Vol 1, No. 1, July 1998

Translated December 1998

*B. I. Birger*

**Schmidt United Institute of Physics of the Earth, Russian Academy
of
Sciences, Bol'shaya Gruzinskaya ul. 10, Moscow, 123810 Russia**

A power-law non-Newtonian fluid is usually assumed to model
slow flows in the mantle and, in particular, convective flows.
However,the power-law fluid has no memory in contrast to a real
material. A new nonlinear model with a memory was recently
proposed recently by * Birger* [1998]. The proposed
model reduces to the power-law fluid model for stationary flows
and to the Andrade model for flows associated with small
strains.

The steady-state convection beneath continents was studied
by * Fleitout and Yuen* [1984], who used a power-law fluid
model
and obtained a cold immobile boundary layer
(the continental lithosphere). In stability analysis of this
layer, the Andrade model must, however, be used. The analysis
shows that the lithosphere is overstable, with a period of
oscillations of about 200 Ma. These thermoconvective
oscillations of the lithosphere are suggested to provide a
mechanism for the formation and evolution of sedimentary
basins on continental cratons [*Birger,* 1998]. The vertical
crustal movement in sedimentary basins can be respresented
as a slow subsidence on which small-amplitude
oscillations are superimposed. The longest period of
the oscillatory crustal movement is of the same order of
magnitude as the period of convective oscillation of the
lithosphere found in the stability analysis. Taking into
account the difference between the depositional and
erosional transport rates we can explain the permanent
subsidence of sedimentary basins, as well as their
oscillation [*Birger,* 1998].

Analysis of convective stability for a horizontal layer involves perturbations harmonically depending of the horizontal coordinate. For a layer with the Andrade rheology, the instability is oscillatory and perturbations can take the form of travelling and standing thermoconvective waves. In this study, we adress the problems on the generation of thermoconvective waves under various initial conditions: the linearized equations for thermal convection in a layer with the Andrade rheology are solved for given initial perturbations of temperature.

A linear rheological model (having a memory) of the lithosphere is described by the integral relationship

(1) |

where
*e*_{ij} and
*t*_{ij} are the deviatoric strain
and stress tensors,
respectively,
*t* is time, and
*K*(*t*) is the creep kernel given by

(2) |

where
*A* is the Andrade rheological parameter. The creep kernel (2)
is introduced so that, in the case of constant stress,
the strain depends on time as
*t*^{1/3} (the Andrade law).

The linearized equations of thermal convection in a horizontal layer heated from below are written as

(3) |

where
*z* is the vertical coordinate,
*x* and
*y* are the horizontal
coordinates,
*v* is the velocity,
*q* and
*p* are the perturbations
of temperature and pressure. The set of equations (3) is
written in the dimensionless form. The length scale is
layer thickness
*d* and the temperature scale is a
temperature drop
*D**T* between the hot lower and cold upper
surfaces of the layer (both surfaces are supposed to be
isothermal). The time scale is
*d*^{2}/*k*, where
*k* is the thermal
diffusivity, and the velocity scale is
*k*/*d*. For a Newtonian
fluid, the stress (and pressure) scale
*k**h**d*^{2}
is usually taken,
and the Rayleigh number is Ra =
*r**g**a**D**Td*^{3}/*h**k*,
where
*r* is the
density,
*a* is the thermal expansion coefficient,
*g* is the
gravitational acceleration, and
*h* is the Newtonian viscosity
having the dimension of Pa s. For the Andrade rheological medium
(the Andrade parameter
*A* has the dimension of Pa s
^{1/3} ), we
introduce a reference viscosity

Then, the Rayleigh number is defined as

(4) |

and the stress scale is
*k**h*_{A}
*d*^{2} = *A*(*d*^{2}/*k*)^{-1/3}.

