Water always finds its own level

"Water always finds its own level." What does this really mean?

According to classical physics, every motion is caused by unbalanced forces, see Newton's laws of motion. So forces are the cause of water flow. The distribution of water determines the shape of the surface. Therefore forces determine the shape of the water surface in equilibrium. As will be demonstrated below on the example of a rotating fish tank, the surface of water is not always flat - water can bend.

Fig 1: Forces acting on a volume element of water

Condition 1: In equilibrium the surface of water is always perpendicular to the sum of all external forces, like gravity $\color{green}{\vec g}$ and centrifugal forces $\color{red}{\vec c}$, acting on every arbitrary small volume of the water. If the water surface is perpendicular to the sum of all external forces, then there are no tangential force components $\color{magenta}{ \vec t }$, and hence water does not flow in any direction.

Condition 2: The surface of water is always continuous.

Condition 3: The surface of water in equilibrium is always a surface of equal potential energy, no matter what shape the surface has.

If however the surface is not perpendicular to the sum of all external forces, like in the middle of Fig 1, then there will be an unbalanced force component $\color{magenta}{ \vec t }$ tangential to the water surface and the water will accelerate in the direction of this component, until condition 1 is reached.

Water molecules at the surface are always supported by the water molecules from below, see $\color{blue}{\vec s}$ vectors (s for support). Due to intermolecular forces the support forces are always exactly as big that all force components perpendicular to the surface cancel each other. That's why there is no motion in the direction vertical to the surface, if we ignore turbulences.

If the surface is perpendicular to the sum of all external forces, gravity $\color{green}{\vec g}$ in Fig 1, all force components cancel each other. So there is no net force component in any direction, which means the water molecules don't move in any direction. No net force, no acceleration according to Newton's $\vec a = \Sigma \vec F / m$; the water is in equilibrium.

If the external forces are not parallel everywhere, like on Earth, where plumb lines at different locations are not parallel to each other, then the water surface can not be flat, but depends on the distribution of this force vectors. Level in this context does not mean flat. It means the surface of water lies on the same level of potential or potential energy. [1] [2] [3]

Note, in reality there is an additional force acting perpendicular onto the surface of water from air pressure. This force is always exactly canceled by part of the support force $\color{blue}{\vec s}$. It can be shown that homogeneous air pressure has no effect on the surface of water, so we can ignore it here for simplicity.

See How Water finds it's own Level farther down for a more detailed description.

Rotating Fish Tank proves that Gravity exists

We can use a rotating fish tank partially filled with water to demonstrate that gravity exists and is an acceleration like the centrifugal acceleration. Note: accelerations and forces are related by F = m · a, so we can convert between forces and accelerations accordingly if neccessary. In the following descriptions I use accelerations rather than forces.

Fig 2: Accelerations determining the water surface

When the fish tank rotates, there are 2 external accelerations acting at each point in space: the gravitatioal acceleration g and the centrifugal acceleration c. Accelerations are vector quantities. They have a direction and a magnitude. Multiple accelerations acting on the same point can be added vectorial to get a single resulting acceleration a. Objects react to the sum of all accelerations (or forces). This is true also for any small volume of a liquid.

The following image shows a side view of a fish tank with some water in equilibrium, rotating once in 1 second. We can clearly see that the surface of the water is curved. That must mean, that the sum of all external accelerations a changes direction from point to point along the water surface. We can even infer the direction of the acceleration a at every point. If water is in equilibrium, it must act perpendicular to the surface everywhere, see How Water finds it's own Level.

Parabolic Water Surface in a rotating Fish Tank at 1 Rotation per Second

Source: Direct Measurement Video of a Rotating Water Tank: Purple Hexane and Parabolic Fish; by Peter Bohacek; on Youtube

Fig 2: Accelerations determining the water surface

The centrifugal acceleration c points always perpendicular away from the axis of rotation. In this experiment it acts everywhere horizontal away from the Z axis. But if water is always perpendicular to the sum of all external accelerations a and there would be no downward acceleration g, then the sum of all accelerations a would be equal to the centrifugal acceleration c and the water surface would not have the shape we see in the image. The water would cling vertically to the outer walls of the container. This can be demonstrated in zero G environments. [4] [5]

So because rotating water behaves different in zero G than sitting on earth, there has to be an additional downward acceleration g when not in zero G.

