Flow over a barrier on a beta plane – Theory

Introduction | Tank – How to |Tank – Examples | Atmospheric – Examples | Theory

Physical Mechanism

See the theory section for Rossby waves for the physical mechanism behind the waves generated in the experiment.

This theory shows that, on the sphere, Rossby waves satisfy the dispersion relation

where c_p is the phase velocity, K the total wavenumber (where K² = k² + l²) and &omega the frequency. Here β = df/dy [where f is the Coriolis parameter (f = 2Ωsinφ, φ the latitude) and y is a coordinate locally increasing northwards), sometimes called the planetary vorticity gradient. In the case of our laboratory experiment on the cone, β is replaced by β’ as defined below.

Rossby waves resulting from flow over a barrier on a beta plane

The flow over a barrier experiment simulates the change in length of the Taylor columns by varying the depth of the rotating tank – the shallow, inner area of the tank represents the (north) pole, and the deeper, outer area represents the equator.

As columns of fluid travel over the barrier, they are forced towards the deep end of the tank (equator) in order to maintain their length. The restoring force due to the conservation of potential vorticity means that this displacement leads to the generation of a Rossby wave.

Since the barrier is stationary in the tank, the forcing from the barrier results in stationary Rossby waves, i.e., waves in which the phase velocity of the Rossby wave exactly balances the rotational flow of the fluid.

As in the Rossby wave theory section, it can be shown that in the presence of the cone, Rossby waves satisfy the dispersion relation (1), with effective β

where Ω is the magnitude of the rotation vector, and H the local depth of the tank.

Now, we are interested in stationary Rossby waves, where ū + c_p = 0. Since the eastwards flow of water is driven by the change in angular velocity (due to slowing the tank down), we look for waves satisfying

where ΔΩ is the change in angular velocity of the table, and r is the distance of the waves from the center of the tank.

So, taking k = l, λ = 2π/k, this becomes

Fig.1. Since the slope of the cone is ∼ 1, it is easy to calculate the depth of the water using the distance from the center of the tank.

If we substitute in the values ∂H/∂r = slope = ≈ 1, H ≈ r – 0.05m (as in Fig.1), Ω ≈ 1.5 rad/s, ΔΩ ≈ 0.1 rad/s, this gives

Hence at r = 0.15m, we have λ ∼ 0.2m = 20cm – this is of the order of the wavelength that we would expect to observe in the experiment.