The lithosphere is characterized by the following
depth-averaged values [*Birger,* 1995]:

(5) |

Equations (3) were written in the Boussinesq
approximation, which is valid if several
dimensionless parameters are small; one of these parameters
is
*a**D**T*. This
parameter is estimated for the lithosphere as
*a**D**T* 0.04. As the zero-order
approximation in the small parameter
*a**D**T*, the
upper
free-deformable surface of the layer, where stresses vanish,
behaves like a "free" boundary; i.e., the condition of zero
normal stress is replaced by a condition of zero vertical
velocity on this boundary. Let us suppose the lower boundary
of the layer to be also "free". Then, the boundary conditions
on the upper and lower surfaces of the layer are

(6) |

Note that the use (6) permits us to find
an exact solution that is not significantly different from
the numerical solutions obtained for more realistic boundary
conditions [*Birger,* 1995, 1998].

The set of equations (1)-(6) has a solution in the form of a thermoconvective wave

(7) |

where
*C* is an arbitrary complex factor (the amplitude of
temperature),
*k*_{x} and
*k*_{y} are the components of wave vector
describing the periodicity in the horizontal directions,
*k* is
the wave number,
*w* is the complex frequency (its imaginary
part describes the wave attenuation), and
*F*(*w*) is the
complex viscosity defined as

(8) |

*K*^{}(*i**w*)
being the Laplace transform of the creep kernel. The
complex viscosity is related to the wave number
*k* by
the dispersion relation

(9) |

This allows to find such a value of the
Rayleigh number Ra
_{m} (called the minimal critical Rayleigh
number) that only a wave with
*k* = *k*_{m} and
frequency
*w* = *w*_{m}
does not attenuate.
For the Andrade rheological
model, the complex viscosity is

(10) |

where
*G*(*x*) is the gamma-function, and Ra
_{m},
*k*_{m} and
*w*_{m} take on the following values:

(11) |

According to the estimated parameters the lithosphere
(5), the Rayleigh number Ra for the lithosphere is on the
same order of magnitude as Ra
_{m}. Thus, the lithosphere is in
the state close to its instability threshold. If
Ra > Ra_{m}, the initial perturbations increase with time,
and at large
*t*, both the linearized equations of thermal convection
(3) and the linear rheological relationship (1) cannot be used.
If
Ra *Ra*_{m}, the solution of
the set of equations (1)-(6), for
a given but not too great initial perturbation of temperature,
completely describes the evolution of the perturbations in the
layer modeling the lithosphere.

The solution (7)-(11) is obtained in the zero-order
approximation in the small parameter
*a**D**T*.
In this approximation,
the vertical displacement
*u*_{z} of the upper surface of the layer is
equal to zero. In the the same approximation in
*a**D**T*, we
find

(12) |

which does not depend on the rheology of the layer.

If the initial perturbation of temperature at
*t* = 0 is given
in the form

(13) |

the factor
*C* in (7) and (12) is defined as
*C* = *q*_{0}.
However, the initial condition (13) does not completely determine
the evolution of the perturbations. Since the wavenumber
*k*,
rather than the components
*k*_{x} and
*k*_{y} of the wave vector,
enters the dispersion relation (9), in addition to the
thermoconvective wave (7), there is a solution with the wave
vector
(-*k*_{x}, -*k*_{y}) describing the wave
that runs in the
opposite direction. The superposition of the waves,
travelling in the opposite directions, forms a standing wave
(thermoconvective oscillation). The solutions of governing
equations (3)-(6) in the form of both running and standing waves
satisfy the initial condition (13). This ambiguity is removed if
the initial perturbation (13) is replaced by a more realistic
initial perturbation that occupies a finite region and tends to
zero for large
*x* and
*y*. Such centered initial perturbations are
treated in the next sections of the paper.

In this case, the initial temperature perturbation will be given as

where
*q*_{0}(*x*, *y*) is not zero
only in a limited area on the plane
*xy*, and the solution is sought for in the form

Thus, the dependence of the solution on z is fixed, and hence,
the problem of three-dimensional distribution of temperature
perturbation in the lithosphere is reduced to a
two-dimensional problem. In the case, when the initial
temperature perturbation does not depend on
*y* and has the form
*q*_{0}(*x*), the problem of two- dimensional
temperature
distribution is reduced to a one-dimensional problem.

