Networked configurations as an emergent property of transverse aeolian ridges on Mars

Transverse aeolian ridges – enigmatic Martian features without a proven terrestrial analog – are increasingly important to our understanding of Martian surface processes. However, it is not well understood how the relationships between different ridges evolve. Here we present a hypothesis for the development of complex hexagonal networks from simple linear forms by analyzing HiRISE images from the Mars Reconnaissance Orbiter. We identify variable morphologies which show the presence of secondary ridges, feathered transverse aeolian ridges and both rectangular and hexagonal networks. We propose that the formation of secondary ridges and the reactivation of primary ridge crests produces sinuous networks which then progress from rectangular cells towards eventual hexagonal cells. This morphological progression may be explained by the ridges acting as roughness elements due to their increased spatial density which would drive a transition from two-dimensional bedforms under three-dimensional flow conditions, to three-dimensional bedforms under two-dimensional flow conditions. Transverse aeolian ridges on Mars develop into networked configurations by formation of secondary ridges, reactivation of primary ridges and a transition from 3D to 2D air flow, according to analyses of HiRISE images from the Mars Reconnaissance Orbiter.

TAR morphotypes are characterized based on crest-line morphology and range from simple with straight and approximately parallel crests without bifurcations to more complex networked closed polygonal forms 1,2,34,35 (Supplementary Table 1, Fig. 1). At least two formative mechanisms have previously been proposed: a dust-deltoid path 6 and barchan-TAR 2 path (Supplementary Fig. 1). Recent work has documented a compound-formation process whereby secondary ridges can form perpendicularly to connect primary ridges 34,35 (Fig. 1a, b). This combination of ridges forms "feathered TARs", a precursor to rectangularnetworked TARs (Fig. 1c) 2,[34][35][36] . While the close spatial relationship between feathered TARs and networked TARs has been noted previously, the developmental relationships between these forms have only been addressed recently [34][35][36] . The connections between sinuous TARs, rectangular networks, and hexagonal networks (Fig. 1d) have been largely unexplored.
We found that primary ridges appear to be reactivated by secondary ridges forming scallops in the primary ridge crest (Fig. 2a-c, Supplementary Fig. 2). These scallops are emergent kink defects in the primary ridge that transform straight crests into sinuous ones, and quadrilateral cells into hexagonal cells. These kinks evolve to form stable-junction angles of~120°. This process is guided by the evolution of asymmetric stress concentration near vertices, which shifts the vertex; over time, symmetry produces a Y junction 37 . We applied metrics that easily differentiate networked morphologies by: 1) the number of vertices per cell and 2) the number of cells per vertex (nodal degree) (Fig. 2d, e) 38 . Rectangular networked TARs average four vertices per cell and a nodal degree of two to four (Fig. 2d). Hexagonal networked TARs have six vertices per cell and a nodal degree of three (Fig. 2e).
Measures of ridge amplitude further differentiate the forms. Immature rectangular networks have at least two prominent and dissimilar ridge amplitudes: wider and taller primary ridges with narrower and shorter secondary ridges perpendicular to the main ridge crests ( Supplementary Figs. 3 and 4). Well-developed Fig. 1 The progression from simple linear TARs to networked TARs. a Secondary ridges form on larger ridges, producing "feathered" TARs 34,35 (ESP_018597_1525). b These secondary ridges become larger and more prominent-often with the formation of tertiary ridge features perpendicular to the secondary crests (ESP_018597_1525). c Secondary ridges enlarge and become principal morphological features, forming regular rectangular cells (ESP_017661_1545). d Rectangular cells form into a hexagonal network (ESP_064491_1950). Coordinates mark the center of each image. North is up in these and the following images. All HiRISE images credit to NASA/JPL/University of Arizona. rectangular networks exhibit cells that have similar amplitudes, while hexagonal-networked ridges have approximate equal amplitudes in all directions ( Supplementary Fig. 4) 34 .
