^{1}

Banded patterns in limestone-marl sequences (“rhythmites”) form widespread sediments typical of shallow marine environments. They are characterized by alternations of limestone-rich layers and softer calcareous-clayey material (marl) extending over hundreds of meters with a thickness of a few tens of meters. The banded sequences are usually thought to result from systematic variations in the external environment, but the pattern may be distorted by diagenetic nonlinear processes. Here, we present a reactive-transport model for the formation of banded patterns in such a system. The model exhibits interesting features typical of nonlinear dynamical systems: (i) the existence of self-organized oscillating patterns between a calcite-rich mode (“limestone”) and a calcite-poor one (“marl”) for fixed environmental conditions and (ii) bistability between these two modes. We then illustrate the phenomena of stochastic resonance, whereby the multistable system is driven by a small external periodic signal (the 100,000 years’ Milankovitch cycle comes to mind) that is too weak to generate oscillations between the states on its own. In the presence of random fluctuations, however, the system generates transitions between the calcite-rich and calcite-poor states in statistical synchrony with the external forcing. The signal-to-noise ratio exhibits many maxima as the noise strength is varied. Hence, this amplification effect is maximized for specific values of the noise strength.

Rocks and minerals often exhibit rhythmic spatial variations in their chemical and/or physical properties over length scales varying from the micrometer to the kilometer [

On the other hand, in a nonlinear nonequilibrium geosystem, the presence of positive feedback may result in a pattern formation that is self-organized even in an external environment that does not change (intrinsic mechanisms) [

However, there exists another rhythmic pattern formation mechanism: stochastic resonance [

In this paper, we study a particular example of a geosystem exhibiting rhythmic patterns: calcareous rhythmites such as limestone-marl or limestone-shale sequences (Figure

Limestone-marl rhythmites from the Vauréal formation (Baie de la Tour, Anticosti Island, Québec), dating from the Katian Stage (Upper Ordovician). The two walkers give an idea of the scale of the formation. Picture provided by André Desrochers.

One of the classic models is due to Ricken [

Schematic representation of Munnecke et al. [

In Böhm et al. [

In this paper, we propose a simple one-dimensional reactive-transport model that implements Munneke’s conceptual model. Our set of partial differential equations is presented in Section

In this section, we establish the basic conservation equations for fluid momentum and components mass, both solid (calcite, aragonite, and terrigenous materials) and dissolved (calcium and carbonate ions).

We use a reference frame with the ^{+2} and carbonate ions

We define the dimensionless compositions of solid components

Dividing (

We now define the concentration of dissolved species

Equation (

Finally, using (

The conservation of momentum takes the form of Darcy law. In our situation, this is

In general [

It is not the object of this work to solve for the full elastic sediment problem. Instead, we will assume that, in the absence of reactions, the porosity field (denoted

Substituting (

As will be clear below, we will use the molecular diffusion coefficient of calcium to choose appropriate length and time scales. Estimates of ^{−1} [

Summarizing, we have eight field variables: three compositions

In this simple model, we use four reactions.

(i) Dissolution of aragonite (reaction DA), A → Ca^{+2} +

(ii) Precipitation of aragonite (reaction PA): Ca^{+2} +

(iii) Dissolution of calcite (reaction DC), C → Ca^{+2} +

(iv) Precipitation of calcite (reaction PC): Ca^{+2} +

The net reaction rates appearing in (

The diagenetic equations for the solid components are first-order partial differential equations. We will fix the composition of the young sediment and the porosity at the water-sediment interface. We will also fix the concentrations of dissolved calcium and carbonate at that interface. On the other hand, one supplementary boundary condition is needed for the dissolved components and the porosity. At the bottom boundary

The densities of the three solid components are fairly close to each other (2.950, 2.710, and about 2.8 g/cm^{3} for A, C, and T, resp. [

Finally, (

It is convenient to express the dynamics of the system in reduced (dimensionless) form. We scale the dissolved species concentrations with

As far as possible, we will adopt conditions corresponding to warm shallow waters (temperature of 25°C, salinity of 35

The geometrical parameters of the system are its length

The inverse of the rate coefficients ^{−1} and 5 a^{−1} are comparable to those used in a recent paleoclimate model [

The parameter ^{−13} cm^{2} for ^{−1} have also been used. Its specific value is not crucial here (as long as the porosity diffusion coefficient is small compared to molecular diffusion). With the choice^{−1}, the porosity field is free from the artificial oscillations often seen in the numerical solution of convective systems in the presence of shock effects (due here to the discrete dissolution zone). For the same reason, the specific value of

In the simulations, the parameters

Parameter values and initial conditions adopted in two base scenarios.

