Coriolis force in 4D

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

Coriolis force in 4D

Postby Teragon » Tue Oct 13, 2015 10:51 pm

As hinted in the discussion about 4D-planets (http://hi.gher.space/forum/viewtopic.php?f=27&t=1573) I rolled up my sleeves and derived a formula for the Coriolis force in 4D.

For me it's an expression of unexpected simplicity and elegance.

φ is the angle along the direction of rotation with angular velocity ω_φ ("solar longitude").
ϕ is the angle along the direction of rotation with angular velocity ω_ϕ ("polar longitude").
ϑ is the angular distance to the equator, ranging between 0 and 90° ("latitude").
Dots above φ/ϕ/ϑ indicate the derivation of the coordinates with respect to the time, in other words the velocity in φ-/ϕ-/ϑ-direction.
φdot > 0 means going in the direction of the solar rotation (east), ϕdot > 0 means going in the direction of polar rotation (marp).
The u's represent the unit vectors in φ-, ϕ- and ϑ-direction.

Image

To really understand weather patterns on 4D planets, I guess it would take numerical simulations of the corresponding differential equations, but knowing how the force looks like, we are able to yield some basic insights.

On a 3D planet, the Coriolis force decreases from the poles to the equator and as a consequence, only near the equator air can flow freely into low pressure zones. No big pressure gradients are possible there, no cyclones, no rotation. Setting ω_ϕ to zero in the 4D case, the Coriolis force takes the same form as in 3D, except there's an additional coordinate which is unaffected by the Coriolis force. At every position on the planet there is an additional direction in which air can move freely and no bog pressure gradients can build up - the direction of the corresponding circle of polar longitude. An exception is of course the equator itself where the polar longitude may not be defined.

Setting ω_φ to zero instead of ω_ϕ yields an analog behaviour. The Coriolis force is now decreasing from the equator to the pole and at any position on the planet there is one direction unaffected - the direction of the equatorial longitude.

In the general case, if both rotational velocities are non-zero, the terms of both special cases add together. It turns out that at any location on the planet, air can flow freely in exactly in one direction (and it's opposite), namely the direction for which ϑ is constant (ϑdot = 0) and the contributions determining the force in ϑ-direction cancel each other. This specific direction is specified by the rotational velocities and is dependent on the latitude ϑ.

Going from south to north, the Coriolis force part due to solar rotation increases with sin(ϑ). At the same time the Coriolis force part due to polar rotation decreases with cos(ϑ). Near the solar equator air can travel freely in φ-direction, because the solar equator is the invariant element of polar rotation. In the local ϕ-ϑ-plane, however, the Coriolis force is strong and air drawn into the intertropical convergence zone is deflected around the solar equator. Near the polar equator air can travel freely in ϕ direction, because the polar equator is the invariant element of solar rotation. As on a 3D planet, air moving is deflected around the pole, with the difference that the pole itself is 2D. So one would expect a donut-shaped cylone at the solar equator and a donut-shaped anticyclone at the pole.

As stated above at any latitude there is a longitudinal direction in which the ϑ-parts of the Coriolis force cancel out: ω_φ sin(ϑ) φdot = ω_ϕ cos(ϑ) ϕdot. These are curves spiraling around both equators, keeping their distance to both of them. The more close to the solar equator, the closer the curves point to φ, the more close to the polar equator, the closer the curves point to ϕ. If both rotational velocities are the same (Clifford rotation), all of these curves are great circles (obviously making up the Hopf fibration).

Essentially the pressure field might be also a 2D one in 4D. In the direction of free movement (a linear combination of phi and psi), pressure might only vary slowly, so pressure systems would either be extensive in one dimension, or rather weak (or this impression deludes). Near the equator and near the pole, where one or another part of the Coriolis force vanishes things are quite clear. However things get very complex in between. I'm having a hard time trying to figure out what is going on there.
Last edited by Teragon on Fri Jan 01, 2016 9:45 pm, edited 2 times in total.
What is deep in our world is superficial in higher dimensions.
Teragon
Trionian
 
Posts: 136
Joined: Wed Jul 29, 2015 1:12 pm

Re: Coriolis force in 4D

Postby PatrickPowers » Thu Dec 03, 2015 5:03 pm

If the two periods of rotation are the same then there should be no coriolis force. Every point is on an equator.
PatrickPowers
Tetronian
 
Posts: 440
Joined: Wed Dec 02, 2015 1:36 am

Re: Coriolis force in 4D

Postby wendy » Fri Dec 04, 2015 12:59 am

Even in clifford or equirotation, you get a coriolis force. It is to do with the complement of roataion, relative to an inertial frame. Need to work out the answer, then find some suitable answer to solve the answer.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Coriolis force in 4D

Postby PatrickPowers » Fri Dec 04, 2015 5:06 pm

wendy wrote:Even in clifford or equirotation, you get a coriolis force. It is to do with the complement of roataion, relative to an inertial frame. Need to work out the answer, then find some suitable answer to solve the answer.


Sorry, I don't know what the complement of rotation is. I'll need a little more of a hint. I would assume that it hints that the path of the surface of the planet is curved, not straight. But on a 4D Clifford rotating planet it would seem that the curve drops straight down, which wouldn't give a Coriolis force as would a sideways curve. Where am I going wrong?
PatrickPowers
Tetronian
 
Posts: 440
Joined: Wed Dec 02, 2015 1:36 am

Re: Coriolis force in 4D

Postby wendy » Sat Dec 05, 2015 11:46 am

The complement of rotation, is to suppose that the heavens are rotating under a force around the earth. This is the measure of force, equal to the acceleration of the ground. There is then a force at ground level, that would provide a screw motion in the wind in the left hand direction of east, which represents the rotation of points around a line, which are moving around the line.

Unfortunately, I can not directly figure it out as yet, as to what kind of vector it is, since it is relative to the observer's acceleration.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: Coriolis force in 4D

Postby Teragon » Sun Dec 06, 2015 9:39 pm

It seems that the most important part - the derived formula itself - was not visible for some time. Hope this one (edited in the original post) is going to stay.
What is deep in our world is superficial in higher dimensions.
Teragon
Trionian
 
Posts: 136
Joined: Wed Jul 29, 2015 1:12 pm


Return to Higher Spatial Dimensions

Who is online

Users browsing this forum: No registered users and 14 guests