The supercritical question for pyroclastic dune bedforms: An overview

Sedimentary structures deposited from dilute pyroclastic currents are largely dominated by the products of metre‐scale dune bedforms containing backset lamination. Since the first seminal works in the 1970s, the vast majority of the literature called these structures antidunes and chutes‐and‐pools. This consensus led to a loose terminology, either as descriptive terms for structures containing backset beds or as an interpretation of hydrodynamic conditions related to Froude‐supercritical flows. Here, the main characteristics of pyroclastic dune bedforms are summarized and classified under four categories. The literature is examined and a discussion is provided regarding a possible interpretation in terms of supercritical‐flow bedforms, including antidunes, chutes‐and‐pools and cyclic steps. The interpretation of pyroclastic dune bedforms as related to supercritical conditions is a possibility among others, yet open questions remain, and under the present state of knowledge alternative interpretations are equally valid. There is consensus that an interpretation of pyroclastic dune bedforms as related to supercritical flows implies deposition from a basal underflow supporting an internal free‐surface formed at a density interface within the overall current structure. Antidunes are likely to be limited to low‐angle, incipient bedforms. Chutes‐and‐pools may occur in the form of series of steep truncations formed by an erosive supercritical ‘chute’, with a depositional signature occurring exclusively in the subcritical ‘pool’ of a Froude jump. Rare examples of cyclic steps may be found as large‐scale undulations with superimposed metre‐scale bedforms, or for short‐scale, fully‐aggrading structures in periodic trains. Additionally, some pyroclastic bedforms may be formed by granular flows passing granular Froude jumps or as frozen granular stationary waves. The dominance of chute‐and‐pool structures over bedforms characterized by relatively more stable morphodynamics (antidunes and cyclic steps) could be the likely result of pulsating flow conditions and abrupt high‐rate deposition, impeding stable flow conditions. In a majority of cases, the growth of pyroclastic bedforms appears triggered by the influence of an inherited bed topography and by bed‐flow feedback effects. Apart from a possible interpretation as supercritical bedforms, other specific dynamics of dilute pyroclastic currents may well explain the characteristics of pyroclastic dune bedform structures. In particular, the inherent turbulence fluctuations, internal organization and highly depositional dynamics may be key to the formation of pyroclastic dune bedforms. In this respect, rather than solely focusing on the Froude dimensionless number of the formative current, theoretical considerations accounting for the Reynolds, Richardson and Rouse dimensionless parameters may be also implemented for the understanding of pyroclastic bedforms.

Here, focus is on sedimentary facies associations containing dune-bedforms (DBs) crossstratification and flow-oriented grain fabrics. These sediments are attributed to currents thought to have low particle concentrations, with clasts supported dominantly by fluid turbulence down to the flow base, and variously termed 'dilute PDCs', 'nu ee ardente', 'ground surge' or 'pyroclastic surge' (e.g. Sparks & Walker, 1973;Sparks, 1976;Valentine, 1987;Branney & Kokelaar, 2002). Estimates of the higher limit of volumetric particle concentrations vary between authors, and could reach 2.8% by volume (e.g. Andrews & Manga, 2012;Weit et al., 2018). Dilute PDCs are often an accompanying part of high concentration 'pyroclastic flows', with a range of allogenic and autogenic parameters influencing their interactions, such as: changes in the current density, temperature, momentum, source parameters and/or changes in slope along transport (e.g. Hoblitt, 1986;Burgisser & Bergantz, 2002;Doronzo & Dellino, 2014). Whereas a large portion of sediments from pyroclastic currents are massive and coarse-grained, ash-dominated and dune-bedded deposits have attracted attention since dilute currents have the capability to overflow topography through flow-stripping from concentrated flows (flow transformation or flow detachment, e.g. Hoblitt, 1986;Wohletz, 1998;Cole et al., 2002;Douillet et al., 2013a).
Here, an overview of the different types of dune bedforms and their interpretation in terms of Froude-supercritical flows is presented and discussed. Not considered here are bedforms related to secondary reworking by aeolian, fluvial or subaqueous processes (e.g. Suthren, 1985;Smith & Katzman, 1991;Manville et al., 2009), extra-terrestrial environments, or subaqueous PDCs (e.g. Mandeville et al., 1996;Pope et al., 2018), although these structures can show some similarities to (Wilson & Hildreth, 1998;Burt et al., 2008), and sometimes be confused with, primary deposits (Nocita, 1988, commented by McPherson et al., 1989. This work is justified by two motivations: (i) The understanding of pyroclastic DBs has the possibility to deliver crucial insights into the dynamics of dilute PDCs. (ii) Conversely, the particular sedimentary environment represented by PDCs provide unique information for the field of process sedimentology.

The specificities of dilute pyroclastic density currents
The sedimentary record of dilute PDCs is in many ways atypical compared to that of most sedimentary environments, and may capture processes that are only transient in aqueous or wind-driven transport (e.g. Wohletz, 1998;Burgisser et al., 2005). In order to understand this uniqueness, the introduction of dimensionless numbers permits the evaluation and comparison of different types of flows in a unified framework (definitions in Table 1). Hypotheses on the flow characteristics of PDCs can be formulated to explain the specificities found in their deposits: effect of gravity. Here the role of particles stands in fundamental contrast to aeolian or fluvial transport, where the fluid phase is set in motion by gravity, independent of its associated particulate load, and can secondarily entrain and transport particles. Hence, in a particulate density current, the momentum (i.e. velocity * mass) is carried by the particle load. When particles are deposited, the flow mass decreases, and so does the momentum. Furthermore, particles are present throughout the flow's thickness and path, without the need of being picked up prior to deposition, resulting in a ubiquitous availability of particles, with material in the upper part of the current being able to feed an underflow (e.g. Gardner et al., 2017). These characteristics are shared with all particulate density currents (for example, turbidity currents and snow avalanches). 2 The density ratio between the particles and the fluid phase (air) is much larger than for any water-supported transport process and the dynamic viscosity of air is two orders of magnitude lower (e.g. Burgisser et al., 2005). Accordingly, the dynamics applying to pyroclastic currents might share little with water-supported sediment transport, and one must consider scaling laws embracing the Rouse and Stokes dimensionless number (Table 1; e.g. Valentine, 1987). This is a major divergence from subaqueous turbidity and hyperpycnal currents. 3 The combination of the availability of particles and their high density-contrast to the supporting fluid may lead to extreme sedimentary dynamics, in particular, through fast sedimentation and draping that may freeze processes otherwise not available in a 'regular' sedimentary record. The growth of some bedforms and associated sedimentary structures may take place within seconds to minutes. 4 The wide particle-size distribution within pyroclastic deposits is a token for the high polydispersity of the parent flows that emplaced these sediments. Polydispersity may be key in PDC dynamics and sedimentology. By fostering the entrapment of air and elevated interclast fluid pressures (pore pressure), polydispersity may facilitate the occurrence of a fluidized boundarylayer zone (e.g. Sparks, 1976;Druitt et al., 2007;Table 1. Overview of the dimensionless numbers and physical quantities listed in the manuscript: u = flow velocity; g = gravity acceleration; g' = reduced gravity; H = characteristic length (here flow thickness); q = flow density; L = characteristic length (particle diameter/flow thickness); µ = dynamic viscosity; d/dz is the derivative to height; w s = fall velocity of particle; K = Von Karman constant, d = characteristic length (of turbulence); f = frequency of vortex emission; L is a characteristic length (e.g. flow thickness); q 0 = bulk flow density.

