Cross-diffusion waves resulting from multiscale, multi-physics instabilities: theory - Solid Earth
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Solid Earth, 12, 869–883, 2021
https://doi.org/10.5194/se-12-869-2021
© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.
Cross-diffusion waves resulting from multiscale, multi-physics
instabilities: theory
Klaus Regenauer-Lieb1 , Manman Hu2 , Christoph Schrank3 , Xiao Chen1 , Santiago Peña Clavijo1 , Ulrich Kelka4 ,
Ali Karrech5 , Oliver Gaede3 , Tomasz Blach1 , Hamid Roshan1 , and Antoine B. Jacquey6
1 School of Minerals and Energy Resources Engineering, UNSW, Sydney, NSW, 2052, Australia
2 Department of Civil Engineering, The University of Hong Kong, Hong Kong
3 Science and Engineering Faculty, Queensland University of Technology, Brisbane, QLD, 4001, Australia
4 CSIRO Deep Earth Imaging Future Science Platform, Kensington, WA, 6151, Australia
5 School of Engineering, University of Western Australia, Crawley, WA, 6009, Australia
6 Department of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA, USA
Correspondence: Klaus Regenauer-Lieb (regenau@gmail.com)
Received: 1 April 2020 – Discussion started: 11 August 2020
Revised: 4 February 2021 – Accepted: 9 February 2021 – Published: 16 April 2021
Abstract. We propose a multiscale approach for coupling 1 Introduction
multi-physics processes across the scales. The physics is
based on discrete phenomena, triggered by local thermo-
hydro-mechano-chemical (THMC) instabilities, that cause The theory presented in this paper grew out of the conference
cross-diffusion (quasi-soliton) acceleration waves. These series dedicated to understanding coupled thermo-hydro-
waves nucleate when the overall stress field is incompati- mechanical-chemical (THMC) in Geosystems (GEOPROC).
ble with accelerations from local feedbacks of generalized The 7th International Conference on Coupled THMC Pro-
THMC thermodynamic forces that trigger generalized ther- cesses was held in 2019 in Utrecht and focussed on earth-
modynamic fluxes of another kind. Cross-diffusion terms in quake and faulting mechanics (as does this special issue).
the 4 × 4 THMC diffusion matrix are shown to lead to multi- Integration of mechanical, hydrodynamical, thermal, and
ple diffusional P and S wave equations as coupled THMC chemical processes covers, however, a much wider field from
solutions. Uncertainties in the location of meso-scale ma- the pore to plate-tectonic scale for a wide range of natural and
terial instabilities are captured by a wave-scale correlation engineering problems in geological systems discussed in fo-
of probability amplitudes. Cross-diffusional waves have un- cus topics at earlier GEOPROC conferences. These problems
usual dispersion patterns and, although they assume a soli- include nuclear waste disposal, coal seam gas, enhanced oil
tary state, do not behave like solitons but show complex inter- and gas recovery, geothermal energy, mineral deposits, tail-
actions when they collide. Their characteristic wavenumber ing dam collapse, landslides, and many others. The individ-
and constant speed define mesoscopic internal material time– ual problems may have their own characteristics. However,
space relations entirely defined by the coefficients of the the common scientific issue of multiscale feedback of THMC
coupled THMC reaction–cross-diffusion equations. A com- processes remains the same.
panion paper proposes an application of the theory to earth- The GEOPROC theme seeks to foster the urgently needed
quakes showing that excitation waves triggered by local re- growth of experimental, numerical, and theoretical studies on
actions can, through an extreme effect of a cross-diffusional multi-physics (THMC) and multiscale framework studies in
wave operator, lead to an energy cascade connecting large earth sciences. The current practice is to still use engineering
and small scales and cause solid-state turbulence. solutions based on empirical material laws to address spe-
cific natural and engineering problems in geological systems
and energy production in geothermal energy, nuclear waste
disposal, reservoir engineering for oil and gas, the formation
Published by Copernicus Publications on behalf of the European Geosciences Union.
870 K. Regenauer-Lieb et al.: Cross-diffusion waves of mineral deposits, induced seismicity, natural hazards, and bations can be found in the short-wavelength regime with CO2 sequestration and utilization. These empirical engineer- divergence growth. ing approaches are often inadequate, as indicated, for exam- While applied mathematical solutions exist, the prefer- ple, in the failure to avoid the 5.5 magnitude earthquake in ence in geosciences is to address the problem of unbounded Pohang, Korea, in November 2017, which was anthropogeni- growth by explicit consideration of additional physics. A cally induced by high-pressure hydraulic injection during the specific case was shown where the infinite response can be previous 2 years (Grigoli et al., 2018). captured by postulating a carefully chosen chemical reac- Part of the reasons for the lack of a wider adoption of cou- tion (Alevizos et al., 2017). A recent contribution has in- pled THMC approaches in the community is a lack of a the- troduced a complex multi-physics approach to compaction oretical basis on which to assess the rich solution space that band formation (Jacquey et al., 2021) by adding a diffusion arises from a coupling of the four (THMC) partial differential mechanism to the carefully chosen reaction term. The solu- reaction–diffusion equations. While parallel numerical tools tion space was explored numerically showing standing waves for modelling fully coupled non-linear systems of THMC that can interfere with a propagating wave and also lead to a equations have become available through pioneering work pattern with spatial periodicity. Both approaches solve the in nuclear engineering (Gaston et al., 2009; Permann et al., ill-posed problem for some cases but a general solution that 2020), the corresponding theory has not progressed as far. uses the physics of internal processes to regularize the prob- The application of the powerful nuclear engineering mod- lem was lacking. This calls for an extension to the theoretical elling tool has been successfully transferred to geosciences work of Hill (1962), which is presented here. and applied to geodynamic modelling (Jacquey and Cacace, The dynamic field is of special interest to the researcher 2020a, b) and the modelling of the non-volcanic tremor and in the area of earthquake and faulting instabilities. The state slip (NVTS) events in the circum-Pacific subduction zones of the art in this field is defined by the influential experi- (Poulet et al., 2014b) as well as applied to geological faulting mental work of Dieterich (1979) including the work on the problems (Poulet et al., 2014a). However, a sound theoretical application of the rate and state variable friction approach to description and interpretation of the local processes resulting earthquakes (Tse and Rice, 1986). The approach based on in the interesting macroscopic phenomena has been lacking. these laboratory-derived constitutive equations has reached The companion article (Regenauer-Lieb et al., 2021) aims a mature stage, and no attempt is made here to compare the at providing a detailed, step-by-step explanation of the new rich field of findings with the present theory. We approach theory used to rectify this shortcoming preceded by a short the problem from an entirely different angle through theoret- introduction into the theory of excitable waves triggered by ical investigation of the mathematical solutions of the sys- THMC reaction terms. tem of coupled