How Do Spiral Troughs Form on Mars?

How do spiral troughs form on Mars?

Jon D. Pelletier Geosciences Department, University of Arizona, 1040 East Fourth Street, Tucson, Arizona 85721, USA

ABSTRACT positive h). The model includes three process- A three-dimensional model for the coupled evolution of ice-surface temperature and es: lateral heat conduction, elevation in the Martian polar ice caps is presented. The model includes (1) enhanced heat Tץ absorption on steep, dust-exposed scarps, (2) accumulation and ablation, and (3) lateral (ϭ␬ٌ2T; (1 Tץ conduction of heat within the ice cap. The model equations are similar to classic equations for excitable media, including nerve ®bers and chemical oscillators. In two dimensions, a small zone of initial melting in the model develops into a train of poleward-migrating accumulation and ablation, troughs with widths similar to those observed on Mars. Starting from random initial conditions, the three-dimensional model reproduces spiral waves very similar to those in h 1ץ the north polar ice cap, including secondary features such as gull-wing±shaped troughs, ϭ T; (2) t ␶fץ bifurcations, and terminations. These results suggest that eolian processes and ice ¯ow may not control trough morphology. and heat absorption on sun-facing, dust- Keywords: Mars, polar, ice cap, ablation, spiral trough. exposed scarps,

Tץ INTRODUCTION cal ice-surface temperature, T, from its equi- ϭ f (T, h), (3) tץ Howard (1978) proposed that the spiral librium value (the temperature at which no troughs of Mars form with an instability in accumulation or ablation takes place), and the which the ice-surface temperatures of steep, depth of local ice-surface topography, h, be- where equator-facing slopes exceed 0 ЊC during low its equilibrium value (i.e., troughs have the summer, melting the ice locally to form steeper, lower-albedo scarps (through exposure of subsurface dust-rich layers) in a self- enhancing feedback. In this model, some of the water vapor released from the equator- facing scarp may accumulate on the pole- facing slope to form a self-sustaining, poleward-migrating topographic wave. The re- lationship between this model and the spiral morphology of troughs has not been fully es- tablished, but Fisher (1993) introduced an asymmetric ice-velocity ®eld into Howard's model and obtained spiral forms. Recent observations from the Mars Global Surveyor, however, suggest that ice ¯ow near the troughs is not signi®cant (Howard, 2000; Kolb and Tanaka, 2001), and it remains unclear pre- cisely how the spirals form and what controls their spacing, orientation, and curvature. In order to establish a clear link between process and form for the spiral troughs, I have constructed a three-dimensional model for trough initiation and evolution. The model is based on the processes in Howard's migrating- scarp model and includes the simplest mathematical descriptions of these processes in order to determine the necessary conditions for realistic spiral troughs. Lateral heat conduction is also included in the model, and this element is crucial for obtaining realistic troughs. The model does not include wind erosion or ice ¯ow. The equations are similar to Figure 1. Diagram of trough evolution and solution to equations (see text) in two those from the ®eld of excitable media, in dimensions. A: Trough initiation involves rapid increase in ice-surface temperature which solitary and spiral waves resulting from to its maximum value as scarp is steepened and dust layers are exposed. During the dynamics of two interacting variables have trough deepening, value of h steadily increases to its maximum value over time scale ␶f ഠ 5 m.y., while temperature decreases in deeper parts of trough due to topographic been studied in detail. shielding. Note that ؊h is plotted instead of h in order to show actual topographic shape (h refers to trough depth, ؊h refers to elevation). B: Solution to equations (see MODEL DESCRIPTION 2 .0.2 ؍ and T0 ,1 ؍ ␶f ,0.05 ؍ km , ␶i 175؍text), using Runga-Kutta integration, with ␬ 8is ؍ The model equations consist of two cou- Initial condition is small pulse with T > T0 at left side of domain. Solution at t1 pled, scaled equations for the deviation of lo- shown with three well-de®ned troughs.

᭧ 2004 Geological Society of America. For permission to copy, contact Copyright Permissions, GSA, or [email protected]. Geology; April 2004; v. 32; no. 4; p. 365±367; doi: 10.1130/G20228.1; 2 ®gures. 365 Figure 2. Numerical model results and north-polar topography. A: Shaded-relief image of Martian north-polar ice-cap digital elevation model (DEM) constructed by using Mars Orbiter Laser Al- timeter topography. White -region north of 88؇N indi cates region with DEM arti- facts. Large-scale close- up indicates examples of gull-wing-shaped troughs, bifurcations, and terminations. Highest elevations are red and lowest elevations are green. B±E: Shaded- relief images of model to- ,10 ؍ pography, ؊h, for (B) t and ,1000 ؍ D) t) ,100 ؍ C) t) starting from 2000 ؍ E) t) random initial conditions. ؍ Model parameters are L 100 (length and width of grid, in nondimensional

␶f ,0.05 ؍ ␶i ,0.3؍units), ␬ -see text). Model evolu) 1 ؍ tion is characterized by spiral merging and alignment in equator-facing direction. Steady state is eventually reached, as shown in E, with uniformly rotating spirals oriented clockwise or counterclockwise depending on random initial conditions.

