Fe tb f Ub f tbUb fin fe tb td

(4) |e —f tb < | ub — I tbub — |m — |e — frozen bed

Up and down arrows represent increases or decreases. Horizontal arrows may be read as "leads to" or "results in".

further, driving the system back toward A (Case 3). Similarly, a decrease in e decreases tbub, thus decreasing m and driving the system to a frozen bed condition (Case 4). In other words, there appear to be two stable states, one in which the bed is frozen and one in which it is at the melting point, m — dr, and Tb < (|)Td.

Raymond (2000) has considered an alternative situation in which a change in m changes dr by changing the thickness of the water layer, 8, separating the ice from the till. An increase in m increases 8 and hence dr. He defines two possible states. In the first, an increase in 8 increases ub more than it decreases tb, so m increases, and conversely. He refers to this state as drainage limited because if the rate of increase in drainage exceeds that in m the situation is stable; otherwise it is unstable. In the second state, the increase in 8 decreases tb more than it increases ub, so m decreases. He refers to this as production limited. This state is always stable.

Also meriting consideration are the relative magnitudes of tbub, G, and Kfto. If the heat produced by straining, tbub, is greater than the heat loss to the overlying ice, Kfto, no geothermal heat is necessary to maintain pressure melting conditions at the bed. This appears to be the case beneath Bindschadler, Kamb, and MacAyeal Ice Streams (Raymond, 2000). However, if tbub < Kfto, geothermal heat is required to maintain sliding. This seems to be the case beneath Whillans Ice Stream. Indeed, at the lower end of Whillans Ice Stream the required value of G is close to the measured value, suggesting that freezing may be occurring there. This could account for the deceleration of this part of the ice stream documented by Bindschadler and Vornberger (1 998).

In some cases, water input from upglacier may be crucial for maintaining a stable state. Consider a unit area of the bed, say 1 km2. Clearly, dr — qout - qm, where qout and q^ are the water fluxes into the control area on its upglacier side and out of it on its downglacier side. If qin decreases without a corresponding decrease in qout, e (or 8) will decrease, thus increasing tb. For small increases in tb, tbub will increase (Case 1), thus increasing m to compensate for the decrease in qin. However, if the decrease in qin is large enough, the increase in m may not be sufficient to compensate for it. In this case, tb would rise above its unstable equilibrium value at B and the ice stream would shut down. Thus, Retzlaff and Bentley's (1993) speculation that the shut down of Kamb Ice Stream might be the result of changes in subglacial drainage in the area where Kamb and Whillans Ice Streams are close together (Figure 5.20) seems well founded.

The dashed curves in Figure 7.27 illustrate the sensitivity of the ice streams to the geothermal flux. If the flux were slightly lower the ice streams would not exist, while if it were higher, they might be even more common.

The extreme sensitivity of ice streams to external conditions is illustrated by a recent experiment at the mouth of Whillans Ice Stream. Bindschadler et al.(2003) used Geographical Positioning Systems (GPS) units to measure the movement of the ice stream at 5-min intervals. They discovered that movement occurred in pulses lasting 10-30minutes separated by periods of quiescence lasting 6-18 h. The pulses were in phase with the diurnal ocean tidal cycle (one high tide per day) often occurring just after high tide and just before low tide. The event just after high tide is attributed to strain accumulated over about 18 h since the last event. The event just before low tide is attributed to reduced back pressure from the sea. Clearly, very small variations in stress can cause failure, either of the ice-till interface or of the till itself.

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