Diffusion Wake: Concepts & Applications

• Diffusion wake is a multi-domain phenomenon defined by the spreading, relaxation, or depletion left behind moving entities, such as fronts, jets, or bluff bodies.
• In hydrodynamics, it governs wake merging, drafting, and lateral interactions, as seen in falling-disk experiments and bluff-body flow studies.
• In turbulent and high-energy contexts, diffusion wakes model stochastic state diffusion, jet-induced hadron depletions, and pattern-forming instabilities.

“Diffusion wake” is a polysemous research term whose meaning depends on domain, but the recurrent structure is a wake whose evolution is governed by diffusive spreading, diffusive relaxation, or a depletion mode left behind a moving body, front, or jet. In moderate-Re bluff-body hydrodynamics it denotes a region of velocity deficit and vorticity that spreads and decays downstream; in turbulent wake modeling it can denote diffusion of a large-scale wake state in low-dimensional phase space; in heavy-ion collisions it denotes the soft-hadron depletion region behind a jet in quark–gluon plasma; and in reaction–diffusion or driven-media settings it denotes the unstable or depleted region left in the wake of a propagating front or obstacle (Brosse et al., 2010, Rigas et al., 2015, Yang et al., 2022).

1. Terminological scope

The term is used in several distinct literatures, and the differences are substantive rather than stylistic. In some contexts diffusion is literal viscous or scalar diffusion in physical space; in others it is turbulent transport across space, stochastic diffusion in a reduced state space, or diffusive hydrodynamic relaxation of a jet-induced hole. This suggests a family resemblance rather than a single canonical definition.

Domain	Meaning of “diffusion wake”	Representative arXiv ids
Falling-disk hydrodynamics	Velocity-deficit and vorticity wake spreading downstream by viscous diffusion and entrainment	(Brosse et al., 2010)
Turbulent axisymmetric wake	Large-scale wake state undergoing diffusion in a 2D stochastic state space	(Rigas et al., 2015)
Turbulent plane wake	Inter-space turbulent diffusion that counteracts the cascade across scales	(Noriega et al., 8 Jan 2026)
Driven lattice gas	Density perturbation field and wake-mediated interaction around impurities	(Kliushnichenko et al., 2020)
Reaction–diffusion fronts	Unstable region left behind a front, where precipitation patterns form	(Thomas et al., 2013)
QGP jet wake	Depletion of soft hadrons behind a jet, distinct from Mach-cone-like enhancement	(Yang et al., 2022, Yang et al., 6 Jan 2025, Yang et al., 6 Aug 2025)
SAR generative modeling	Ship-wake images synthesized by latent diffusion models from physics-based data	(Kamirul et al., 28 Apr 2025)

A second terminological distinction is internal to each field. In the QGP literature, for example, “Mach-cone-like wake” and “diffusion wake” are explicitly said to be not the same as the “sound mode” and “diffusion mode” in linear hydrodynamics; in turbulent axisymmetric wakes, “diffusive wake” refers not to scalar diffusion in physical space but to diffusive wandering of the wake’s large-scale state (Yang et al., 28 Sep 2025, Rigas et al., 2015).

2. Bluff-body hydrodynamics at moderate Reynolds number

For two identical, axisymmetric disks falling in tandem in a quiescent fluid at $Re \approx 100$, the wake of a single disk is in the steady, axisymmetric regime, with a near-wake recirculation bubble, a velocity deficit behind the disk, and downstream widening produced by entrainment and viscous diffusion of vorticity. In that usage, a diffusion wake is the persistent region behind the body where vorticity spreads outward and a tracer broadens according to advection–diffusion, consistent with

$$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$

The trailing disk enters the low-momentum wake of the leading disk, experiences reduced drag, accelerates, and catches up; the wake therefore acts not as a passive trail but as the mediator of drafting and entrainment (Brosse et al., 2010).

