Graphically interpreting how incision thresholds  influence topographic and scaling  properties of modeled landscapes

Abstract. We examine the influence of incision thresholds on topographic and scaling properties of landscapes that follow a landscape evolution model (LEM) with terms for stream-power incision, linear diffusion, and uniform uplift. Our analysis uses three main tools. First, we examine the graphical behavior of theoretical relationships between curvature and the steepness index (which depends on drainage area and slope). These relationships plot as straight lines for the case of steady-state landscapes that follow the LEM. These lines have slopes and intercepts that provide estimates of landscape characteristic scales. Such lines can be viewed as counterparts of slope–area relationships, which follow power laws in detachment-limited landscapes but not in landscapes with diffusion. We illustrate the response of these curvature–steepness index lines to changes in the values of parameters. Second, we define a Péclet number that quantifies the competition between incision and diffusion, while taking the incision threshold into account. We examine how this Péclet number captures the influence of the incision threshold on the degree of landscape dissection. Third, we characterize the influence of the incision threshold using a ratio between it and the steepness index. This ratio is a dimensionless number in the case of the LEM that we use and reflects the fraction by which the incision rate is reduced due to the incision threshold; in this way, it quantifies the relative influence of the incision threshold across a landscape. These three tools can be used together to graphically illustrate how topography and process competition respond to incision thresholds.

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Graphically interpreting how incision thresholds influence topographic and scaling properties of modeled landscapes

10.5194/esurf-2020-45 ◽

2020 ◽

Author(s):

Nikos Theodoratos ◽

James W. Kirchner

Keyword(s):

Peclet Number ◽

Power Laws ◽

Dimensionless Number ◽

Stream Power ◽

Linear Diffusion ◽

Scaling Properties ◽

Péclet Number ◽

Straight Lines ◽

And Diffusion ◽

Characteristic Scales

Abstract. We examine the influence of incision thresholds on topographic and scaling properties of landscapes that follow a landscape evolution model (LEM) with terms for stream-power incision, linear diffusion, and uniform uplift. Our analysis uses three main tools. First, we examine the graphical behavior of theoretical relationships between curvature and the steepness index (which depends on drainage area and slope). These relationships plot as straight lines for the case of steady-state landscapes that follow the LEM. These lines have slopes and intercepts that provide estimates of landscape characteristic scales. Such lines can be viewed as counterparts of slope–area relationships, which follow power laws in detachment-limited landscapes, but not in landscapes with diffusion. We illustrate the response of these curvature–steepness-index lines to changes in the values of parameters. Second, we define a Péclet number that quantifies the competition between incision and diffusion, while taking the incision threshold into account. We examine how this Péclet number captures the influence of the incision threshold on the degree of landscape dissection. Third, we characterize the influence of the incision threshold using a ratio between it and the steepness index. This ratio is a dimensionless number in the case of the LEM that we use, and reflects the fraction by which the incision rate is reduced due to the incision threshold; in this way, it quantifies the relative influence of the incision threshold across a landscape. These three tools can be used together to graphically illustrate how topography and process competition respond to incision thresholds.

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Scaling and similarity of a stream-power incision and linear diffusion landscape evolution model

Earth Surface Dynamics ◽

10.5194/esurf-6-779-2018 ◽

2018 ◽

Vol 6 (3) ◽

pp. 779-808 ◽

Cited By ~ 4

Author(s):

Nikos Theodoratos ◽

Hansjörg Seybold ◽

James W. Kirchner

Keyword(s):

