Vol. 6, No. 6, December 2004

*S. M. Molodensky*

**Schmidt Institute of Physics of the Earth, Russian Academy of Sciences,
Moscow, Russia**

The phenomenon of nearly diurnal free nutation of a planet with an ellipsoidal
homogeneous ideal incompressible liquid core and a solid shell was discovered
and examined simultaneously and independently in 1909-1910 in classical works by
Hough, Sludsky, and Poincare (e.g. see
[*Lamb,* 1945;
* Poincare,* 1910]).
This phenomenon reduces to the fact that, as the frequency of the diurnal tidal wave
approaches the resonance frequency equal to

(1) |

(here
*w* is the angular frequency of the Earth's diurnal
rotation,
*e*_{liq} is the geometric flattening of the liquid core, and
*A* and
*A*_{2} are the equatorial
moments of inertia of the whole Earth and the solid shell, respectively), the
amplitude of the differential rotation of the liquid core relative to the shell
increases unboundedly, which leads to an unbounded increase in the amplitude of
forced nutation of the shell in space excited by this wave.

In the case of a more realistic model of the Earth including an elastic or
inelastic mantle, a nonideal liquid core, and ocean, resonance amplitudes of
forced nutation and tidal strain of the shell have finite values determined by
the total amount of energy absorbed over the tidal cycle. Therefore, correct
estimation of the resonance amplitudes should take into account not only the
mantle elasticity effect on the value of
*s*_{p} but also all main dissipative
effects.

The nearly diurnal resonance of the liquid core, influencing significantly
precise measurements of amplitudes of forced nutation and diurnal earth tides,
has recently been studied rather extensively. Earth models fitting best the
entire set of modern astrometric and tidal gravity data were constructed
[*Molodensky,* 2004]
in order to describe dynamic tidal effects of the real Earth
including a heterogeneous compressible viscous electrically conductive liquid
core, inner solid core, viscoelastic shell, and ocean. In particular, analysis
of these models showed that the best fit between theoretical and observed
nutation amplitudes can be attained with the following values of parameters:

(2) |

where
*e*_{liq} and
*e*_{sol} are the effective dynamic flattenings of the
outer liquid and inner solid cores, respectively, and
*K*_{m} is the dimensionless coefficient of
effective inelasticity of the mantle defined as the ratio of the coefficient of
mantle effective rigidity
*l*_{2} (with respect to volume forces of
tidal type) for
diurnal oscillations to its value for oscillations at a period of 200 s (the
PREM model was constructed for this period).

The value
*e*_{liq} = 0.002736 exceeds the hydrodynamic equilibrium dynamic
flattening
of the liquid core
*e*_{hyd} = 0.00256 by about 8%, and the value
*e*_{sol} = 0.0053
is greater than the equilibrium flattening of the inner solid core by about two
times. As was shown in
[*Molodensky,* 2001;
* Molodensky and Groten,* 2001],
these divergences can be related to rheological properties of the mantle (due to
their
effect, the contemporary flattening of the liquid core can be conformable to the
daily rotation velocity existing at the time 280-230 Ma, when its value and
the
equatorial ratio of the centrifugal force to the gravitational force was
higher), as well as to the effect of the electromagnetic coupling of the liquid
core with the mantle and the inner solid core, which produces an additional
force moment acting on the inner solid core in the same way as the moment
arising due to the ellipticity of the solid and liquid cores. With electrical
conductivity values characteristic of metals at temperatures and pressures of
the Earth's core, the magnetic field diffusion time is many orders greater than
the period of nutation-induced oscillations of the solid core relative to the
liquid core, implying that both electromagnetic and inertial coupling does not
lead to tidal energy dissipation and a phase shift of the moment of
electromagnetic forces relative to the moment of hydrodynamic pressure forces.
Therefore, the fact that
*e*_{sol} is significantly greater than the hydrostatic
flattening of the inner solid core
*e*_{h} does not indicate that the real
(geometric) flattening of the solid core differs from
*e*_{h} significantly.

In the recent epoch, the period of the nearly diurnal free nutation in space determined by values (2),

(3) |

amounts to 434 sidereal days, which differs significantly from the solar year.
However, due to a higher ratio of the centrifugal force to the gravitational
force in the interval 280-230 Ma, hydrostatic flattenings of the liquid core
and
the mantle were also considerably higher. As is evident from formulas (1) and
(3), the period of nearly diurnal free nutation decreases with an increase in
*e*_{liq}.