Symbol Definition
Formula for flows  Smith et al., 2018;Breard et al., 2018Breard et al., , 2019. In particular, dilute PDCs are likely to grow basal underflows dominated by a transitional layer of traction carpets, fluidized granular flows and metastable early sediment (e.g. Sheridan & Updike, 1975;Druitt, 1992;Sohn, 1997;Douillet et al., 2015), as for turbidity currents (e.g. Sanders, 1960). In the quasi-absence of direct observations on the internal structure of dilute PDCs, their sedimentary signatures are the largest source of natural observation (exception: Scharff et al., 2019).

Supercritical flows
A flow where more energy is stored as kinetic energy than potential energy is said to be in the Froude-supercritical regime, this relation being expressed by the dimensionless Froude number (Fr;  Simons et al., 1961;Alexander et al., 2001;Cartigny et al., 2014). The supercritical regime also corresponds to flow conditions where the current runs faster than the group-velocity of surface gravity waves propagating at its upper free boundary (respectively, subcritical, slower). When a current moves at similar speed as gravity waves (transcritical regime), upstream propagating surface waves appear to be almost stationary over the sediment bed, and may interact with the bed to form a bedform: an antidune (e.g. Gilbert, 1914;Kennedy, 1963). 'Antidune' is thus, from a sedimentology point of view, intrinsically interpretative and this term does not contain any descriptive connotation (see Nomenclature section for descriptive terms). One of the advantages of interpreting a sedimentary structure as originating from an antidune is that it allows to apply a simple equation frame to extract quantitative flow parameters. The rationale is that an antidune would be the mould of a stationary surface gravity wave in a current having a Froude number equal to or close to unity (e.g. Prave, 1990; for the complete development of the framework, see Douillet, 2015, chapter 2). Unfortunately, there has been a tendency in the literature to use the word antidune as a descriptive term.
In the supercritical regime, surface gravity waves are too slow to travel upstream in reference to the bed. Since gravity waves act as the agent of transfer for changes in flow depth, a supercritical flow cannot 'foresee' a change in height, as for example resulting from an obstacle or a dam (hence supercritical is also termed 'shooting'), whereas a subcritical flow does. The passage from super to subcritical can thus lead to an abrupt flow 'jamming', forming a so-called 'Froude jump'. Whereas the term 'hydraulic jump' is more commonly used for this phenomenon, the absence of water in a PDC is here considered an obvious reason to adopt the more generic term 'Froude jump' in this manuscript ('pneumatic jump' and 'granular jump' have also been used for pyroclastic currents; e.g. Branney & Kokelaar, 2002;Douillet et al., 2013aDouillet et al., , 2019Smith et al., 2020). Both sedimentation and/or erosion may occur (alone or concurrently) at a Froude jump, and may lead to the development of bed configuration known as a chute-and-pool (e.g. Jopling & Richardson, 1966). Just like 'antidune' specifically applies to a bedform related to a stationary gravity wave, the term 'chuteand-pool' is used for any bed configuration related to a Froude jump. Again, that term thus appears as solely interpretative from a sedimentological point of view, and does not contain descriptive sedimentary characteristics. When a flow develops a train of repetitive, long-wavelength Froude jumps, the corresponding bed configuration is called 'cyclic steps'.

Nomenclature
The term 'surge' is abundant in the literature, yet avoided here due to its potential ambiguity, since it refers both to a dilute PDC in volcanology and to a retreating Froude jump in hydrodynamics. The term 'dune bedform' (DB) refers to a roughly metre-scale bed undulation forming an individual sedimentary structure (or non-planar bedsets, Fig. 1). Here, DB is not restricted to a superficial shape but includes the stratification participating in the formation of the outer shape, i.e. the associated sedimentary structure ( Fig. 1A; e.g. Carey, 1991;Douillet et al., 2013). The stratal patterns record the succession of bed shapes with time. Dune bedform is preferred over the variety of other terms applying to the same objects because of its absence of genetic connotation, yet the pyroclastic literature abounds with such terms as 'wave ', 'sandwave', 'megawave', 'wave-form', 'megaripple', 'dune', 'dune-like', 'antidune', 'bedform' or 'dune bedform'. The terms 'antidune' and 'chute-and-pool', which have been pervasively used as descriptive terms by volcanologists, are reserved here to their hydrodynamics meaning only (see above).
Deposits with stratification include any type of strata, including 'laminations' used for millimetre-scale strata, whereas 'layers' are massive, pluri-centimetric thick strata. A 'laminaset' is a set of several laminae forming a consistent pattern, intrinsically understood as related to a single accretion phase; laminasets are separated by set boundaries, either erosive or conformable (stratasets refer similarly to groups of strata). 'Cross-stratification' is used with general reference to any strataset with angular relationship with respect to its bounding surface. The 'stoss' face of a bedform faces upstream, and the 'lee' faces downstream. Stratasets deposited on a stoss side are called 'backsets' (not to be confounded with 'backsteps', which correspond to a stratigraphic transgressive stratal architecture pattern), as opposed to the term 'foresets', denoting stratasets dipping downcurrent along a bedform lee side. Stratasets that show continuous deposition from the stoss to the lee side are called 'fullyaggrading'. Fully-aggrading stratasets that show thicker strata toward their stoss entail a general upstream shift of the bedform crest and are called 'regressive', whereas stratasets with a downflow equivalent trend are called 'progressive'.
• Common co-existence of peculiar finer-scale structures ( Fig. 3; Douillet et al., 2015Douillet et al., , 2018Douillet et al., , 2019, and references therein) such as: ○ lineations (i.e. non-massive deposits with coherent lines that cannot confidently be attributed to depositional laminae); ○ shark fins (  These structures are generally attributed to a combination of high sedimentation rates, high bedload with granular-based flows and traction carpets, and metastable sediments during the early phase of deposition. The quasi absence of ripples with preferential downstream aggradation (i.e. with exclusion of 'backset ripples') is characteristic of the depositional environment of pyroclastic DBs. Rare reported occurrences might be an exception (e.g. Dellino & Volpe, 2000), yet often represent wind reworking (e.g. Smith & Katzman, 1991;Douillet et al., 2018).

Dimensions
Although referred to as dune-bedforms, pyroclastic DBs range from ripple-sized (backset stratasets of a few centimetres length and thickness) up to over two orders of magnitude greater (ca 30 m length, ca 3 m thickness; Fig. 4). They are thus much smaller than most aeolian or submarine dunes, and rather correspond to the size of 'megaripples' (as termed in Self & Sparks, 1978). A swarm of datapoints of thickness/ lengths (sometimes reported as wavelength and amplitude) is built from deposits of several eruptions, with emergence of a vague linear increase, roughly contained within a ratio of one-third to one-thirtieth ( Fig. 4; note that the methods of measurement are not standardizedfrom a surface bedform or lamination patterns).

