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Appendix A Dyads and Vector Identities

This Appendix is partially based on Appendix A-Algebra and Calculus of in Ben-Menahem and Singh (2000), to whom we refer for details. We recall herein only the basic definition of a dyad, and provide those vector identities that are necessary to expand in spherical harmonics the equation. The dyad formulation is useful since these vector identities are valid for any . The most general dyad is the juxtaposition of any two vector a and b defined as  ab = aα bβ eα eβ (A.1) α, β where α and β vary from 1 to 3 and eα, eβ denote the unit vectors along the respective xα, xβ axes; Eq. (A.1) is known as the algebric product between a and b, different from the , a · b, and cross, a × b, products. The first vector of the dyad is called antecedent and the second one the consequent; if we reverse the order of the vectors in the dyad we obtain its , herein indicated by the superscript T , as for example the transpose of the dyad entering the definition of the strain Eq. (1.26), as for the following f vector and the ∇

(∇ f )T = f ∇ (A.2) where the symbol ⊗ of the algebraic product is now on omitted, to make easier the comparison with Appendix A of Ben-Menahem and Singh (2000) where the former is also omitted. We now list the vector identities for deriving Eq. (1.83)

∇(uv) = (∇u)v + u(∇v) (A.3)

∇ f − f ∇ =−1 × (∇ × f ) (A.4)

© Springer Science+Business Media Dordrecht 2016 337 R. Sabadini et al., Global Dynamics of the Earth: Applications of Viscoelastic Relaxation Theory to Solid-Earth and Planetary Geophysics, DOI 10.1007/978-94-017-7552-6 338 Appendix A: Dyads and Vector Identities where 1 denotes the unitary dyad.

∇ · (u D) = u(∇ · D) + (∇u) · D (A.5) where D denotes a dyad.

∇ · (u1) = ∇u (A.6)

∇ · (∇ f ) = ∇2 f (A.7)

∇ · (∇ f )T = ∇(∇ · f ) (A.8)

A.1 Divergence and Volume Changes

The divergence of the spherical harmonic vectors yields 2 ∇ · R = Y (A.9) m r m ( + 1) ∇ · S =− Y (A.10) m r m

∇ · T m = 0 (A.11) when the following identities are considered

∇ · (u f ) = u∇ · f + (∇u) · f (A.12)

∇ × (u f ) = u∇× f + (∇u) × f (A.13) 2 ∇ · e = (A.14) r r ∇ · ( f × g) = (∇ × f ) · g − f · (∇ × g) (A.15)

It is noteworthy that toroidal deformations have no radial components and does not involve volume changes . In fact the divergence of the toroidal part of the displacement, which is given by Eqs. (1.63) and (1.66), yields, on the of Eq. A.5  ∇ · uT = (∂r Wm er · T m + Wm∇ · T m) = 0 (A.16) m

On the contrary, the divergence of the spheroidal part of the displacement, which is given by Eqs. (1.62), (1.64) and (1.65), yields, on the basis of Eq. A.5 Appendix A: Dyads and Vector Identities 339  ∇ · uS = (∂r Um er · Rm + Um∇ · Rm m  + ∂r Vm er · Sm + Um∇ · Sm) = χm Ym (A.17) m with χm given by 2  ( + 1) χ = ∂ U + U − V (A.18) m r m r m r m The spherical harmonic expansion of the volume change  can thus be written as follows   = χm Ym (A.19)

A.2 Laplacian Entering the Divergence of the Cauchy

From

∇2 ( f a) = a ∇2 f + f ∇2a + 2 ∇ f · (∇a) (A.20) 1   ∇e = eθ eθ + eφ eφ (A.21) r r 2 ∇2e =− e (A.22) r r 2 r the Laplacian of the spherical harmonic vector Rm reads

2 1 ∇ R = [2 (S − R ) − ( + 1) R ] (A.23) m r 2 m m m Making use of

∇2∇ f = ∇∇2 f (A.24) we obtain the Laplacian of the spherical harmonic vector Sm ( + ) 2 1 ∇ S =− (S − 2 R ) (A.25) m r 2 m m 340 Appendix A: Dyads and Vector Identities

From

∇2∇ × f = ∇ ×∇2 f (A.26)

∇r = 1 (A.27)

∇×(∇ f ) = 0 (A.28) we obtain the Laplacian of the spherical harmonic vector T m ( + ) 2 1 ∇ T  =− T  (A.29) m r 2 m