partial differential THMC equations that de- Before discussing a possible application of the new theory liver wave solutions with short-wavelength instabilities. In to the processes of earthquakes and faulting in our compan- the course of developing the new approach, we describe wave ion article (Regenauer-Lieb et al., 2021), here we present a physics phenomena that have previously not been reported in transdisciplinary approach to bridging the gap between ob- the solid earth community but are well known in a range of servations of instabilities from the molecular scale to the different fields from quantum systems to ocean waves (Za- very large scale. The theory in this paper is written using kharov et al., 2004). It is fair to say that the theory is rather in approaches familiar to the theoretical and applied mechan- its infancy, and special care needs to be taken before consid- ics community. The original work is based on the 1960’s ering a direct application to the aforementioned systems. The work (Hill, 1962) building the foundation of theoretical ap- first part therefore presents the theoretical derivation, and the proaches to localization criteria, via the so-called acous- second part delves into possible applications and proposed tic tensor criterion, widely used in the engineering com- experiments to test the applicability of the theory. munity (Rudnicki and Rice, 1975). The approach focusses In this paper, we introduce the classical approach of ac- on standing-wave quasi-static solutions based on vanishing celeration waves in plasticity theory to the seismology com- speeds of acceleration waves which, without consideration of munity by starting with the Helmholtz decomposition of the additional length scales, leads to infinite values of variables seismic wave equation into P and S waves (see Sect. 3.1). We on the localization bands such as infinite strain rate in shear show how plasticity theory can be integrated into the equa- or infinite pressure (Veveakis and Regenauer-Lieb, 2015) for tions via Hill’s acceleration waves. This approach leads di- volumetric localization bands. Surprisingly, little effort has rectly to the unbounded short-wavelength growth described been made to explore the rich wave field of the corresponding by Benallal and Bigoni (2004), which cannot be solved with- travelling-wave solutions, probably because dynamic events out further assumptions. The innovation proposed in the two are only of academic interest to the engineering plasticity papers is to appeal to the multiscale nature of the THMC community that focusses mainly on developing safety stan- coupled problem. We regularize the problem by embed- dards as well as limit analysis and design. A notable excep- ding an open-system thermodynamic multiscale theory with tion is the work of Benallal and Bigoni (2004), who found unbounded solutions into a closed-system macro-scale ap- that under dynamic conditions, unbounded growth of pertur- Solid Earth, 12, 869–883, 2021 https://doi.org/10.5194/se-12-869-2021
K. Regenauer-Lieb et al.: Cross-diffusion waves 871 proach that describes the emergence of standing-wave solu- the form of “cross-diffusion” waves. The emergence of cross- tions. diffusion waves can be perhaps best understood from a chem- There are two opposite starting points for the deriva- ical viewpoint (Regenauer-Lieb et al., 2021) where propa- tion of the approach. Here, we investigate the meso-scale gating chemical waves have been studied in detail (Vanag from the conventional mechanical quasi-steady-state (infinite and Epstein, 2009). In chemical systems, cross-diffusion is timescale) solution of the macro-scale. Although the present defined as the phenomenon in which a gradient in the con- paper uses the macro-scale perspective, i.e. the classical me- centration of one species induces a flux of another chem- chanical viewpoint for the investigation of the physics of ac- ical species. In the present context thermodynamic forces celeration waves (Hill, 1962), in variance to the classical ap- and fluxes are generalized THMC fluxes defined in Table 1. proach we acknowledge that both meso-scale processes and Before discussing cross-scale coupling of thermodynamic the macro-scale behaviour have an effect on each other. The forces and fluxes in Sect. 3.3, it is useful to briefly review companion article (Regenauer-Lieb et al., 2021) describes insights into the formation of discrete dissipative structures. the meso-scale view, which is the classical viewpoint of a chemist. Both viewpoints are objective descriptions of the coupled THMC problem and should deliver the same out- 2 Dissipative structures come. In order to define the separation between the meso- and The concept was introduced first in chemical and biologi- macro-scale of a coupled THMC problem, we propose that cal systems where morphogenic patterns (Turing, 1952) were the scale for each of the THMC processes is defined by identified as solutions to the underlying reaction–diffusion its own characteristic diffusion timescales and length scales equations (see Fig. 1). These discrete patterns were later on (Regenauer-Lieb et al., 2013b). The THMC diffusion length named Turing patterns. A review of Turing patterns in nature scale is thereby related to the timescale of a considered can be found in Ball (2012). THMC process as defined by the proportionality to the We propose here a generalized approach to cross-diffusion square root of the diffusivity multiplied by the process time. that is known in bio-physics as taxis (Heilmann et al., 2018). For simple problems progress can be made by studying ther- For example, the pufferfish and the siltstone in Fig. 1 show modynamic equilibrium states in isolated closed systems. similar patterns caused by a fundamental mechanism known Likewise closed, coupled, far-from-equilibrium THMC sys- as taxis. This is a process that forces components of a pat- tems that feature irreversible behaviours can be modelled by tern to organize as an ensemble in reaction to changes in a thermomechanics approach (Collins and Houlsby, 1997), the environment, and so to move towards, or away, from also called a thermodynamics with internal variables ap- a perturbation. The process eventually leads to the forma- proach (Maugin and Muschik, 1999) or a hyperplastic ap- tion of a new energetically stable pattern. The fundamen- proach (Houlsby and Puzrin, 2007). This theory is, however, tal pattern-forming taxis mechanism can be caused by adhe- only applicable to faults that have reached a thermal steady sive (hapto-taxis), hydrodynamic (gyro-taxis), gravitational state as implied by a standing-wave solution of acceleration (gravi-taxis), light-intensity (photo-taxis), or chemical driv- waves. This approach prevents modelling of dynamic phe- ing forces (chemo-taxis), as shown in Fig. 1. Non-biological nomena. Modelling of earthquakes and faulting is hence one patterns are generally formed by THMC reactions, which can of the most difficult topics to address using a self-consistent also include electrical and biological drivers. In mathemati- thermodynamic approach. cal biology, taxis models are used to understand and quantify A particular challenge for deriving dynamic THMC cou- a variety of complex problems, ranging from nerve pulse re- pled wave solutions is the discrete nature of the cascade of sponses and spreading of diseases (Zemskov et al., 2017) to steady-state solutions defined by the standing-wave solutions predicting the spatio-temporal patterns of predator–prey sys- of thermomechanics, which leads to a discrete material be- tems. haviour as discussed in the next section. Standard probability However, except for the seminal early work by Ortol- theory is therefore