Formation of Kinneyia via shear-induced instabilities in microbial mats
Katherine Thomas, Stephan Herminghaus, Hubertus Porada, Lucas Goehring
FFormation of Kinneyia via shear-induced instabilities in microbial mats
Katherine Thomas, ∗ Stephan Herminghaus, Hubertus Porada, and Lucas Goehring Max Planck Institute for Dynamics and Self-Organization, Am Fassberg 17, D-37077 G¨ottingen, Germany University G¨ottingen, Geowissenschaftliches Zentrum, Goldschmidtstr. 3, D-37077 G¨ottingen, Germany (Dated: September 13, 2018)Kinneyia are a class of microbially mediated sedimentary fossils. Characterised by clearly defined ripplestructures, Kinneyia are generally found in areas that were formally littoral habitats and covered by microbialmats. To date there has been no conclusive explanation of the processes involved in the formation of these fossils.Microbial mats behave like viscoelastic fluids. We propose that the key mechanism involved in the formationof Kinneyia is a Kelvin-Helmholtz type instability induced in a viscoelastic film under flowing water. A ripplecorrugation is spontaneously induced in the film and grows in amplitude over time. Theoretical predictionsshow that the ripple instability has a wavelength proportional to the thickness of the film. Experiments carriedout using viscoelastic films confirm this prediction. The ripple pattern that forms has a wavelength roughly threetimes the thickness of the film. This behaviour is independent of the viscosity of the film and the flow conditions.Laboratory-analogue Kinneyia were formed via the sedimentation of glass beads, which preferentially depositin the troughs of the ripples. Well-ordered patterns form, with both honeycomb-like and parallel ridges beingobserved, depending on the flow speed. These patterns correspond well with those found in Kinneyia, withsimilar morphologies, wavelengths and amplitudes being observed.
I. INTRODUCTION
Biofilms are ubiquitous. Found in environments ranging from Antarctic ice [28, 78] to the walls of deep sea hy-drothermal vents [82] and the human body [36], biofilms represent one of the earliest life forms on Earth. Recentdiscoveries provide evidence that fossils of microbial origin date back to 3,450 million years ago [3], suggesting thatbiofilms have been present throughout the majority of the Earth’s history.Microbial mats can be seen as thick biofilms. They range in thickness from a few millimetres to a few centimetres,depending on the habitat, growth conditions and bacteria that form them [32, 45]. One of the most successful matforming bacteria is cyanobacteria due to its large morphologic variability and capacity for biostabilisation (increasingsediment stability or reducing erosion) [30]. However, mat formation is not limited to cyanobacteria and can resultfrom organisms including bacteria, archaea [83], protozoans, algae [21] and fungi [79]. The term ‘microbial mat’ willbe used here to describe any macroscopic cohesive microbial community growing on, adhering to, or enmeshing aninorganic substrate [38].Modern microbial mats are commonly found growing on rock, soil and granular carbonatic and siliciclastic surfacesin water-based habitats and are preferentially observed to grow in the intertidal to lower supratidal zones of riverine andmarine environments [30, 32] as well as hypersaline lagoons [2]. Microbial mats can enhance stabilisation of sandysubstrata via the secretion of adhesive mucilages, which glue the sediment grains together, reducing the erodabilityof the sediment [29, 56, 58, 59, 69]. The prevalence of biofilm systems, in particular cyanobacteria, have made themsuccessful at leaving traces in sediments. This is readily observed in the host sediments of modern microbial mats[30].Microbially induced sedimentary structures (MISS) can be grouped depending on the role that the microbial mat hadin the formation process [48] and arise from physical processes including biostabilisation [18], ba ffl ing and trapping[8, 64], binding [55] and growth [5, 32]. The patterns and structures arise due to the presence of microbial mats.In the absence of a mat no structures are expected to form. MISS are highly diverse [49], with features rangingfrom the millimetre to metre length scales. They include wrinkle structures, palimpsest ripples, roll up structures andlaminar structures [50, 56]. Recently Bouougri and Porada [13] documented various structures including crack patternsand crumpled layers, which form when strong winds blow over microbial mats. While chemical processes are not ∗ Electronic address: [email protected] a r X i v : . [ n li n . PS ] F e b considered here, recent studies on modern microbial mats have shown that microbially mediated mineral precipitationis important for the preservation of sedimentary structures [17, 20]. In siliciclastic environments precipitation ofhydrated alumino-silicates from the mats may also result in cementation and lithification of the surrounding sediments,potentially also playing a stabilizing role in fossil preservation [4, 39, 44]. A. Fossilised microbial mats
