Re: Moon's synchronism (not not very long)

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

Here we go again. :-(.  It seems that every time the adjective 'tidal'
appears on this list Jim Green immediately jumps to the conclusion that
it necessarily refers to the actual oscillations of the height of the
water level on the shoreline of some sea coast (averaged over the
the high frequency oscillations of the individual breakers on a time
scale of a few hours).   This is *not* what I have been talking about. 
Tidal bulges have precious little to do with tides (in the sense used
just above).  For instance my discussion of the instantaneous equilibrium
response of an ideal fluid planet subject to the tidal influences of
another astronomical body had *nothing* to do with 'tides' (used in
the above vernacular sense).  As a matter of fact, the above hypothetical
planet would experience *no tides whatsoever*.  Rather, the shape of the
planet would merely be in instantaneous equilibrium with its geopotential
at all times.  This would be a nearly undetectable effect.  If the planet
of interest was tidally locked to the other astronomical body then this
distortion would be permanent and undergo no time dependent motion
(relative to the body of the planet) whatsoever.  This would not have any
effect that would show up at the surface other than a slightly weird
geography for the tiny changes in location-dependent values for the
planet's surface gravity.  The local surface g-value would be smaller at
the tip of the tidal bulges than at the valleys of the tidal sags.  If the
other astronomical body had some diurnal motion relative to the planet of
interest (because the planet's sideral spin rate was not locked in a 1:1
way with the orbital motion of the other body) then the planet would 
periodically flex according to that diurnal motion as the planet spins
under the tidal field gradient.  This would show up in tiny oscillations
of the local surface g-value which are synchronized at the semi-diurnal
rate of the other astronomical body.  A flexing planet is not the same
as sloshing water (i.e the kind of tides for which Jim is seemingly fixated.)

In a *nonideal* planet whose construction is more complicated than that
of an ideal, instantaneously responding fluid so described, then it is
possible (likely) that the flexing action of a time dependent driving
field gradient would drive a host of dissipative modes in the planet.
Some of these would be very overdamped responses, say from a semiplastic
planetary interior whose response would be like that of kneaded Play-Doh
(TM).  Since the necessary flexing of any thin, hard, surface continental
crustal plates would would be spread out over a wavelength of 1/2 of
the planet's circumference--yet with an amplitude of, at most, of the
order of meters, it is easy to see that such tiny slight flexions of the
crustal plates would not cause hardly any seismic stress/strain in those
plates.  (For an illustration of what I mean here consider a flat slab of
solid aluminum.  It takes a strong local stress to flex it over a radius
of curvature comparable to its thickness.  Now consider a square sheet
of aluminum foil that is flexed over a radius of curvature of a few
meters or so by a very tiny amount. This is so easy to do that it is
actually hard to limit such flexions to such a tiny amount, and such
gradual bends in the foil do not introduce any localized stress/strain in
the foil to speak of.)  It would seem that bulk characteristic planetary
flexing response times would be slow compared to the driving frequency of
a(n Earth-like) semi-diurnal tidal forcing action.  For instance, as I
recall the surface of Northern Canada is still rebounding upward in
response to the lifting of the ice sheet from the last glaciation some
10^4 years ago.  (Of course this response is somewhat limited by the
higher shear stiffness/viscosity of the athenosphere compared to the
more flexible outer core.)

Besides the overdamped bulk response of a semi-plastic planetary
interior.  There are also highly underdamped responses of any (low
viscocity) ocean and atmosphere that may be present.  A liquid ocean
surface would support low dissipation propagating surface modes
(Rayleigh gravity waves) that would be the main relaxational mechanism
of transporting, i.e. radiating, ('radiating' here in the sense of outward
propagation of a solution of a wave equation) the tidal energy between the
tidal bulges and valleys.  Eventually, after such Rayleigh waves propagate
across the ocean on top of convective/advective ocean currents, get
twisted by the Coriolis effect, reflect off of continents, excite seiche
resonances in bays, gulfs & estuaries, interact frictionally with the
air and the sea bottom (esp. near a shore with a shallow continental
shelf), etc., they eventually dissipate the energy carried in them.
These propagating underdamped responses give rise to the (mostly)
standing wave motions (with a wide range of local characteristic phase
shifts relative to the driving forces) that are commonly called tides
(and in which Jim Green is so interested).  They are not the tidal
bulges to which I had referred.  The tidal bulges are pulled up and
down/back and forth/to and fro *in situ* by a time-dependent
gravitational force synchronized with the lunar/solar positions; they are
*not* the free propagating Rayleigh waves which are excited *in response
to* the underlying bulk bulge motions and direct tidal forces imposed.
Jim seems to confuse the complicated underdamped propagating surface
relaxation responses with the local bulk forced motions of the bulges
themselves.  Admittedly the complication of all these surface standing
waves on top of the tidal bulges makes the presence of the oceanic
contribution to the bulges hard to see.  It does not make them not exist,
however.  The (oceanic contribution to the) bulges are expected to be
invisible/nonexistent near continental shores.  The best place to observe
them would seem to be in deep mid-ocean far from the complicating effects
of large land masses.  Satellite observations (thanks for the reference,
Leigh) of the ocean surface would seem to be ideal for making these
measurements for the oceanic bulges.  Possibly, (if they are sensitive
enough) they could also detect the underlying bulk solid planetary bulges
as they raise and lower the crust sweeping across the continents as well.
We've been through all this last year.

