A few integral physical quantities of the system

In general, one studies the deformations and the dynamics of the Earth in terms of a model of continuum characterized by an internal distribution of mass not specified for the moment, in other words by a mass density ρ, in a volume B delimited by a surface ∂B. In the problem which occupies us, this border ∂B is not given a priori , but must be given by supposing that it corresponds to the shape of balance of the deformable body. Moreover, we will suppose for the moment that there exists a field speed v_i compared to the selected frame of reference but not yet specified, as well as a density of kinetic moment internal ℓ_i, a voluminal force f_i, a Stress field (or field of tensions ) T_ij, one moment of voluminal force m_i and surface moment of force M_ij. All these quantities can be explicit or implicit functions time T. Except different indication, we will use a Cartesian frame of reference whose origin (0, 0,0) is at the point O. We will indicate the space coordinates by x_i and the successive derivative space ∂/∂x_i, ∂ ²/∂x_i∂x_j,… by ∂_i, ∂_i∂_j,… (I, J = 1,2,3). In general, we adopt here the indicielle notation for the vectors and the tensors, as well as the rule of summation of Einstein on the repeated dumb indices.

The total mass of the model is

M = ∫_B ρ dτ,

where dτ = dx₁dx₂dx₃ is an element of volume. The total impulse is

P_i = ∫_B ρ v_i dτ,

and the total kinetic moment is

L_i = ∫_B (ε_ijkx_jρ v_k + ℓ_i) dτ,

where ε_ijk is the operator of permutation of Levi-Cività. The resulting from the forces applied to volume B of the body is

F_i = ∫_B f_i dτ + ∫_∂Bn_jT_ij dσ,

where dσ is an element of surface ∂B and (n₁, n₂, n₃) a normal unit vector on the surface ∂B indicates and pointing out of volume B. Likewise, manner, the moment of force resulting applied to the body B is

M_i = ∫_B (ε_ijkx_jf_k + m_i) dτ + ∫_∂B (ε_ijkx_jn_qT_kq + n_qM_iq) dσ.

These formulas are valid in any circumstance. If we admit moreover that the surface-border ∂B of B is regular and that the fields of the tensions and the surface moments of force are differentiable in B, one can apply the Théorème of the divergence of Gauss and write

F_i = ∫_B (f_i + ∂_jT_ij) dτ,

M_i = ∫_B + m_i + ∂_q (ε_ijkx_jT_kq + M_iq) dτ.

Laws of conservation

The equations governing a continuous medium can be deduced, either by using a “principle of correspondence” to adapt the laws of the mechanical discrete points of Newton to the material points of the mechanical of the continuous mediums, or by expressing in a direct way the fact that certain physical attributes, such as the mass, the impulse, the kinetic moment, the kinetic energy, internal energy, total energy, etc, contained in an arbitrary volume B (T) becoming deformed during time T, cannot change in an arbitrary way, but that their precise changes are governed by laws of conservation . These last can be obtained by expressing the rate of change in time in an arbitrary volume of matter by means of the theorem of transport of Reynolds.

That is to say B (T) an unspecified part becoming deformed during time of material volume B (T). It is supposed that its border ∂b (T) is regular, and one directs the unit normal N (T) in an unspecified point of ∂b (T) towards outside. For a physical quantity arbitrary Q associated with a material point pertaining to volume B (T) or its border ∂b (T), one has then

D_tQ = ∫_{b (T)} ∂_tq dτ + ∫_{∂b (T)} n_jqv_j dσ.

The notations used here are the following ones: Q = ∫_b Q dτ is an arbitrary physical quantity, and Q the voluminal density of the corresponding field; the D_t symbols and ∂_t are the total derivative (material) d/dt and partial (local) ∂/∂t compared to time, respectively; ∂b is the closed regular surface which limits volume B, of unit external normal n_i (i=1,2,3). While again using the theorem of the divergence of Gauss, it comes

D_tQ = ∫_{b (T)} + ∂_j (qv_j) dτ.

If ϰ is the local rate of production or destruction of the quantity Q, i.e. D_tQ = ∫_b ϰ dτ, then the preceding formula becomes

∫_{b (T)} + ∂_j (qv_j) - ϰ dτ = 0

or, since volume B (T) is arbitrary and can thus be taken arbitrarily small:

∂_tq + ∂_j (qv_j) = ϰ.