1 step, the ice-surface temperature is assumed to trough deepening. Here I assume that ␶i ϭ f (T, h) ϭ [T(T Ϫ T0)(1 Ϫ T) Ϫ h] (4) ␶i increase from its threshold value for melting, 0.05␶f ϭ 0.25 m.y., which implies that ϳ25 T0, to a maximum value of 1 over a time scale m of ablation must occur before the incipient if ␶i. This instability is represented mathemati- scarp is suf®ciently steepened for subsurface cally in the term f(T, h) by a cubic polynomial dust to be exposed and the maximum ice- h ϫ rÃ Ն 0, (5) that generates a negative feedback if T is be- surface temperature to be reached. The modelٌ low the melting temperature, T0, and a posi- behavior is not sensitive to the precise value and tive feedback for larger temperatures, bounded of ␶i as long as it is small compared with ␶f. by a maximum value of 1. This dynamical be- The threshold temperature for melting, rela- 1 havior represents the triggering of melting at tive to the equilibrium and maximum temper- f (T, h) ϭϪ h (6) ␶i a threshold temperature and the positive feed- atures, is between 0 and 1 and varies as a back between ablation, slope gradient, and al- function of latitude. At 75Њ latitude, the max- if bedo as the scarp is formed. In the second imum summer temperature of a nearly ¯at, step, the temperature remains near its maxi- high-albedo slope is ϳϪ10 ЊC (Howard, h ϫ rÃ Ͻ 0. (7) mum value of 1 as the trough deepens over a 1978). The temperature difference betweenٌ longer time scale, ␶f. Trough deepening does equator-facing troughs and nearby ¯ats is ϳ40 In the equations, ␬ is the thermal diffusivity not continue inde®nitely, however, because ЊC (Howard, 1978). If these values are used, of ice, ␶i and ␶f are the time scales for initial deep portions of equator-facing scarps do not T0,or0ЊC, is equal to 0.2. I assume in the and ®nal scarp formation, respectively (de- absorb as much solar energy as shallow por- model that this value applies to the entire ice scribed below), T0 is the threshold temperature tions due to partial shielding from the pole- cap for simplicity. The equations can be fur- for melting, and rÃ is the radial unit vector facing slope. The heat-absorption term f(T, h) ther simpli®ed by scaling time to the value of away from the pole. The model does not in- includes a term, Ϫh, to represent this negative ␶f. With this rescaling, the model is reduced clude diurnal or annual cycles; these are as- feedback. As the trough deepens and h in- to three independent parameters: ␬ϭ175 2 sumed to be included in the equilibrium tem- creases in value, the temperature eventually km , ␶i ϭ 0.05, and T0 ϭ 0.2. perature. Deviations from the equilibrium falls below the melting temperature and ac- The equations herein, without the direction- temperature initiate ablation and accumulation cumulation begins, eventually returning the al dependence in the term f(T, h) are one ex- at any time, t, in the model, although accu- ice surface to its equilibrium elevation. The ample of the FitzHugh-Nagumo (FHN) equa- mulation and ablation of water ice on Mars cubic term in f(T, h) applies only to equator- tions, well-studied equations that describe are understood to be restricted to the summer facing slopes (i.e., ٌh ϫ rÃ Ն 0). excitable media. Examples of excitable media months. The equations include four parameters: ␬, to which the FHN equations have been ap- The initiation and deepening of troughs is ␶i, ␶f, and T0. The thermal diffusivity of ice is plied include predator-prey systems (Murray, assumed to take place in two steps, illustrated ␬ϭ35 m2/yr. To estimate the vertical ablation 1989), electrochemical conduction in the heart in Figure 1A, and controlled by the two time rate on sun-facing slopes, Howard (1978) used (Winfree, 1987), chemical oscillators (Fife, scales ␶i and ␶f in the model. In the ®rst step, Viking observations of summer atmospheric 1976), and electrical transmission lines (Na- slope steepening and dust exposure are initi- water-vapor concentration to obtain 10Ϫ4 m/ gumo et al., 1965). The FHN equations come ated from an undissected ice surface through yr, implying ␶f ഠ 5 m.y. for an average trough in several forms that all reproduce solitary a positive feedback involving ice-surface tem- of 500 m depth. The time of scarp initiation waves in two dimensions [i.e., h(x)] and spiral perature, slope gradient, and albedo. In this is uncertain, but may be small compared with waves in three dimensions [h(x,y)] for an ap-