Structures similar to those observed in modern living mats are also observed in the ancient geologic record[12, 30, 48]. Sedimentary features in the clastic sedimentary rock records from the Precambrian era point towardsthe prevalence of microbial mats during this part of the Earth’s history, particularly in storm-a ff ected subtidal environ-ments and the intertidal zone [42, 48]. The abundance of MISS on Precambrian, particularly Proterozoic, siliciclasticbedding planes suggests that microbial mats were widespread at that time [10]. The absence of metazoan grazing andbioturbation may also have made it more likely that mats from this period were preserved [71]. Microbially inducedstructures are also observed in Phanerozoic siliciclastic sediments [42], although not as widely as in the Proterozoicgeologic record [26, 48, 57, 61, 66].The presence and influence of microbial mats can be inferred from sediment properties that are uncharacteristic ofsand and mud deposited via a purely physical process [6, 11, 17, 25, 68]. In ancient siliciclastic biolaminates, formermicrobial mats are indicated by darker layers rich in iron-oxides, black carbonaceous materials [12, 48, 52] and pyrite[60]. These compounds are produced in modern mats by the metabolic activity of micro-organisms living in andbelow the mat [24, 67, 70]. In sediments below and above the mat layers, isolated sedimentary grains and mica flakessurrounded by sericitic layers have been attributed to the presence of microbial mats and bacteria, which trapped andbound the grains [8, 23, 53]. Isolated and orientated sediment grains can indicate biofilms, which initially formedaround the grains and have subsequently grown into full mats, rotating and trapping grains in the process [49, 51].Other microbial mat signatures include the presence of tunnel patterns by undermat miners [16, 33], structures relatedto mat growth and destruction processes, such as shrinkage cracks [68] and wavy laminae. In some cases fossilisedmicrobial filaments [14, 54], biomat fragments [76] and microbial death masks can also act as indicators [20]. B. Physical properties of modern microbial mats
The presence of microbial mats has been preserved in the ancient geologic record. However, deriving or deducingtheir exact material properties from fossil findings is not possible. This makes clear-cut statements about the behaviourand properties of ancient microbial mats problematic. Modern microbial mats are dense, cohesive organic layers thatact as a single unit. They are formed of microbes held together by extracellular polymeric substances (EPS) [81],which participate in the binding of cells and the formation of microbial aggregates. EPS act as a cohesive gel-likenetwork, provide a sca ff old for the cells and make up 50-90% of the total organic material in the film [27]. EPS havea number of functions [27]. They are responsible for adhesion of the mat to surfaces [22] and provide mechanicalstability for the microbial colony [43]. EPS can also stabilise clastic sediment surfaces [69].The composition, chemical and physical properties of EPS, and therefore microbial colonies, can vary widely. Re-cent investigations of modern microbial mats however, have revealed some well conserved features. Modern microbialmats behave like viscoelastic films [40, 65, 72, 80]; they undergo both reversible elastic responses and irreversible de-formation. For example, microbial colonies grown for long periods under turbulent flow conditions have been knownto display a rippled surface texture [77]. Lieleg et al. [40] have shown that biofilms display elastic-like responsesfor high frequency stimuli and viscous-fluid responses when low frequency stimuli are applied. In the context ofmicrobial mats, this means that for short term exposure of the mat to shear stress the mat responds elastically; whenthe applied stress is removed the mat returns to its original shape. For sustained exposure to shear stress, on the otherhand, internal physical stresses are dissipated through viscous flow. This reduces the possibility of structural failureand uncontrolled detachment of the mat under shear. The shear modulus G and viscosity η of microbial mats are seento vary over seven orders of magnitude [72]. However, the stress relaxation time τ = η/ G is strikingly well conservedacross a wide range of biofilms and found to be approximately 18 min. The timescale τ is essentially the time overwhich a biofilm, deformed by external forces, will ‘forget its original shape’. (b)(a) (c)(e) (f )(d) FIG. 1: Proterozoic Kinneyia structures from (a-c) Haruchas farm and (d-f) Neuras farm, Namibia. (a, d) Storm deposits onwhich Kinneyia are observed. (b) Ripple structures found below the storm deposit. These have a much larger wavelength than theKinneyia and a di ff erent morphology. (c) Example Kinneyia from Haruchas. Arrow shows inclination direction of the outcrop. (e)At Neuras Kinneyia were observed over areas larger than 1 m . (f) Close up of (e) showing a clearly defined structure. Scale barsall 10 cm. From these observations two general conclusions about the properties of microbial mats can be made [72]. Firstly,microbial mats behave as viscoelastic films, with little or no influence from biological processes on short time scales.This suggests that the physical properties are solely determined by the EPS. Secondly, the relaxation time of theviscoelastic medium is universally about 18 mins, the typical lifetime of the temporary crosslinks in the EPS.Comparison between modern and ancient microbial structures suggest that ancient microorganisms existed with thesame diversity as today [48]. It is not unreasonable therefore that ancient and modern microbial mats shared similargeneral material characteristics. For the purpose of this paper it is assumed that ancient microbial mats can be modelledas viscoelastic polymeric materials with viscosities and relaxation times similar to those of modern microbial mats.The properties of the polymer films are given in section III A.
