How to find convergence in vector field?

Question

I am trying to find areas or points where there is convergence in a vector field.

I have used the code below to generate the following plot:

import matplotlib.pyplot as plt
import numpy as np


def generate_fake_data():
    return -(np.sin(X) * np.cos(Y) + np.cos(X)), -(-np.cos(X) * np.sin(Y) + np.sin(Y))


x = np.arange(0, 2 * np.pi + 2 * np.pi / 20, 2 * np.pi / 20)
y = np.arange(0, 2 * np.pi + 2 * np.pi / 20, 2 * np.pi / 20)

X, Y = np.meshgrid(x, y)

u, v = generate_fake_data()

fig, ax = plt.subplots(figsize=(7, 7))

# quiveropts = dict(headlength=0, headaxislength=0, pivot='middle', units='xy')
# ax.quiver(X, Y, u, v, **quiveropts)
ax.quiver(X, Y, u, v)

ax.xaxis.set_ticks([])
ax.yaxis.set_ticks([])
ax.axis([0, 2 * np.pi, 0, 2 * np.pi])
ax.set_aspect('equal')
ax.axis("off")
plt.gca().set_axis_off()
plt.subplots_adjust(top=1, bottom=0, right=1, left=0,
                    hspace=0, wspace=0)
plt.margins(0, 0)
plt.gca().xaxis.set_major_locator(plt.NullLocator())
plt.gca().yaxis.set_major_locator(plt.NullLocator())
plt.savefig("mock_data.png", bbox_inches='tight', pad_inches=0)

Ideally, what I am after is to find where there is convergence in this vector field in the top right and bottom right of the image.

I hope that the curl values could be used to achieve this, but any method is good to use.

Also, this is just a proof of concept, generate_fake_data will be replaced with a function reading in data from elsewhere that's changeable.

Answer 1

For converging points the divergence of the vector field is < 0.

from functools import reduce
conv = reduce(np.add,np.gradient(u)) + reduce(np.add,np.gradient(v))

(see Compute divergence of vector field using python)

We only need negativ divergence:

conv[conv>=0] = np.nan

plt.imshow(conv): the darker the more converging is the field:

Finding the absolute minimum (in the top right) is easy:

absmin = np.unravel_index(np.nanargmin(conv), conv.shape)
print(absmin, conv[absmin])
#(0, 15) -0.6669774724547413

Finding relative minima is harder, it should be possible with argrelmin, but frankly speaking I couldn't get it work correctly to return the second local minimum at (19,15). Using this answer I get

lm = detect_local_minima(conv)
list(zip(lm))
[(0, 15), (1, 0), (19, 15), (20, 0)]

which also includes both minima in the left corners (which is mathematically correct, but not desired in our case, so maybe we could just exclude corners).