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Geomorphology 91 (2007) 311 ­ 331

Complex systems in aeolian geomorphology

Andreas C.W. Baas

Department of Geography, King's College London, Strand, London WC2R 2LS, UK Received 25 September 2006; accepted 30 April 2007 Available online 1 August 2007

Abstract Aeolian geomorphology provides a rich ground for investigating Earth surface processes and landforms as complex systems. Sand transport by wind is a classic dissipative process with non-linear dynamics, while dune field evolution is a prototypical self-organisation phenomenon. Both of these broad areas of aeolian geomorphology are discussed and analysed in the context of complexity and a systems approach. A feedback loop analysis of the aeolian boundary-layer-flow/sediment-transport/bedform interactions, based on contemporary physical models, reveals that the system is fundamentally unstable (or at most meta-stable) and likely to exhibit chaotic behaviour. Recent field-experimental research on aeolian streamers and spatio-temporal transport patterns, however, indicates that sand transport by wind may be wholly controlled by a self-similar turbulence cascade in the boundary layer flow, and that key aspects of transport event timeseries can be fully reproduced from a combination of (self-organised) 1/f forcing, motion threshold, and saltation inertia. The evolution of various types of bare-sand dunes and dune field patterns have been simulated successfully with self-organising cellular automata that incorporate only simplified physically-based interactions (rules). Because of their undefined physical scale, however, it not clear whether they in fact simulate ripples (bedforms) or dunes (landforms), raising fundamental cross-cutting questions regarding the difference between aeolian dunes, impact ripples, and subaqueous (current) ripples and dunes. An extended cellular automaton (CA) model, currently under development, incorporates the effects of vegetation in the aeolian environment and is capable of simulating the development of nebkhas, blow-outs, and parabolic coastal dunes. Preliminary results indicate the potential for establishing phase diagrams and attractor trajectories for vegetated aeolian dunescapes. Progress is limited, however, by a serious lack of appropriate concepts for quantifying meaningful state variables at the landscape scale. State variables currently used in the bare-sand models are far from capturing the rich 3D topography and patterns and are not sufficiently discriminative to distinguish different attractors. The vegetation component in the extended model, and indeed ecogeomorphic systems in general, pose even graver challenges to establishing appropriate state variables. A re-examination of older concepts, such as landscape entropy, perhaps complemented by recent developments in information theory, may be a potentially fruitful avenue for research, although the outlines of such an implementation are still rather vague. © 2007 Elsevier B.V. All rights reserved.

Keywords: Complexity; Self-organisation; Aeolian; Sediment transport; Dunes; Streamers; Vegetation; Cellular automaton

1. Introduction Aeolian sediment transport and landscape development by wind presents a rich mixture of complex systems at various spatial and temporal scales, as motions in the

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atmospheric boundary layer interact with the Earth's surface to dissipate energy derived from synoptic scale pressure systems to frictional heat and rearrangement of sedimentary material. Form and process in aeolian geomorphology have in the past largely been treated separately, landforms and dune field evolution principally being understood by

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means of (static) morphometric descriptions while sediment transport is captured in reductionist (dynamic) process models. Recently, advances in numerical modelling have facilitated investigation of dune field evolution and dune migration dynamics as process (either with reductionist simulation models or with self-organising cellular automata), and advances in field experimentation and instruments have allowed recognition of form in sediment transport, such as aeolian streamers and patterning in spatial and temporal transport variability. A re-interpretation of both land-form and transportprocess in the context of complex systems may lead a way forward to an integrated framework of common principles and language that can, for example, relate turbulence characteristics in the mixing layer via interactions between sediment transport and sparse vegetation elements to the development of a nebkha field in a predictive physical model. This may be one step toward an Earth System Science that is analogous in explanatory power to what statistical mechanics has achieved for understanding physical systems with large number of particles, such as molecular gases. The lack of such an analogy may be perceived as the main weakness of the much touted Earth System Science paradigm (e.g. Richards and Clifford, 2006). This paper presents an interpretation of aeolian sediment transport and dune landscape dynamics as complex systems, discussing new theoretical approaches and empirical results, indulging in a little speculation, and identifying key research questions for future lines of investigation. Complex systems usually involve three fundamental properties: first, an open and dissipative system with fluxes of energy and/or matter supplied from an external source flowing through the system and expended to an energy sink in a degraded form, usually as heat. Second, a system involving a large number of particles or elements that, third, are dynamically related via non-linear interactions and feedbacks. The phenomenology of such systems include emergent behaviour and dissipative structures in the form of spatial or temporal patterns, evolving through self-organisation. In such cases, some fundamental micro-level variable (e.g. a property of the elementary particles) can project its control to a characteristic geometry on the synoptic level (the scale of the system as a whole). On the micro-level though, the nonlinear dynamics may generate chaotic behaviour, a sensitivity to initial conditions that progressively scuttles any precise predictability. This micro-level chaotic behaviour is often balanced, however, by distinctive pattern formation on the synoptic level, such as spatial organisation toward a fractal pattern, self-similar statistical

distributions (spatial and temporal) ­ e.g. 1/f signatures where event magnitude is inversely proportional to frequency of occurrence ­ or by fractal (`strange') attractors in the system's phase-space framed by state variables. A brief exposition of these concepts and their application in geomorphology in general may be found in Baas (2002) and references therein. A fitting prototype of such a system can be considered the (idealised) model of a sand pile subject to avalanching due to addition of grains at its top. This system clearly comprises a large number of elements in the form of sand grains, that interact non-linearly through friction and collision during an avalanche to dissipate directed gravitational energy into the degraded form of heat. It exhibits the classic signatures of a complex system, in that avalanching maintains a critical slope via self-organisation ­ or self-organised criticality according to Bak et al.'s (1988) and Bak's (1996) terminology ­ individual avalanching is unpredictable because of sensitivity to initial conditions and timeseries are chaotic but display a 1/f (i.e. self-similar) statistical frequency-size distribution. In this case it is the size and angularity of the sand grains (the microlevel) that control the angle of repose. An explicit fractal aspect of this prototype may be found in the spatial patterning of avalanches of various sizes around the pile. 2. Transport processes Aeolian sediment transport by wind in desert environments, coastal dunes, and on beaches can be treated as a classic complex system: it is a strongly dissipative system where the advective and turbulent kinetic energy of the boundary layer flow is dissipated by the initiation and (primarily) saltation of grains extracting momentum from the flow, which is subsequently transferred to the bed through inter-granular friction and collision (dissipated as heat) upon grain impact. Additionally, dissipation of turbulent kinetic energy in the flow occurs through the eddy-cascade, converted ultimately to heat by molecular viscosity. The sediment transport system further comprises a large number of elementary objects, i.e. the collection of sand grains in the bed and in the saltation layer, though it is also worthwhile to consider the cascade of eddies in the shear flow and ­ at a larger scale ­ aeolian streamers as manifold elementary objects. Sediment transport finally includes a suite of non-linear interactions and feedbacks between the bed, saltation, and airflow components, thereby meeting all the requirements for a complex system as such. Accordingly, there are a number of related phenomena that have been discovered and