C. Kinneyia
Wrinkle and ripple-like structures are perhaps the most documented microbially mediated sedimentary structures[42, 56, 57, 61, 63]. Observed wrinkle-structures have widely varying morphologies, characteristics, formation pro-cesses and preservation modes. Patterns can develop in positive or negative relief [7] and can occur on or within therock bed [56]. There are many di ff erent hypotheses as to how wrinkle structures formed, but all agree that a cohesivemat is present at the sediment surface. This is confirmed by the observation of a large variety of wrinkle structures inmodern microbial environments [31, 46–48].The focus of this paper is Kinneyia. To avoid confusion with other types of wrinkle structures, here only Kinneyia asdefined by Porada et al. [62] are considered. Kinneyia are a sub-class of sedimentary fossils with clearly defined ripplefeatures (e.g. figure 1). Kinneyia are dominantly linear structures found on the upper bedding planes of sandstone orsiltstone layers. These bedding layers are much thicker than the amplitude of the Kinneyia pattern and are interpretedas storm or flood event deposits (figure 1a). The upper layer of the event deposit is generally observed to be coveredwith a thin topset veneer thought to have sedimented after the event deposit at high flow velocity and shallow waterdepths. Thin sections made from Kinneyia from Sweden [62] show that the depth of the veneer is proportional to theamplitude of the ripple structure, with the bottom of the veneer layer coinciding with the troughs of the Kinneyia.The Kinneyia pattern is characterised by an undulating ripple-like structure (figure 1c) with a wavelength on themillimetre to centimetre scale (2-20 mm). The ripple shape deviates slightly from a sinusoidal wave, with flattenedcrests and rounded troughs most commonly described in the literature. However, rounded and even sawtooth-shapedcrests are also seen [62]. The relative widths of the troughs and the ridges varies widely from sample to sample[59]. The ripple-pattern is generally well ordered with the crests orientated parallel to one another. Honeycomb-likepatterns with round or elongated pits are also seen with the two morphologies often coexisting on di ff erent areas of thesame sample [35, 61]. The Kinneyia patterns often have one preferred crest orientation. In Kinneyia from ¨Oland, thisorientation was observed to run perpendicular to the paleocurrents from which the event layers were deposited [41].However, in some examples a second orientation direction is observed at 40 − ◦ to the first [62]. This led Porada etal. to suggest that some interference phenomena is involved in their formation [62]. D. Suggested mechanisms for Kinneyia formation
Initially, it was assumed that an abiotic mechanism was responsible for the ripple-like patterns observed in Kinneyia.Proposed mechanisms range from perturbation of cohesive sediments in shallow water by wind [74], to erosion ofsediments by waves [73] and foam induced patterns [1]. However, none of these mechanisms satisfactorily explainthe observed Kinneyia ripple patterns. Hagadorn and Bottjer [34, 35] were the first to suggest that the formationof wrinkle-structures and Kinneyia may in some way involve microbial mats. This assumption was made due to theobserved decrease in grain density near the surface of the wrinkled sedimentary fossils, which suggested that sedimentswere bound at the surface by an organic matrix. The ripples of Kinneyia are often seen to have very steep sides, whichit is suggested would be unstable without some ’glue’ or microbial mat to bind the grains together once a criticalslope is reached. Hagadorn and Bottjer’s conclusion was also made due to the discovery of similar wrinkle patterns inmodern microbial mats found growing in the Great Basin, USA [35]. The modern patterns had a wavelength of a fewmillimetres with both rounded and pointed ridges, which extend for tens of centimetres, being observed.Pfl¨uger went on to propose a microbially mediated gas bubble model for the formation of Kinneyia [59]. Hesuggested that gas rising from a sedimentary substrate can be trapped by a microbial mat, and collect to form bubbles.Using a mixture of water-saturated sand and sodium bicarbonate, Pfl¨uger was able to directly observe a processwhereby the bubbles destabilize the sediment leaving trace patterns. Gas domes, polydisperse round shapes up to tensof centimetres in diameter, are observed in modern and ancient microbial mats [30]. However, gas domes are no longerthought to be responsible for the formation of Kinneyia, as their patterns do not correspond well with the well-definedelongated ridge and troughs structures of Kinneyia.The most recent model for the development of Kinneyia is that of Porada et al. , who proposed that the pattern formsin liquefied sediment confined beneath microbial mats. Like Pfl¨uger [59], Porada et al. assumed that the microbialmats acts as a barrier to gas and groundwater trapped in the underlying sediment. Here however, the bacteria areassumed to form dispersed colonies in the sediments, where the mats have adhered and grown downwards. The gasthen accumulates in pore spaces between the colonies, leading to a local anisotropy in the water saturated sediment onthe microscale. Cyclic stressing, from oscillatory water flow, of the overlying microbial layer, causes the underlyingsediments to liquefy. This is due to an oscillating pore pressure induced by the change in water depth with tidecycling. The liquefied sand layer is assumed to be several centimetres thick with the overlying mat being 3 cm thick.The oscillatory pressure changes induce ripple structures in both the liquefied sand and microbial mat. The wavelengthof the ripples is on the order of 1 m for the sand and a few centimetres for the mat. The ripple pattern in the mat inducesfurther local variations in the pore pressure causing seepage and grain lifting. If the liquefied sediment layer becomesvery thin, then replication of the overlying small wavelength microbial ripples can occur in this underlying liquefiedlayer. This condition is satisfied periodically at the seaward boundary of the mat during each tidal cycle. This modelrequires a very specific number of criteria to be met, along with significant grain lifting, for the formation of Kinneyiaon the scale of a few millimetres to centimetres to occur. When these conditions are not met, ripple patterns on themetre scale are instead predicted.In the model of Porada et al. the small-scale ripple patterns are induced first in the overlying mat and then transferredinto the underlying sediment. Observation of modern microbial mats, however, suggest that it is not always possibleto make a clear distinction between these two layers. Mats are often seen to develop from biofilms which initiallyform