Steady State ◽

Peclet Number ◽

Degrees Of Freedom ◽

Landscape Evolution ◽

Initial Conditions ◽

Evolution Model ◽

Model Parameters ◽

Stream Power ◽

Linear Diffusion ◽

Characteristic Scales

Abstract. The scaling and similarity of fluvial landscapes can reveal fundamental aspects of the physics driving their evolution. Here, we perform a dimensional analysis of the governing equation of a widely used landscape evolution model (LEM) that combines stream-power incision and linear diffusion laws. Our analysis assumes that length and height are conceptually distinct dimensions and uses characteristic scales that depend only on the model parameters (incision coefficient, diffusion coefficient, and uplift rate) rather than on the size of the domain or of landscape features. We use previously defined characteristic scales of length, height, and time, but, for the first time, we combine all three in a single analysis. Using these characteristic scales, we non-dimensionalize the LEM such that it includes only dimensionless variables and no parameters. This significantly simplifies the LEM by removing all parameter-related degrees of freedom. The only remaining degrees of freedom are in the boundary and initial conditions. Thus, for any given set of dimensionless boundary and initial conditions, all simulations, regardless of parameters, are just rescaled copies of each other, both in steady state and throughout their evolution. Therefore, the entire model parameter space can be explored by temporally and spatially rescaling a single simulation. This is orders of magnitude faster than performing multiple simulations to span multidimensional parameter spaces. The characteristic scales of length, height and time are geomorphologically interpretable; they define relationships between topography and the relative strengths of landscape-forming processes. The characteristic height scale specifies how drainage areas and slopes must be related to curvatures for a landscape to be in steady state and leads to methods for defining valleys, estimating model parameters, and testing whether real topography follows the LEM. The characteristic length scale is roughly equal to the scale of the transition from diffusion-dominated to advection-dominated propagation of topographic perturbations (e.g., knickpoints). We introduce a modified definition of the landscape Péclet number, which quantifies the relative influence of advective versus diffusive propagation of perturbations. Our Péclet number definition can account for the scaling of basin length with basin area, which depends on topographic convergence versus divergence.

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A graphical method to interpret how incision thresholds influence topographic and scaling properties of landscapes

10.5194/egusphere-egu2020-10307 ◽

2020 ◽

Author(s):

Nikos Theodoratos ◽

James W. Kirchner

Keyword(s):

Graphical Method ◽

Numerical Models ◽

Drainage Area ◽

Stream Power ◽

Simple Method ◽

Linear Diffusion ◽

Scaling Properties ◽

Curvature Lines ◽

Straight Lines ◽

The Relationship

Theoretical analysis of the governing equations of numerical models can reveal relationships between topographic properties, such as drainage area, slope, and curvature, in simulated landscapes. These relationships are testable predictions; they can diagnose whether real-world landscapes could potentially arise from similar mechanisms. For example, the stream-power incision model is consistent with drainage area and slope data that plot as straight lines on logarithmic axes.Here we graph theoretical relationships between topographic curvature and the steepness index, which depends on drainage area and slope. These relationships plot as straight lines for steady-state landscapes that have evolved according to a model that combines stream-power incision, linear diffusion, and uplift. Further, they link topography (drainage area, slope, and curvature) to characteristic length scales of the landscape, which depend on the competition between the processes of incision, diffusion, and uplift.Adding an incision threshold to the model changes the relationship between the steepness index and topographic curvature. We examine these changes graphically and we show that they shed light on how incision thresholds influence topographic and scaling properties of landscapes. Specifically, we present a graphical method that consists of plotting steepness index&#8211;curvature lines and of tracing their intersections with each other and with the coordinate axes. This simple method reveals both how topography and process competition are influenced by the incision threshold, and how these influences vary within a given landscape and across different landscapes.

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Effect of Diffusion on Dispersion

SPE Journal ◽

10.2118/115961-pa ◽

2010 ◽

Vol 16 (01) ◽

pp. 65-77 ◽

Cited By ~ 17

Author(s):

Raman K. Jha ◽

Steven L. Bryant ◽

Larry W. Lake

Keyword(s):