Under the hypothesis that hexagonal networks develop from rectangular networks, there is an expectation that a spatial correlation should exist where both forms occur. At these locations (Fig. 3), hexagonal networked TARs display residual rectangular forms (i.e., relict-identifiable primary/secondary ridges) at the margins of the field ( Supplementary Fig. 5), whereas the converse pattern-rectangular TARs surrounded by hexagonal TARs-was not observed at any location. Previous studies have noted that TARs have an affinity for topographic depressions 2,3,9,30 , and we found that this pattern was especially true for hexagonalnetworked TARs ( Supplementary Fig. 6, Supplementary Table 2).

Results and discussion
Relating TAR morphologies in a new evolutionary path. Forked TARs represent simple TARs that have developed the defects that would be common in similarly well-developed fields of dunes or ripples. Bifurcation defects are widely accepted to be emergent properties of dune and ripple fields on Earth that develop as the field ages [39][40][41][42] . As such, classifying forked TARs as a disparate morphology from simple TARs is an incorrect distinction and one that would not be drawn in a dune or ripple field on Earth.
Feathered TARs are intermediate forms between simple/ forked TARs and sinuous or networked TARs 34,35 . Cratering superpositioning shows that the secondary ridges characteristic of feathered TARs usually develop some time after the original crest has been stabilized 34,35 , suggesting that feathered TARs form after simple TAR formation. The physics underlying the development of secondary ridges remains poorly understood, but analogous features have been observed in ripples where trailing helical vortices are generated by the main transverse crest 43,44 .
Barchan-like TARs form a bent two-pointed shape (Supplementary Fig. 7). They were originally envisioned as a possible precursor morphology from which other TAR forms could be derived 2 ( Supplementary Fig. 8). However, well-developed barchan-like TAR features are very rare on Mars, and we argue that a generalized model for TAR development should not be based on a rare edge-case morphology. Rather, we suggest that barchan-like TARs develop from simple transverse forms that experience a combination of unusual windflow conditions and/or sediment supply.
Sinuous TARs were first hypothesized as the product of merging barchan-like TARs 2 . We instead argue that emergent kink defects initiated by secondary ridges on primary ridges create this sinuous morphology (Fig. 4). This conceptualization places the evolution of sinuous TARs after linear simple TARs and before networked TARs.
Rectangular-networked TARs form as secondary ridges grow between nonkinked simple TARs ( Fig. 4a-c) 34 . We hypothesize that hexagonal networked TARs develop from sinuous TARs (Fig. 4e-h) or from rectangular TARs that develop kinking (Fig. 4c, d). The original primary/secondary ridges can still be differentiated until the hexagonal network is fully formed (Supplementary Fig. 5). This change from rectangular to hexagonal is supported by the observed spatial patterning outlined above. If rectangular networks evolved from hexagonal forms, we would expect to see primitive hexagonal TARs surrounding well-developed rectangular cells, but the opposite was observed. This correlation is not concrete evidence of this change, but the spatial relationships combined with observations of incomplete kinking are supportive of the pathway outlined here.
Networked TARs as the most-evolved end member. We hypothesize that networked TARs represent the endpoint of TAR evolution. We believe that this evolution is caused by the interaction between the ridges and flow transitioning from a 3D interface to a 2D one (Fig. 5a). Under a 3D-flow regime, the topography of the ridges themselves plays an important role in shaping the flow over the ridges and in generating turbulence that creates subsequent features. Conversely, under a 2D-flow regime, well-connected TARs act as roughness elements in the flow, generating a skimming-flow regime.
Skimming-flow regimes exist when roughness elements are spaced so densely that their wakes completely overlap across the surface, creating stable vortices and causing the flow to pass smoothly above the roughness elements [45][46][47] . As such, the spatial density and consistent amplitudes of the ridges present in networked TARs make them ideal roughness elements. This notion that bedforms can act as roughness elements themselves is well-established 48,49 .