> Despite ample evidence to the contrary, David continues to believe in "tidal
> bulges" on the Earth. There are no such things, the Great Newton (or the great
> David) notwithstanding. Nor does High Water always lead (or lag) the Moon or
> even the Sun/Moon combination. In deed there is no correlation between High
> Water and the position of the Moon. There is no prediction by anyone of the
> relationship between High Water and the position of the Moon or Sun/Moon based
> on a classical mechanical relationship.

Jim is looking at trees and claims there is no forest.

>                                        And, it is not true that the Earth's
> oceans can "rapidly respond to the forces acting on [them]."

I disagree, here.  Try tilting a pan of water by, say one degree, over
a time scale of 6 hours and see if the level of the surface coincides
with the pan, or with an equi-geopotential surface.  It does not take
very long for water to seek and find its local level.  (It does take
longer, though, when we have to contend any with local sloshing that was
induced by the tilting action, but this sloshing is *on top of* the new
level established by the tilting action.)

> In last year's discussion David seemed to concede this, but held onto the
> possibility that superimposed on the ocean's tides, there is a set of
> mini-tidal bulges. However, I have yet to see any documentation in support of
> this.

The earth's spin *is* loosing angular momentum to the moon's orbit.
This is a measured effect.  The lunar recession is also a measured
effect.  Tidal effects of the Sun and the Moon exert a mean torque of
about 7.8 x 10^16 Nm on the Earth's mean tidally-induced quadrupole
moment.  Such a quadrupole moment is due to an induced mass
redistribution of the planet that we conventionally call tidal bulges.
If there were no such bulges then there would not be such an induced
quadrupole moment and there would not be this mean braking torque acting
on the earth's spin. (Note, the mean orientation of the induced quadrupole
moment needs to be rotated out in front of the average orientation of the
tidal field gradient in order for the torque to exist.  A moment in its
equilibrium orientation wrt its inducing field gradient experiences no
torque.)  We have been through all this last year. 

Regarding my calculations for the tidal/spin bulges on a hypothetical
ideal equilibrium fluid *lunar* surface Jim writes:
> Now as to the Moon's tides: David neglects that the tides are really damped
> driven oscillators and as such their amplitude depends on how close the various
> driving frequencies are to the natural resonances of the Moon's waters.  These
> resonances are not calculable except in the case of a totally aqueous Moon and
> then they would depend of the Moon's ocean depth -- which is not given. 

To the extent that the Moon's motion around the Earth *is* tidally locked
to its spin and its spin axis is perpendicular to its orbital plane (it
actually seems to be off by some 6.68 deg.), to that extent the induced
tidal bulges are *not* driven systems.  They are static wrt the lunar
surface.   The already much-discussed libratory motions will place a
slight time dependence with a two week period and a few degrees on the
orientation of the field gradient.  Compared to the calculated heights
of the bulges these tiny librations would be small.  Again, I was
interested in calculating the equilibrium bulge heights, not any
propagating relaxational response to a tiny rocking of the bulge-
producing forces.

> Well, if one were also to assume that the Moon's water has nil viscosity and
> that a traveling tidal wave really could keep up with the driving forces, this
> is not all that bad.

Propagating tsunamis have nothing to do with the effect I was calculating.
Besides, the driving forces have a period of 2 weeks on small planet and
only oscillate by a few degrees one way or the other.

>                      However, remember that for normal water any wave
> phenomenon is limited by the speed at which a wave can move in the water.