Exceptionally large dimensions have been
reported, yet the associated structures might be related to the stratigraphic backstepping of depositional profiles (length of 60 m in Self & Wright, 1983; 100 m in Sigurdsson et al., 1987;200 m in Brand &Heiken, 2009; 100 to 250 m in Marshall et al., 2015; 520 m long in Brown & Branney, 2004). Spatially, the size of DBs is found to decrease downstream within a deposition zone (see Main types of patterns section). Greater dimensions are largely interpreted as reflecting higher flow velocities or 'energy' (e.g. Druitt, 1992;Brand & Clarke, 2012), yet there may also be a fundamental influence of sedimentation rate (or Rouse number) as well as event duration (Table 1, e.g. Valentine, 1987;Douillet et al., 2013a). Interestingly, identical structural patterns can be observed with length scales ranging over two orders of magnitude (for example, Fig. 8A versus 8E, Fig. 6B versus 10C).
Relatively small structures (<1 m length) are commonly flatter than larger ones, with dip angles on stoss and lee sides generally below 20°. Internally, they comprise subplanar foreset to low-angle backset lensoidal layers, or aggrading structures resembling hummockycross-stratifications (HCS). Low-angle DBs are dominant in deposits from monogenetic, phreatomagmatic eruptions, but occur from any eruption style (for example, at Ubehebe crater: maximum of 23°on stoss and 12°on lee, reported by Crowe & Fisher, 1973).
Steep-sided structures span the entire dimensional range (from centimetric to metric scales). Whereas lee side laminae rarely exceed 20°to 30°dip angles, the steepness is systematically better expressed on stoss sides, which are commonly reworked by steep truncations, up to vertical cuts. The covering truncation-based backset beds mimic the underlying morphology, reaching surprising dip angles up to the vertical on bedsets up to tens of centimetres in thickness (70°reported by Schmincke et al., 1973;60°by Giannetti & Luongo, 1994;68°by Brand et al., 2016;90°by Douillet et al., 2019). Notably, when very steep structures are present, they coexist with low-angle structures. In that case, the structures drastically evolve from subhorizontal to near-vertical during growth, until truncations and aggradation sometimes lead back to a subsequent low-angle configuration. At Tungurahua, where dip angles could be measured on >300 pristine surficial bedforms, angles were reported from 0°to 35°for both stoss and lee faces, with average slopes of 10°to 15° (Douillet et al., 2013b). Thus the surface expressions of bed configurations at the end of an event are much smoother than the internal patterns preserved during the flow.

Main types of patterns
End-member types of pyroclastic DB structures have been variously drawn for specific eruptions, either regarding their internal lamination patterns (five types in Schmincke et al., 1973; three types in Tokunaga & Yokoyama, 1979; five other types in Cole, 1991;eight groups in Wohletz & Sheridan, 1979) or including surficial shapes (four other groups in Douillet et al., 2013b). Based on the literature and numerous observations, four main types of patterns are recognized here (Fig. 2

):
Type A (Figs 2A and 5): Subplanar to backset or foreset lensoidal patterns: These are characterized by relative flatness (e.g. type 5 in Schmincke et al., 1973;Colella & Hiscott, 1997;Cole et al., 2001;or 'elongate' in Douillet et al., 2013b). Such patterns occur either isolated growing from flat beds, or forming the base of a subsequently steepening structure or, conversely, towards the top of a structure, when the patterns flatten again.
Type D (Figs 1A, 2D and 8): Steep truncations covered with steep backset strata (e.g. types 1 and 2 in Schmincke et al., 1973; types d and e in Cole, 1991;Tokunaga & Yokoyama, 1979;Brand et al., 2016). Often occurring in upcurrent propagating series (retrogression), where the backset beds against a previous truncation are partially cut by a subsequent truncations more upstream. Eventually, backset bedsets fill up the troughs and flatten the bedform.
These categories represent end-members, and many structures straddle between two of the described types (e.g. Trough cross-beds and tabular beds are notably rare, yet not totally absent (e.g. Wohletz & Sheridan, 1979;Sohn & Chough, 1989).

Surface geometries
Based on datasets from a number of relatively small historical eruptions, DBs could be documented from surface outcrops and surficial shapes (Figs 1 and 9, eruptions from B arcena in 1952;Taal in 1966;Mount St. Helens in 1980;El Chich on in 1982;Pinatubo in 1991, Tungurahua in 2006. Pyroclastic DBs generally occur in swarms of tens to hundreds of bedforms in surface outcrops (Figs 1B and 9). For PDCs funnelled within ravines, DBs are best documented as grouped within limited deposition zones (hundreds of metres wide) on overbanks of ravines, due to the overspill of turbulent currents (Douillet et al., 2013a). In unconfined settings, DB deposition zones spread from a distance from the eruptive vent and up to several kilometres downstream. Interestingly, a zone of erosive furrows can precede a DB field, with an abrupt transition from erosive to depositional zones (plate 24 in Richards, 1959;see Fisher, 1977, for discussion). The dimensions of DBs decrease downstream away from the eruptive centre in simple radial fashion, or along individual deposition zones when deposits are funnelledi.e. not directly with distance from the crateror in zones of lowered topographic gradients (e.g. Moore, 1967;Sigurdsson et al., 1984;Rowley et al., 1985;Douillet et al., 2013a;Lube et al., 2014). In-train patterns are rare (counterexamples: Sigurdsson et al., 1984;'two-dimensional DBs' in Douillet et al., 2013b). Groups of DBs can show a spatial organization in terms of external shape types. Hoblitt et al. (1981) reported a downstream evolution from irregularly shaped hummocks to transverse DBs at Mount St. Helens. At Tungurahua, the shape of DBs evolved from steep 'transverse' shapes in upstream zone of DB fields, with a downstream increase in 'lunate' shapes i.e. crescent pointing upcurrent (Douillet et al., 2013b). On downstream edges of deposition zones and on upslope transport zones, steep and wide (flowperpendicular) 'two-dimensional' DBs occurred in periodically repetitive bedform trains with short wavelength ( Fig. 9; Douillet et al., 2013aDouillet et al., , 2013b, an observation corroborated at El Chich on (Sigurdsson et al., 1984). Notably, these spatial trends in bedform shapes follow a pattern observed in analogue flume experiments with supercritical density currents by Spinewine et al. (2009). Sigurdsson et al. (1987) noted a superimposition of 'larger elongate transverse bedforms up to 100 m in length and smaller (metre-scale DBs) with short and curved crests' in the 1982 El Chich on deposits, similar to the superimposition of short-period 'chutes-andpools' and long period 'cyclic steps' in analogue flume experiments by Cartigny et al. (2014).
In terms of type, Schmincke et al. (1973) suggested that DB structures of types C and D  Giordano et al., 2008); (B) and (C) Ubehebe Crater (see Crowe & Fisher, 1973;Fierstein & Hildreth, 2017). The structure in (B) is low-angle and fully-aggrading (i.e. mixed type A/B), pencil for scale is ca 15 cm. A train of two DBs is visible in (C) and transitional to draped truncation type (mixed type A/C), hammer head for scale is 15.5 cm wide. (D) Mount Leura (see Joyce, 2004;Cas et al., 2017), stack of low-angle bedsets, hammer for scale is 28.5 cm long. (E) Large-scale, composite low-angle structure at Pululahua Volcano (see Papale & Rosi, 1993). For all images, flow orientation roughly from left to right; photographs (B), (C) and (D) have been mirrored for consistency. would occur in more proximal zones than those of types A and B. At Mount St. Helens, Druitt (1992) observed that the aggradation of DBs formed preferential foreset stratasets where currents passed on the upstream face of ridges, whereas low-amplitude backset or foreset stratasets could develop on the downstream side of hills. For the same deposit, Fisher (1990) observed that crests aggrading uphill and upcurrent are primarily found on lee side slopes >20°, whereas down-current migration would occur on flat zones. These latter observations contrast with recent field data proving the coexistence of any pattern within a single DB  Rowley et al., 1985;Brand et al., 2016), low-angle to regressive to progressive trend. Pen for scale is 14 cm long. (B) Tungurahua, backset ripples in sediment plates/lacquer peels (see Douillet et al., 2018Douillet et al., , 2019. For all images, flow orientation roughly from left to right; photograph (A) has been mirrored for consistency.  Dellino et al., 2004), relatively low-angle and truncated beds, pen for scale is 14 cm long. (D) Villarica, likely 'P2C member' (see Parejas et al., 2010), truncated structure draped by fallout and subsequent flow deposits. Person for scale is ca 1.55 m tall. (E) Laacher See (see Schmincke et al., 1973), truncated structure transitional to type D (mixed type C/D), hammer for scale is 28.5 cm long. For all images, flow orientation roughly from left to right; photographs (A), (B) and (E) have been mirrored for consistency.  Rowley et al., 1985;Brand et al., 2016), zoom in (B) into coarse-enriched backset beds. (C), (D) and (E) Tungurahua (see Douillet et al., 2013bDouillet et al., , 2019. Complete dune-bedform in (C) with zoom into truncation trains in (D). (E) Same structure at centimetre-scale and silt-size material, visible only after epoxy impregnation in sediment plates (Douillet et al., 2018). For all images, flow orientation roughly from left to right. Photographs (A) and (B) have been mirrored for consistency. structure in between parallel cuts separated by tens of centimetres (Douillet et al., 2019). Within a single structure, several types of patterns can coexist (types B, C and D), as well as fractions with dominance of foreset versus backset laminasets, suggesting that the observations of Schmincke et al. (1973) and Druitt (1992) may reflect a tendency only, or that they may simply be coincidental. An organization in repetitive periodic trains of two to five DB structures may occur for low-angle DBs (type A, for example Figs 5C and 7A), steep truncations (for example, Fig. 8A, D and E), or fully-aggrading patterns (for example, Fig. 10; e.g. Cagnoli & Ulrych, 2001;Douillet et al., 2019). Where several cross-stratified beds are superimposed, DBs have a strong tendency to build up over the shape of former structures, resulting in vertically stacked DBs at a fixed location, thus forming composite structures ( Fig. 11; e.g. 'perched dunes' in Giannetti & Luongo, 1994;Sigurdsson et al., 1987;Douillet et al., 2013b).

