I was surprised to see that the new BCAB tides module includes description of the effects of lunar perigee, apogee and perihelion on tides. I guess it's useful to understand that some spring tides are bigger than others, but I'm not sure that knowing the specifics of why that's the case is terribly useful, and I've given these topics only a cursory description in my notes. To compensate, here's a more detailed explanation for the dedicated tidal masochist. A brief warning - this is only going to make sense if you already have a clear understanding of how the moon drives the tides and the spring-neap cycle - perhaps read this first!
Elliptical orbit
The orbit of the moon is not circular, but slightly elliptical. The image below shows the earth, moon and moon's orbit to scale. For comparison, a circular orbit is shown in grey:
(to make things clearer, we'll exaggerate the elliptical orbit and show the earth, sun and moon not to scale in the diagrams that follow!).
The maximum distance from the earth to the moon is 405500 km. This highest point in the orbit is known as 'apogee'. The minimum distance is 363300 km, known as 'perigee'. Clearly, the tidal forces exerted by the moon will be less when it's further away at apogee, and more when it's close to the earth at perigee. Whilst the ratio of distances is only 1:0.9, the tidal forces scale with the cube of the distance, so the tidal forces at apogee are around 72% of those at perigee.
The earth's orbit around the sun is also elliptical, although less so than the moon's with a maximum distance ('aphelion') of 152,100,000 km and a minimum distance ('Perihelion') of 147,095,000 km. The distance ratio is 1:0.97, and we'd thus expect the tidal forces at aphelion to be about 90% of those at perihelion. Perihelion, the closest approach of the sun occurs around January 3rd, whilst aphelion, when the sun is furthest away happens around July 4th. This has a small effect on the tidal range between winter and summer, but we won't consider this further... so, back to the moon...
Interaction with springs-neap cycle
As a result of the elliptical orbit of the moon, we might expect bigger tidal ranges at perigee than at apogee. However, the change in tidal force from perigee to apogee is not, of course, the main change in tidal forces that occurs over the course of the moon's monthly orbit. Going from springs to neaps due to the relative position of the sun alters the tidal forces by 46%, so the springs-neap cycle is the dominant effect on tidal ranges. The question is how this interacts with the apogee-perigee cycle. Let's consider two extremes:
1) Line of apsides aligned with sun
The 'line of apsides' is the line between the further parts of the elliptical orbit, so when this line is aligned with the sun, the earth-sun-moon system looks like this:
At +0 days the moon is at perigee and the sun is aligned with the moon to create a spring tide. The tidal forces are increased by both the sun's alignment (springs) and the moon being close to the earth (perigee). The large tidal ranges that result are called 'perigean spring tides'.
15 days later, the sun, moon and earth are again aligned for a spring tide, but the moon is now at apogee - its furthest point from the earth. The result will be a tidal range rather smaller than that of the previous spring tide - an 'apogean spring tide'.
The result is a modulation where successive spring tides are of different heights:
2) Line of apsides not aligned with sun
What happens when the elliptical orbit is not aligned to the sun? This situation will occur 3 months later due to the earth moving around the sun:
Now both spring tides occur with the moon at a similar distance from the earth. We'd expect a similar range at springs in both cases, with a range lying somewhere between the apogean and perigean spring ranges.
Apsidal precession
So, we'd expect the spring tides to fluctuate between 2 states:
Successive spring tides having different and extreme ranges (apogean and perigean springs with apsides aligned with the sun)
Successive spring tides having similar and medium ranges (apsides not aligned to the sun)
On about a 6 monthly cycle. Actually, the line of the apsides moves with time, a phenomenon known as 'apsidal precession', so that the length of the cycle is closer to 7 months, as shown in the images below of successive perigean spring tides:
[Note that a full moon perigean springs (left hand image) is followed by a new moon perigean springs (middle image). Whilst it takes about 7 months for the apsides to align with the sun, the period between perigean springs is thus 6.5 or 7.5 months.]
So, if we plot just high water heights over a year, we'd expect to see this sort of a pattern:
The real world
So much for the theory - how does this look for real tides? As an example, let's look at predicted tides for Newlyn over a period of a few years.
We do, indeed appear to see the pattern shifting in the way we expect. To make things easier to see, the spring high waters are each labelled with a dot. Blue dots indicate where successive spring tides are of very different ranges, red dots indicate where successive spring tides have similar ranges. We can see that in blue dotted areas, the larger perigean spring tides are followed by much smaller apogean spring tides. And in the red dotted areas, the tidal range lies somewhere between these extremes. And it looks credible that the red-blue-red cycle happens on about a 7 month period.
To try and make things clearer, let's plot out how the ratio of successive spring tides (set up to be a number <1) and the range of the tides (taking the larger of each pair of spring tides to make the perigean springs easier to see) varies over time:
The data here (grey dots) are a bit noisy, as there are obviously lots of other tidal effects going on, so I've plotted a smoothed curve (black line) to make things clearer.
The blue dashed lines show the approximately 7 month cycle that we expect perigean spring tides to recur on. The inequality of successive spring tides seems to follow the expected pattern. And at perigean spring tides, the ratio is close to the predicted value of 0.72. In general, the largest spring tides occur at the times of perigean springs, although this isn't the case in 2021/2022.
Indeed, it appears that another cycle, perhaps with a period of around 4.5 years may be affecting the tides at Newlyn... but more on that in a future post....