Dissipative Point Vortex Model

• The dissipative point vortex model is a finite-dimensional system that augments classical Hamiltonian dynamics with irreversible energy decay via friction, splitting cascades, and nonlocal memory effects.
• It employs varied mechanisms—such as rotated-velocity friction, vortex–sound coupling, and non-Markovian drag—to simulate complex vortex interactions in quantum fluids and superconducting arrays.
• Analytical reductions and numerical experiments reveal practical insights into energy decay, vortex splitting, and coarsening processes that govern the evolution of quantum turbulent systems.

A dissipative point vortex model is a finite-dimensional vortex dynamics in which the Hamiltonian point-vortex equations are supplemented by irreversible mechanisms while the vortices remain represented as singular or effectively point-like defects. Across the literature, the dissipative ingredient takes several distinct forms: reverse three-vortex collapse used as a splitting rule that dissipates interaction energy, mutual-friction terms of rotated-velocity type for quantized vortices, phenomenological vortex-sound coupling and dipole annihilation, non-Markovian memory kernels in superconducting arrays, and mixed symplectic–gradient flows on periodic domains (Leoncini et al., 2010, Billam et al., 2014, Durkin et al., 2015, R. et al., 26 Apr 2026). This suggests that the expression denotes a modeling class rather than a unique canonical equation: the shared structure is a Hamiltonian point-vortex skeleton together with dissipation, coarsening, annihilation, or effective reduced irreversibility.

1. Hamiltonian skeleton and dissipative deformation

The starting point is the classical point-vortex Hamiltonian system. In continuum superfluids, one convenient form is

$$H \;=\; -\,\frac{\rho_s \hbar}{2\pi}\sum_{i<j}s_i\,s_j\,\ln\!\bigl(\tfrac{|{\bf r}_i-{\bf r}_j|}{\xi}\bigr),$$

with canonical motion

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$

Equivalent periodic-box forms, complex-coordinate forms, and Green-function formulations are used depending on geometry (Tattersall et al., 14 Mar 2025, R. et al., 26 Apr 2026).

The minimal dissipative deformation in finite-temperature superfluids is the rotated-velocity prescription. Billam, Reeves and Bradley write

$$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$

where $\gamma$ is the dimensionless dissipation rate inherited from the dPGPE (Billam et al., 2014). Closely related forms appear in the vortex-fluid description of chiral clusters,

$$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$

and in Hall–Vinen form,

$$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$

where $\alpha$ and $\alpha'$ are the longitudinal and transverse mutual-friction coefficients (Stockdale et al., 2019, Grani et al., 29 Aug 2025).

On the flat torus, dissipation can be written as a mixed symplectic–gradient flow. For $N$ vortices of circulations $\kappa_j=\pm1$,

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$0

and the Hamiltonian obeys

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$1

Monotonic energy decay is therefore explicit in the finite-dimensional dynamics (R. et al., 26 Apr 2026).

Not all dissipative point-vortex models use a local rotated-velocity term. In superconducting arrays, Durkin et al. introduce a generalized Langevin equation with a history-dependent dissipation kernel,

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$2

so the vortex drag depends on past motion rather than only instantaneous velocity (Durkin et al., 2015). A plausible implication is that “dissipation” in this context includes both Markovian mutual friction and non-Markovian effective drag.

2. Reverse collapse, splitting cascades, and irreversible offspring dynamics

A distinct dissipative point-vortex construction is developed in “Offsprings of a Point Vortex” (Leoncini et al., 2010). The model starts from one parent vortex of circulation $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$3 and replaces it by three offspring through time reversal of a three-point-vortex collapse. The splitting stipulations are total-circulation conservation,

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$4

together with the scale-free condition

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$5

Equivalently,

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$6

The three strengths can be parametrized by a single angle $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$7, and the offspring are placed on a small expanding triangle centered on the parent-vortex location.

The dissipative character appears at the Hamiltonian level. If one vortex of strength $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$8 is split into three offspring at mutual distances $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$9, then

$$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$0

From the collapse conditions, $$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$1, hence $$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$2. Each splitting therefore irreversibly dissipates interaction energy.

Iterating the rule produces a vortex population

$$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$3

after $$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$4 splittings. Descendant circulations are multiplicative in the splitting angles,

$$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$5

with $$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$6 chosen i.i.d. uniform in $$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$7. The generation occupancy satisfies the mean-field recursion

$$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$8

leading to a self-similar depth distribution peaked at

$$\frac{d\mathbf r_i}{dt} \;=\; \mathbf v_i \;+\;\mathbf w_i,\qquad \mathbf w_i \;=\; -\,\gamma\,\kappa_i\,\hat{\mathbf e}_z\times\mathbf v_i,$$9

The resulting vorticity PDF is a weighted sum over generations, and numerically the model displays “a broad, self-similar spectrum of $\gamma$0, with its peak slowly drifting to smaller circulations as $\gamma$1 increases.” In the continuum limit, the summary states that $\gamma$2 decays logarithmically,

$\gamma$3

with $\gamma$4 in the numerical experiments (Leoncini et al., 2010).