not suitable as this assumes a continuum eva and co-workers (Dewers and Ortoleva, 1990; Ortoleva, of wave functions (Cohen, 1988). In order to solve this issue 1993, 1994) developments of taxis models in earth and ma- we use a transfer of knowledge from classical quantum me- terial sciences have lagged. A review of the progress made in chanics to characterize any system at a larger scale. The in- this field as well as a specific case study of rhythmic band- formation on multiple internal material timescale and length- ing in marls can be found in the recent work of L’Heureux scale processes disperses each at characteristic velocities in (2018, 2013). The present work develops the key ideas into the form of acceleration waves. a geomechanical perspective building on an initial approach The nucleation mechanism of these waves relies on the proposed for hydro-mechanical coupling (Hu et al., 2020; meso-scale open-system behaviour where the overall macro- Alevizos et al., 2017; Regenauer-Lieb et al., 2016, 2013a; scale thermodynamic forces can become incompatible with Veveakis and Regenauer-Lieb, 2015). accelerations from local thermodynamic fluxes. These in- By analogy to the mathematically similar biological and compatibilities radiate wave energy away from its source in chemical systems, we propose here that earthquake instabil- https://doi.org/10.5194/se-12-869-2021 Solid Earth, 12, 869–883, 2021
872 K. Regenauer-Lieb et al.: Cross-diffusion waves
ities are preceded (and followed in the post-seismic stage) tail in the companion article (Regenauer-Lieb et al., 2021),
by propagating THMC dissipative waves which could en- where experiments that are very close to the stationary mode
able new detection methods if they can be resolved by sen- are also introduced (Barraclough et al., 2017). The station-
sors. We will discuss such possible precursor phenomena for ary mode allows development of a robust thermomechan-
earthquakes in the companion article (Regenauer-Lieb et al., ics theory which offers a modular thermodynamically self-
2021). In chemical systems, propagating waves stemming consistent approach for modelling earth instabilities (Jacquey
from reaction–diffusion processes are very well documented and Regenauer-Lieb, 2021).
and the hypothesis here is that this applies generally to all Experiments with highly porous carbonates have been
THMC coupled processes. A review and update of the for- performed (Chen et al., 2020). These produced stationary
mulation for chemical systems can be found in Vanag and and non-stationary compaction bands under uniaxial loading.
Epstein (2009). Unfortunately, these experiments have the opposite problem
While chemical oscillations thus appear to be well under- in that the exact analysis of the dynamic evolution did not
stood, the phenomenon of an oscillatory response is less well reach sufficient resolution in space and time to convincingly
established in other THMC systems. However, these sys- detect the wave phenomenon described in the granular brittle
tems show the same transitions from a simple continuum re- matter. We, therefore, explore in this contribution theoreti-
sponse to a highly localized state. The existence of a discrete, cal predictions of the dynamic wave propagation based on an
particle-like nature has also been discovered in fluids when extension of the thermomechanics approach for wave propa-
they are driven far from equilibrium. If driven far from equi- gation in dissipative materials (Coleman et al., 1965).
librium by surface forces, fluids clearly show (see Fig. 2) a We propose here that the dissipative wave phenomenon
highly dissipative, sharp transition from a continuum state is universal for THMC reaction–diffusion systems that are
to one of a highly localized, propagating, particle-like state driven far from equilibrium. The approach allows an inter-
(Lioubashevski et al., 1996). pretation of observations in nature and the laboratory in terms
For the case of deforming geomaterials, a theory for lo- of propagating particle-like states which emerge as stationary
calization phenomena, characterized by a sudden transition Turing patterns for long-timescale standing-wave solutions
from continuum deformation behaviour to a highly localized of a THMC cross-diffusion formulation. In order to recover
state, is well established (Rudnicki and Rice, 1975). Fig- the dissipative wave equations, we present in the following
ure 3 shows a periodic set of localized deformation struc- the standard constitutive assumptions for any generic ther-
tures formed as a result of a compressive tectonic regime. modynamic fluid or solid mechanical system and describe
Similar standing-wave-like features are encountered in many how the physics of THMC feedbacks can be implemented to
geological systems (L’Heureux, 2013; Ball, 2012). However, resolve the phenomenon of propagating dissipative waves in
direct experimental evidence for precursory transient trav- these systems.
elling solitary states is largely unknown and has only been
shown recently based on mathematical considerations (Hu
et al., 2020). In the supplementary material of the companion 3 Wave equations
article (Regenauer-Lieb et al., 2021) we will discuss possi-
3.1 Constitutive assumptions
ble experimental tests of the precursor phenomena. The lack
of experimental evidence can be explained by the challeng- The fundamental equation of motion is
ing task of dealing with the large length scale of the geome-
chanical phenomena and the long timescales of observation ∇ · σ + f = ρ a, (1)
required to mimic natural processes in the laboratory (Pater-
son, 2001). where σ = σij is the Cauchy stress tensor, ρ is the density, f
The dynamics of the formation of these mechanical dissi- is a body force (e.g. gravity), and a is the acceleration. This
pative patterns can therefore only be investigated using ana- equation does not stipulate a constitutive law but with con-
logue materials in the laboratory. Analogue experiments have stitutive assumptions it becomes the master equation for the
been performed in granular, brittle matter compressed uni- theory of elastic waves, fluid mechanics and continuum me-
axially (Guillard et al., 2015; Einav and Guillard, 2018). chanics. In elasticity, wave equations directly result from the
A propagating compaction wave phenomenon has been ob- equation of motion defining the wave characteristics by us-
served. Acoustic bursts have been registered when the waves ing the Helmholtz decomposition, in terms of shear (S wave)
interact with interfaces leading to the conversion of their and compressional (P wave) wave velocities, which is a con-
kinetic energy into acoustic emissions. The phenomenon venient description for the purpose of this paper.
has been compared to ice-quakes in ice sheets (Einav and For an isotropic elastic medium, for instance, accelerations
Guillard, 2018). While these experiments allow some in- in Eq. (1) are only allowing elastic displacements described
sight into the precursor phenomena of stationary compaction by u. In this case the material can be characterized by just
bands, the experiments themselves never reached the sta- two velocities: the elastic P-wave velocity vp and the elastic
tionary mode. This aspect will be discussed in more de- S-wave velocity vs , and we obtain from Eq. (1) the elastic-
Solid Earth, 12, 869–883, 2021 https://doi.org/10.5194/se-12-869-2021K. Regenauer-Lieb et al.: Cross-diffusion waves 873
Figure 1. Dissipative patterns in chemical and biological systems: simulated and real patterns on pufferfish (Sanderson et al., 2006) and
strangely patterned siltstone (Zebra Rock, Ranford formation in the Kununurra district in the East Kimberley region of Western Australia).