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investigated both in the past and more recently, such as the Owen effect, saltation overshoot, hysteresis, motion threshold variability, intermittent transport, time-series signatures of both wind and sand transport, streamer patterns, and impact ripple development, that may be reinterpreted in the context of complexity in terms of manifesting aspects of emergent behaviour, self-organisation, self-organised criticality, sensitivity to initial conditions (chaos), and spatial and/or temporal selfsimilarity. 2.1. Physical models Aeolian saltation models traditionally relate transport rates to a cubic term of shear velocity U in the boundary layer flow (Bagnold, 1936), or excess shear above some threshold Ut (Kawamura, 1951; Lettau and Lettau, 1977), thus resulting in a non-linear forcing. The cubic relation is based both on dimensional analysis and on fundamental assumptions concerning certain proportionalities between shear stress on the bed and saltation trajectory length scales. While usually a logarithmic velocity profile is assumed for boundary layer flow above a smooth surface (the law-of-the-wall), Bagnold recognised that the flow must adjust to an increase in roughness created by active transport over the surface, i.e. it must accommodate the additional drag created by the extraction of momentum from the flow by the saltating grains. Bagnold modelled this adjustment by incorporating a fixed `focal point' in the vertical profile and an offset equivalent to the threshold wind velocity. Owen (1964) reasoned that the `apparent' or `aerodynamic' roughness created by the saltation layer is proportional to its thickness and hence 2 correlates with U , echoing Charnock's relationship for aerodynamic roughness of rippled water surfaces (Charnock, 1955; Sherman, 1992). Owen further argued that the shear stress on the bed under saltation was reduced to equal the motion threshold and hence the shear velocity at any height within the saltation layer attains a constant (impact-) threshold level. This feedback and adjustment between the fluid flow and sediment transport is now usually termed the Owen effect (Gillette, 1999), and has been reproduced by a range of numerical so-called `self-regulating cloud models' of aeolian saltation (Ungar and Haff, 1987; Werner, 1990; Anderson et al., 1991; McEwan and Willetts, 1993) which probably best capture the contemporary understanding of the system. More recent research has probed the interaction further (McEwan, 1993; Gillette et al., 1998), and the characteristics of the velocity profile within and above the saltation layer are

still under scrutiny (Spies et al., 1995; Butterfield, 1999; Li et al., 2004; Bauer et al., 2004). The current numerical and analytical flow-saltation models apply to steady-state equilibrium solutions only, and furthermore are largely based on wind tunnel observations and parameterisations. Indeed, Shao and Raupach (1992) reported a significant overshoot effect in initial saltation response to a newly imposed wind speed in their wind tunnel and the temporal response of the saltation system to fluctuations in the wind forcing has subsequently been investigated in a wind tunnel by Butterfield (1993) and modelled by Spies et al. (2000; Spies and McEwan, 2000). These studies have reported a significant saltation inertia limiting the one-dimensional response to fluctuations in forcing to a time-scale of 1 to 2 s. The saltation inertia is thus responsible (together with threshold variability, see below) for the distinct hysteresis effects noticeable in transport intensity response to near-surface gusting. This hysteresis has been observed in the wind tunnel studies and numerical models mentioned above as well as more recently detected in field experiments. Fig. 1 shows a time-trajectory of transport intensity at 4 cm above the bed, measured with a saltation flux impact responder (Baas, 2004), responding to the rise and fall in shear velocity (see Baas and Sherman, 2005 for further experimental detail). The arrows indicating the temporal sequence outline the complicated hysteresis pathways. The autofeedback between saltating grains impacting the bed and releasing ejecta that are entrained in the saltation layer led Bagnold (1941) to specify two types of threshold of initiation of grain motion: a clean-fluid threshold at the first instance of transport and a `dynamic' or impact threshold for subsequent continuation of saltation. It seems increasingly clear, however, that either threshold can not be treated as a local constant, but instead varies over a considerable and effectively unpredictable range. Not only is the threshold very sensitive to minute changes in surface moisture conditions (Namikas and Sherman, 1995; Cornelis et al., 2004), but Nickling (1988) showed that for naturally graded sand a gradual transition occurs from rolling and creeping to entrainment that often takes the form of a sudden cascade. Indeed, McMenamin et al. (2002) have reported on the chaotic nature of transport events or bursts of activity in a wind tunnel near the threshold condition and present a 1/f frequency-size distribution that they attribute to the grain-packing at the surface developing toward a self-organised critical state, much like the avalanching slope in the sand-pile model. This interpretation will be revisited later. A final component of the aeolian sediment transport system is the deformation of the bed surface in the shape

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Fig. 1. Time-trace of transport response to shear velocity at a resolution of 5 Hz, though based on 2-second smoothed data. Collected in the Windy Point experiment (Baas, 2003).

and migration of impact ripples. This aspect is not routinely included in contemporary numerical or analytical self-regulation saltation models, yet it is the only component that has been modelled and analysed in the context of complex systems behaviour. A stability analysis readily shows that a horizontal deformable surface is unstable under a shearing stress, such that small perturbations are amplified over time (McLean, 1990). In terms of physical mechanism, aeolian ripples had long been thought to be a direct result of the impact of saltating grains on the bed. Bagnold (1941) suggested that ripples were harmonically related to the mean saltation length, while Sharp (1963) proposed the growth of impact zones and shadow zones related to the usually very shallow impact angle of saltating grains over slight undulations. In the 1980s high-speed photography experiments and numerical simulations of single-grain impacts on granular beds by Willetts and Rice (1986a,b) and Werner and Haff (1988), respectively, discerned a population of low-energy ejecta, now called reptation, associated with the ballistic impact of a saltating grain. Subsequently Anderson (1987) proposed that the development and translation of impact ripples results from spatial variations in this reptation process and its related splash-function, rather than the ballistic saltating grains. Furthermore, the eventual ripple wavelength is a result of the complex merger of smaller

and larger ripples until a quasi-stable ripple-size is achieved by means of self-organisation (Anderson, 1990). These ripple dynamics have been successfully modelled with cellular automata (Werner and Gillespie, 1993; Landry and Werner, 1994; initiated by P.K. Haff), including size segregation and internal stratigraphy (Anderson and Bunas, 1993). These self-organising CA models have confirmed the fundamental control of grain-size over ripple geometry ­ known from wind tunnel and field experiments ­ something that traditional continuum models do not predict. Over the past two decades a large number of ripple simulation models have been developed, both based on self-organisation principles as well as on reductionist physical models (numerical or analytical), some with and some without an explicit reptation component (Nishimori and Ouchi, 1993; Hoyle and Woods, 1997; Hoyle and Mehta, 1999; Valance and Rioual, 1999; Werner and Kocurek, 1999; Yizhaq et al., 2004). Perhaps surprisingly, a complete physical numerical or analytical model that integrates all aspects of the aeolian sediment transport system described above has (to the author's knowledge) not been constructed: the self-regulation saltation models appear to ignore the development of ripples on the bed, while reductionist ripple models appear to neglect the feedback of the increased surface roughness on the wind forcing.