A.3 Vector Product Entering the Divergence of the

Making use of

∇ × ∇ × f = ∇(∇ · f ) −∇2 f (A.30) we obtain

Um er × (∇×(Um Rm)) = Sm (A.31)  r  Vm er × (∇×(Vm Sm )) =− + ∂r Vm Sm (A.32)  r  Wm e × (∇×(W T  )) =− + ∂ W T  (A.33) r m m r r m m Appendix B Analytical Functions

A function f (z) in the complex is called analytical in a point z = z0 if f (z) is differentiable in z = z0 and in a small surrounding area. Mathematically stated: the derivative df δ f f (z + δz) − f (z) = lim = lim (B.1) dz δz→0 δz δz→0 z + δz − z should exist. If we split the function f into real and imaginary parts as f = u + iv and z into z = x + iy, then we get δ f δu + iδv = (B.2) δz δx + iδy

Setting (δx → 0,δy = 0) gives:   δ f δu δv ∂u ∂v lim = lim + i = + i (B.3) δz→0 δz δx→0 δx δx ∂x ∂x while setting (δx = 0,δy → 0) gives:   δ f δu δv ∂u ∂v lim = lim −i + =−i + (B.4) δz→0 δz δy→0 δy δy ∂y ∂y

Existence of df/dz thus leads to the following two conditions: ∂u ∂v ∂u ∂v = and =− (B.5) ∂x ∂y ∂y ∂x which are called the Cauchy-Riemann conditions. The Cauchy theorem states the following: if f (z) is analytical inside a region bounded by the closed contour C, then

© Springer Science+Business Media Dordrecht 2016 341 R. Sabadini et al., Global Dynamics of the Earth: Applications of Viscoelastic Relaxation Theory to Solid-Earth and Planetary Geophysics, DOI 10.1007/978-94-017-7552-6 342 Appendix B: Analytical Functions  f (z)dz = 0(B.6)

C

We can proof the Cauchy theorem by using the Stokes’ theorem for converting a line integral over a closed contour into a surface integral:    f (z)dz = (udx − vdy) + i (vdx + udy) (B.7)

C C C

The Gauss lemma states that   ∂ f f (x, y)dx =− dxdy (B.8) ∂y C S   ∂ f f (x, y)dy = dxdy (B.9) ∂x C S where the surface S is contoured by C. Applying the Gauss lemma to Eq. (B.7)we obtain     ∂v ∂u f (z)dz = − − dxdy ∂x ∂y C S    ∂u ∂v + i − dxdy (B.10) ∂x ∂y S

Applying the Cauchy-Riemann conditions to the above equation we obtain  f (z)dz = 0. (B.11)

C

B.1 Cauchy Integral Representation

Let us assume that f (z) is an analytical function within the domain R. We can show that, if C is a closed curve in R, the following Cauchy integral representation holds

  1 f (z )  f (z) = dz (B.12) 2πi z − z C for any z internal to C. Appendix B: Analytical Functions 343

We consider the quantity

 f (z ) − f (z) (B.13) z − z

Since f is a continuous function, for any >0 we can find δ( ) > 0 such that for

 |z − z| <δ( ) (B.14) we have

 | f (z ) − f (z)| < (B.15)

 C denotes a circle in the z plane centered on z of radius r <δ( )expressed by

 z = z + reiθ (B.16)

The modulus of the integral of equation (B.13) over C satisfies

 (  ) − ( ) f z f z  < π .  dz 2 r (B.17) z − z r C

 In fact, over C we have that |z − z|=r which results into

  (  ) − ( ) (  ) − ( ) f z f z  = f z f z  .  dz dz (B.18) z − z r C C

The Darboux inequality for functions of complex variables states that the above modulus of the integral over C is always smaller than the maximum of the value attained by the function which is integrated, r in our case, multiplied by the length of the arc along which the function is integrated, 2πr in our case, leading to equation Eq. (B.17) from Eq. (B.18). If we take the limit → 0, the right part of Eq. (B.17) vanishes, leading to

  f (z ) − f (z)  dz = 0 (B.19) z − z C

We thus obtain

    f (z )  dz dz = f (z) . (B.20) z − z z − z C C 344 Appendix B: Analytical Functions

From Eq. (B.16) we obtain

   dz = i dθ = 2πi (B.21) z − z C C that finally leads to

  1 f (z )  f (z) = dz (B.22) 2πi z − z γ which finally proves the result, since the curves γ and C are equivalent for the Cauchy Theorem, being obtained one from the other by continuous deformation within the domain R where f (z) is analytical. From the Cauchy integral representation of the function f (z) it is straightforward to obtain the following representation of the nth derivative of f (z)  n  d f (z) n! f (z )  = dz . (B.23) dzn 2πi (z − z)n+1 γ

