## Submergence and emergence velocities

Earlier (Equation (5.1)), we gained insight into the magnitude of the horizontal velocity by considering a glacier in a steady state, such that its surface profile remained unchanged. Let us now use this idealization to study vertical velocities. In such a steady state, the surface in the accumulation area must everywhere be sinking at a rate that balances accumulation, and conversely in the ablation area. Thus, the vertical velocity at the surface, ws, is clearly related to the net balance rate, bn. Remembering that a point on the surface is also moving with a horizontal velocity, us, and that the surface has a slope, a,we findthatthe appropriate relation is (Figure 5.10):

Here, we have taken the x-axis as horizontal and positive in the downglacier direction, and the z-axis as positive upward. Thus, in the accumulation area, both ws and a are negative and, owing to the relative magnitude of the two terms on the right-hand side of the equation (see Figure 5.10a), the minus sign in the equation makes the right-hand side positive. In the accumulation area, the right-hand side is called the submergence velocity.

Equation (5.26) also applies in the ablation area (Figure 5.10b), except that here ws is positive so both terms on the right-hand side take on negative values. Thus bn is negative, reflecting ablation. Here, the right-hand side is called the emergence velocity.

Clearly, the submergence and emergence velocities are defined for any point on a glacier surface. However, they equal bn only in the idealized steady-state situation that we have specified. This is because bn varies from year to year, and because, even averaged over several years, glaciers are rarely in a steady state. Put differently, if the accumulation rate consistently exceeds the submergence velocity and the ablation rate consistently falls short of the emergence velocity, the glacier is becoming thicker and will advance, and conversely.

Other possibilities can also be visualized. For example, if the equality in Equation (5.26) holds everywhere except in the lower part of the ablation area where the ablation rate exceeds the emergence velocity, the glacier may be in the final stages of adjustment to a climatic warming. The implication of such a situation would be that the accumulation area has essentially adjusted to the warming, but the glacier is still retreating slightly.

We have shown that on a glacier that is in a steady state and that has a balanced mass budget, the velocity field at the surface is related to bn. It is instructive to consider in greater detail the physical mechanisms behind this relation. In this case, bn is the independent variable, and the velocity field is the dependent variable. (In a larger system involving glacier-climate interactions, bn would be dependent upon the climate.) The physical mechanism by which bn and the velocity field are related is viscous flow, in which the flow rate increases with the driving stress, pgha. If the velocities are, say, too low (in absolute value), the submergence velocity will be less than the accumulation rate so the glacier will become thicker in the accumulation area (Figure 5.10a). Similarly, the emergence velocity will be less than the ablation rate, so the glacier will become thinner in the ablation area (Figure 5.10b). The slope of the glacier surface thus increases. The increase in slope, coupled with the increase in thickness in the accumulation area, increases the driving stress and hence us. Because u = 0 at the head of the glacier and at the terminus, an increase in us in the middle makes du/dx more extending in the accumulation area and more compressive in the ablation area. Thus, by the arguments leading to Equation (5.26), |ws| increases. The increases in both us and |ws| tend to restore the steady state.

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