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

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

<p>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.</p><p>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.</p><p>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–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.</p>

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

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.

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

2021 ◽  
Vol 9 (6) ◽  
pp. 1545-1561
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.

scholarly journals How concave are river channels?

2018 ◽  
Vol 6 (2) ◽  
pp. 505-523 ◽  
Author(s):  
Simon M. Mudd ◽  
Fiona J. Clubb ◽  
Boris Gailleton ◽  
Martin D. Hurst
Keyword(s):  
Numerical Models ◽  
Theoretical Models ◽  
Structural Heterogeneity ◽  
Drainage Area ◽  
Tectonic Activity ◽  
River Channels ◽  
Stream Power ◽  
Channel Incision ◽  
Rock Types ◽  
Channel Steepness

Abstract. For over a century, geomorphologists have attempted to unravel information about landscape evolution, and processes that drive it, using river profiles. Many studies have combined new topographic datasets with theoretical models of channel incision to infer erosion rates, identify rock types with different resistance to erosion, and detect potential regions of tectonic activity. The most common metric used to analyse river profile geometry is channel steepness, or ks. However, the calculation of channel steepness requires the normalisation of channel gradient by drainage area. This normalisation requires a power law exponent that is referred to as the channel concavity index. Despite the concavity index being crucial in determining channel steepness, it is challenging to constrain. In this contribution, we compare both slope–area methods for calculating the concavity index and methods based on integrating drainage area along the length of the channel, using so-called “chi” (χ) analysis. We present a new χ-based method which directly compares χ values of tributary nodes to those on the main stem; this method allows us to constrain the concavity index in transient landscapes without assuming a linear relationship between χ and elevation. Patterns of the concavity index have been linked to the ratio of the area and slope exponents of the stream power incision model (m∕n); we therefore construct simple numerical models obeying detachment-limited stream power and test the different methods against simulations with imposed m and n. We find that χ-based methods are better than slope–area methods at reproducing imposed m∕n ratios when our numerical landscapes are subject to either transient uplift or spatially varying uplift and fluvial erodibility. We also test our methods on several real landscapes, including sites with both lithological and structural heterogeneity, to provide examples of the methods' performance and limitations. These methods are made available in a new software package so that other workers can explore how the concavity index varies across diverse landscapes, with the aim to improve our understanding of the physics behind bedrock channel incision.

scholarly journals Dimensional analysis of a landscape evolution model with incision threshold

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).

Layman's method to verify Goldbach's conjectures

2013 ◽  
Vol 10 (4) ◽  
pp. 401-404
Author(s):  
Y. Chang
Keyword(s):  
Graphical Method ◽  
Number Line ◽  
Prime Numbers ◽  
Coordinate Frame ◽  
Horizontal Line ◽  
Analytical Geometry ◽  
Mathematical Problems ◽  
Straight Lines ◽  
Relationship Of ◽  
The Relationship

Goldbach conjecture of prime numbers is one of the unsolved mathematical problems. Many trial solutions appeared in the literature, but so far none has been accepted by the mathematics societies. This paper describes a graphical method devised by me to explain the mystery of the said conjecture. My method based on the teachings of analytical geometry using a rectangular coordinate frame with even numbers as ordinates and prime numbers as abscissas. Straight lines with 45 degree slop and intercepets of varying prime numbers on the ordinate are drawn to meet all the vertical straight draw grom the abscissas. These diagonal lines are designated as separation lines and identified by its intercept number. The intersection of vertical abscissa line, the separation line and a horizontal line drawn from the ordinates shows the relationship of an even number and its pair of prime numbers. These intersections vividly appear on the horizontal even number line and can be easily seen. This method is a graphical version of binary combination of prime numbers and can locate the prime-pairs of any even nuber by drawing a family of separation lines.

scholarly journals Dimensional analysis of a landscape evolution model with incision threshold

2020 ◽  
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).

scholarly journals How concave are river channels?