B.2 Residue Theorem

If f (z) is analytical within the domain D, except for a number of isolated singular- ities, the Residue Theorem holds, which states that the integral of f (z) extended to any closed curve γ in D, not passing through any singular point of f (z), is equal to 2πi the sum of the residues of the singularities of f (z) internal to γ . Let us consider the function g(z) f (z) = k (B.24) (z − z0) where g(z) is analytical; f (z) is singular with a pole of kth order in z = z0.Onthe basis of the integral representation of the derivatives of an analytical function, we obtain  g(z) 2πi dk−1 dz = g(z)| = . (B.25) k k−1 z z0 (z − z0) (k − 1)! dz γ

By definition,  1 |Resf (z)| = = f (z)dz (B.26) z z0 2πi γ Appendix B: Analytical Functions 345 which gives on the basis of equation (B.23)  1 g(z) |Resf (z)| = = dz. z z0 k (B.27) 2πi (z − z0) γ

Taking into account the Cauchy representation of the kth derivative of an analytical function g(z), the above expression becomes

1 dk−1g(z) |Resf (z)| = = | = (B.28) z z0 (k − 1)! dzk−1 z z0 or

k−1 1 d k |Resf (z)| = = lim (z − z ) f (z) (B.29) z z0 → k−1 0 (k − 1)! z z0 dz

If we have a simple pole in z = z0, the above expression becomes

| ( )| = ( − ) ( ) Resf z z=z0 lim z z0 f z (B.30) z→z0 Appendix C Icy Moons

˜ ˜ C.1 Derivation of the Propagator Matrices W1 and W2

The presence of an internal liquid ocean divides the propagation process into three regions of application, namely: the silicate mantle, the liquid ocean and the ice shell. Here, we combine the propagation within each of these separate regions into one single propagator by explicit application of the boundary conditions given by Eqs. (9.19) and (9.21). We start the combination process by relating the conditions at the free (unforced) surface to the solution vector at the base of the ice shell, i.e.

⎛ ⎞ ⎛ ( − ) ⎞ (1) ˜ n 1 ( ) U˜ (R) U rn  ( − ) ⎜ ˜ (1) ⎟ ⎜ ˜ n 1 ( ) ⎟ ⎜ V (R) ⎟ ⎜ V rn ⎟ ⎜ ⎟ ⎜ (n−1) ⎟ 0 si ⎜ σ˜ , (rn) ⎟ ⎜ ⎟ = B˜ ⎜ rr ⎟ (C.1) ⎜ ⎟  ⎜ σ˜ (n−1)( ) ⎟ ⎜ 0 ⎟ ⎜ rθ, rn ⎟ ⎝ ˜ (1)( ) ⎠ ⎝ ˜ (n−1) ⎠  R  (rn) ˜ (n−1) 0 Q (rn)

˜ si where B is the ice propagator matrix within the icy layers from surface of Europa to the bottom of the ocean, according to the scheme of Fig. 9.2, based on Eqs. (2.9), ˜ (2.10) where the fundamental matrix Y is that of an incompressible viscoelastic material given by Eq. (2.42). Equation (C.1) introduces three constraints to the propagation problem, as both ˜ stress elements (σ˜rr, and σ˜rθ,) and the so-called potential stress (Q) are by definition equal to zero at the surface in the free surface case. Here, we recall that the free surface case is used to determine the normal modes or free oscillations of our interior model. The determination of these modes and their corresponding relaxation times and strengths is a very important step in the calculation of the viscoelastic response of an icy moon to tidal , because these modes describe the effect of viscoelastic

© Springer Science+Business Media Dordrecht 2016 347 R. Sabadini et al., Global Dynamics of the Earth: Applications of Viscoelastic Relaxation Theory to Solid-Earth and Planetary Geophysics, DOI 10.1007/978-94-017-7552-6 348 Appendix C: Icy Moons relaxation on the response at the surface. Consequently, we proceed our discussion by only taking into account the constrained part of the solution vector, i.e. ⎛ ⎞ ˜ (n−1) U (rn) ⎜ ˜ (n−1) ⎟ ⎜ V (rn) ⎟ ⎜ (n−1) ⎟ si ⎜ σ˜ (r ) ⎟ 0 = P B˜ ⎜ rr, n ⎟ (C.2) 1  ⎜ σ˜ (n−1)( ) ⎟ ⎜ rθ, rn ⎟ ⎝ ˜ (n−1) ⎠  (rn) ˜ (n−1) Q (rn) where the projector operator P1 is given by ⎛ ⎞ 001000 ⎝ ⎠ P1 = 000100 (C.3) 000001