366 GEOLOGY, April 2004 propriate range of parameter values. Excitable from random. In these calculations I used the argued that certain secondary features of the media are composed of two variables: an ``ac- Barkley (1991) approximation to equation 1, spiral troughs, including gull-wing shapes, tivator'' (T in the equations) and an ``inhibi- which uses a semi-implicit algorithm to inte- provide further evidence for eolian control. tor'' (h in the equations). Triggering or exci- grate the term f(T, h) in order to gain com- The model troughs of this paper, however, re- tation of the medium occurs when a positive putational speed. This Barkley (1991) algo- produce poleward orientations and gull-wing feedback is initiated above a threshold value rithm enables spiral evolution to be shapes, even though no eolian processes are of the activator. On Mars, this process is the investigated over long time scales. The early- present in the model. This result suggests that feedback between ablation, slope gradient, and time evolution of the model is characterized eolian processes may not control trough mor- albedo that occurs above 0 ЊC. Local trigger- by the initiation, growth, and merging of spi- phology, although eolian and/or mass- ing may initiate the triggering of adjacent rals. It should be noted that heat conduction movement processes must certainly be present zones through diffusive spreading of the ac- in the neighborhood of an equator-facing scarp in order to transport dust from the scarp tivator (i.e., diffusion of heat laterally through may initiate melting in nearby regions, even (Howard, 2000). The agreement between ob- the ice cap). The value of the inhibitor increas- if those regions face away from the equator. served and modeled troughs also suggests that es during excitation and eventually returns the The late-time evolution of the model is char- ice ¯ow is not required to generate realistic system to the unexcited state through the neg- acterized by continued spiral merging, align- spiral forms. Instead, Howard's original scarp- ative coupling between h and T in the equa- ment of troughs to face the equator, and a migration model, augmented by lateral diffusion, is suf®cient to reproduce the basic fea- tions. On Mars this process is the topographic gradual increase in trough spacing as ``de- tures of the polar troughs. shielding of deep portions of the equator- fects'' are removed from the system. Mature facing scarp by the pole-facing slope. The spirals face the equator at low latitudes, but REFERENCES CITED FHN equations exhibit both excitable behav- curve toward the pole at higher latitudes, just Barkley, D., 1991, A model for fast computer sim- ulation of waves in excitable media: Physica ior, characterized by wide swings in h and T as those on Mars. In the model, this poleward D, v. 49, p. 61±70. when ␶i K ␶f, and oscillatory behavior, char- orientation evolves as part of the development Fife, P.G., 1976, Pattern formation in reacting and acterized by smaller ¯uctuations when ␶i ഠ ␶f of steady-state, rigidly rotating spiral forms. diffusing systems: Journal of Chemical Phys- (Gong and Christini, 2003). Although both re- By developing poleward orientations, high- ics, v. 64, p. 554±564. Fisher, D.A., 1993, If Martian ice caps ¯ow: Abla- gimes may have application to Mars, here I latitude scarps face perpendicular to the equation mechanisms and appearance: Icarus, focus on the excitable regime. tor, thereby reducing ablation rates and migra- v. 105, p. 501±511. tion speeds. Decreased migration speed offsets Gong, Y., and Christini, D.J., 2003, Antispiral the smaller distance required for high-latitude waves in reaction-diffusion systems: Physical MODEL RESULTS Review Letters, v. 90, article 088302. The solution of the equations in two di- scarps to complete a revolution, enabling spi- Howard, A.D., 1978, Origin of the stepped topog- mensions is given in Figure 1B, starting with rals to rotate rigidly and preserve their steady- raphy of the Martian poles: Icarus, v. 84, a narrow pulse of localized melting on the left state form. The model also reproduces a num- p. 2581±2599. ber of secondary features of polar troughs on Howard, A.D., 2000, The role of eolian processes side of the domain (T ϭ 0.25 in the pulse, T in forming the surface features of the Martian ϭ 0 elsewhere). The pulse initiates a self- Mars, including gull-wing±shaped troughs, bi- polar layered deposits: Icarus, v. 144, sustaining train of solitary waves that