Grain-size distributions
Pyroclastic DB structures have grain-size distributions characterized by poorer sorting and spanning a wider range of particle sizes compared to most aeolian and waterlaid bedforms, ranging from silt to gravel sizes (for example, Fig. 8B versus 8E versus 12, Walker, 1984;Love et al., 2018). Within pyroclastic deposits, dunebedded material tends to overlap in terms of sorting versus mean particle diameter on the finer and better sorted fields of 'granular flows' and 'fallout' end-members (Walker, 1971;Crowe & Fisher, 1973;Sparks, 1976;Walker, 1984).
Accretionary pellets are common in DB deposits (e.g. Schumacher & Schmincke, 1991;Dellino et al., 2004). Polymodal distributions are common and deconvolution of signal could help with unravelling superimposed transport mechanisms (e.g. Wohletz et al., 1989;Genc ßalio glu-Kus ßcu et al., 2007;Douillet et al., 2013a;Mackaman-Lofland et al., 2014;Love et al., 2018). In wind-tunnel experiments with pure winds, the surface roughness and saltation threshold for pyroclasts were found to be roughly similar to those of particles of similar density, providing an estimate for transport and depositional flow velocities (Douillet et al., 2014). The Sauter mean diameter, defining a characteristic length scale for fluid-particle interactions, may represent the most relevant parameter for PDC transport and deposition (e.g. Breard et al., 2019).
Interestingly, the smallest documented pyroclastic structures (few centimetres length) can consist of fine silts, a characteristic absent from larger DB structures (tens of metres length), which are commonly dominated by lapilli/pebble sizes. However, many examples show that grain size is not smoothly distributed with dimensions of DBs (for example, Figs 5D, 6B and 7C), but rather depends on the source material (eruption style). Many DB structures of any type have stoss face backset strification or lenses visually fairly enriched in coarse fractions, but depleted in fines (for example, Figs 5B, 6B, 7C and 8B). In samples from DB structures of type B at Tungurahua, stoss side bedsets were on average better sorted and coarser-grained than lee side ones (Douillet et al., 2013b).

Seminal work and early interpretations as supercritical bedforms
Pyroclastic DB cross-stratifications were recognized as primary features after the observation of undulating bedforms in deposits of artificial explosions (for example, Bikini 1946 and Sedan 1962 nuclear 'test' bombs; Carlson & Roberts, 1963), or mantling the surface of pyroclastic deposits from historic eruptions (plate 24 in Richards, 1959;Moore et al., 1966;Moore, 1967). From the 1966 eruption at Taal Volcano (Philippines), Moore described cross-stratification with both stoss and lee side laminae, episodic scouring of the stoss side, steeper stoss than lee angle, and evidence for a slight downcurrent migration of the bedform crests. Cross-strata in ancient successions have initially been termed antidunes as a descriptive term for "low-amplitude bed waves with stoss-and leeside slopes inclined at angles less than the angle of repose" (Fig. 5B and C ;Fisher & Waters, 1969). Similarly, Fisher & Waters (1970) 'referred' to PDC DBs as antidunes because of their typical lowangle lamination contrasting with the steeper lee side dip angles (at or near the angle of repose) of subcritical bedform laminae evidenced by Simons & Richardson (1966). Waters & Fisher (1971) were the first to use antidune in its hydrodynamical meaning, i.e. as an interpretative term for structures formed under a standing wave at a relatively sharp density interface between the moving bed load and the overriding PDC cloud. Their argument was that the shape and internal cross-stratifications resembled antidunes produced in flume experiments by Middleton (1965), and described from turbidite outcrops interpreted as antidunes by Walker (1967). Waters & Fisher (1971) noted that only a high depositional rate would permit preservation of antidune cross-stratifications, likely to occur in PDCs. Crowe & Fisher (1973), re-using the illustration of Fisher & Waters (1969) and Mattson & Alvarez (1973), followed the antidune interpretation of Waters & Fisher (1971) for several other volcanoes, applying equations for stationary surface-gravity waves and supercritical flows for palaeohydraulic calculations. Schmincke et al. (1973) observed decametre scale structures with backset and foreset stratification (fully-aggrading) and instances of preserved upstream migration of the crest (regressive), but with steep stoss and lee sides (Figs 7E, 10C and 11A). Even if the primary evidence of Fisher & Waters (1970) for facies interpretation was not observed (i.e. stoss and lee sides below the repose angle), Schmincke et al. (1973) interpreted these structures as the products of antidunes. These authors thoroughly focused on structures made from stoss-aggrading lensoidal layers deposited in front of steep erosive surfaces that they attributed to more proximal areas. As such, these structures were regarded as representing higher-energy bedforms than antidunes, and thus interpreted as 'chutesand-pools' (Figs 11A and 12B).
The first detractor of the supercritical interpretation was Allen (1984), who considered it 'bolder' in his general sedimentology textbook. Allen (1984) suggested that upstream aggradation and migration was a plastering effect due to the cohesion of particles in the presence of liquid water in PDCs, whereas downstream migrating bedforms would be limited to 'dry surges' (see also Carey, 1991). This process model was disproved by two in-depth studies by Cole (1991) and Druitt (1992), which showed that both upstream and downstream-accreting bedforms could occur during the same eruptive event. Hence, aggradation patterns were not related to water content, but to local changes in the flow characteristics. The dimensions of DB structures produced in wet versus dry eruptions are similar (Fig. 4), accretionary pellets may occur in both, and no distinctive characteristics in stratification patterns were identified in this review.
Alternatively, it has been suggested that backset stratasets may actually represent foresets deposited under reversed flows. Fisher (1990) suggested a decoupling of the transport and depositional system, so that secondary basal granular flows could flow downslope against the direction of the main current (locally travelling upslope), to form the final structures. Andrews & Manga (2012) proposed that the uplift of a phoenix cloud in the proximal part of a flow could suck back a PDC front, thus creating a local flow reversal that might have a sediment signature with apparently regressive DBs. Although such reversed-flow processes may occur in particular cases, there is various evidence that most pyroclastic DBs are purely primary features related to a relatively consistent flow direction, based on the orientation of clast imbrications, truncations and sheared structures (e.g. Schmincke et al., 1973;Douillet et al., 2013aDouillet et al., , 2015Douillet et al., , 2018. Still, using reasonable flow relations in an antidune equation framework, most pyroclastic bedforms appear too steep to support a supercritical-flow origin (Charland & Lajoie, 1989;Douillet et al., 2013b).

