9, and (k/N) is a viscous scale height.

Reference Schubert, Schubert, Turcotte and OxburghSchubert and others (1969) show how the viscous critical Rayleigh number (Ra)_η* varies with the ratio (K/N for both rigid and free boundary conditions in fluid convection. For polar ice sheets, rigid boundaries exist when

and the bed is frozen whereas free boundaries are approximated when and the bed is thawed, with basal boundary conditions being more important than surface boundary conditions (Reference HughesHughes, I972[c]). Figure to shows the variation of (.Ra)η * with (K/N) for both frozen and thawed beds. Figure 10 can be applied to the initiation of ice-sheet convection since it begins as viscous flow according to Equation (7) and Figure 9.

Fig. 9.

A polynomial-exponential flow law fitted to creep data for polycrystalline ice in which hard glide dominates. Plotted is the variation of octahedral shear stress τ with octahedral shear strain-rates y at various temperaturesΘ. A best-fit of the flow law at various homologous temperatures T/T_M (thin lines) is given to creep data from laboratory experiments and glacier studies (dashed lines). Note that τ when τ < 0.5 bars, τ³ when τ > 5.0 bars, and increases tenfold as T/T_M → 1. This figure is modified from Rudd (1969, fig. 2.2).

Fig. 10.

The variation of the critical Rayleigh number for initiating convection with the viscous scale height in the convecting layer. Rayleigh numbers (Ra) (no) can be calculated for the effective viscosity no averaged through a convecting layer of thickness d, or Rayleigh numbers (Ra) (nb)can be calculated for the effective viscosity n\b, at the base of the convecting layer of thickness d(N)K). Solid lines are for a coupled bed and dashed lines are for an uncoupled bed. This figure illustrates the Reference Schubert, Schubert, Turcotte and OxburghSchubert and others (1969) treatment modified for polar ice sheets (Reference HughesHughes 1972[c]).

The minimum ice thickness needed to initiate convection is obtained by solving Equation (26) for h after setting

and using Figure 10 to relate (k/N)and (Ra)η*. This gives

Equation (66) will be applied to the region of the Wilkes Land ice dome in East Antarctica, where

and radio-echo sounding suggests regions having both frozen and thawed beds. The thawed beds are sometimes above subglacial lakes deep enough to substantially uncouple the ice sheet from the bed (Reference Oswald and Robin.deOswald and Robin, 1973). Equation (63) can be applied to this region by setting T _b = T _M to obtain for a thawed bed and setting obtain T _b for a frozen bed, 3

The resulting temperature profiles are compatible with N = 3 in Equation (64) for the lower portion of the ice sheet where convection would be active. Taking K = 25.3 from Figure 9 gives (N/K) = 8.4, for which (Ra)η* = 25 for a thawed bed and Ra)η*= 32 for a frozen bed according to the curves showing (Ra) as a function of η_b in Figure 10, From Equation (65):

Where τ = √2σ_z' from Equation (57) for

σ_z' = σ_T being given by Equation (15). However, n = 1 for convection initiated as transient creep so that τ⁽ⁿ⁻¹⁾ = 1 and B = B_t For a temperate ice layer (Reference LliboutryLliboutry, 1966) above a subglacial lake but for a frozen bed using the B _t values in Figure 9. Solving Equation (66) gives h = 1 km for the thawed bed and h = 3 km for the frozen bed. Since 4 km km near the Wilkes Land ice dome, convection can theoretically begin anywhere.

Rayleigh numbers for maintaining convection

Convection begins as transient creep which creates a broad upward bulge in the T _M isotherm at the top of the temperate ice layer covering a thawed bed or in the T _b isotherm at the bottom of the cold ice layer covering a frozen bed. This is accompanied by a downward bulge of isotherms in the flanking ice, and the net result is a tiny fraction of one convection overturn so that the convective perturbation of conductive heat flow takes place, as shown in Figure 6 for

In dike-sill convection, however, the broad, low, upward bulge rapidly transforms into a narrow, high, up-thrusting dike. During this transformation, Stage II transient strain also transforms to Stage V steady-state strain. As shown in Figure 6, convection transports heat faster than conduction during State V strain and the question arises: can convection be maintained at these higher convection velocities? It can if the dike narrows with increasing convection velocity in the dike such that heat is not transported upward from the bed faster than it is provided at the bed by shear deformation and the geothermal flux. Equation (54) expresses this requirement in quantitative terms.

The Rayleigh number for maintaining dike-sill convection is obtained from a consideration of Stage V steady-state strain. Let (Ra),η* be the viscous critical Rayleigh number which must be exceeded to initiate convection as Stage II transient strain, and Ra), _n * be the visco-plastic critical Rayleigh number which must be exceeded to maintain convection as Stage V steady-state strain. The thermal buoyancy stress driving convection can be expressed in terms of Ra),η* and (Ra) _n * by setting

in Equation (27), setting in Equation (26), and substituting these equations into Equation (22):

Substituting for η_o, using Equation (24) and substituting for

using Equation (7), where gives:

Note that

when is small and is large. As seen in Figure 6, a small is sufficient to initiate convection as Stage II transient strain when but a larger is required to maintain convection as Stage V steady- state strain when Equation (69) is plotted in Figure 11 for the values of B _t and B _sin Figure 9.Note that bar is taken as the yield stress for ice when using plasticity theory.

The minimum ice thickness for maintaining convection is obtained by solving Equation (66) when η_b is given by:

Here

from Equation (56), assuming all components are important and approximately equal to each other at the heads of ice streams where dike-sill convection becomes ice-stream convection. These are regions near the ice-sheet margin where advecting flow will transport dike-sill convecting systems initiated under central ice domes and where dike-sil! convection must therefore terminate. Since dike-sill convection is in Stage V steady-stale strain, for Stage III steady-state strain controlled by hard glide and for Stage V steady-state strain controlled by easy glide (Fig. 9 and unpublished data by M. Nakagawa and T. J. Reference HughesHughes, 1971). Setting from Equation (27), gives (K/N) = 1.4 for a thawed bed in Figure 10. The bed is undoubtedly thawed at the head of an ice stream, so that N = 18 for K = 25.3 in Figure 9. Equation (64) then predicts that T_η = [N/(N +1)]T_b = 0.95Tb, in the shear zones flanking the ice stream, where T_b = T_M This is what happens if the shear zones behave like dikes in dike-sill convection. However, if the entire ice stream is buoyed up by σ_T then the flanking shear zones do not behave like dikes and ice-stream convection replaces dike-sill convection. Setting ≠T = 25 deg as a value appropriate for the beads of ice streams, Equation (66) predicts that dike-sill convection persists until d = 685 m, where d replaces h because d h is invalid near ice-sheet margins. Since ice-stream convection is expected when ice streams are observable on the surface at h ≈ 1 000 m, dike-sill convection “rolls” can merge at the heads of ice streams and be transformed into ice-stream convection “slabs” (see Fig. 8).

Fig. 11.

The variation of the ratio of the final critical Rayleigh number to the initial one with the thermal buoyancy stress driving convection. Note that bars. Compare with Figure 9 .