Theoretical Constraints on True Polar Wander Victor C' Tsai1 and David J' Stevenson2 1' vtsaifas'har

1
Theoretical Constraints on True Polar
WanderVictor C. Tsai1 and David J. Stevenson2
1. vtsai_at_fas.harvard.edu, Department of Earth
and Planetary Sciences, Harvard Univ., Cambridge,
MA 021382. djs_at_gps.caltech.edu, Division of
Geological and Planetary Sciences, Caltech,
150-21, Pasadena, CA 91125
U33A-0025
5. Results Note The expressions below are
symmetric in all indices. Case 1 (only changes in
the diagonal components of cij) The expression
for mi/mj (6) The maximum TPW velocity
is given below and is attained halfway between
switch in principal axes (e.g. j 45
degrees) or (7) With sinusoidal
driving force c33-c22(t)(c33-c22)cos(2pt/tTPW)
we can write a simple expression for the maximum
TPW angle or (8)
where Case 2 The maximum TPW
velocity is given by (9) The approximate
maximum TPW angle is a complicated function. It
is similar to the expression (7) but is not
especially enlightening. _________________________
_________________________________
Figure 4b Two possible TPW response functions
due to the forcing in Figure 4a. The red curve
is the maximum TPW curve. Note that in both
cases the high frequency perturbation is
filtered out. (Red begins at j67, Blue
begins at j27)

• 1. Main Points
• TPW can be faster that APW (plate tectonic
  motions)
• For much of geologic time, the evidence supports
  less TPW than APW
• The timescale for large amplitude TPW is larger
  than the natural viscous relaxation time by the
  ratio of the bulge to the geoid anomaly driving
  the TPW
• The timescales for TPW and convection scale in a
  similar way with mantle viscosity, but with TPW
  predicted to be somewhat faster
• TPW is a low pass filter. Rapid changes in
  moment of inertia produce subdued and delayed
  responses. The biggest driver of TPW is long
  timescale changes.
• Inertial interchange TPW (IITPW) does not
  produce faster TPW than conventional TPW, and is
  delayed in time by the gradual growth of the
  moment of inertia driving force.
• The factors causing present-day TPW to be less
  important than simple models predict are
  (a) Unusual or surprising coherence of mantle
  convection/plate tectonics, (b) Very viscous
  lower mantle, (c) Possible triggers for TPW are
  too fast or too small

