The “T term” in the longitudinal stress-equilibrium equation for glacier mechanics, a double y-integral of ∂2
τ
xy
/∂x
2 where x is a longitudinal coordinate and y is roughly normal to the ice surface, can be evaluated within the framework of longitudinal flow-coupling theory by linking the local shear stress τ
xy
at any depth to the local shear stress τ
B at the base, which is determined by the theory. This approach leads to a modified longitudinal flow-coupling equation, in which the modifications deriving from the T term are as follows: 1. The longitudinal coupling length is increased by about 5%. 2. The asymmetry parameter σ is altered by a variable but small amount depending on longitudinal gradients in ice thickness h and surface slope α. 3. There is a significant direct modification of the influence of local h and α on flow, which represents a distinct “driving force” in glacier mechanics, whose origin is in pressure gradients linked to stress gradients of the type ∂τ
xy
/∂x. For longitudinal variations in h, the “T force” varies as d2
h/dx
2 and results in an in-phase enhancement of the flow response to the variations in h, describable (for sinusoidal variations) by a wavelength-dependent enhancement factor. For longitudinal variations in α, the “force” varies as dα/dx and gives a phase-shifted flow response. Although the “T force” is not negligible, its actual effect on τ
B and on ice flow proves to be small, because it is attenuated by longitudinal stress coupling. The greatest effect is at shortest wavelengths (λ 2.5h), where the flow response to variations in h does not tend to zero as it would otherwise do because of longitudinal coupling, but instead, because of the effect of the “T force”, tends to a response about 4% of what would occur in the absence of longitudinal coupling. If an effect of this small size can be considered negligible, then the influence of the T term can be disregarded. It is then unnecessary to distinguish in glacier mechanics between two length scales for longitudinal averaging of τ
b, one over which the T term is negligible and one over which it is not.
Longitudinal flow-coupling theory also provides a basis for evaluating the additional datum-state effects of the T term on the flow perturbations Δu that result from perturbations Δh and Δα from a datum state with longitudinal stress gradients. Although there are many small effects at the ~1% level, none of them seems to stand out significantly, and at the 10% level all can be neglected.
The foregoing conclusions apply for long wavelengths λ
h. For short wavelengths (λ
h), effects of the T term become important in longitudinal coupling, as will be shown in a later paper in this series.