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1. The energetics of pilot-wave hydrodynamics

   https://doi.org/10.1017/jfm.2025.168  

   Durey, Matthew; Bush, John WM (April 2025, Journal of Fluid Mechanics)

   A millimetric droplet may bounce and self-propel across the surface of a vertically vibrating liquid bath, guided by the slope of its accompanying Faraday wave field. The ‘walker’, consisting of a droplet dressed in a quasi-monochromatic wave form, is a spatially extended object that exhibits many phenomena previously thought exclusive to the quantum realm. While the walker dynamics can be remarkably complex, steady and periodic states arise in which the energy added by the bath vibration necessarily balances that dissipated by viscous effects. The system energetics may then be characterised in terms of the exchange between the bouncing droplet and its guiding or ‘pilot’ wave. We here characterise this energy exchange by means of a theoretical investigation into the dynamics of the pilot-wave system when time-averaged over one bouncing period. Specifically, we derive simple formulae characterising the dependence of the droplet’s gravitational potential energy and wave energy on the droplet speed. Doing so makes clear the partitioning between the gravitational, wave and kinetic energies of walking droplets in a number of steady, periodic and statistically steady dynamical states. We demonstrate that this partitioning depends exclusively on the ratio of the droplet speed to its speed limit. 

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2. Extreme Energy Spectra of Relativistic Electron Flux in the Outer Radiation Belt

   https://doi.org/10.1029/2022JA031038  

   Mourenas, D.; Artemyev, A. V.; Zhang, X. ‐J.; Angelopoulos, V. (November 2022, Journal of Geophysical Research: Space Physics)

   Abstract Electron diffusion by whistler‐mode chorus waves is one of the key processes controlling the dynamics of relativistic electron fluxes in the Earth's radiation belts. It is responsible for the acceleration of sub‐relativistic electrons injected from the plasma sheet to relativistic energies as well as for their precipitation and loss into the atmosphere. Based on analytical estimates of chorus wave‐driven quasi‐linear electron energy and pitch‐angle diffusion rates, we provide analytical steady‐state solutions to the corresponding Fokker‐Planck equation for the relativistic electron distribution and flux. The impact on these steady‐state solutions of additional electromagnetic ion cyclotron waves, and of ultralow frequency waves are examined. Such steady‐state solutions correspond to hard energy spectra at 1–4 MeV, dangerous for satellite electronics, and represent attractors for the system dynamics in the presence of sufficiently strong driving by continuous injections of 10–300 keV electrons. Therefore, these analytical steady‐state solutions provide a simple means for estimating the most extreme electron energy spectra potentially encountered in the outer radiation belt, despite the great variability of injections and plasma conditions. These analytical steady‐state solutions are compared with numerical simulations based on the full Fokker‐Planck equation and with relativistic electron flux spectra measured by satellites during one extreme event and three strong events of high time‐integrated geomagnetic activity, demonstrating a good agreement. 

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3. Dynamics and Scaling of a Small River Discharging into the Surf Zone

   https://doi.org/10.1175/JPO-D-24-0072.1  

   Lou, Yingzhong; Horner-Devine, Alexander R; Derakhti, Morteza; Giddings, Sarah N; Spydell, Matthew S; Rodriguez, Angelica R; Simpson, Alexandra J (August 2025, Journal of Physical Oceanography)

   Abstract We use an idealized numerical model to investigate the dynamics and fate of a small river discharging into the surf zone. Our study reveals that the plume reaches a steady state, at which point the combined advective and diffusive freshwater fluxes from the surf zone to the inner shelf balance the river discharge. At a steady state, the surf zone is well mixed vertically due to wave-enhanced vertical turbulent diffusion and has a strong cross-shore salinity gradient. The horizontal gradient drives a cross-shore buoyancy-driven circulation, directed offshore at the surface and onshore near the bottom, which opposes the wave-driven circulation. Using a scaling analysis based on momentum and freshwater budgets, we determine that the steady-state alongshore plume extent (L_p) and the fraction of river water trapped in the surf zone depend on the ratio of the near-field plume length to the surf-zone width (L_nf/L_sz) across a wide range of discharge and wave conditions and a limited set of tidal conditions. This scaling also allows us to predict the residence time and freshwater fraction (or dilution ratio) in the steady-state plume within the surf zone, which ranges from approximately 0.1 to 10 days and from 0.1 to 0.3, respectively. Our findings establish the basic dynamics and scales of an idealized plume in the surf zone, as well as estimates of residence times and dilution rates that may provide guidance to coastal managers. Significance StatementSmall rivers and estuaries often carry pollutants, sediments, and larvae into the coastal ocean, where wave action in the surf zone can trap them near the shore. This process can play an important role in the flux of material into and out of the nearshore ecosystem and presents a potential risk to swimmers when materials are harmful. The present study uses a numerical model to investigate the fate of freshwater discharged from small rivers into the surf zone and the processes through which trapped riverine freshwater escapes from the surf zone. These results establish a basis for predicting the fate of river-borne materials from coastal rivers and understanding the exchange between the surf zone and the inner shelf. Additionally, this work provides a theoretical framework for predicting the residence time and concentration of river-borne material trapped in the surf zone. 

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4. Speed oscillations in classical pilot-wave dynamics

   https://doi.org/10.1098/rspa.2019.0884  

   Durey, Matthew; Turton, Sam E.; Bush, John W. (July 2020, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   We present the results of a theoretical investigation of a dynamical system consisting of a particle self-propelling through a resonant interaction with its own quasi-monochromatic pilot-wave field. We rationalize two distinct mechanisms, arising in different regions of parameter space, that may lead to a wavelike statistical signature with the pilot-wavelength. First, resonant speed oscillations with the wavelength of the guiding wave may arise when the particle is perturbed from its steady self-propelling state. Second, a random-walk-like motion may set in when the decay rate of the pilot-wave field is sufficiently small. The implications for the emergent statistics in classical pilot-wave systems are discussed. 

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5. Tidally-induced nonlinear resonances in EMRIs with an analogue model

   https://doi.org/10.1088/1361-6382/acfcfe  

   Bronicki, David; Cárdenas-Avendaño, Alejandro; Stein, Leo C (October 2023, Classical and Quantum Gravity)

   Abstract One of the important targets for the future space-based gravitational wave observatory Laser Interferometer Space Antenna is extreme mass ratio inspirals (EMRIs), where long and accurate waveform modeling is necessary for detection and characterization. Modeling EMRI dynamics requires accounting for effects such as the ones induced by an external tidal field, which can break integrability at resonances and cause significant dephasing. In this paper, we use a Newtonian analogue of a Kerr black hole to study the effect of an external tidal field on the dynamics and the gravitational waveform. We have developed a numerical framework that takes advantage of the integrability of the background system to evolve it with a symplectic splitting integrator, and compute approximate gravitational waveforms to estimate the timescale over which the perturbation affects the dynamics. Comparing this timescale with the characteristic time under radiation reaction at resonance, we introduce a tool for quantifying the regime in which tidal effects might be included when modeling EMRI gravitational waves. As an application of this framework, we perform a detailed analysis of the dynamics at one resonance to show how different entry points into the resonance in phase-space can produce substantially different dynamics, and how one can estimate bounds for the parameter space where tidal effects may become dominant. Such bounds will scale as $\epsilon \gtrsim Cq$, whereɛmeasures the strength of the external tidal field,qis the mass ratio, andCis a number which depends on the resonance and the shape of the tide. We demonstrate how to estimateCusing our framework for the 2:3 radial to polar frequency resonance in our model system. This framework can serve as a proxy for proper modeling of the tidal perturbation in the fully relativistic case. 

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