around individual sediment grains and subsequently grow together into a thick layer. The sediments are thus anintegral part of the microbial mat. This is particularly the case for mat growth after storm events.Here we propose a simple model for the formation of Kinneyia ripple patterns, where the key mechanism is a Kelvin-Helmholtz instability (KHI) induced in the mat under flowing water. A KHI naturally gives rise to an undulatingstructure on the lengthscales typical of Kinneyia. Evidence from analogue experiments is presented and compared data1data2data3 a = ( z − tanh z − z tanh z )( z tanh ( z ) − z ) a = az − a = az − z = qh h v FluidMicobial MatSediment (a) (c)(b) (d) xy h v FluidMicobial MatSediment (a) (c)(b) (d) xy x y (e) data1data2data3 FIG. 2: Schematic representation of the generation of Kinneyia from a hydrodynamic instability. The microbial mat grows in aquiescent environment and is suddenly subject to significant flow in the overlying water (a). This results in a Kelvin-Helmholtzinstability, giving rise to a ripple pattern at the surface of the mat (b). Once the amplitude of the ripple reaches a certain threshold,eddies will from in the valleys giving rise to enhanced sedimentation (c). Rupture of the film can occur when the amplitude of thetroughs become comparable to the film thickness (d). The predicted relative growth rate (e) of the ripple pattern is sharply peakedaround a dominant wavelength where qh (cid:39) with detailed measurements of fossilised Kinneyia. It should be noted that our model does not contradict that ofPorada et al. , but is in fact indirectly implied, although not discussed. II. FORMATION OF KINNEYIA VIA A KELVIN-HELMHOLTZ TYPE INSTABILITY
Kinneyia are generally found on upper bedding planes in littoral environments that have experienced recent stormdeposits [62]. For the purposes of this model we shall therefore consider a planar microbial mat, or biofilm, on asolid substrate subject to some flow in the overlying fluid (figure 2a). The microbial mat is considered to behave asa viscoelastic fluid [72]. The system can be approximated by two immiscible fluids (water and microbial mat), withdi ff erent viscosities and flowing at di ff erent velocities over a rigid substrate. The question is then: what happens at theinterface between these two fluids?It is well known that spontaneous destabilisation of a fluid-fluid interface may occur in a two fluid system where thelayers respond di ff erently to shear. A well defined instability forms giving rise to a harmonic interfacial corrugation(figure 2b). This KHI occurs ubiquitously in nature [15, 37, 75] for example in cloud layers [19]. Typically, a KHI isstudied using fluids of di ff erent densities, however, di ff erences in viscosity also lead to instability.The system being studied is sketched in figure 2. Water flows with far-field velocity V in the x -direction over aviscoelastic film of thickness h . A flat film / water interface is a solution of the pertinent hydrodynamical equations,but an unstable one. Consider a small periodic perturbation at the interface between the two layers. Qualitatively thestream lines are compressed at the peaks and expanded at the valleys of the perturbation (figure 2b). Due to massconservation, this requires an increased flow velocity at the peaks and a decreased velocity at the valleys. Accordingto Bernoulli’s law this gives rise to a decrease in pressure at the crests and an increase in pressure in the valleys. Thepressure di ff erences drive flow in the film from troughs to peaks. Hence small thickness variations in the microbial matcan be amplified over time. For a small perturbation of the interface h ( x , t ) = h + ε ( x , t ), where ε ( x , t ) = ε ( t ) cos qx and q is the wave number of the perturbation. The amplitude ε is assumed to vary slowly, such that the fluid dynamics canbe treated quasi-statically. The velocity, v = ( v x , v y ), within the viscous layer of the mat is given by Stokes’ equation ∆ v = η ∇ p (1)where η is the viscosity of the microbial mat and p ( x , y , t ) the pressure within it. The elastic response is neglectedhere, since only long time dynamics are being considered. The film is taken to be incompressible, such that ∇ · v = ∆ p =
0. The base of the mat y = v ( x , , t ) =
0, while the upper boundary is taken to be free, such that ∂ y v x ( x , h , t ) = p = ε f ( y ) = ε ( t ) cos qx ( P cosh qy + P sinh qy ) , (2)where P and P are constants, which will be determined. The components of the velocity field that are consistentwith the boundary conditions are v = (cid:0) ε ( t ) U ( y ) sin qx , ε ( t ) V ( y ) cos qx (cid:1) (3)for some functions U ( y ), V ( y ). For an incompressible fluid the velocity field can be expressed by derivatives of a scalarstream function ψ as v = ∇ × ψ . From this, and Eqn. 1, it is derived that U = − ∂ y V / q and ∆ ψ =
0. The solution tothis latter biharmonic equation yields V = ( A + B y ) cosh qy + ( A + B y ) sinh qy , (4)where A , A , B and B are constants. The no-slip lower boundary condition V (0) = ∂ y V (0) = A = A = − B / q . Combining Eqns. 1 – 4 the constants B and B are found to be B = P / η and B = P / η . Thefree boundary condition at y = h gives ∂ yy V ( h ) = B ( q sinh qh + q h cosh qh ) + B (2 q cosh qh + q h sinh qh ) = . (5)Thus B = − B (cid:18) qh + tanh qh + qh tanh qh (cid:19) = − g ( qh ) B . (6)If the pressure at the interface is now taken to be p ( x , h , t ) = − P ( q ) ε ( x , t ) (7)then, using Eqns. 2 and 6, B = P ( q )2 η ( g ( qh ) sinh qh − cosh qh ) . (8)Introducing this back into Eqn. 4 gives V = P ( q )2 η q (cid:34) qy cosh qy − (1 + g ( qh ) qy ) sinh qyg ( qh ) sinh qh − cosh qh (cid:35) . (9)The equation of motion for the interface is ∂ t ε = v y ( x , h , t ). Thus, ∂ t ε = ε V ( h ) and ε ( t ) ∝ exp( α t ). The growthrate α ( q ) is α ( q ) = P ( q ) h η (cid:34) qh − tanh qh − qh tanh qhqh (tanh qh − (cid:35) . (10)The most rapidly growing mode is given by the maximum of α ( q ). Since the expression in brackets is sharply peakedaround qh ≈
2, the maximum will not be strongly dependent on the exact form of P ( q ).To be more specific, however, an estimate for P ( q ) needs to be found. Calculating the pressure distribution above theperturbation exactly, requires the full boundary layer theory to be considered. A treatment of this problem was carriedout by Bordner [9]. He found that the pressure scales as ε/δ , where δ is the thickness of the disturbance sublayer inthe fluid flowing above the mat. The expression for δ is given by δ = (cid:32) µ q π ρτ (cid:33) / (11)where τ is the mean surface shear stress at the boundary. µ and ρ are the viscosity and density of the flowing liquidrespectively. P ( q ) is then given by P ( q ) ∝ (cid:32) π ρτ h µ (cid:33) / ( qh ) − / . (12)The maximum of α is then qh ≈ .
