When is the sea warm enough for our swimming club? A sinusoidal model of sea temperature

I swim with a youth open-water club that trains in the pool all year and in the sea in summer. Our coach has two rules: the club only runs sea sessions when the sea is at least 14 °C, and juniors may swim without wetsuits only when it is at least 16 °C. Every year the committee books sea sessions and lifeguards for fixed calendar dates — this year 1 June to 30 September — and every year some sessions are cancelled in early June because the water is too cold, while in October, when the sea still feels warm, we are back in the pool. I offered to use the sea temperature records to suggest better dates for next season.

Sea temperature obviously goes up and down every year, which is why I expected a periodic model to fit. What I did not know was how well a simple sine or cosine curve would describe it, and how much I could trust dates read off a curve.

Aim. I will (1) build a sinusoidal model for the weekly sea temperature at our beach from four years of records, first from the maxima and minima and then by linear regression, (2) test the model on a fifth year that was not used to build it, and (3) use it to predict when the sea is at least 14 °C and at least 16 °C, and decide how far those dates can be trusted before I recommend them to the committee.

A coastal monitoring station on the pier about 400 m from where we swim publishes the sea surface temperature every half hour. I downloaded five years of readings (2020–2024) and used a spreadsheet to calculate the mean for each week, numbering the weeks \(k = 1, 2, \ldots, 52\) from the first week of each year (I left out the few days of the 53rd, incomplete week).

Data note: the temperatures in this exemplar are illustrative, simulated by IB Math Revision to behave like real coastal records; a student must use real records for their own beach and cite them.

I used weekly means because the half-hourly values change with the tide and the time of day, which does not matter for planning a season. I kept 2024 aside and did not look at it closely until Section 6, so that I could test the model on data it had not seen.

The pattern repeats every year: coldest in late winter, warmest in late summer. Plotting the four years against the week of the year (Figure 2) shows that the years lie almost on top of each other, which suggests one model for all of them.

In AI SL a sinusoidal model has the form \(T = a\cos(bk) + d\), where \(a\) is the amplitude, \(d\) is the principal axis and the period is \(\frac{360}{b}\) (in degrees). The cosine form has its maximum when \(bk = 0\), but our maximum is in late summer, not at \(k = 0\). Instead of shifting the graph, I measured time from the warmest week: \(t = k - t_0\), where \(t_0\) is the week of the maximum, so that \(T = a\cos(bt) + d\) has its maximum at \(t = 0\).

Period. One year is \(365.25 \div 7 \approx 52.18\) weeks, so \(b = \frac{360}{52.18} \approx 6.899\) degrees per week. As a check on the shape, a cosine curve takes exactly half a period (26.1 weeks) to rise from its minimum to its maximum. In the data the rise took on average 23.5 weeks and the fall 28.7 weeks — the sea warms up faster than it cools down. I noted this as a possible weakness of any cosine model and came back to it in Section 5.

Amplitude and axis. From Table 1, \(a = \frac{16.93 - 7.65}{2} \approx 4.64\) °C and \(d = \frac{16.93 + 7.65}{2} \approx 12.29\) °C, and the maximum came on average in week \(t_0 \approx 33.25\). So Model 1 is

\[ T \approx 4.64\cos(6.899t) + 12.29, \quad \text{where } t = k - 33.25. \]

To compare models I used the mean absolute error (MAE) — the average size of the differences between the recorded and predicted temperatures, ignoring sign. Model 1 has an MAE of 0.68 °C over the 208 weeks. That is already small, but when I drew it over the data, Model 1 peaked a little too early and too high. The trouble is that it is built from only eight weekly values out of 208: the week of a maximum is easily moved by one unusually warm week — in 2021 the highest value was reached in both week 31 and week 34.

A sinusoidal model \(T \approx 4.38\cos(6.899t) + 12.29\), with \(t\) measured in weeks from week 35.0, fitted by turning it into a linear regression, describes the weekly sea temperature at our beach with a mean absolute error of about 0.3 °C, and it predicted 2024, a year it had not seen, with a similar error. Using it, the sea is at least 14 °C from about week 25 to week 44 (using the bias-corrected end of the season, which the records support), so the club's season should start later and finish later than it does now; the 16 °C season is short and varies a lot from year to year.

The main limitation is the shape of the model: a cosine curve is symmetrical and the sea is not — it warms faster than it cools — which shows up as a double wave in the residuals and is the reason I checked whether the dates needed correcting. Other limitations are that the station is 400 m from where we swim, and shallow water near the beach may be warmer on sunny days; that weekly means hide cold days after storms, which are exactly the days sessions get cancelled; and that four annual means (12.28, 12.04, 12.31, 12.58 °C) are far too few to judge whether the sea is getting warmer, which would move every date.

If I did this again, I would put a thermometer in the water at our beach for a summer and compare it with the pier station, and I would model spring and autumn separately. What I learned most is that a model that fits well overall can still be wrong in exactly the place you need it — in my case, the start and end of the season.