Chapter 38: Coriolis and the Euler Acceleration

There are two accelerations other than the Centrifugal force created by the motion of particles within rotating systems: Coriolis and the Euler acceleration.

Coriolis is, as mentioned previously, one of those ‘fictitious forces’ that physicists don’t like to recognize but that plays a very important role in the actual happenings on our planet and in our universe. They say that in a non-rotating, inertial system, coriolis does not truly exist, or need to be accounted for as a part of the system forces that control the interaction of the objects within them. Balderdash.

First, there are no ‘non-rotating inertial systems’. Think this through; this ‘non rotating inertial system’ concept is a fallacy, a feint, a mirage. Inertial systems are systems that, by definition, have mass, like stars and planets and things like that. They aren’t flat (we discussed that), and furthermore, they are all rotating. All of them are. There is no evidence that there is any body or construct in the entire known universe that is not rotating; so to define a perspective that is not rotating is to start out by saying that “first, we will define a position that is not related to anyplace in our perceived reality”, and then draw our conclusions from that perspective.

Coriolis is thought to be responsible for initiating the motion that characterizes hurricanes, but for different reasons than shown in the illustration in the previous chapter. Assuming you have created a low pressure area, normally as a result of local heating, there is a difference in the density of the atmosphere between the heated and non-heated areas and a pressure gradient has been created. As the air from the surrounding atmosphere travels toward that low pressure area to equalize the pressure difference, it is deflected away from its path to the center of the low pressure by a force (Coriolis) that is perpendicular to that path, forcing the flow of air into the characteristic logarithmic spiral pattern.  Hurricanes swirl and every sufficiently sizeable wind velocity must swirl because the spinning of the Earth requires that you must operate under the rules of the Spiral Dimension wherein Coriolis exists and affects the interactions within the system.

When objects are moving within a rotating system the Coriolis effect acts in several ways. First, and most importantly for earthly dynamics, it causes objects within the system to counter rotate. That is, if you have a frictionless object that is within a rotating system, it will not stay in one place, even if the forces within it, namely gravity and the centrifugal force, are perfectly balanced. An object placed on a ‘flat’ field (meaning flat, force-wise) will rotate within what is referred to as an ‘Inertial Circle’, the radius of which, in the case of the Earth, is determined by the force of gravity and the speed of its rotation. These Inertial Circles are the result of the effects of Coriolis and are believed to be responsible for the direction and form of the ocean’s currents and the atmospheric  jet stream. It should be noted that these Inertial Circles rotate in opposition to the direction of rotation seen in hurricanes.

So, in the Spiral Dimension, there is no such thing as ‘static’.

The second thing is (and many discussions regarding the topic dismiss this aspect), that Coriolis has an out of plane component that is different from the Euler Force.

The Euler force is named after its historical discoverer Leonhard Euler, a brilliant Swiss mathematician and physicist who lived during the 18th century. Unlike Coriolis, the Euler force acts in the plane of the rotation. In some respects, the Euler force can be seen as a planar phenomenon, and therefore might be more appropriately addressed in the chapter that discusses rotation, But while it does act in the plane of rotation, its effects are really more volumetric than flat, so its discussion has been included here.  A description of the Euler acceleration, and consequently, the force, can be roughly stated as follows (again, from Wikipedia) http://en.wikipedia.org/wiki/Euler_force:

The Euler force will be felt by a person riding a merry-go-round. As the ride starts, the Euler force will be the apparent force pushing the person to the back of the horse, and as the ride comes to a stop, it will be the apparent force pushing the person towards the front of the horse. The Euler force is perpendicular to the centrifugal force and is in the plane of rotation.


Notice that the right side of the equation has a negative sign in front of it. In this case, it means that the resultant acceleration and force is in the opposite direction as the rotation. This is why it is better to launch rocket ships eastward. They weigh less when they are travelling in that direction. And since they weigh less, there can be only one conclusion. In the Spiral Dimension, when one is moving or accelerating in the direction of the system rotation, there is an acceleration vector that in this case, is in opposition to the force of gravity.