This is the general form eulérienne of a law of conservation in physics of the continuous mediums. The Lagrangian form equivalent, valid for each individual material point, is

D_tq = Q ∂_jv_j = ϰ.

One easily deduces this Lagrangian form from the form eulérienne using the identity (see the article basic Concepts in theory of the continuous mediums )

D_tq = ∂_tq + v_j ∂_jq.

Conservation of the mass, or equation of continuity

In integral form, the conservation of the mass is expressed by

D_t M = D_t ∫_{b (T)} ρ ( X , T) dτ = ∫_{b (T)} (D_tρ + ρ ∂_kv_k) dτ = 0.

This formula provides the equation of continuity , expressing the conservation of the mass in differential form, while making tighten arbitrary volume B (T) to tend towards a point:

D_tρ + ρ ∂_kv_k = 0.

The form eulérienne of this last equation is

∂_tρ + ∂_k (ρ v_k) = 0.

It is noticed that it is well here about a particular form of the law of general conservation ∂_tq + ∂_j (qv_j) = ϰ, where Q = ρ and ϰ = 0. The fact that ϰ = 0 indicates well that there is neither production nor destruction of mass, which is thus preserved.

Conservation of the impulse, or equation of motion

In a similar way, hardly more complicated, one can establish the conservation equation of the impulse governing the movement of a material point, by posing Q = ρ v_i and by admitting explicitly that the mass is preserved too. One obtains

ρ D_tv_i = f_i + ∂_jT_ij (ξ, T).

Conservation of the kinetic moment, or equation of rotation

The equation expressing the conservation of the kinetic moment, which is paramount in the studies treating of the rotation of a deformable body like the Earth, can be obtained in a way similar to that expressing the conservation of the impulse. By supposing that there are conservation of the mass and conservation of the momentum, one obtains

D_tℓ_i = m_i + ∂_kM_ik + ε_ijkT_jk.

Symmetry of the tensor of the tensions

If, moreover, we suppose that the kinetic moment intrinsic ℓ_i of an unspecified material point does not change during the movement of deformation, in other words if

D_tℓ_i = m_i + ∂_kM_ik,

then the tensor of the T_jk tensions is symmetrical:

ε_ijkT_jk = 0 ⇔ T_jk = T_kj.

Conservation of energy

The equation expressing the conservation of energy can be obtained method consequently that employed for the mass, the momentum and the quantity of rotation. One finds:

∂_t (½ ρv ² + U) + ∂_j + U) v_j - v_kT_kj + H_j = v_kf_k + Γ.

The symbol U indicates here internal energy per unit of volume, H_i is the vector density flux of heat, and Γ is production or the intrinsic destruction rate of energy per unit of volume, chemical reactions, radioactivity, or any other internal process. Symbol v ² represents the square v_kv_k of course speed, i.e. twice the kinetic energy per unit of mass.

Summary table of the principal laws of conservation

The table below summarizes the principal general, in form eulérienne, useful laws of conservation to solve the problems arising from the mechanics of the continuous mediums, in particular when it is a question of studying the total deformations of the Earth following a seism, following a variation in the length of the day, following the application of a potential of tide or following the application of a distribution of mass on external surface. The notations employed in the table are those of this text. Let us recall that the integral

Q = ∫_B ϕ dτ

is the physical quantity which is preserved; it represents, according to the cases, the total mass of the body B, its total momentum P_i, its kinetic moment total L_i, or its total energy E. the general form eulérienne of a law of conservation is

∂_tϕ + ∂_jk_j = Q.