In the recent epoch, the day duration is known to increase by about 2
milliseconds per century (mainly due to dissipation of energy in shallow ocean
areas). However, estimates for remote geologic epochs (based on calculations of
cotidal maps showing the then-existing land-sea distributions) yield an
appreciably larger value. If one assumes, in accordance with
[*Zharkov et al.,* 1996],
that an increase in the day duration over the last 280-230 Myr lie within
the range
*d**T*_{day} (4-6) 10^{3} s and the
value of the liquid core-mantle
electromagnetic coupling was nearly the same as presently, the period of nearly
diurnal free nutation in that epoch (with
*d**T*_{day} 5 10^{3} s) should have
been
exactly equal to the solar year, and the frequency of retrograde annual nutation
excited by the tidal wave
*Y*_{1} should have coincided with the nearly
diurnal
resonance frequency of the liquid core.

Resonance excitation of nearly diurnal free nutation abruptly increases the
amplitude of nearly diurnal oscillations of the liquid core relative to the
mantle. This should give rise to two observable effects: (1) a change in the
geomagnetic dynamo regime (particularly in the turbulent boundary layer at
liquid core/mantle interface, where the velocity gradient of nearly diurnal
oscillations is highest) and (2) an abrupt increase in tidal strain amplitudes
in the mantle (determined by the dimensionless Love ( *h* and
*k* ) and Shida ( *l* )
numbers.

The first effect can be associated with the period of anomalously rare changes in the orientation of the magnetic dipole relative to the Earth's geographic poles.

Movements of the Earth's magnetic poles (defined by the orientation of the
dipole that characterizes the main part of the geomagnetic field) are
reconstructed from paleomagnetic data. The movements of geomagnetic poles are
most clearly determined from profile data on the geomagnetic field recorded
across the strike of the Mid-Atlantic Ridge, where mantle material rises and,
upon cooling, moves horizontally eastward and westward at an average velocity of
about 2 cm yr
^{-1}; during this process, rocks retain in their "memory'' the magnetic
field corresponding to the Curie point (the temperature at which ferromagnetic
domains are pinned). Profile data obtained at various latitudes coincide,
implying that the maximum distance between the geomagnetic and geographic poles
never exceeded a few degrees. Polarity of the magnetic dipole experienced
jumplike changes. The time of constant orientation of the magnetic dipole varies
from a few tens of thousands to 50 million years (the longest period of the
Earth's history during which the orientation did not change coincides with the
interval 280-230 Ma). The northern magnetic pole was close to the northern
geographic pole in about half of polarity change cases; the northern (southern)
magnetic pole approximately coincides with the southern (northern) geographic
pole in the other half of cases. The approximate coincidence of geographic and
magnetic poles is accounted for by a strong effect of the Coriolis force on the
convective motions in the liquid core. According to the Taylor-Proudman theorem
(e.g. see [*Greenspan,* 1969]),
stationary (geostrophic) flows of a rotating liquid meet the condition
**v**/
*z*=0, where
* v* is the velocity vector and the

An abrupt rise in mantle amplitudes of the tidal strain in the resonance time
interval can lead to observed effects (e.g. the effects that are associated with
the rise in the amplitude of oceanic tides excited by the wave
*Y*_{1} and can be
studied from data on daily and yearly nodes of fossil corals). Below, to obtain
their estimates for the resonance epoch, I calculate the Love number
*h* (determining the ratios of vertical displacements of the Earth's surface
to the
vertical displacement of the equipotential surface of the summarized
gravitational potential of the Earth and the tide-generating potential at its
surface), the Love number
*k* (determining the ratio of the potential change due
to the tidal mass redistribution inside the Earth to the tide-generating
potential at its surface), and the Shida number
*l* (determining the horizontal
tidal displacements of the Earth's surface).

To estimate the effectiveness of the mechanism of resonance excitation of nearly
diurnal free nutation, one should evidently take into account the effects of
tidal energy dissipation due to which the resonance amplitude tends to a certain
finite limit. For this purpose, resonance effects of tidal energy absorption are
considered below (1) in oceans, (2) in an inelastic mantle, and (3) in
a viscous
core with regard for the electromagnetic coupling of the liquid core and mantle.
The inelasticity of mantle in the tidal frequency range, the viscosity of the
liquid core, and the core-mantle electromagnetic coupling were estimated with
the use of new data on amplitudes and phases of forced nutation obtained in
[*Molodensky,* 2000, 2004;
* Molodensky and Groten,* 2001].

