Figure 2 is a Radarsat-1 image provided by the Canadian Space Agency [23]. It shows tributary ice streams that get packed together as they converge on Byrd Glacier to become “flow stripes” that continue onto the Ross Ice Shelf. If thermal convection rolls are aligned with these tributaries, do the rolls continue under flow stripes within the glacier and onto the ice shelf? This needs to be investigated. Flow stripes are not caused by lateral compression because Byrd Glacier generally widens as it

moves through its fjord, Barne Inlet, and enters the Ross Ice Shelf in Figure 2. Flow stripes may be slowly descending limbs of convection rolls, with faster rising limbs confined to narrow shear zones between flow stripes [21].

The suggestion that ice-stream tributaries are the surface expression of underlying convection rolls [5,21] is supported by temperature profiles obtained from hotpoint drilling to the bed, showing warmer ice in Whillans Ice Stream and colder ice in Ice Ridge B1-B2, informally called the “Unicorn” [24]. In between is a lateral shear zone, which would add warmer ascending convecting ice to advective ice flow from the ice ridge to the ice stream. Frictional heating and an “easy glide” ice fabric that facilitate both advective and convective ice flow are produced in the lateral shear zone [21,25]. Winter cold air sinking into crevasses made ice in the top 30 m of the shear zone colder than deeper ice [26].

Figures 1 and 2 suggest two field experiments to test whether ice-stream tributaries are the surface expression of thermal convection rolls: 1) tributary surfaces should lower faster than predicted by advective thinning if they are the sinking limbs of convection rolls, and 2) ice crossing lateral shear zones should move faster if it is augmented by ice in the rising limbs of convection rolls, see Figure 3" target="_self"> Figure 3. These experiments must be designed very carefully by both ice-sheet modelers and field glaciologists to make sure data are reliable and can constrain models. Deep drilling to get the full strain-rate tensor and physical properties to the bed will be required.

4. THEORY

In the classic theory of thermal convection in a Newtonian fluid heated from below, steady-state convection begins when the Rayleigh number Ra attains a critical value Ra*, where Ra is defined as follows [1]:

(1)

Here, is the ascending velocity of thermal convection in an incompressible fluid layer of height h_C where r_C = k/h_C is the ascending rate of thermal conduction, k is the thermal diffusivity in the fluid, is the strain rate for convective ice flow due to gravitational driving stress (s_G)_C = Drgh_C where density r decreases with depth by Dr = ra_VDT_C due to thermal expansion from temperature increase DT_C for volume coefficient of thermal expansion a_V, and h is the viscosity of the fluid.

Superficially, it seems reasonable that thermal convection can begin when heat transported upward from convection and conduction are the same, so u_C = r_C and Ra* = 1 in Eq.1. However, thermal convection occurs when thermal buoyancy is attained, so temperature must be

included in the derivation of Ra*. The derivation requires a force balance that includes all stresses resisting s_G see Figure 4, where. Therefore, when convection begins. Weertman [4] and Hughes [5] included temperature variations linked to extending and compressive deviator stresses, respectively, at the top and bottom of rising convection columns, and vice versa for sinking columns, with shear stresses at the boundaries of vertical columns and their horizontal extensions in Figure 4. Take Cartesian axes with x along advective flow, y transverse to flow, and z vertical, using deviator stresses with i, j = x, y, z in conventional tensor notation. For tributary ice streams partly decoupled from slow sheet flow by warm ascending convection flow in roll boundaries, is s_C for compressive converging flow at the bottom and s_T for tensile diverging flow at the top. For descending convection flow within rolls, is s_C for compressive converging flow at the top and s_T for tensile diverging flow at the bottom. For shear zones between vertical ascending and descending flow, s_yz is s_S for side shear, with s_yz also between upper and lower horizontal convective flow in opposite directions. This force balance gives u_C = 280 r_C, so Ra* = 280 for Newtonian flow when these stresses are allowed to warp the top and bottom surfaces of convection rolls as shown in Figure 4.