Stoss-aggradation unrelated to supercritical conditions
A pervasive belief in pyroclastic literature is that foreset bedding would be diagnostic for subcritical-flow conditions at time of deposition, and backset for supercritical conditions. A field study of parallel cuts within a single pyroclastic DB has recently demonstrated the co-existence of steep erosive-based backset laminasets (type D), fully-aggrading stoss-aggrading laminasets (type B) together with stoss-eroded lee bedsets within the same structure (Douillet et al., 2018(Douillet et al., , 2019. The crude shortcut of a simplistic interpretation based purely on the dip of stratasets is thus not applicable, since it would imply the unlikely synchronous co-existence of a chuteand-pool, antidune and subcritical dune bedform in a single sedimentary structure. Foreset strata are neither uncommon in supercritical-flow structures, nor diagnostic for subcritical conditions. Stoss-eroded foreset beds may well represent remnants of a formerly fullyaggrading, progressive or regressive structure. It has been demonstrated in analogue experiments and highlighted in several studies that supercritical conditions may lead to foreset stratification and downstream migrating structures (e.g. Carling & Shvidchenko, 2002;Spinewine et al., 2009;Ito, 2010).
It is here considered that backset bedding should not necessarily be ascribed to supercritical conditions and can occur in a variety of settings. Sigurdsson et al. (1987) related the formation of pyroclastic DBs to a combined effect of saltation-bedload transport, together with a suspension component (see also Cole & Scarpati, 1993, and the 'simultaneous fallout  (Douillet et al., 2018). For all images, flow orientation roughly from left to right. and surge' in Valentine & Giannetti, 1995). Building on this idea, a mechanism of 'differential-draping' was suggested for fully-aggrading structures with continuous stoss to lee side laminae, including preferential upstream aggradation (Douillet et al., 2013b). 'Differentialdraping' is unrelated to supercritical conditions, and would correspond to a fallout-modified process over an inherited topography, affected by a slow-moving current (i.e. below or close to the saltation threshold) and high aggradation rates. Wind-tunnel measurements with pure wind and pyroclasts show that the threshold for saltation can be 50% higher on a 20°upslope incline (stoss side analogue) than on a À25°downslope face (lee side analogue) (Douillet et al., 2014). This result has been suggested to explain preferential aggradation on stoss faces as well as upstream-aggrading bedforms unrelated to antidunes, and is supported by field evidence (see fig. 6C in Douillet et al., 2019).

Antidunes
The argument to interpret certain sedimentary structures as the products of antidunes in the seminal literature was related to the flatness or low relief of bedforms, as observed from openchannel water flows (e.g. Waters & Fisher, 1971;Crowe & Fisher, 1973;Self & Wright, 1983;Druitt, 1992). Rapid and shallow supercritical currents are unlikely to create steep bedforms. Any antidune growing too steep would lead to a chute-and-pool configuration. Taking into account the previous consideration of nonuniqueness of backset/foreset formation, flatness is here considered a necessary argument to infer antidunes in pyroclastic DBs. Thus, only patterns of 'type A' would be eligible for an interpretation as antidunes in pyroclastic DBs ( Fig. 5; e.g. Douillet et al., 2013b), in accordance with the initial interpretation (Waters & Fisher, 1971;Crowe & Fisher, 1973).
Steep sided, stoss-aggrading, fully-aggrading DB structures (type B, Fig. 6) are largely taken as antidunes as well in the literature. However, field observations of pyroclastic DB stratification of type B (and C and D) share very few attributes with deposits from most experimental work on antidunes, independent of a deposition rate or environment (e.g. Alexander et al., 2001;Spinewine et al., 2009;Cartigny et al., 2014). Furthermore, the measured dimensions and steepness (>35°slopes) are not compatible with antidunes and are disqualified by a quantitative analysis, since the parent underflows should be less than 15 cm deep with a velocity <1 m s À1 (Charland & Lajoie, 1989;Douillet et al., 2013b). Additionally, the uncommon occurrence of series of periodic, in train patterns, and the important spatial stability of pyroclastic DBs are contrasting with the dynamics of free-surface gravity waves. Further, fully-aggrading DB structures can exhibit preferential downstream or upstream aggradation, yet lee side erosion is virtually absent. Given that an antidune is a morphological bed reflection of a gravity wave (a 'mould'), when backset bedding occurs, it should be associated with concurrent lee side erosion due to the mould being translated upstream (Douillet et al., 2013b(Douillet et al., , 2019.
Some authors have suggested that DB bedding of type C (i.e. low-angle truncated structures with draping beds, Fig. 7) could represent antidunes (e.g. Schmincke et al., 1973;Wohletz & Sheridan, 1979). Besides the aforementioned arguments, in that case, the associated gravity wave would be moulded by a pre-existing bedform, so that it would be driven by the basal topography rather than by the free-surface. As such, this rules out antidunes in the sense of a a bedform moulded by a free-surface wave. Existing estimations of emplacement velocities using an antidune equation framework for pyroclastic bedforms >100 m length are extremely high (52 to 73 m s À1 , Brand & Clarke, 2012). Whereas such high velocities may occur during transport (e.g. Scolamacchia & Schouwenaars, 2009), they might be too high to lead to deposition, since they are far above the threshold for saltation for pyroclasts (6 to 8 m s À1 near-bed velocities e.g. Douillet et al., 2014Douillet et al., , 2019.