Both pattern formations can be modelled by a reaction–diffusion–advection type instability due to chemo-taxis.
wave equation:
∂ 2u
= v 2 ∇(∇ · u) − v 2 ∇ × (∇ × u). (2)
∂t 2 | p {z } | s {z }
P wave S wave
Similarly, by allowing the material to deform in a viscous
manner, acceleration can be monitored by a local change
in velocity v, and the Helmholtz decomposition identifies a
scalar P-wave and a vectorial S-wave potential field. The ma-
terial constants are the dynamic shear η and bulk ζ viscosities
to obtain the generalized Navier–Stokes equation:
∂v
ρ + v · ∇v = −∇p + 2∇ 2 (η˙ 0 ) + ∇(ζ (∇ · v)) + f, (3)
∂t
where
1
˙ 0 = ∇v + (∇v)T − ˙ 0
2
is the deviatoric viscous strain rate, with
1
˙ 0 = (∇ · v)I
3
where I is the identity matrix.
We emphasize here, that although the Helmholtz decom-
position can be performed in a similar way to derive volu-
Figure 2. Dissipative patterns in fluid systems: water molecules ex- metric and shear moduli that describe dissipative material be-
hibit a discrete quantum-like solitary state when forced by a me- haviour, their response to infinitesimal perturbation is gener-
chanical shaker at a critical condition (here 41 Hz). Periodic finger- ally to dampen propagating elastic waves. One could, there-
like solitary states travel from right to left at a constant velocity. fore, come to the erroneous conclusion that their contribution
Each snapshot shows 20 ms intervals. Unlike classical solitons their to precursory wave phenomena and to macroscopic failure is
appearance is particle-like. They can pass through each other with an overall suppression of instabilities.
a slight loss of amplitude, or “collide” to create a new state whose
Coleman and Gurtin (1965) have shown that this conclu-
direction of propagation is at an angle to that of the original states or
sion is wrong using the concept of materials with fading
disintegrate upon collision (image from Lioubashevski et al., 1996
with copyright permission for the American Physical Society arti- memory conceptualized by rational thermodynamics. We are
cle “Dissipative Solitary States in Driven Surface Waves”; identifi- using a simpler approach and are introducing fading mem-
cation number RNP/20/OCT/032299). ory through THMC dissipation processes based on the non-
equilibrium thermodynamics approach of deGroot (1962).
Therein, non-equilibrium conditions are seen as a concate-
nation of thermostatic equilibrium processes. We, therefore,
https://doi.org/10.5194/se-12-869-2021 Solid Earth, 12, 869–883, 2021874 K. Regenauer-Lieb et al.: Cross-diffusion waves
Figure 3. (a) Photograph of three generations of shear-enhanced compaction bands and pure compaction bands in silt- and sandstones of
the Miocene Whakataki Formation, south of Castlepoint, North Island, New Zealand. These deformation bands formed during the slow
compression of the Hikurangi subduction wedge (see Elphick et al., 2021, for details). The positive relief of the metre-long bands is caused
by the significant porosity reduction relative to the host rock, which renders them less susceptible to erosion. The dominant microphysical
deformation mechanism for inelastic volume loss is grain crushing (b). The bands were possibly formed by hydro-(chemo-)mechanical
coupling, which can explain the relatively short (diffusive) length scales (Hu et al., 2020). The regular, constant spacing may be indicative
of the standing-wave phenomenon (Regenauer-Lieb et al., 2013a; Veveakis and Regenauer-Lieb, 2015). (b) Scanning-electron micrograph,
recorded with the backscattered electron detector, of a compaction band (region above the white dashed line) and its host rock in fine-grained
sandstone of the Whakataki Formation. Pores appear black (the white arrow marks an example). The compaction band displays a marked
reduction in porosity and contains a much larger proportion of crushed grains than the host rock.
can use the local equilibrium definition of the pressure as over the entire space to form dissipative structures (Vasil’ev,
p = − ∂U∂V , where U is the internal energy and V the volume, 1979).
and explore the emergent dynamics through investigating the They are an entirely different class of waves as they are
stability of small perturbations from individual equilibrium based on dissipation in active kinetic systems in contrast to
states. In the present context pressure is therefore defined as waves in conservative systems. For simplicity, we only dis-
p = 13 tr(σ ) and is negative for compression. cuss the slow viscoplastic wave phenomenon allowing the
For the elasto-viscoplastic case, we have the equiva- investigation of conservative and dissipative waves as differ-
lent fourth-order elastic-viscoplastic stiffness tensor C char- ent processes. For decoupling elastic and dissipative waves,
acterizing material stiffness and the corresponding elasto- we need to assume large differences in the propagation speed
viscoplastic bulk viscosity ζ to give of the waves. This is done by assuming Maxwellian rheol-
ogy, implying a separation of elastic and viscoplastic wave
∂v timescales in the context of an additive strain-rate decompo-
ρ + v · ∇v = −∇p + 2∇(C˙ 0 ) + 3∇(ζ ˙0 ) + f. (4)
∂t sition of Eqs. (2) and (4). To recover dissipative waves from
the above-discussed Navier–Stokes equation, modified for
In this case ˙ 0 denotes the deviatoric strain rate which in the inclusion of elasto-viscoplastic behaviour, we introduce
the purely elastic case before yield is ˙ 0 = ˙ 0e , becoming local thermodynamic THMC feedback processes that change
post-yield the elasto-viscoplastic strain rate defined by ˙ 0 = the instantaneous fourth-order elastic-viscoplastic stiffness
˙ 0e + ˙ 0vp , where the subscripts e and vp refer to the elas- tensor C.