Porous Media ◽

Peclet Number ◽

Plug Flow ◽

Nonlinear Dependence ◽

Hydrodynamic Dispersion ◽

Direction Transverse ◽

Péclet Number ◽

Wide Range ◽

And Diffusion ◽

Stream Splitting

Summary It is known that dispersion in porous media results from an interaction between convective spreading and diffusion. However, the nature and implications of these interactions are not well understood. Dispersion coefficients obtained from averaged cup-mixing concentration histories have contributions of convective spreading and diffusion lumped together. We decouple the contributions of convective spreading and diffusion in core-scale dispersion and systematically investigate interaction between the two in detail. We explain phenomena giving rise to important experimental observations such as Fickian behavior of core-scale dispersion and power-law dependence of dispersion coefficient on Péclet number. We track movement of a swarm of solute particles through a physically representative network model. A physically representative network model preserves the geometry and topology of the pore space and spatial correlation in flow properties. We developed deterministic rules to trace paths of solute particles through the network. These rules yield flow streamlines through the network comparable to those obtained from a full solution of Stokes’ equation. Paths of all solute particles are deterministically known in the absence of diffusion. Thus, we can explicitly investigate purely convective spreading by tracking the movement of solute particles on these streamlines. Then, we superimpose diffusion and study dispersion in terms of interaction between convective spreading and diffusion for a wide range of Péclet numbers. This approach invokes no arbitrary parameters, enabling a rigorous validation of the physical origin of core-scale dispersion. In this way, we obtain an unequivocal, quantitative assessment of the roles of convective spreading and diffusion in hydrodynamic dispersion in flow through porous media. Convective spreading has two components: stream splitting and velocity gradient in pore throats in the direction transverse to flow. We show that, if plug flow occurs in the pore throats (accounting only for stream splitting), all solute particles can encounter a wide range of independent velocities because of velocity differences between pore throats and randomness of pore structure. Consequently, plug flow leads to a purely convective spreading that is asymptotically Fickian. Diffusion superimposed on plug flow acts independently of convective spreading (in this case, only stream splitting), and, consequently, dispersion is simply the sum of convective spreading and diffusion. In plug flow, hydrodynamic dispersion varies linearly with the pore-scale Péclet number when diffusion is small in magnitude compared to convective spreading. For a more realistic parabolic velocity profile in pore throats, particles near the solid surface of the medium do not have independent velocities. Now, purely convective spreading (caused by a combination of stream splitting and variation in flow velocity in the transverse direction) is non-Fickian. When diffusion is nonzero, solute particles in the low-velocity region near the solid surface can move into the main flow stream. They subsequently undergo a wide range of independent velocities because of stream splitting, and, consequently, dispersion becomes asymptotically Fickian. In this case, dispersion is a result of an interaction between convection and diffusion. This interaction results in a weak nonlinear dependence of dispersion on Péclet number. The dispersion coefficients predicted by particle tracking through the network are in excellent agreement with the literature experimental data for a broad range of Péclet numbers. Thus, the essential phenomena giving rise to hydrodynamic dispersion observed in porous media are (1) stream splitting of the solute front at every pore, causing independence of particle velocities purely by convection; (2) velocity gradient in pore throats in the direction transverse to flow; and (3) diffusion. Taylor's dispersion in a capillary tube accounts only for the second and third of these phenomena, yielding a quadratic dependence of dispersion on Péclet number. Plug flow in the bonds of a physically representative network accounts only for the first and third phenomena, resulting in a linear dependence of dispersion on Péclet number. When all the three phenomena are accounted for, we can explain effectively the weak nonlinear dependence of dispersion on Péclet number.

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ESD Ideas: The Peclet number is a centerstone of the orbital and millennial Pleistocene variability

10.5194/esd-2020-53 ◽

2020 ◽

Author(s):

Mikhail Y. Verbitsky ◽

Michel Crucifix

Keyword(s):

Time Scales ◽

Peclet Number ◽

Physical Phenomenon ◽

Péclet Number ◽

Entire Spectrum ◽

Temperature Advection ◽

And Diffusion

Abstract. We demonstrate here that a single physical phenomenon, specifically, a naturally changing balance between intensities of temperature advection and diffusion in the viscous ice media, may influence the entire spectrum of the Pleistocene variability from orbital to millennial time-scales.

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Dimensional analysis of a landscape evolution model with incision threshold

Earth Surface Dynamics ◽

10.5194/esurf-8-505-2020 ◽

2020 ◽

Vol 8 (2) ◽

pp. 505-526

Author(s):

Nikos Theodoratos ◽

James W. Kirchner

Keyword(s):

Dimensional Analysis ◽

Governing Equation ◽

Dimensionless Parameter ◽

Landscape Evolution ◽

Initial Conditions ◽

Uplift Rate ◽

Stream Power ◽

Linear Diffusion ◽

Scaling Properties

Abstract. The ability of erosional processes to incise into a topographic surface can be limited by a threshold. Incision thresholds affect the topography of landscapes and their scaling properties and can introduce nonlinear relations between climate and erosion with notable implications for long-term landscape evolution. Despite their potential importance, incision thresholds are often omitted from the incision terms of landscape evolution models (LEMs) to simplify analyses. Here, we present theoretical and numerical results from a dimensional analysis of an LEM that includes terms for threshold-limited stream-power incision, linear diffusion, and uplift. The LEM is parameterized by four parameters (incision coefficient and incision threshold, diffusion coefficient, and uplift rate). The LEM's governing equation can be greatly simplified by recasting it in a dimensionless form that depends on only one dimensionless parameter, the incision-threshold number Nθ. This dimensionless parameter is defined in terms of the incision threshold, the incision coefficient, and the uplift rate, and it quantifies the reduction in the rate of incision due to the incision threshold relative to the uplift rate. Being the only parameter in the dimensionless governing equation, Nθ is the only parameter controlling the evolution of landscapes in this LEM. Thus, landscapes with the same Nθ will evolve geometrically similarly, provided that their boundary and initial conditions are normalized according to appropriate scaling relationships, as we demonstrate using a numerical experiment. In contrast, landscapes with different Nθ values will be influenced to different degrees by their incision thresholds. Using results from a second set of numerical simulations, each with a different incision-threshold number, we qualitatively illustrate how the value of Nθ influences the topography, and we show that relief scales with the quantity Nθ+1 (except where the incision threshold reduces the rate of incision to zero).

