1D and 2D Singular Wavefronts

As one of the most impressive nonlinear internal wave trains so far observed, the transbasin oceanic internal waves [photo link], which propagate for many hundreds of kilometers across basins and interact with one another, can be described by the weak solutions [1] of the Euler–Poincaré differential equation (EPDiff equation) on the diffeomorphism group of , or of an -dimensional manifold . In the case of one spatial dimension, the EPDiff equation can be derived from the Camassa–Holm (CH) shallow water wave equation [, 3], which has "peakon" solutions and possesses one order of accuracy in the asymptotic expansion beyond the Korteweg–de Vries (KdV) equation [4], which has "soliton" solutions that interact by the exchange of momentum in unidirectional elastic collisions. However, as pointed out in [5], "EPDiff" distinguishes "CH" because the Euler–Poincaré (EP) equations describe geodesic motion with respect to any metric that defines a norm on the vector space of the Lie algebra of a Lie group, whereas CH only refers to the shallow water wave equations for the case of one spatial dimension. For the purpose of a mathematically as well as historically concise perspective of the role played by the EPDiff equation when it comes to numerical singular wavefront simulation, the papers [6,] and the references therein are recommended.

Using vector notation, the EPDiff equation in the -dimensional case on reads

(1) ,

where the momentum and velocity are vector functions of time and the spatial variables . They are related by the second-order Helmholtz operator ,

(2) ,

with positive length scale factor . By using the Einstein summation convention and writing

(3)

which regards as a one-form density and as a vector field, the EPDiff equation becomes

(4) .

It can be solved with a particle approximation method [1], which yields solitary wave structures. The (approximate) weak solution of Eq. (1) and Eq. (2) can be expressed as

(5)

with

(6)

As to Eq. (5), is the total number of particles, is a Green's function for the Helmholtz operator and can take many forms [], the symbol denotes convolution and for . The solutions are vector-valued functions supported in on a set of surfaces or curves of codimension with , and . When evaluated along the curve , it can be proved that

(7) ,

which means (or more specifically, ) are equivalent to the Lagrangian coordinates. The solution is the momentum map for singular solutions of the EPDiff equation.

As to Eq. (6), the symbol denotes the inner product, is the Hamiltonian function, and the sets satisfy canonical Hamiltonian dynamics.

Incidentally, if we use to denote the Lie algebra of vector fields on an -dimensional manifold , let be a given Lagrangian and let denote the space of one-form (momentum) densities on , then the EPDiff equation can be concisely written as [6]

(8)

where denotes the Lie derivative of momentum with respect to velocity , and is defined as the functional derivative of the Lagrangian with respect to fluid velocity .

If we denote partial derivatives as subscripts, then in the case Eq. (4) and Eq. (2) become

(9)

Adopting a particle method [1], its approximate numerical solution takes the form of a linear combination of Dirac distributions

(10) ,

where represents the location of particle and the corresponding weight at time . The evolution of and obey the equations

(11)

with . If we take Green's function as

(12) ,

then from the particle method we have [9]

(13)

By solving Eqs. (10), (11), and (13) we can obtain and . In this Demonstration we show only the evolution of the velocity field .

In the case , Eq. (4) and Eq. (2) become

(14)

where , , and . Adopting the particle method [1], its approximate numerical solution also takes the form of a linear combination of Dirac distributions

(15) ,

where represents the location of particle and the corresponding weights at time . The evolution of and obey the following equations

(16)

On the other hand, from the particle method we have [9]

(17)

where is the modified Bessel function (which serves as a Green's function) and is its derivative. Considering that is a singularity, it can be predicted that Eq. (17) may yield relatively very high values if two particles approach each other. So it is necessary to regularize the solution values at each step. To do this, a cutoff function is usually adopted, which is taken as a smooth approximation of the Dirac delta function and thus satisfies

(18) .

In practice, a further modified function is often used, which is defined as

(19) .

