Convergent Thermal Waves: Models & Applications

• Convergent thermal waves are non-Fourier phenomena where thermal energy propagates in a wave-like, self-focusing manner due to hyperbolic and nonlinear transport models.
• They emerge from advanced frameworks such as Dual-Phase-Lag, Cattaneo–Vernotte, and Boltzmann transport equations, exhibiting oscillatory patterns and finite propagation speeds validated by experiments and simulations.
• These waves enable innovative applications in thermal metamaterials, nanoscale heat management, and atmospheric science, offering new avenues for efficient energy transport.

A convergent thermal wave is a phenomenon in which thermal energy, instead of diffusing irreversibly and isotropically as in classical Fourier transport, propagates in a wave-like, often spatially focusing or self-organizing manner. This concept arises across disciplines: in the quantum description of thermal states, in non-Fourier continuum models, within periodic and composite media, in atmospheric and astrophysical contexts, and in advanced materials. Convergent thermal waves typically feature finite propagation velocities, oscillatory or coherent spatial-temporal patterns, and regimes where energy density or flow self-focuses due to nonlinear, nonlocal, or geometrical effects.

1. Mathematical Foundations: Models and Physical Scenarios

Convergent thermal waves are consistently found where the governing models deviate from the parabolic Fourier law and admit hyperbolic or wave-type behavior. Prominent mathematical structures include:

• Dual-Phase-Lag (DPL) and Cattaneo-Vernotte (CV) Models:

The DPL model introduces two delay times in the constitutive law for heat flux:

$$q(\mathbf{r}, t + \tau_q) = -\kappa_t \nabla T(\mathbf{r}, t + \tau_T)$$

with $\tau_q$ the lag of the flux and $\tau_T$ the lag of the temperature gradient. Substitution yields dispersion relations:

$$\tilde{k}^2 (1 + iZ\tilde{\omega}) = \tilde{\omega}^2 (1 - i/\tilde{\omega}), \quad Z = \tau_T/\tau_q$$

supporting oscillatory, underdamped (i.e., wave-like and potentially convergent) solutions when the modal quality factor $Q = |\omega_1|/\omega_2 > 1$ (Gandolfi et al., 2019).

• Phonon Boltzmann Transport Equation (BTE):

The semiclassical kinetic approach tracks the phonon distribution:

$$\frac{\partial f(\mathbf{r}, \mathbf{k}, t)}{\partial t} + \mathbf{v}_g(\mathbf{k}) \cdot \nabla_{\mathbf{r}} f = -\frac{f - f^{(0)}(T)}{\tau(\mathbf{k},T)}$$

allowing the emergence (in hydrodynamic or ballistic regimes) of heat waves that exhibit oscillatory and/or convergent character depending on scattering, grating period, and phonon dispersion (Zhang et al., 2021).

• Thermal Wave Model (TWM) in Plasmas:

The TWM uses a complex beam wave function $\Psi$ governed by a Schrödinger-like equation (with time-like parameter $\xi = z - ct$) self-consistently coupled to a wake potential,

$$i\epsilon\frac{\partial\Psi}{\partial\xi} = \mathcal{H} \Psi$$

predicting beam self-focusing ("squeezing"), collapse, vortex formation, and solitonic states in charged particle-plasma systems (Fedele et al., 2011).

• Convergent Beam Envelope Dynamics:

In both linear and nonlinear regimes, envelope equations of the form

$$\frac{d^2\sigma}{d\xi^2} + K\sigma + \frac{\eta}{\sigma} - \frac{\epsilon^2}{\sigma^3} = 0$$

admit focusing (convergent) behavior under suitable nonlinearity (e.g., effective squeezing) (Fedele et al., 2011).

• Refraction and Focusing in Heterogeneous Media (Tangent Law):

When heat flux crosses material boundaries, the path of least thermal resistance yields

$$\frac{\tan\theta_1}{k_1} = \frac{\tan\theta_2}{k_2}$$

analogous to Snell's Law, so heat "rays" converge in the higher conductivity medium (Ciotola et al., 2014).

These modeling approaches define the necessary conditions and quantitative measures for convergent wave behavior, including oscillatory regimes, amplitude focusing, and wavefront phase relationships.

2. Regimes and Mechanisms of Convergence

Convergent thermal wave phenomena typically arise in the following scenarios:

• Non-Fourier and Hyperbolic Transport:

Delay-based models (DPL, CV) admit wave propagation when certain dimensionless groups (ratios of relaxation times and excitation frequency) exceed thresholds. Explicitly, under conditions

$\tau_T \ll 1/|\omega| \ll \tau_q$

underdamped thermal oscillations are supported, and a modal Q-factor analysis reveals parameter bands with convergent wave behavior (Gandolfi et al., 2019).