Thereafter, we apply the set of boundary conditions at the ocean-ice interface (Eq. (9.19)) to the right side of Eq. (C.2). This step leads to the following expression ⎛ ⎞ ˜ (n)  (rn ) ⎜ g(rn ) ⎟ ˜ ˜ R1 ⎜ ⎟ ds = B ˜ (n) (C.4) ⎝  (rn) ⎠ ˜ (n) Q (rn)

˜ R1 where the matrix B is defined as ⎛ ⎞ 100 ⎜ ⎟ ⎜ 000⎟ ⎜ ⎟ ˜ R1 ˜ si ⎜ 000⎟ B = P1B ⎜ ⎟ (C.5) ⎜ 000⎟ ⎝ 010⎠ 001

˜ and the vector ds as ⎛ ⎞ ˜ si −R, K − B K ⎜ 3 4 ,32 5 ⎟ d˜ = ⎝ − − ˜ si ⎠ s R,4 K4 B,42 K5 (C.6) − − ˜ si R,6 K4 B,62 K5 with R,y (y ∈{3, 4, 6}) defined by

= ˜ si + ρ(n) ( ) ˜ si + π ρ(n) ˜ si R,y B,y1 0 g rn B,y3 4 G 0 B,y6 (C.7) Appendix C: Icy Moons 349

In Eqs. (C.6) and (C.7) the subscripts refer to an individual element of the ice ˜ si propagator matrix B . By convention, the first digit in the subscript indicates the row and the second digit the column. The next step is to express the right hand side of Eq. (C.4) in terms of the conditions at the bottom of the ocean layer. Substitution of Eqs. (9.19) and (9.20) into Eq. (C.4) yields ⎛   ⎞ 1 ˜ f ˜ (n) ˜ f ˜ ∗(n) −  ( + ) − ( + ) g(r ) B,11  rn 1 B,12 Q rn 1 R1 ⎜ n ⎟ ˜ ˜ ⎜ f (n) f ∗(n) ⎟ ds = B ˜ ˜ ( ) + ˜ ˜ ( ) (C.8) ⎝ B,11  rn+1 B,12 Q rn+1 ⎠ ˜ f ˜ (n)( ) + ˜ f ˜ ∗(n)( ) + B,21  rn+1 B,22 Q rn+1 J

˜ f where B is defined in Eq. (9.18) and where the auxiliary J is defined by   (n)    + πGρ f f = 1 − 4 0 ˜ ˜ (n)( ) + ˜ ˜ ∗(n)( ) J B,11  rn+1 B,12 Q rn+1 (C.9) rn g(rn)

˜ (n) ˜ ∗(n) Moreover, the radial functions  (rn+1) and Q (rn+1) at the bottom of the ocean can be expressed in terms of the conditions at the CMB by applying the set of boundary conditions at the mantle-ocean interface (Eq. 9.21) and the viscoelastic propagation through the silicate mantle. Then, we can write Eq. (C.8) as follows ⎛ ⎞ G,11 G,12 G,13 ˜ ⎝ ⎠ ˜ ds = G,21 G,22 G,23 Cc, (C.10) G,31 G,32 G,33 where the elements G,vw (row 1 ≤ v ≤ 3 and column 1 ≤ w ≤ 3) are defined by

˜ R1   B,v1 ˜ f ˜ sm ˜ f G,vw =− B B + B Z,w ( ) ,11 ,5w ,12 g rn   + ˜ R1 ˜ f ˜ sm + ˜ f B,v2 B,11B,5w B,12 Z,w   + ˜ R1 ˜ f ˜ sm + ˜ f B,v3 B,21B,5w B,22 Z,w   (n)   R1  + πGρ f f + ˜ 1 − 4 0 ˜ sm + ˜ B,v3 B,11B,5w B,12 Z,w (C.11) rn g(rn) in which   (n) sm πGρ  + sm πG sm = ˜ + 4 0 − 1 ˜ − 4 ˜ Z,w B,6w B,5w B,3w (C.12) g(rn+1) rn+1 g(rn+1) 350 Appendix C: Icy Moons