tic and viscoplastic components. The same approach is used
for decomposing the equivalent elasto-viscoplastic volumet- 3.2 Acceleration waves and classical theories of
ric strain rate ˙0 . While the emergence of elastic P and localization
S waves for any infinitesimal disturbance is a well-known
physics phenomenon, dissipative processes are commonly The simplest implementation of the non-equilibrium ap-
known to dampen elastic waves. Long-range wave propa- proach of deGroot (1962) is the theory of internal vari-
gation in dissipative materials was therefore contested for a able thermodynamics, which unifies the kinetic descrip-
long time (Coleman and Gurtin, 1965). Ideal elastic waves tion of chemical reaction–diffusion processes and the
without damping propagate without loss of energy as they above-described elastic-viscoplastic formulation (Maugin
are based on the conservation of energy. However, the dissi- and Muschik, 1999). Perturbations to the local equilibrium
pative chemical and biological diffusion waves are known to assumption of the non-equilibrium thermodynamic theory
propagate as autonomous wave sources, spontaneous oscil- of internal variables can lead to conditions of violation of
lations, and quasi-stochastic waves which are synchronized smoothness on surfaces in a body, where one or more in-
Solid Earth, 12, 869–883, 2021 https://doi.org/10.5194/se-12-869-2021K. Regenauer-Lieb et al.: Cross-diffusion waves 875
ternal variables from the lower scale suffer jump disconti-
nuities, owing to locally reaching a critical dissipation. In a
classical thermodynamic sense, this can be viewed as sud-
denly switching on a micro-engine somewhere in the system
that disturbs the overall stress field. This is the physical rea-
son for the formation of acceleration fronts, where the diffu-
sive length scales are linked to the convective velocity of the
step function on dissipation waves. While fluid- and solid-
wave phenomena occur when the above equation includes
inertial forces, so-called “acceleration waves” (Hill, 1962),
caused by local surfaces of acceleration (see Fig. 4), can also
occur in the creeping flow limit when no acceleration due to Figure 4. Acceleration waves can originate at a body surface when
a (gravity) potential is present and f = 0. These acceleration the existing internal stress gradient is dynamically incompatible
waves are defined as geometric surfaces (here assumed to be with accelerations imposed on particles of the surface. A propagat-
plane waves) that move relative to the material. ing plane-wave front is shown here for reference, but a plane-wave
Acceleration waves can be described in two ways. One can is not a necessary restriction. Across these surfaces particle acceler-
use two coordinate systems, one for the reference state and ations and spatial gradients of velocity are momentarily discontinu-
one for the current state. A more elegant way is to consider ous while the velocity itself is continuous.
convective coordinate systems by formulating the constitu-
tive law in terms of stress rate. For this we consider the space Acceleration waves form the basis of localization criteria
derivative normal to the moving wavefront (see Fig. 4) indi- in plasticity theory. The criterion for instability is derived
∂
cated by ∂s . Waves are travelling concerning a background from the equivalent theory in elastodynamics where for an
Lagrangian moving material reference frame. elastoplastic body the acoustic tensor 0 is defined by
Considering the traction (load per unit area) in the direc-
tion normal to the wavefront as F and choosing the magni- 0 = n · C · n. (9)
tude of the velocity of the moving wavefront as c, the jump In elastodynamics, the eigenvalues of 0 divided by the mass
condition indicated by the square Iverson brackets can be density represent the square of the elastic wave propagation
advected along c. This leads to Hadamard’s jump condition speed in the direction of the unit normal vector n. In elasto-
where the true traction rate along the advected coordinates is plasticity, the equivalent dynamic stability criterion is de-
∂F fined by Eq. (8) which in terms of acoustic tensor implies
Ḟ = −c . (5) that
∂s
∂v ∂v
Hadamard’s jump condition applies to all internal variables, 0 = ρc2 . (10)
∂s ∂s
and the acceleration across the wavefront is constrained by
Dynamic system stability can be evaluated through assess-
∂F ing the eigenvalues of the acoustic tensor, thus determining
[ρ v̇] = . (6) the speed of the acceleration waves which must be real and
∂s
defined by the square root of the instantaneous modulus C
Combining Eqs. (5) and (6) we obtain divided by the instantaneous density ρ (Coleman and Gurtin,
1965). Mathematically, Eqs. (2–4) can be represented by the
[Ḟ] + c[ρ v̇] = 0. (7) addition of two functions, a scalar field and the curl of a vec-
tor field. The scalar field without curl or rotation identifies
Substituting v̇ = −c ∂v
∂s into Eq. (7) we obtain dissipative compressional P waves and the curl features zero
∂v
divergence and corresponds to isochoric sinistral and dextral
[Ḟ] = c2 ρ . (8) dissipative shear S waves.
∂s
These waves are interpreted as stationary (standing) waves
Hill’s formulation of acceleration waves in Eq. (8) expresses when the determinant of the acoustic tensor, and conse-
the energetics of the acceleration waves by the square of quently the wave speed is zero:
the material velocity c times the mass of the characteristic
det(0) = 0, (11)
segment defined by ∂v ∂s . This provides a simple formulation
where the energetics of the material is solely described by which is the standard condition for localization in plastic-
Eq. (1) and the meso-scale mass exchange rates on acceler- ity theory (Vardoulakis and Sulem, 1995). Accordingly, the
ation waves by Eq. (8). The material velocity c, being the formation of localized shear bands out of homogeneous plas-
velocity of acceleration waves, becomes a material constant tic flow is assumed, when the velocity of the wavefront van-
for the propagation of acceleration waves. ishes. Hill (1962) was discussing shear acceleration waves
https://doi.org/10.5194/se-12-869-2021 Solid Earth, 12, 869–883, 2021876 K. Regenauer-Lieb et al.: Cross-diffusion waves
in an ideal linear, time-independent elasto-plastic material functions of space. The compatibility condition relates jumps
where two families of characteristics (dextral and sinistral in rates of change of internal variables to jumps in gradients
slip lines) feature a jump in strain rate at the wavefront ac- for all internal variables µ (Duszek-Perzyna and Perzyna,
companied by one in stress rate (but not in stress). This in 1996) and implies that the jump in the gradient of pressure
turn implies a related jump in stress gradient. Later work ex- inside the acceleration wave is constrained by
tended the theory to formulate acceleration waves as the basis
1
of modern criteria for localization in plastic media (Rudnicki [∇p] = − [ṗ]. (12)
and Rice, 1975; Rice, 1976). In those theories the possibility c
of volumetric acceleration waves was, however, neglected, Acceleration waves consider a step function (Eq. 5) where
and volumetric deformation was parameterized by an em- the stress rate is discontinuous along the surface. The stress
pirical dilatancy angle. Another shortcoming of the local- is, however, continuous across the wavefront and the stress
ization criterion for the application to THMC instabilities is self-diffusion coefficient is also constant outside of the wave.