The solution in the form of thermoconvective wave is valid for a sufficiently large time elapsing from the moment of the perturbation onset. Under this condition, the stresses harmonic in time induce strains also harmonic in time in the rheological model (1), and the complex analog of viscosity (10) depends only on the frequency, rather than on time.

When the initial temperature perturbation is given in the form (13), we seek a solution of convection equations in the form

Thus, the coordinate of the temperature and velocity remains in
the same form as (7) and the problem reduces to
the determination of the time dependence. We seek the solution
for
*k* = *k*_{m} and Ra = Ra
_{m}, whose the values are found in (11).

THe Laplace transformation of the basic equations (1)
and (3) and the elimination of all of the physical variables,
except the temperature, yields the Laplace transform of the
desired function
*q*(*t*)

(14) |

The viscosity analog is now the function

(15) |

Rearranging the denominator of the expression in the right-hand side of (14),

(16) |

where
*s*_{1} = *s*/(*k*^{2}_{m} + *p*^{2}).

Since

we may assume that the expression (16) is zero when
*s*_{1} = *i*3^{1/2},
*s*_{1} = -*i*3^{1/2} and
*s*_{1} = -1/3.
However, expressions (10) and (15) for
the complex viscosity imply that only the first value of the
root, for which the argument of the complex number s satisfies
the condition
0 arg *s* < 2*p*,
must be used.
Then,

and hence, (16) is zero for
*s*_{1} = *i*3^{1/2} but
not for
*s*_{1} = -*i*3^{1/2} and
*s*_{1} = -1/3.
The Laplace transform (14) can be rewritten as

(17) | |

This expression has two singularities: at the
branch point
*s* = 0 and at the pole
*s* = *i*3^{1/2}(*k*^{2}_{m} + *p*^{2}).
In the vicinity of point
*s*=0,

(18) |

and in the vicinity of point
*s* = *i*3^{1/2}(*k*^{2}_{m} + *p*^{2}),

(19) |

Using the theorem on the asymptotic behavior of Laplace originals
[e.g., * Von Doetsch,* 1967], we find the following asymptotic
solution for large
*t*

(20) |

where
*w*_{m} = 3^{1/2}(*k*^{2}_{m}
+ *p*^{2}),
which corresponds to (11). The
first aperiodic term in the right-hand side of (20) is much
smaller than the second, periodic term, even for
*t* equal to the oscillation period
2*p*/*w*_{m} 1/5.
Thus, for
*t* 2*p*/*w*_{m}, the solution takes the form

(21) |

Since
|*C*_{1}| 2, the amplitude of steady-state
thermoconvective wave
is two times greater than the initial amplitude of
temperature perturbation
*q*_{0}. The argument of complex number
*C*_{1} defines a phase difference. In the next sections of the paper,
the factor
*C*_{1}, which
appears in the transient process, is omitted for brevity.

Taking
*k*_{y} = 0 and
*k*_{x} = *k*, we first consider only
one-dimensional perturbations independent of the coordinate
*y*. Expanding the
complex
frequency into the Taylor series in the neighborhood of
*k* - *k*_{m},

(22) |

where the coefficient
*V* means the group velocity of a packet
of thermoconvective waves. Substituting (22) into the
dispersion relation (9), we find the values of the
coefficients in (22) for the Andrade model

The group velocity
*V* of thermoconvective waves in a medium with
the Andrade rheology is slightly lower than the
phase velocity
*w*_{m}/*k*_{m}
= 7*p*/2.

The initial temperature perturbation
*q*_{0}(*x*) is represented in
the form of the Fourier integral

(23) |

where
*F*(*k*) is the Fourier transform of the initial
temperature

(24) |

The solution satisfying the initial condition is represented as

(25) |

Substituting the expansion of frequency (22) into (25) and
denoting
*k* - *k*_{m} = *u*, we obtain

(26) |

where the integration is actually taken over a small vicinity
of the point
*u* = 0
(*k* = *k*_{m}) and the following notation is introduced

(27) |

The integral in (26) can then be evaluated by the saddle
point method [e.g., * Copson,* 1965].
The stationary point
*u*_{0} is found
from the condition

(28) |

Equations (27) and (28) give

(29) |

In the vicinity of point
*u*_{0}, function
*f*(*u*) is
represented by the series

(30) |

Since
*f*^{}(*u*_{0}) = 0
and
*f*^{}(*u*_{0})
= -2*a*,
(30) takes the form

(31) |

where only two first terms of expansion are retained.