Because only vectors of the same kind (e.g. accelerations) can be added and there must be a downward acceleration g added to the centrifugal acceleration c to get the total external acceleration a that determines the slope of the water we observe, the shape of the water surface in this experiment proves that such a downward acceleration exists. We call this acceleration Gravity.

Equation for the Surface of the rotating Water

From the conditions mentioned at Water always finds its own level we can derive an equation for the surface of the water, taking gravity into account. Then we can calculate the height of the water for any location, and test, whether the prediction of the equation matches the experiment.

We get the following quadratic equation. This means that the surface of the water in the tank builds a paraboloid.

Note: if we set the rotation $\omega = 0$ then we get $z(x) = z_0$, which means the surface height is the same everywhere. On the other hand, if we set gravity $g = 0$, we get a division by zero, which means that this equations is not valid for zero gravity. This means that without gravity the shape of the water surface would not be a paraboloid. And because a flat plane is a special case of a paraboloid, without gravity the surface of water would not be flat either.

Prediction and Reality

Lets predict what the water height is at 24 cm from the center of rotation. Lets set z₀ = 0 so we can use the grid in the image to make measurements. We know that the tank in the experiment rotated once every second, so f = 1/s.

So our equation, based on gravity and centrifugal acceleration, predicts for the experiment in the image, that at 24 cm from the axis of rotation the water height will be 11.6 cm above the water height at the rotation axis (the lowest horizontal grid line). That's exactly what we see in the image!

This confirms that equation (1), which contains a gravity term g, is correct. If gravity would not exists, like Flat Earthers often claim, we could not derive an equation that correctly describes the shape of the water surface in the rotating fish tank. So obviously gravity exists.

But lets test the equation for the whole surface using a model:

Model of the Rotating Water Surface

The following interactive model can be used to predict the surface of rotating water depending on some parameters. To predict the water height for the example above, set Revolution = 1 s⁻¹, Gravity = 9.806 m/s² and X = 24 cm and read the absolute water height at that position at z, or the water height relative to z₀ at z−z₀ = 11.6 cm.

Note: the Gravity slider snaps to 1.621 m/s², which is the gravitational acceleration on the moon, and to 9.806 m/s², the mean gravitational acceleration on earth.

Measuring Gravity

We can calculate the magnitude of the downward acceleration by measuring the slope of the water at a distance r from the center of rotation and calculating the centrifugal acceleration $c = \omega^2 r$ at that location. I measured a slope angle at r = 22 cm of about φ = 41.5°. The magnitude g of the downward vector can be calculated using some trigonometry:

The result is very close to 9.81 m/s², which is the mean value of the gravitational acceleration at the surface of the earth. We can repeat this measurement for all points of the surface. It will show, that the downward acceleration g is the same everywhere. So a rotating fish tank is another method to measure the gravitational acceleration of Earth.

How Water finds it's own Level

Fig 3: Forces acting on a volume element of water

Forces are vector quantities. Vectors can always be decomposed into components acting in different directions. The vector sum of the components must always be the original vector. That is on the slope in Fig 3 the vector sum of the components $\color{red}{ \vec p }$ and $\color{magenta}{ \vec t }$ is the gravity vector $\color{darkgreen}{ \vec g }$.

Note: If we decompose the gravity vector $\color{green}{\vec g}$ on the horizontal part into a tangential and a perpendicular component, the tangential component $\color{magenta}{\vec t}$ is zero and the perpendicular component $\color{red}{\vec p}$ is equal the gravity vector $\color{green}{\vec g}$.

Level: Lets regard a small volume of water at the horizontal part of the surface. It is pulled down by gravity $\color{green}{\vec g}$ and supported from below $\color{blue}{\vec s}$. The tangential force component $\color{magenta}{\vec t}$ is zero. So in equilibrium there is only the force of gravity $\color{green}{\vec g}$ and the support force from below $\color{blue}{\vec s}$ which cancel each other, so the water volume stays at rest.

Slope: When we decompose the gravity vector $\color{green}{\vec g}$ into a component tangential to the sloped surface $\color{magenta}{ \vec t }$ and a perpendicular component $\color{red}{ \vec p }$ we find, on a slope the tangential force component $\color{magenta}{\vec t}$ is not zero anymore.

