^{1}and Simon R. Clarke

^{1}

### Abstract

We consider the propagation of stratified fluid through a contraction near resonance, for which the governing asymptotic equation is the forced Korteweg–de Vries equation. Steady solutions of this equation are sought that have constant but differing amplitudes upstream and downstream of the contraction, and for which leading-order dispersive effects are retained. A numerical algorithm is outlined to obtain such solutions, yielding a parametric relationship between the normalized velocity perturbation and the normalized width perturbation.Matrix methods are then used to investigate the linear stability of these solutions.

I. INTRODUCTION

II. PROPERTIES OF THE FORCED KdV EQUATION

III. NUMERICAL PROCEDURE AND PARAMETRIC RELATIONSHIP

A. ODE integration

B. Minimization algorithm

C. Solution refinement

D. Branch following algorithm

IV. PARAMETRIC RELATIONSHIP FOR DISPERSIVE HYDRAULICSOLUTIONS

V. LINEAR STABILITY ANALYSIS

A. Stability formulation

B. Matrix stability method

VI. CONCLUSION

### Key Topics

- Hydraulics
- 48.0
- Solution processes
- 28.0
- Eigenvalues
- 21.0
- Perturbation methods
- 18.0
- Topography
- 15.0

## Figures

(Color online) Numerical solution to (3) for both the (i) original solution region defined over with and (ii) reflected solution region defined over with , subject to , , .

(Color online) Numerical solution to (3) for both the (i) original solution region defined over with and (ii) reflected solution region defined over with , subject to , , .

(Color online) Top panel: Using , a sketch of homoclinic orbits of the KdV equation with center at and saddle point at corresponding to far-field upstream and downstream amplitudes. Bottom panel: sketch of corresponding potential where and showing the center and saddle points.

(Color online) Top panel: Using , a sketch of homoclinic orbits of the KdV equation with center at and saddle point at corresponding to far-field upstream and downstream amplitudes. Bottom panel: sketch of corresponding potential where and showing the center and saddle points.

(Color online) Top panel: Numerical solution to (3) over original region defined over , where , , , before applying exponential decay (solid line), after the application of exponential decay (dashed), and the point of the application of the exponential decay denoted by asterisk. Bottom panel: Semilog plot of residue solution , in the original region defined over , to (3), , , , and . The point of application of the exponential decay on the numerical solution is denoted by an asterisk.

(Color online) Top panel: Numerical solution to (3) over original region defined over , where , , , before applying exponential decay (solid line), after the application of exponential decay (dashed), and the point of the application of the exponential decay denoted by asterisk. Bottom panel: Semilog plot of residue solution , in the original region defined over , to (3), , , , and . The point of application of the exponential decay on the numerical solution is denoted by an asterisk.

(Color online) Plot of vs using and . This describes the locus of dispersive hydraulic solutions for (3). Corresponding absolute value of relative error (bottom panels) between our computed and that in GS, to verify the (a) narrow forcing limit and (b) wide forcing limit, both using and .

(Color online) Plot of vs using and . This describes the locus of dispersive hydraulic solutions for (3). Corresponding absolute value of relative error (bottom panels) between our computed and that in GS, to verify the (a) narrow forcing limit and (b) wide forcing limit, both using and .

Numerical solution of (3) after solution refinement with and using (a) , (b) , (c) , (d) , (e) , (f) , (g) , (h) , and (i) . Solutions (a)–(i) have each been displaced such that the upstream level corresponds to zero.

Numerical solution of (3) after solution refinement with and using (a) , (b) , (c) , (d) , (e) , (f) , (g) , (h) , and (i) . Solutions (a)–(i) have each been displaced such that the upstream level corresponds to zero.

Plots of eigenvalue spectra, vs , formed by combining the eigenvalue plots using (a) , and (b) , . is the eigenvalue with the largest real part, which can be obtained after solving (20). The smaller box embedded in the bottom panel is a zoomed plot of the spectra for .

Plots of eigenvalue spectra, vs , formed by combining the eigenvalue plots using (a) , and (b) , . is the eigenvalue with the largest real part, which can be obtained after solving (20). The smaller box embedded in the bottom panel is a zoomed plot of the spectra for .

Plots of the and vs using of the two leading eigenvalues, obtained after solving (20), ranked according to the size of the real components. [(a) and (b)] and [(c) and (d)] .

Plots of the and vs using of the two leading eigenvalues, obtained after solving (20), ranked according to the size of the real components. [(a) and (b)] and [(c) and (d)] .

(Color online) Plots of and : real and imaginary components of the eigenfunction associated with , the eigenvalue with the largest real part which can be obtained after solving (20). [(a) and (b)] , ; [(c) and (d)] , ; [(e) and (f)] , ; [(g) and (h)] , ; (i) , . The solutions all have mean value zero but have been displaced for clarity.

(Color online) Plots of and : real and imaginary components of the eigenfunction associated with , the eigenvalue with the largest real part which can be obtained after solving (20). [(a) and (b)] , ; [(c) and (d)] , ; [(e) and (f)] , ; [(g) and (h)] , ; (i) , . The solutions all have mean value zero but have been displaced for clarity.

(Color online) Logarithmic scale plot of perturbation momentum vs time over the combined region , using and for (a) , and (b) , . The dotted lines show the exponential growth due to . Normalized numerical solutions to (17) with and for (c) , and (d) , .

(Color online) Logarithmic scale plot of perturbation momentum vs time over the combined region , using and for (a) , and (b) , . The dotted lines show the exponential growth due to . Normalized numerical solutions to (17) with and for (c) , and (d) , .

(Color online) Logarithmic scale plot of perturbation momentum vs time over combined region , using and for (a) , and (b) , . The dotted lines show the exponential growth due to . Normalized numerical solutions to (17) with and for (c) , and (d) , .

(Color online) Logarithmic scale plot of perturbation momentum vs time over combined region , using and for (a) , and (b) , . The dotted lines show the exponential growth due to . Normalized numerical solutions to (17) with and for (c) , and (d) , .

(Color online) A depiction of the parameter space for solutions of (1) for . The dashed lines and text denote asymptotic regimes for the solutions of (1) for the trivial initial condition. The solid lines and shading depict allowable boundaries for the dispersive hydraulic solutions considered here.

(Color online) A depiction of the parameter space for solutions of (1) for . The dashed lines and text denote asymptotic regimes for the solutions of (1) for the trivial initial condition. The solid lines and shading depict allowable boundaries for the dispersive hydraulic solutions considered here.

(Color online) Numerical solution of (3) after solution refinement with , along with topographic perturbations using , , and .

(Color online) Numerical solution of (3) after solution refinement with , along with topographic perturbations using , , and .

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