˜ sm where B denotes the propagator within the mantle layers from the bottom of the ocean to the core mantle boundary, according to the scheme of Fig. 9.2, based on Eqs. ˜ (2.9), (2.10) and (1.164) where the fundamental matrix Y is that of an incompressible viscoelastic material given by Eq. (2.42) and the core-mantle boundary conditions are based on Eq. (1.150), but for the core of Europa. As can be seen from Eq. (C.10), we only applied three constraints to a problem having six unknowns (K1 to K6). The additional constraints can be obtained from the radial functions that cannot be propagated through the ocean layer, but that are related to the CMB-constants K1 to K3 through the set of boundary conditions at the mantle-ocean interface. The first additional constraint is defined by taking into account continuity of radial stress at the mantle-ocean boundary. We can express the constant K6 in terms of the CMB-constants,   1 ˜ sm ˜ sm ˜ sm ˜ K =− B B B C , (C.13) 6 ρ(n) ( ) ,31 ,32 ,33 c 0 g rn+1 thereby reducing the number of unknowns to five (K1 to K5). A second additional constraint can be introduced by taking into account continuity of tangential stress at the mantle-ocean boundary by the following expression:   = ˜ sm ˜ sm ˜ sm ˜ 0 B,41 B,42 B,43 Cc, (C.14)

The third and last additional constraint can be obtained from the boundary con- dition regarding the radial displacement at the mantle-ocean interface. We can write the following   ˜ 0 = L,1 L,2 L,3 Cc, (C.15) where the elements L,w are defined by

= ˜ sm − 1 ˜ sm + 1 ˜ sm L,w B,1w ( ) B,3w B,5w (C.16) ρ n ( ) g(r + ) 0 g rn+1 n 1

Finally, combination of Eqs. (C.10), (C.14) and (C.15) allows us to write out the ˜ propagator matrix W1 that relates the five defined constraints to the five unknowns, i.e. ⎛ ⎞ 00L,1 L,2 L,3 ⎜ ˜ sm ˜ sm ˜ sm ⎟ ⎜ 00B,41 B,42 B,43 ⎟ ⎜ ˜ si ⎟ ˜ = ⎜ R, B G, G, G, ⎟ W1 ⎜ 3 ,32 11 12 13 ⎟ (C.17) ⎝ ˜ si ⎠ R,4 B,42 G,21 G,22 G,23 ˜ si R,6 B,62 G,31 G,32 G,33 Appendix C: Icy Moons 351 which satisfies the characteristic equation ˜ ˜ 0 = W1C (C.18)

˜ T where C = (K1 K2 K3 K4 K5) . In a similar way as for the constrained part of Eq. (C.1), we can express the unconstrained parameters at the surface directly in terms of the unknown constants K1 to K5. After some analytical manipulation we obtain ⎛ ⎞ ˜ U(R, s) ˜ ⎝ ˜ ⎠ ˜ ˜ X(s) = V(R, s) = P35W2C (C.19) ˜ (R, s)

˜ ˜ where X(s) is defined as the unit impulse response, W2 is the propagator matrix, ˜ C is the vector of unknown constants and P35 is a projection matrix that filters out ˜ ˜ ˜ the first two elements of the product between W2 and C. The propagator matrix W2 itself is defined by ⎛ ⎞ 00L,1 L,2 L,3 ⎜ sm sm sm ⎟ ⎜ 00B˜ B˜ B˜ ⎟ ⎜ ,41 ,42 ,43 ⎟ ˜ ⎜ ˜ si    ⎟ W2 = ⎜ R,1 B,12 G,11 G,12 G,13 ⎟ (C.20) ⎜ ˜ si    ⎟ ⎝ R,2 B,22 G,21 G,22 G,23 ⎠ ˜ si    R,5 B,52 G,31 G,32 G,33

 where the elements G,vw (row 1 ≤ v ≤ 3 and column 1 ≤ w ≤ 3) are defined by

˜ R2    B,v1 ˜ f ˜ sm ˜ f G =− B B + B Y,w ,vw ( ) ,11 ,5w ,12 g rn   + ˜ R2 ˜ f ˜ sm + ˜ f B,v2 B,11B,5w B,12Y,w   + ˜ R2 ˜ f ˜ sm + ˜ f B,v3 B,21B,5w B,22Y,w   (n)   R2  + πGρ f sm f + ˜ 1 − 4 0 ˜ ˜ + ˜ B,v3 B,11B,5w B,12Y,w (C.21) rn g(rn) with ⎛ ⎞ 100 ⎜ ⎟ ⎜ 000⎟ ⎜ ⎟ ˜ R2 ˜ si ⎜ 000⎟ B = P2B ⎜ ⎟ (C.22) ⎜ 000⎟ ⎝ 010⎠ 001 352 Appendix C: Icy Moons and P2 given by ⎛ ⎞ 100000 ⎝ ⎠ P2 = 010000 (C.23) 000010

