Hills and valleys
Many of the spatial models that people study assume flat, homogeneous domains. In this example, we’ll numerically explore what happens if we replace flatness with all of real life’s roughness.
Hillside vegetation
In our page on vegetation patterns, we explore the Klausmeier model, which can be stated as
\[\begin{aligned}\pd{w}{t} &= a-w -wn^2+v\pd{w}{x} + \nabla^2w,\\ \pd{n}{t} &= wn^2 - mn + \nabla^2n,\end{aligned}\]when written in terms of water $w$ and plant biomass $m$. There are many extensions of this model to include varying, real-world topography, some of which are neatly summarised in a 2018 paper.
We’ll use the simplest possible model, which modifies the original to become
\[\begin{aligned}\pd{w}{t} &= a-w -wn^2+ D\nabla^2w + V\vnabla \cdot (w\vnabla T),\\ \pd{n}{t} &= wn^2 - mn + \nabla^2n,\end{aligned}\]where $T(x,y)$ is the spatially varying height of the landscape. The parameters $D$ and $V$ capture the relative sizes of the water transport terms.
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Load this Klausmeier simulation, which implements the modified PDE for a given $T(x,y)$. Watch the vegetation invade into water-rich regions in the valleys, and seemingly travel uphill following rainfall, which then get used up and dry out.
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Try varying the parameters $a$ and $m$ to see how they impact the structure of patterns.
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What effect does reducing $V$ have on the patterns formed? Do they resemble those in our flat-domain example?
Rainfall on the hilltops
Let’s take a more detailed look at water. We could adapt the shallow water equations that we used in our Visual Story on waves, but instead we’ll use a simpler model that’s really more suited to glaciers and lava than water.
For water of height $h(x,y,t)$ above topography of height $T(x,y)$, this model of gravity-driven flow over terrain reduces to the scalar PDE
\[\pd{h}{t}=D\vnabla \cdot(h^3\vnabla(h+T)),\]where $D$ represents the relative strength of gravity to the stickiness of the fluid.
This equation (with some helpful numerical tricks) is encoded in an rainfall simulation that lets us click to introduce water to the rolling landscape.
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Try clicking to introduce spots of water and watch as they flow down into the valleys.
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Make it rain by dragging over the terrain, gradually filling up the riverbed.
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Try modifying the equation for $h$ by including a constant rainfall (0.001 should be enough) and see the landscape slowly fill.
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We’ve picked an example of real-world topography for you to explore, but you can swap this out for your local area by swapping out the topographical map found by clicking → Images.
Springing to life
Of course, rivers aren’t only filled by rain. Let’s see what happens if we introduce a spring to the hillside.
This spring simulation captures the same river system as before, but now with a spring emerging from one of the hilltops.
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Watch as the spring spreads down the hillside and gradually fills up the large riverbed.
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Try speeding up the process by clicking to add in additional springs.
Looking for high-resolution versions of these simulations? Try out:
These simulations might stretch your device to its limits!