Dispersive Kinetic Energy (DKE) Overview

• DKE is the kinetic energy arising from spatial or spectral inhomogeneity and mesoscale organization in physical systems.
• It is quantified using triple decomposition methods in atmospheric flows, band dispersion in correlated electron systems, and dispersive processes in plasma turbulence.
• DKE provides a unified diagnostic to link mesoscale heterogeneity with energetic conversion, offering practical insights for improved model closure schemes.

Dispersive Kinetic Energy (DKE) refers to kinetic energy contributions arising from spatial or spectral inhomogeneity—typically associated with mesoscale organization, dispersive wave effects, or nontrivial band dispersion—across a wide range of physical systems. The term encapsulates phenomena where kinetic energy arises or is modified via spatial structure, wave–particle or turbulence–heterogeneity interactions, or multi-orbital band effects. DKE has been rigorously formalized and measured in contexts spanning Earth’s atmospheric boundary layer, strongly correlated electrons in ferromagnetic metals, and collisionless plasma turbulence.

1. Theoretical Formulations of Dispersive Kinetic Energy

DKE is systematically defined through the energy decomposition of a field variable into components associated with mean, dispersive, and turbulent or incoherent motions. In mesoscale atmospheric flows, Waterman et al. (Waterman et al., 7 Nov 2025) use a triple decomposition of the velocity field $u_i$:

$$\tfrac12\langle u_i u_i\rangle = \underbrace{\tfrac12\,\langle\overline{u_i}\rangle\,\langle\overline{u_i}\rangle}_{\mathrm{MKE}} + \underbrace{\tfrac12\,\langle \overline{u_i}''\,\overline{u_i}''\rangle}_{\mathrm{DKE}} + \underbrace{\tfrac12\,\overline{\langle u_i' u_i'\rangle}}_{\mathrm{TKE}}$$

• MKE (Mean Kinetic Energy): Kinetic energy from the spatially and temporally averaged flow.
• DKE: Kinetic energy from spatial variance of the time-averaged flow (i.e., stationary mesoscale perturbations or organized circulations).
• TKE (Turbulent Kinetic Energy): Kinetic energy from local, rapidly fluctuating turbulence.

In the context of ferromagnetic bcc Fe, Borghi et al. (Borghi et al., 2013) define kinetic energy via the expectation of the hopping operator in the Gutzwiller wavefunction, with DKE manifesting as the net kinetic energy gain associated with dispersion of itinerant $t_{2g}$ holes when local moments are ferromagnetically aligned.

In nonlinear plasma dynamics, DKE is the portion of the original wave/packet energy (magnetic + bulk–kinetic) irreversibly transferred by dispersive and kinetic processes into particle internal energy, as shown in Alfvén wave packet dynamics (Tenerani et al., 2023). Here, DKE is operationally tied to specific mechanisms—plasma compression, phase space mixing, and resonance heating—mediated by Hall and kinetic effects.

2. DKE in Atmospheric Boundary Layer Heterogeneity

Waterman et al. (Waterman et al., 7 Nov 2025) provide a systematic framework for quantifying DKE in the atmospheric boundary layer using Doppler LiDAR profilers:

• The spatial variance of mean wind across a LiDAR network is evaluated at each vertical level.
• DKE profile at height $k$: $$DKE(k) = \frac{1}{2}[\sigma_{\overline{u},k}^2 + \sigma_{\overline{v},k}^2 + \sigma_{\overline{w},k}^2]$$
• Vertically integrated DKE: $$\mathrm{DKE}_v = \frac{1}{2\,\overline{\rho}\,z_{\mathrm{top}}} \sum_{k=1}^{N_z} \rho_k \left[ \sigma_{\overline{u},k}^2 + \sigma_{\overline{v},k}^2 + \sigma_{\overline{w},k}^2 \right] \Delta z_k$$

A dimensionless DKE fraction, $DKE_f = {DKE_v}/{MKE_v}$, quantifies the energetic dominance of mesoscale (heterogeneity-driven) flows over the mean background wind. DKE and $DKE_f$ are empirically correlated with surface heterogeneity metrics (standard deviation and correlation lengthscale of land surface temperature), with $\rho \approx 0.5$ between the heterogeneity metric and DKE after synoptic and convection contamination filtering. Large Eddy Simulation (LES) studies reproduce the observed DKE statistics, validating both the metric and network sensitivity (as few as 3–5 distributed profilers suffice).

The practical significance of DKE in this context lies in diagnosing and parameterizing mesoscale flows (such as thermally-driven breezes and boundary layer convection) that are poorly captured in traditional turbulence closure schemes. The clear surface–DKE linkage suggests a robust route for including land–atmosphere heterogeneity effects in next-generation weather and climate models.

3. DKE in Strongly Correlated Electron Systems

Within ab-initio electronic structure theory, dispersive kinetic energy underpins the mechanism of double exchange in strongly correlated ferromagnets. Borghi et al. (Borghi et al., 2013) show that in bcc Fe, Gutzwiller+LDA calculations yield:

$$\begin{aligned} \hat T &= \sum_{i,j}\sum_{\alpha,\beta}\sum_{\sigma} t_{ij}^{\alpha\beta} c^\dagger_{i\alpha\sigma} c_{j\beta\sigma} \ E_{\rm kin} &\equiv \langle\Psi_G|\,\hat T\,|\Psi_G\rangle \ E_{\rm kin}^{e_g} &= \sum_{k\sigma\in e_g} \epsilon_k^{e_g} n_k^{e_g}\ E_{\rm kin}^{t_{2g}} &= \sum_{k\sigma\in t_{2g}} \epsilon_k^{t_{2g}} n_k^{t_{2g}} \end{aligned}$$

• $e_g$ states are nearly flat and highly localized.
• $t_{2g}$ states are broad-band and highly dispersive.
• The energetic hallmark of DKE is that ferromagnetic ordering reduces $E_{\rm kin}$ (LDA+G: $-2.9$ eV/atom) rather than increasing it as in LDA/GGA.

