Spiral troughs on Mars

**IMPORTANT corrections to typographical errors in Geology paper described below

The spiral troughs on Mars are some of the most puzzling landforms anywhere. They are big features - approximately 10 km in width and up to 1km in depth. The shaded-relief image below illustrates the spiral troughs on the North Polar ice cap with MOLA (Mars Orbital Laser Altimeter) data. Color represents elevation in this image, with red representing the highest elevations of the ice cap and green representing the sand dunes and cratered plains around the ice cap. The straight-lined grooves near the pole are not actual features in the ice cap - north of 88 degrees the MOLA instrument data has position errors. I have covered this region with a transparent circle to emphasize that this part of the image has errors. Below the image of the ice cap is a subset region blown up to show some of the details in the troughs, including gullwings (two troughs joining at odd angles), bifurcations, and terminations.

The basic model for how the troughs form has been well established for decades. Small cracks in the ice surface can initiate localized sublimation of ice because the side of the crack that is oriented toward the equator will absorb more solar radiation, heat up, and sublimate if the temperature rises above the sublimation point of ice (0 degrees Celsius). The ice goes directly to vapor under the low-pressure conditions of Mars'atmosphere, so sublimation is the correct term, not melting. Much of the water vapor refreezes on the opposite side of the crack or trough, which is especially cold since it faces away from the equator. This model for trough formation is due to Alan Howard and has been termed "accublation" by David Fisher. Direct evidence for this process comes from satellite observations of high concentrations of water vapor in the troughs during the Martian summer.

Scientists have had a rich debate about how the accublation model actually creates spiral forms. Both eolian processes (wind erosion) and ice flow have been suggested as the controlling processes for how the troughs came to be spiral in shape rather than circles or some other form. My contribution to this problem was to show that the accublation model itself generates spirals without the need for other processes except for conduction of heat within the ice cap. The image below shows a time sequence of a simplified numerical model of how the accublation process works starting from a random distribution of sublimation on the surface. I assumed a circular ice cap with a diameter of 600 km. The incipient troughs grow at their ends and join up with other troughs over time, preferentially orienting themselves to face the equator at low latitudes. At the center of the ice cap, I assumed a small, permanently-frozen region that is too cold to sublimate. Without this region, the model eventually produces a single spiral winding ever tighter around the pole. The presence of the permanently-frozen region is responsible for creating multiple spiral arms (similar to the spiral geometry on Mars) rather than just a single spiral.

The basic equations I used to explain the spirals have also been used to model spiral shapes that arise in other fields including biology (bacteria growth in a petri dish) and chemistry (reaction-diffusion equations). This general set of equations is known as the Fitzhugh-Nagumo equations, which in one specific form is written as

In a recent paper in Geology, I argued that the processes acting on Mars could be described using these equations, with a small change to include the angle of the trough with respect to the equator. In two dimensions, the relationship between the ice-surface temperature and topography is especially clear. The figure below graphs a solution to the Fitzhugh-Nagumo equations given above. Even though the heat in the ice cap tends to spread out in all directions, the coupling between the ice-surface temperature and the shape of the trough creates a self-sustaining wave that migrates toward the poles. Also, just a single, small pertubation is sufficient to create an entire train of solitary waves. This remarkable feature is called a solitary wave.

Although this numerical model gives realistic-looking spirals, many questions remain that require further study. First, wind erosion and ice flow may still be important, but because they are harder to quantify it may take scientists longer to prove their respective roles. Second, what were the initial conditions that led to spiral-trough development on Mars? In the Geology paper, the spirals merged from an initially-random distribution of sublimation, but cracks on Mars probably originated near the edge of the ice cap. Finally, how do the spirals respond to climate changes, such as those driven by changes in the tilt of Mars' axis? Do they change form or just evolve and migrate faster? Answering this questions will require a more complex model, but one that is a more precise representation of the actual processes that occur on Mars.

**Important note to Geology readers:
In the Geology paper, I used the term melting to refer to the ablation of ice. However, this mistakenly implies that there is liquid water, which there is not. The ice goes directly to vapor, and sublimation is the correct term. The final, printed version of the Geology paper also contains two typographical errors in the equations mistakenly introduced in final copyediting. The errors are only in how the equations were printed - they do not affect the science or the results of the paper.
Equation (1), the diffusion equation, should read

instead of

Also, the cross product should be a dot product everywhere in paper:

instead of

The equations of the paper can be written most compactly as

I apologize for these confusing errors in the paper. The corrections will be printed in Geology in next month's issue and are included as a link in the version of the paper now online.