Figure 2 An example of TPW for a step
function change in moment of inertia (h1022
Pas) Figure 3 The maximum TPW as a
function of X where X is as described in Sections
5 and 6 Figure 4a Fluctuations in the
moment of inertia (cij) given by the sum of two
sinusoids of different periods but similar
amplitudes
3. A Heuristic Argument It is often useful to
develop an understanding of a dynamical process
through energy considerations, and TPW is no
exception. Of course, one must also analyze the
dynamical equations to confirm the heuristic
picture, as we do in Sections 4-6. J angular
momentum C moment of inertia dC moment of
inertia perturbation h average viscosity W
angular velocity tTPW characteristic time of
reorientation R Earth radius M Earth
mass r mantle density e strain associated
with moving the rotational bulge g
GM/R2 DEchange in energy associated with moment
of inertia change tvisc viscous relaxation
timeh/(rgR) Conserving J, we have (1)
This reduction in energy is balanced by the
viscous relaxation associated with moving the
rotational bulge. The strain associated with
this is eW2R3/GM0.003oblateness The viscous
energy loss per unit time per unit volume is
approximately h(e/tTPW)2. Equating the energies
then gives (2) Introducing the
natural viscous relaxation time tvh/(rgR), then
to order of magnitude (3) Physical
interpretation The characteristic time for large
amplitude TPW is larger than the natural viscous
relaxation time of the system by the ratio of the
rotational bulge to the geoid anomaly driving the
TPW. Further comments - Any sized geoid
anomaly can drive large TPW but the timescale
becomes extraordinarily long - Less obviously,
the TPW resulting from rapid changes is reduced
significantly. Thus, TPW is a low pass filter,
allowing slow variations to yield full TPW and
filtering out fast variations (see Section 5) -
There is no major difference between the rate of
IITPW (inertial interchange TPW) and regular TPW.
Both processes are driven and inhibited by the
same processes We can do a similar comparison
with the timescale of mantle convection.
Performing standard mantle convection scaling we
obtain the result that (4) - The
strength of TPW relative to convection is
primarily due to viscosity structure, not
absolute viscosity
Figure 1 Two ways of describing TPW
2. Introduction What is True Polar Wander
(TPW)? - TPW is when the geographic coordinates
of the rotation axis change - See Figure 1 for
graphical description of TPW - Throughout our
analysis, we use the geographic frame of
reference Why does TPW occur? - The lowest
rotational energy state corresponds to rotating
about the axis of largest moment of inertia -
TPW is driven by the energy associated with this
change in rotation axis - The hydrostatic
(equatorial) bulge slows TPW since the bulge must
relax before significant TPW occurs - Even at
times longer than the relaxation time of the
bulge, TPW does not always track the axis around
which the non-hydrostatic moment of inertia is
largest. It lags behind the desired
axis. What do we contribute to the current
knowledge of TPW? - The basic theory for TPW is
well established (Munk MacDonald, 1960) and
detailed numerical calculations of TPW exist,
(e.g. Richards et al, 1999) - However, there is
little analysis of an intermediate kind, firmly
based in theory but devoted to scaling arguments
and constraints - Our intent is to fill this
gap, at least partially - The intent is to guide
both geologists and geodynamicists as to what
parameters are important for TPW

• 4. Our Model Analysis
• Wi Wmi the components of the angular velocity
  vector in the geographic frame
• mi unit vectors for the components of the
  angular velocity vector
• j angle of angular velocity vector relative to
  some axis generically, tan(j)mi/mj
• cij components of the perturbations in moment
  of inertia (e.g. caused by mantle convection)
• C-A difference between the maximum and minimum
  moment of inertia (due to the equatorial bulge)
• In our analysis we make some simplifying
  assumptions
• - Maxwell Earth (primarily viscous mantle with
  one average viscosity)
• - Times in question are much longer than the
  viscous relaxation time
• with then (5)
• (See Munk MacDonald, 1960 for details.)
• We analyze the following cases
• Only diagonal components of cij changing, in some
  geographic frame, not necessarily the same as the
  rotation frame. Note that at any given time, we
  can rotate to a frame such that this is true
  instantaneously
• Adding a non-zero c23 term but still with zero
  c12 and c13 terms

6. Conclusions - With viscosity h 1022
Pasec, sinusoidal forcing period of tTPW 100
Myr, and Dcc33-c2210-5C (typical convection
anomaly) then X1 - Reading off the graph
(Figure 3), X1 corresponds to Djmax90
(actually 88) - X is a linear function of tTPW
and (c33-c22) and inversely proportional to h
- In units used above, the following are examples
of maximum TPW - Maximum TPW only occurs
when axis is already in the process of
reorienting. (Compare red and blue curves on
Figure 4b.) Because of this fact, the largest
TPW may be significantly smaller than the maximum
values possible - Maximum TPW velocity is
roughly 10 times the average TPW velocity and is
independent of timescale of forcing (tTPW) - See
Main Points (Section 1) for more conclusions
7. References 1. Goldreich, P and A Toomre.
Some Remarks on Polar Wandering, Journal of
Geophysical Research, 74 (10), 2555-2567,
1969. 2. Munk, WH and GJF MacDonald. The Rotation
of the Earth, Cambridge University Press,
Cambridge, 1960. 3. Richards, MA, Y Ricard, C
Lithgow-Bertelloni, G Spada, and R Sabadini. An
Explanation for Earths Long-Term Rotational
Stability, Science, 275 (17 Jan 1997), 372-375,
1997.