The geometry of the disks controls the post-catch-up regime. For thick disks with $d/t=3$, the disks lose their initial wakes, separate laterally, and evolve to a repulsive side-by-side configuration with a typical horizontal separation of about $\sim 6$ disk diameters. For thin disks with $d/t=10$, the two wakes merge into a single wake that remains attached to both bodies, and the disks adopt a stable Y-configuration. In more than $90\%$ of cases this Y-configuration is observed and remains stable; the pair falls at about $1.15$ times the velocity of an isolated disk and drifts horizontally by $4\%$ to $10\%$ of the vertical displacement. At somewhat higher $Re$, the Y-configuration persists but develops regular transverse oscillation and periodic vortex shedding; at even higher $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$0, where isolated disks would follow a periodic zigzag path, the Y-configuration remains but with fluctuating relative distance and inclination (Brosse et al., 2010).

Within this classical hydrodynamic meaning, the wake’s diffusion sets the spatial extent of the velocity-deficit corridor, the entrainment that pulls a trailing body inward, and the lateral pressure gradients that determine whether a coupled configuration becomes attractive or repulsive. The distinction between side-by-side separation and stable shared-wake pairing is therefore a geometry-dependent consequence of how two diffusion wakes interfere or merge.

3. Turbulent wakes: state-space diffusion, inter-space transfer, and curled-wake diffusion

In turbulent axisymmetric wakes, “diffusive wake” acquires a low-dimensional stochastic meaning. The large-scale wake state is represented by the instantaneous centre of pressure $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$1 on the base of an axisymmetric bluff body and is modeled by a nonlinear two-dimensional Langevin equation,

$$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$2

with additive Gaussian white noise. In polar variables, the deterministic part yields a ring of symmetry-broken states at $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$3, while the stochastic forcing produces free diffusion in angle and confined diffusion in radius. The stationary density has the form

$$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$4

a Mexican-hat potential whose flat angular direction explains statistical restoration of rotational symmetry. Using base-pressure data at $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$5, the fitted parameters are $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$6, $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$7, and $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$8; the model reproduces the PDFs, MSDs, PSDs, and conditional reorientation statistics of the wake state (Rigas et al., 2015).

A different turbulent-wake meaning appears in plane wakes, where two-point turbulent diffusion remains dynamically important across the decay region. There, the inter-space transfer rate $$\frac{\partial \omega}{\partial t} + (\mathbf{u}\cdot\nabla)\omega = \nu \nabla^2 \omega, \qquad \frac{\partial c}{\partial t} + (\mathbf{u}\cdot\nabla)c = D \nabla^2 c.$$9 is positive while the horizontal part of the inter-scale transfer $d/t=3$0 is negative, so turbulent diffusion counteracts the forward cascade down to scales smaller than the Taylor length. In the inertial range, except at the near-field edge of the decay region, the sum satisfies an approximate proportionality

$d/t=3$1

showing that spatial transport and scale transfer form a joint equilibrium rather than the homogeneous-turbulence balance $d/t=3$2. This suggests a “diffusion wake” in which non-homogeneity does not vanish at small scales but persists through inter-space transfer at scales down to $d/t=3$3 (Noriega et al., 8 Jan 2026).

In yawed-wind-turbine wakes, diffusion enters through the decay of the streamwise-vorticity sheet and the widening of the momentum deficit. A vortex-sheet model describes the curled wake produced by yaw-induced and rotation-induced vorticity,

$d/t=3$4

and yields a universal curled-wake shape in suitable dimensionless variables. For turbulent boundary-layer inflow, the decay of vortex-sheet circulation due to turbulent diffusion is included, and the resulting shape and deflection are embedded in a modified Gaussian wake model to calculate wake profiles behind yawed turbines (Bastankhah et al., 2021).

4. Driven media and pattern-forming wakes

In a driven lattice gas with static impurities, the wake is the steady-state perturbation $d/t=3$5 of the mobile-particle density generated by scattering of a gas stream from impurities. The macroscopic kinetics are governed by

$d/t=3$6

and, after linearization, by an anisotropic screened Green’s function

$d/t=3$7

This wake mediates non-equilibrium correlations and non-reciprocal effective interactions between impurities. Disorder strongly amplifies the effect: for random clusters the scattered-field measure scales as $d/t=3$8, giant local fluctuations of $d/t=3$9 appear inside the cluster, and the force distribution acquires heavy tails. A central conclusion is that effective-medium descriptions in terms of a penetration index or effective diffusion coefficient break down because local fluctuations dominate transport and force transmission (Kliushnichenko et al., 2020).