2018 ◽  
Author(s):  
Simon M. Mudd ◽  
Fiona J. Clubb ◽  
Boris Gailleton ◽  
Martin D. Hurst
Keyword(s):  
Numerical Models ◽  
Theoretical Models ◽  
Structural Heterogeneity ◽  
Drainage Area ◽  
Tectonic Activity ◽  
River Channels ◽  
Stream Power ◽  
Channel Incision ◽  
Rock Types ◽  
Channel Steepness

Abstract. For over a century geomorphologists have attempted to unravel information about landscape evolution, and processes that drive it, using river profiles. Many studies have combined new topographic datasets with theoretical models of channel incision to infer erosion rates, identify rock types with different resistance to erosion, and detect potential regions of tectonic activity. The most common metric used to analyse river profile geometry is channel steepness, or ks. However, the calculation of channel steepness requires the normalisation of channel gradient by drainage area. This relationship between channel gradient and drainage area is referred to as channel concavity, and despite being crucial in determining channel steepness, is challenging to constrain. In this contribution we compare both slope–area methods for calculating concavity and methods based on integrating drainage area along the length of the channel, using so-called ``chi'' (χ) analysis. We present a new χ-based method which directly compares χ values of tributary nodes to those on the main stem: this method allows us to constrain channel concavity in transient landscapes without assuming a linear relationship between χ and elevation. Patterns of channel concavity have been linked to the ratio of the area and slope exponents of the stream power incision model (m/n): we therefore construct simple numerical models obeying detachment-limited stream power and test the different methods against simulations with imposed m and n. We find that χ-based methods are better than slope–area methods at reproducing imposed m/n ratios when our numerical landscapes are subject to either transient uplift or spatially varying uplift and fluvial erodibility. We also test our methods on several real landscapes, including sites with both lithological and structural heterogeneity, to provide examples of the methods' performance and limitations. These methods are made available in a new software package so that other workers can explore how concavity varies across diverse landscapes, with the aim to improve our understanding of the physics behind bedrock channel incision.

Identifying Physico-Chemical Laws from the Robotically Collected Data

2019 ◽  
Author(s):  
Liwei Cao ◽  
Danilo Russo ◽  
Vassilios S. Vassiliadis ◽  
Alexei Lapkin
Keyword(s):  
Experimental Data ◽  
Numerical Models ◽  
Predictor Variable ◽  
Physical Models ◽  
Training Data ◽  
Mixed Integer ◽  
Physico Chemical ◽  
Data Points ◽  
Future Work ◽  
The Relationship

<p>A mixed-integer nonlinear programming (MINLP) formulation for symbolic regression was proposed to identify physical models from noisy experimental data. The formulation was tested using numerical models and was found to be more efficient than the previous literature example with respect to the number of predictor variables and training data points. The globally optimal search was extended to identify physical models and to cope with noise in the experimental data predictor variable. The methodology was coupled with the collection of experimental data in an automated fashion, and was proven to be successful in identifying the correct physical models describing the relationship between the shear stress and shear rate for both Newtonian and non-Newtonian fluids, and simple kinetic laws of reactions. Future work will focus on addressing the limitations of the formulation presented in this work, by extending it to be able to address larger complex physical models.</p><p><br></p>

Estimation of the Area of Rubber Leaves (Hevea brasiliensis Muell. arg.) Using Two Leaflet Parameters

1972 ◽  
Vol 8 (4) ◽  
pp. 311-314 ◽  
Author(s):  
T. M. Lim ◽  
R. Narayanan
Keyword(s):  
Hevea Brasiliensis ◽  
Simple Method ◽  
Different Ages ◽  
The Relationship

SUMMARYA rapid, simple method is described for estimating the area of rubber leaves from two measurements on the middle and one of the side leaflets. The relationship between the area of a leaflet (A) and its length × breadth (LB), described by the expression A = 0.654 LB, does not vary between the three leaflets or between leaves of different ages, and clonal differences are slight.

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