Glen's flow law

Glen's flow law
Name	Glen's flow law
Field	Glaciology, Rheology, Cryophysics
Discovered	1955
Discoverer	John W. Glen
Equation	\dot{\varepsilon} = A \tau^n
Keywords	Ice dynamics, Viscous flow, Non-Newtonian

Glen's flow law Glen's flow law is an empirical constitutive relation that describes the nonlinear viscous behavior of polycrystalline ice under deviatoric stress. Developed in the mid‑20th century, it links strain rate to stress through a temperature‑dependent rate factor and a stress exponent, and it underpins modern models of Antarctica, Greenland, Alps, and planetary ice shell dynamics. The law is central to interpretations of Pleistocene ice sheet evolution, glacier surge mechanics, and comparative studies of icy satellites such as Europa and Enceladus.

Introduction

Glen's flow law was formulated by John W. Glen in 1955 to account for laboratory observations of steady‑state creep in polycrystalline ice samples. The law supplanted simple linear viscous descriptions for many glaciological problems and became standard in numerical ice‑sheet models used by Intergovernmental Panel on Climate Change contributors and regional studies of the Greenland Ice Sheet and West Antarctic Ice Sheet. Its development drew on earlier experimental studies from institutions like the Cold Regions Research and Engineering Laboratory and theoretical frameworks advanced by figures associated with Royal Society publications.

Mathematical Formulation

Glen's flow law is commonly expressed in tensorial form relating the deviatoric stress tensor σ' to the strain‑rate tensor ε̇ via a nonlinear relation with exponent n and rate factor A(T,p): - Scalar form for shear: ε̇ = A(T,p) τ^n, where τ is characteristic shear stress. - Tensorial representation uses effective stress σ_e and effective strain rate ε̇_e: ε̇_ij = A(T,p) σ_e^{\,n-1} σ'_{ij}. Parameters include the stress exponent n (often ≈3 for temperate ice), the temperature‑dependent rate factor A (Arrhenius or activation energy formulations), and pressure‑dependent corrections that appear in high‑pressure glaciological contexts such as Antarctic Peninsula ice streams and deep ice cores from Vostok Station.

Physical Basis and Derivation

The physical basis of Glen's law arises from crystal‑plastic deformation mechanisms in hexagonal ice Ih, principally basal slip on the {0001} plane and dislocation glide controlled by thermal activation. Microphysical derivations relate A and n to dislocation density, activation energy, and impurity effects; these concepts were developed alongside studies at Cambridge University, University of Oslo, and laboratories affiliated with National Research Council programs. Glen's empirical fit captures power‑law creep regimes observed in secondary creep experiments and integrates with theoretical treatments from Orowan and Cottrell on dislocation dynamics.

Laboratory and Field Measurements

Determination of A(T) and n has relied on uniaxial and torsion creep tests, triaxial deformation apparatus, and borehole deformation measurements from deep drilling projects like EPICA and North Greenland Eemian Ice Drilling (NEEM). Laboratory campaigns at facilities such as Scott Polar Research Institute and Lawrence Berkeley National Laboratory quantified temperature, grain‑size, and impurity dependencies; field validations used surface velocity inversions of Pine Island Glacier, satellite radar interferometry from missions like ERS-1 and RADARSAT, and GPS measurements across mountain glaciers in the Himalayas.

Applications in Glaciology and Planetary Science

Glen's law is embedded in continuum ice‑flow models of ice sheets and glaciers employed by groups at British Antarctic Survey, NASA, and European Space Agency, informing projections of sea‑level contribution from Greenland Ice Sheet and Antarctic Ice Sheet mass balance. It is used in coupled models of ice‑ocean interaction for tidewater glaciers and outlet glaciers such as Jakobshavn Isbræ and Thwaites Glacier. In planetary science, the law (or its modified forms) models convective and viscous relaxation of icy lithospheres on Titan, Ganymede, and Mars polar caps, and informs interpretation of surface morphology observed by missions like Voyager and Cassini.

Limitations and Extensions

Limitations include sensitivity to grain size, fabric (crystallographic preferred orientation), impurities, and transient, damage‑related processes such as firn compaction and crevassing. Glen's original formulation does not explicitly account for anisotropic flow due to lattice preferred orientation, transient stress‑dependent damage, or rate‑weakening/strengthening linked to meltwater lubrication beneath glaciers. Extensions include anisotropic rheologies developed by researchers affiliated with ETH Zurich, non‑local damage mechanics from groups at Imperial College London, grain‑size sensitive flow laws (e.g., dislocation‑diffusional composite models), and temperature‑ and pressure‑dependent Arrhenius formulations used in ice‑core interpretations at Dome C and Dome Fuji.

Numerical Implementation and Modeling Considerations

Implementations of Glen‑type rheology appear in finite‑element and finite‑difference ice‑flow codes such as Elmer/Ice, PISM (Parallel Ice Sheet Model), ISSM (Ice Sheet System Model), and bespoke models developed at institutions like University of Toulouse and University of Washington. Numerical considerations include regularization for n>1, stabilization of nonlinear solvers, coupling with basal sliding laws (e.g., Weertman or Coulomb friction formulations), and parameter estimation via inverse methods using adjoint techniques pioneered in collaborations with Princeton University and ETH Zurich. Grid resolution, timestep constraints, and treatment of boundary layers near grounding lines (as studied for Ross Ice Shelf and Amundsen Sea Embayment) are critical for accurate projection of ice dynamics.

Category:Glaciology