The onset of thermal convection in Newtonian viscous fluids heated from below begins at critical Rayleigh numbers Ra* = 657 when unwarped top and bottom surfaces are free, Ra* = 1100 when one is rigid and one is free, and Ra* = 1708 when both are rigid [27]. In the Antarctic Ice Sheet, the first case corresponds to thermal convection from the bed to the surface, where low surface accumulation rates that put the density inversion near the ice surface coexist with a subglacial lake at the bed; the second case corresponds to either a density inversion far below the surface, so cold overlying ice is a relatively rigid upper boundary, or a frozen bed that provides a rigid lower boundary; and the third case corresponds to a density inversion at depth and a frozen bed,

both providing rigid boundaries.

Values of Ra* for initiating thermal convection in Newtonian fluids may not apply for polycrystalline glacier ice in which elastic deformation with an infinite strain rate becomes transient creep with a decreasing strain rate over time that stabilizes as slow steady-state creep independent of time until replaced after recrystallization by fast steady-steady creep, see Figure 5. Recrystallization from a hard-glide random crystal fabric to an easy-glide crystal fabric produced by simple shear begins at strain e » 0.1 for both applied stresses, and increases e by a factor of 12.8, from 1.88/a to 24.1/a for s_C = 117 kPa at −3˚C. Relating to gravitational stress (s_G)_C driving convective flow employs the flow law of ice [28] for both transient and steady-state creep combined to give creep strain e_C as the sum of transient strain e_T and steady-state strain e_S:

(2)

where t is time since thermal convection began, m = for transient creep, n = 3 for steady-state creep, respective ice hardness parameters A_T and A_S are for ice fabrics and temperatures during transient and steady-state creep, and R_C is a scalar that takes account of all strain rates associated with convective and advective ice flow [21]. Then Ra is:

(3)

Note that Ra is infinite at t = 0, so there is no critical Ra* for initiating thermal convection in ice sheets. As t increases, transient creep is replaced by steady-state creep so Ra becomes finite and may fall below Ra* for ice sheets [5,21]. Transient creep does not exist in Newtonian fluids, so Eq.3 reduces to Eq.1 where m = n = 1 for Newtonian creep and h =A_S /R_C is the Newtonian viscosity, for which R_C = 1. At Byrd Station in West Antarctica where s_C = 13.7 kPa for h_C = 1103 m below the density inversion about halfway down the corehole, R_C = 3 for convection rolls, A_T = 2.5 × 10³ kPa·a^1/3, and A_C = 140 kPa·a^1/3 before recrystallization gives t_C = 4.7 years as the time when steady-state creep overtakes transient creep. This shortens to t_C = 0.2 years for A_C = 59.8 kPa·a^1/3 after recrystallization [21].

Gravitational driving stresses for sheet flow, convective

flow, and shelf flow generally have comparable magnitudes. The gravitational driving stress (s_G)_A for slow advective sheet flow spreading along x from interior ice domes is linked to vertical shear stress  [21,29,30].

(4)

where h_I is the ice thickness, q is the ice surface slope, r_I is ice density and g is gravity acceleration. The gravitational driving stress (s_G)_C for convective ice flow in rolls along x is the vertical buoyancy stress given by [21]:

(5)

where Dr_I is the change of ice density in height h_C below the density inversion. The compressibility of ice partly offsets thermal expansion of ice at depth, reducing Dr_I from 4.0 kg/m³ to 1.2 kg/m³ at Byrd Station. This reduces (s_G)_C from 45.0 kPa to 13.7 kPa for h_C = h_I, compared to 45.6 kPa for (s_G)_A [21]. However this reduction is largely removed if h_C ≈ h_I, a situation that cannot be discounted when surface accumulation rates are low for slow sheet flow [31]. In fast stream flow, increasing temperature, hence decreasing density, begins close to the surface of Whillans Ice Stream in West Antarctica [24]. The two stresses in Eqs.4 and 5 are nearly identical if Dr_I ≈ r_Iq and h_C ≈ h_I. This is the case if tributary ice streams are the surface expression of convection rolls. Faster surface ice in these tributaries correlates with a wet bed that reduces ice-bed coupling, and hence reduces Ra* for convecting ice rolls. Ice-bed coupling vanishes when the floating fraction of ice beneath tributaries approaches unity. The gravitational driving stress (s_G)_F for freely floating ice along x is [29,30]:

(6)

where r_W is the density of water. Eqs.4 through 6 are all gravitational driving stresses having comparable magnitudes close to the yield stress of ice, but each one applies to a specific flow regime based on the strength of ice-bed coupling.

As seen in Figure 6, the flow law for both slow and fast steady-state creep before and after recrystallization can be normalized with respect to plastic yield stress s_O = 100 kPa [12] at strain rate as follows for 1 £ n £ ¥ in the viscoplastic creep spectrum:

(7)

where n = 1 for viscous flow, n = ¥ for plastic flow, n =

3 for ice, and when s = s_O for all values of n. For n = 3 in Figure 6, the maximum stress-curvature yield criterion gives a viscoplastic yield stress for ice of s_V = 0.386 s_O = 38.6 kPa and the stress-intercept yield criterion for the tangent line at gives s_V = 0.667s_O = 66.7 kPa, see Ice Sheets, Chapter 8 [15]. The viscoplastic viscosity h_V obtained from Eq.7 for steadystate creep is:

(8)

At s = s_O where in Eq.7, viscoplastic viscosity h_V compared to Newtonian viscosity for n = 1 is:

(9)

The Rayleigh number in Eq.3 for steady-state creep now becomes:

(10)

Eq.10 for steady-state creep gives Ra* = 280 for n = 1 [4,5]. This rises to Ra* = 840 when n = 3, which is within the range for Newtonian fluids.

For the Antarctic Ice Sheet, typical values in Eqs.4 through 6 are h_I = 3 km paired with q = 0.002 for sheet flow, giving (s_G)_A = 60 kPa in Eq.4, h_C = h_I = 3 km and Dr_I = 1.2 kg/m³ for convection rolls underlying tributaries, giving (s_G)_C = 42 kPa in Eq.5, and r_I/r_W = 0.9 with h_I = 100 m at the calving front of ice shelves, giving (s_G)_F = 50 kPa in Eq.6. Then gravitational driving stresses (s_G)_A » (s_G)_C » (s_G)_F » s_V » 50 kPa for the Antarctic Ice Sheet. As seen in Figure 6, creep in ice occurs at s_V < s_O. Creep is significantly slower below s_V than creep above s_V. For West Antarctic ice, s_V ≈ 45 kPa is common [32]. It may be higher for colder East Antarctic ice.

The dependence of (s_G)_C on h_I is important. For example, grounded ice 2 km thick at the head of ice streams can thin to floating ice 200 m thick at the calving front of an ice shelf, a reduction of 90 percent. Since about 10 percent of ice floats above water, this is a reduction in ice elevation of 99 percent, ice 2000 m high lowers to 20 m. Given the strong dependence of Ra on h_C in Eq.3, it is unlikely that steady-state thermal convection under ice-steam tributaries for h_C » h_I can be sustained in ice streams when ice thickness is reduced so drastically. Somewhere along ice streams convection must shut down.