The formula for the Coriolis force is:

ac = -2 x v  where: =  * direction and v is the velocity of the object.

Since this formula is a cross product, and the resultant Coriolis acceleration, ac, must have a line of action at right angles to the plane created by the vectors  and v and therefore, for the right combination of those vectors, the result is a force that is out of the rotational plane, and the magnitude of that force is proportional to , the rate of rotation.

Unlike all of the other forces that are the result of the interaction of mass and space, Coriolis creates an out of plane acceleration. It is Coriolis that creates the giant eye wall of the hurricane, and the spout associated with the tornado. Coriolis creates a linear acceleration vector that is perpendicular to the plane of the rotation. The direction of that vector can be controlled by manipulating the orientation of the rotating field and  that acceleration vector can be directed in opposition to the direction of the force of gravity, or in any other direction one desires, really, only dependent on the relative motions of the system.

And because Coriolis is a resultant force created in the Spiral Dimension, it does not obey Newton’s Third Law of Motion, the one that states: ‘for every action there is an equal but opposite reaction’. In the Spiral Dimension, an acceleration in the rotational field creates a perpendicular reaction in a magnitude equal to -2 x v.

There will be those that, while understanding these forces and descriptions, will argue that the Coriolis force is only applicable to the rotating system that generates it, that the forces generated do not affect the larger world outside of that system, and therefore, that the rotating system cannot affect any mass that is outside of that system.

This is a typical argument used regarding rotating systems and their effects.

This is not the case, however. A rotating gyroscope, for example, clearly has more inertia, at least in specific directions than a non-rotating one, as evidenced by the arguments put forth in Chapter 33.

The case for Coriolis is a little less obvious, especially since it is difficult to generate and control, but fortunately, an excellent example of the actual use of this force does exist.

Within the realm of engineering, there has been a requirement developed to automatically regulate the flow of gasses and liquids to very precise levels driven by the needs, especially, of the chemical industry. In the days before the invention of modern electronics, this was accomplished by carefully regulating the pressures of the system and utilizing precise valves and/or small holes called ‘orifices’ to stabilize and regulate those flows. However, should any of the pressures vary, these types of systems cannot automatically adjust, and a small variance in  those inputs will change the flow considerably and alter the amount of fluid or gas transferred significantly.

This all changed with the invention of the Mass Flow Controller (MFC), and it’s twin, the Mass Flow Meter (MFM). The early devices created depended on the ability of the mass passing through the device to cool a heated element, and by measuring the temperature of the fluid as it passed through the device, one can determine the amount of mass that passed through. Although this concept required that one knows the thermal conductivity properties of the substance being metered, the reading obtained from the device can be interpreted after the data has been collected and adjusted accordingly. This information is utilized then, through the use of modern electronics, to control a variable valve to regulate that flow. However, since the measurement must be interpreted by the assumed thermal properties of the substance being measured, which must be known to a high degree of accuracy, some inaccuracy is inherent in this system.

Several years ago, however, a new line of MFC’s and MFM’s was developed that is based on the principle of Coriolis. Devices such as this all share a common concept. In them, the fluid to be measured is sent through a pair of small tubes that are configured with small partial loops, that force the linear flow of the substance to curve around in a short, semicircular path. These curved tubes are made to vibrate at a given frequency, and that oscillation is monitored by a force transducer. What happens is this: the Coriolis force generated by forcing the fluid to pass through the curved tubes alters the vibration frequency of those tubes, causing them to oscillate at a different frequency, and this can be sensed by the force transducers. Since the force created is directly proportional to the mass that passes through the curved section and not the properties of the chemicals, this change in frequency can be related to the amount of mass that went through and a measurement can be accomplished.

If Coriolis created a simultaneous and equal but opposite acceleration as Newtonian physics predicts it should, this device would not work. You can’t drive a spaceship by beating it with a hammer from the inside, all you’ll do is hurt your ears.