Parameters | Scenario A | Scenario B | Unit |
---|---|---|---|

| 100.09 | 100.09 | g/mol |

| 18.45 | 18.45 | g/mol |

| 2.950 | 2.950 | g/cm^{3} |

| 2.710 | 2.710 | g/cm^{3} |

| 2.8 | 2.8 | g/cm^{3} |

| 1.023 | 1.023 | g/cm^{3} |

| 131.9 | 131.9 | cm^{2}/a |

| 272.6 | 272.6 | cm^{2}/a |

| 0.1 | 0.01 | cm/a |

| 5 | 10 | (kPa)^{−1} |

| 10^{−6.19} | 10^{−6.19} | M^{2} |

| 10^{−6.37} | 10^{−6.37} | M^{2} |

| 1.0 | 0.01 | a^{−1} |

| 0.1 | 0.001 | a^{−1} |

| 2.48 | 2.48 | - |

| 2.80 | 2.80 | - |

| 50 | 50 | cm |

| 100 | 100 | cm |

| 500 | 500 | cm |

| 0.1 | 0.01 | cm/a |

| 0.6 | 0.6 | - |

| 0.3 | 0.3 | - |

| 0.326 | 0.326 | mM |

| 0.326 | 0.326 | mM |

Starting with an initial uniform distribution for the five variables ^{−6}), the algorithm is stable, convergent, and relatively fast. With a 2.53 GHz Intel® Core™ i5 CPU, it takes about 50 seconds to generate a solution for a dimensionless time

Figure

Numerical solution for scenario A with

Steady state

One can understand the general features of the profiles as follows. The dissolved species concentrations reach a steady state profile relatively early in the simulation. Porosity decreases slightly with position as a result of compaction. Recall that, whereas calcite dissolution can occur wherever the system is locally undersaturated, dissolution of aragonite occurs only in the ADZ,

For scenario A, there exists a combination of initial and boundary porosities for which the large-time solution is oscillatory. Thus, it appears that this simple reactive-transport implementation of Munneke’s conceptual model is propitious to diagenetic self-organized periodic solutions. Figure

Numerical solutions for scenario A illustrating self-organized oscillations. The calcite composition at the bottom of the system (

Moreover, the model exhibits another interesting nontrivial signature of nonlinear behaviour: multistability. This occurs when distinct steady state solutions are obtained for different initial values of the variables, leaving all parameters unchanged otherwise. To illustrate this feature for scenario A, Figure

Numerical solutions for scenario A illustrating bistability. All variables are taken at the bottom of the system (

Phase diagram in (

One can understand the nature of the steady states as follows. A state 0-C (as illustrated in detail in Figure

Figure

Phase diagram in (

In the presence of an external stochastic signal, nonlinear dynamical systems may exhibit interesting nontrivial phenomena, such as noise-induced transitions [

Three conditions are necessary to obtain stochastic resonance: (i) the deterministic (noiseless) dynamical system must exhibit multistability, whereby more than one attractor coexists, (ii) the system is driven by a weak deterministic periodic signal, and (iii) the external signal also includes a stochastic contribution. In the absence of an external signal, one can think of the dynamical system as moving in a multiwell potential. It evolves to one attractor or another, depending on its initial condition. In the presence of the weak external periodic signal, the perturbations to the system are too small to cause the system to transit from one potential well to another. The signal just oscillates slightly around one attractor as the potential barriers separating the basins of attraction fluctuate slightly. However, in the presence of noise, the fluctuations may induce the system to cross the potential barriers. If the noise parameters are chosen appropriately, the barrier-crossing dynamics can occur in stochastic synchrony with the weak periodic signal: these noise-induced transitions occur more easily as the potential barrier is smaller, and this happens once every cycle of the periodic signal. The result consists in a series of large transitions from one attractor to another, with a higher probability of this occurring once per cycle, even though the external periodic signal alone (without noise) is too weak to induce these transitions.

To illustrate this qualitative description, one can use the standard case of a single-variable dynamical system

Figure

(a–c) Time series illustrating the evolution of a variable

Another feature of stochastic resonance is shown by plotting the residence time distribution. This is a histogram of the time between transitions. One example is illustrated in Figure

In this section, we apply the concept of stochastic resonance to our multistable model: we investigate the effect of a weak periodic external signal driving the model, superposed to a source of random fluctuations. The weak external periodic signal could represent a fluctuating climate proxy, such as the Milankovich cycle, which is believed to be instrumental in inducing ice ages [

Not surprisingly, if the precession and obliquity forcing is strong enough, a periodic small-scale sedimentary sequence is generated. But, in order to investigate the potential impact of the stochastic resonance mechanism on rhythmite formation, we rather consider the weakest orbital forcing term in our model: the eccentricity signal at 100 ka. Notwithstanding the fact that postdepositional compaction is not modeled in our system, a pattern generated at that period would correspond to large-scale variations in the thickness of layer bundles [