C.2 Auxiliary Variables in Stress Equations

The elements of the diurnal stress tensor at the surface, which are mathematically defined by Eqs. (9.56)to(9.61), depend on the Love numbers (i.e. on the properties of the interior) and the co-latitude through the Beta-functions. These functions are listed below     θθ 3 e e 3 e e β , (θ) = 3h − 10l cos(2θ)+ h − 2l (C.24) 2 0 4 2 2 4 2 2 θθ, 3   3   β j (θ) = 3hv − 10lv cos(2θ)+ hv − 2lv (C.25) 2,0 4 2 j 2 j 4 2 j 2 j   θθ 3 e e β , (θ) = 3h − 10l sin(2θ) (C.26) 2 1 2 2 2 θθ, 3   β j (θ) = 3hv − 10lv sin(2θ) (C.27) 2,1 2 2 j 2 j     θθ 3 e e 9 e e β , (θ) =− 3h − 10l cos(2θ)+ h − 2l (C.28) 2 2 2 2 2 2 2 2 θθ, 3   9   β j (θ) =− 3hv − 10lv cos(2θ)+ hv − 2lv (C.29) 2,2 2 2 j 2 j 2 2 j 2 j ϕϕ 3   3   β (θ) = 3he − 8le cos(2θ)+ he − 4le (C.30) 2,0 4 2 2 4 2 2 ϕϕ, 3   3   β j (θ) = 3hv − 8lv cos(2θ)+ hv − 4lv (C.31) 2,0 4 2 j 2 j 4 2 j 2 j ϕϕ 3   β (θ) = 3he − 8le sin(2θ) (C.32) 2,1 2 2 2 ϕϕ, 3   β j (θ) = 3hv − 8lv sin(2θ) (C.33) 2,1 2 2 j 2 j ϕϕ 3   9   β (θ) =− 3he − 8le cos(2θ)+ he − 4le (C.34) 2,2 2 2 2 2 2 2 ϕϕ, 3   9   β j (θ) =− 3hv − 8lv cos(2θ)+ hv − 4lv (C.35) 2,2 2 2 j 2 j 2 2 j 2 j βθϕ(θ) = e (θ) 2,1 3l2 sin (C.36) βθϕ,j (θ) = v (θ) 2,1 3l2 j sin (C.37) βθϕ(θ) = e (θ) 2,2 3l2 cos (C.38) βθϕ,j (θ) = v (θ) 2,2 3l2 j cos (C.39) Appendix C: Icy Moons 353

In a similar way, the NSR stress tensor at Europa’s surface (Eqs. (9.67)to(9.72)) depends on the Love numbers and co-latitude through the following alpha-functions     αθθ (θ) =−3 ˆe − ˆe ( θ)+ 9 ˆe − ˆe 2,2 3h2 10l2 cos 2 h2 2l2 (C.40) 2   2   αθθ,j (θ) =−3 ˆv − ˆv ( θ)+ 9 ˆv − ˆv 2,2 3h2 j 10l2 j cos 2 h2 j 2l2 j (C.41) 2   2  αϕϕ (θ) =−3 ˆe − ˆe ( θ)+ 9 ˆe − ˆe 2,2 3h2 8l2 cos 2 h2 4l2 (C.42) 2   2   ϕϕ, 3 9 α j (θ) =− 3hˆv − 8lˆv cos(2θ)+ hˆv − 4lˆv (C.43) 2,2 2 2 j 2 j 2 2 j 2 j αθϕ (θ) = ˆe (θ) 2,2 3l2 cos (C.44) αθϕ,j (θ) = ˆv (θ) 2,2 3l2 j cos (C.45)

ˆe ˆe ˆv ˆv where the elastic Love numbers h2 and l2, and modal strengths h2 j and l2 j refer to the tidal response of interior models in which the silicate mantle has been assumed to behave as a fluid with respect to NSR (see Sect.9.6.2). Index

