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Tsunami Precursors: Mathematical Model and Applications

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A mathematical model of seismo-hydro-electromagnetic (EM)-temperature geophysical field interaction has been formulated, according to the principles of the magneto-thermo-elasticity and magneto-hydrodynamic theories. Signals of different physical natures, arising in a lithosphere–ocean–atmosphere domain as a result of submarine seismic excitation, were computed. On the basis of satisfactory results of comparisons with measurements, a multidisciplinary vertically distributed (from the seafloor up to the ionosphere) tsunami precursors detection system is described.


  • Tsunami
  • Precursors
  • Detection
  • Mathematical model

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Fig. 3.1
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Appendix A: SHEMTI Equations

3.1.1 String–Diffusion Structure of the SHEMTI Model

Weak seismic excitations (see Sects. 3.1 and 3.2), supplying energy for the SHEMTI process of tsunami precursor formation, may be described by operator \( \mathcal{H} \) of the linear dynamical theory of elasticity

$$ {\displaystyle \begin{array}{l}\mathcal{H}\mathbf{u}\equiv \rho \frac{{\mathit{\partial}}^2\mathbf{u}}{\mathit{\partial}\;{t}^2}-\mathrm{Div}\kern0em \widehat{\sigma},\\ {}\widehat{\sigma}={\left({\sigma}_{ij}\right)}_{i,j=1}^{i,j=3},\kern0.36em {\sigma}_{ij}=2\mu \left(\mathbf{x}\right){\varepsilon}_{ij}+\lambda \left(\mathbf{x}\right){\delta}_{ij}\mathit{\operatorname{div}}\mathbf{u},\\ {}{\varepsilon}_{ij}=\frac{1}{2}\left(\frac{\partial {u}_i}{\partial {x}_j}+\frac{\partial {u}_j}{\partial {x}_i}\right),i,j=1,2,3\end{array}} $$

So, the stress tensor σij is supposed to obey the Hook strain–stress relation with the Cauchy strain tensor εij and the Lame elastic parameters μ and λ, and Div is used for the divergence of the stress tensor:

$$ \mathrm{Div}\;\widehat{\sigma}={\left({\left(\mathrm{Div}\;\widehat{\sigma}\right)}_1,{\left(\mathrm{Div}\;\widehat{\sigma}\right)}_2,{\left(\mathrm{Div}\;\widehat{\sigma}\right)}_3\right)}^{\mathrm{tr}},{\left(\mathrm{Div}\;\widehat{\sigma}\right)}_i=\sum \limits_{j=1}^3\frac{\partial {\sigma}_{ij}}{\partial {x}_j}\kern0.34em ,\kern0.5em i=1,2,3\kern0em $$

In these notations, the system of partial differential equations (PDEs) of the dynamic theory of elasticity takes the form:

$$ \mathcal{H}\mathbf{u}=\rho \mathbf{g}, $$


$$ \mathbf{u}={\left({u}_1\left(t,\mathbf{x}\right),{u}_2\left(t,\mathbf{x}\right),{u}_3\left(t,\mathbf{x}\right)\right)}^{\mathrm{tr}},\mathbf{x}=\left({x}_1,{x}_2,{x}_3\right)=\left(x,y,z\right),\mathbf{g}={\left({g}_1,0,0\right)}^{\mathrm{tr}} $$

are the vectors of the elastic displacements, coordinates, and acceleration of gravity, respectively; indexes 2 and 3 are for the horizontal directions and 1 is for the vertical one; and ρ = ρ(x) denotes the density.

Operator \( \mathcal{H} \) is determined formally, i.e., without functional spaces (Tahery 2015), because this chapter is not a mathematical one. This operator belongs to the known hyperbolic type (the operator of a string is the simplest example): the energy of deformation processes described by similar operators does not disappear. But this is not true for rocks, because temperature (T) and EM fields are generated by deformations (see Sects. 3.1 and 3.2). Therefore, operator \( \mathcal{H} \) should be combined with operator \( \mathcal{P} \) of a parabolic type (see Ladyzhenskaya et al. 1967), describing dissipative processes.