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2.2. Complex systems analysis The components and dynamics as described above may, at first instance, be recast in the phenomenology of complex systems: the interactions between the airflow and the saltation layer and the overshoot and delayed feedback response, for example, could behave analogous to a dampened (i.e. dissipative) oscillator, which under certain conditions exhibits chaotic response to periodic or random forcing. Indeed, the transport timeseries of one of Spies et al. (2000) periodic forcing experiments (Fig. 15 therein) shows a positively chaotic character. The variable nature of the threshold of motion adds a further complexity: the continuous rearrangement of surface grains (and small changes in moisture content for that matter) may help to either buffer or conversely amplify fluctuations in the saltation layer, or as suggested by McMenamin et al. (2002) may selforganise to a hair-trigger state where any small wind speed fluctuation may initiate a saltation avalanche. The latter may be more plausible if the velocity profile within the saltation layer is reduced to at or around the threshold velocity according to Owen's (1964) hypothesis. Such complexity provides one possible explanation for the occurrence of streamers (also known as sandsnakes) and patterns in aeolian sediment transport commonly observed in the field (Fig. 2): in terms of a characteristic emergent behaviour and spatial organisa-

tion of the system, with length and time scales that may be traced to fundamental system variables such as the grain-size and/or settling velocity. The next section demonstrates, however, that this approach is at odds with the current thinking about the formation and behaviour of aeolian streamers. The development of the rippled bed-surface, meanwhile, increases complexity further: the self-organising ripple field may ultimately induce a harmonic effect on the saltation layer, which again may result in either a chaotic or, in contrast, a correlated transport response. In turn, fluctuations and spatial variability in saltation may initiate crest termination and maintain bedform defect densities. A rudimentary analysis of the stability of the aeolian sediment transport system can reveal some insights into the relative effects of the interactions involved and a way forward to quantifying the dynamics. Systems analysis of this kind involves a linearisation of the set of differential equations that govern the interactions between the components via a Taylor series expansion, solving for the eigenvalues in the interaction matrix. The real parts of the eigenvalues are the Lyapunov exponents, which indicate potential for chaotic system behaviour if at least one is positive (Phillips, 1999). Such a stability analysis has been applied previously to river hydraulic geometry (Slingerland, 1981). An equivalent approach is the loop analysis of system `digraphs' presented by Puccia and Levins (1985). Loop

Fig. 2. Aeolian streamers moving toward the observer over a sandy beach surface at Camber Sands, England (with increased colour contrast).

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analysis is based on identifying the positive and negative feedback loops on all levels and to determine the relative strength of positive feedback loops over negative ones. If the former are stronger than the latter, the system will be unstable to perturbations and (in the case of non-linear interactions) likely subject to chaotic behaviour. Feedback at level k of a system, Fk, represents the mutual influence among elements in loops of k components: Fk ¼

k X m¼1

forcing, U, saltation flux, S, reptation/splash, R, and the bedform development, B. Interactions include proportional relationships (positive links) depicted by arrows, and negative links (inverse relationships) indicated by solid circles. These include: - shear stress from the boundary layer flow driving saltation, (+) - saltating grains draining momentum from the flow, saltation roughness: (-) z0s - saltation impacts driving reptation through splash, (+) s - reptation inducing ripple formation, (+) r - bedform development limiting reptation (if only in aerial extent), (-) b - bed morphology affecting shear stress through form drag, (-) z0f and two feedbacks: - a positive feedback for the saltation component through ejection and entrainment, (+) e, and - a negative feedback on ripple formation as gravity and angle of repose limit ripple formation, (-) g It is assumed that reptation is not energetic enough to maintain itself (no autofeedback on R), the reptation process is detached from the wind forcing (as shown by Haff and Anderson, 1993) and no direct effect of ripple development on saltation exists. With respect to the latter, however, it is also conceivable that bedforms, B, have a negative impact directly on saltation S (indicated by the dotted ? link) rather than or besides reptation only. Assuming the system description of Fig. 3 and retaining the signage of the interactions, the RouthHurwitz stability criteria demand: F1 ¼ e À gb0 F2 ¼ Àsz0s À br þ egb0 F3 ¼ Àsz0s g þ rbeb0 F4 ¼ Àssrz0f À sz0s brb0 and F1 F2 þ F3 ¼ Àsz0s e þ gbr þ e2 g À eg 2 N0 ð4Þ ð3aÞ ð3bÞ ð3cÞ ð3dÞ

ðÀ1Þmþ1 Lðm; k Þ


where L(m, k) is the product of the interaction equations of m disjunct loops (loops not sharing the same components) with a cumulative k components. For a system of 3 or 4 components two stability criteria can be established (so-called Routh-Hurwitz criteria): Fi b0 for all i ð2aÞ ð2bÞ

F1 F2 þ F3 N0

For systems with more than 4 components the RouthHurwitz criteria become cumbersome and are omitted here since they are not needed in the analysis that follows. Using the above criteria, an analysis of the signs of the interactions between components (i.e. positive or negative) is sufficient to yield information on stability of the system. A signed interaction diagram (digraph) of the aeolian sediment transport system is shown in Fig. 3, which includes the components and interactions described previously. The four system components include wind

Fig. 3. Digraph of the aeolian sediment transport system, including bedform development. Positive links indicated by arrows, negative links depicted by solid circles. The dotted ? link indicates a potential negative feedback of bedform development on saltation.

An exact evaluation of these criteria would require quantification of the relative strengths of the interactions, and this does not appear currently feasible. Some basic assumptions and scaling arguments, however, can lead to a tentative outcome. It may be argued, for example, that the ejection and the gravitational autofeedbacks (e and g)

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are relatively weak: saltation impact studies show that saltating grains lose significantly more energy to reptating grains than to new ejecta that are entrained in saltation (e.g. Rice et al., 1995) and ripple morphology only approaches the angle of repose (and hence gravity limitations) on the lee-side. If e and g are assumed correspondingly small, F1 is close to zero and F2 is likely negative. F3 is only negative if interactions between wind forcing and saltation flux are stronger than interactions between reptation and ripple formation. F4 is always negative regardless of the strength of interactions. The second Routh-Hurwitz criterion, meanwhile, can only be met if the fluid-flow/saltation subsystem is weaker than the reptation/bedform subsystem (the quadratic terms in e and g are small and effectively cancel out). This limited evaluation reveals that the system as a whole must be unstable or (at most) meta-stable since the requirements for F3 are contradictory to those for F1F2 + F3 N 0, and the two can not both be met at the same time, i.e. unstable and chaotic behaviour is likely. A more extended analysis of the potential level of chaotic sensitivity could be achieved by determining the actual magnitude of the largest Lyapunov exponent. Although the interactions between U, S, and R, can likely be quantified in some way, it is not evident how the bed morphology component can be specified appropriately, especially with regards to the 3D patterns formed by ripples and bedform defects. Thus, it appears that a fully quantified interaction matrix can not be achieved at present.