• Confined and Periodically Structured Media:

In thermal wave crystals—periodic multilayers described by CV or DPL—the interplay of periodicity and Bragg scattering leads to frequency stop-bands and pass-bands. Bloch theory and the transfer matrix method yield dispersion relations; within certain frequency ranges, thermal waves are strongly attenuated, while in others they propagate, sometimes preferentially focusing energy in the structure (Chen et al., 2017, Li et al., 2019).

• Quantum and Statistical Thermal States:

Quantum statistical treatments decompose the thermal density operator into a convex sum of localized wave packets. Upon time evolution, these packets may spread, overlap, and reconverge at specific spacetime regions, modeling convergent propagation in a quantum-coherent setting (Chenu et al., 2016).

• Nonlinear and Collective Self-Focusing:

In regimes where nonlinear self-interaction is strong (e.g., intense beams in plasma), nonlocal nonlinearities in NLS-type equations yield soliton formation, self-pinching equilibria, and collapse-averse convergence in the collective thermal/particle wave profile (Fedele et al., 2011).

• Atmospheric and Planetary Convection:

In geophysical and planetary systems, convergent thermal waves manifest as planetary-scale tides or as boundary-forced waves giving rise to organized, momentum-carrying flows (e.g., zonal jets on Venus; see Section 5) (Giles et al., 2022).

3. Experimental and Computational Evidence

Robust verification and characterization of convergent thermal waves come from a diverse array of experiments and numerical simulations:

• Thermal Modulation in Turbulent Convection:

In Rayleigh-Bénard cells, harmonic modulation of boundary temperatures launches thermal waves that propagate into the turbulent interior. The profile of amplitude $$A_S(z, t) = A_T \exp(-z/\delta_S) \sin(2\pi f_m t - z/\delta_S)$$, with the Stokes layer thickness $\delta_S = (\pi f_m/\kappa_{\text{eff}})^{-1/2}$, matches theory for moderate frequencies and amplitudes. Notably, in the bulk, increased local Nusselt number results in "superconducting" wave propagation with minimal attenuation, and increased modulation amplitude dramatically enhances heat transport (Nu) (Urban et al., 2021).

• Classroom Demonstration of Thermal Lensing:

In a two-conductor setup, visualization of isotherms confirms that heat flux "bends" at contacts, consistent with the Tangent Law. Simulations and experiment match predicted angle relationships, illustrating thermal convergence in accessible contexts (Ciotola et al., 2014).

• Transient Thermal Grating (TTG) and Boltzmann Simulations:

TTG experiments can generate and monitor thermal waves in materials such as graphene and silicon. Boltzmann transport calculations show that, depending on temperature and grating period, oscillatory (hydrodynamic or ballistic) heat waves emerge and converge transiently, but are suppressed by increased scattering at room temperature (Zhang et al., 2021).

• Protoplanetary Disk Modeling:

1+1D radiative transfer models demonstrate that irradiation-driven surface perturbations ("bumps") locally heat the disk and initiate surface thermal waves that propagate radially and converge toward the star. In massive, optically thick disks, these waves remain confined to the upper layers, with the bulk (midplane) shielded from convergence, emphasizing the role of radiative coupling and geometry (Pavlyuchenkov et al., 2022).

• Venus Atmospheric Observations:

3D temperature maps derived from CO$_2$ infrared absorption reveal planetary-scale thermal waves with zonal wavenumbers 2--4, amplitude increasing from ~3 K (65 km) to ~6 K (75 km), and an eastward wavefront tilt of 8–15° per km. These signatures illustrate upward and convergent propagation of thermal tides (Giles et al., 2022).

4. Structures and Material-Specific Realizations

The manifestation of convergent thermal waves is highly dependent on material structure and composition:

• Thermal Wave Crystals:

One-dimensional superlattices—repeating A/B layer stacks with differing thermal conductivities, densities, or relaxation times—create band gaps and pass-bands in the thermal wave spectrum. Bragg scattering conditions,

$$f_{\text{center}}(m) = \frac{m}{2(l_A/C_{\text{cv},A} + l_B/C_{\text{cv},B})}$$

set frequencies for strong/weak transmission. The mismatch in thermal impedance enhances convergence (focusing) or blocking of thermal energy, directly enabling thermal diodes, imaging components, and waveguides (Chen et al., 2017, Li et al., 2019).

• Quantum Materials (Nanotubes):

Carbon nanotubes present a unique case where, due to the interplay of quantum statistics and anharmonic scattering channels from flexural/optical phonon modes, thermal conductivity converges to a finite value only at macroscopic lengths (millimeter scale). At mesoscopic lengths, apparent divergence masks the eventual convergence associated with nonlocal and multidimensional scattering (Barbalinardo et al., 2021).