propa- furcations, and terminations (Figs. 2A, 2C, p. 267±288. gate from left to right with the trough wave and 2D). Kolb, E.J., and Tanaka, K.L., 2001, Geologic his- The steady state of the model after t ϭ 2000 tory of the polar regions of Mars based on trailing the temperature wave. The width of Mars Global Surveyor data: Icarus, v. 154, is a set of seven rigidly rotating spirals. The the troughs is given by ␭ ഠ ͙␬␶f,or13km p. 22±39. for the parameters characterizing the Martian sense of spiral orientation has an equal prob- Murray, J.D., 1989, Mathematical biology: New polar ice caps. This value is in good agree- ability of being clockwise or counterclockwise York, Springer-Verlag, 767 p. in the model, depending on the random- Nagumo, J., Yoshizawa, S., and Arimoto, S., 1965, ment with observed trough widths on Mars Bistable transmission lines: IEEE Transactions (Zuber et al., 1998). This behavior contrasts number seed used to generate the initial con- on Circuit Theory, v. 12, p. 400±412. with that of previous models, in which troughs ditions. The spiral orientation of Figure 2E is Smith, D., Neumann, G., Arvidson, R.E., Guiness, opposite to the orientation of the spirals E.A., and Slavney, S., 2003, Mars Global Sur- widened inde®nitely without reaching a steady veyor Laser Altimeter Mission Experiment state. Another similarity with the observed to- around the north pole of Mars, but 50% of the Gridded Data Record: NASA Planetary Data pography of spiral troughs is trough asym- random initial conditions I constructed for the System, MGS-M-MOLA-5-MEGDR-L3- metry: both in the model and on Mars, equa- model lead to spirals that rotate in the same V1.0. sense as those on Mars (i.e., clockwise away Winfree, A.T., 1987, When time breaks down: The tor-facing scarps are about twice as steep as three-dimensional dynamics of electrochemi- pole-facing scarps (Zuber et al., 1998). from the pole). This behavior suggests that a cal waves and cardiac arrythmias: Princeton, Spiral formation in equations 1±4 and 6 is uniform sense of spiraling need not represent New Jersey, Princeton University Press, illustrated in Figures 2B±2E for an idealized an asymmetrical process of trough evolution. 339 p. Instead, it could be the result of a minor asym- Zuber, M.T., Smith, D.E., Solomon, S.C., Abshire, circular ice cap with a permanently frozen re- J.B., Afzal, R.S., Aharonson, O., Fishbaugh, gion near the pole and starting from random metry in the initial pattern of melting, as in K., Ford, P.G., Frey, H.V., Garvin, J.B., Head, initial conditions. A shaded-relief image of a the model. J.W., Ivanov, A.B., Johnson, C.L., Muhleman, polar-stereographic digital elevation model D.O., Neumann, G.A., Pettengill, G.H., Phil- lips, R.J., Sun, X., Zwally, H.J., Banerdt, (DEM) of Mars Orbiter Laser Altimeter DISCUSSION AND CONCLUSIONS W.B., and Duxbury, T.C., 1998, Observations (MOLA) data (Smith et al., 2003) is given in Several recent studies have argued that of the north polar region of Mars from the Figure 2A for comparison with the model re- wind erosion and ice ¯ow exert a signi®cant Mars Orbiter Laser Altimeter: Science, v. 282, p. 2053±2060. sults. Random initial conditions were chosen control on trough morphology. Howard for this paper to illustrate how spirals self-or- (2000), for example, proposed that the corre- Manuscript received 21 September 2003 ganize in this system from structureless initial lation between wind directions and trough ori- Revised manuscript received 14 December 2003 Manuscript accepted 17 December 2003 conditions; the actual initial conditions on entations at high latitudes is the result of eo- Mars are unknown but may have been far lian processes. In addition, Howard (2000) Printed in USA

GEOLOGY, April 2004 367 CORRECTION

How do spiral troughs form on Mars?: Correction The cross product should be a dot product everywhere it appears Jon D. Pelletier (equations 5 and 7 and in the text). For example, equation 5 should read Geology, v. 32, p. 365±367 (April 2004) (Several errors appear in the text of this article. They are: ٌh ´ rÃ Ն 0 (5 Equation 1 should read instead of Tץ (ϭ␬ٌ2T, (1 (t ٌh ϫ rÃ Ն 0 (5ץ rather than Two additional errors were made in the manuscript text. First, the Barkley (1991) approximation was applied to equation 4, not to equa- T tion 1 as stated. Second, the term melting was incorrectly used to referץ (ϭ␬ٌ2T, (1 .T to the ablation process of the ice cap. Sublimation is the correct termץ

384 GEOLOGY, May 2004