The unbalanced tangential force component $\color{magenta}{\vec t}$ causes the water molecules to accelerate in the direction of the vector. The perpendicular component $\color{red}{\vec p}$ is balanced by the support force $\color{blue}{\vec s}$ from below, so our molecule can only flow along the surface. The water on the slope starts to flow towards the lower level. The lower level raises, the higher level sinks until the surface of the water is perpendicular to the sum of all external forces at every point so that there are no tangential force components left anywhere.

That's the mechanism of how water finds it's level.

Level

Fig 4: Blue lines are Equipotential Levels

In Physics and Geodesy a Level Surface is a surface that is continuous and perpendicular to the sum of all external forces or accelerations at every location of the surface. [3] We can describe a force field in terms of a scalar Potential Field, a field describing the potential energy at every point in space. The force vector at any point is then the negative Gradient of the potential energy field. That is the force vector points in the direction of the biggest change in potential energy and the magnitude is proportional to this change. Points of equal potential energy build a continuous surface, called a Level Surface.

In Fig 4 we see a couple of equipotential surfaces for the rotating fish tank. Water in equilibrium occupies such a Level of equal potential, because the tangential components $\color{magenta}{\vec t}$ of the total external accelerations a on a specific level are always zero. Hence water occupying the same level does not flow in any direction. It has "found it's own Level".

A Level surface is not necessarily flat, as Fig 4 shows. A level surface can have any shape, depending on the forces acting at each point in space. A level surface is only flat in the special case of a force field with only parallel force vectors. In the rotating fish tank example, every Level is a parabolic shaped surface.

There is an infinite amount of Level surfaces of equal potential. Such Levels do never cross each other and are continuous. Depending on the amount of water present, the water surface in equilibrium occupies exactly one of this Levels.

Only if the earth were non-rotating and all gravity vectors were exactly parallel to each other, all Level surfaces would be flat. Using theodolites it can be measured (simultaneous reciprocal zenith angle measurements), that plumb lines are not parallel but angle 1° every 111.2 km. So gravity vectors are not parallel. Hence earth's Level surfaces are not flat planes but approximately ellipsoids. The particular level surface that would be occupied by the oceans, if there were no additional forces like wind and tides, is called Mean Sea Level and has the shape of the Geoid.

Force Fields

If we can assign a force to every point in space, we call this a force field. A force field has a magnitude and direction at every point in space. This force field defines Level surfaces and hence the surface of water at equilibrium. There are commonly multiple force fields acting at the same time at every location. They can be added vectorial to a total force field that defines how objects move via Newton's second law $F = m \cdot a$, or solved for the acceleration a and multiple dimentions and multiple forces: $\vec a = \Sigma \vec F / m$.

In our fish tank example one force field is the gravitational force field g, the other is the centerifugal force field c caused by the rotation of the fish tank. Together they form a force field a that has an infinite set of Equipotential Level surfaces with parabolic shapes.

Blue Arrows: Water flow due to tangential force components of the tidal forces of moon+sun (black arrows)

The force field of the Globe Earth is the sum of the force field due to earth's attraction plus the force field of the centrifugal acceleration due to earth's rotation. In the attractive force field all force vectors point towards the center of the earth. In the centrifugal force field all vectors point perpendicular away from the axis of rotation. Ellipsoids are the only equipotential Level surfaces where all tangential force components of the combined field vanish everywhere.

Additional force fields from sun and moon modulate earth's force field and cause tides, explained by the same mechanism. Not by pulling on the water surface (the pull is way too small to lift water), but by creating regions where the tangential force components are not zero so the water flows in the corresponding directions, creating the tides.

If mathematic is not of interest to you, you can stop reading here. Below I will derive the equation of the parabolic Level surfaces and the Potential Field. To understand this fully, you need a basic understanding of Calculus and some Vector Algebra.

Derivation of the Equation

Because the forces are symmetric to the axis of rotation of the fish tank, we can either use a polar coordinate system with the coordinates (r, θ, z) or a cartesian coordinate system with the coordinates (x, y, z) with origin at the bottom of the fish tank at the axis of rotation. The Level height z is then a function of the other coordinates. Because the fish tank has a rectangular bottom, I chose cartesian coordinates.

Due to the rotational symmetry we can simplify the calculations and only calculate the cross section parabola in the X/Z plane and later extend the parabola to a 3D paraboloid.

The way to find the shape of the parabola is as follows:

We can calculate the sum of the accelerations acting at every point in space. Because a Level surface is always perpendicular to the sum of the accelerations (or forces) and we can calculate the direction of the vector sum at any location, we know the slope of the Level at any location. So we can express the slope of the Level surfaces for every point in space or in the X/Z plane as a function of the x coordinate.