that it is not directly applicable to the rate-dependent elasto- Therefore, for modelling acceleration waves in a homoge-
viscoplastic case. The inclusion of rate effects implies a pos- neous material we can simplify Eq. (1) further and assume
itive wave speed different from zero (Duszek-Perzyna and constant bulk and shear viscosity outside the wave and as-
Perzyna, 1996). The thermomechanics approach (Jacquey sume continuity of stress across the acceleration wave. Note
and Regenauer-Lieb, 2021) allows incorporation of a quasi- that the traction in the direction of the normal vector n
static wave speed related to the internal variable that quanti- on the acceleration wavefront is F = n · σ . It follows from
fies the rate dependence. Eq. (7) that the jump in stress rate on the acceleration wave
However, to date, no generally accepted localization cri- is (Duszek-Perzyna and Perzyna, 1996)
teria for the transition from a dynamic to quasi-static rate-
dependent solution of Eq. (4) exists, although stationary n · [σ̇ ] = −c[ρ v̇]. (13)
and wave-like propagating localization phenomena for rate-
Substituting the stress rate for the acceleration v̇ from
sensitive materials (Barraclough et al., 2017) are observed in
Eq. (13) and the pressure rate for the gradient of pressure
the laboratory and nature. The method of choice to date is to
from Eq. (12) and inserting the jump condition into Eq. (4) it
use all field equations and perform a numerical stability anal-
follows that
ysis. A discussion on an extension to the above-discussed cri-
terion has been presented recently (Pisanò and Prisco, 2016), 1 1
[n · [σ̇ ]] = − [ṗ] − [C∇(˙ 0 )] − [ζ ∇ ˙0 ]. (14)
and energy-based criteria that successfully model the adia- c c
batic limit have been revisited many times over the past 30
If we define the magnitude of the wave speed in the normal
years (Paesold et al., 2016). The present approach provides
reference system as w = w · n, then c = w − v · n is the local
an alternative path to systematically analyse the full system
particle velocity of THMC accelerations in the acceleration
of field equations.
wave relative to the normal material velocity.
Equation (14) allows us to draw some important conclu-
3.3 THMC acceleration waves
sions for acceleration waves. (1) The first term on the right
shows that the pressure rate divided by the wave velocity or
We assume creeping flow in Eq. (4), and there is, therefore,
the equivalent gradient of pressure plays an important role
no effect of inertia (F = 0) but there still can be effects of
in acceleration waves. (2) The second term on the right im-
“gravitational acceleration”, i.e. creeping flow in a gravity
plies that gradients of deviatoric strain rates are related to rate
field. We will show that the meso-scale formalism identifies
changes of the stiffness tensor as implied by the jump con-
an alternate internal force density from within the consid-
dition of the internal variable inside the propagating wave.
ered material volume stemming from a local thermodynamic
Recall that the jump condition (Eqs. 5 or 12) advects jumps
THMC force (e.g. ∇p). This internal force integrates over
in gradients of the internal variable around the propagating
the accelerations aM of the micro-processes inside the contin-
wavefront through a jump in the rate of change of the vari-
uum element by multiplying them with the average volume
able. (3) The last term implies that the same is true for the
density. These accelerations stem from dissipative mecha-
volumetric strain rates and the rate of change of bulk viscos-
nisms (e.g. volume changes by phase transitions, fracture) in-
ity.
side the representative volume element (Hu et al., 2020). For
critical conditions, they can cause acceleration waves propa-
gating as creeping waves. Hadamard’s jump conditions need 4 Multiscale cross-diffusion model
to be extended for internal THMC variables µ such as tem-
perature, porosity, permeability, viscosity, etc. So far we have only discussed the mechanical reaction–
Hadamard’s jump conditions state that if time derivatives diffusion equation, where the shear and bulk viscosities con-
(Ḟ, v̇, µ̇) and gradients (∇F, ∇v, ∇µ) have jump disconti- trol the diffusion of stress. For the multi-physics implemen-
nuities across the wavefront then F, v and µ are continuous tation, it is convenient to think of diffusion of momentum
Solid Earth, 12, 869–883, 2021 https://doi.org/10.5194/se-12-869-2021K. Regenauer-Lieb et al.: Cross-diffusion waves 877
and use the momentum diffusivity (kinematic viscosity) in- lower scale. We propose that this information is contained
stead of the dynamic viscosity. We, therefore, denominate ζM in so-called “cross-diffusion coefficients” that are hidden at
as the volumetric diffusion coefficient of pressure (kinematic macro-scale in the reaction term Ri . This innovation regular-
viscosity). In the following, we first formulate the reaction– izes the ill-posed problem for all couplings and was first in-
diffusion equation in a classical way. That is to say that troduced for HM coupling in Hu et al. (2020). A very impor-
meso-scale cross-diffusion effects are neglected. We iden- tant aspect is that the inclusion of the cross-diffusion terms
tify THMC Turing patterns as multiscale energy eigenstates into the reaction–diffusion equation of Table 1 leads to a new
of the reaction–diffusion equations, thus characterizing Pri- form of soliton-like waves as clearly shown by Tsyganov
gogine’s dissipative structures if they emerge. et al. (2007) for a mathematically similar system of equa-
In these formulations, the viscous (M) mechanical pres- tions.
sure diffusion equation finds its counterparts in the equivalent The discussion of these waves and their unusual proper-
thermal (T) Fourier, (H) Darcy, and (C) Fick diffusion laws ties will be the subject of the remainder of the paper. For
where the diffusion coefficients are indicated by the associ- introduction and completeness, however, the arguments for
ated THMC subscript. The corresponding reaction rates are non-local reaction–diffusion equations with cross-diffusion
the local hidden-variable reaction rate RT , RH , RM , and RC , terms (Hu et al., 2020) are repeated in the next section and
respectively. It is common practice to ignore the meso-scale generalized from poromechanics HM problems to all THMC
cross-diffusion kinetics introduced in the classical theories of processes.
localization. We emphasize therefore the difference between For the following discussion, we also simplify further and
large-scale reaction rates Ri and meso-scale reaction rates ri neglect the deviatoric terms in Eq. (14) and retain only the
of the multiscale theory which considers the important effect scalar volumetric terms and reduce the equations to 1-D.
