[Paper Review] Numerical simulation of the evolution of glacial valley cross sections

Hakime Seddik, Ralf Greve+2 more|ArXiv.org|Jan 9, 2009

Cryospheric studies and observations21 references18 citations

TL;DR

This study develops a 2D numerical model coupling ice flow dynamics with a sliding velocity–dependent erosion law to simulate the formation of U-shaped glacial valleys. It demonstrates that lateral shear stresses are critical for developing the characteristic U-shape, and shows that U-shaped profiles can form within 50,000 years, though the model primarily simulates deepening rather than widening, limiting its ability to replicate observed Patagonia-Antarctica-style valley evolution.

ABSTRACT

A numerical model was developed for simulating the formation of U-shaped glacial valleys by coupling a two-dimensional ice flow model with an erosion model for a transverse cross section. The erosion model assumes that the erosion rate varies quadratically with sliding speed. We compare the two-dimensional model with a simple shallow-ice approximation model and show the differences in the evolution of a pre-glacial V-shaped valley profile using the two models. We determine the specific role of the lateral shear stresses acting on the glacier side walls in the formation of glacial valleys. By comparing the model results with field data, we find that U-shaped valleys can be formed within 50 ka. A shortcoming of the model is that it primarily simulates the formation of glacial valleys by deepening, whereas observed valleys apparently have formed mainly by widening.

Motivation & Objective

• To investigate the role of lateral shear stresses in the development of U-shaped glacial valleys using a 2D ice flow model.
• To compare a full 2D ice flow model with a shallow-ice approximation model to assess differences in stress distribution and erosion patterns.
• To constrain the basal sliding coefficient using field data from the Tian Shan Mountains to validate model output.
• To evaluate whether the model can reproduce observed U-shaped valley morphologies within realistic timescales (e.g., 50 ka).
• To identify limitations in simulating valley widening, which is dominant in some real-world glacial valleys (e.g., Patagonia-Antarctica model).

Proposed method

• A 2D finite-difference model solves the momentum balance for ice flow in a transverse valley cross section, assuming steady-state flow and no variation along the glacier's length.
• The model uses Glen’s flow law with a non-Newtonian viscosity, incorporating a rate factor A = 214 MPa⁻³ a⁻¹ and flow-law exponent n = 3.
• Basal sliding is modeled via a linear relationship between sliding velocity and basal shear stress (u_b = -c τ_b), with c = 50 m a⁻¹ MPa⁻¹.
• An alternative sliding law is tested using u_b = k τ_b^p / N^q with p = 3, q = 2, and k adjusted to match observed velocity ranges.
• Erosion rate is modeled as quadratically dependent on sliding velocity, with erosion rates computed at each grid point based on local velocity and stress.
• The model is initialized with a pre-glacial V-shaped valley and simulates evolution over 50,000 years, comparing results with field data from the Tian Shan.

Experimental results

Research questions

• RQ1 How do lateral shear stresses influence the development of U-shaped glacial valley cross sections compared to a shallow-ice approximation?
• RQ2 Can a 2D ice flow model coupled with a sliding velocity–dependent erosion law reproduce observed U-shaped valley profiles within 50,000 years?
• RQ3 What is the role of effective pressure and basal sliding law variations (e.g., power-law vs. linear) on erosion patterns and valley morphology?
• RQ4 Why does the model primarily simulate valley deepening rather than widening, and how does this compare to observed valley evolution in regions like Patagonia and Antarctica?
• RQ5 To what extent can field data from the Tian Shan Mountains be used to constrain the basal sliding coefficient in the model?

Key findings

• The inclusion of lateral shear stresses in the 2D model produces a more realistic basal sliding velocity profile with a central minimum, which is essential for U-shaped valley development.
• Compared to the shallow-ice model, the 2D model generates a more accurate erosion pattern due to proper stress distribution, particularly lateral drag.
• U-shaped valley profiles can be formed within 50,000 years using the calibrated sliding coefficient, matching field observations from the Tian Shan Mountains.
• The form ratio (b) of the simulated valley cross section is 2.19 when using the effective pressure-dependent sliding law, indicating a pronounced U-shape.
• The model's erosion rate and sliding velocity distributions show a central minimum, consistent with the Rocky Mountain model of valley formation via deepening.
• Despite successful reproduction of U-shape formation, the model fails to simulate valley widening, which is dominant in real-world Patagonia-Antarctica-type valleys, indicating a limitation in the current erosion law formulation.