Assumptions and laws to describe the deformations of the Earth

In the problems of theoretical geodynamics, one in general considers that the resultant of the external forces per unit of f_i volume is composed of a voluminal gravific force f_i ^(G) caused by the autogravitation and the gravific attraction of external bodies, of a voluminal electromagnetic force f_i ^(EM) , of inertial voluminal forces (Coriolis and axifuge) f_i ⁽ⁱⁿ⁾ , of forces of tide f_i ^(m) , and of voluminal forces caused by external loads f_i ^(CH) . In addition, the surface forces T_i ( X , T, N ) depend on the orientation N which contribute in general to the tensor of the tensions T_ij ( X , T) via the relations of Cauchy

T_i ( X , T, N ) = T_ij ( X , T) n_j ( X , T, N ),

are the elastic forces which generate a tensor of the elastic tensions T_ij ^(el) , the electromagnetic surface forces which generate a tensor of Maxwell T_ij ^(EM) , viscous forces which generate a tensor of Reynolds a tensor of the elastic tensions T_ij ^(v) , forces of friction of nonviscous contact which generate a tensor of the forces of friction a tensor of the elastic tensions T_ij ^(F) , and of any other surface force generating a tensor of the tensions a tensor of the elastic tensions T_ij ^(S) .

The force of gravity per unit of mass G , more simply called gravific acceleration or revolved , drift of a potential gravific Φ ^(G) which we will compare here to gravific potential energy:

F ^(G) = ρ G = - ρ ∇Φ ^(G) .

The potential Φ ^(G) is determined by the Poisson's equation

∇ ² Φ ^(G) = 4πGρ,

where G is the Constante of gravitation. It should be noted that the properties of the potential gravific imply that Φ ^(G) and G is continuous functions everywhere in space, and in particular on the borders between two different continuous mediums.

With regard to the field of force produced by electromagnetic interactions, one usually admits that the material points do not transport an electric charge, i.e. they are electrically neutral . This way, one supposes that no Force of Lorentz applies directly to a mass point ξ being driven in the magnetic field of the Earth. However, the terrestrial material conducts electricity to differing degree, in other words has an electric conductivity σ_e finished nonnull (σ_e ≠ 0). Consequently, when this material moves at a speed v (≠ 0), it interacts with the geomagnetic field interns and generates a density of induced electric current J which modifies the initial magnetic field by giving rise to a variable magnetic induction field B (ξ, T), while the variable flow of this current produces an electric induction field E (ξ, T). The result of all these interactions of terrestrial material moving with the geomagnetic field B is the birth of an induced electromagnetic field of force

F ^(EM) = c^-1 J X B , J = σ_e ( E + c^-1 v X B ),

which modifies in its turn the movement. One used here electrostatic units of Gauss, C being speed of light in the vacuum. It is advisable to notice that in dielectric perfect (σ_e = 0), the electromagnetic force would be null = 0. On an electromagnetic interface of external normal N , the quantities N x' E' , N x' H' and N ⋅ B is continuous. When one studies the elastic strain of the Earth, in particular the propagation seismic waves, the oscillations free, the deformations of tide, etc, one generally neglects to consider such magnetohydrodynamic or magnetoelastic phenomena, while admitting simply which the elastic forces of recall are much more important than the electromagnetic forces. It is necessary however to keep in mind that these forces exist and are essential when one considers the generation of the geomagnetic field interns in the external core of the Earth by effect of self-excited dynamo. They can possibly become important when the deformations and oscillations of the core are studied, but are negligible in practice when one considers total deformations of the Earth of low amplitude.

The inertias can be formulated as follows:

F ⁽ⁱⁿ⁾ = ρ ∇ (½ | Ω X R | ) ² - 2 ρ Ω X v ,

where Ω is the vector instantaneous rotation, R the vector position of the material point ξ , and v the instantaneous speed of this point. The first term represents the force axifuge and the second term represents the force of Coriolis per unit of volume.

The deformations due to the tides are produced by forces

F ^(m) = - ρ ∇ W ^(m)

also deriving from a potential, the Potential of tide W ^(m) . In the case of the terrestrial tides , taking into account the sensitivity of the detector receiving sets currently on the market, it is necessary to consider the quadrupolar terms (N = 2) and octupolaires terms (N = 3) for the Moon and the Sun in the general development of the potential of tide in Multipôle S:

W ^(m) = Σ_A G M_A d_A^-1 Σ_n=2→∞ (r/d_A) ⁿ P_n (cos z_A).