If our evolutionary sequence is correct, there must be some process that forces the system toward networked configurations 42 . Empirical measurements comparing flow over "2D straight dunes" 49,50 (analogous to simple/forked TARs), to 3D forms, suggest such an evolution toward networked forms. This transition proceeds via (1) "Full-width saddle" (barchan TARs), (2) "Fullwidth lobe" (a reversed barchan form, unobserved in nature), (3) "Sinuous crests" (sinuous TARs), and (4) "Irregular crests"comparable to hexagonal networked TARs in their plan-view irregularity 50 (Supplementary Fig. 9). Note that these forms are not true dunes, and that they are simply analogous in their crest-line morphologies, this is not to contradict the conceptualization of TARs as megaripples 33 .
In these studies, upstream bedforms generated stacked wakes that composed the bulk of the observed turbulent flow 49 , consistent with previous predictions 51 . The barchan-like configurations increased boundary shear stress by 38% over the simple/ forked 2D forms, the sinuous configuration generated stresses comparable to the simple/forked morphology, and irregular (network-like) configurations generated 20% less stress than the simple/forked form 42 . 3D bedforms reducing flow resistance relative to 2D forms are consistent with previous work conducted over mobile beds 52 . Other studies found that both turbulence intensity and Reynolds stress declined as 3D bedforms developed, with 2D forms naturally evolving toward 3D forms and reaching a stable configuration 52 . This tendency toward 3D forms could explain why TARs change from one morphology to another.
Further, 2D dune forms (analogous to simple TARs) have been documented spontaneously developing into 3D forms (sinuous and networked TARs; see Figure 9 of the study 48 ). Other work has observed the development of perpendicular ridges on transverse ridges (termed "bedform spurs" 43,44 , similar to secondary ridges documented on TARs 34,35 ) created by vortices shed by the primary transverse ridge 28,29 . Interpretation of the above findings suggests that (1) the scarcity of barchan-like TARs may be the result of barchan configurations generating higher shear stresses, making them more susceptible to activation/reactivation by wind, more transient, and less likely to be stabilized and preserved than other TAR morphologies; (2) more complex TAR morphologies are lower-energy configurations than simple crests, supporting our contention that simple TAR morphologies must be Reactivation of the primary ridge forms kinked sinuous primary ridges, directly from a to e. Rectangular networked TARs are formed by the growth of secondary ridges between primary ridges that do not experience kinking (a-c). Hexagonal-networked TARs can develop from rectangular networks that develop kinking (c, d) or from sinuous TARs (e-g). In this conceptualization, d and g are essentially equivalent forms with disparate origins. Over time, the hexagonal networks become more irregular (h) and the original primary/secondary ridge structure is lost. precursors to composite ones: the system should transition from higher to lower shear stress.
Further research using computational fluid-dynamic simulations of TARs is warranted. Such simulations could measure the shear stresses on fully accurate TAR configurations, including the rectangular and hexagonal network forms that have only been roughly approximated in the current literature. Such a quantitative study could explicitly connect the hypothesized TAR development pathway and the forcing mechanism outlined here. Further, a global survey of TAR morphologies and distributions could provide insight into the developmental relationships between TAR morphologies. There has been some work to automate the mapping of TARs in general using neural networks 53 , but there must first be a consensus around the characteristics of each specific morphology as this study attempts to clarify.
If a global database of TAR locations and morphologies were created, it would enable scientists to examine both correlations between local contexts and TARs, such as between the surrounding terrain age or elevation and the predominant TAR morphology. A TAR database could also help characterize the overall abundance of each TAR morphology. Assuming the hypothesized pathway in this paper is correct, the abundance of each TAR morphology could thus be used as a demographic profile of relative TAR development. For example, if a plurality of TARs on Mars were simple, it would suggest that a combination of (1) TARs being relatively young and have not having had time to develop into more complex morphologies, and (2) the conditions required for further TAR development being rare.