This construction differs from finite-temperature mutual-friction models because dissipation is attached to topological multiplication rather than frictional drift. It nevertheless fits the dissipative point-vortex category because the irreversible step acts directly on point-vortex degrees of freedom and decreases the interaction energy at every event.

3. Quantum-turbulent formulations: mutual friction, sound coupling, and coarsening

In two-dimensional compressible superfluids, well-separated vortices on an essentially uniform background density can be reduced to an effective point-vortex model. Billam, Reeves and Bradley derive a dissipative periodic-box model with phenomenological vortex-sound interactions and dipole annihilation (Billam et al., 2014). The Hamiltonian part is the classical periodic point-vortex velocity, while the dissipative Peach–Koehler part is first order in $\gamma$5. To mimic increased acoustic damping at small inter-vortex separations, the damping is locally enhanced according to

$\gamma$6

with $\gamma$7 and $\gamma$8. Vortex–antivortex pairs with separation below one healing length $\gamma$9 are removed from the simulation.

The model is used to define an incompressible kinetic-energy spectrum and an unambiguous spectral flux associated with vortex motion. The transfer function is split as

$$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$0

where $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$1 is scale-to-scale transport due to Hamiltonian vortex interactions and $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$2 is the direct dissipative loss. Three dynamical regimes are reported. For strong dissipation, $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$3, there is a “dissipative collapse” with $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$4 comparable to $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$5 and negligible net transport of energy in scale space. For weak dissipation, $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$6, and a small box, the long-time dynamics again becomes a dissipative collapse toward few vortex dipoles. For weak dissipation and a large box, $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$7, a regime of hydrodynamic vortex turbulence emerges: the spectrum shows an approximate $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$8 range for $$\dot{\mathbf r}_i = \mathbf v_i \;-\;\kappa_i\,\gamma\;\bigl(\hat{\mathbf z}\times\mathbf v_i\bigr),$$9, the Hamiltonian flux $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$0 is negative and roughly constant over an inertial-range of scales, and dissipative losses are negligible in that window (Billam et al., 2014).

Tattersall et al. compare a dissipative point vortex model against Fokker–Planck evolution for annihilation-driven coarsening (Tattersall et al., 14 Mar 2025). Their periodic system contains $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$1 vortices and $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$2 antivortices in a square of side $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$3, with dipole annihilation when separation falls below $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$4. The main conclusion is sharply regime-dependent: the dissipative point vortex model is well approximated by the Fokker–Planck dipole model only when the configuration is overwhelmingly dominated by tightly bound dipoles, namely for initial energies per vortex approximately less than $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$5. The best agreement occurs near $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$6, whereas above $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$7 many free or clusterable vortices appear and the Fokker–Planck assumption breaks down. The dynamical critical exponent $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$8 also crosses over: for $$\dot{\mathbf r}_i=(1-\alpha')\,\mathbf v_{s,i}-\alpha\,\sigma_i\,\hat{\mathbf z}\times\mathbf v_{s,i},$$9, $\alpha$0 independent of initial conditions, while for $\alpha$1,

$\alpha$2

with $\alpha$3 extracted from the initial dipole-length power-law exponent (Tattersall et al., 14 Mar 2025).

These results make clear that dissipation in quantum-vortex point models does not have a single phenomenology. Depending on box size, damping, and initial energy, the same formal structure can yield inverse transport, dipole-dominated collapse, or non-universal coarsening.

4. Finite-temperature clusters and vortex binaries

For like-signed chiral clusters, the dissipative point-vortex equations admit a coarse-grained vortex-fluid description with a universal long-time attractor. In free space,

$\alpha$4

and dissipation is again introduced through

$\alpha$5

When the bulk density gradients become negligible, the vortex density obeys

$\alpha$6

hence

$\alpha$7

Conservation of vortex number then yields an expanding Rankine vortex with

$\alpha$8

and

$\alpha$9

Simulations with $\alpha'$0 and $\alpha'$1, together with quasi-2D BEC experiments with $\alpha'$2, support this universal expansion dynamics and the emergence of frustrated lattice correlations (Stockdale et al., 2019).

For minimal few-vortex systems, dissipative point-vortex models often become analytically solvable. In a disk with one antivortex pinned at the center and one mobile vortex, Hall–Vinen dynamics with $\alpha'$3 gives closed expressions for the radius and angle in terms of the Lambert-$\alpha'$4 function. This framework is used to infer $\alpha'$5 and $\alpha'$6 from trajectory data, and the analysis shows that fluctuations in the initial positions can bias the inferred coefficients. For $\alpha'$7 and $\alpha'$8, the reported radial bias at $\alpha'$9 is

$N$0

while for $N$1, $N$2, and the same noise level,

$N$3

The paper’s conclusion is that sub-healing-length control of the initial configuration is required for precision inference of mutual-friction parameters (Grani et al., 29 Aug 2025).

Two-vortex dissipative reductions also provide exact collapse and chirp laws. On the flat torus, the two-vortex dynamics closes on the relative coordinate $N$4. In the local regime $N$5, equal same-sign vortices spiral outward with

$N$6

whereas equal opposite-sign dipoles obey

$N$7

so the planar limit exhibits finite-time collapse. For unequal opposite-sign pairs, contraction and rotation combine to produce a finite-time nonlinear chirp with

$N$8

On the torus, the dipole angle is no longer invariant because geometry induces a slow angular drift (R. et al., 26 Apr 2026).