The subcritical zone of a Froude jump/erosive chute and depositional pool
Even if DB structures of type B, C and D seem unrelated to antidunes and direct supercritical conditions, one can envisage the formative process to be related to the pool of a Froude jump. Pyroclastic DB stratification of type D has been interpreted as the product of chutes-and-pools since the seminal manuscript of Schmincke et al. (1973). Indeed, the typical pyroclastic DB patterns, with alternation of purely depositional phases with stoss face scouring and subsequent infill with coarse-grained backset bedding, share similarities with experimental deposits and some coarse-grained sedimentary structures interpreted as formed by chutes-and-pools in deposits of other environments (e.g. Postma et al., 2014;Massari, 2017;Lang et al., 2021). This interpretation is strengthened by the common fines-depletion and coarse-enrichment within the toe of backset beds, suggesting an increased turbulence and loss of capacity, coherent with a Froude jump or a basal-blocking process (Figs 8B and 12, e.g. Schmincke et al., 1973;Cole & Scarpati, 1993;Douillet et al., 2019). Additionally, truncation surfaces may become almost vertical and show overturned laminae at the truncation front (type D, Fig. 8). This latter result was reproduced experimentally by eroding a bedform with a short-lived air burst used to mimic turbulent pulses, yet it may be related to the oscillation of the shooting part of a Froude jump as well (Douillet et al., 2017). A similar signature with truncations underlined by overturned laminae has been observed in experimental Froude jumps, and may develop from natural density currents .
For a chute-and-pool interpretation to be realistic, a shooting (supercritical) flow should impact the stoss of a bedform, erode it, and a Froude jump would have to retreat rapidly upstream, allowing the formation of backset beds. In this frame, the backsetting would be the signature of deposition in the subcritical pool, and not related to supercritical conditions, the latter being limited to the erosion phase. Steep erosive fronts can occur in a series of several truncations cutting one another and stacking in an upstream direction (Fig. 8). Again, this is compatible with an upstream-retreating Froude jump, yet it could also result from internal short-lived bursts or pulses related to turbulence or coherent flow structures near the bed. In such a scenario, the Froude jump would rather be a consequence of a bed topography than occur spontaneously; however, necessary process evidence is still lacking for further discussion.
Additional to type D, it is here considered that patterns of type B and C could well represent the signatures of chutes-and-pools as well. Type C may be envisioned as a transitional form to D, having only lower-angle truncation, and as transitional form to type B, sharing the same internal organization apart from the occurrence of a truncated stoss side. Central in this interpretation is that patterns of type B, C, and D coexist laterally within a single DB structure, and during the same time interval. They are thus likely to represent the same phenomenon, yet with more or less competence (i.e. ability to erode) depending on whether the exposure belonged to the centre or side of the DB structure. Indeed, aside from the truncation, the aggrading parts of these structures, often showing preferential stoss side aggradation and steep backset beds, share most of the characteristics of backset beds within type D patterns (Figs 6  and 10). A sudden drop in competence of the currents, as in a Froude jump, may well explain such structures. Further, citing Chough & Sohn (1990), the often encountered 'smooth and rounded crests without brinkpoints suggest that the bedforms did not entail flow separation', a result that could be attributed to the conditions in the 'pool' of a Froude jump. The mechanism of 'differential-draping' suggested for patterns of type B (Douillet et al., 2013b), which corresponds to a fallout process influenced by a slow moving current over an inherited topography, is fully compatible with an occurrence in the subcritical part of a Froude jump. As such, pyroclastic DBs of type B, C, and D would thus all be related to chutes-and-pools, the different types solely being influenced by the advance or retreat of the erosive/depositional front of the Froude jump.
Interestingly, besides the interpretation of DBs, large-scale sediment bodies, being either lithic breccias (e.g. Macıas et al., 1998) or entire deposition zones covered with DBs (Douillet et al., 2013a), have been postulated to result from Froude jumps of the entire flows at break in slopes. As such, these features could represent giant chutes-and-pools.

Possible rare cyclic steps
There has been an enthusiastic emphasis on cyclic steps in recent sediment literature, especially in the context of sedimentation from density currents (Parker et al., 1996;Cartigny et al., 2011;Dietrich et al., 2016;Slootman & Cartigny, 2020). Cyclic steps are trains of generally asymmetrical bedforms separated by stable, migrating Froude jumps, characterized by a transition from transcritical to subcritical flow over their stoss face to supercritical flow over their lee face. Documented cyclic steps are notorious for their periodic in-train repetition and their upstreamdirected migration concomitant with prevalent deposition on gently dipping stoss faces and erosion on the lee sides. The growing recognition of cyclic steps formed by subaqueous density currents legitimises the question of their possible identification from pyroclastic deposits. However, most pyroclastic DB structures share little with those documented from cyclic steps. In particular, lee side erosion is rare, bedforms are fairly irregular, repetitive in-train patterns are uncommon (although not rare), and backset strata are extremely steep. The occurrence of cyclic steps, however, may be hypothesized for several cases. Sigurdsson et al. (1987) noted a superimposition of "larger elongate transverse bedforms up to 100 m in length and smaller [m-scale DBs] with short and curved crests" at the surface of the 1982 El Chich on deposits. This observation fits with the superimposition of short-period 'chutes-and-pools' and long period 'cyclic steps' observed in analogue flume experiments by Cartigny et al. (2014), and might represent a rare documented case of cyclic steps in pyroclastic deposits. In such a case, large-scale undulations would represent the preservation of cyclic steps. Another insight could come from surface shapes observed at Tungurahua, where DBs exhibit a downstream transition in shape from straightcrested 'transverse' to crescent shaped 'lunate' morphology, ending with wide, straight-crested 'two-dimensional' bedforms ( Fig. 9; Douillet et al., 2013aDouillet et al., , 2013b. This spatial morphological succession is strikingly similar to experimental results with subaqueous density currents of supercritical flows interpreted as representing cyclic steps (see figs 17 to 19 in Spinewine et al., 2009). This would imply that all of these DBs could actually represent cyclic steps. Further data is needed to validate hypotheses on the recognition of cyclic steps, in particular regarding the crests' knickpoints, the steepness of bedforms, and the consequent implications for the scale and stability of the parent flows; however, this interpretive possibility remains open.
Another possible occurrence of cyclic steps in pyroclastic DBs is represented by in-train, repetitive patterns of fully climbing structures of type B (Figs 10 and 11). If DB stratification of type B may represent the pool of Froude jumps, as suggested above, then trains of fully-aggrading DB structures may represent small-scale cyclic steps. These structures further represent an uncommon case of stable deposition from periodic bedform trains, and their geometry resembles that of some submarine sediment waves, all of these characteristics being compatible with those of cyclic steps (see e.g. Cartigny et al., 2011;Pope et al., 2018;Slootman & Cartigny, 2020).
In this framework, thick, stoss-aggrading, lensoidal backset layers were understood as being formed by the basal blocking of a granular-based PDC in correspondence with bed irregularities (Fig. 12;Freundt & Schmincke, 1985;Douillet et al., 2013bDouillet et al., , 2019. Such an interpretation is supported by granular flow experiments and complemented by dynamical observations that show the development of a phenomenon similar to a Froude jump, referred to as a 'granular jump' (e.g. Boudet et al., 2007;Johnson & Gray, 2011;Viroulet et al., 2017;Smith et al., 2020).
Field and experimental evidence concur to suggest that the formation of such bedforms is initiated in the 'subcritical' part of a granular Froude jump triggered by the pre-existing bed topography (Brennen et al., 1983;Freundt & Schmincke, 1985;Douillet et al., 2019;Smith et al., 2020). Pyroclastic DBs could thus contain a rare natural example of bedforms created by Froude jumps occurring in granular flows.
Besides, for many DB structures of type C, a stratified formset structure is overlain by massive deposits infilling and interacting with the bed (e.g. Fisher et al., 1983). These thus represent a form of granular flow interacting with a pre-existing bedform, and participating in its development (see also elongate DBs in Douillet et al., 2013b). Rowley et al. (2014) showed experimentally that a succession of stacked surfaces deposited from fluidized flow could result from progradational or retrogradational accretion and result in patterns similar to many natural pyroclastic DB structures. Such patterns could correspond to forms of granular Froude jumps, or even granular antidunes. An interpretation of DBs as products of basal granular flows would imply an extremely short timescale for their growth, in the order of seconds. Further, the basal granular flows would be largely dominant in shaping the bedform; in other words, the thin depositional system would not contain a record of the transport system, and thus of the vast majority of the upper parts of the carrying currents.

