Warning - another post that probably contains too much maths for the average reader...
We had a curious experience paddling in the Cuan Sound this summer. We were playing on some fast flowing water (flowing right to left in the images) by the beacon. As you can see in the upper photo, there was a nice eddy line, but not much else going on. To be honest, it was a bit tame for the group.
A few minutes later, a big motorboat came through, generating a fair bit of wake as it ploughed up the sound against the flow. The race we'd been playing on was transformed - for several minutes it was full of rough water... and became a bit too much for most of the group (lower image - my kayak bow just visible).
Whilst we often see the result of waves interacting with tide in a race, it's unusual to see the waves suddenly 'switched on' in the way that we did on that day. It raises all sorts of questions about how tide races work. The waves were travelling at the speed of the boat - which was going faster than the stream - so why did they become trapped in the race, unable to move into the flat water to the right of the lower photo? And why did the race remain rough for several minutes, especially given the boat had just turned, such that it's wake had been directed in a different direction until shortly before it entered the race?
Why are tide races rough?
By far the best information I know of on tide races is Ken Endean's book 'Coastal Turmoil' - sadly now out of print, but still available as an ebook. He outlines several processes that create rough water when waves are opposed by fast flowing tide:
The waves are compressed by the flow, becoming shorter and much steeper - and often breaking
Refraction occurs at the edges of the flow (the part of the wave in the slower water can travel faster, so the wave turns into the flow), concentrating and containing the waves in the strongest flows
Waves with initial (still water) speeds less than 4 times the flow speed are stopped by the flow, and can be reflected back down the race, increasing the chaos
All these processes seem to be in play in the lower image above. There's clearly a lot of breaking waves caused by the flow compressing and steepening them. It's possible that the wake of the boat may have been trapped in the fast flow by refraction, even as the flow turned a corner, so that waves kept coming into the race for some time after the boat had passed. And waves seem to be trapped in the race - dissipating energy by breaking, but not travelling into the faster flow to the right of the lower image.
The last of these processes seems mysterious - why would a wave be stopped by a flow of only a quarter of its initial speed? Endean gives a reasonable explanation, but notes that 'the mathematics is a little complicated'. Sadly, he doesn't give any references.
Stopping velocity
It turns out that the result isn't widely discussed, and the papers that do mention it treat it as a side note and make it difficult to trace the reasoning. The analysis below is, as far as I know, original, although it draws heavily on a 1976 paper "Interaction of Water Waves and Currents" by D. Howell Peregrine. I've tried to preserve as much physical intuition as possible and rely less on the algebra.
Let's imagine some waves running from an area of still water over some distance into an area with flow in the opposite direction of speed U. Conveniently, a pier runs alongside our waves, on which stands Simon, who observes the waves from his stationary position on the pier. On the water, bobbing along at the same speed as the flow is Kiera the kayaker:
Initially, in still water, the waves move at a speed C0 and have a wavelength L0. As they move into the flow, they slow down to a slower speed C. This happens because:
They are moving against a flow that tends to slow them down
The waves are 'compressed' by the flow, so that their wavelength L on the flow is less than the initial wavelength L0. The speed at which water waves move is affected by their wavelength - shorter waves have a slower speed.
The wave period - i.e. the time between successive waves peaks passing Simon is simply the wavelength divided by the wave speed: L0/C0 in still water and L/C in flowing water. Say Simon measures this wave period at two positions along the pier - call them A and B, with the waves moving from A to B. Let's consider some options as to what Simon might see:
If the period at B is less than A (e.g. 6s vs. 10s), that means 1 wave is leaving the zone between A and B every 6s, but a wave is entering that zone only every 10s. So in a minute, 10 waves leave and 6 waves enter. That means that 4 waves are disappearing every minute! This doesn't seem possible.
If the period at B is more than A, the opposite happens - more waves are entering the zone between A and B every minute than are leaving. Again, this doesn't seem sensible.
So, the only option that makes sense is for the wave period to remain the same all along the pier. So we know that L0/C0=L/C.
We said that the speed of the waves depends on the wavelength. For waves in deep water (which we assume these are), the speed is 1.25 times the square root of the wavelength. I won't attempt to explain where that comes from here - if you're interested, these lecture notes are pretty good. That relation works for waves in still water, so it's fine for relating L0 to C0. We'll need to think a bit harder to relate the wavelength and speed of the waves in the flow.
