Dynamic model of a spiral-vortex phase field

Introduction

The study of vortex and spiral-wave structures is one of the important areas of modern physics of nonlinear phenomena, optics and hydrodynamics. Of particular interest are two-dimensional phase fields in which the superposition of two fundamental topological modes is realized: radial spiral modulation and azimuthal phase shift, characterized by an integer azimuthal mode number. Such structures are described in the framework of continuous models, where the key parameters are the amplitude of the radial component, the frequency of time evolution, and the magnitude of the regularization parameter that eliminates the singularity in the coordinate center.

Visualization is carried out both in a static three-dimensional representation (surface) and in a dynamic form, which allows observing the evolution of the field over time. This makes it possible not only to qualitatively assess the emerging spatial structures, but also to investigate their dependence on the control parameters of the model.

Initial data

Let's define a two-dimensional computational grid on a plane.

x = range(-2, 2, length = 400)
y = range(-2, 2, length = 400)
X = [i for i in x, _ in y]
Y = [j for _ in x, j in y]

Phase field function

We define the vortex wave field using the function:

Where:

— the amplitude of the radial phase modulation;

is a regularization parameter that prevents the singularity of the function at the point r = 0;

— angular velocity of phase change over time;

— azimuthal mode number;

— the time.

For the effect of the polar coordinate system, we set the condition: Z[R .> 2] .= NaN.

function Z(t, A, ε, ω, m)
    R = sqrt.(X.^2 .+ Y.^2)
    Z = sin.(A ./ (R .+ ε) .+ t*ω .+ atan.(Y, X)*m)
    Z[R .> 2] .= NaN 
    return Z
end

Z (generic function with 2 methods)

Wave parameters

Let's define the initial parameters of the phase field.

t = 1
A = 10
ε = 0.1
ω = 0.15
m  = 5

Visualization

3D visualization

Let's display the phase field as a three-dimensional shape.

p = surface(x, y, Z(t, A, ε, ω, m), alpha=0.9, c=:viridis)

Animation

Let's imagine a vortex wave field in a dynamic form.

anim = @animate for t in 1:300
    heatmap(x, y, Z(t, A, ε, ω, m), aspect_ratio = :equal,  axis = nothing, color = :viridis,       
    title = "", framestyle = :none, size = (500, 500), background_color = :transparent)
end

gif(anim, "Vortex.gif", fps = 15)

Warning: detected a stack overflow; program state may be corrupted, so further execution might be unreliable.
Warning: detected a stack overflow; program state may be corrupted, so further execution might be unreliable.
[ Info: Saved animation to /user/Vortex/Vortex.gif

Conclusion

The visualization results confirm that the proposed mathematical model generates stable vortex patterns with pronounced spiral symmetry. Animation of the time evolution makes it possible to trace the rotation and radial propagation of phase fronts, which is typical for systems with a helical dislocation of the wavefront.

The presented model and the methods of its visualization are of direct practical importance in a number of scientific and technical fields.:

Optics and photonics — in the analysis and synthesis of optical vortices, beams with orbital angular momentum, as well as in the modeling of interference patterns in systems with phase singularities.
Hydrodynamics and plasma physics — to describe spiral wave structures that occur in nonequilibrium media, including eddy currents and density waves.
Signal and image processing — in the development of algorithms for phase filtering, singular point detection, and texture recognition based on spiral harmonics.
Biophysics and nonlinear dynamics of excitable media in models reproducing spiral waves in cardiac tissue, neural ensembles, and reaction-diffusion systems.

Thus, the constructed computational scheme can serve as an educational and demonstration tool for studying the topological properties of phase fields, as well as a basic element of more complex engineering and research calculations in these areas.