Internal density stratification and Richardson number
The sedimentary dynamics of a mobile bed in the supercritical regime are understood to be strongly correlated to the upper free surface of the flow dictating the growth of bedforms. The Froude regime being related to a restoring gravity force, this free surface must be a density interface. Further, the metric scale of individual pyroclastic DBs contrasting with the hundreds of metres in thickness of the visible envelope of a PDC, the free surface interacting with the bed during deposition must be an internal interface within the lower part of the current, probably governed by density stratification (e.g. Schmincke et al., 1973;Valentine, 1987). Whether this density interface is a sharp or a smooth gradient is unresolved, yet PDCs are thought to be vertically stratified in density and experiments support the existence of a basal dense region (e.g. Burgisser & Bergantz, 2002;Breard et al., 2016). In an interpretation in terms of antidune, the depth-dependent Brunt-Vaisala frequency (e.g. Vallis, 2006, p. 92) would represent the relevant length scale for an 'internal gravity wave' (e.g. Valentine, 1987;Carey, 1991). The flow depth at which the relevant interface would occur is unclear, but should lie somewhere below the (wave)length of DBs. It has been stressed that such underflows should be very thin (<15 cm) and slow (<1 m s À1 ) to create metre-scale DBs (Charland & Lajoie, 1989;Douillet et al., 2013b). However, the existence and stability of such an oscillating density interface represented by a slight contrast of particle concentrations seem questionable for PDCs. Would it not be erased by turbulent mixing or temperature buoyancy effects before any undulation interacts with the bed? In other words, the Richardson gradient numbers (ratio of buoyancy over shear/turbulent mixing, Table 1) in a PDC may not allow a stable density interface to develop and create supercritical underflows. Even if these interfaces occur and are stable over the timescale of a bedform's growth (possibly seconds to hours), the presence of the underflow might form a depositional system partly decoupled from the transport system, and thus informing only on basal flow-boundary processes.

Rarity of stable bedforms
In an interpretation of DBs as upper-flow regime bedforms, antidunes would represent only a marginal portion of pyroclastic DB deposits, whereas the vast majority would be related to the pools formed by Froude jumps. Similarly, cyclic steps would only be a rare end-member confined to large structures. A first explanation may be that antidunes and cyclic steps may be essentially erosional and unrecognized, or have very little preservation potential in a PDC environment. Three specific reasons tied to PDCs can be considered to explain the dominance of chutes-and-pools over cyclic steps and antidunes: sedimentation rate, pulsation and unstable interfaces.
1 At high sedimentation rates, the onset of an antidune will be rapid and abruptly destabilize the flow, leading to the growth of Froude jumps and 'chutes-and-pools'. Thus, if any, signs of antidunes could be preserved as incipient structures (small-scale DB patterns of 'type A'), and they would be directly overrun and possibly reworked by chutes-and-pools. Additionally, the later growth of steep-sided chutes-and-pools may in turn impede any transition to the cyclicstep stability field. 2 Pyroclastic density currents may be far more pulsating, turbulent and short-lived than most natural density currents (e.g. Vazquez & Ort, 2006;Andrews & Manga, 2012;Pedrazzi et al., 2013;Andrews, 2019). As a consequence, it may be impossible to maintain flow conditions for a prolonged enough time within the stability regime for stable trains of antidunes and cyclic steps to grow and be preserved. A minimal timescale might never be reached, apart from rare sustained eruptions with long runout and long duration (longer than the minutes to hours of a typical eruptive pulse). 3 As mentioned above, the stability of a density stratified PDC may be questioned, the growth of a wave interface for antidunes may be impeded by the transient nature of the density contrasts, and only allow for Froude jumps to grow. A detailed physical study of the relevant equation frames would be needed to provide further evaluation on this hypothesis.

Supercritical can be low-energy and waning
Based on grain-size distribution data, Walker (1984) concluded that dune-bedded deposits correspond to 'weak' events, in agreement with measurements of the saltation threshold for typical pyroclastic grains (Douillet et al., 2014) and structural characteristics (Colella & Hiscott, 1997;Douillet et al., 2013b). In parallel, the observation that steep backset beds grow up to the uppermost laminae at the surface of recent pristine deposits, show that these must correspond to the final stage of events (Douillet et al., 2013b(Douillet et al., , 2019, an observation in agreement with the transition of DB structures into draping patterns (Dellino et al., 2004;commented by Le Roux, 2005). Backset laminasets thus at least partly record waning and/or depleting dynamics, in agreement with low-energy 'weak' events, and highly depositional dynamics has been emphasized by most authors.
Although supercritical flows are often envisioned as 'highly energetic', in fact the flow regime is unrelated to the total amount of energy in a flow, and the Froude number represents a dimensionless ratio involving flow velocity and thickness (or kinetic over potential energy, Table 1, see e.g. Prave, 1990;Douillet, 2015). Thus, low-energy and waning conditions may be fully consistent with supercritical flows, and do not preclude any flow-regime interpretation. Possibly, PDCs might pass from subcritical to supercritical conditions on steep volcano flanks when decreasing their (under)flow thickness during a waning flow phase, together with a decrease in total energy of the current. Pyroclastic DBs might thus represent low-energy bedforms formed during the waning phase of a PDC into supercritical conditions.

