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    Re: Navigating Around Hills and Dips in the Ocean
    From: George Huxtable
    Date: 2003 Aug 19, 19:53 +0100

    Keith Williams asked the following question-
    >George says: "It's possible to go from one value of g to another, always
    >> an equipotential path, and there will be changes in elevation above
    >> the Earth's centre along that path, but being an equipotential path,
    >> there will be no work done against gravity. None at all. In fact, this
    >> is exactly what
    >> happens when a ship travels from one latitude to another and from one
    >> value
    >> of g to another, over the oceans."
    >Can we have further explanation of this? My brain tells me that moving
    >along an isobar (to make a simple analogy) can't move you to an area of
    >higher pressure. Are equipotential shells not analogous to isobars?
    George replies-
    I will try to explain a bit further.
    In the case of isobars, they are surfaces of equal pressure, as Keith says,
    and so moving along an isobar can not move you to an area of higher or
    lower pressure. That's simple enough, and well understood.
    Similarly, an equipotential shell is a surface of equal gravitational
    energy, so in just the same way moving along that shell involves no change
    at all in gravitational energy. That's EXACTLY the same as Keith's isobar
    With isobars, there's another quantity you can consider, however, the
    CLOSENESS of their spacing, and anyone familiar with weather-maps knows
    that when the isobars are closely-spaced then that means there's a high
    pressure-gradient, and when there's a high pressure-gradient there's always
    a lot of wind. There are some differences here with our potential-shell
    analogy, in that we're used to seeing a pressure-contour map as a section
    sliced through the atmosphere at or near surface level. Really, though, the
    pressure distribution is three-dimensional, a series of isobar shells
    (though I wouldn't like to guess just how the closeness of those shells in
    the vertical direction affects the wind).
    But I think the picture is clear: you can follow an isobar on a weather-map
    and you can find it will lead you from a region of sparsely-spaced isobars
    to other areas where they are closely spaced; from a weak wind to a storm,
    all along that same isobar, at an identical pressure throughout.
    Now the gravity analogy should be getting clear. The strength of the
    gravity-field at any point; that is, the local value of g, is defined NOT
    by which equipotential you are on at that point, but by how closely those
    equipotentials are spaced there. If they are unusually close, then when you
    put in a certain amount of work to move a certain mass from one
    equipotential to another, you are doing it over a shorter distance. We know
    that work = force x distance, and the work is the same, so that means that
    the force must be greater. That force is the force of gravity on that mass,
    or g. Where those same equipotentials are more widely spaced, the work in
    moving between them is just the same, but the force, and therefore the
    value of g is correspondingly less.
    So g is not the same along an equipotential, it simply shows the closeness
    of the spacings between them, and a map of the shells of equal-g would be
    rather different from the map of equipotentials (and rather less useful).
    There is some economic importance in this. The oil industry prospects for
    deposits by looking for small changes in g from a survey vessel, which (of
    course) travels along the sea-level equipotential. If g remained contant
    over an equipotential, they would never observe any fluctuation.
    However, we have to keep a sense of proportion here. On weather maps,
    isobars can wander all over the place. Conversely, if the Earth was a
    uniform sphere, the equipotential shells would be exactly spherical, and
    so, as it happens, would be our equal-g contours. Even the ellipsoidal
    effects caused by the Earth's spin give rise to changes which are less than
    1%. And the hills and dips, that we've been discussing in such detail, are
    molehills on that ellipsoid. So the gravity variations that we have been
    discussing are small, and the surfaces of equal potential and the surfaces
    of equal gravity, though not spherical shells, are pretty damn near to
    Has that helped, Keith?
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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