A Chemical boundaries, 25 Adjustment of the equatorial bulge, 102 Chemical stratification, 65 Adriatic, 234–236, 238, 239 CMB, see core-mantle boundary , 90 Coble creep, 2 Angular , 90 Complex contour integration, 42, 118, 309 Antarctica, 149, 151, 195, 203, 233 analytical functions, 341 Apennines, 234, 237, 238, 278 poles, 38, 42 Apparent Polar Wander, see polar wander Compressibility, 11, 211, 217, 270, 331 Apulia, 236, 240 Continental drift, 99 Aquileia, 236, 239, 241 Convolution, 227, 257, 258, 260, 262 Core, 88 Core-mantle boundary, 26, 29 B boundary conditions, 26, 350 Bernese software, 282 interface matrix, 29, 54 Bothnic Gulf, 226, 233, 237 Correspondence Principle, 3, 11 Boundary conditions, 21 Crete, 242 centrifugal, 24 Crust, 88 external, 21 lower crust, 277 internal, 19 transition zone, 277 internal forcing, 43, 45 upper crust, 277 surface, 21, 32 Cycloidal cracks, 294, 329 tidal, 24 Bromwich path, 36 Bulk modulus, 8 D Darboux inequality, 343 Decoupling, 303 C Delta function, 43, 229 Calabrian Arc, 237 Density stratification, 173 Canada, 149, 171 PREM, 96 Cauchy integral representation, 342 Dislocation sources Cauchy-Riemann conditions, 341 forcing terms, 45 Cauchy theorem, 341 Displacements, 5, 16, 29, 206, 213 Cavitation, 25 Diurnal stresses, 294, 313, 322 Centrifugal potential, inertia perturbations, Diurnal tides, 293, 310, 314 94 Dyadic formulation, 7, 337 Chandler wobble, 88, 102 Dynamic form factor J2, 126 frequency, 104 changes, 149 © Springer Science+Business Media Dordrecht 2016 355 R. Sabadini et al., Global Dynamics of the Earth: Applications of Viscoelastic Relaxation Theory to Solid-Earth and Planetary Geophysics, DOI 10.1007/978-94-017-7552-6 356 Index

Dynamic topography, 142, 246, 251 GPS campaigns, 280 GRACE, 202, 214, 219 Gravitational constant, 95 E Gravitational potential field, 189 Earth’s models, 62 Gravitational seismology, 189, 215, 218 31-layer models, 161, 166 Gravity, 28, 40 56-layer models, 164 Gravity anomalies five-layer models free-air (GIA), 231 fixed-boundary contrast, 64, 163 Green functions, 82 volume-averaged, 164 gravitational potential, 227, 228 half-space models, 237 radial displacement, 228 PREM, 156, 269 Greenland, 151, 195, 203 Earthquakes Irpinia (1980), 284 leveling campaigns, 284 H seismic moment, 284 Heaviside function, 104, 156, 229 Sumatran (2004), 211 Himalayas, 100 Tohoku-Oki (2011), 218 Hooke’s law, 12 Umbria-Marche (1997), 277 Horizontal displacements (GIA), 206 GPS campaigns, 282 Hot-spot reference frame, 99, 101 Earth’s , 87 Hudson Bay, 226, 233 Eccentricity, 296 Hydrostatic equilibrium, 258 Egnatiæ, 236, 240, 241 Hydrostatic , 5 Equatorial bulge, 102 readjustment time scale, 125 Euler equation, 90, 264 I Eulerian free precession, 102 Ice Age cycles, 171 frequency, 98, 104 Ice Ages, 87 Europa, 293, 301, 329 Ice mass changes Antarctica, 196, 198 F Greenland, 196, 201 Finite-element models, 237 ICE-3G, 155, 203, 231 Fourier approach, 302 present-day, 195 Fundamental solution matrix saw-tooth function, 154 compressible case, 73 Ice sheets incompressible case, 61 Antarctica, 155, 157, 164 inverse, 62 Fennoscandia, 155, 157, 164, 226 Laurentide, 155, 157, 164, 171, 226 Ice shell decoupling, 295, 297 G ICE-3G, see ice mass changes Galileo, 293 Icy moons, 293 Gauss lemma, 342 Incompressibility, 123, 217 Gauss theorem, 22 Incompressible models, 57 Geographical frame, 101, 267 spheroidal solution, 61 Geoid, 228, 246 Inertia geoid anomalies, 212, 227, 231, 247 moments of inertia, 97 long-wavelength components, 189 products of inertia, 103 GIA, see Glacial Isostatic Adjustment Inviscid core, 26, 27, 29, 30, GIPSY software, 282 Istria, 239, 241 Glacial Isostatic Adjustment, 2, 202, 206, 231, 235 Global change, 189, 195 J GPS, 269, 270 Jupiter, 293 Index 357