This allows replacing with its modification , with which the solutions can be regularized as

(20) ,

where again denotes convolution. The Gaussian function, which is regarded as one of the simplest examples of a cutoff function, is used here,

(21) ,

so the modification in our case is determined as [9]

(22) ,

where is the Bessel function of the first kind. By solving Eqs. (15), (16), and (17) and replacing with in Eq. (17), we can obtain and . As and its derivative are numerically difficult to calculate (considering the property of and the infinite integral limit), we use their approximate interpolation function instead, which was directly copied in the initialization code to shorten the processing time. Again, we show only the evolution of the velocity field here.

Provided that at least several hundred particles are used in this case, the EPDiff equation can yield very nice results representing the motion of internal wave trains. But as a real-time simulation, the upper limit of the particle number cannot be set that high as it would take too long to calculate, so up to eight particles can be initially distributed in this Demonstration.

The "process data" function was tested on a computer with an Intel core i7 860 processor and sufficient DDR3 1333MHz memory. It will take only several seconds (from less than 0.001s with the minimum settings up to about 6s with the maximum settings) to complete in the 1D case. However, in the 2D case, it will take about 0.5–1s, 2–3s, and 60–80s for the minimum, default, and maximum settings, respectively.

Depending on the speed of the computer, the simulation may not run smoothly when the number of particles is large. We offer the option of exporting animations to video, which can provide much smoother animations and a better understanding of the dynamics at the cost of a longer export time. In order to do this, remove the comments from three code segments in the source code and then rerun the program (you must use Mathematica not CDF Player):

1. In the main code part, (* Export to video *),

2. In the “Visible Controls” part, (* The control of video export *),

3. In the TrackedSymbols:>{} part, uncomment (*,toVideo*).

The frame rate is set as 20 frames/second for both the 1D and 2D cases. The video files will be named as either "peakons1D.avi" or "peakons2D.avi" and will be saved in the directory containing this source code file (any older file with the same name will be replaced). Based on our tests, with the default settings and on the same computer, it will take about 3 minutes and 5–6 minutes for the export process to complete the 1D and 2D cases, respectively, and the sizes of the corresponding video files will be 135.21MB and 45.37MB. Increasing the number of particles or the mesh quality in the 2D case may greatly increase the export time. After the video file is exported, your system will automatically open it using the default application. If the video cannot be played smoothly, you can manually open it with another player installed on your computer.

More details, including several nice 2D and 3D evolution videos, can also be accessed on Martin Staley's personal page, http://math.lanl.gov/~staley/.

[1] P. A. Raviart, "An Analysis of Particle Methods, Numerical Methods in Fluid Dynamics, in Lecture Notes in Mathematics, Vol. 1127, Berlin: Springer, 1985 pp. 243–324.

[2] R. Camassa and D. D. Holm, "An Integrable Shallow Water Equation with Peaked Solitons," Physical Review Letters, 71(11), 1993 pp. 1661–1664.

[3] R. Camassa, D. D. Holm, and J. M. Hyman, "A New Integrable Shallow Water Equation," Advances in Applied Mechanics, 31, 1994 pp. 1–33.

[4] E. M. de Jager, "On the Origin of the Korteweg–de Vries Equation," arXiv:math/0602661v1, 2006.

[5] C. J. Cotter and D. D. Holm, "Discrete Momentum Maps for Lattice EPDiff," arXiv:math/0602296, 2006.

[6] D. D. Holm and J. E. Marsden, "Momentum Maps and Measure-Valued Solutions (Peakons, Filaments, and Sheets) for the EPDiff Equation," Progress in Mathematics, 232, 2005 pp. 203–235.

[7] D. D. Holm and R. I. Ivanov, "Smooth and Peaked Solitons of the CH Equation," Journal of Physics A: Mathematical and Theoretical, 43(43), 2010 pp. 434003 (18pp).

[8] O. B. Fringer and D. D. Holm, "Integrable vs. Nonintegrable Geodesic Soliton Behavior," Physica D: Nonlinear Phenomena, 150(3–4), 2001 pp. 237–263.

[9] A. Chertock, J. Marsden, and P.D. Toit, "Integration of the EPDiff Equation by Particle Methods," submitted (Mar 17, 2011) http://www4.ncsu.edu/~acherto/papers/ChDTuMa.pdf.