of cross-diffusion. The two rates are identical in the infinite This allows us to investigate the poorly known volumetric
timescale limit as cross-diffusion can be eliminated adiabati- dissipative waves which must exist in addition to the dissi-
cally (Biktashev and Tsyganov, 2016). pative shear waves discussed by Hill (1962). To introduce
In the adiabatic limit we obtain similar reaction–diffusion the meso-scale considerations we identify wave-scale reac-
equations across a vast range of THMC diffusion length tive source terms rT , rH , rM , rC of the hidden variable rates
scales as tabulated in Table 1. The reaction rates most often RT , RH , RM , RC as the actual terms that trigger acceleration
stem from different micro-processes at lower scale inside the waves. These meso-scale source terms stem from a jump in
considered continuum element which introduces cross-scale the thermodynamic force (gradient of the variable) into a
diffusion fluxes as shown in the next section. jump in the thermodynamic flux (rate of the variable, i.e. tem-
In order to generalize the approach, we propose that all perature, fluid, and solid pressure and concentration). The
reaction–diffusion equations in Table 1 are strongly coupled. important volumetric coupling is overlooked in the classi-
We construct a composite multiscale THMC diffusion wave cal localization theory (Rudnicki and Rice, 1975). The wave-
operator ĤTHMC from the four reaction–diffusion equations scale source term provides the convected pressure rate built
in Table 1. up by internal accelerations. It relates to the local mass ex-
change processes according to Eq. (8).
N
X
ĤTHMC = −ζi ∇ 2, (15)
i=1 5 Cross-diffusion as a multiscale theory for localization
∂2 ∂2 ∂2
where ∇ 2 = ∂x 2 + ∂y 2 + ∂z2 and i refers to the individual In the following we generalize the discussion of the meso-
thermodynamic THMC process. scale THMC mass exchange processes using mixture theory
The wave operator ĤTHMC defines the asymptotic dy- applied to HM coupling as presented in Hu et al. (2020). We
namic state of the coupled system of THMC reaction– show that the physics of cross-diffusion follows from a re-
diffusion equations by mapping excitations from reactions active source term at the macro-scale that requests a cross-
into a new waveform. It therefore selects the wave that we diffusion term at the meso-scale for thermodynamic consis-
can expect from the discrete interactions at lower scale (e.g. tency. The full derivation is found in Hu et al. (2020). Here
atomic or molecular scale reactions in the chemical exam- we summarize the main conclusion from the mixture theory
ple). If we thus excite waves with the reaction term Ri , the analysis for convenience.
approach can lead to unbounded instabilities because the sta- We consider two mass fractions A and B for mass ex-
tistical information from the lower-scale interactions, such as change denoted by the ith and j th phase as an example.
a diffusional length scale that limits unbounded reactions on We identify ξ̇iREV as the large-scale representative elemen-
the lower scale, is missing. A particular strategy to include tary volume (REV) for averaging of mass transfer rate from
such information is to use non-local reaction–diffusion equa- the phases A to B where VREV , VA , VB denote the REV vol-
tions (Rubinstein and Sternberg, 1992). We therefore argue ume and the volume of the ith and j th phase, respectively.
that the wave operator must include information from the ξ̇iREV defines the REV-scale averaging of the mass exchange
https://doi.org/10.5194/se-12-869-2021 Solid Earth, 12, 869–883, 2021878 K. Regenauer-Lieb et al.: Cross-diffusion waves
rate between the phases where the REV-scale source term of In a saturated porous medium, a straightforward interpre-
mass is obtained from the other species: tation of ρA and ρB may be the density of the fluid phase and
Z that of the solid phase, respectively (Hu et al., 2020). The
1
ξ̇iREV = ξ̇jlocal dVREV , (16a) interpretation of the incorporation of the effects of chemi-
VREV cal and thermal processes may not be as straightforward for
VREV
Z the observer as they act via Eq. (15) as a linear convolution
REV 1
ξ̇j = ξ̇ilocal dVREV , (16b) operation. If we interpret the time-domain convolution op-
VREV eration of THMC waves in the frequency domain, then the
VREV
chemical and thermal waves can be seen as filters for HM
where ξ̇ilocal and ξ̇jlocal denotes the mass exchange rate from coupling, sharpening or smoothing the waves. The interpre-
the A to B phase and vice-versa. In the meso-scale formalism tation of THMC waves in terms of a sharpening or smoothing
we need to consider information from the local-scale pro- filter analogue is discussed in detail in the companion article
cesses in the THMC diffusion matrix and decompose the pro- (Regenauer-Lieb et al., 2021).
cesses leading to the local mass production ξ̇ilocal and ξ̇jlocal . To illustrate the point of choosing a particular time–space
In order to specify this further we define the global volume scale of observation of THMC waves we first consider the
fraction of the A phase as simple homo-entropic flow assumption and choose a clas-
sical mechanics point of view. Density is then defined as
VA VB
φ= = 1− . (17) a function of pressure, temperature, and chemical concen-
VREV VREV trations by the equation of state. The coupling in Eq. (3)
Mass conservation at global scale for the phases A and B leads to the possible nucleation of hydro-mechanical cross-
gives diffusion pressure waves (Hu et al., 2020). Considering that
possible thermal processes, ρA , and ρB at the solid–fluid in-
∂[ρA VA ] ∂[ρA VA vA ]
+ = ξ̇A VREV , (18a) terface may be affected by the local temperature changes,
∂t ∂x we identify new time-dependent processes at the solid–fluid
∂[ρB VB ] ∂[ρB VB vB ] boundary. The process of heat transport sets two new internal
+ = ξ̇B VREV . (18b)
∂t ∂x timescales changing the fluid and solid pressure, respectively.
ρA and ρB identify the density of the respective phases, while Therefore, the thermal process acts as a “convolution filter”
vA and vB are their velocities in the direction of x and ξ̇A added to the pressure evolution of each phase. Conversely,
and ξ̇B represent the volume averaged mass generation in the the pressure diffusion process in the solid and fluid phase
REV. Following an approach presented in Hu et al. (2020) triggers two additional timescales in the change of tempera-
using the generalized THMC mass exchange processes we ture. Now a 3 × 3 thermo-hydro-mechanical cross-diffusion
arrive at formulation can be obtained following the same steps of up-
scaling from local to a global scale, and the additional four
∂[ρA φ] ∂[ρA φvA ]
+ timescales correspond to the newly introduced four cross-
∂t | ∂x
{z } diffusion coefficients. One can arrive at a similar conclusion
Self-diffusion by considering the interplay between the evolution of pres-
∂[ρB (1 − φ local )vB ]
Z
1 sure distribution and internal chemical reactions.