A reaction–diffusion-front wake is different in mechanism but similar in function. In the precipitation system $\sim 6$0, the front obeys diffusion-limited reaction–diffusion dynamics,

$\sim 6$1

and advances as

$\sim 6$2

Behind the front, the product concentration $\sim 6$3 is approximately uniform but unstable or metastable, so pattern selection occurs in the wake via a Cahn–Hilliard-type dynamics with source and conserved noise,

$\sim 6$4

Liesegang bands, helices, and helicoids are selected in this unstable wake. The probability $\sim 6$5 of a helical structure is controlled by the competition between the front time scale and the unstable-mode time scale; the paper derives the condition

$\sim 6$6

Experimentally, $\sim 6$7 for $\sim 6$8, with $\sim 6$9, then rises above threshold, reaches a maximum around $d/t=10$0 for $d/t=10$1, and decreases again as more complex multi-helix structures proliferate (Thomas et al., 2013).

These two literatures share a precise structural feature: the wake is the nonequilibrium region in which transport or front propagation prepares a state that then mediates forces, correlations, or pattern selection.

5. Jet-induced diffusion wake in quark–gluon plasma

In high-energy heavy-ion collisions, a diffusion wake is the depletion region of soft hadrons and depleted fluid density that forms behind a supersonic jet traversing the QGP. The microscopic picture is based on jet–medium scattering: recoil partons carry positive energy–momentum into the medium, while the removed thermal partons are represented as negative partons, or holes. In coupled transport–hydrodynamic descriptions such as CoLBT-hydro, the medium evolves according to

$d/t=10$2

where $d/t=10$3 contains positive contributions from recoil partons and negative contributions from holes. The Mach-cone-like wake is the enhancement channel, whereas the diffusion wake is the depletion channel. The literature explicitly notes that these terms refer to the general medium response along the jet direction and the depletion of medium behind the jet, respectively, and are not the same as the sound mode and diffusion mode of linear hydrodynamics (Yang et al., 2022, Yang et al., 28 Sep 2025).

The experimentally relevant manifestation is a valley in jet–hadron correlations. In $d/t=10$4-jet and $d/t=10$5-jet events, two-dimensional correlations in $d/t=10$6 show a valley in the trigger direction on top of an MPI ridge; after projection onto rapidity this produces a double-peak structure with a dip in between. In dijets without a rapidity gap, the diffusion wake of one jet overlaps with the medium-induced hadron enhancement of the other, so the signal is largely reduced to a reduction of enhancement. A finite rapidity gap between the two jets shifts the wake valley away from the other jet’s enhancement region and exposes it as a rapidity asymmetry in jet–hadron correlations. A background-free generalization fixes the trigger rapidity and varies the associated-jet rapidity, so the background cancels identically in the difference of hadron rapidity distributions. In CoLBT-hydro simulations this asymmetry is largest for soft hadrons, weakens as hadron $d/t=10$7 increases, and is negligibly changed by subtraction of a $d/t=10$8 baseline (Yang et al., 6 Jan 2025, Yang et al., 6 Aug 2025).

The 3D structure matters. In coordinate space, hydrodynamic calculations show a positive-energy-density front and a negative-energy-density region trailing the jet. In momentum space, the rapidity position of the depletion tracks the jet rapidity, and the wake also contributes to azimuthally asymmetric jet shapes when coupled to transverse and longitudinal medium gradients. The depth and shape of the wake are sensitive to the jet energy loss as characterized by $d/t=10$9-jet asymmetry, and they also respond to the QGP equation of state and shear viscosity. In the CoLBT-hydro studies, the diffusion wake valley deepens with stronger jet energy loss, is modified by changing $90\%$0, and becomes shallower for an equation of state with a first-order phase transition than for the $90\%$1 crossover equation of state (Yang et al., 2021, Yang et al., 2022).

6. Observables, measurements, and current empirical status

The measurement problem is dominated by background. Soft hadrons from the underlying event, flow, and MPI can obscure a depletion that is intrinsically small. This has produced a heterogeneous experimental record across channels and analysis strategies.

