2D Debris Development Model

Debris development

Debris-covered glaciers play an important role in high mountain environments, where layers of rock and sediment strongly influence how ice melts and evolves. Understanding how this debris develops is key for predicting glacier behaviour, meltwater supply, and long-term stability.

Here I present a simple two-dimensional MATLAB model designed to explore the main controls on debris development.

Let’s define a 1D array, where each item represents a pixel of finite size (length $\Delta x$, width $\Delta y$). This array will store the volume of debris within each pixel. The debris thickness is simply:

The development of debris cover has two primary controls (Anderson and Anderson, 2018):

1. Velocity driven
2. Meltout driven

Velocity driven

Debris is moved down-glacier simply by ice flow. Thus, we must define the debris flux through each pixel boundary.

The volumetric debris flux through each boundary is:

where $U$ is the ice surface velocity.

The change in debris volume inside pixel $i$ is therefore the flux intering through its up-glacier face minus the flux leaving its down-glacier face:

Assuming the valley width remains constant in our model, it can be seen that when the glacier velocity is constant across regions, $dV/dt = 0$, and hence, there is no debris development.

Mountain glaciers typically decline in velocity monotonically toward the termius. Therefore, under such regimes, the debris flux into a given region is always greater than the flux out. Consequently, debris cover thickens down-glacier.

Meltout driven

Debris also emerges from the meltout of englacial septa, which adds additional debris from velocity driven debris development. Therefore, we must define the volume of ice melted, which itself is a function of the overlying debris thickness, and the ratio of debris meltout to ice melt.

Debris attenuates the energy transfer from the atmosphere to the ice surface (Østrem, 1959; Brock et al., 2010). The melt rate at the ice surface increases as debris thickness increases until a critical thickness (known as the ’effective thickness’), $H_c$, is reached, which represents the maximum ablation rate, owing to the lower albedo of the debris surface compared to clean ice and snow (Østrem, 1959). Beyond this threshold, the attenuation of thermal energy from the atmosphere through the debris negates the increased energy absorption at the surface by the darker debris, resulting in debris insulating the ice surface.

Let’s represent this using a piecewise function to define a dimensionless debris factor: how much debris multiplies melt compared to a clean-ice surface. We’re not using real physical relationships at the moment.

In our model, the maximum ablation is reached at 0.1 units thick, where debris cover doubles ablation relative to a clean ice surface. Note above that whilst we defined the effective thickness arbitarily at 0.1 units thick, debris still enhances ablation relative to a clean ice surface for a period after it. In our model above, debris only reduces ablation beyond ~0.55 units thick. In the real world, these two values are tyically 2 and 3 cm (Mattson et al., 1993).

We now have a way of defining debris’ impact on ice melt compared to a climatologically equivilant clean ice surface. But what is the melt of a climatolocially equivilant clean ice surface? Ablation is complex even on a clean ice mountain glacier ablation. But for the sake of our simple model, lets say that ablation [m s$^{-1}$] is a simple decreasing function of elevation (because temperature decreases as you increase elevation):

and $a$ and $b$ are constants with units chosen so that elevation is converted into melt thickness per unit time. Let’s also assume that elevation increases linearly up-glacier (our glacier is on a smooth slope):

Therefore, the sub-debris melt volume at position $i$ per unit time is:

To connect ice melt to debris emergence, we define a dimensionless debris fraction $d_f$, which expresses how much debris volume is revealed per unit ice volume melted. The corresponding debris volume source in pixel $i$ is

Bringing it all together

Finally, we need to consider the input of debris from avalanches or rockfalls from mountainsides. Let’s set this to a constant, $L$ [m$^3$ s$^{-1}$], for now (variable input to come later).

The full conservation equation for debris volume in pixel $i$ is then:

Numerical Model

Let’s now write this in Matlab.

Let’s define our constants arrays:

pixel_width = 1;
num_pixels = 100;


debris_volume = zeros(1, num_pixels);
ice_velocity = []  % Monotonically decreasing  
elevation = []

References