What influences the tides in Croatia? - 14 April 2023

Skip if you are allergic to mathematics :-)

In the previous post, I presented the tide chart for Rovinj for the 17 days starting from 17 April as shown on previous post.

 

In an attempt to better understand the interactions that result in such an interesting mixed tidal pattern (semi-diurnal and diurnal periods alternating), I digitized the 17 day series in Excel so as to study it further...

Digitised tide level forecast in Rovinj, 17 days from 17 April 2023

So what factors were found to contribute to the pattern?

I used spectral analysis to identify the patterns. This is a method often used in epidemiologic surveillance to set alert thresholds for communicable diseases that have a seasonal pattern. The example below shows diarrhea in France, with a fitted model over the number of cases by week (blue dashed line). The method consists of fitting one or more sinusoidal curves to a dataset using the least squares method (minimizing the sum of the squares of the differences between the data and the model using the Excel solver tool) and in turn obtaining the characteristics of the seasonality: when it peaks and what is the magnitude of the seasonal variation.

The example below is fairly simple, considering a single seasonality.

When it comes to tides, the situation is more complex, given the interactions of several independent parameters presented in the previous posts. I used the same approach as for seasonality, using a series of hourly sinusoids until I found a combination that fit the data correctly.

At the end of the exploration, I was left with the following sine curves:

• A yearly sine curve, i.e. repeating after 8760 hours (365 days * 24 hours)
• A daily curve of 24 hours and a half daily curve of 12 hours (first harmonic)
• A half lunar daily curve of 12.42 hours (the moon takes 24.84 hours to return to the same position over the earth). The choice of the half lunar day is because the moon attracts the water when it is over the sea, but as a result it also raises the water when it is opposite (see the animation on the NOAA site).

The picture below shows the model generated by the combination of these curves over the tides in Rovinj. As you can see, it fits pretty well with the model that "explains" 95% (r square statistic) of what drives the value in the tide levels. It explains the observed mixed tidal pattern, when the effect of the moon and the sun combine into a single tidal cycle in one day. It is normal for a model not to explain 100% of the variation in sea level because other factors, not necessarily cyclical, affect the magnitude of the tides.

On the graph below, we can see the different components that affect sea level.

• The blue line (calendar year) shows an increase of a few centimeters in the pattern of sea level in April.
• The green line (moon day) shows that the rotation of the moon around the earth has an effect of about 50 cm (-25 to +25). 
•  The red and violin lines represent the Sun's contribution in terms of gravity, heat, and pressure. Since the height of the tide tends to be higher in the evening and at night than during the day, it is necessary to use the harmonic of 12 hours, which increases during the night, when the peaks coincide, and decreases during the day, when the peaks are in opposition.

In short, it confirms that in a sea like the Adriatic, which is small in size, the daily cycles related to the Sun's effect on the Earth contribute as much as the Moon's effect (purely gravitational), resulting in this interesting pattern. In the open oceans, the gravitational and centrifugal effects of the Moon and Sun contribute to a semi-diurnal pattern that is only marginally influenced by other forced factors.