4. Sediment deposited from the flowing water will accumulate in the troughs of thecorrugation. The corresponding pressure contribution is then proportional to the wavelength, such that P ( q ) ∝ / q .This would yield a maximum of α at qh ≈ .
2. The expected value of qh should lie between these values, and dependon the relative strength of the contributions. These growth rate predictions are shown in figure 2e. In terms of thewavelength of the most unstable mode, an instability of wavelength approximately 4–5 times the mat thickness h ispredicted.A Kelvin-Helmholtz type instability of a microbial mat therefore leads robustly to the formation of a ripple insta-bility under any shear flow. The most unstable wavelength of the instability is predicted to be proportional to, and afew times, the film thickness, but is insensitive to either the fluid flow speed or mat viscosity. This complies well withthe observation that Kinneyia patches often exhibit reduced wavelengths at their boundaries, where the mat can beassumed to have been thinner. Once the pressure variation is firmly established viscous flow within the film leads toan increasing corrugation amplitude. When the amplitude of the ripple reaches a certain threshold, eddies will form inthe valleys of the undulation (figure 2c). Some of the clastic sediments carried by the overlying fluid as it flows, willsettle in the troughs due to the back flow and stagnation points arising from the eddies. Accumulation of sediments inthe troughs may also act as an additional driving force for the instability; the sediment is denser than the microbial matand the surrounding water. The growth of the ripple pattern is thereby accelerated and steep slopes develop betweenthe troughs and the crests. As the microbial mat dies, remains of the EPS glue the sediments together, preserving theKinneyia structures that are found in the geologic record. III. EXPERIMENTAL PROCEDURE
To test the hypothesis that Kinneyia arise from KHIs in microbial mats, analogue experiments were carried outwhere the microbial mat was replaced with an abiotic polymeric viscoelastic film and subjected to flowing water.
A. Materials and methods
Experiments were carried out using a viscoelastic cross-linked poly(vinyl alcohol) (PVA) film. The PVA films weremade by fully dissolving PVA (Sigma-Aldrich, molecular weight 145 kg / mol) in de-ionized (Millipore) water at 90 ◦ C.The mixture was cooled and crosslinked using sodium borate solution (Borax, Sigma-Aldrich). The ratio of PVA tosodium borate was kept constant at 10:1 w / w. White paint (titania-based) was added to the PVA (2 g /
100 ml) prior tocrosslinking to create opaque films. PVA solutions of 3, 4 and 5% by weight were mixed. The rheological properties ofthe films were characterised, at T = ± ◦ C, using a parallel-plate rheometer (Stresstech, Rheosystems). Viscositieswere measured in the linear regime for applied stresses of 0.01-100 Pa. The viscosities of the cross-linked solutionswere measured to be 25 ± ±
10 Pa s and 398 ±
20 Pa s for 3, 4 and 5% solutions respectively. The relaxationtime of the PVA solutions were found to be 15-18 s, 115-338 s and 1200-1450 s respectively. The variation arises fromsmall di ff erences in PVA and sodium borate concentrations between subsequent batches of the solutions. A relaxationtime of 1200-1450 s corresponds well with the properties of modern microbial mats, which are universally observedto have a relaxation time of around 1020 s [72]. The viscosities of modern mats are seen to vary from 10-10 Pa s.Two di ff erent flow setups were built to probe the behaviour of the films. A schematic diagram of a deformed PVAfilm subjected to water flow can be seen in figure 3a. The first setup consisted of a small flow cell (9 × × x × z × y ) connected to a fluid reservoir via a centrifugal pump. The PVA film was placed onto a porous glass substrate(Robu Glas) in the flow cell and left for 1 hr to relax. The height of the substrate could be altered, allowing thethickness h of the film to be varied. A valve directly in front of the pump allowed the flow speed to be adjusted. Flowspeeds of 0.05, 0.12, 0.18 and 0.24 m / s were used. Glass windows in the flow cell allowed optical observation of theripple formation. The profile of the film was monitored in situ using an oblique 532 nm laser sheet and imaged every Film h λ v (a) (b)(c) xy z y xxz FIG. 3: (a) Deformed PVA film subject to water flow. h is the initial thickness of the film and λ is the wavelength of the instability.The dashed line represents the deviation of the laser sheet due to the deformed film. (b) Top view of instabilities indicating anelongated sinusoidal pattern. (c) Side view of instabilities observed in the PVA after 30 mins of water flow. A sinusoidal pattern isobserved. Scale bars in (b,c) are 10 mm.
60 s using a Nikon D5100 digital SLR camera mounted directly above the flow cell. The position of the laser line wasextracted from the resulting images with a resolution of ± .
04 mm in the x -direction and ± .
02 mm in the z -direction.The wavelength was calculated from the first non-zero peak of the 1D autocorrelation function, the inverse fouriertransform of the power spectrum, of the extracted laser line. The temperature of the reservoir varied from 18 − ◦ Cbetween experiments, but was kept at ± . ◦ C during each experiment using a Julabo FT402 temperature controller.All flow experiments were carried out at room temperature ( T = ± ◦ C).The second setup was a larger flow trough, allowing for a bigger sample (20 ×
20 cm) to be observed. This setup wasused to qualitatively observe how the patterns developed when sediments were added into the system. The sedimentsused were glass beads with a diameter of 0 . − .
