zaterdag 23 augustus 2014

The tale of the two tides

Last week, I was sitting on the beach looking at the "Oosterscheldekering", one of the largest engineering constructions on Earth. It was designed to protect Zeeland from flooding during large storms and extreme high tides. As a boy I was always remembered of the power of the water (my complete family is from the island Schouwen-Duiveland, where the storm of 1953 hit hard). It gave me a sense of aw and pride, that engineers designed and build these large constructions. Maybe even, it gave me motivation to go into engineering.

As I was admiring the view, I was thinking about tides and their cause. One of the most common questions about tides is: Why are there two high (and low) tides a day? If tides are caused by the Moon, due to its gravitational attraction, there should only be one tide a day, because the Earth rotates only ones a day, right? The answer to this problem is: Correct Frame of Reference!

Since we, humans, know the Earth is round, we tend to place our point of origin in the center of the Earth. We can stand on the surface of the Earth, because gravity is pulling us (and sea water) towards this center of mass. This is why you do not fall off at the other side of the globe (yes even in Australia, everything falls down, I tested this myself!). This same force, together with the angular momentum of the Moon, keeps the Moon rotating around the Earth.

The Moon also has its own gravitational attraction, pulling other mass particles towards it, even our own Earth. If we look at a mass particle at an arbitrary location around the Earth, it experiences several forces. The gravitational attraction of the Earth is counteracted by the normal forces of the surface (me) or centrifugal force due to rotating motion (our mass particle). Another force is the gravitational attraction of the Moon (red arrow). However, both the Earth and the particle feel this force. This is where the correct frame of reference is coming into play.

The Earth is pulled a little bit, which means that the center of mass shifts a little bit towards the direction of the Moon. But as I told you, we like to think that our point of origin is at the center of the Earth. We live on the Earth, that is our frame of reference, that is how we experience tides.  Therefore, we (mathematically) have to correct for this by adding a resulting force (green arrow) to the Earth and any other mass particle (you, sea water, or our mass particle in an arbitrary location), same in magnitude as the gravitational attraction of the Moon exerting on the Earth, but in the opposite direction. This correction sets us back in the reference frame that we like, Earth centered.

This correction introduces an additional force on our mass particle (and on you and sea water, you get the point). With elementary vector addition (black striped lines), we obtain the true tide force (black arrow) that the particle experienced, as seen in the reference frame we live in, Earth centered. Theoretically we can do this for any particle, so also for the sea water on the surface of the Earth. The following sea level curve (which should be an ellipsoid, but my paint skills are not excellent) is observed due to the attraction of the Moon:

As you can see there are two bulges, instead of one. Lets take a better look at both locations. At location 1 (closest to the Moon) the gravitational attraction of the Moon is slightly larger than that at the center of the Earth. Gravitational attraction looses strength quite fast with distance (this is a good thing, otherwise black holes would be devastating and walking on the Moon, even more difficult). This means that the resulting tidal force is towards the Moon, as we all expected. Location 2 is experiencing the opposite, the gravitational attraction at the center of the Earth is larger than the force exerted on the sea water. This results in a tidal force, similar in magnitude as location 1, but in opposite direction, away from the Moon. However, Living on the surface of the Earth (as we all do), both tidal forces look similar, so therefore we experience two tides a day. 

Of course, the complete problem is much more elaborated and complex, but than you have to follow courses like Planetary Sciences, Astrodynamics or even Satellite Orbit Determination. Or post a comment on this blog and I will try to see if I can explain some more.

donderdag 14 augustus 2014

Gravity Expeditions at Sea: Promotion movie

Several blogpost are about this historic project I am working on. We, together with people of the library of the TUDelft, are describing the work and voyages of one of the most adventurous scientist and professor of my university, Professor Vening Meinesz. In the beginning of previous century, professor Vening Meinesz measured Earth's gravity field onboard several submarines of the Dutch Navy. Me, being a geoscientist with experience in gravity field modelling, was asked to explain his work and relate it to Solid Earth Science. Diving into his work and stories, I became very enthusiastic and motivated for the project.