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Scaling and similarity of a stream-power incision and linear diffusion landscape evolution model

10.5194/esurf-2018-33 ◽

2018 ◽

Author(s):

Nikos Theodoratos ◽

Hansjörg Seybold ◽

James W. Kirchner

Keyword(s):

Steady State ◽

Degrees Of Freedom ◽

Characteristic Length ◽

Initial Conditions ◽

Model Parameters ◽

Length Scale ◽

Characteristic Length Scale ◽

Stream Power ◽

Linear Diffusion ◽

Characteristic Scales

Abstract. Scaling and similarity of fluvial landscapes can reveal fundamental aspects of the physics driving their evolution. Here we perform dimensional analysis on a widely used landscape evolution model (LEM) that combines stream-power incision and linear diffusion laws. Our analysis assumes that length and height are conceptually distinct dimensions, and uses characteristic scales that depend only on the model parameters (incision coefficient, diffusion coefficient, and uplift rate) rather than on the size of the domain or of landscape features. We use a previously defined characteristic length scale, but introduce new characteristic height and time scales. We use these characteristic scales to non-dimensionalize the LEM, such that it includes only dimensionless variables and no parameters. This significantly simplifies the LEM by removing all parameter-related degrees of freedom. The only remaining degrees of freedom are in the boundary and initial conditions. Thus, for any given set of dimensionless boundary and initial conditions, all simulations, regardless of parameters, are just re-scaled copies of each other, both in steady state and throughout their evolution. Therefore, the entire model parameter space can be explored by temporally and spatially re-scaling a single simulation. This is orders of magnitude faster than performing multiple simulations to span multi-dimensional parameter spaces. The characteristic length, height, and time scales are geomorphologically interpretable; they define relationships between topography and the relative strengths of landscape-forming processes. The characteristic height scale specifies how drainage areas and slopes must be related to curvatures for a landscape to be in steady state, and leads to methods for defining valleys, estimating model parameters, and testing whether real topography follows the LEM. The characteristic length scale is roughly equal to the scale of the transition from diffusion-dominated to advection-dominated propagation of topographic perturbations (e.g., knickpoints). We introduce a modified definition of the landscape Péclet number, which quantifies the relative influence of advective versus diffusive propagation of perturbations. Our Péclet number definition can account for the scaling of basin length with basin area, which depends on topographic convergence versus divergence.

Download Full-text

Dimensional analysis of a landscape evolution model with incision threshold

10.5194/esurf-2019-80 ◽

2020 ◽

Cited By ~ 1

Author(s):

Nikos Theodoratos ◽

James W. Kirchner

Keyword(s):

Dimensional Analysis ◽

Governing Equation ◽

Dimensionless Parameter ◽

Landscape Evolution ◽

Initial Conditions ◽

Uplift Rate ◽

Stream Power ◽

Linear Diffusion ◽

Scaling Properties

Abstract. The ability of erosional processes to incise into a topographic surface can be limited by a threshold. Incision thresholds affect the topography of landscapes and their scaling properties, and can introduce non-linear relations between climate and erosion with notable implications for long-term landscape evolution. Despite their potential importance, incision thresholds are often omitted from the incision terms of landscape evolution models (LEMs) to simplify analyses. Here, we present theoretical and numerical results from a dimensional analysis of an LEM that includes terms for threshold-limited stream-power incision, linear diffusion, and uplift. The LEM is parameterized by four parameters (incision coefficient and incision threshold, diffusion coefficient, and uplift rate). The LEM's governing equation can be greatly simplified by recasting it in a dimensionless form that depends on only one dimensionless parameter, the incision-threshold number Nθ. This dimensionless parameter is defined in terms of the incision threshold, the incision coefficient, and the uplift rate, and it quantifies the reduction in the rate of incision due to the incision threshold relative to the uplift rate. Being the only parameter in the dimensionless governing equation, Nθ is the only parameter controlling the evolution of landscapes in this LEM. Thus, landscapes with the same Nθ will evolve geometrically similarly, provided that their boundary and initial conditions are normalized according to appropriate scaling relationships, as we demonstrate using a numerical experiment. In contrast, landscapes with different Nθ values will be influenced to different degrees by their incision thresholds. Using results from a second set of numerical simulations, each with a different incision-threshold number, we qualitatively illustrate how the value of Nθ influences the topography, and we show that relief scales with the quantity Nθ + 1 (except where the incision threshold reduces the rate of incision to zero).

Download Full-text