2.3. Streamers and transport patterns Our understanding of the aeolian sediment transport system is limited by the fact that many aspects of the saltation models have been developed and parameterised strictly in wind tunnel studies, as many of the grain-scale observations such as splash/reptation and fluid velocity profiles within the saltation layer can only be achieved under carefully controlled laboratory conditions. The wind tunnel environment, however, is fundamentally restricted in the type of flow and depth of boundary layer that can be achieved. Wind tunnels simply cannot reproduce the full spectrum of near-surface airflow turbulence over the range of spatial and temporal scales that are usually encountered in the outdoor atmospheric environment (Gerety, 1985; Kind, 1990; Anderson et al., 1991; Greeley et al., 1996) and there are critical scaling issues associated with wind tunnel calibrations of transport equations (Farrell and Sherman, 2006, in press). From recent advances in field instrumentation and experiments it is now becoming increasingly clear that atmospheric turbulence has a significant if not overwhelming control on sand transport in the atmospheric environment. Extensive field experiments by the author indicate that aeolian sand transport occurs primarily in the form of streamers (Fig. 2) that are driven by distinct eddies travelling down toward the surface through the boundary layer and scraping across the bed while exciting saltation along their trail (Baas, 2003; Baas and

Fig. 4. Wavelet power spectra of sand transport intensity and collocated streamwise wind speed, normalised by effective period. Slopes for selfsimilar response and theoretical Kolmogorov inertial subrange (turbulence dissipation) are indicated.

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Sherman, 2005). The experiments showed that bed surface control, such as differentiation in moisture or grain-size, hence local variability in motion threshold, is not a necessary condition for the formation of streamers, and that bed-induced coherent flow structures, such as burst-sweep events and streamwise vortices are not associated with the passage of these so-called sandsnakes. Instead, the notion that streamers are initiated by turbulence imposed from higher regions of the flow accords well with the top-down turbulence model for high Reynolds number boundary layer flow proposed by Hunt and Morrison (2000). Further insights into the relations between nearsurface turbulence and sediment transport in the form of streamers have been deduced from wavelet spectral analysis (Baas, 2006). Fig. 4 shows (normalised) spectra of sand transport intensity and collocated wind speed ­ the same type of data as in Fig. 1 ­ as a function of event period or duration (rather than the frequency or wave-number as is usual in Fourier spectral analysis). The graph also includes trend lines corresponding to the theoretical Kolmogorov inertial subrange (power f - 5/3 ) and to self-similarity (power f - 1). The streamwise wind speed spectrum shows a self-similar range for gusts lasting between 1 and 7 s and potentially extending up to 100 s underlying two external forcings at roughly 60 and 20 s. This self-similar range appears to correspond

with the process of surface blocking of eddies impinging on the bed, as the f - 1 signature is predicted by the Hunt and Morrison (2000) topdown turbulence model, and this hints at possible selforganisation in the eddy breakdown. The wind speed spectrum also suggests a roll-off to the inertial subrange (as would be expected) at the smallest effective periods (b 0.3 s). The sand transport spectrum indicates two temporal ranges of interest here: 1) a spectral response for events lasting less than approximately 0.8 s that trends between self-similarity and a white (random) noise, with a best-fit slope of f - 0.8, and 2) a range from roughly 1 to 7 s with a slope proportional to f - 1.5. The first range corresponds well with the saltation inertia reported from numerical and wind tunnel studies (discussed previously), with an upper limit of around 1 Hz for response to wind speed fluctuations, and is associated with the dissipation of saltation momentum. The second range captures the formation of streamers and the organisation of transport patterns in response to top-down turbulence forcing from the corresponding temporal range in the wind speed spectrum. This direct relationship can be demonstrated by transformation of idealised (synthetic) time-series. Fig. 5a presents such a time-series for wind speed, u (with normally distributed values around a mean and with a standard deviation identical to those of the field data of Fig. 4),

Fig. 5. Transformation of a synthetic wind speed time-series (upper-left panel) exhibiting self-similar spectral response (f- 1, upper-right) to a derivative sand transport time-series (lower-left) with a spectral response (f - 1.5, lower-right) matching that of the experimental field data shown in Fig. 4.

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and with an imposed power spectrum f - 1, as indicated by the Fast-Fourier-Transform in the righthand panel. This time-series is transformed to a transport sequence via the fundamental Bagnold-type relationship: q ¼ ð u À ut Þ 3 and q ¼ 0 if ubut ð5Þ

3. Dune field evolution The regularity and patterning of desert dune fields has naturally inspired many models to simulate their development by self-organisation using cellular automata. Similar to the sediment transport system, dune field development results from the dissipation of advective kinetic energy in the atmospheric boundary layer but on a larger scale: the winds of global circulation and regional pressure systems drive the formation and migration of dunes in transport corridors, valleys, and internal-basin sand seas (e.g. Wilson, 1971; Mainguet, 1984). The origin and development of various types of bare-sand dune fields, such as barchans, transverse, seif, linear, and star dunes, have long been established as a result of two principal factors: available sediment supply and wind regime variability (Wasson and Hyde, 1983; Cooke et al., 1993). These factors are usually quantified in terms of equivalent sediment thickness (EST) and RDP/DP ­ the ratio of resultant sand drift potential (vector sum) to total sand drift potential (scalar sum) ­ and the approximate requisite ranges of these factors have been collected for many of these dune types in a phase-state diagram. The airflow characteristics and transport patterns involved in the formation of various sand dunes are also relatively well understood. Numerous detailed field studies monitoring wind and transport directions as well as analyses of internal sedimentary structure have established the general formation mechanisms of many desert dunes (Pye and Tsoar, 1990; Lancaster, 1995). The development of some of these dune structures has been simulated numerically using coupled airflow and sand transport models (Wipperman and Gross, 1986; Weng et al., 1991; Van Dijk et al., 1999; Andreotti et al., 2002, amongst others). The modelling of entire dune fields, however, is hampered by the complex airflow dynamics and interactions between multiple dunes and associated computational challenges, while current analytical models are unable to reproduce the kind of stable dune migration patterns found in transport corridors (Hersen et al., 2004). A complex systems approach using cellular automaton modelling has already established a rich and productive alternative that will be discussed in detail in the first half of the section below. Current understanding of the effects of vegetation on dune landscape development is largely limited to descriptive models (Lancaster, 1995, pp.177). Although various investigations have uncovered details regarding the small-scale interactions between vegetation elements and aeolian sediment transport, on the larger scale of dune landscapes the role of vegetation is identified only on a

where ut is a constant threshold. Its value is derived from field data by the percentile of the wind speed distribution that is associated with the temporal transport activity detected by the saltation sensors (for this experiment 25%). Subsequently an exponential decay function is imposed to capture the dissipation of saltation inertia as: hðt Þ ¼ eÀ0:5t ð6Þ