This innovation has made a dramatic change in the accuracy and capabilities of this class of devices, since a Coriolis based MFM measures the mass of the substance directly, and does not rely on interpretations based on the thermal properties of the fluid and the heat transfer that it creates during flow. This is more accurate and effective not just because it is extremely difficult to measure temperature to a high degree of accuracy, but because, by measuring the effect that the Coriolis has on the outside system, one is measuring a true property of the mass, not just the secondary properties of the specific chemical involved.

This practical, real world example shows that, not only is Coriolis a real force that can be utilized, but that it can and does affect the larger world outside of the rotating system.

There is one more phenomenon of volumetric motion that ought to be discussed  before we leave the subject of the Spiral Dimension , and that has to do with flow in a tube. Surprisingly, this applies to both water and electricity. You probably believe, as most do, that water flows through a pipe as ‘plug flow’ that is, that water that flows into and out of a pipe flows much like marbles that would roll though a pipe in ‘blob’ fashion. This is not the case. Once flow has been established, what happens is that instead of flowing down the tube like marbles, the fluid is concentrated in the section of the internal area that is closest to the outside wall, in a spiral fashion. The center of the tube plays only a small role in the transfer of the fluid from one end of the pipe to the other in steady state flow.

The flow of electrons through a wire has similar characteristics. Most of the current carried by a wire is in the layer just inside the outer boundary of the wire; as with fluids, the center only participates mildly. These flows are also spiral, which is why the current flowing in a wire will create a circular, directional magnetic field around itself, the strength of which is proportional to both the current and the applied voltage. This field is a circular one (well, spiral, really) that is perpendicular to the nominal direction of the current flow, and follows the right hand rule, that is, if the current is flowing into the page, the direction of the rotation would be clockwise, like the inertial circles in the northern hemisphere.

These three dimensions: Linear, Rotational and Spiral, have interesting relationships with electromagnetic phenomena, some of which will be explored in later chapters, others that will need to be left to future books (if any). The subject was touched upon here to emphasize that in most energetic systems, where matter is moving from one potential to another (usually from high density to low) that motion self organizes in the form of a spiral, and that system must obey the rules that govern that dimension.

If you suffer from math-o-phobia, you can relax; we’ve finally finished putting together the framework for our new geometry, and there will be no more complex equations that you have to either try to understand or just gloss over.

Mostly, they were put in here to attempt to convince some of the skeptics that, yes, in fact, there is a theoretical basis for all of this weird stuff. And it is weird.

The idea that one can create an acceleration that does not have a Newtonian equal but opposite reaction is one that even I, quite frankly, have trouble internalizing. But if you look at the equations, it’s quite plain. Because of the mass property of Coriolis, a rotating body can create an acceleration that is out of the plane of action, and can, must define a space that is volumetric in nature.

A vortex, by its very nature, cannot exist in a ‘2-D’ system, any more than a cube can. And in many respects, the vortex is the velocity analog to the cube. For what form better contains the fundamental aspects of Euclidean geometry than a cube? A cube is formed by the intersection of perpendicular lines, just as the basic structure of the Euclidean axes and gives volumetric form to the system as a whole.

There’s just one problem. There are no true cubes in nature, excepting some crystals. Details, details.

Vortexes on the other hand, are all over the place. Galaxies, hurricanes, tornadoes, plumbing drains, waterspouts, even the weather in general shows this form with the presence of high and low pressure systems that rotate around a core. But we, collectively, tend to marginalize these observations because they don’t fit neatly into our scheme of how things are put together.

In the next few chapters, we’re going to discuss some of the more practical aspects of this new geometry that we’ve created and offer some new descriptions for gravity and light. Don’t worry, for although there will be equations, they’ll be simple and brief.

And that, I think, is a good thing, because a good explanation should be simple, brief and easy to understand, all characteristics that I tried to use as a basis for this book. But in so doing, we’re going to include explanations for properties of matter, space and time that have eluded us, collectively, for centuries. I think that Dr. Einstein would be pleased, but then, I never got to know him.

Hold on, the train’s about to leave the station.