The physical origin is that $t_{2g}$ hole motion is coherent only when the $e_g$ moments are aligned, giving rise to a kinetic energy lowering upon FM alignment—consistent with double-exchange physics.

This mechanism is distinct from conventional Stoner-Wohlfarth (mean-field) arguments and illustrates how DKE—here quantified via orbital-selective band dispersion—mediates phase stability in correlated metals.

4. DKE in Plasma Turbulence and Wave–Particle Interactions

The development and dissipation of DKE in plasma systems is intimately tied to the transition from fluid (MHD) to dispersive and kinetic regimes at small scales. In solar wind turbulence (Verscharen et al. (Verscharen et al., 2012)), the system transitions from MHD (Kolmogorov inertial range) to a dispersive regime as $k\ell_p \gtrsim 1$, where $\ell_p = c/\omega_p$ is the proton inertial length.

While the explicit kinetic energy spectrum $$E_{\mathrm{kin}}(\mathbf{k}) = \frac{1}{2} m_p \langle | \tilde{\mathbf{v}}_p(\mathbf{k}) |^2 \rangle$$ is not calculated, the magnetic and density power spectral densities (PSDs) and the observed energy break at $k\ell_p \sim 1$ are proxies for the onset of dispersive effects and the associated kinetic energy redistribution.

In the context of kinked Alfvén wave packets, Brunetti et al. (Tenerani et al., 2023) explicitly connect DKE to energy irreversibly transferred to internal/thermal particle energy via Hall-dispersive and kinetic effects. The characteristic time for DKE conversion is

$$\tau^* = \tau_a \ell/d_i,\quad \tau_a = \ell_x/v_A,\, d_i = v_A/\Omega_{ci}$$

The mechanisms include:

• Plasma compression by Hall-dispersive mode coupling.
• Resonant phase-space mixing leading to anisotropic heating (e.g., $\langle p_{xx}\rangle$ increase).
• The rate and efficiency of DKE transfer scale with $d_i/\ell_x$.

Quantitative results (§5 of (Tenerani et al., 2023)) indicate up to $40\%$ loss of wave energy and corresponding increases in internal energy on this timescale for strong dispersion.

A plausible implication is that DKE provides a pathway for dissolution of magnetic switchbacks and heating of solar wind ions, with rates compatible with in situ observations.

5. DKE as a Diagnostic and Model Parameterization Tool

DKE offers a scalar, physically transparent diagnostic for mesoscale organization, energetic conversion, and the efficiency of coupling between spatial inhomogeneity and internal energy across research fields:

• Atmospheric Science: DKE (or grid-analog Mesoscale Kinetic Energy, MsKE) quantifies subgrid variability. Its correlation with land surface heterogeneity provides a basis for improved turbulence closure schemes—capturing mesoscale organization missing from bulk TKE approaches (Waterman et al., 7 Nov 2025).
• Plasma Physics: DKE marks the channel by which dispersive and kinetic processes in turbulence, or wave-packet dissipation, heat particles and shape ion distributions (Verscharen et al., 2012, Tenerani et al., 2023).
• Condensed Matter: DKE identifies instances where collective electronic/magnetic order is not driven by potential energy minimization, but by kinetic energy gain associated with band dispersion under constraints (e.g., double exchange) (Borghi et al., 2013).

In each domain, DKE is inherently tied to spatial (or spectral) organization and associated energetic conversion—offering an essential metric for theoretical, numerical, and observational diagnostics.

6. Comparative Table of DKE Definitions and Contexts

Physical System	DKE Definition/Metric	Core Energetic Role
Boundary Layer Flow	Spatial variance of mean wind (triple decomposition)	Quantifies mesoscale circulation energy due to heterogeneity (Waterman et al., 7 Nov 2025)
Correlated Electron Solid	Kinetic energy gain due to band dispersion (Gutzwiller)	Drives ferromagnetism via double-exchange (band/dispersion gain) (Borghi et al., 2013)
Plasma Turbulence/Waves	Wave energy irreversibly transferred to internal energy	Hall and kinetic effects dissipate wave energy to plasma heating (Tenerani et al., 2023)

7. Scaling Laws, Quantitative Results, and Limitations

• In atmospheric measurements, the DKE–surface heterogeneity correlation is maximized when synoptic/convective effects are filtered ($\rho \simeq 0.5$). Fractional DKE exceeding 0.1 denotes days of strong mesoscale circulation relative to mean wind.
• In plasma simulations, the rate of DKE conversion scales as $\tau^* = \tau_a \ell/d_i$, with up to 40% energy conversion to internal modes when $\ell_x/d_i \le 10$.
• For bcc Fe, DKE lowering upon FM ordering is on the scale of $-2.9$ eV/atom versus the PM state in LDA+G calculations.

A plausible implication is that DKE-based diagnostics can bridge across model hierarchies and disciplinary boundaries, providing a unified metric for mesoscale or spectrally organized kinetic energy, and suggesting pathways for improved model closures and energetic estimates. However, the physical interpretation of DKE and its full separation from turbulent or mean-flow contributions requires careful application of decomposition and model assumptions, and is system-dependent.