where h(t) defines the fraction of transport flux remaining at time t after the initial transport magnitude. The resultant q time-series is shown in Fig. 5b and has a power spectrum f - 1.5 (FFT shown in right-hand panel), equivalent to the `streamer' range in the field data of Fig. 4. The spectral response is dependent on the threshold and on the decay function used: in this case the coefficient of -0.5 was derived by fine-tuning to yield the desired spectral response. The response is not particularly sensitive to this parameter though: a decay coefficient of -0.3 yields a spectral response f - 1.8 and a coefficient of -0.7 yields a spectral response f - 1.4). The selected decay coefficient of -0.5 used for Fig. 5b is equivalent to a `half-life' of 1.4 s for the intrinsic saltation flux decay, a value which is not unrealistic in light of the numerical and wind tunnel studies discussed previously. The results and analysis presented above indicate that aeolian sand transport may be modelled as in direct response to a self-organised, fractal break-down of eddies in the boundary layer flow impinging on the bed, combined with an exponential decay or dissipation of saltation inertia within the resulting streamers. This is quite different from the traditional saltation models that assume an inherently uniform and semi-steady shear stress forcing. It assigns boundary layer turbulence in near-complete control of sediment transport, and suggests that local details such as variability in motion threshold and the Owen effect only play a subordinate role. Further progress on understanding and predicting aeolian sand transport is most likely found in the complexity of the fluid flow dynamics. This is a vast and vibrant field of research in fluid mechanics; a comprehensive discussion of this literature is, however, beyond the scope of the present paper.

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qualitative basis and vegetation controls and effects on dune type have not been quantified to the same extent as with bare sand dunes (Thomas and Tsoar, 1990). This is a shortcoming in our understanding of dune landscapes that is particularly significant as the large majority of aeolian environments include various degrees of vegetation cover and are located in areas of considerable human and economic interest and natural resources, such as coastal areas and semi-arid regions. Ecogeomorphic systems in general provide a less explored but fertile area of crossdisciplinary research (Naylor et al., 2002) that lends itself well to a complex systems approach (Stallins, 2006). The small-scale physical interactions between biological dynamics and geomorphic processes and landforms are multitudinous and varied, in many cases not quantified or quantifiable, and elude the possibility of analytical or numerical modelling in a reductionist framework to such an extent that simulating simplified non-linear dynamics in cellular automata is particularly attractive and productive. A comprehensive overview of the use of CA-models in various ecogeomorphic systems has recently been catalogued by Fonstad (2006). The second half of the following section presents the development and preliminary results of a discrete ecogeomorphic aeolian landscape model (DECAL) that incorporates the effects of and interactions with various types of vegetation to simulate, amongst others, the evolution of nebkhas and parabolic dunes. 3.1. Bare-sand dunes The family of dunescape cellular automata has its roots in Werner's (1995) original model. It simulates a topography composed of stacks of sand slabs on a cellular grid with periodic boundaries. Slabs are picked up and moved a certain distance across the grid ­ mimicking transport by wind ­ and erosion and deposition of slabs are governed by probabilities. Two additional rules enforce the angle of repose via avalanching and a `shadow zone' in the lee of topography where slabs are not susceptible to erosion. Without modelling the complex airflow and sand transport dynamics, this simple model is capable of simulating fairly realistic barchans, transverse dunes and linear dunes, although the stoss slopes of dune features are generally too steep. The model can also reproduce the phase-state diagram of dune types as a function of sediment availability and wind regime variability (Bishop et al., 2002). Fig. 6 illustrates these achievements with a simulated dune field developing under a unidirectional wind over a gradient of sediment thickness overlying a solid substratum, yielding barchans that gradually morph into transverse dunes where

Fig. 6. Development of barchan and transverse dunes in the bare-sand CA model. Upper and lower regions of the model space were initiated with different amounts of average sediment availability. Simulation after 1500 iterations, slab height 0.1, transport direction from left to right.

sediment availability increases. The Werner model has been replicated by the author as part of a precursor to the vegetated dunescape CA (Baas, 1996, 2002) and also by Momiji (2001), who enhanced the model by including a wind speed-up factor to increase transport rates over elevated parts of the model topography (yielding more realistic stoss slopes) and by an improved grid rotation algorithm for simulation of varying wind directions (see also Bishop et al., 2002). A similar, though semi-discrete, simulation model was developed by Nishimori et al. (1998), while Narteau et al. (2006) recently presented a version that includes an explicit CA fluid flow component. The model has revealed some of the mechanisms of

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pattern formation through the migration and merger of crest terminations or `bedform' defects (Werner and Kocurek, 1999) and is being employed to investigate the historical evolution of compound and multi-episodic dune fields by Kocurek and Ewing (2005). Despite the successes of replicating various dune patterns, less progress has been made on the systematic investigation of non-linear dynamics and potential trajectory and attractor behaviour of the modelled system. Indeed, Werner (1995) presents an attractor/ trajectory diagram for linear dune development, using dune orientation (relative to resultant transport direction) and number of crest terminations to quantify the landscape state, but concludes that these state variables are not sufficiently discriminative to describe differing dunescapes and furthermore exhibit multiple trajectory realisations that appear to be unconfined in the statespace. To date most results have only reported on semiequilibrium `final' state landscapes, and it is not known whether the model may exhibit sensitivity to initial conditions in some part of the phase-space, nor how dunescapes may evolve toward different end-member attractor states. The problems with establishing phasespace and trajectory diagrams is discussed further in Section 3.2 below. The second issue raised by bare-sand cellular automata is that of scale. The spatial dimensions are defined in terms of the elementary grid-cell size, , and temporal development is marked by iterations, i, where an iteration is defined as a fixed number of slabmovements -- usually equal to the number of cells present in the model-space. Average volumetric rate of sand transport, q, in the bare-sand model (with periodic boundaries) can then be calculated from the model parameters as: q ¼ per

l Án hs L X À hs L 1 1 À pdep u q ¼ per i n¼0 i pdep


where: hs is slab height, L is transport distance, i is iteration period, and per and pdep are erosion and deposition probability respectively (Baas, 1996). Specifying spatial variables, hs and L, in terms of yields a non-dimensional transport rate in 3- 1i- 1.This means that spatial and temporal correspondence to real-world scales can be manipulated freely to adjust transport rates and model space dimensions so that it can represent both ripples (bedforms) and dunes (landforms). A related issue is that under unidirectional transport the model invariably develops toward a final state where all the sediment is collected into a single migrating shape if the simulation is run sufficiently long, fundamentally

