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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. 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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3. 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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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. A High‐Order Accurate Summation‐By‐Parts Finite Difference Method for Fully‐Dynamic Earthquake Sequence Simulations Within Sedimentary Basins

   https://doi.org/10.1029/2022JB025357  

   Harvey, Tobias W.; Erickson, Brittany A.; Kozdon, Jeremy E. (April 2023, Journal of Geophysical Research: Solid Earth)

   Abstract We present an efficient numerical method for earthquake sequences in 2D antiplane shear that incorporates wave propagation. A vertical strike‐slip fault governed by rate‐and‐state friction is embedded in a heterogeneous elastic half‐space discretized using a high‐order accurate Summation‐by‐Parts finite difference method. Adaptive time‐stepping is applied during the interseismic periods; during coseismic rupture we apply a non‐stiff method, enabling a variety of explicit time stepping methods. We consider a shallow sedimentary basin and explore sensitivity to spatial resolution and the switching criteria used to transition between solvers. For sufficient grid resolution and switching thresholds, simulations results remain robust over long time scales. We explore the effects of full dynamics and basin depth and stiffness, making comparisons with quasi‐dynamic counterparts. Fully‐dynamic ruptures generate higher stresses, faster slip rates and rupture speeds, producing seismic scattering in the bulk. Because single‐event dynamic simulations penetrate further into sediments compared to the quasi‐dynamic simulations, we hypothesize that the incorporation of inertial effects would produce sequences of only surface‐rupturing events. However, we find that subbasin ruptures can still emerge with elastodynamics, for sufficiently compliant basins. We also find that full dynamics can increase the frequency of surface‐rupturing events, depending on basin depth and stiffness. These results suggest that an earthquake's potential to penetrate into shallow sediments should be viewed through the lens of the earthquake sequence, as it depends on basin properties and wave‐mediated effects, but also on self‐consistent initial conditions obtained from seismogenic cycling. 

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