Estimates in an inelastic mantle were obtained with the use of the power-law
function of creep with an exponent
*a* [*Smith and Dahlen,* 1981];
in this case, the ratios of the real and imaginary parts of the shear modulus at
frequencies
*w* and
*w*_{0} obey the relations
(e.g. see [*Zharkov et al.,* 1996])

(4a) |

(4b) |

where
*Q*_{m} are mechanical quality parameters,
*l* is the depth to a spherical
layer, and
*f*(*w*, *w*_{0})
is a coefficient depending on rheological properties of
the medium. In the particular case
*a* 0, this dependence
includes the
Lomnitz function of creep:

(5a) |

(5b) |

In the more general case of a power function with
*a*0, the functions
*f*(*w*, *w*_{0} )
and
*g*(*w*, *w*_{0} )
are given by the expressions

(6a) |

(6b) |

As was shown in
[*Molodensky,* 2004],
the optimal value of
*a* in the range of tidal periods is
*a* 0.04,
and the quality factor
*Q*_{m} in the lower
mantle (for the oscillation period 200 s) is

(7) |

The values of electromagnetic and viscous coupling between the liquid core
and
the shell are determined by the expressions
[*Molodensky,* 2004]

(8a) |

(8b) |

where

(9a) |

(9b) |

(9c) |

(9d) |

|*k*_{1}| and
|*k*_{2}| are the moduli of roots of the quadratic equation

(10a) |

corresponding to different signs of its right-hand side and
*f* = *s*/(2*w*).

(10b) |

*n* is viscosity;

(10c) |

(10d) |

*H*_{0} is the component of the magnetic field normal at the solid/liquid
core and
liquid core/mantle boundaries; and

is the signum function. Here
*C*_{c} is the principal moment of inertia of the
liquid core and the indices "in'' and "out'' denote components of moments,
respectively, coinciding in phase with the moments of the tidal forces (in-phase
components) and lagging behind them by
*p* /2 (out-off-phase components).

Very few data are presently available on the angular distributions of
*H*_{0} in
the core. However, since the moment of electromagnetic forces
*C*_{c} *t*^{(e-m)}_{in}
(** e**_{y} cos *s**t* -
** e**_{x} sin *s**t*) coincides
in
phase with the moments of tidal forces and forces of the hydrodynamic pressure
acting on the solid/liquid core and liquid core/mantle ellipsoidal boundaries,
the
*H*_{0} distributions can be taken into account by introducing parameters
of
the effective dynamic flattening of the solid and liquid cores. Because the
coefficients
*t*^{(e-m)}_{in} determined by relation (9b)
are negative for any distributions
of the functions
*x*(, *j*) (10b), the influence of this term reduces to the
fact that the effective dynamic flattenings of the inner and outer cores
*e*_{sol} and
*e*_{liq} become greater than their geometric flattenings. Because
of deficient
data, I assume below that the values of the antiferromagnetic and viscous
coupling did not significantly varied over the last 230 Myr and, therefore,
the
deviations of
*e*_{liq} and
*e*_{sol} from their hydrostatic values remained constant.

The resonance excitation of nearly diurnal free nutation including effects of the mantle inelasticity and electromagnetic and viscous coupling between core and shell can be described in terms of the Liouville equation for the liquid core

(11) |

where

(12) |

is the
*i* th component of the angular momentum of the liquid core,
*I*_{ik} is the
product of its moments of inertia,
*t* is its volume, and
* L* is the total
moment of forces acting on the liquid core; the latter includes the moments of
viscous (

In the bulk of the liquid core, I describe the displacement field by expressions of the simplest (Poincare) type:

(13) |

where
*y*_{0} is a dimensionless resonance parameter
determining the nutation
amplitude of the liquid core relative to the shell and
*e* is the
nutation amplitude relative to the moving (Tisserand) coordinate system. The
substitution of (13) into (12) yields

(14) |

where
*A*_{1} and
*C*_{1} are the principal moments of inertia of the liquid core;
*I*_{1} is the variation amplitude of inertia products determined by
the relations

(15) |

(16) |

and
*e*_{d} = (*C*_{1} - *A*_{1})/*C*_{1}
is the dynamic flattening of the liquid core.