Substituting (31) into (26),

(32) |

The point
*u*_{0} is the saddle point for the complex function
*f*(*u*).
According to the saddle point method, the path of integration on
the complex plane is chosen in such a way that it passes through
point u and, in the small vicinity of this point, the path
is a
segment of the staight line on which the function
*f*(*u*)-*f*(*u*_{0}) is
real
and negative. In the vicinity of
*u*_{0},

(33) |

where

It is clear from (33) that the straight line for which
*b* = -*a*/2 must
be chosen as the path of integration. On this straight line,
the function
*f*(*u*)-*f*(*u*_{0}) takes on real negative values
and the
integral in (32) reduces to the simple expression

(34) |

In transformation (34), the integral is reduced to the real
error integral, and the transition to the infinite
limits of
integration can be made under the condition
|*a*|*t* 1. Since
|*a*| 10,
the result given by (34) is holds true for
*t* 1/10.

Thus, the solution in the case of an arbitrary initial perturbation has the form

(35) |

This is a wave packet moving to the left. To obtain the total
solution, a term corresponding to the wavenumber
*k* = -*k*_{m} must be
added to the right-hand side of (35). This term, which is
readily
obtained by changing the sign of
*k*_{m} and
*V* in the right-hand
side of (35), describes a wave packet moving to the right.

The quantity
*x* = *x* + *Vt* can be interpreted as
a coordinate in
the reference frame that moves together with the wave packet at
the group velocity
*V*. The right-hand side of (35) goes to zero at
*x*^{2}/4|*a*|*t* 1. The width of the wave packet can be considered to be
of the order of
2(2|*a*|*t*)^{1/2} ; i.e., the wave packet exists only
for
|*x*| (2|*a*|*t*)^{1/2}.
Under this condition, it follows from (29)
that

Since
|*a*|*t* 1, we find that
|*u*_{0}| *k*_{m}.
Consequently, in
the expression
*F*(*k*_{m} + *u*_{0})
in (35) can be replaced with
*F*(*k*_{m}).

In the case when the initial perturbation of temperature
takes place in a fixed point
*x* = 0, function
*q*_{0}(*x*) and its
Fourior transform are written as

(36) |

where
*d*(*x*) is the delta-function satisfying the
relationship

Note that (36) retains its form in the dimensional variables
since the delta-function has the dimension of reciprocal
length and the quantity
*Q*_{0} has the dimension of temperature
multiplied by length.

As follows from (35), the asymptotic
(|*a*|*t* 1) solution
under the initial condition (36) takes the form

This solution represents two wave packets running in the
opposite directions from the point
*x* = 0, where the initial
perturbation takes place. The distributions of temperature at
fixed moments of time are shown in Figure 1.

Let the initial perturbation be represented by the sum of two delta functions

(37) |

The Fourier transform of (37) is

but it cannot be used in the problem with the
initial condition (37) because its solution is
a simple superposition of two solutions obtained for the initial
perturbations specified at points
*x*=*l* and
*x*=-*l*. Four
wave packets move away from these points: two of them move From
*x* = *l* toward the left and the other two move from
*x* = -*l* toward the right. Their velocities are
*V*, respectively. At the
moment
*t*_{0} = *l*/*V*, the centers of the packets, moving in the
opposite directions, meet each other at point
*x*=0. The perturbation of temperature at
-*l* *x*
*l* is the superposition of two wave packets

(38) |

Denoting
*t*_{1} = *t* - *t*_{0} ( *t*_{1} is positive after
the meeting and negative before it), substituting
*t* = *t*_{0} + *t*_{1} into the right-hand side of
(38) and observing that

after algebraic transformations, we obtain

(39) |

Thus, (39) represents the perturbation of temperature as the
superposition of a standing wave and two running waves moving in
the opposite directions. The amplitudes of the running waves
is zero for small
*x* and
*t*_{1}, and in the vicinity of
*x* = 0, the
solution (39) reduces to