15 mm. Again the height of the substrate could be altered to changethe thickness of the film. The upper water surface was in this case free. The thickness of the flowing water layer was2 − / s. Sedimentwas deposited uniformly (0.05 g / cm) over the PVA surface 60 s after the water flow had started. A camera directlyabove the sample was used to monitor the development of the instabilities. B. Fossil measurements
For comparison with the experimental data, Kinneyia were studied from two sites in Namibia. Both sites date tothe terminal Proterozoic Vingerbreek Member, Schwarzrand Subgroup, Nama Group. The first site was located atHaruchas farm, Namibia [24 ◦
21’ 46.3” S; 16 ◦
24’ 21.6” E] [52]. The outcrop is approximately 3 × to around 400 cm (e.g. figure 1c). The Kinneyiacover 50-60% of the outcrop. The outcrop sits in a modern dry river bed on top of an event deposit 15 −
20 cm thick(figure 1a). This deposit is thought to have arisen from a storm event [62] and overlies an older rippled substrate,which indicates the flow direction. The older ripples run parallel to the Kinneyia ripples, but have a larger wavelength(10-20 cm) and a smoother sinusoidal undulation (see figure 1b). Examples of Kinneyia from this specific locality atHaruchas have previously been presented in the literature [62].The second fossil site was located at Neuras farm, Namibia [24 ◦
24’ 11.3” S; 16 ◦
15’ 8.7” E]. The Kinneyia hereare found on two isolated rock outcrops located on a small cli ff overlooking a river bed. In this case the event depositon which the Kinneyia were observed was ∼
35 cm thick. The structures observed at Neuras were more extensivethan those at Haruchas and were seen to cover areas in excess of 1 m . The source of the Kinneyia at Neuras couldnot be directly observed. However, both Neuras and Haruchas are located along the same storm-a ff ected Proterozoicshoreline.Replicas of the fossils were made to allow detailed measurements to be made without destroying or removing theoutcrops. The areas were cleaned and then cast with Mold Star ® pourable silicone rubber (Smooth-on). The rubber λ ( mm ) (mm) h (a) λ / h Flow speed (m/s) λ / h Viscosity (Pa s) (b) (c)
FIG. 4: Saturation wavelength of the ripple instability induced in PVA films subject to flow. The wavelength does not vary with theflow speed or film viscosity. (a) All data. PVA ( (cid:4) ) η =
124 Pa s, ( (cid:8) ) η =
25 Pa s and ( (cid:78) ) η =
398 Pa s at ( (cid:4) ) 0.12 m / s, ( (cid:4) ) 0.18 m / sand ( (cid:4) ) 0.24 m / s. Data were measured using the flow cell. ( (cid:70) ) The wavelength of the patterns which develop when sediments areadded to the system. Data were measured using the flow trough. The line shows a fit to the data for λ = . ± . h , where h is thethickness of the initial PVA film. The flow rate and viscosity have no e ff ect on the resulting dominant wavelength that develops.(b) λ/ h as a function of flow speed for PVA η =
124 Pa s. (c) λ/ h as a function of viscosity for a flow speed of 0.12 m / s. A m p li t ude ( mm ) t (s) A m p li t ude ( mm ) t (s) A m p li t ude ( mm ) (mm) h (a) (b) (c) S a t u r a t i o n a m p l i t u d e ( mm ) A m p l i t u d e ( mm ) A m p l i t u d e ( mm ) FIG. 5: Time evolution of the peak to trough amplitude of ripples induced in PVA films subject to flow. (a) Amplitude as a functionof film thickness for ( (cid:8) ) 1mm, ( (cid:4) ) 2mm, ( (cid:72) ) 3mm and ( (cid:78) ) 4mm thick films. Grey and black symbols show di ff erent runs usingthe same experimental conditions. Lines show the exponential fits to the data with relaxation times of between 145 and 400 s. Thiscorresponds to the measured relaxation time of the PVA of 115-338 s. The variation here is seen for di ff erent batches of the PVAmixture. (b) The saturation amplitude is not dependent on the flow rate of the water. The graph shows data for 1 mm thick films atflow rates of ( (cid:8) ) 0.12 m / s, ( (cid:13) ) 0.18 m / s and ( (cid:9) ) 0.24 m s − . The saturation amplitude is only dependent on the film thickness (c). was left to cure for 45 min before being peeled back to create a negative replica mold of the underlying structure. Solidpositive replicas were produced from these molds using Smooth-Cast ®
300 (Smooth-on) in the laboratory.Surface profile measurements of the Kinneyia were carried out using a stylus profilometer (Dektak XT Bruker).Areas 400-2500 mm were scanned to create 3D height maps. The 3D map was created by stitching together a seriesof 2D line traces taken at 50 µ m intervals. The resolution in the line traces was 3 µ m in the horizontal direction and10 nm in the vertical direction. The 2D auto-correlation functions of the topographs were computed using Matlab. Anangle-dependent auto-correlation function was found by averaging over any azimuthal angle θ ± . ◦ . The orientationof the pattern was defined as the value of θ at which the radial auto-correlation function was at a maximum and the firstnon-trivial peak of the radial auto-correlation function as the wavelength of the Kinneyia ripples. The peak-to-troughamplitude of the ripples was measured directly from the line scans. IV. RESULTS AND DISCUSSIONA. Lab-made Kinneyia