In order to calculate the total moment of forces exerted by the liquid core on the shell, I calculate the moment of hydrodynamic pressure forces:

(17) |

The substitution of relations (8a), (8b), (14) and (17) into (11) yields

(18) |

where

- is the effective dynamic flattening of the core/mantle boundary.

The second relation connecting the parameters
*y*_{0} and
*e* can be obtained by using
the continuity condition for the normal component of displacements at the
core/mantle ellipsoidal boundary. The normal to this surface can be represented
as

where
*e*_{g} is the geometric flattening of this surface. In the
case of diurnal
tidal waves, the dependence of the normal component of displacements of the
core/mantle inelastic boundary on the angular variables ( , *f* )
and time
*t* can
be represented as

(19) |

where
*h*^{(m)} is the complex amplitude independent of ( , *f*, *t* ).
Then, I obtain

(20) |

Equations (18) and (20) form a closed system of two algebraic equations (with
complex coefficients) with respect to the two unknown complex parameters
*e* and
*y*_{0} determining amplitudes and phases
of the shell nutation and tidal
oscillations of the core relative to the shell near the resonance. The values of
*I*_{1} and
*h*^{(m)} represented as coefficients in these equations were
calculated for
rheological models (4)-(6) in [*Molodensky,* 2004].

Below, I demonstrate that, with these values of parameters, the amplitude of resonance oscillations of the liquid core is about one and a half order of magnitude higher than their amplitude in the recent epoch and, consequently, the hypothesis that the geomagnetic variation pattern and the resonance excitation of nearly diurnal nutation were interrelated in the geologic epoch under consideration is fairly well substantiated.

For the numerical simulation of the resonance excitation of nearly diurnal free
nutation in the interval 280-230 Ma, I used values of viscosity of the core
and
quality factor of the mantle from admissible ranges estimated in
[*Molodensky,* 2002, 2004].
Because of lacking data on the temporal variation in the core-mantle
electromagnetic coupling, I used its contemporary value determined by the
real part of the contemporary dynamic flattening of the liquid core (see formula
(2)).

Amplitude ratios of forced nutation of the real Earth to nutation of the Earth's
solid model
^{e/e}0,
resonance parameters
*Y*_{0}, and dynamic values of the Love
numbers
*h* and
*k* and the Shida number
*l* calculated by formulas (18) and (20)
are presented in the Table 1 for geologic epochs corresponding to various values
of the dynamic flattening of the liquid core
*e*_{liq}. The first value
*e*_{liq} = 2.736
10
^{-3}, presented in this table, is the effective dynamic flattening of the
liquid core (including its electromagnetic coupling with the mantle) in the
recent epoch, whereas the value
*e*_{liq} = 3.0615
10
^{-3} corresponds to an epoch of
approximately 280-230 Ma (more accurate dates are not presented due to
determination uncertainties in values of the secular slowing-down of the Earth's
daily rotation, although one may evidently assume that, in the first
approximation,
*e*_{liq} decreases linearly with time).

As seen from the table, the Love and Shida numbers of the resonance epoch exceed
their contemporary values by about two orders of magnitude, and resonance values
of the factor
*Y*_{0} (proportional to the amplitude of
diurnal oscillations of
the liquid core relative to the shell) are about 300 times greater than its
contemporary values for the same retrograde annual component. The values of
*Y*_{0} are about an order of magnitude
higher than the amplitude of
contemporary nearly diurnal oscillations of the core (excited by the main
lunisolar component
*K*_{1} ). Taking into account that, given turbulent friction of
the core against the shell, the force of viscous friction is proportional to the
second power of the resonance factor, and the absorbed energy is proportional to
its third power, it is evident that the dissipation rate of tidal energy at the
core/mantle boundary should have increased by a about
10^{3} times. Therefore, the
hypothesis on the interrelation between the geomagnetic variation pattern and
the resonance excitation of nearly diurnal nutation 280-230 Ma is fairly
plausible. Moreover, the estimate of the contribution of this effect to the
history of the Earth's thermal balance is of significant interest.

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