controlled only by the angle of repose. Even the wind speed-up factor introduced by Momiji et al. (2000) merely slows the growth of dune height proportional to the square root of elapsed time and yields no upper limit. Combined with the free scalability of the spatiotemporal dimensions any size bedform pattern or wavelength can be replicated by a convenient selection of the elapsed simulation time. The search for basic rules and processes of pattern development is initiating a larger cross-disciplinary debate about the fundamental nature of current ripples, aeolian dunes, and subaqueous dunes. Aeolian bedforms are termed impact ripples, with an explicit recognition that they are generated by saltation impacts driving the reptation as described by Anderson (1987). Aeolian literature has generally accepted the existence of a clear scaling gap between aeolian (impact) ripples and (aerodynamic) dunes, particularly transverse or crescentic types, when normalised by grain-size (Wilson, 1972; Ellwood et al., 1975), although a second scale separation between dunes and the larger-scale draas (also proposed by Wilson) has generally not been supported by later studies (Lancaster, 1995, pp.181). For subaqeuous environments, Allen (1968) argued a similar fundamental difference between current ripples and river dunes, but recent experiments appear to indicate that morphology can show a continuum from ripples to dunes as a function of flow depth and discharge, and can apparently be treated as functionally similar (Jerolmack et al., 2004; Jerolmack and Mohrig, 2005). Are current ripples and aeolian dunes fundamentally the same type of sedimentary deformation under a fluid flow and can they be compared as such? This question has come to a head because recent small-scale flume studies that report barchan-shaped current ripple dynamics (Endo et al., 2004) have been used to justify a type of numerically simulated subaerial dune interaction (Schwämmle and Herrmann, 2003, 2005) that does not accord with our contemporary understanding and empirical evidence of aeolian dune dynamics in the atmospheric environment (Livingstone et al., 2005). The similarity argument is also assumed as part of a universal scaling law for dunes forming in a variety of materials and fluids, including current ripples (Hersen et al., 2002; Claudin and Andreotti, 2006). The confusion of the present state is exemplified by a recent study that explicitly compares the dune models of Werner and Nishimori with surface (impact) ripples in a wind tunnel experiment (Hatano et al., 2004). Although Endo et al. (2005) have recognised differences in shape between subaerial dunes and current ripples, other quarters believe that experimental water flumes do indeed

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Fig. 7. Three types of growth function used in the DECAL model, representing growth response to burial and/or erosion conditions of plant species comparable with marram grass and woody shrubbery (for parabolic dune development in coastal environments), and mesquite bush (for nebkha evolution in semi-arid environments).

produce "dunes that are downsized copies of desert dunes" (Kroy and Guo, 2004). At the root of the debate must lie a fundamental question of sediment transport mechanisms: the relative roles of suspension, saltation, and creep in subaqeuous vs aeolian environments, as well as the impact of the fluid and fluid flow properties (e.g. viscosity, flow depth) and its interaction with the morphology. The debate thus affects the current CA dunescape models, as the algorithm is intended to simulate the principal transport mechanism in its most simplified form: does the sand-slab displacement mimic aeolian saltation or subaqueous bedload transport? It therefore remains an open question whether the Werner model and its simulated patterns and behaviours represent flow ripples or aeolian dunes and/or whether these can be assumed identical. 3.2. Vegetated dunes The discrete ecogeomorphic aeolian landscape model, DECAL, traces its origins to work by the author (Baas, 1996, 2002) that introduced ecogeomorphic interactions in the Werner model by: 1) changing sand-slab erosion and deposition probabilities depending on the amount of vegetation present in a cell, and 2) seasonal vegetation adjustments in response to the local erosion/deposition balance. Comparable attempts to simulate vegetated

dunes by de Castro (1995) and Nishimori and Tanaka (2001) have employed a similar strategy. Current work by Nield and Baas (2006a,b; Baas and Nield, 2007) has developed the algorithm further to include multiple types of vegetation. The influence of plants on the sediment transport process is quantified by a `vegetation effectiveness' variable for each grid-cell which can be interpreted as a coverage density or frontal area index (FAI), ranging in magnitude from 0 to 1 (or 0­ 100%). The vegetation effectiveness modifies the local sand-slab erosion and deposition probabilities linearly from their bare-sand values, so that erosion probability is reduced from its bare-sand maximum (usually 1.0) at 0% vegetation to no erosion possible (per = 0.0) at 100% vegetation effectiveness, while deposition probability is increased from the bare-sand value pdep to 1.0 (immediate deposition) at a fully vegetated cell. Each grid-cell can contain multiple types of vegetation in differing amounts, e.g. a pioneer grass species such as marram grass (Ammophila arenaria) at an effectiveness of 70%, as well as an amount of successor or climax species such as creeping willow (Salix repens) at 15%, and their effectiveness is combined cumulatively for modifying transport probabilities. The development of the vegetation in turn is controlled by so-called `growth functions' that periodically relate the erosion/deposition balance at each grid-cell to a seasonal increase or decrease of vegetation

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effectiveness. The growth functions define and distinguish the types of vegetation in the model environment, as they reflect differences in tolerance to burial and/or erosion and differences in seasonal growth rates. Fig. 7 shows typical growth functions used in DECAL to represent marram grass and woody shrubs for the development of parabolic dunes in a temperate coastal environment. Marram requires fresh input of sediment to reduce the impact of soil pathogens and parasites (Van der Putten et al., 1993; Moore, 1996), and this is reflected in its growth function, which shows a positive response to sand burial and a decline under neutral or negative sedimentation balance. The second growth function represents a more conservative type of vegetation much less tolerant of erosion and deposition and with a comparatively slower growth rate. With these two types of vegetation in the environment the model is capable of developing seemingly realistic-looking parabolic dunes starting from bare patches on an otherwise fully vegetated flat surface, as shown in Fig. 8. The resultant landforms exhibit some of the defining features of classic parabolic dunes, such as trailing ridges formed by the colonisation and stabilisation of the lateral edges of the migrating dune, a deflation plane that exposes the erosion base (corresponding to the phreatic level in this case), and vigorous grass coverage on the lee of the depositional

lobe. These simulations are also showing initially unexpected emergent behaviour such as local competition between the types of vegetation in response to changing geomorphic conditions. Indeed, the model requires no rules to constrain the relative proportion of the different vegetation types in a cell (i.e. no imposed ceiling exists on their combined effectiveness), because they react differently to the sedimentation conditions and occupy different ecogeomorphic niches. The introduction of vegetation in the model adds another set of feedback links between the spatial and temporal scales ­ in addition to the spatio-temporal linkage embodied by the transport rate ­ that constrains the scalability of the model. When defining the parameters that control the transport rate and vegetation interactions, care must be taken to match them appropriately to represent realistic conditions. Imposing a specific transport rate environment, for example, requires a decision on absolute spatial and temporal scales of and i in real units in order to define a realistic vegetation response. Fig. 9 illustrates this crucial difference with the bare-sand model by showing the impact of changing the absolute length represented by on simulating the development of a nebkha dune field in the presence of a mesquite-like vegetation (third growth function in Fig. 7) while keeping the transport rate and vegetation response constant (in real

Fig. 8. Parabolic dune development in the DECAL model after 50 seasons. Green gradation indicating grass `density' (vegetation effectiveness), spacing and size of red sticks indicating woody shrubbery `density'. Started from a flat fully vegetated surface with a few bare circular patches. Transport direction from lower-left to upper-right (unidirectional).