Here, G indicates the constant of gravitation (G = 6,673 X 10^-11 m ³ s^-2 kg^-1), M_A is the mass of the generating star of the tide, d_A is the distance from the center of mass of the Earth in the center of mass from has, z_A is the zenith distance from has and R is the distance from the center of mass of the Earth at the point where the tide is measured. The P_n symbol indicates, as usual, the polynomial of Legendre of degree N. The first summation in the expression above extends in theory on all the stars has being able to generate an effect of tide on the Earth; in practice, it is limited to the Moon, the Sun and, in very precise calculations, to Venus and Jupiter.

Law of Hooke

In many geophysics applications, particularly in Sismologie, the Earth is treated like a material linearly elastic and isotropic, which means that the Lagrangian tensor of the elastic tensions is expressed by the law of Hooke. In a Cartesian reference frame, one thus has

T_ik ( ξ , T) = (κ - ⅔ μ) ∂_ju_j δ_ik + μ (∂_iu_k + ∂_ku_i),

where U is the field of displacements and where the material κ constants and μ are the modules of compression and shearing, respectively. In the problems which relate to durations spreading out approximately between a fraction of a second and a few hours, i.e. typical amounts of time of the periods of the seismic waves, free oscillations and usual tides, the module of compression to be considered is the module of adiabatic compression , evaluated for isentropic conditions. For the modulus of rigidity, Leon Brillouin showed in 1940 that it was not necessary to distinguish between a shearing carried out under isentropic conditions or isothermal conditions. In addition, one can take account of a low non-elasticity while making depend the coefficients κ and μ on the frequency; it is the approach adopted in particular by Kanamori and Anderson and Dziewonski and Anderson.

Conditions at the borders

In order to be able theoretically to study the total deformations of the Ground (or a telluric Planet), it is necessary to associate with the differential equations which describe the movement of deformation the conditions which apply to the center and to the external surface of the Earth, as of the adequate conditions which apply to the various internal borders between different continuous mediums. Thus, for the gravific continuity of the potential and the force of gravity, it is necessary to take account of two general types of conditions to apply when a simple interface is met: dynamic conditions kinematics and conditions.

Conditions kinematics

The conditions kinematics are obtained easily by expressing the fact mathematically that the material points belonging to a surface-border F = const at a given moment must still belong to the same surface at another moment, i.e.

δF^{milieu 1} = δF ² medium

or, in form eulérienne:

∂F ¹ medium + U ^{medium 1} ⋅ ∇F ¹ medium = ∂F ² medium + U ^{medium 2} ⋅ ∇F ² medium.

As we limit ourselves to regular surfaces, the function F admits derivative partial continuous compared to time and the space coordinates. It follows that the normal component of the field of displacement relating to the interface must be continuous, in other words

N ⋅ U ¹ medium = N ⋅ U ² medium.

These conditions do not exclude the possibility of slip from a medium compared to the other, and one will call a border on which such a slip is possible a “interface slipping” (in English: slipway boundary ). On the other hand, if we want to exclude the possibility of slip, we must impose continuity through the border at the same time components normal and tangential of displacement:

U ¹ medium = U ² medium.

A border of this type is called “welded interface” or “interfaces not-slipping” (in English: slipway-free boundary ).

Dynamic conditions

To establish the dynamic conditions, one considers small a parallélipipède right (a " box of allumettes") of height h. One supposes this " boîte" partitionnée by a surface-border in two volumes V₁ and V₂ filled of material of the continuous medium 1 and material of the continuous medium 2, respectively. The " boîte" is limited above by a S₁ surface of external unit normal N ₁, below by a S₂ surface of external unit normal N ₂, and on the sides by a S₃ surface of external unit normal N ₃. One applies to this material volume V = V₁ + V₂ limited by closed surface S = S₁ + S₂ + S₃ the conservation equation of the impulse in integral form, that is to say

D_t∰_{V (T)} ρ v_i dτ = ∰_{V (T)} f_i dτ + ∯_{S (T)} T_ik n_k dσ.

By supposing that the voluminal forces are continuous through the border, and while letting tighten H towards zero, we obtain the dynamic conditions of interface

T_ik ^{medium 1} n_k = T_ik ^{medium 2} n_k,

by taking high note owing to the fact that for H = 0, we have V = 0, S₃ = 0, and N ₁ = - N ₂. Thus, the normal components of the tensor of the tensions as a whole must remain continuous with through the interface.