Emergent structures in reaction-advection-diffusion systems on a sphere

We demonstrate unusual effects due to the addition of advection into a two-species reaction-diffusion system on the sphere. We find that advection introduces emergent behavior due to an interplay of the traditional Turing patterning mechanisms with the compact geometry of the sphere. Unidirectional advection within the Turing space of the reaction-diffusion system causes patterns to be generated at one point of the sphere, and transported to the antipodal point where they are destroyed. We illustrate these effects numerically and deduce conditions for Turing instabilities on local projections to understand the mechanisms behind these behaviors. We compare this behavior to planar advection which is shown to only transport patterns across the domain. Analogous transport results seem to hold for the sphere under azimuthal transport or away from the antipodal points in unidirectional flow regimes.

Figure 1

A diagram of the three regions analyzed in Sec. 3, showing how they relate to unidirectional flow on the sphere. We consider a local disk tangent to the sphere at each antipode, and then a cylindrical region along the center of the sphere.

Figure 2

Transient formation of labyrinthine patterns in the Schnakenberg model (52) with $R=100$, $a=0.1$, $b=0.9$, ${\delta }_1=1$, ${\delta }_2=10$, and $A_x=4$ for $\mathbf{A}=\mathbf{B}=A_x[-sin\left(\theta \right)sin\left(\varphi \right),cos\left(\theta \right)cos\left(\varphi \right)]$. The three columns are different views of source-side sink, and each row corresponds to $t=40,60,100$.

Figure 3

Formation and transport of labyrinthine patterns in the Schnakenberg model (52) with $R=40$, $a=0.01$, $b=1.2$, ${\delta }_1=1$, ${\delta }_2=10,$ and $A_x=0.1$ for $\mathbf{A}=\mathbf{B}=A_x[-sin\left(\theta \right)sin\left(\varphi \right),cos\left(\theta \right)cos\left(\varphi \right)]$. The three columns are different views of source-side sink, and each row corresponds to $t=200,400,600$.

Figure 4

Long-time ($t=1000$) dynamics of labyrinthine patterns in the Schnakenberg model (52) with $R=100$, $a=0.1$, $b=0.9$, ${\delta }_1=1$, ${\delta }_2=10$, and $A_x=4$ for $\mathbf{A}=\mathbf{B}=A_x[-sin\left(\theta \right)sin\left(\varphi \right),cos\left(\theta \right)cos\left(\varphi \right)]$. The three columns are different views of source-side sink.

Figure 5

Long-time movement of spot patterns in the Schnakenberg model (52) with $R=20$, $a=0.1$, $b=0.9$, ${\delta }_1=1$, ${\delta }_2=20$, and $A_x=0.5$ for $\mathbf{A}=\mathbf{B}=A_x[-sin\left(\theta \right)sin\left(\varphi \right),cos\left(\theta \right)cos\left(\varphi \right)]$. The three columns are different views of source-side sink, and each row corresponds to $t=300,310,320$.

Figure 6

Final patterns in the Schnakenberg model (52) with $a=0.1$, $b=0.9$, ${\delta }_1=1$, and $A_{\varphi }=1$ for $\mathbf{A}=\mathbf{B}=(0,A_{\varphi })$. On the left we take ${\delta }_2=10R=100$, and on the right ${\delta }_2=20R=20$. The view is such that the point $(\pi /4,\pi /4)$ is at the center of the image.

Figure 7

Transient formation of spot patterns in the Schnakenberg model (52) with $R=100$, $a=0.1$, $b=0.9$, ${\delta }_1=1$, ${\delta }_2=40$, and $A_{\varphi }=1$ for $\mathbf{A}=\mathbf{B}=(0,A_{\varphi })$. These are at times $t=20,30,40,50$. The view is such that the point $(\pi /4,\pi /4)$ is at the center of the image.

Figure 8

Long-time movement of spot patterns in the Gierer-Meinhardt model (53) with $R=20$, $a=2$, $b=20$, ${\delta }_1=1$, ${\delta }_2=100$, and $A_x=0.5$ for $\mathbf{A}=\mathbf{B}=A_x[-sin\left(\theta \right)sin\left(\varphi \right),cos\left(\theta \right)cos\left(\varphi \right)]$. The three columns are different views of source-side sink, and each row corresponds to $t=1010,1015,1020$.

Figure 9

Steady state patterns in the FitzHugh-Nagumo model (54) with $R=20$, $a=1.2$, $b=1.1$, ${\delta }_1=1$, ${\delta }_2=20$, and $\mathbf{A}=\mathbf{B}=(0,0)$. The three columns are different views corresponding to $t=1000$, although in the absence of advection there is no qualitative difference between these regions.

Figure 10

Long-time movement of wave patterns in the FitzHugh-Nagumo model (54) with $R=20$, $a=1.2$, $b=1.1$, ${\delta }_1=1$, ${\delta }_2=20$, and $A_x=1.8$ for $\mathbf{A}=\mathbf{B}=A_x[-sin\left(\theta \right)sin\left(\varphi \right),cos\left(\theta \right)cos\left(\varphi \right)]$. The three columns are different views of source-side sink, and each row corresponds to $t=1005,1010,1015$.

Figure 11

Long-time movement of wave patterns in the FitzHugh-Nagumo model (54) with $R=20$, $a=1.2$, $b=1.1$, ${\delta }_1=1$, ${\delta }_2=20$, and $A_x=1.9$ for $\mathbf{A}=\mathbf{B}=A_x[-sin\left(\theta \right)sin\left(\varphi \right),cos\left(\theta \right)cos\left(\varphi \right)]$. The two columns are different views with the source on the left and sink on the right, and each row corresponds to $t=1004,1006,1008$. Note that the views have been slanted slightly to observe the formation of the wave at the source, and its disappearance at the sink.