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Fig. 9. Simulating nebkha dune fields at different spatial resolutions after 20 seasons. Top-centre panel shows standard nebkha development at 1.0 m resolution. Lower left-hand sequence shows results under increasing resolutions (0.5 and 0.1 m cell-size, respectively), lower right-hand sequence shows results under decreasing resolution (2 and 10 m cell-size, respectively). Note changing grid sizes.

terms). At an absolute scale where is equivalent to 1 m the nebkhas attain a diameter on the order of 10 m after a simulated period equivalent to 20 seasons (Fig. 9, topcentre). This size remains unchanged when the spatial scale is coarsened to 2 and 10 m (Fig. 9, lower right-hand sequence); at the lowest resolution of 10 m the hummocks are captured by single cells or pixels as this is the minimum possible representation of a single nebkha. At smaller length scales, however, the model reveals a breakdown of the expected landscape development (Fig. 9, lower left-hand sequence). At a resolution of = 0.1 m the topography does not seem to develop toward isolated nebkhas at all but instead forms narrow chevron-shaped strips or ripples of sediment that resemble

rather the effect of a series of sand screens where the `vegetation' seems to act like a regularly maintained stockade of porous fencing. Conceptually then, the vegetation and its associated ecogeomorphic interactions appear to impress a characteristic scale on the system so that size and shape of vegetated dunes are fundamentally controlled by the ecological attributes of the plant species in the environment. This may be relevant to Dietrich and Perron's (2006) quest for a signature of life on the morphology of the Earth's surface. They conclude that in the context of hill slope erosion at least there seems to be no such distinction between surfaces on Earth and on Mars. Whereas bare-sand dunes are found over several orders of magnitudes in size and can not be distinguished

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Fig. 10. Phase-diagram of different types and transitions of dunes developing in the vegetated environment of DECAL, dependent on the peak seasonal growth rates for the two types of vegetation. Classification criteria of the dune types are listed in Table 1. The purple circle indicates the combination of growth rates that yielded the parabolic dunescape shown in Fig. 8.

between different planets, dunes developing under the influence of vegetation, i.e. life, may exhibit a clear restriction in size-range and shape that is fundamentally related to the biological limits of physiological and photosynthetic potential. It is instructive, for example, to consider the question what kind of vegetation would hypothetically be needed to produce giant parabolic dunes: it may require an impossible combination of plant characteristics such as a simultaneous high seasonal growth rate and exceptional burial tolerance that cannot be achieved within known possibilities of vegetative development. Alternatively, is it possible to identify abiotic processes that could replicate the effects of a vegetation species like marram grass in forming parabolic-shaped landforms? Besides raising these rather philosophical questions, the DECAL model is proving successful in simulating nebkha dune fields and blow-outs as well as parabolic dunes (Nield and Baas, 2006b), though more specific results on the various dunescapes must await later publication. It is becoming exceedingly clear, however, that a comprehensive exploration of the phase-space and trajectory/attractor dynamics, as well as the parameter sensitivities requires a systematic and statistical approach. Conceptually the model dynamics can be thought of as a functional mapping from an n-dimensional parameter space to an (m + 1)-dimensional state-variable space, where the +1 dimension constitutes time. From this

fundamental mapping can be derived two representations. First, phase diagrams (in the traditional meaning of the term) identify different types of outcomes or states (this can be a qualitative or categorical assessment) as a function of changing parameter values. The phase diagram for the different types of desert dunes as a function of sand availability and wind regime complexity by Wasson and Hyde (1983) is a good example. Second, state-space plots or trajectory diagrams indicate the development over time of the system (subject to a specific parameter setting) as a point moving in a space whose axes are defined by state-variables (also sometimes confusingly referred as the phase-space). These trajectories may be random, may converge on a static equilibrium point attractor, may move in a dynamic equilibrium periodic attractor, or may evolve on a chaotic and/or fractal (`strange') attractor, such as the classic (butterfly) Lorenz attractor (1963). The two basic representations of the fundamental mapping can be linked by defining phase diagram types or regions as corresponding to certain combinations of state variable outcomes or attractors. The development of phase diagrams and state-space plots for the vegetated aeolian dune model is difficult because of the number of parameters involved (even for the simplistic CA approach) and because of a lack of suitable state-variable descriptors that can accurately capture various unique aspects of different vegetated dunescapes. A first attempt at a meaningful phase

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diagram is shown in Fig. 10, which presents a visual assessment on the type of landscape that results after 100 seasons as a function of changing the peak seasonal growth rate of the two types of vegetation in the model, variant of the parameter settings used for the parabolic dune simulation of Fig. 8. The diagram shows a sharp transition from transverse ridges to parabolic dunes as the peak seasonal growth rate (the height of the maximum in Fig. 7) increases for the first vegetation type (representing the marram grass dynamics). Increasing the peak growth rate of the second vegetation type (shrub-type) quickly leads to a shutdown of sediment movement and a stabilisation of the landscape. The phase diagram includes various transitional landscapes, such as barchanoid transverse dunes and break-away dunes (see Table 1). The dunescape types do not occupy exclusive contiguous regions and in some areas of the phase space the different types intermingle, indicating potentially chaotic behaviour. The phase diagram suffers from two critical limitations: first, it represents the outcomes as a function of only 2 parameters (the peak seasonal growth rates of the two vegetation types) while the definition of a full growth function alone involves at least 5 separate parameters ­ not to mention the other model parameters that can be altered ­ and second, the specification of the outcome is based on a visual and partly subjective assessment, rather than criteria related to quantifiable state variables. These then present the two significant challenges that are faced not only by the DECAL model but will indeed apply to ecogeomorphic models in general. The number of parameters that can be considered must be restricted or parameters need to be combined into non-dimensional groups, much like a Buckingham theorem approach (comparable to e.g. the Reynolds number); and meaningful and quantifiable state-variables need to be established that can discriminate between various

complex morphologies as well as spatial distributions and correlations of vegetation patterns. The combination of a number of model parameters into a transport rate in Eq. (7) is a first step on the way to developing nondimensional parameter groupings and some research has been conducted on establishing the `allowable' ranges of parameter values that yield the same landscape under the same non-dimensional transport rate. Work is currently underway to develop a suite of numerical state variables that capture various aspects of the vegetated dunescape. This includes such calculations as the sum of the local products of directional gradient (along the transport direction) and vegetation effectiveness of the first type (`marram grass') to capture the growth of the pioneerspecies on the slip-face of dunes, and the ratio between the integral length scales of the topography (or semivariance range) in the transport-parallel versus the lateral direction to capture the different shapes and orientations of the main dune mass for various types of dunescape. Developing and researching these state-variables and the parameter space is a laborious and time-consuming process that might be aided by statistical tools such as multi-variable correlation analysis (to determine intrinsic correlations between state-variables), Principle Component Analysis (to identify potential groupings of parameters), and cluster analysis (to identify natural boundaries between different landscape types in the state space). Without appropriate discriminatory state variables it is at present still difficult to apply numerical techniques for analysing, comparing and predicting nonlinear dynamics, such as attractor reconstruction (Takens, 1981), correlation dimensions (Grassberger and Procaccia, 1983), space-state plot distances (Murray, 2001), and artificial neural networks (Sivakumar et al., 2002). Moreover, if the model is to be compared or applied to real-world landscapes, the state variables that are developed must be capable of being derived from

