Re: Bulges

[David Bowman <dbowman@tiger.gtc.georgetown.ky.us>]

Jim Green recently wrote:

> I think it clear that the Newtonian tidal bulges are a myth.  But Newton
> assumed a rigid Earth.  I need to look more closely at the crustal motion,
> which likely *does* obey the equilibrium theory and *does* have "tidal
> bulges".  I thank the crustal motion is of the order of a few feet (20ft
> ???) -- If this *is* the case and if this is what David Bowman holds, then
> the story changes.  You see David is worried about the theory of some
> planetary geophysicists that it is a "tidal bulge" which applies torque to
> the Moon and slows it's orbit.  If David, et al are willing to say that it
> is the crustal bulge which does this then I relent.  If however there are
> those who think there are Newtonian water tidal bulges which cause this
> torque, then I must continue to say myth.  (And I don't see the geo people
> saying this.)

I've tried to stay out of this bulge discussion on this time around in public
and have confined my comments to a few private notes, but Jim's invoking of
my name here has drawn me out.

First of all I do *not* think the earth's tidal bulges are *anywhere near*
20 ft in size as Jim guesses above.  A few months ago in a previous tidal
discussion with Jim I calculated the size of the earth's tidal bulge
necessary to account for the moon's measured recession velocity of about
3.7 cm/yr.  My calculation made all sorts of drastic assumptions like
assuming the moon's orbital plane coincided with the earth's equator,
neglecting any out-of-roundness effects of the earth other than the lunar-
induced prolate tidal distortion, assuming the tidal distortion was exactly
that of a prolate spheroid, neglecting any effects due to the sun, and
assuming the earth had a uniform mass density.  The result of the calculation
was that each tidal bulge caused a prolate stretching of the earth by 11 cm
wrt a background spherical shape on each side.  Now none of these assumptions
is correct in practice.  The uniform density assumption is the most drastic
in terms of giving an inaccurate prediction.  The density of the earth is
much greater in its iron core than near its rocky and water surface.  If we
assume that most of the tidal bulge is due to flexing in the earth's outer
(upper mantle and above) layers where the lower pressures presumably
correspond to a lower bulk modulus then the size of the tidal bulges would
need to be significantly increased to produce the needed mass quadrupole
moment that couples the earth's spin to the moon's orbit and causes the moon
to recede at the measured rate.  The lower density of the earth's outer
layers could conceivably cause the needed tidal bulge to be 2 or 3 times the
11 cm value that I calculated.  I recently remember reading someone else's
post on tidal bulges that mentioned that these bulges are some 30 cm high.
If so, it is in the right theoretical ballpark.

As far as the relative contribution of the tidal bulge from each of the
earth's parts goes, I don't think that the crust is much of a contributor.
The crust is quite rigid and would not be expected to show hardly any tidal
thickening or thinning.  If the crust moved relative to itself on the order
of a few cm every 12 hours we would be having earthquakes all along the all
of the crustal plate boundaries twice a day.  The typical tectonic drifting
motions of the crustal plates are on the order of a couple of cm per year and
that causes the seismic events that we see.  A comparable amplitude motion
twice a day would seem to certainly show up in continuous earthquakes.  If
the crust doesn't contribute much to the bulges then what does?  It seems to
me that the main contributor would be the mantle, followed by the liquid outer
core, the surface oceans, and least of all the atmosphere.  The atmosphere
would presumably move the most easily in response to the tidal stresses
because of its ultralow viscosity and fluid nature, but the atmosphere has
such a small mass that it can't contribute much to the earth's lunar-induced
mass quadrupole moment. This leaves the mantle, the outer core and the oceans
as the remaining suspects.  Now any tidal flexing of the earth's interior
would result in the crust's shell shape being distorted as the underlying
plastic material flexes, but this crustal motion is presumably innocuous
regarding tectonics because here the crust's thickness doesn't change and
there is no local crustal motion of one part of the crust wrt another nearby
part.  All nearby parts tend to move together as the underlying mantle gently
flexes vertically by a fraction of a meter over a lateral length scale of
over than 10^4 km.  Now the oceans are much more flexible than the mantle,
so I would expect their relative response to be much greater than that of the
mantle, but the mantle and outer core are so much thicker than the oceans
that even if each part of the earth's flexible interior is much more sluggish
than the oceans, the fact that the interior is so thick compared to the oceans
would presumably cause the interior to be the majority contributor to the
earth's lunar-induced quadrupole moment, with the oceans being a relatively
minor player.  So in conclusion, I would expect a minority fraction of the
earth's tidal bulge (which altogether is itself a fraction of a meter) to be
due to the oceans.  Therefore it is no wonder that Jim doesn't see any
evidence of a tidal bulge in the oceanic tidal data that he has ammassed.
The contribution of the (highly underdamped) oceans to the tidal bulge is too
small to show up in the time series for the ocean level at a typical
coastline which is dominated by other undamped resonant/sloshing and
continental reflection effects as well as extraneous wind effects, seasonal
ocean current effects, etc.  I suspect that if the time series for the ocean
height in the middle of an ocean was Fourier analyzed and its power spectrum
taken, that there would be a significant spike located at the lunar driving
frequency which would correspond to the presence of at least some of the
tidal bulge being present in the oceans.

> ...                                              .  Besides David Bowman has
> privately damaged my speed-of-ocean-waves arguments, so I need to go back to
> the drawing board anyway. (:-)

Sorry about that.  Actually, the ocean-wave-speed argument's invalidity was
present independent of my criticisms.  I just pointed out its intrinsic
problems.

David Bowman
dbowman@gtc.georgetown.ky.us