Stress-gradient Coupling in Glacier Flow: Iii. Exact Longitudinal Equilibrium Equation*

The "vertically" integrated, exact longitudinal stress-equilibrium equation of Budd ( 1970) is developed further in. such a way as to yield an equation that gives explicitly and exactly the contributions to the basal shear stress made by surface and bed slope, surface curvature, longitudinal stress deviators, and longitudinal stress gradients in a glacier flowing in plane strain over a bed of longitudinally varying slope. With this exact equation, questions raised by various approximate forms of the longitudinal equilibrium equation can be answered decisively, and the magnitude of errors in the approximations can be estimated. To first order, in the angle 6 that describes fluctuations in the surface slope a from its mean value, the exact equilibrium equation reduces to (I + 2sina)r 8 = pghsina + 2G + T + B + K where G and T are the well-known stress-deviator-gradient and "variational s tress• terms, K is a "longitudinal curvature" term, and B is a "basal drag" term that contributes a resistance to sliding across basal hills and valleys. Except for T, these terms are expressed in simple form and evaluated for practical situations. The · bed slope a (relative to the mean slope) is not assumed to be small, which allows the effects of bedrock topography to be determined, particularly through their appearance in the B term. RESUME. Coup/age du graJiem de conlraime dans /"ecoulcment des glaciers: III. Equation e:mcte de l'equilibre longitudmal. L'equation d'equilibre en contrainte longitudinalement exacte de Budd (1970), integre venicalement, est devetoppe plus completement, de fa<;on a Mtir une equation qui donne explicitement et exactement les contributions a Ia contrainte basale de cisaillement produites par les pentes de Ia surface et du lit, de Ia courbure de surface, du deviateur longitudinal de contraintes, et des gradients tongitudinaux des contraintes pour un glacier secoutant en c isaillement plan sur un lit de pente variable tongitudinalement. Grace a cette equation exacte, des questions soulevees par differentes approches de !'equation d'equilibre longitudinal trouvent des n!ponses decisives, et l'ordre de grandeur des erreurs des approximations peut etre estimee Au premier ord re, lorsque !'angle 6 decrit les fluctuations de Ia pente de 13 surfa.::e autour de sa moyenne, !'equation exacte d'equilibre se reduit a: (J + 2sin2a)r 8 = pgh sina + 2G + T + 8 + K