K Mega-wobble, 141, 143, 144 Kelvin-Voigt model, 103 Mercury, 139 Keplerian elements, 299 Milankovitch cycle, 153 function, 313 Momentum equation, 3, 4, 6, 17 Moon, 88, 139 L Lamé parameters, 3, 8, 108 Laplace domain, 11, 104, 259 N Laplace equation, 22, 28 Non-Hydrostatic bulge contribution, 115 Laplace resonance, 297 Non-synchronous rotation (NSR), 293, 318 Laplace transform, 11 Normal modes Laplace variable, 11 modes, 39 Late Cretaceous, 246 compositional modes, 76 Layering, 118 dilatational modes, 75 Legendre polynomials, 14 high degree modes, 277 Length of day variations, 102 transient modes, 39 Libration, 296 Numerical integration, 19 Linearized Liouville equations, 97 Nutation, 88 Linearized rotation theories, 103 Liouville equation, 90, 91, 98 O Lithosphere Obliquity, 294 thickness, 126, 139, 191 Ocean function, 229 Loading, 153 Oxygen isotopes, 153 ice sheets, 149 Love numbers, 82 elastic limit, 107 P fluid limit, 96, 105, 259 PGR and GRACE data, 202 gravitational potential, 229, 243 Phase-change boundaries, 25, 243, 247 radial displacement, 229, 243 Planets, 138 tangential displacement, 347 Pleistocene deglaciation, 227 tidal Love number, 118 Poisson equation, 4 fluid limit, 260 Polar shift, 102 Low- layers, 293 Polar wander, 87, 99, 124, 127, 149 Apparent Polar Wander, 99 M terrestrial planets and Moon, 138 MacCullagh’s formula, 91 True Polar Wander, 99, 257 Mantle, 2 path, 135, 171 Mantle convection, 1, 3, 89, 100, 171, 174, , 132, 265 231, 247, 259, 265 Poles, see complex contour integration Mantle stratification, 170, 269 Post-glacial rebound, see Glacial Isostatic Mantle viscosity, 2, 151, 156 Adjustment convex, 270, 271 Post-seismic deformation, 215, 269 ˙ from TPW and J2, 151 global, 269 lower mantle, 195 shallow earthquakes, 277 multi-branch solutions, 65, 165, 193, Precession, 88 206, 231 PREM, see Earth’s models two-layer profile, 167 Propagator matrix, 56 uniform, 111 Pseudo-spectral technique, 231 upper mantle, 168 Mars, 138 Maxwell model, 3 R Maxwell time, 2 Radial displacement (GIA), 206 Mediterranean, 233–235 Ravenna, 235, 236, 239, 241 358 Index

Rayleigh-Taylor instabilities, 38, 40, 76 Secular , 37, 38, 307 Reference frame, 101 Seismic moment, 211, 270 Relaxation modes, 39, 65, 76 Self-compression, 9 C0 mode, 66 Self-gravitation, 6, 211 L0 mode, 66 Shear relaxation function, 8 M0 mode, 66 SLR, see Satellite Laser Ranging M0 rotation mode, 109 Solution vector, 19, 303 analytical formula, 112 spheroidal, 19 M1andM2 modes, 64 toroidal, 20 M1 mode, 243, 259, 260 Spello, 282 total number, 39 Spherical coordinates, 13 Relaxation times, 62 Spherical harmonics, 13 tidal forcing, 106 Spheroidal equations, 19 Residue theorem, 344 Stiffness, see fundamental solutions matrix Rheological models, 1 Strain rate, 3 Rheologies, 1 Strain tensor, 7 constitutive law, 2 Stratification, 156 non-linear, 2 Stress, 2 Rigidity, 2 non-hydrostatic, 6 Roman ruins, 235, 236 Stress-strain relations, 7, 313 Root-solving procedure, 40 Stress tensor, 6, 314 bisection algorithm, 40 Subduction, 257 complex numbers, 111 distribution of slabs, 251 grid-spacing, 40 single sinking slab, 249 Rotation slab distribution, 251, 265 Earth, 87 terrestrial planets, 127, 138 Rotational deformation T excitation functions, 98 Tectonic processes, 235 forcing function, 103 Thin shell approximation, 294 rigid Earth, 246 Tibetan Plateau, 100 Rotational number, 126, 138, 141 Tidal deformation, 293 Rotation equation Tidal energy, 293 long-term behavior, 124 Tidal locking, 297 Rotation frequency, 97 Tidal potential, 296, 298 Rotation theories Tidal stresses, 293 comparison, 108 Tides, 294 linearized, 103 Toroidal equations, 20 non-linear, 257 , 90 unification, 114 Transition zone Runge-Kutta propagation, 42, 211, 265 high-viscosity models, 67 True Polar Wander, see polar wander S Satellite Laser Ranging, 189 Scandinavia, 149 V Sea-level changes, 225, 227, 228, 231, 246 Venice, 235, 236, 241 eustatic, 226 Venus, 139 induced by polar wander, 242 , 12, 270, 295, 312 induced by subduction, 246 Viscosity, 1 relative sea-level changes, 230 steady-state, 3 self-gravitation, 225 Viscous response, 42 third-order cycle, 244 VLBI, 237, 270 Sea-level equation, 226 Volume-averaged models, 156