In the case of seismic EM generation in conductive rocks with a minimal electrical conductivity of the order of 0.0001 S/m, these processes include propagation of a low-frequency (from 0.01 up to 100 Hz) diffusive field of magnetic induction:

$$ \:B\left( {t{\text{,}}x} \right) = {\left( {{B_1}\left( {t{\text{,}}x} \right){\text{,,}}{B_2}\left( {t{\text{,}}x} \right){\text{,,}}{B_3}\left( {t{\text{,}}x} \right)} \right)^{tr}} $$

and a thermal field with deviation θ(t, x) of the temperature ϴ(tx) from the initial temperature θ0 existing at t = 0. Let us denote these dissipative processes by w(t, x):

$$ w\left( {t{\text{,}}x} \right) \equiv {\left( {H\left( {t{\text{,}}x} \right){\text{,}}\theta \left( {t{\text{,}}x} \right)} \right)^{tr}},H\left( {t{\text{,}}x} \right) = {\left( {{H_1}\left( {t{\text{,}}x} \right){\text{,,}}{H_2}\left( {t{\text{,}}x} \right){\text{,,}}{H_2}\left( {t{\text{,}}x} \right)} \right)^{tr}} = {\left( {{\mu _0}} \right)^{ - 1}}\:B\left( {t{\text{,}}x} \right), $$

where μ0 is the magnetic permeability of vacuum. The medium is supposed to be uniform in regard to magnetic permeability, and its value is μ0.

So, the diffusion in the medium under consideration is described by the parabolic vector equation (i.e., the system of PDEs of the parabolic type):

$$ \mathcal{P}\mathbf{w}=\mathbf{q}, $$


q = (0, 0, q(x))tr is the density of the power of heat sources (e.g., the radioactivity of the rocks),

$$ \mathcal{P}\mathbf{w}\equiv \left(\begin{array}{c}\frac{\mathit{\partial}\;\mathbf{B}}{\mathit{\partial}\;t}+\mathrm{rot}\left({\sigma}_e^{-1}\left(\mathbf{x}\right)\mathrm{rot}\mathbf{H}\right)\\ {}\rho \left(\mathbf{x}\right){c}_V\frac{\mathit{\partial}\;\vartheta }{\mathit{\partial}\;t}-\operatorname{div}\left(\kappa \left(\mathbf{x}\right)\operatorname{grad}\vartheta \right)\end{array}\right) $$

Here, the first line, which is a vector with three components, determines the operator of magnetic diffusion (the low-frequency displacement currents are negligible as compared with the ones of conductivity) by electrical conductivity σe, and the second line determines the scalar diffusion operator of heat transfer by heat conductivity κ; heat capacity is denoted by cV.

Contrary to observations in seismically active regions, the classical hyperbolic and parabolic vector Eqs. (3.1) and (3.2) are not able to describe generation of magnetic (H) and temperature (θ) field disturbances by seismic excitation of a medium, i.e., magnetic and temperature precursors of earthquakes (see Sect. 3.1).

Along with the above assumptions, let us suppose that polarization (magnetic, electrical) and thermoelectric processes are negligible and a change in the physical characteristics of rocks is impossible under the influence of a weak precursory excitation.

For the conditions mentioned, we apply the magneto-thermo-elasticity (MTE) theory (Maugin 1988) to describe seismo-EM-thermal interaction in the lithosphere. Eq. (3.1) is replaced by (3.3) and Eq. (3.2) by (3.4). So, a coupled system is obtained:

$$ \mathcal{H}\mathbf{u}+\mathcal{A}\left(\mathbf{w}\right)=\rho \mathbf{g} $$
$$ \mathcal{P}\mathbf{w}+\mathcal{B}\left(\mathbf{u},\mathbf{w}\right)=\mathbf{q} $$

with the interaction operators in the form of an additional force \( \mathcal{A}\left(\mathbf{w}\right) \) and field source \( \mathcal{B}\left(\mathbf{u},\mathbf{w}\right) \),

$$ {\displaystyle \begin{array}{c}\mathcal{A}\left(\mathbf{w}\right)\equiv \operatorname{grad}\left(\beta \left(\mathbf{x}\right)\vartheta \right)+\mathbf{B}\times \mathrm{rot}\mathbf{H}\\ {}\mathcal{B}\left(\mathbf{u},\mathbf{w}\right)\equiv \left(\begin{array}{l}-\mathrm{rot}\left[\mathbf{v}\times \mathbf{B}\right]\\ {}\beta \left(\mathbf{x}\right){\vartheta}_0\operatorname{div}\mathbf{v}-{\sigma}_e^{-1}\left(\mathbf{x}\right){\left(\mathrm{rot}\mathbf{H}\right)}^2\end{array}\right)\end{array}} $$