+ dVREV
VREV ∂x
VREV 5.1 Formulation of the THMC cross-diffusion matrix
| {z }
Cross-diffusion
∂[ρB (1 − φ local )] The concept of cross-diffusion is well known in chemistry.
Z
1
=− dVREV , (19a) In a chemical system with just two species A and B, for in-
VREV ∂t
VREV stance, cross-diffusion is the phenomenon in which a flux
∂[ρB (1 − φ)] ∂[ρB (1 − φ)vB ] of species A is induced by a gradient of species B (Vanag
+ dVREV and Epstein, 2009). In more general THMC terms, cross-
∂t ∂x {z
| } diffusion is the phenomenon where a gradient of one gen-
Self-diffusion
Z local
eralized thermodynamic force drives another generalized
1 ∂[ρA φ vA ] thermodynamic flux. Staying with the chemical example of
+ dVREV
VREV ∂x species A and B, we have in 1-D
VREV
| {z }
Cross-diffusion ∂CA 2 2
= ζA ∂∂xC2A + LAB ∂∂xC2B + rB ,
∂[ρA φ local ] ∂t
Z
1
=− dVREV . (19b)
VREV ∂t 2 2 (20)
VREV
∂CB
∂t = ζB ∂∂xC2B + LBA ∂∂xC2A + rA ,
Solid Earth, 12, 869–883, 2021 https://doi.org/10.5194/se-12-869-2021K. Regenauer-Lieb et al.: Cross-diffusion waves 879
Table 1. Generalized thermodynamic fluxes and forces in a THMC coupled system.
Type Force Flux Reaction–diffusion equations
T FT = ∇T qT = − DT
Dt
DT = ζ ∇ 2 T + R
Dt T T
H FH = ∇pH qH = − Dp
Dt
H
- Dp H 2 0
Dt = ζH ∇ pH + η˙ − RH
DpM DpM 2
M FM = ∇pM qM = − Dt - Dt = ζM ∇ pM + ∇(C˙ 0 ) − RM
C FC = ∇C qC = − DC
Dt
DC = ζ ∇ 2 C + R
Dt C C
where rA and rB are the local source terms using the meso- 5.1.1 Criterion for nucleation of cross-diffusion waves
scopic self-diffusion and cross-diffusion decomposition in
Eq. (3). A detailed discussion of the criterion for nucleation of
Following Eq. (15) we can now generalize (20) to include cross-diffusion waves and their waveforms can be found in
the full cascade of internal accelerations through multiscale Tsyganov and Biktashev (Tsyganov and Biktashev, 2014).
coupling. Cross-diffusion allows coupling of accelerations Here, we first summarize the basic method that is well es-
from one classical REV-scale reaction–diffusion system, de- tablished in the fields of mathematical biology and chem-
istry and follow on with a discussion of other communi-
p
fined by the (self-)diffusive length scale ζ(T,H,M,C) t, to an-
other. The time t is initially defined in this paper from a ties, where the phenomenon of cross-diffusion waves is well-
macroscopic perspective. It is the time for which the macro- documented under different names.
scale applied boundary conditions drives a given series of The criterion for nucleation of cross-diffusion waves re-
THMC processes. The individual THMC processes may re- lies on assessing the dispersion relation of the eigenvalues
act by a feedback loop between the respective physics of of the characteristic matrix of a perturbed cross-diffusion–
reaction and self-diffusion, often leading to oscillatory be- reaction equation (Vanag and Epstein, 2009). The eigenval-
haviour through tight coupling of the system dynamics, ues are functions of the square of the wavenumber of the
thus adding new internal material timescales. The reaction– perturbed state and identify the growth rate of the perturba-
diffusion problem is initially fully dynamic, but often af- tions. This approach for deriving the mathematical criterion
ter sufficient time has elapsed, it can reach a quasi-static for nucleation of acceleration waves is hence evaluated from
macroscopic response resulting from an oscillatory, or a a small plane-wave perturbation of Eq. (21) with
steady-state equilibrium, between the reactive source term
C̃(x, t) = C0 (1 + )eλt+i(kx) . (22)
and its associated self-diffusion process. In this case the in-
ternal material timescale is the time it takes to reach this The characteristic matrix of the thus perturbed Eq. (21) al-
macro-scale equilibrium (Regenauer-Lieb et al., 2013b, a). lows assessment of the stability of the system. Accordingly,
The cross-diffusion coefficients, introduced in this paper, en- all eigenvalues of the characteristic matrix must be real and
rich the tightly coupled cross-scale self-diffusion-controlled positive, and hence the determinant of the matrix must be
reaction–diffusion processes by linking the gradient of a ther- larger than zero. For determinants smaller than zero, cross-
modynamic force Cj of one THMC process to the flux of diffusion waves are expected to propagate as quasi-solitons
another kind and thus significantly increases the potential (Tsyganov and Biktashev, 2014). A working example for
for feedback through additional coupling across scales. The hydromechanical cross-diffusion waves can be found in Hu
wave operator ĤTHMC is now expanded through a fully pop- et al. (2020).
ulated diffusion matrix that includes self-diffusion (diagonal)
and cross-diffusion (off-diagonal) coefficients as in
6 Soliton versus quasi-soliton solutions
ζT LTH LTM LTC
DC LHT ζH LHM LHC ∇ 2 C + ri . Since cross-diffusion waves in geomaterials are largely un-
= (21)
Dt LMT LMH ζM LMC explored due to the extreme length scales and timescales en-
LCT LCH LCM ζC countered in a geosystem, an appreciation of their complex
characteristics can be obtained from mathematically similar
The cross-diffusion processes formulate the link between systems such as waves in oceans, lasers, and ice. There is
different THMC processes. The cross-diffusion coefficients an important difference between solitonic waves and quasi-
thereby introduce new cross-scale coupling length scales solitonic cross-diffusion waves. We follow Zakharov et al.’s
and timescales which are often much smaller than the self- (Zakharov and Kuznetsov, 1998) definition of solitons and
diffusion scales. This is not always the case (Manning, quasi-solitons and identify solutions to the perturbed Eq. (22)
1970). Hu et al. (2020) show normal examples where cross- of the type
diffusion length scales are much smaller than the self-
diffusion length scales. ψ(x, t) = β(x − vt)eit , (23)
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