This is only for *unforced* phenomena.  For a force moving with a speed
over a medium whose characteristic wave speed is different from the
force's speed, the medium responds in two ways.  First there is the local
response at the location of the force's application and moving with the
speed of the force, and second there is the "wake" response propagating
at the medium's wave speed and radiating outward into the medium from
the current location of the force.  Consider Cherenkov radiation, the
wake behind a motor boat, a sonic boom, etc.  We've been through all of
this last year.

> On the Earth tidal motion can't keep up with the Moon in latitudes <~60o.

Why not?  The speed of the Rayleigh waves whose ringing responds to a
tidal force and its attendent bulge, is not the speed of the force/bulge
itself.  If forced wave behavior could only result in a medium whose wave
velocity exactly matched the speed of the disturbing force, then hardly
any wave motions could be induced.  Even time-dependent but stationary
forces have a force-speed of zero, and this does not match the speed of
the waves which is non-zero.  Do you really think that the speakers
connected to your stereo have to move across the room at 345 m/s in order
for sound to propagate through the air from them?

> Oh, yes! The Moon/Sun do work on the Earth's oceans and, likewise, the Earth
> on the Moon. The energy of the ocean system increases and that of the Moon
> decreases --.

No.  A time-dependent (due to the mismatch of earth's spin rate and the 
relevant orbital motions) interaction, the involving the field
gradients' of the Sun and the Moon does work on the oceans.  The energy
supplied is nearly all eventually dissipated as (~4 x 10^13 W) heat (yes,
heat).  The rotational kinetic energy of the earth provides the energy for
this dissipating work *as well as* provides the extra energy required to
lift the Moon to a higher orbit as it acquires the angular momentum lost
from the decreasing spin of the earth.  The orbital energy of the Moon
*increases*, not decreases.

> ...
> Alas, it is true that many of the oceanographers that I have spoken to _do_
> accept he idea of tidal bulges -- and with a similar positiveness that
> biologists ascribe to evolution. More's the pity.

I guess a their blind belief in the conservation of angular momentum is
misplaced.

> Now crustal tidal motion I don't know about.  That will be a wholly different
> phenomenon .  I think that the crustal motion can be as much as a few feet.  I
> don't know if that motion is oscillatory in the same sense as the ocean's
> waters.   And I don't know how this motion correlates with the tides.

Its a very similar phenomenon to the one I and others were/are discussing.
Jim, you were the one that keeps bringing up the sloshing of the water
tides (vernacular meaning).  Any tidal motion of the height of the crust
will be expected to be less than that oceans because of the greater mass,
shear stiffness and shear viscosity of the system (esp. of the underlying
mantle) would make its characteristic response time much slower than
that of liquid water (and probably slow, too, compared with the needed
12 1/2 hour time scale needed to get much of a tidal response).

BTW, I found an interesting book by Allan Cook on the subject of the lunar
motions in the library entitled _The_Motion_of_the_Moon_ (IOP Publishing,
Ltd., 1988).  Among the other neat stuff in this book are some parameters
relevant for the resonant frequency of lunar libration problem.  In
particular, this book claims that the important relative differences of the
Moon's princial moments of inertia actually have the values:
(I_yy - I_xx)/I_zz = (2.28022 +- 0.001) x 10^(-6) and
(I_zz - I_xx)/I_yy = (6.31787 +- 0.00132) x 10(-6).

For the case of the rocking mode about the Moon's spin axis this means the
period of resonant vibrations comes out as T = 9.1 x 10^7 s = 2.9 yr.
This is still 38 times slower than the 27.3 day driving frequency for this
mode however, but it is significantly faster than the 8.6 yr period this
mode would have if the Moon's shape was determined by it being in
approximate equilibrium with the Earth's tidal field gradient.  Also the
ratio of the above relative differences in the moments of inertia should
be approximately 3:4 for a tidally locked Moon in equilibrium with the
Earth's field gradient.  Since this ratio is more like 1:2.77 it seems
apparent that when the moon 'froze up' and became too stiff to further
relax to the changing terrestrial field gradient that it found itself in
that it was *not* tidally locked, and still possessed a significant spin
rate greater than its orbital angular velocity.  Another interesting
parameter I found from this book is that the avg. lunar crustal rock
density seems to be approximately 2.95 g/cm^3.  This would give a q-value
density ratio of q = 2.95/3.34 = 0.88 rather than the 0.84 value that I
guessed for this value in my last monster post on this subject.  Changing
this value has little effect on the answers obtained.

David Bowman
dbowman@gtc.georgetown.ky.us