Minimum-energy states. Figure 5a illustrates the hypothesized transition from 3D to 2D flow. In (I), two simple TARs (red) form in a flow (dashed line) generating downstream turbulence. In (II), secondary ridges (orange) begin to fill the interridge area, reducing turbulence. In (III), secondary ridges extend crest-tocrest between the primary ridges. Scalloping shifts the primary ridge laterally (blue arrow). Finally, in (IV), the networked configuration interridge amplitude (A 1 ) is nearly equal to the original primary ridge amplitude (A 2 ). The flow skims over the networked TARs without much interaction and shear stress and turbulence are minimized. Figure 5b illustrates a conceptual model of TAR development on Mars (red line) compared with Earth (blue line). The dashed green arrow marks the point where megaripples on Earth are flattened as they increase in height 27,50,51 . On Mars, megaripples are more stable and can grow larger since the lower wind pressures do not impose such a finite limit 27 . The Fig. 5b inset illustrates TAR morphological evolution. Simple/forked TARs develop first; occasionally unstable barchan types form; feathers form on simple TARs and develop into rectangular networks; some rectangular networks finally form hexagonal networks. Figure 5c shows all possible convex tiling combinations. Note the separation of hexagonal patterning (3) from rectangular forms (1,2). For comparison, natural fracturing phenomena are indicted in red. Of these, basalt fracturing (A) and to a lesser degree, permafrost (B) show similarities to hexagonal TAR forms. Mud cracks (C) and surface fractures in granite (D), while exhibiting space-filling tessellations, are more similar to rectangular TAR networks.
Similarities between TAR geometries and patterns of tiling and fragmentation 45 span a spectrum between "Platonic" quadrangles 54 (those tiling patterns with four vertices per cell) and Voronoi hexagons 55 (a mathematical method for partitioning space) that mirrors the rectangle/hexagon patterning observed in networked TARs (Fig. 5c). The visual similarity between networked TARs and Voronoi tessellations 56,57 is striking (Supplementary Fig. 10).
Our results suggest a connection between fragmentation dimensionality and the flow dimensionality observed with TARs. Quadrangle patterning dominates in 3D fragments, while hexagonal patterning dominates 2D fracturing 54 . We observe that 3D fragments share patterning with TARs in 3D flow (six vertices per cell, nodal degree of three), and 2D fragments share patterning with TARs in 2D flow (four vertices per cell, nodal degree of 2-4). The connection between these concepts is abstract, likely in the evolution of both systems toward lowerenergy configurations and in the efficient partitioning of space (e.g., during the cooling of basalt or the evaporation of water from mud). While at the moment speculative, we feel that this could be a fruitful avenue for future theoretical geomorphic research.

Methods
HiRISE images were downloaded as map-projected JPEG2000 files using the 'random image' tool on www.uahirise.org. Images containing examples of welldeveloped networked TARs or other relevant features were integrated into an ArcGIS Pro 2.7.2 file 60 . If an image contained large scan-line artifacts, earlier or later images of the same area were downloaded to mitigate errors from these artifacts. If interesting TARs were near the edge of an image or were cut off in the original images, adjacent images were searched for and downloaded where available. The distribution of the selected images can be seen in Supplementary Fig. 11, and a complete list of images used is found in Supplementary Table 3.
Ridge crests were manually traced using polylines in a geodatabase. Where DTMs were available, amplitude transects were derived via the 'interpolate shape' tool in ArcGIS Pro (pro.arcgis.com/en/pro-app/latest/tool-reference/3d-analyst/ interpolate-shape.htm). Where DTMs were not available but stereoimage pairs were, visual estimations were made using the full-resolution anaglyphs. Examples of primary ridge reactivation by secondary ridge scalloping were traced and saved as a line file in the geodatabase. Hanging-ridge segments were excluded from our networked TAR metrics: for nodal degree, only completely closed polygons were included and for "number of vertices per cell", only vertices that were part of a completely closed polygon were included.
The Voronoi diagram in Supplementary Fig. 10 was generated by the Create Theissan Polygons tool in ArcGIS Pro (pro.arcgis.com/en/pro-app/latest/toolreference/analysis/create-thiessen-polygons.htm). The centroid points of the manually traced cells were extracted via geometry calculation, and subsequently given as input to create the synthetic cells.

Data availability
All HiRISE images analyzed in this study are publicly available from NASA/JPL/ University of Arizona: https://www.uahirise.org/.
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