(40) |

Equation (40) holds for
|*x*| *D**x*
and
|*t*_{1}| *D**t*,
where
2*D**x* is the width of the standing wave zone,
2*D**t* is the lifetime of the standing wave.
These quantities are estimated as

(41) |

where
|*a*| 10. The typical dimension of a craton
is on
the order of 2000 km, which corresponds to
*l* 10. Then,
*t*_{0} = *l*/*V* 1,
the width of standing wave zone is
2*D**x*
5, and its lifetime is
2*D**t*
1/2. Since the wavelength is
2*p*/*k*_{m} 2 and the period of
convective oscillation is
2*p*/*w*_{m} 1/5,
the zone of standing wave
envelops two wavelengths, and two periods of oscillation
occur for the lifetime of the standing wave.

Consider another example of the initial perturbation

(42) |

The Fourier transform for function (42) is

(43) |

With transform (43), the solution is

(44) |

Since the dominant contribution to the integral in (44) comes
from small vicinities of points
*k* = *k*_{m} and
*k* = -*k*_{m}, (44) can
be rewritten in the form

(45) |

where
*w* is a function of
*k* and
*w*(*k*_{m}) = *w*_{m}.
Each of the four
integrals in (45) describes a wave packet associated with
the initial perturbation in the form of
*d* -function. The
first packet moves from point
*x* = -*l* toward the left,
the second one moves from
*x* = -*l* toward the right, the third one moves from
*x* = *l* toward
the left, and the fourth one moves from
*x* = *l* toward the right. The first
and fourth packets move outward from the initial region of
temperature perturbation, and the second
and third packets move inward this initial region. When they
meet, the standing wave is formed in the
vicinity of
*x* = 0, like in the problem with
the initial condition given by the sum of two delta functions.

Considering two-dimensional perturbations, we
introduce, for brevity, the notation
*k*_{x} = *p* and
*k*_{y} = *q*. Then,

(46) |

The complex frequency
*w* is expanded in the power series

(47) |

where

(48) |

The solution, satisfying the initial conditions, is written as

(49) |

where
*F*(*p*, *q*) is the Fourier transform
of the initial temperature
distribution
*q*_{0}(*x*, *y*)

Substituting (47) for
*w* into (49), we get

(50) |

where

The stationary point
(*u*_{0}, *v*_{0}) of the function
*f*(*u*, *v*) is defined by
the condition

which allows us to find

(51) |

Since

the function in the neighborhood of point
(*u*_{0}, *v*_{0}) is
represented as

(52) |

where

Substituting expansion (52) into (50),

(53) |

where

Using the substitution

*M* is represented in the form

Evaluating the integrals
*M*_{1} and
*M*_{2} by the saddle point
method, we reduce (53) to

(54) |

where

The wave vector components
*p*_{m} and
*q*_{m}, satisfying
condition (46), can be represented as

(55) |

where
*j* varies from 0 to
2*p*. Since all the directions of
the wave vector
(*p*_{m}, *q*_{m}) are equivalent, the integration
over
*j* is implied
in the right-hand sides of (53) and (54).

If the initial perturbation takes place at point (0, 0), then

and we must substitute in (54) the Fourier transform of this function

(56) |

It is convenient to introduce the polar coordinates
*r* and
*y*

(57) |

As follows from (55) and (57), the factor
*E*_{m} in (54) becomes

Thus, the temperature perturbation
*q* induced by the initial
point-concentrated perturbation is given by the equation

(58) |

where the dependence of integrand on
*j* is defined by equations
(48) and (55). Since it is clear that
*q* depends on
*r* rather
than on
*y*, for the case of the initial point-concentrated
perturbation,
we can take
*y* = 0 (and hence
*x* = *r*,
*y* = 0 ) in the integrand. Then,

(59) |

where

The integral
*I*_{1}(*r*, *t*) at large
*r* is calculated using the
saddle point method

(60) |

(61) |

After simple algebraic transformations, we find an asymptotic
solution (valid only for sufficiently large
*t* and
*r* ) of the
problem with the initial condition specified at the given point:

(62) |

where

Note that parameter
*a* has already appeared in the
one-dimensional
problem. Solution (62) describes a cylindrical wave
because
the dependence on coordinates reduces to the dependence only on
*r* = (*x*^{2} + *y*^{2})^{1/2}.