v = ∂u/∂t, v2 = v12 + v22 + v32 < < c2, c is the light velocity in a vacuum and the inequality represents the known assumption of slow motion, β = (2μ + 3λ)αθ, αθ is the coefficient of thermal expansion, αθ = 5 × 10−6 K−1. The first term in the expression for interaction operator \( \mathcal{A}\left(\mathbf{w}\right) \) describes the thermo-stress and the second one originates from the Lorentz force. The first (vector) line in the expression for interaction operator \( \mathcal{B}\left(\mathbf{u},\mathbf{w}\right) \) arises by applying the operator rot to the Maxwell equation (without displacement currents; see above): rot H = j, j = σe( E+[v × B]   ), j is the density of the conductivity current in a slowly moving medium, and E is the electrical intensity. The first and second terms of the second (scalar) line of the operator \( \mathcal{B}\left(\mathbf{u},\mathbf{w}\right) \) describe the heat production due to motion and electrical currents, respectively.

System (3.3), (3.4) of PDEs of the seismo-EM-thermal interaction is not of a classical type, because of the different, namely string and diffusion, physical natures of processes u, w and, correspondingly, different mathematical types of operators \( \mathcal{H} \), \( \mathcal{P} \) (see above). Let us note:

$$ \mathcal{A}\left(\mathbf{w}\right)\to \mathbf{0},\mathcal{B}\left(\mathbf{u},\mathbf{w}\right)\to \mathbf{0}\kern1em \mathrm{by}\kern1em \mathbf{w}\to \mathbf{0},\mathbf{u}\to 0 $$

In other words, if the physical fields under consideration are weak enough, then the string–diffusion (S.-D.) system of PDEs (partial differential equations) takes the form of Eqs. (3.1), (3.2) of the classical types. By going over to the seismo-hydro-EM-thermal interaction in the lithosphere–ocean medium, we apply the oceanology model of “shallow water ” with its hyperbolic operator of motion (e.g., see Landau and Lifshits 1986), instead of one from the theory of elasticity described above. So, the resulting mathematical structure of the three-dimensional (3D) model of the SHEMTI process is of the S.-D. type, i.e., represented by hyperbolic and parabolic operators coupled with field interaction operators (see above) and contact conditions at the lithosphere–ocean interface. To reduce the number of written equations, we describe the full S.-D. form of the SHEMTI model for the two-dimensional (2D) case only (see next section) because the 3D case was computed for the seismo-EM interaction only.

3.1.2 Initial Boundary Value Problem of the 2D SHEMTI Model

We denote the seismic disturbance of the “scalar magnetic potential” as A, that of the magnetic field intensity as H = (H1, H2)tr, and that of magnetic induction as B = (B1, B2)tr, H1 = −∂A(t, x1, x2)/∂x2, H2 = ∂A(t, x1, x2)/∂x1.

Also, we use the following 2D thermo-mechanical notations: λ = λ(x1, x2) and μ = μ(x1, x2) are parameters of Lame, u = (u1(tx1x2), u2(tx1x2))tr is the elastic displacement at point (x1, x2) at time t, vi = ∂ui/∂t, i = 1, 2, are the components of velocity; g = (g,0)tr is the vector of acceleration of gravity. The evolution of the interacting thermo-mechanical and EM fields in the lithosphere part of the medium according to the 2D version of the MTE model (3.3), (3.4) obeys:

$$ {\displaystyle \begin{array}{c}\rho \frac{{\mathit{\partial}}^2{u}_i}{\mathit{\partial}\;{t}^2}=\frac{\mathit{\partial}}{\mathit{\partial}{x}_k}\left(\mu \left(\frac{\mathit{\partial}\;{u}_i}{\mathit{\partial}\;{x}_k}+\frac{\mathit{\partial}\;{u}_k}{\mathit{\partial}\;{x}_i}\right)\right)+\frac{\mathit{\partial}}{\mathit{\partial}\;{x}_i}\left(\lambda \frac{\mathit{\partial}\;{u}_k}{\mathit{\partial}\;{x}_k}\right)-\frac{\mathit{\partial}\left(\beta \vartheta \right)}{\mathit{\partial}\;{x}_i}+\rho {g}_i,\kern0.5em i=1,2\\ {}\frac{\mathit{\partial}\;A}{\mathit{\partial}\;t}=\frac{1}{\mu_0{\sigma}_e}\Delta A-{v}_k\frac{\mathit{\partial}\;A}{\mathit{\partial}\;{x}_k}+{v}_2{H}_{10}-{v}_1{H}_{20}\\ {}\rho {c}_V\frac{\mathit{\partial \vartheta }}{\mathit{\partial t}}=\operatorname{div}\left(\kappa \cdot \operatorname{grad}\vartheta \right)-{\beta \vartheta}_0\operatorname{div}\mathbf{v}+\frac{1}{\sigma_e}{\left(\Delta A\right)}^2+q\end{array}} $$

where ρ = ρ(x1, x2) is the density, σe = σe (x1, x2) is the electrical conductivity, κ = κ(x1, x2) is the thermal conductivity, q = q(x1, x2) is the density of the power of heat sources, and H0 = (H10, H20)tr is the stationary (unperturbed) magnetic field; here and below there is a summation over repeated subscripts which take at 1 and 2, subscript 1 corresponds to the vertical direction and 2 to the horizontal one. The EM term is omitted in the equations of motion, i = 1, 2, because of its negligible influence, according to our calculations.

At the sea bottom the normal component vn of the elastic velocity v equals the normal component wn of the water velocity w = (w1, w2)tr:

$$ \left(\mathbf{\mathsf{v}}\cdot \mathbf{\mathsf{n}}\right)=\left(\mathbf{\mathsf{w}}\cdot \mathbf{\mathsf{n}}\right) $$

where n and τ (see Eq. (3.7)) are the local normal and tangent to the sea bottom (the normal is directed inwards the water). Then, the normal component of force fn acting from the sediment layer equals the water pressure p at this point, while the tangent component of this force fτ vanishes because we neglect shear stresses in the water:

$$ {f}_n={\sigma}_{ij}{n}_i{n}_j=p,\kern1em {f}_{\tau }={\sigma}_{ij}{\tau}_i{n}_j=0, $$

where the Hook’s relation is used in the form:

$$ {\sigma}_{ij}\equiv \mu \left(\frac{\mathit{\partial}{u}_i}{\partial {x}_j}+\frac{\mathit{\partial}{u}_j}{\partial {x}_i}\right)+{\delta}_{ij}\lambda \frac{\mathit{\partial}{u}_k}{\partial {x}_k} $$

We describe the water dynamics in the framework of the nonlinear “shallow water” theory (e.g., Landau and Lifshits (1986)), which operates the horizontal velocity w2 and elevation of the sea surface η(tx2):

$$ {\displaystyle \begin{array}{c}\frac{\mathit{\partial}}{\mathit{\partial t}}\left(\eta +{h}_u\right)+\frac{\mathit{\partial}}{\mathit{\partial}{x}_2}\left(\left(\eta +h\right){w}_2\right)=0,\kern1em \\ {}\frac{\mathit{\partial}{w}_2}{\mathit{\partial t}}+{w}_2\frac{\mathit{\partial}{w}_2}{\mathit{\partial}{x}_2}+g\frac{\mathit{\partial \eta }}{\mathit{\partial}{x}_2}=0\end{array}} $$

The sea bottom is a moving line:

$$ {\mathit{\mathsf{x}}}_{\mathsf{1}}=\mathit{\mathsf{h}}\left(\mathit{\mathsf{t}},{\mathit{\mathsf{x}}}_{\mathsf{2}}\right)={\mathit{\mathsf{h}}}_{\mathsf{0}}\left({\mathit{\mathsf{x}}}_{\mathsf{2}}\right)+{\mathit{\mathsf{h}}}_{\mathit{\mathsf{u}}}\left(\mathit{\mathsf{t}},{\mathit{\mathsf{x}}}_{\mathsf{2}}\right), $$

where h0(x2) is the stationary profile and hu(tx2) is its perturbation due to elastic deformations caused by seismic excitation (3.1) in the upper mantle. In seawater, we neglect the seismic disturbance of the temperature field and the Lorentz force, so the interaction operator \( \mathcal{A} \) = 0 there.