Table 1 Criteria and descriptions used in the visual classification of simulated dunescape types for establishing the phase-diagram shown in Fig. 10 Dune type Transverse (0) Barchanoid (1) Barchanoid transition (2) Static (3) Breakaway transition (4) Small parabolic (5) Large parabolic (6) Description A single transverse dune completely covering the landscape perpendicular to the wind direction. Partial transverse dunes, not covering the landscape completely perpendicular to the wind direction, single barchan or multiple barchan dunes joined together. Small chaotic mounds of sand spread throughout the landscape, not parallel to the wind direction and not in line with the original bare patches. Small lumps of sand in breakaway lines of varying length from the original bare patches, parallel to the wind, overall little movement has occurred. Some parabolic shaped dunes, up to two of these have had breakaways, where the nose of the dune has moved away from the main body of sand that had been stabilised. Small parabolic shaped dunes. Large parabolic shaped dunes, with a body of sand at least the size of the original bare patch, thick trailing arms, chevron shaped nose.

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measurable field data. If this can be achieved, the cellular automaton may be employed to help investigate the principles behind such processes as the activation and stabilization episodes in dune fields on the Northern Great Plains (Hugenholtz and Wolfe, 2005), hysteresis in dune vegetation and re-activation (Tsoar, 2005), and the development of wind-swept `streets' in semi-arid shrub lands (Okin and Gillette, 2001). 4. Concluding remarks In this paper I have attempted to approach both process and form in aeolian geomorphology from a common perspective of complex systems. In the case of sediment transport processes it is still rather limited, confined to a qualitative assessment of the overall boundary-layer-flow / sediment-transport/bedform system behaviour, while recent experimental evidence suggests that the impact of topdown turbulence may exert an overriding control. The latter implies that spatio-temporal transport patterns are an imprint of potentially self-organising processes in boundary layer flow, which may lend itself well to a complex systems approach. What impact this turbulence control may have on bedform development is as yet unclear. It could be argued that small-scale sediment transport dynamics, such as streamers, are decoupled from the ripple field evolution because of the different time-scales involved, i.e. the ripple field may only respond to a timeaveraged forcing in which all traces of the complex nonlinear dynamics on shorter time-scales have been filtered. Considering the great variety of ripple patterns that can be found on natural sand surfaces, however, it is not unreasonable to suppose that evolution of 3D bedforms is affected by some kind of macro-scale characteristic related to the local transport dynamics (e.g. an average streamer density or transport intermittency) which could ultimately be attributed to the turbulent airflow. At the landscape scale level of dune fields, the complex systems approach appears more viable, particularly in the context of modelling self-organising patterns with cellular automata. Such models serve to investigate the development of various dune types and behaviours, and also to determine what are the most critical and fundamental physical processes involved in their most rudimentary form. In this sense a model that fails to replicate a certain real-world phenomenon is perhaps more informative as it reveals that a vital physical process or element is missing or has been overlooked. Previous attempts at simulating parabolic dunes with just one vegetation type, for example, failed to replicate the characteristic trailing ridges (Baas, 1996), whereas the new DECAL model with two vegetation types can indeed achieve this. Nonetheless, much work still lies

ahead to investigate the parameter space sensitivities and develop a complete understanding of the non-linear behaviour in phase-space. A common feature of all the systems discussed in this paper is that of pattern, be it in the form of streamers, ripples, dunes, or vegetation, and it is this aspect of the complex system approach that remains particularly problematic. Most attempts at quantifying pattern are restricted to relatively simple statistical descriptions, such as average wavelength or streamer density. Even with more advanced statistical or mathematical manipulations, such as the state-variables that are being developed for the DECAL model, a doubt remains that these do not quite capture the full uniqueness of the landscape features and patterns on the system level. Ecological literature, for example, presents a veritable host of spatial statistics that can be applied to vegetation maps in attempts to quantify patterns (McGarigal and Marks, 1995; Bowersox and Brown, 2001), but these are largely based around simple binary representations and are not mutually exclusive or uniquely discriminative. Furthermore, such macro-scale descriptors are developed specifically for one type of situation and can not easily be transferred or applied to other complex systems. It is, for example, not obvious how a state variable such as defect density in ripple fields can be applied ­ with the same functionality ­ to a streamer pattern, or how a macro-scale descriptor such as plant-species connectivity can be translated to a comparable state-variable referring to turbulence in boundarylayer flow. As with the analogy of statistical mechanics mentioned in the introduction, where macro-scale variables such as temperature and pressure of a gas can be physically related to aspects of the statistical distribution of molecular speeds (i.e. properties of the constituting elements), it may be worthwhile to revisit the concept of entropy in its geomorphic application, exploring links between the older and largely neglected notions of landscape entropy in the thermodynamic sense (Leopold and Langbein, 1962) and recent applications of information theory or Shannon-entropy in sediment transport and flow hydraulics (Singh and Krstanovic, 1987; Chiu, 1992; Wang et al., 1992), as well as in various morphological analyses (Culling, 1988; Fiorentino et al., 1993; Claps et al., 1996; Tate, 1998), possibly combined with object-based landform identification (Dragut and Blaschke, 2006). Even so, entropy only quantifies degrees of disorder in a system and is likely insufficient to capture all unique aspects of an evolving pattern; hence, other universal descriptors should be developed, though it is unclear what form these could take. It may well be that a deterministic or analytical description of spatio-temporally extended complex

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geomorphic systems can not be achieved in closed form. Some kind of interactive IT-based knowledge system (Loudon, 2000) could then be a potential approximation for integrating, investigating, and updating our continuously evolving understanding of the geomorphic system within a consistent framework. Acknowledgements Many of the thoughts and speculations in this paper have been influenced by discussions with a number of colleagues, including Bernie Bauer, Charlie Bristow, Philippe Claudin, Jean Ellis, Eugene Farrell, Mark Fonstad, Gary Kocurek, Victor Loudon, Brad Murray, Doug Sherman, and Giles Wiggs. Mark Fonstad and an anonymous reviewer are thanked for constructive comments and suggestions that helped improve the manuscript. Figs. 6­10 were prepared with the invaluable help of Joanna Nield. Data presented in Figs. 1 and 4 were collected in field experiments supported by the US National Science Foundation, Geography and Regional Sciences Program (0002471), with kind permission from the Bureau of Land Management Palm Springs Field Office, while research and development of the DECAL model is made possible by the UK Natural Environment Research Council (NE/D521314/1). References

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