In glacier science and the study of materials, the Glen–Nye flow law is a rule that describes how glacial ice moves. This law is based on observations and experiments and is used to model how ice behaves under pressure. It treats ice as a fluid that is thick, cannot be compressed, has the same properties in all directions, and does not follow the same rules as water. The viscosity of ice, or how thick it feels, depends on a relationship between how fast it stretches (strain rate) and the force applied (stress). This relationship is shown by the equation:

ϵ ˙ e = A τ e n

The effective strain rate (how quickly ice stretches, measured in seconds) and effective stress (the force applied, measured in Pascals) are connected to values calculated from the properties of their related mathematical structures. The constants A and n are numbers determined through research and experiments. The number n has no units, and A has units of Pascals times seconds. The Glen–Nye flow law simplifies the complex force calculations in ice to a single value, called dynamic viscosity, which depends on properties of the stress and strain rate in the ice.

When ice is under constant pressure, it flows like a liquid. Changes in the force applied cause changes in how ice flows, but these changes are not directly proportional. This behavior, which the Glen–Nye flow law explains, happens through a process called creep, which is the main way glaciers move.

Viscosity definition

The constitutive relation is developed as a generalized Newtonian fluid, where the deviatoric stress and strain tensors are connected by a viscosity scalar:

τ = 2μϵ˙

Here, μ is the viscosity (measured in Pa·s), τ is the deviatoric stress tensor, and ϵ˙ is the strain rate tensor. In some calculations, λ = (2μ)^−1 (measured in Pa·s) is used instead of μ.

This model assumes several conditions:

Although glacial ice is often treated as incompressible, it can also be anisotropic, meaning its properties vary depending on direction. In general, the strain rate may not align with the direction of the principal stress.

Under these assumptions, the stress and strain rate tensors are symmetric and have a trace of zero. These properties simplify the calculation of their invariants and squares.

The deviatoric stress tensor is related to an effective stress through its second principal invariant. Einstein notation, which implies summation over repeated indices, is used in these derivations.

The same approach applies to an effective strain rate.

From this, it follows that:

Viscosity is a scalar value and cannot be negative (a fluid cannot gain energy as it flows). Therefore, μ can be expressed using the invariant effective stress and effective strain rate:

μ = τ / (2ϵ˙) = (1/2)τₑϵ˙ₑ⁻¹

The Glen–Nye flow law allows substituting either τₑ or ϵ˙ₑ, enabling μ to be defined in terms of effective strain rate or effective stress alone:

μ = A⁻¹/ⁿ / (2ϵ˙ₑ^(n−1)/n) = τₑ^(1−n) / (2A)

Here, B = A⁻¹/ⁿ (measured in Pa·s¹/ⁿ) is sometimes used in place of A.

Parameter values

The Glen–Nye model describes two key values, A and n, that help explain how ice flows.

The rate factor A is influenced by temperature and is often explained using an Arrhenius relationship, which shows how temperature affects ice deformation. This relationship includes Q, the activation energy; R, the universal gas constant; and T, the absolute temperature. A base value, A₀, may depend on the ice’s crystal structure, impurities, or other characteristics. Estimates of A can differ greatly and are often calculated from A₀ or by comparing real-world glacier measurements and experiments. In some cases, A is treated as a field of values determined through mathematical analysis of ice flow equations at specific locations.

Viscous ice flow is an example of shear thinning, which occurs when the flow parameter n is greater than 1. Research across different methods and locations suggests that n typically ranges between 2 and 4, with n = 3 being the most commonly used value. However, n can also change depending on stress levels, reflecting different ways ice deforms under pressure.

Improving the accuracy of these parameters remains an active area of scientific study.

Limitations

The word "law" used to describe the Glen-Nye model of ice movement might make it seem simpler than it is. This model does not fully explain how different factors affect how ice flows under pressure, even within one glacier. It also does not account for the many assumptions and simplifications made in the model itself.

The model treats ice as a fluid with general properties, but this approach may not capture how ice actually changes at the level of individual ice crystals. Ice crystals in glaciers can grow from millimeters to 10 centimeters in size. As these crystals adjust to changes in stress inside the ice, they cause large differences in how much the ice stretches over the same distance. Additionally, individual ice crystals are not the same in all directions, and their arrangement is not random. These features change over time as the ice moves and adjusts to stress, making them difficult to include in a model.

The Glen-Nye model also does not show all ways ice responds to stress. These include bending (elastic deformation), breaking (such as cracks or crevasses), and short-term changes in how ice flows under pressure.