Ripple patterns are observed to form at the film / water interface in flat PVA films subjected to water flow conditions(figure 3b,c). These instabilities are visible within tens of seconds of flow initiation. The growth rate of the instabilities0depends on the viscosity of the film. Figure 4a shows the wavelength of the instability that forms as a function of thefilm thickness h . The wavelength λ was monitored using a laser sheet (figure 3a) and was found from the first non-zeropeak of the 1D autocorrelation function of the deflected laser line. The data points in figure 4a are an average of thewavelengths calculated from successive images taken every 60 s over a 20-45 minute period. The error bars indicatethe standard deviation in the wavelength measured from these images over the course of the experiment. For films withinitial thicknesses h of 1 to 4 mm, the wavelength of the ripple pattern that forms is ∼ λ = . ± . h . No dependencyof the wavelength on either the viscosity of the film or the flow rate is observed (figure 4b,c). This is in agreementwith the theory presented here, which predicts that the wavelength is only dependent on the thickness of the film.The growth of the interfacial ripples as a function of time is indicated in figure 5a. The peak-to-trough amplitudeof the ripples was measured directly from the laser line. The amplitude grows with time, eventually saturating. Theamplitude saturates on the same timescale as the relaxation time of the polymer. The saturation timescale was notobserved to be dependent on the film thickness or the flow rate (figure 5a,b). The saturation amplitude varies linearlywith the film thickness (figure 5c). A small increase in the saturation amplitude is observed with increasing flowrate (figure 5b). However, this increase is within the experimental error. The data shown in figure 5 were obtained byfollowing the growth of individual ripples along the instability. While the amplitude should not be taken as an absolutevalue for all ripples formed under the given conditions, they are representative and show the trends observed in theexperimental data.In our model of Kinneyia formation eddies in the ripple valleys lead to enhanced sedimentation (figure 2c). To testthe e ff ects of sedimentation in our PVA films, glass beads were added to the system while the water was flowing. Theglass beads used were large enough for sedimentation to occur. As the instabilities developed, the beads collected inthe troughs of the pattern (figure 6). The resulting, experimentally produced, PVA Kinneyia (hereafter lab-Kinneyia)that form are well-ordered and exhibit an interconnected pattern of elongated ridges. The same thickness-wavelengthrelationship observed in the other experiments (figure 4a) is found. Overspill of the PVA, from the sample chamber,leads to the development of patterns in regions with two di ff erent thicknesses on the same sample. The patterns thatdevelop have the same morphology, but di ff erent wavelengths (inset figure 6b,c). This shows that Kinneyia patternswith di ff erent wavelengths can arise under the same flow conditions due to thickness variations across the sample.Figure 6e, f shows the lab-Kinneyia that formed in 6 mm thick films at low and high flow speeds. The pattern infigure 6e is qualitatively similar, but with a longer wavelength, to that shown in figure 6c for the 2 mm thick film. Thepatterns in figure 6c and 6e were formed at the same flow speed. Increasing the flow speed appears to result in anincrease in the order of the pattern, with a transition from honeycomb-like patterns to parallel ridges perpendicularto the flow direction. The increase in order is observed in the 2D autocorrelation functions of the resulting patterns(figure 6g,h). The patterns generated in the lab-Kinneyia shown in figure 6 qualitatively resemble the Kinneyia foundin Namibia e.g. figures 1 and 7. The results suggest that the di ff erent patterns may arise from variations in the flowconditions under which the microbial mat or film is deformed. B. Kinneyia
Figure 7 shows representative height profiles of the Kinneyia replicas made at Neuras and Haruchas. In each caseimages of the fossil (figure 7a-c) and map scan (figure 7d-f) are shown alongside the 2D autocorrelation function ofthe map scan (figure 7g-i) . Morphologies range from small circular-shaped patterns (figure 7a) to elongated ridgesextending across the whole fossil (figure 7c). Intermediate patterns containing circular and elongated ridges are alsoseen (figure 7b).Analysis of the profilometry measurements from the Kinneyia can be seen in figure 8. The points in figure 8a givethe wavelength and corresponding amplitude for each area scanned. The data show that as the wavelength of theripple increases so to does the amplitude. This is in agreement with the trends observed in the analogue PVA flowexperiments (figures 4 and 5). The amplitude measured from the fossils however, can only be taken as a lower limitfor the amplitude of the instability when it first formed. Compaction after burial may have considerably decreasedthe amplitude of the ripples. In addition, the outcrops at Haruchas and Neuras are unprotected from the elementsand subject to erosion. It is therefore likely that the original amplitude was higher. It is not possible however, toquantify the e ff ects of either compaction or erosion. The amplitude of the Kinneyia ripples (figure 8a) are higher thanthose for the PVA experiments (figure 5). The PVA experiments from figure 5 do not take into account the e ff ect1 z (mm) x ( mm )
