Deriving shallow water equations: why is the vertical velocity equal to the material derivative of the surface level?

Let’s dive into the shallow water equations, those fascinating tools we use to describe how water flows in relatively shallow environments. Think rivers, lakes, or even the coastline – places where the depth is significantly less than the horizontal reach. These equations are derived from some pretty heavy-duty physics, the Navier-Stokes equations, but with a few clever simplifications that make them manageable.

So, what are these simplifications? Well, for starters, we assume the pressure increases linearly with depth, a concept known as hydrostatic pressure. Basically, the pressure at any point is just the weight of the water pushing down from above. This holds true when vertical accelerations are small compared to gravity. Another key assumption is that the horizontal velocity is pretty much uniform from the surface to the bottom. Imagine a river flowing; we’re saying the water’s moving at roughly the same speed at the top as it is near the riverbed. Finally, and perhaps most importantly, we assume the water is shallow! That is, the vertical scale (the depth) is much smaller than the horizontal scale (the length or width).

Now, how do we actually get to these shallow water equations? The basic idea is to take the fundamental laws of physics – conservation of mass and momentum – and average them over the entire depth of the water column. But here’s where things get really interesting: the connection between the vertical velocity at the surface and how the surface itself is changing.

This connection is captured by something called the kinematic boundary condition. It’s a fancy name for a pretty intuitive idea: a water particle on the surface stays on the surface. Think of a leaf floating on a pond; it moves with the water and remains on the surface. Mathematically, we express this as: W = Dh/Dt at z = h(x, y, t).

Let’s break that down. W is the vertical velocity – how fast the water is moving up or down. h(x, y, t) is the surface elevation – the height of the water surface at a given location and time. And D/Dt is the material derivative, which is just a fancy way of saying “the rate of change as seen by a water particle as it moves around.”

You can expand the material derivative as: D/Dt = ∂/∂t + u ∂/∂x + v ∂/∂y, where u and v are the horizontal velocity components. So, putting it all together, the kinematic boundary condition becomes: W = ∂h/∂t + u ∂h/∂x + v ∂h/∂y at z = h(x, y, t).

What does this all mean? Well, imagine you’re standing by a lake. If the water level is rising (∂h/∂t is positive), then the water at the surface must be moving upwards (W is positive). Makes sense, right? And if the wind is pushing the water, causing the surface to change shape (u ∂h/∂x + v ∂h/∂y is non-zero), then the water at the surface has to move vertically to stay on that changing surface.

This might seem like a detail, but it’s absolutely crucial for deriving the shallow water equations. It’s the key that locks everything into place. By using this kinematic boundary condition, along with the fact that the water can’t go through the bottom (the vertical velocity at the bottom is zero), and integrating the mass conservation equation, we get an equation that tells us how the surface elevation changes over time. Combine that with equations describing how the water’s momentum changes, and you’ve got the full set of shallow water equations!

So, next time you see a river flowing or waves crashing on the beach, remember that these seemingly simple phenomena are governed by equations that rely on this fundamental connection between the vertical velocity and the changing water surface. It’s a beautiful example of how math can capture the essence of the natural world.