Now let us consider the seismo-hydro-EM interaction in the ocean–atmosphere system. The potential A in the water obeys:

$$ \frac{\mathit{\partial A}}{\mathit{\partial t}}=\frac{1}{\mu_0{\sigma}_e}\Delta A-{w}_k\frac{\mathit{\partial A}}{\mathit{\partial}{x}_k}+{w}_2{H}_{10}-{w}_1{H}_{20} $$

The vertical component w1 of the hydrodynamic velocity is calculated from w2 and elastic velocity using Eqs. (3.7), (3.9), and the incompressibility condition:

$$ \frac{\partial {w}_1}{\partial {x}_1}+\frac{\partial {w}_2}{\partial {x}_2}=0 $$

Electrical conductivity is assumed to be continuous across the sea bottom (i.e., a thin transition layer is used in the computations).

The equation for A above the sea (where x1 < 0) and the conjugation conditions at the sea surface x1 = 0 are:

$$ {\displaystyle \begin{array}{l}\Delta A=0\kern1em \mathrm{for}\ {x}_1<0,\kern1em 0<{x}_2<{L}_2,\kern1em t>0;\\ {}A\left|{}_{x_1=+0}\right.=A\left|{}_{x_1=-0}\right.,\kern1em {\left.\frac{\partial A}{\partial {x}_1}\right|}_{x_1=+0}={\left.\frac{\partial A}{\partial {x}_1}\right|}_{x_1=-0}\end{array}} $$

The initial conditions were chosen so that before the moment t = 0, the system was at rest; thus, the water velocities and elevation of the sea surface vanish, the elastic displacements equal the stationary configuration under the gravity force, and the initial magnetic field is uniform in space: H10 = 30, H20 = 35 A/m. For all fields, the normal derivative vanishes at the left (x2 = 0) and right (x2 = L2) boundaries. The remaining “outer” boundary conditions are (see LD, L1, and other input in Appendix B):

$$ {\displaystyle \begin{array}{l}{\left.\partial A/\partial {x}_1\right|}_{x_1=-{L}_D}={\left.\partial A/\partial {x}_1\right|}_{x_1={L}_1}=0,\\ {}{\left.\vartheta \right|}_{x_1={h}_0\left({x}_2\right)}=0,\kern1em {\left.\partial \vartheta /\partial {x}_1\right|}_{x_1={L}_1}=0,\\ {}{\left.\mathbf{u}\right|}_{x_1={L}_1}={\mathbf{u}}_0\left(t,{x}_2\right)\end{array}} $$

This approach to the theory of the seismo-hydro-EM-temperature processes, including the existence and uniqueness theorems for the solution of the initial boundary value problem of the system of PDEs with the coupled operators of the hyperbolic and parabolic types, e.g., (3.53.13), and the corresponding discrete approximations, is a further development of Novik (1995), Novik et al. (2001), Novik and Ershov (2001), Ershov et al. (2001, 2006), and Ershov and Novik (2009). Other approaches to the seismo-electromagnetics and bibliography may be found in the references.

Appendix B: Data Applied to the Calculations in Sect. 3.2

We consider a 2D problem of seismic generation and propagation of elastic, hydrodynamic, EM and temperature (generated by dissipation in rocks) field disturbances in a domain including a vertical cross-section of the central part of the Sea of Japan down to the upper mantle at a depth of L1 = 37 km below the sea level, the sea, and the atmosphere up to the lower boundary of the ionosphere domain D at the height x1 = −LD, LD = 70 km.

The cross-section, with a length of 425 km, runs from the Bay of Peter the Great through the Pervenets Rise and the Yamato Basin, and ends at the North Yamato Rise. The coordinates x = (x1, x2) are chosen so that x1 is directed downward with x1 = 0 at the sea surface, and x2 is directed horizontally, 0 ≤ x2 ≤ L2 = 425 km.