00 10105 5 z (mm) x ( mm ) z (mm) x ( mm ) (a) (b) (c) (d)(e) (f) (g) (h) FIG. 6: Development of lab-Kinneyia in a 2 mm thick PVA film at a flow speed of 0.024 m / s. Prior to flow the film is initiallyflat (a). The black colour comes from the underlying substrate. After 60 s of flow sediment is deposited. Images (b) and (c) showthe pattern formation 120 s and 180 s after the flow started. All images are 6 × × ◦ at a wavelength of 5.2 mm, where 0 ◦ is defined to beparallel to the flow direction. The inset in (d) shows the 2D autocorrelation function of the inset in (c). The first non-zero peak isobserved at 0 ◦ for a wavelength of 2.9 mm. (e-h) Development of lab-Kinneyia in 6 mm thick films for flow speeds of (e) 0.024 m / sand (f) 0.18 m / s. (g, h) 2D autocorrelation function of (e, f). The wavelengths are 19.4 mm and 21 mm in (e) and (f) respectively.An increase in the order is observed with increasing flow speed. (e, f) are the same size as (a-c). The colour maps for the 2Dautocorrelation functions show the intensity in arbitrary units. of sedimentation, which was qualitatively observed to enhance the growth and break up of the ripples (figure 2d).Comparison of figure 8a and figure 4a suggests that the microbial mats involved in Kinneyia formation were ∼ . − ff ects of erosion or compaction. Erosion could presumably also lead to flattened crestsbecoming more rounded.The orientation of the ripples was measured from the autocorrelation function of the three-dimensional map scans(figure 7g-i). For well-ordered ripples peaks are observed at 0 ◦ and 90 ◦ (e.g. figure 7h,i). Here 0 ◦ is defined tobe perpendicular to the ripples ( x -axis), while 90 ◦ runs parallel to the ripples ( z -axis), as indicated by the arrows infigure 7g. In well-ordered patterns the peaks at 0 ◦ and 90 ◦ correspond to the wavelength and mean extent of the ripplesrespectively. For more disordered ripples (e.g. figure 7a) additional peaks are observed at other angles θ in the two-dimensional autocorrelation function. The angle θ at which the additional peaks are observed defines an additionalorientation direction of the ripples and is indicated by the arrow in figure 7g. θ varies slightly from sample to sample,but lies between 30 ◦ and 50 ◦ with a mean value of 33 ± ◦ . A secondary orientation was also noted by Porada et al. [62]. A similar secondary orientation is observed in the lab-made Kinneyia shown in figure 6.2 x ( mm ) z (mm) x ( mm ) z (mm) x ( mm ) z (mm) (a)(b)(c) (f)(e)(d) (i)(h)(g) FIG. 7: Photos of Kinneyia stuctures found at (a, c) Neuras farm and (b) Haruchas farm. The photographs in (a-c) have the samemagnification and are 10 x 10 cm. (d-f) Three-dimensional map scans taken from casts of (a-c) respectively. The scans are (d)22.5 × ×
55 mm. The inset in (d) shows the map scan in (d) at the same magnification as (e) and (f). (g-i)Two-dimensional autocorrelation functions of (d-f). The colour maps show the relative intensity of the peaks in arbitrary units. Thearrows in (g) indicate the orientation of the ripples.
There are a number of possible origins for this secondary orientation, including multi-directional flows arising fromtidal rips or di ff erences in the forward motion of the tide and the subsequent backwash, changes in the flow directionor the presence of obstacles in the flow path. Another possibility is that the secondary orientation is induced bythe ripples themselves. Pinning of the upper air / water surface to the PVA ripple crests is observed in the analoguePVA / sediment experiments when the amplitude of the ripple becomes comparable to the height of the flowing waterlayer. This induces small wakes, at the air / water interface, locally changing the flow direction of the water. Over timethis secondary flow could influence the orientation of the ripples that form in the film. However, a full experimentaland theoretical study of the 2D stability of Kinneyia ripples is beyond the scope of the current paper.The robustness of the mechanism put forward here suggests that Kinneyia type structures should be abundantlyfound in modern biomats. However, wrinkle patternss are only very occasionally observed in modern biomats andgenerally have morphologies that di ff er from Kinneyia patterns. To resolve this apparent contradiction, one should bearin mind that, for some so far unknown reason, Kinneyia fossils are intimately linked to storm deposits, which representrather scarse events. In fact, Kinneyia is a comparably rare fossil. In addition, the omnipresence of grazing animalsand rapid bioturbation since the Cambrian may have rendered the conditions for Kinneyia formation increasinglyunfavorable, to the point of their virtual absence today.Another, very speculative but nevertheless interesting, line of thought concerns the rheological properties of biomats.Although modern biofilms may appear to be viscoelastic media to a rheometer, we know that active mechano-responseis well conserved at least in modern eucaryotes. Strictly speaking, biomats must therefore be viewed as active mat-ter, possibly stalling the instability observed in abiotic films. As we do not know the history of mechano-responseevolution, the mechanical properties of biomats may have been, subtly but distinctly, di ff erent when Kinneyia fossilsformed.3 A m p li t ude ( mm ) λ (mm) H e i gh t ( mm ) X (mm) H e i gh t ( mm ) X (mm) H e i gh t ( mm ) X (mm) H e i gh t ( mm ) X (mm) (a) (b) (c)(d) (e) x xxx
FIG. 8: (a) Amplitude of the Kinneyia ripples as a function of their wavelength for fossils from ( (cid:4) ) Haruchas ( (cid:3) ) Neuras. (b-e)Example line traces for fossils from (b), (d) Haruchas and (c), (e) Neuras. Line traces are for wavelengths from ∼ V. CONCLUSIONS
Destabilisation of viscoelastic films under shear by flowing water results in the formation of sinusoidal ripple-likestructures. The key process involved in the ripple formation is a Kelvin-Helmholtz instability, which arises from thespontaneous destabilisation of a fluid-fluid interface due to shear, when the two fluids have di ff erent viscosities. Thewavelength of the ripples is dependent on the thickness of the film, but not on the film’s viscosity or the flow speed.Changes in the flow speed result in changes in the ordering of the ripples, with both honeycomb-like patterns andwell-ordered parallel ripples being observed. The wavelength and morphology of the ripples corresponds well withthe patterns seen in Kinneyia structues. The experimental results from the lab-Kinneyia suggest that the microbialmats involved in Kinneyia formation were ∼ . − Acknowledgments
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