This is where Kiera comes in. Remember, that she's bobbing along at the same speed as the flow. As anyone who's paddled in tide knows, it might not be apparent to Kiera that she's moving at all. In fact, she's perfectly entitled to say that she and the water she's sitting on are stationary, and it's the pier that is moving. And this means that she can apply the still water wavelength-speed relation and compute the speed of the waves relative to her as 1.25 times the square root of their wavelength.
The problem is that the speed that Kiera perceives the waves as moving at is different to the speed that Simon sees them moving at. Remember that, in reality, the flow is sweeping Kiera into the waves, so she'll see them moving towards her faster than Simon on the pier sees them moving. The difference in apparent speeds is simply the flow speed U. So the speed of the waves relative to Simon is 1.25 times the square root of the wavelength minus U.
So we've now got expressions that links speed to wavelength for both the still water and the moving water:
Now we can put those values for L0 and L into our constant period equation from earlier: L0/C0=L/C:
And some simple algebra turns that into our final result:
This is what we wanted. It relates the ratio between the initial speed of the waves to the flow speed (U/C0) to the proportion by which the waves are slowed down by the flow (C/C0). Hopefully a plot will help - I find it more intuitive to plot C0/U:
Let's take a quick example to understand this plot. Say we have a 6 and a half second period swell that moves at 20 knots in still water. This swell encounters a 4 knot tidal stream running in the opposite direction - so C0/U=20/4=5. From the graph C/C0 is thus about 0.5, so we expect the swell to slow to half its speed in the tidal stream - to about 10 knots.
The interesting thing about the plot is that there the line only goes down to C0/U=4 - i.e. there are no solutions for a wave that has an initial speed less than 4 times the flow speed. So, such waves can't exist on the flow. A flow speed of one quarter the initial wave speed is known as the 'stopping velocity' - it's the flow speed that literally stops the wave.
Tracking a wave up a race
That all makes sense, but it seems a bit abstract, relying on the fact that solutions to an equation don't exist. Fortunately, the approach outlined above allows us to track what happens to the wave as it moves up the tide race to the point where it eventually gets stopped.
Let's imagine looking at some 5s period waves on an increasing tidal stream over a 2km distance. For the first 200m, the waves are moving in still water at about 15 knots. The flow speed then increases gradually over the next 1800m to 4 knots.
As the flow speed increases, the wavelength drops, as does the wave speed (as seen both by Simon on the pier and Kiera floating along with the flow).
At this point, we need to introduce the group speed. Waves tend to move in sets, with the sets moving at the 'group speed', which is half that of the wave speed. It's a bit hard to explain, but here's an animation that gives a feel for how this works:
The key thing to know is that the energy in the waves moves at the group speed rather than the wave speed.
The group speed, as seen by Kiera is simply half the wave speed that she observes. And we can find the group speed relative to Simon by subtracting the flow speed. As the flow increases, the group speeds reduce in the same wave that the wave speed does.
As the waves work their way up the race, their speed and group speed both drop relative to the flow (i.e. as seen by Kiera). The flow continues to increase, until the relative group speed becomes the same as the flow speed. Or, to put it another way, as seen by Simon, the group speed drops to zero.
The waves cannot, of course, get ahead of the energy that forms them, so they must stop progressing up-flow when the group speed drops to zero. We might see the strange phenomenon of waves building at the back of a set, becoming larger, than fading away at the front of a set. Whilst this process happens continually in open water, it's hard to observe as it occurs over big distances. As the waves, and the set, are compressed in a tide race, we can often see this happening in a fairly short distance.
Many waves
Of course, in reality, the sea state is rarely a single simple swell. Instead, waves come in many different periods and sizes. Waves of each period will have a different stopping speed and hence will stop at a different point in the race as the flow accelerates. The result is that:
In the lower, slower, part of the race, many wave periods are present, and there are likely reflected waves coming back down the race too. The waves are all being compressed and steepened, so the result is a chaotic mess of breaking waves
As we move up the race, only the longest wavelengths are left. As a result, the front waves of the race are much more regular and 'clean' - this is the best part of the race to surf
Above the front waves, the race may continue to accelerate to a speed greater than the stopping velocity of any of the waves. None of the waves can penetrate onto this fast flow, so the water surface will be smooth. The smooth water may continue above the race when the flow speed decreases.
And so...
That was a pretty deep dive into some aspects of tide races. Obviously, there's a lot going on in a race, but the patterns and phenomena outlined here are often observable - so hopefully some of this may help you make sense of what often seems like a chaotic environment.
Like my last post, this captures some of the detail that isn't going to make it into the revised notes on sea kayak navigation - which I promise won't have equations in it! Hopefully if I stop falling down rabbit holes like this one, I can get the revisions done soon....