Froude versus Reynolds
Whereas an important focus in the literature has been put on analyzing pyroclastic DBs in regard to the Froude number of formative flows, their growth patterns might actually be driven by the dynamics of turbulent structures, and thus reveal more about parameters such as the Reynolds number (indicative of the ratio of inertial to viscous forces internal to the flow) or the Strouhal number (indicative of the ratio of inertial forces due to local flow acceleration over those due to convective acceleration) than Froude number (Table 1; Douillet et al., 2019). Particle-laden density currents are subjected to intermittency, pulsation and large turbulent fluctuations (e.g. Sequeiros et al., 2018;Sovilla et al., 2018). Pyroclastic currents may similarly be pulsating, characterized by high values of the Reynolds number (up to 6*10e9) and by large coherent turbulence structures up to hundreds of metres internally (e.g. Walker, 1984;Vazquez & Ort, 2006;Andrews & Manga, 2012;Lube et al., 2015;Andrews, 2019;Brosch & Lube, 2020).
The vertical truncations observed in DB structures of type D, together with overturned truncation planes, have been reproduced experimentally by means of short-lived (<<1 s) air bursts (at 20 to 40 m s À1 ; Douillet et al., 2017Douillet et al., , 2019. The origin of such short-lived pulses in a natural flow is unknown, but it could represent the signature of coherent turbulent structures in the basal portions of PDCs or the occurrence of fluctuating Froude jumps. Concurrently, extreme rates of sedimentation are needed to preserve steep to subvertical backset laminasets, since the preservation of laminae with dip values well above the angle of repose in the sediment record implies rapid burial within a timespan of seconds. Interestingly, pulses of erosion and sedimentation waves are triggered by transient segregations of particles into mesoscale clusters around turbulent vortices (Burgisser et al., 2005;Andrews & Manga, 2012;Breard et al., 2016;Breard & Lube, 2017). It thus remains currently unclear whether pyroclastic DB structures of type D, but also B and C, can be explained by dynamics based on high Froude numbers or through a combination of turbulent fluctuations, high Reynolds numbers and high deposition rates.
Bed shape controla flow-bed feedback The predominance of individual bedforms rather than tabular sets, widespread occurrence of fully-aggrading DB structures (type B), and the general vertical stacking of DBs one above the other all point towards a strong spatial stability of pyroclastic DBs and towards an absence of migration (i.e. absence of translation of the entire structure). Such a spatial stability suggests that DBs are related to a bed configuration rather than a flow's free-surface. In parallel, Druitt (1992) noted that DB stratification nucleated against obstacles at Mount St. Helens, and the role of truncations in the nucleation and growth of backset bedding for DB structures of type C and D is illustrated here (Figs 7,8,11A and 12; see also Valentine et al., 1989). Similarly, Allen (2001) attributes DBs to "internal waves generated by interaction with obstacles" and Freundt & Schmincke (1985) emphasized the role of topographic control in the emplacement of coarse-enriched backset beds. A simplistic equation framework also suggests that backset strata could grow in the supercritical regime as a simple feedback effect between an existing topography and the current, without being related to antidunes, chutes-and-pools or cyclic steps (Douillet, 2015). Additionally, several mechanisms of bed control for the growth of backset bedding unrelated to supercritical flows have been suggested, including differential draping, basal blocking, or the influence of local gradient on saltation (Douillet et al., 2013b(Douillet et al., , 2014(Douillet et al., , 2019. Thus, there seems to be a fundamental role of the bed shape in the growth of backset beds, under supercritical conditions or otherwise. Whereas some initial bed configurations may be driven by supercritical flow dynamics, this seems largely contained in an erosive signal, or in low-angle structures. The vast majority of sediments deposited as pyroclastic DBs might be a sole consequence of these initial shapes and positive bed-flow feedback mechanisms.

CONCLUSIONS AND SUMMARY
Pyroclastic dune bedforms form peculiar sedimentary structures with widespread occurrence of fully-aggrading stratasets, steep truncations on stoss faces, and steep-sided backset beds. Since their recognition, these sedimentary structures have largely been associated with antidunes and chutes-and-pools, either as a descriptive term or an interpretation of flow dynamics, yet an understanding of their origin remains equivocal.
Here, it is stressed that backset stratification is not a sufficient feature for the attribution of such structures to supercritical flow conditions; likewise, foreset stratification is not a sufficient criterion to justify an interpretation of subcritical flow conditions. Most of the sedimentary signature of supercritical flows is likely represented by erosional truncations or low-angle bedforms. In that frame, an interpretation of low-angle bedforms as antidunes seems a possibility. Steep-sided backsetbearing structures similarly may represent a record of the subcritical 'pool' of Froude jumps, whereas associated underlying steep truncations would represent pulses of a retreating supercritical 'chute'. Cyclic steps seem rare to absent and, if present, they may be expressed as long-wavelength bedforms superimposed by metre-scale dune bedforms, or as repetitive trains of fully-aggrading, regressive structures. A prevalence of intermediate-stage structures (chutes-and-pools) over stable antidunes and cyclic steps could be attributed to the unstable and highly depositional nature of the flows. However questions remain to be addressed and alternative explanations are possible.
In an interpretation as Froude supercritical bedforms, the free flow interface shaping structures must be originated by a basal layer within a density-stratified current (i.e. flow stratification induced by vertical gradient in particle concentration). Thus, if the bedforms are interpreted as Froude supercritical, they would essentially inform about deposition under a thin, basal flow layer, providing little information about the majority of the overlying transport system. It is presently unclear whether such a density interface is stable enough with regard to internal turbulence and buoyancy. The role of turbulent flow structures may also be fundamental in shaping truncations and backset beds, in particular when particles are segregated at the edge of mesoscale turbulent vortices. Turbulent cells and fluctuations alone may offer an alternative interpretation for the formation of pyroclastic structures unrelated to supercritical flows.
Independent of any possible interpretation in terms of flow regime, the bed-inherited shape also seems to play a fundamental role in the development of pyroclastic dune bedforms, which appears largely related to a flow-bed interaction process. Three explanations exist for the formation of backset stratification unrelated to supercritical flow conditions, all of them related to an existing topography as the trigger: (i) differential draping by suspended material; (ii) differential saltation capacity regulated by local gradient; and (iii) basal blocking of a bedload. Finally, most authors emphasize that pyroclastic bedforms are deposited under extremely high deposition rates, which is a central factor in the preservation of a variety of peculiar sedimentary structures observed in this kind of deposit, but so far unreported in clastic successions from other sedimentary environments.

ACKNOWLEDGEMENTS
Data was collected over numerous field campaigns, and short courses between 2009 and 2020, a PhD and post-doc project. I thank the two reviewers Benjamin Andrews and Martin Jutzeler as well as the editor Dario Ventra for their constructive reviews that improved the manuscript. U. Kueppers and D.B. Dingwell are acknowledged for continuous support over the years at LMU Munich and F. Schlunegger at University of Bern. Funding through several grants, including SNF-Ambizione, DFG-DO1953/ 1-1, DFG-KU2689/2-1, ERC Advanced Grant Evokes (247076), LMU-Excellent and Elitnetzw-erkBayern. Numerous colleagues enabled great field work including A. Abolghassem, J. Battaglia, B. Bernard, B. Brand, J. Bustillos, J.R. Dujardin, C. Mato, D. Pacheco, N. Pollock, P. R amon, R. Reschetizka, C. Robin, and E. Tsang-Hin-Sun, as well as the members of IG-EPN in Ecuador. The literature review would not have been possible without access through Sci-hub. Open Access funding enabled and organized by Projekt DEAL.

DATA AVAILABILITY STATEMENT
All images from the author. All figures available upon personal request. Length versus thickness data are available in the cited literrature.