A 2024 ATLAS search in photon–jet events in 0–10% central Pb+Pb collisions at $90\%$2 TeV measured jet–track correlations for $90\%$3 in the hemisphere opposite the jet, binned in the jet-to-photon ratio $90\%$4. No diffusion wake signal was observed within the current sensitivity, and upper limits at 95% confidence level were reported. For the CoLBT-hydro benchmark width $90\%$5, values more negative than $90\%$6 were excluded at 95% confidence level, while the CoLBT-hydro prediction $90\%$7 lay within the 68% confidence interval (Collaboration, 2024).

A 2026 CMS dijet analysis in PbPb and $90\%$8 collisions at $90\%$9 TeV used dijet–hadron correlations and their dependence on dijet pseudorapidity separation to isolate the wake. By comparing large-gap and small-gap dijets, and then PbPb relative to $1.15$0, the analysis established a negative modification of low-$1.15$1 charged-particle yields at negative $1.15$2 in central PbPb collisions. The wake had a significance greater than 5 standard deviations for charged particles in the range $1.15$3, thereby firmly establishing the presence of a jet diffusion wake in that observable (Collaboration, 23 Feb 2026).

Perspective and synthesis papers interpret recent $1.15$4-jet, $1.15$5-jet, and dijet measurements together as evidence that the diffusion wake is now experimentally accessible through observables that exploit its rapidity structure: the valley on top of the MPI ridge, the double peak in rapidity, and rapidity asymmetry with a finite dijet gap. A neutral reading is that the field has moved from model-based expectation to channel-dependent empirical access: some observables remain statistics-limited or sub-percent level, while others, especially gap-sensitive dijet asymmetries, have crossed into observation (Yang et al., 28 Sep 2025, Yang et al., 6 Aug 2025).

7. Related extension: diffusion models for SAR ship wakes

A distinct use links diffusion not to wake physics but to wake synthesis. In SAR ship-wake generation, the physical wake is a Kelvin wake, while the diffusion is the probabilistic denoising process of a latent diffusion model. The training data are generated by the AssenSAR physics-based simulator, which superposes a frozen wind-driven sea surface with a vessel-generated Kelvin wave pattern and maps that surface to SAR intensity through a two-scale model. A Stable Diffusion v1.4 latent diffusion model is then trained on 3,450 simulator-generated images paired with text prompts encoding ship speed, heading, wind speed, wind direction, fetch, and incidence angle (Kamirul et al., 28 Apr 2025).

The model uses the standard DDPM objective in latent space,

$1.15$6

with text conditioning through the CLIP encoder of Stable Diffusion. Here the wake remains the physical Kelvin wake, but the generative mechanism is diffusion in model space rather than hydrodynamic diffusion. Quantitatively, the generated images achieve SSIM $1.15$7, PSNR $1.15$8 dB, and LPIPS $1.15$9, while inference becomes increasingly favorable at higher resolution, with a reported speed gain of $4\%$0 at $4\%$1 (Kamirul et al., 28 Apr 2025).

This usage is not equivalent to the hydrodynamic or heavy-ion meanings. It is instead a terminological extension in which “diffusion wake” denotes a wake image synthesized by a diffusion model whose conditioning variables encode physical wake parameters. A plausible implication is that the term now spans both physical transport phenomena and learned surrogate representations of wakes.

8. Conceptual synthesis

Across these literatures, a diffusion wake is never merely a trail. In falling-disk hydrodynamics it is a low-momentum corridor that determines drafting, pairing, and lateral stability; in turbulent wake theory it can be a stochastic large-scale state diffusing on a symmetry manifold or an inter-space transfer that remains active down to $4\%$2; in driven media it is the perturbation field that mediates non-equilibrium forces and invalidates coarse effective parameters; in reaction–diffusion systems it is the unstable region where bands and helices are selected; and in the QGP it is the depleted soft-hadron region inseparable from jet-induced medium response (Brosse et al., 2010, Rigas et al., 2015, Kliushnichenko et al., 2020, Thomas et al., 2013, Yang et al., 2022).

The common structure is therefore precise but broad: diffusion wakes are wakes whose dynamical significance lies in how a disturbance spreads, relaxes, depletes, or reorganizes a surrounding medium. The specific diffusion operator, state variable, and observable vary sharply by discipline, but the wake is consistently the region through which transport and response become measurable.