We are in the middle of the project and are asking people to help us in any way they can. For this purpose (and because we live in a media-type world), we have made a promotional video. And I wanted to share this first version of the film. Please be aware! You will see me talking science :). If you like the movie and the project, please share it among your friends. Maybe we find somebody, that can help us in our quest:

dinsdag 5 augustus 2014

The Waddensea Experiment: combining sailing and science

Last week, I read in the papers that the Danish part of the Waddenzee is also put on the World Heritage List. This means that the complete Waddenzee (Dutch, German and Danish) is a protected nature site. Me, as a strong admirer of this region, makes this news happy and content! It also gave me new inspiration for a blogpost. In this blogpost I will look at an experiment that I did on the Waddenzee. At the end I will prove that the Earth is round (or more precisely, not flat).

The Waddensea is an area north of Holland, north-west of Germany and west of Denmark. when inspecting any atlas or google Earth, you can spot a row of islands from west to north-east. In between these islands and the mainland is the Waddensea, with all its beauty! 

The Waddensea is the area between the mainland and the small islands. The Dutch, German and Danish part can be seen.
Special about this region is the large influence of Earth's tides on the landscape and nature, the Waddensea is a very shallow sea. At high tide, the area is completely covered with water, but at low tide, large parts of the area become dry. This creates a very special eco-system thriving with life, see Wiki (The English page is quite short, but if you can read Dutch go to that page for more information). I know the Waddensea mostly onboard a "platbodem", or flat-bottom sailing boat. This is the best way of visiting this area and really get to experience the nature, people and water. Speciality of this ship is that at low-tide it can rest on the ocean floor, without tipping over. You just have to wait 6 hours, before continuing your journey.

We made the trip with the platbodem "de Overwinning".
Onboard one of these voyages I did my first GPS and navigation experiments. I just got my Garmin GPS receiver and I wanted to play with it. So, during a trip between two islands (from Ameland to Vlieland) I turned on the Garmin receiver and recorded the data. It was quite a nice trip, because at the end the wind started to increase and the captain put out all his sails to maximise the speed. We will see this later in the data. The track we sailed is shown in the next figure:

During this voyage, I challenged the "schipper" (captain) to go as fast as possible, because he was all the time bragging about the speed of his vessel. So, he put on all the sails he got and we flew over the water. I experienced this at the front of the ship, where I was responsible for the Fok. The speed and excitement is felt best at this location. The GPS receiver was turned on and recorded the whole trip, also the velocity of the ship.

Recorded velocity of the ship
As you can see, our averaged velocity was 6 knots, but at a certain point we touched the 11 knots (set-up of the many sails). We couldn't maintain this velocity, because our destination port was close. However, the trip was a good test-run for my GPS receiver. That night, enjoying a proper "schippersbitter" (sailing booze), I post-processed the data. One of my goals was to see how accurate the velocity was that was given by the GPS receiver. To find out, I used the more frequent position measurements, which I numerically differentiated. As we have learned in high school and again on university, the derivative of position/path/length/voyage is velocity. This is what I obtained:

First try at computing the velocity of the ship. Blue line is the recorded velocity. Red line is the computed velocity.
What was this about? My computed velocity was much higher then the GPS receiver presented. What did I do wrong? Because of a storm, we could not sail out. We had to stay in the harbour, which gave me time to explore the island (Vlieland, the second island from the west, see figure 1 and 3) and to clear my head. While enjoying the scenery of violent waves and stormy weather, I found my bug/error/stupidity. After getting back to the ship, grabbing a hot coffee to heat up, I jumped behind my computer and changed my code. "The Earth is round!", I stated. "Yes", said the captain, "it will be like that for some years, they say!".

I miscalculated the length of our voyage by using math for a flat Earth. I told my software that latitude was y and longitude was x, before calculating the length between two points by the following equation:

s = sqrt{(x2-x1)^2 + (y2-y1)^2}]

Which is correct on a flat surface, but not on the round Earth. Luckily, I was following a course on spherical geometry, which enabled me to calculate the distance between two points on a spherical surface.

s = 111.317*rad2deg(acos(cos(lat(2))*cos(lat(1)) + sin(lat(2))*sin(lat(1))*cos(lon(2)-lon(1))))

(Its less complicated than it looks). Inserting this relation in my code, for the previous one would give me the correct result.

The correct computed velocity in red overlies the recorded velocity in blue.
You can clearly see that the computed velocity has a higher time resolution, due to the high resolution in position recordings. Nevertheless, I found that the presented velocity of the Garmin receiver was as accurate as the position estimates. But more important, I rediscovered that the Earth is round and will be for many years to come (because, a "schipper" never lies).