The function
*f*_{1}(*j*) introduced in (59)-(61),
in addition to
the saddle point
*j*_{0} = *p*,
has the second saddle point
*j*_{0} = 0. The use of the latter would
lead to an additional
term in the right-hand side of (60). This term describes a wave
running to the point
*r* = 0 from outside and includes the factor

which becomes zero for sufficiently large positive
*r* and
*t*.

In the center of the wave packet, i.e., at
*x* = 0 (*r* = 3*p**t*),
equation (62) violates. When
*x* = 0,
the second derivative
*f*^{}_{1}(*j*_{0}) is zero (the third derivative is also
zero).
In this case, the saddle point method leads to

(63) |

instead of (60). Note that
*G*(1/4) 4.
The derivation of the
asymptotic relation (63) that holds for large
*r*, is analogous to
that of (30)-(34), with the exception that the
Taylor series is trancated at the term including the fourth
derivative, and furthermore, the integral

is used instead of the error integral.

Substituting the value of the fourth derivative

into (63), we find the solution valid in the small vicinity of
the wave packet center, i.e., at
*r* 3*p**t*,

(64) |

A simplified approach of this section to the problems
under consideration is not to expand
*w* in the power series but to
set
*w* = *w*_{m}
for values of
*k* = (*p*^{2} + *q*^{2})^{1/2} close to
*k*_{m} in equation (49). Then,

(65) |

where

Here the polar coordinates are used and the integral is taken over
a ring on the plane
*p*, *q*. The radius of the ring is equal to
*k*_{m}, but
its thickness
*D**k* is indeterminate in the framework of
the
simplified approach
used now. In the interval
0 *j*< 2*p*,
function
*f*(*j*, *y*)
has the following stationary points

(66) |

For the initial perturbation in the form of
delta function (an elementary radiator of thermoconvective
waves), substituting
*F* = *Q*_{0}/4*p*^{2} into (65), using the saddle
point method, and omitting the solution corresponding to the saddle point
*j*_{0} = *y*
(a wave
arriving at point
*r* = 0 from outside), we find the
solution for large
*r*

(67) |

In contrast to solution (62), which takes into account the
wavenumber dependence of
*w* (i.e., dispersion), equation (67)
describes a wave with nonmodulated amplitude. It is noteworthy
that the simplified approach gives the factor
*r*^{-1/2} in (67),
which determines attenuation of any cylindrical wave
[e.g., * Whitham,* 1974]. Comparison of solution (67) with
solutions (62) and (64) shows that, in order to take the
dispersion into account, it is enough to substitute into (67) the following
expressions for
*D**k*

(68) |

We then use the same simplified approach for the case when the initial perturbation has the form

that is, the uniform temperature perturbation
*q*_{0} at the
initial moment is given within a square with a side
2*l*. For
such initial condition,

(69) |

It is convenient to rewrite (69) in the form

(70) |

Substituting (70) into (65),

(71) |

Consider separately the integral corresponding to the first term.
The vector ( *x* + *l*,
*y* + *l* ) connects the apex
(-*l*, -*l*) of the square with
the point
(*x*, *y*). Introducing a polar coordinate system whose
origin is located at this apex,

(72) |

Then,

(73) |

Using the saddle point method and omitting the term that
describes the wave running to the apex from outside, we obtain
the expression for
*J*_{1} at sufficiently large
*r*_{1}

(74) |

Equation (74) is valid when
*y*_{1} is outside of a small
vicinity of points 0,
*p*/2,
*p*, and
3*p*/2. Substituting these
values of
*y*_{1} into (73), we
readily verify that the integral
*J*_{1} goes to zero for these
values of
*y*_{1}.