The seawater and the sedimentary, granite, basaltic, and upper mantle layers are denoted by W, S, G, B, and M, respectively (Fig. 3.4 and Table 3.1). The heat capacity is cV = 660 J/(kg × K) everywhere and the magnetic permeability is μe = μ0 = 4π × 10−7 (in SI units). The components H10 and H20 of the stationary geomagnetic field H0, H0 = (H10, H20)tr, are supposed to be uniform in the considered domain.

Table 3.1 Physical parameters of the medium: values (in SI units) of density ρ, elastic wave velocities vp and vs, electrical and thermal conductivities σe and κ, and densities of heat flow q

To model the seismic excitation due to the arrival of a seismic wave from a deeper source at the lower external boundary x1 = L1 = 37 km, we use the vector u0 = (u01, u02)tr as a nonstationary boundary condition for elastic displacements u1(t, x1, x2) and u2(t, x1, x2):

$$ {\mathit{\mathsf{u}}}_{\mathsf{1}}\left(\mathit{\mathsf{t}},{\mathit{\mathsf{L}}}_{\mathsf{1}},{\mathit{\mathsf{x}}}_{\mathsf{2}}\right)={\mathit{\mathsf{u}}}_{\mathsf{01}},{\mathit{\mathsf{u}}}_{\mathsf{2}}\left(\mathit{\mathsf{t}},{\mathit{\mathsf{L}}}_{\mathsf{1}},{\mathit{\mathsf{x}}}_{\mathsf{2}}\right)={\mathit{\mathsf{u}}}_{\mathsf{02}} $$


$$ {\displaystyle \begin{array}{c}{u}_{01}={a}_1\times {te}^{- bt-\alpha {\left({x}_2-{x}_{20}\right)}^2}\cos \left(\omega t\right)\cos \frac{2\pi {x}_2}{d},\kern2em \\ {}{u}_{02}={a}_2\times {te}^{- bt-\alpha {\left({x}_2-{x}_{20}\right)}^2}\sin \left(\omega t\right)\sin \frac{2\pi {x}_2}{d}\end{array}} $$

Here: a1 = 7 cm, a2 = 3.5 cm, b = 1/3 s−1, α = 5 × 10−11 m−2, 0 < t < T, ω = 5/3 s−1, x20 = L2/2 = 212.5 km, d = 150 km, T = 20 s.

The duration of this model seismic excitation (varied in different simulations) is several seconds, the amplitude is about 5 cm, the main frequency is about several tenths of a hertz (the spectrum of u02 is the same), and the excitation decays with time and far from the center of the domain.

Let us formulate a few remarks about the dependence of the characteristics of the computed SHEMTI process on the variations of the data in its string–diffusion model.

Our simulation algorithms allow us to compute evolution of fields under a wide class of initial and boundary conditions, including volumetric a seismic excitation instead of a boundary excitation. Indeed, a computed field configuration for some moment may be considered as the initial one for later moments. In the frame of the 2D string–diffusion model (3.53.13) of the SHEMTI process, we have varied the parameters of the excitation and other conditions at the outer boundaries x1 = L1, x1 = −LD and x2 = 0, x2 = L2, as well as the characteristics of the medium (taking into account the geophysical and petrophysical restrictions), but the main features of the evolution of the model seismic perturbations of fields, especially the relation between seismic excitations in the earth depth under the seafloor and EM signals at the sea surface and above, have remained the same. These computations show that the values of the parameters of the seismic excitation (i.e., the nonstationary condition at the lower boundary of the domain considered) are of the most importance, as compared with other data, in forming (a) the seismo-EM process in the lithosphere part of the medium (i.e., the energy supply of the tsunami), and (b) EM signals above the sea. Our 3D computations of the seismo-EM process with the string–diffusion model (3.13.4) for the case of a geological structure with a weak dependence of its properties on one of the horizontal coordinates (lineament, e.g., a domain near a central cross-section of a structure) support the 2D computations; for example, the computed long EM wave inherits the time and space characteristics of the seismic wave for both 2D and 3D geometries. Also, the hydrodynamic component of the SHEMTI process has been computed with the help of the hydro-acoustic operator instead of the “shallow water” one (both are of the hyperbolic type). The pictures of the computed SHEMTI processes are similar in terms of the main details.

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Novik, O., Smirnov, F., Volgin, M. (2019). Tsunami Precursors: Mathematical Model and Applications. In: Electromagnetic Geophysical Fields. Springer, Cham.

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