Tides explained

On the origin of a global ocean phenomenon

What are Tides?

The general definition of the term tide according to Morrison and Owen [3] is that:

"A tide is a distortion in the shape of one body induced by the gravitational pull of another nearby object".

Specifically on Earth the term tide is also used to describe the rise and fall of sea levels caused mainly by the combined effects of the gravitational forces exerted by the Moon and the Sun. This is a bit misleading since focusing on the sea level is ignoring the fact that tides also affect the solid earth crust. Moreover the observable rise and fall of the sea level is influenced strongly by shoreline topography, ocean currents and the distribution of the continents on earth. For now We will investigate the tides based on the definition given by Morrison and Owen and ignore the effects of topography and ocean currents that lead to more complex tidal cycles on Earth. These effects will be explained later in the section Tidal Cycles on Earth.

This web page is based partially on another article about the "Computation of Tides" by Paolo Sirtoli [1] as well as the article "Tidal Misconceptions" by Donald E. Simanek [4]. I will extend the ideas brought forward by those two articles with up to date visualization. We will start be first looking into the gravitational pull that the Moon is exerting on the Earth.

An image of the Bay of Fundy at high and low tide
The port of Alma, New Brunswick at high and low tide by Samuel Wantman [2].

Gravitational pull of the Moon

The theoretical basis for the computation of the tidal effects caused by Moon and Sun is the same. For simplicity we will first examine the effect of the Moon. We will also ignore Earths rotation in our models since the rotation of Earth has no influence on the existence of the tidal bulges. So lets start by looking at the Earth, the Moon and the law of gravity.

Newtonian Law of Gravity: $$\vec F_{1} = \gamma \cdot m_1 \cdot m_2 \cdot { { \vec r_2 - \vec r_1 } \over |\vec r_2 - \vec r_1|^3} = - \vec F_2$$ where:
  • \(\vec F_1\) and \(\vec F_2\) are the forces acting on the two masses
  • \(\vec r_1\) and \(\vec r_2\) are the position vectors of the two masses
  • \(m_1\) and \(m_2\) are the masses in \(kg\)
  • \(\gamma\) is the gravitational constant ( \(6.674 * 10^{11} { m^3 \over {kg*s^2} }\) )

At first we will exclusively look at the acceleration caused by the gravity of the Moon on objects lying at the surface of Earth. For the moment we also assume both Earth and Moon are pinned down in space and not orbiting one another. By substituting the force in Equation 1 with Newtons second law of motion (\(\vec F = m \cdot \vec a\)). We can quickly rearrange this equation to calculate the acceleration of a body lying on the earths surface due to the gravitational pull of the moon (Equation 2):

$$\vec a(\vec r) = \gamma \cdot m_m \cdot { { \vec r_m - \vec r } \over |\vec r_m - \vec r|^3}$$ Where:

By doing this for a variety of points at the Earths surface as well for its center of mass we get the following result:

Computing the acceleration caused by the gravity of the moon. The Moon is drawn unrealistically close to earth. Use the mouse to drag the moon and investigate the acceleration vectors.

The sizes of Earth and Moon in the image above are to scale. You may use the button to toggle between a visualization that is correct with respect to distance and one that shows the moon unrealistically close to Earth. In latter setup you can grab the moon with the mouse and move it around to see how the acceleration vectors change. For the sake of completeness i have also added a marker to denote the common center of mass of the system Earth-Moon. There are a few things you will notice:

In the visualization above we focus exclusively on the gravitational forces caused by the moon. Earths own gravity is ignored. If you do the math you will realize that the acceleration caused by the moon is by orders of a magnitude lower that the acceleration caused by earths own gravity (3.3e-5 m/s² vs. 9.8 m/s²). It is also slightly lower on the side of earth opposing the moon.

This concludes our look into the gravitational pull Moon is exerting onto Earth. In the next chapter we will look into the motion of the Earth and Moon as they circle one another and what role this motion plays in the computation of tidal accelerations.

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