The total solution is obtained as the sum of four integrals

(75) |

Here, the angles
*y*_{3} and
*y*_{4} corresponding to the apexes
(*l*, -*l*) and
(-*l*, *l*) are measured clockwise, whereas
*y*_{1} and
*y*_{2} are
measured counter-clockwise. Substituting

into (67), we write the solution in the form

where

(76) |

Equation (76) is valid for all
*x* and
*y* except the points located
on the lines
*x* = *l*,
*y* = *l* and in the neighborhood of
these lines
having the width of the order of the wavelength
2*p*/*k*_{m}.

Introducing the polar coordinates
*x* = *r* cos*y*,
*y* = *r* sin*y* and
using the simple relationship

we obtain that, for
*r* *l* (the great distance from the
initial square), solution (76) reduces to

(77) |

where

Equation (77) describes the wave running outside from the
initial square. Note that
*G* and hence the right-hand side of
(77) does not depend on
*y* only under the condition
*k*_{m}*l* 1 (the length
of the side of the initial square is small in comparison with
the thermoconvective wavelength). Under this condition, the
initial pertur-bation given in the square acts like an initial
pointwise perturbation.

Function
*G*(*y*) is plotted for various values of
*l* in
Figure 2.
The directivity pattern of the thermoconvective radiation
is defined by the amplitude factor
|*G*(*y*)|.
As follows from the plots in Figure 2,
the directions of the most intense radiation are
*y* = 0,
*y* = *p*/2,
*y* = *p*,
*y* = 3*p*/2. When
*l* 2, the real function
*G*(*y*) periodically changes its sign, and
hence,
arg *G*(*y*) takes on the
values 0 or
*p*. Thus, the phase of the wave described by (77),
as well as the
amplitude, is direction-dependent.

In the theory of radiation, a radiator whose dimensions are
much smaller than the radiated wavelength is called simple.
The simple radiator has an omnidirectional radiation. A composite radiator
is one whose dimensions are not small compared to the
wavelength and which emits a directional radiation. Thus, the
square where the temperature is initially perturbed is a
simple radiator of thermoconvective waves
if
*k*_{m}*l* 1, but this
square is a composite radiator if
*k*_{m}*l* > 1.

In order to account for the dispersion that leads to
the amplitude and phase modulation, it is enough to
substitute the
expressions (68) for
*D**k* into (77).

For
*r* *l* (in the neighborhood of the center
of the
initial square), the following relationship takes place

where
2^{1/2}*l* is the distance between the apex and the center of
the square. By using this approximate equation, we reduce
(76) for
*r* *l* to the form

(78) |

where

As is seen from (78), a standing wave with square cells is
formed in the central part
(*r* *l*) of the initial square. The
sides of the cells are parallel to the sides of the initial square
and the length of the cell side is
2^{1/2}*p*/*k*_{m}.
The same standing wave is formed
in the neighborhood of the point (0,0) in the case when the
initial temperature perturbation is given at four points:
(*l*, *l*),
(*l*, -*l*),
(-*l*, *l*), and
(-*l*, -*l*) i.e. when the initial perturbation
square is replaced by the pointwise perturbation at its apexes.

Note that in order to obtain the solution (77), valid for
*r* *l*,
we could use the expression (69) for
*F*. We represents
*F* in the
form (70) to find the solution for
*r* *l* i.e. in the neighborhood
of the center of the initial square. The result (78), obtained
by the saddle point method, is holds true under the condition
*k*_{m}*l* 1.
(length of the side of the initial square is much greater
then thermoconvective wave length). This condition is satisfied when
the initial temperature perturbation covers a craton as a
whole, and therefore,
*l* 10. for
*k*_{m} = 2.7.

The distribution of temperature in the lithosphere is represented in the form

where
*q*(*x*, *y*, *t*) can be interpreted
as a temperature
perturbation in the middle of the lithosphere
(*z* = 1/2).
The temperature
perturbation
*q*(*x*, *y*, *t*) is found under
a few simple initial
conditions: the initial perturbations
*q*_{0}(*x*, *y*) are given on
the straight line
(*x* = 0), in the strip
(-*l* *x*
*l*), at the point ( *x* = 0,
*y* = 0 ), at
four points, or within a square.

