*Warning: **this post delves into physics, maths and obscure details of tidal flow. It even has an equation in it. I won't be offended if you choose to skip past it!*

In my job, I spend a lot of time trying to understand why things behave in the way that they do. Experiments can be useful, as can computer simulations, but often a simple mathematical model is invaluable. Although it might not capture all the details of the specific situation, a good 'pencil and paper' model describes the behaviour in a clear and simple way and, most importantly, provides insight into why it happens.

Oceanographers have tried to create such simple models for tidal flow. Strangely, they have tended to focus on areas of interest for tidal energy or large shipping, rather than the best venues for moving water sea kayaking!

Some of the most interesting tidal flows for sea paddlers are in short straits or channels where the flow is accelerated to speeds that are much higher than those either side of the channel. Think of Kyle Rhea, The Falls of Lora, The Grey Dogs, The Gulf of Corryvreckan, Bardsey Sound, Ramsey Sound or Jack Sound (the channel east of Skomer, image above).

A model of the flow in such a short strait needs to include:

The fluctuating difference in water levels at either end of the strait, caused by the tide

The energy dissipated by friction and turbulence within the strait

The energy needed to accelerate the water into the strait from the slower water at the upstream end

The maths needed isn't complicated if you're into that sort of thing, but involved enough to put off the uninitiated, so I'll stick with just stating the key equation:

The first term is all about the water levels that change with time - remember those sine curves that go up and down. The second bit describes the friction and the acceleration of the water. And the bit to the right of the equals sign says that the things on the left determine how fast the speed of the water changes. The funny backward h thing (lambda) is a measure of how long the strait is, and we'll come back to that. Anyhow, maths lesson over.

Unfortunately, solving that equation turn out to be a pain, so I got a computer to work out the details. Here's a plot of flow speed against time:

The vertical axis is a relative flow speed - don't worry too much about the numbers, as I've played around a bit with them. The horizontal axis is time, in tidal cycles - i.e. multiples of 12 and a half hours - don't worry about the large numbers, it simply means that the simulation has had about 2 years to settle down. The plot covers one full tidal cycle.

There's a few interesting things about the plot:

Slack water (zero flow speed) occurs

*after*the time when the height distance across the strait is zero - this is shown by the short red lines. This means that the water is flowing*uphill*for while before it changes direction. This seems counterintuitive, but remember that the fast moving water in the strait has a lot of momentum.The curve is flatter at the top than we might expect when compared to the smoothly fluctuating water level. The reason is simple - the drag forces are increasing as the square of the speed - i.e. if the water goes twice as fast, the drag forces increase by a factor of four. As a result, there comes a point where it's hard to make the flow go any faster

The curve is skewed - the flow speed increases quickly and reduces slowly. Because of the slack water lag, the height difference across the strait becomes large when the flow speeds (and hence the drag) are low. This means that the flow speed increases very quickly just before and after slack water. Conversely, the height difference becomes small when the flow is still going quickly, so that the flow slows down at a much more leisurely rate.

These phenomena all occur in real tidal streams:

It is fairly typical for water to flow uphill for a time before slack water - Ken Endean comments on this in his excellent book 'Coastal Turmoil'. There aren't many places where there's sufficient data available to see this, but the tidal diamonds in the Pentland Firth, together with tidal constants for Duncansby Head and Scrabster provide a helpful example (which is discussed in Endean's book). By my interpretation of the data, the flow here goes uphill for about 78 minutes before each slack water. That's pretty close to the 71 minutes that the simple mathematical model predicts.

Tidal flows are often reported to increase to close to maximum in the early hours of the flow, faster than the 50/90 rule would predict. Flattened tidal stream curves can be seen by plotting out the data from many tidal diamonds in areas with rapid flow - one example is Hurst Point narrows at the western end of the Solent.

Skewed tidal stream curves also seem to be fairly common, although they are often reported to occur with tidal flow in one direction only. For example, it is said that the Ebb flow on the North Coast of Anglesey increases rapidly in speed in the first hour of the flow. The tidal diamonds in the Little Russel and Big Russel channels around Herm in the Channel Islands also show this effect.

It's interesting to see how the flows predicted by the model are different for short straits and longer channels. 'Short' here means that the time taken for water to transit the channel is a small fraction of the tidal cycle - so for short read 'short channel / fast flow'. By changing the value of that lambda term, the model can be applied to channels of increasing length:

For very short channels, flows are fast, and the flattened top of the curve is clear. However, because there's not a lot of fast water in the channel, the momentum isn't that high and there's almost no lag between the time of zero height difference and slack water - so the curve is symmetric. As the channel gets longer, there's more water in it, and the slack water lag and resulting asymmetry starts to appear. As the channel gets very long, the lag tends to ~3 hours, flows are slower and the curve becomes more symmetric and more similar to the variation in tidal height. This is similar to what is observed in open water where the tide moves as a wave - for example along the south coast of the UK.

Like all models, this one is far from perfect, and it's certainly not useful as a predictive tool. However, it's helped me understand some oddities of tidal stream behaviour, and I hope you've found it interesting too!