5. CONCLUSIONS

Thermal convection in ice sheets and Newtonian fluids are fundamentally different. Atoms in fluids are not attached to crystal lattice sites, so heat transport by mass transport is much less inhibited. In polycrystalline glacial ice, thermal conduction increases as heat increases the vibrational amplitudes of atoms at their lattice sites. Deformation is elastic, therefore recoverable as temperature drops. With increasing temperature, atoms begin to break free from their lattice sites and move as chains, called dislocations, through the crystal lattice, allowing heat transport by mass transport. This begins as transient creep having the initial infinite strain rate of elastic deformation, but over time the strain rate falls as dislocations pile up at crystal grain boundaries for a random distribution of optic c-axes, one for each grain. This allows only slow steady-state creep in hard glide across high-angle grain boundaries. Strain energy adds to thermal energy at grain boundaries until new grains can be nucleated at pileups. The optic axes of new grains are aligned to produce an easy-glide ice fabric of low-angle grain boundaries that allow dislocations to pass much more easily from grain to grain. This is the process of recrystallization. It allows fast steady-state creep, as shown in Figure 5. Recrystallization begins at a creep strain of about 10 percent for a given applied stress. The strain rate, initially infinite, gives an infinite Rayleigh number. Both decrease over time and the Rayleigh number can fall below its critical value before slow steadystate creep is established. Then convective flow stops. If the Rayleigh number remains above its critical value, recrystallization can take place and allow fast steadystate creep to produce a higher Rayleigh number that allows faster convective flow. This component of icesheet flow can therefore turn on and off, and thereby regulate the 90 percent of ice discharged by ice streams in the Antarctic Ice Sheet. In the extreme, the faster discharge may allow the ice sheet to self-destruct, if more ice leaves than can be replaced by precipitation over the surface.

If thermal convection rolls underlie the ice-stream tributaries in Figure 1, virtually the entire Antarctic Ice Sheet consists of convective flow superimposed on advective flow. Since convective flow may turn on and off as a Rayleigh number rises above and falls below a critical value for sustaining thermal convection, this may have a major impact on stability of the ice sheet, especially on the discharge of ice streams. This could be a factor in the ability of ice sheets to rapidly self-destruct, thereby terminating a glaciation cycle of the present Quaternary Ice Age [21,29,30] and perhaps initiate a new glaciation cycle [8,33].

Fast transient creep at the onset of thermal convection in ice sheets can last for months, possibly ascending as transient pipes along ice divides where advective flow is small and as transient curtains aligned with advective flow downslope from the ice divides. Transient creep is initiated first where top and bottom boundaries are almost free surfaces, meaning low surface accumulation rates and a wet bed, ideally a subglacial lake. These lakes are widespread under the Antarctic Ice Sheet [34]. A few months of transient creep are sufficient to cause vertical displacements of a few centimeters. This is the sampling interval for extracting climate records from oxygen isotopes and ionic impurities in ice cores deposited during precipitation over the ice sheet [20]. Transient creep would randomly alter these records by altering the stratigraphy. That could render these records invalid. With improved radar technology, the widespread “echo-free zone” beneath the density inversion of the Antarctic Ice Sheet is now a zone with irregular and warped radioecho radar horizons. This alerts us to the possibility that transient thermal convection is active in this zone, and it distorts climate records.

The Antarctic Ice Sheet can be considered as a miniature mantle if advective and convective ice flow combine to produce major ice streams that discharge 90 percent of Antarctic ice. This has applications to thermal convection in Earth’s mantle [35-37]. Ice drainage basins in Figure 1 may be the equivalent of moving tectonic plates that generate crustal rifts, trenches, and mountain ranges where moving plate boundaries meet. As in rifted crustal ridges, ice pulls away from ice divides, but ice is too slow and soft to produce rifts. Ice converges on ice streams that supply ice shelves, which calve instead of collide like crustal plates. This understanding of ice-sheet dynamics would draw wide interest, and be a great boon to glaciology. The glaciology of ice sheets would undergo a Scientific Revolution comparable to plate tectonics if convection rolls were shown to underlie ice-stream tributaries.