A GPE-extracted two-vortex model gives a related overdamped separation law. With a Rayleigh dissipation function $N$9 and interaction force $\kappa_j=\pm1$0, the relative separation satisfies

$\kappa_j=\pm1$1

In the large-$\kappa_j=\pm1$2 regime, the leading solution is the familiar annihilation scaling

$\kappa_j=\pm1$3

described in the summary as a universal signature of vortex–antivortex annihilation (Shanquan et al., 2020).

5. Memory, inertia, confinement, and instability suppression

The dissipative point-vortex framework extends beyond mutual friction in homogeneous superfluids. In superconducting superconductor-normal-superconductor arrays, Durkin et al. model dilute vortex transport by a generalized non-Markovian Langevin equation with pinning, Lorentz driving, and a memory kernel (Durkin et al., 2015). In the overdamped dilute limit,

$\kappa_j=\pm1$4

with

$\kappa_j=\pm1$5

For steady flow at constant velocity, the effective drag becomes $\kappa_j=\pm1$6, leading to

$\kappa_j=\pm1$7

The model explains two anomalies emphasized in the summary: a broad linear $\kappa_j=\pm1$8–$\kappa_j=\pm1$9 region with an extrapolated intercept at the de-pinning current, and the absence of a differential-resistance peak near de-pinning. The added dissipation is therefore history dependent rather than purely local.

A second extension incorporates finite vortex-core mass, mutual friction, and boundaries. In the generalized model for the superfluid Kelvin–Helmholtz instability, each vortex carries an inertial mass $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$00 and experiences a friction force

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$01

For a straight row of equally spaced vortices, the massless limit gives the classic growth rate

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$02

whereas large $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$03 changes the scaling from $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$04 to $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$05. In the massless dissipative theory,

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$06

or, in $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$07 notation,

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$08

Confinement adds a further stabilization mechanism: in an annulus, the maximum growth rate vanishes below the critical thickness

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$09

The model is compared against cold-atom measurements of vortex-necklace instabilities; using $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$10 and fitting $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$11, the reported agreement is obtained for $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$12 (Caldara et al., 2024).

These variants show that the dissipative point-vortex idea is compatible with nonlocal drag, inertial second-order dynamics, and geometry-dependent stabilization. A plausible implication is that the minimal rotated-velocity law is only one member of a broader effective-theory family.

6. Rigorous justification, reduced closures, and limiting regimes

Several works address when dissipative point-vortex descriptions are mathematically justified or how dissipative reduced models emerge from conservative many-vortex dynamics. For modified surface quasi-geostrophic equations with fractional dissipation,

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$13

the point-vortex ansatz formally leads to

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$14

with $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$15 arising from moment-of-inertia decay of small patches. The main theorem applies for $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$16, $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$17, identical-sign circulations $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$18, and dissipation scaling satisfying

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$19

Under these assumptions, smooth mSQG solutions converge weak-* to the point-vortex measure uniformly on bounded time intervals (Rosenzweig, 2019). The same-sign hypothesis is essential because mixed signs can produce finite-time collapse.

A different route to dissipation is optimal closure for Hamiltonian vortex ensembles. In “Reduced models of point vortex systems,” the exact Liouville dynamics is projected onto trial densities parametrized by resolved spatial moments. The cost functional is the time integral of the squared Liouville residual, and near equilibrium this produces a quadratic lack-of-fit Lagrangian,

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$20

A matrix Riccati equation determines the value function, and the closed reduced dynamics becomes

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$21

Here $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$22 is the reversible part and $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$23 is symmetric positive-semidefinite; the summary explicitly interprets $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$24 as a “dissipation matrix” controlling the decay of relative entropy (Maack et al., 2018).

Regularized Euler–Poincaré point-vortex dynamics provides a further, more singular, route to effective dissipation. For three vortices satisfying the collapse condition

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$25

the regularized dynamics develops self-similar collapse with exponent $$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$26. The associated regularized enstrophy variation obeys

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$27

and in the sense of distributions

$$\dot{\bf r}_i \;=\; s_i\,\hat{\bf z}\times\nabla_{\bf r_i}H.$$28

Even though the point-vortex ODE is Hamiltonian and time reversible, the inviscid-regularization limit therefore exhibits a finite enstrophy drop at collapse (Gotoda et al., 2017).

The regime of validity of dissipative point-vortex models remains sharply conditional. The summaries repeatedly require well-separated cores, uniform background density, or vanishing-viscosity scaling; they also note breakdown when density fluctuations become large, when vortex cores overlap, or when strong vortex–sound coupling destroys the point-vortex character at early times (Billam et al., 2014, Rosenzweig, 2019, Grani et al., 29 Aug 2025). A common misconception is to treat all such models as interchangeable. The literature instead supports a more specific view: each dissipative point-vortex model is tied to a particular asymptotic regime, dissipation mechanism, and observable sector.