I found a tutorial here that could be quite good, except that I find a lot of confusions in the wording, changes in variables between equations and text, an occasional mis-used term (angular velocity, for instance), unlabeled changes of units, etc. that make reading it hard. As it stands, it really isn't up to college class quality. Cleaned up, it could be the best tutorial I've seen as a compromise between oversimplification or skipping over important stuff, versus the dense math and lack of examples in the book by Hoffman-Wellenhof and Moritz.
Today's issue: Halfway down, under Equipotential Surfaces item 5.b. a statement is made that is counter to my intuition. Does everyone believe that a gravity equipotential is "Everywhere convex (implies no concave dips in surface (that is holes)" ?
In a local region of low gravity, the geoid or another equipotential surface certainly dips down relative to the ellipsoid. I guess the issue is whether it ever dips enough that in the big picture it goes concave, or whether it just gets a little less convex than the ellipsoid.
I think that it would be theoretically possible for it to go concave, but I don't know if it ever happens on earth.
Anyone?
I think so, too (that it's possible for a 'concavity' to exist). Finding it and proving its existance would be challenging.
In '93 or '94 I listened to a fellow named Dr. Michael Jackson speak about the gravity models and their relationship to the geoid. During one of the breaks he asked me if I was following him.
I said, "Maybe. If I heard you right, you basically said that somewhere on the earth water could theoretically run uphill."
He grinned and told me, "You were listening."
> Does everyone believe that a gravity equipotential is "Everywhere convex (implies no concave dips in surface (that is holes)" ?
I think it depends on how you define "concave." If you mean a place in which liquid will puddle, then by definition there can't be any concavity, because the gravity potential is the same everywhere. If you define it as a portion of the surface that would be closer to the planetary center of mass than a planar surface connecting surrounding points of the equipotential surface (think of a sheet of metal laid across a dimple on a golf ball), then concavity it possible.
That's my take, anyway.
>I think it depends on how you define "concave." If you mean a place in which liquid will puddle, then by definition there can't be any concavity, because the gravity potential is the same everywhere.
By that thinking, it couldn't be convex either, because water doesn't run away from every point. The liquid surface is by definition "flat".
So we're using the dimple in the golf ball approach, and asking if any of those dimples on the geoid fall below a plane in 3D space.
By the word "plane", are you meaning the equipotential force of gravity?
No, a geometric plane in physical 3-space. That's the only way I see to make the idea of convex and concave meaningful. The plane is the divide between those conditions.
Is there some reason why you don't define concavity in the geometric sense of there being no line between any two points on the surface arbitrarily close such that the surface lies "above" the line? "Above" probably needs a definition for a general case but in the case of the geoid probably not.
Howdy,
FWIW, Dr. Ghilani's web pages are in support of his class lectures. I don't think they were intended to be "stand alone." His program is for undergraduates studying surveying. Geodesy is covered at a basic level.
Some more rigorous material is available from these sites:
http://www-gpsg.mit.edu/12.201_12.501/BOOK/chapter2.pdf
http://web.ics.purdue.edu/~ecalais/teaching/eas450/Gravity2.pdf
As for your specific question, the assertion that equipotential surfaces are always concave is probably related to a low-order spherical harmonic representation. I would rather describe the characteristics as:
continuous, never crossing, not parallel, with only smooth changes (therefore at high-frequencies both convex and concave).
BTW, there is a new (4th) edition of the classic text Geodesy by Torge (now with Jurgen Muller). It is rather advanced (even labeled "Graduate') but more accessible than your Physical Geodesy text by Mortiz.
For those seeking a less detailed grasp of these ideas, check out the definitions in the NGS Geodetic Glossary. The text "Geodesy for Geomatics and GIS Professionals" by Elithorp and Findorff (ISBN 13: 978-1-58152-658-5, ISBN 10: 1-58152-658-X) is a good introduction without much advanced maths. I just took a look at this text and noted that it also included the assertion that equipotential surfaces "...are convex everywhere, they do not possess troughs..." The authors reference the text Geodesy: The Concepts by Vanicek and Krakiwsky. Unfortunately, I gave my copy away when I retired.
A lot of good material is available in the form of reports and lecture notes from places like The Ohio State University, University of Calgary, University of New Brunswick, and others. Nothing like reading a nice Ph.D dissertation on physical geodesy to help you get to sleep 🙂
HTH,
DMM
Thanks for the reply. I'll be looking at some of your lectures also, found through the link after your name.
It's good to see a citation that says the statement is true. I think "that's what we measured due to the earth's density" and not "it mathematically has to be," but I'm not certain. I guess it really isn't one of the more important facts for the student or working professional to know, anyway, but I just got hung up on it.