Without strong bedrock control, tributaries and any underlying convection rolls can sweep back and forth sideways so the entire cold ice ceiling sinks slowly into the warm ice basement over time in all of the ice catchment areas shown in Figure 1. The lowering ice surface results from reduced ice-bed coupling in the wet bed underlying ice-stream tributaries. Taking h_C = h_I and a viscoplastic yield stress s_V between 38.6 kPa and 66.7 kPa depending on what yielding criterion is used in Figure 6, and also depending on ice temperatures in East and West Antarctica, s_V ≈ (s_G)_A ≈ (s_G)_C ≈ (s_G)_T is common from the grounded interior to the floating margin of the Antarctic Ice Sheet. Strain rates and ice velocities will be comparable if these stresses are comparable, otherwise the flow law for ice [28] must be discarded. Strong ice-bed coupling in Eq.4 for grounded ice weakens under convection rolls in Eq.5 and vanishes for floating ice in Eq.6. This can reduce ice elevations by over 99 percent, from interior elevations exceeding 2000 m thinning to under 20 m at the calving fronts of ice shelves. The 90 percent reduction in ice thickness is accompanied by boundary conditions that proceed from rigid to free in ice-stream tributaries, so Ra* is progressively reduced.

Radar reflecting horizons will not be altered much by this kind of thermal convection, since horizons are preserved in ice sinking slowly en masse in and between tributaries. Two field experiments are now possible to test this model in a conclusive way, using the highly accurate surface elevation and velocity data that revealed the tributaries in Figure 1. 1) Surface lowering in tributaries can be measured to millimeters of accuracy using InSAR satellite data, so lowering due to advective ice thinning can be separated from lowering in the broad sinking limb of a convection roll. 2) Surface velocities measured to centimeters of accuracy across the narrow lateral shear zones of tributaries using GPS technology can detect whether lateral advective flow is augmented by rising convective ice flow in the shear boundaries that is diverted into the tributaries by advective flow.

If thermal convection is indicated by experiments (1) and (2), shutting down convective flow may be studied by conducting these experiments on the flow stripes in Figure 2 as ice thins downstream, so decreasing h_C in Eq.3 causes Ra to fall below Ra*. The field experiments to test this possibility would be daunting. A surface strain network, a grid of coreholes, radar profiling, and seismic sounding would be needed both along and across a tributary to obtain ice temperatures, fabrics, and deformation (borehole tilt) at depth so the full strain-rate tensor could be obtained and the convection roll could be mapped, both for geometry and velocities, and then linked to advective flow. Reproducing this pattern in computer models would be equally daunting.

As an example, take 200 m/a as the advective velocity difference between fast flow in tributaries and slower flow between tributaries in Figure 1. Add another 50 m/a from convective flow rising in the shear zone and diverted into the tributary. Strain rates are then 0.100/a and 0.025/a for advective and convective ice flow, respectively, across a lateral shear zone 1 km wide. For ice 3 km thick and a tributary 5 km wide between lateral shear zones, ice rises 3 km in 60 years in the shear zones and sinks 3 km in 150 years in the tributary. Surface ice moving from rising to sinking flow averages 25 m/a over 2.5 km. The convective circuit of 11 km is completed in 410 years. During this time, the tributary has moved 82 km downslope. Hence, convective flow spirals downstream.

Ice downdrawn by fast advective flow will give tributaries a lower surface than slower ice advecting into tributaries from flanking ice ridges, as seen in Figure 3. When tributaries pack together at the head of an ice stream, this flanking advective ice flow vanishes and only convective flow rising in lateral shear zones contributes to sinking flow in tributaries, as seen in Figure 4. The proposed field experiments should be able to detect this change.

6. ACKNOWLEDGEMENTS

Eric Rignot and Kenneth Jezek kindly provided Figures 1 and 2, respectively. I thank Kenneth Jezek and Robert Jacobel for critiquing an earlier draft of this manuscript. Kenneth Jezek suggested field experiment (1). Input from two anonymous referees greatly improved the presentation. The Center for Remote Sensing of Ice Sheets (CReSIS) at the University of Kansas supported this work.

REFERENCES

NOTES

^*Professor emeritus.