For a plane thermoconvective wave with the wave vector
(*k*_{x}, *k*_{y}),
the displacement of the upper surface of the layer
(lithosphere) is related to the temperature perturbation by (12).
This relation is easily generalized to the case of a packet of
thermoconvective waves with the wave numbers close to
*k*_{m}

When more real boundary conditions on the upper and lower surfaces
of the lithosphere are considered [*Birger,* 1988],
the dependence of temperature perturbation on the vertical
coordinate
*z* is not determined by the function
sin *p**z* and the wave number
*k*_{m} is
not equal to 2.7. However, the displacement
*u*_{z}(*x*,*y*,*t*) remains
equal to the temperature perturbation
*q*(*x*,*y*,*t*) multiplied by a
constant whose value depends on the boundary
conditions. This is also valid for the case when the depth dependence
of physical parameters (in particular, the Andrade rheological
parameter) of the lithosphere is taken into account. Thus,
functions
*q*(*x*, *y*, *t*), found above under
various initial
conditions, describe the
vertical displacements of upper surface of the lithosphere,
with accuracy to a constant factor. These
displacements determine sedimentary processes.

The initial temperature point-concentrated perturbation
can be treated as a source of thermoconvective waves in the
lithosphere. When the initial perturbation occupies a finite area,
thermoconvective waves propagate outward from this
area and thermoconvective oscillations (standing waves) are settled
inside the area. The thermoconvective oscillations create a
system of convective cells in the lithosphere. Over the
convective cells, the surface of the lithosphere subsides (or
rises) forming sedimentary basins. Since the Rayleigh number for
the lithosphere is not greater than Ra
_{m}, the shape of
cells and hence the shape of basins is determined by initial
perturbations. The initial perturbation in the square
considered above leads to the appearance of square cells.
However, other initial conditions may lead to rectangular,
hexagonal, and other shapes of cells and basins.

Thermoconvective oscillations may be considered as a
mechanism inducing the formation and evolution of sedimentary
basins on continental cartons [*Birger,* 1998].
In this paper, we have studied
the excitation of thermoconvective waves and oscillations by
initial perturbations of temperature. Note that these waves are
also generated from initial vertical displacements of the
Earth's surface (relief perturbations).

Packets of thermoconvective waves propagate in the
lithosphere with the velocity
3*p**k*/*d* 0.15 cm/year and create
sediment-filled depressions on the upper
surface of the lithosphere. The age of sediments increases
proportionally to the
distance from the front of the wave to its source.
Thermoconvective
waves may be related to the development of peripheral
depression on a craton. In this process known in geology
[*Beloussov,* 1978; * Khain,* 1973],
the initial depression occurring in a
geologically active area ("geosyncline") adjacent the craton
slowly develops within the craton.

Beloussov, V. V., * Endogenous Regimes of Continents,*
Nedra, Moscow, 1978 (in Russian).

Birger, B. I., Linear and weakly nonlinear problems of the
theory of thermal convection in the earth's mantle, * Phys. Earth Planet. Inter.,
50,* 92-98, 1988.

Birger, B. I., On the thermoconvective mechanism for
oscillatory vertical crustal movements, * Phys. Earth Planet. Inter., 92,* 279-291,
1995.

Birger, B. I., Rheology of the Earth and thermoconvective
mechanism for sedimentary basins formation, * Geophys. J. Inter., 134,* 1-12,
1998.

Copson, E. T., Asymptotic Expansions, University Press, 170 pp., Cambridge, 1965.

Fleitout, L., and Yuen, D. A., Steady state, secondary
convection beneath lithosphere plates with temperature- and
pressure-dependent viscosity, * J. Geophys. Res., 89,* 9227-9244, 1984.

Khain, V. E., * General Geotectonics,* 512 pp., Nedra, Moscow, 1973 (in
Russian).

Von Doetch, G., Anleitung zum praktischen Gebrauch der Laplace-Transformation und der Z-Transformation, Springer, 365 pp., Munich, 1967.

Whitham, G. B., * Linear and Nonlinear waves,* John Wiley and
sons, New York, 1974.