– O. Penurmind

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6 thoughts on “Chapter 38: Coriolis and the Euler Acceleration

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    • Re: Coriolis acceleration

      Let’s consider a point moving on a curved trajectory with a tangential velocity (V) as a function of the curvature radius (R) and angular velocity (α’):

      V = R α’ as a vector and
      |V| =| Rα’ | as a scalar.

      Let’s calculate the tangential acceleration as a time derivative:
      V’ = R’ α’ + R α’’ as a vectors sum and
      | R’ α’| =|R’ α’| + |R α’’ | as a sum of scalar values,

      where
      R’ α’ is a type of Coriolis acceleration and
      R α’’ is the Euler acceleration.

      Since the above equations are valid both for vectors and scalar quantities, then all the vectors, the type of Coriolis acceleration included, must be collinear on the tangent to the curve. Is that right?

      • Re: Coriolis acceleration
        Message from 19 January 2015

        Please read | Rα’ |’ =|R’ α’| + |R α” | instead of | R’ α’| =|R’ α’| + |R α’’ |.

      • Dear Ilie,
        I’m sorry for the tardy response, and thanks for writing. Although you have the right equations for the Euler force, and the acceleration for that is co-linear, Coriolis is determined by the cross product of the velocity and radial velocity, or from Wiki: In non-vector terms: at a given rate of rotation of the observer, the magnitude of the Coriolis acceleration of the object is proportional to the velocity of the object and also to the sine of the angle between the direction of movement of the object and the axis of rotation.
        The paper should read: “ac = -2 Omega x v where: Omega = omega * direction and v is the velocity of the object.” I’m not sure why it doesn’t. Since Coriolis is a cross product, the tangential velocity only reacts with the components of the radial velocity that are perpendicular to its line of action, and creates a force that is, by necessity, perpendicular to both. In the case of a vortex which exhibits both radial and rotational acceleration simultaneously, that direction must be out of plane and the magnitude will be proportional to the tangential velocity times the radial velocity.

  2. Thanks for your answer.
    Let’s consider a movement in polar coordinates with

    polar radius R,
    polar angle Ө and
    transverse acceleration at.

    We have
    scalar at = 2R’Ө’ + RӨ”

    Note that for a movement under the influence of a central force (gravity for ex.), at = 0.

    In the case of polar coordinates, if RӨ” is the Euler acceleration, what would be in your opinion the denomination for the 2R’Ө’ acceleration type?

    Kind regards

    • Ilie,
      One of the problems of being an author like I am, not associated with a learning institution or a close coterie of like-minded individuals, like Dr. Einstein was, is that my work does not get critically reviewed. Consequently, much of my writing is not as thoroughly thought through as it might be. Apparently, this is especially true of my chapter on Coriolis. For, while I believe that Coriolis is a relatively undiscovered mass property that will allow us to create a revolution in space travel and maybe even energy conversion, I have to admit that I hadn’t really considered all of the possible velocity and acceleration cases that can create Coriolis.
      So, I’ve been giving that a lot of thought since I got your letter. I’m a pretty slow thinker, so it’s taken me a while to coalesce my thoughts. I do more thinking than research, which both hinders and liberates me, and the limited amount of research that I’ve done on the subject does not provide any insight at all.
      So, in thinking about the natural examples of vortexes that are common: tornadoes, drain vortexes and hurricanes, it would seem that it does not matter whether there is acceleration of either of the individual vectors that combine to make a vertex flow, just that both are in motion. The Coriolis expression is an instantaneous energy equation, much like F=ma, and therefore, only considers the instantaneous velocity of the system to quantify the instantaneous acceleration of the whole system. So, it would be my opinion that for any of the cases: R’ and q’ each velocities, only, R’’ and q’ with R accelerating and q as a velocity, and for R’’ and q’’, and where R’’ and q’’ are both accelerations, would all create Coriolis, with the direction dependent the relative directions of the constituent vectors.
      Thanks for your questions, and interest. I think that I had better revise Chapter 38.

      O. Penurmind

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