Concretely, we assume that the small eccentricity driving signal translates in weak periodical variations in the sediment deposition conditions. For simplicity, we keep all model parameters constant as well as the chemical composition of the incoming sediment but vary its structural properties, that is, its porosity. We thus modify our diagenetic model by using the following time-dependent porosity boundary condition:

Figure ^{5} rd/a (corresponding to a driving frequency

(a) Dimensionless time series illustrating the deterministic evolution of the calcite composition ^{5} a. The parameter values are those of case 2. The transients are omitted. (b–d) Calcite composition

^{−3}

^{−3}

^{−3}

Again, we can define a signal-to-noise ratio SNR as the ratio of the power spectrum at the driving frequency

In this contribution, we propose a simple reactive-transport model for the formation of limestone-marl sequences, implementing the conceptual ideas of Munnecke et al. [

Coupling the system to a weak external periodic signal superposed to a random contribution presents the necessary ingredients for stochastic resonance to occur in the system. For illustration purpose, we chose a weak 100,000 years Milankovitch signal that affects the physical properties of the incoming sediment (porosity). We found that, for a noise strength sufficiently high, switches between a calcite-rich state and a terrigenous-rich one are obtained, in statistical synchrony with the weak external signal.

The analysis presented here is only preliminary. Other parameter values need to be explored; the coupling between the system and its environment does not need to involve only the surface sediment porosity and other types of stochastic resonance patterns could be obtained, particularly in the regime where the system is bistable between a steady state and a limit cycle. But these findings open the way to a more thorough understanding of the nontrivial effects of noise on the formation of limestone-marl sequences influenced by a weak external signal, such as the Milankovitch 100,000 years’ cycle.

Space-independent parameter in (

Sediment compressibility ((kPa)^{−1})

Mass of component

Initial homogeneous values of

Values of

Molar concentration of dissolved species (M)

Initial homogeneous values of

Values of

Average grain diameter (cm)

Damköhler number (-)

Molecular diffusion coefficient of ion ^{2}/a)

Molecular diffusion coefficient of ion ^{2}/a)

Porosity diffusion coefficient (cm^{2}/a)

Correction factor in the hydraulic conductivity (see (

Acceleration of gravity (cm^{2}/s)

Response function

Thickness of the aragonite dissolution zone (cm)

Hydraulic conductivity (cm/a)

Solubility of component ^{2})

Rate coefficients (a^{−1})

Length of the system (cm)

Reaction order for aragonite precipitation and dissolution (-)

Mass of component

Total mass of dry sediment (g)

Number of discrete spatial steps (-)

Reaction order for calcite precipitation and dissolution (-)

Fluid pressure (kPa)

Reaction rate for solid component

Reaction rate for ion

Reaction rates for precipitation and dissolution (g/a-

Reaction rate for water (mol/a-

Sedimentation rate (cm/a)

Time (a)

Time scale (a)

Solid matrix velocity (cm/a)

Scaled solid matrix velocity (-)

Volume of solid component ^{3})

Total volume of solid components (cm^{3})

Pore water velocity (cm/a)

Scaled pore water velocity (-)

Position from the water-sediment interface (cm)

Position of the top edge of the aragonite dissolution zone (cm)

Position scale (cm)

Dynamical variable in standard stochastic resonance (-)

Amplitude of the periodic driving signal (scaled) (-)

Constant in the hydraulic conductivity (see (

Random number sampled from a normal Gaussian (-)

Time step (-)

Spatial step (-)

Random variable in the Ornstein-Uhlenbeck process (-)

Viscosity of water (Pa-s)

Characteristic function for the dissolution zone (-)

Ratio of rate coefficients in (

Molar mass of water (g/mol)

Molar mass of calcite or aragonite (g/mol)

Ratio of rate coefficients in (

Random white noise process (-)

Density of component ^{3})

Total solid density (g/cm^{3})

Initial value of the total solid density (g/cm^{3})

Density of water (g/cm^{3})

Noise intensity (scaled unit) (-)

Effective stress (kPa)

Scaled correlation time of the Ornstein-Uhlenbeck process (-)

Porosity (-)

Initial homogeneous porosity (-)

Stationary unreactive system porosity in (

Porosity at the water-sediment interface (-)

Parameter in (

Bistable potential in standard stochastic resonance (-)

Frequency of the periodic driving signal (scaled) (-)

Oversaturation or undersaturation factors (-).

The author declares that there are no conflicts of interest regarding the publication of this paper.

This work was funded by a grant from the Natural Sciences and Engineering Research Council of Canada. The author acknowledges André Desrochers for providing the picture in Figure