• Thermoelastic Media with Microtemperatures:

Strongly elliptic, isotropic solids described by Green–Naghdi entropy balance equations support finite-speed, non-dispersive thermal waves even in the presence of microstructural (microtemperature) degrees of freedom. In these settings, convergent propagation arises in joint thermal-elastic Rayleigh modes (Passarella et al., 2021).

• Planetary and Astrophysical Contexts:

In planetary atmospheres (Venus), convergent thermal waves (thermal tides) are large-scale, coherent, vertically-propagating temperature structures, linked to momentum deposition and angular momentum transport in superrotating atmospheres (Giles et al., 2022).

5. Nonlinear, Quantum, and Collective Effects

Nonlinearities, quantum statistics, and collective interactions play a central role in the emergence and structure of convergent thermal waves:

• Nonlocal and Nonlinear Self-Interactions:

Models with strong nonlinear terms (e.g., nonlinear Schrödinger equations for beam-plasma systems) admit soliton solutions, collapse (arrested by dispersion or nonlocality), and convergent focusing equilibria (Fedele et al., 2011).

• Quantum Wave Packet Decomposition:

The statistical mixture of spatially localized, momentum-distributed wave packets in quantum equilibrium (thermal density operators) points to the quantum analog of convergent thermal waves: upon evolution, packets spread but reconverge regionally due to phase and amplitude interference, with particle statistics (Pauli exclusion or bosonic bunching) further shaping the convergence properties (Chenu et al., 2016).

• Collective Beam Effects and Wakefields:

Beam-plasma coupling through self-consistent wakefields leads to convergent (focusing) or divergent (halo formation) beam structures, with stability determined by conserved virial integrals and envelope equations. Thin plasma lenses exploit rapid phase modulation, preconditioning incoming beams for downstream convergence (Fedele et al., 2011).

6. Applications and Relevance

Convergent thermal waves underpin a growing array of technological and scientific applications:

Application Area	Key Role of Convergent Thermal Waves	Reference(s)
Thermal Wave Crystals & Metamaterials	Band gap engineering, thermal diodes, waveguides, and cloaking	(Chen et al., 2017, Li et al., 2019)
Nanoscale Heat Management	Controlled pulse propagation, quantum-limited conductivity, logic elements	(Zhang et al., 2021, Barbalinardo et al., 2021)
Advanced Accelerators	Plasma lensing, beam pinching, and halo suppression for relativistic beams	(Fedele et al., 2011)
Atmospheric & Planetary Science	Interpreting thermal tides, zonal flows, and energy transport in planetary atmospheres	(Giles et al., 2022, Reiter et al., 2020)
Protoplanetary Disk Evolution	Modeling irradiation-driven instabilities, surface wave transport, disk morphology	(Pavlyuchenkov et al., 2022)
Experimental Heat Transfer	Enhanced Nusselt number in turbulent convection via boundary-driven thermal waves	(Urban et al., 2021)

7. Outlook and Open Challenges

Continued research aims to expand theoretical, computational, and experimental frontiers:

• Complexity and Nonlinearity:

Stability, collapse, and control of solitonic and vortex modes in nonlinear and nonlocal wave models remain open questions, with strong implications for advanced accelerators and optical/thermal devices (Fedele et al., 2011).

• Optimal Thermal Device Design:

The modal Q-factor and bandpass filter paradigm suggest routes for tuning device response by engineering relaxation times, crystal size, and material composition for targeted spectral selectivity (Gandolfi et al., 2019, Li et al., 2019).

• Quantum–Classical Bridging:

Further development of quantum statistical descriptions (wave packet-based density matrix decompositions) and their connection with observable, macroscopic convergent thermal phenomena is anticipated to clarify the limits and transition regimes between quantum and classical wave-like heat transport (Chenu et al., 2016).

• Cross-disciplinary Synthesis:

The unification of variational principles (e.g., Fermat's principle and the principle of least resistance) across thermal, optical, and mass transport highlights the deep, multidisciplinary nature of convergent thermal wave phenomena (Ciotola et al., 2014).

• Material Innovation and Planetary Observations:

As new materials and instruments emerge, the direct observation and manipulation of convergent thermal waves in laboratory, terrestrial, and planetary environments are expected to advance both applications and fundamental understanding (Giles et al., 2022, Urban et al., 2021).

In summary, convergent thermal waves represent a central concept binding together quantum, nonlinear, materials science, and astrophysical paradigms, with a rigorous mathematical and physical foundation established by delay-laden, nonlocal, and wave-based transport equations, and substantiated across theoretical, simulation, and experimental domains.