But we don't want to know the slope of the Level surface, but the height z of a Level surface as a function of x. Now the slope is equal to the derivative of the unknown function. So if we integreate the slope function, we get the function for all possible Level surfaces. Then the amount of water in the tank determines the Level surface we can observe.

So lets calculate the acceleration field in the X/Z plane.

The sum of the accelerations is then:

We need the direction of the slope. The acceleration $\vec a$ is perpendicular to the slope. So we get the slope direction by rotating the acceleration vector 90° counter clockwise. A 90° vector rotation can be achieved in 2D by exchanging the components and multiplying the first component of the result by -1:

The vector $\vec t$ has now the direction of the slope of our unknown function. As this direction in the X/Z plane is the same as the direction of the slope of our unknown function and the slope is the the derivative of z with respect to x, we can write:

We can see that the slope is a linear function of the distance x from the center of rotation with a proportionality factor of $\omega^2/g$.

If we integrate the derivative we get our function for the Level surface in the X/Z plane:

This integral is simple to solve:

z₀ is determined by the amount of water and the size of the container. z₀ is the height of the water at the axis of rotation. It's the lowest height of the water surface. But it is only defined, if there is enough water in the tank, so that the parabolic shape is not clipped by the bottom of the container.

How to extend the parabola to the 3D parabolic shape of the Level surface and how to calculatue the value z₀, see Calculating the Parabolic Water Surface.

Potential Field

To finish I show the potential field $\phi(x,z)$ in the coordinate system of the rotating fish tank. The potential field is the potential energy V per unit mass at every point in space: $\phi(x,z) = V(x,z) / m$. The potential field can be calculated from the acceleration field by integration.

This yields:

Gravitational potential is a relative value with respect to a certain arbitrary potential $\phi_0$. Therefore we can add $\phi_0$ to the equation without changing the corresponding force field. We assumed the gravitational acceleration g as approximately constant along the height of the fish tank.

The acceleration field $\vec a$ is the negative gradient of the potential field which is also the force field $\vec F$ per unit mass:

This is just the acceleration we have calculated at (5). The nabla symbol $\nabla$ is the gradient operator which means we have to take the partial derivatives of the operand $\phi$ and each derivative yields one component of the acceleration vector.

Lets test whether the potential along a level surface is really constant. For that we insert in (11) for z the function (9) for the paraboloid surface we have found:

So the potential on a parabolic surface is:

The symbols g, z₀ and $\phi_0$ are all constants. So we see that along the parabolic surface the potential is constant, the surface is at the same potential Level, it is Level but NOT Flat. This means there are no tangential force components along this surface and water will be in equilibrium on such a surface.

Refecences

[1]

Geopotential; Wikipedia
Gravity is defined as the resultant force of gravitation and the centrifugal force caused by the Earth's rotation. Likewise, the respective scalar Potentials can be added to form an effective potential called the Geopotential. Global mean sea surface is close to one of the isosurfaces of the geopotential. This equipotential surface, or surface of constant geopotential, is called the Geoid.
https://en.wikipedia.org/wiki/Geopotential

[2]

Geoid; Wikipedia
Being an equipotential surface, the geoid is a surface to which the force of gravity is everywhere perpendicular. That means that when traveling by ship, one does not notice the undulations of the geoid; the local vertical (plumb line) is always perpendicular to the geoid and the local horizon tangential to it. Likewise, spirit levels will always be parallel to the geoid.
https://en.wikipedia.org/wiki/Geoid

[3][a][b]

Definitions (Physical Geodesy); What-When-How; Online resource for In depth Information and Tutorials on many topics
A level surface is a continuous surface that is always perpendicular to the local plumb line. Due to the Earth's curvature and variations of density within the Earth, the direction of the plumb line changes as one moves from point to point on or near the surface of the Earth. The Geoid is an equipotential surface most closely represented by mean sea level in equilibrium all over the world.
http://what-when-how.com/the-3-d-global-spatial-data-model/definitions-physical-geodesy-the-3-d-global-spatial-data-model/

[4]

What Happens to a Whirlpool in ZERO-G?; The Action Lab; Youtube
https://www.youtube.com/watch?v=qCzf2VdpikQ

[5]

Having Fun in Space!
Rotating water bottle on the ISS
https://youtu.be/Otb_IHfMj_w?t=85