The flow of a superfluid past a solid boundary can generate vortices, with a key criterion being that the local flow speed reaches the speed of sound (Landau). However, the physical mechanism underlying this vortex generation remains unresolved. We consider the canonical problem of uniform flow past an impenetrable planar wall, to examine the vortex nucleation process generated by a solid obstruction moving through a superfluid. A linear stability analysis is conducted of this uniform two-dimensional flow governed by the Gross–Pitaevskii equation (GPE). This shows that the flow is marginally stable to two-dimensional, normal-mode perturbations. Direct numerical solutions of the GPE confirm this finding for small perturbations of finite amplitude. They also reveal the existence of a critical amplitude above which instabilities, in the form of vortices, are nucleated at the solid wall in finite time. Once nucleated, these vortices convect away from the solid wall with infinite initial speed. The physical mechanism underlying this nonlinear phenomenon is discussed, and a proposal is advanced to deepen its understanding.
By considering the stability of potential flow of a superfluid around large obstacles of size R, we derive an analytical result for the critical velocity that is of order ${v}_{c}\ensuremath{\sim}\ensuremath{\Elzxh}/mR,$ scaling inversely with obstacle size, in contrast to what is obtained from a Landau criterion. Our results are compared with numerical solutions of the Gross-Pitaevskii equation and with recent measurements of the critical velocity in Bose-Einstein condensates of dilute atomic gases.
When a superfluid flows past an obstacle, quantized vortices can be created in the wake above a certain critical velocity. In the experiment by Kwon et al. [Phys. Rev. A 91, 053615 (2015)], the critical velocity
vc was measured for atomic Bose-Einstein condensates (BECs) using a moving repulsive Gaussian potential, and
vc was minimized when the potential height
V0 of the obstacle was close to the condensate chemical potential $\ensuremath{\mu}$. Here we numerically investigate the evolution of the critical vortex shedding in a two-dimensional BEC with increasing
V0 and show that the minimum
vc at the critical strength ${V}_{0c}\ensuremath{\approx}\ensuremath{\mu}$ results from the local density reduction and vortex-pinning effect of the repulsive obstacle. The spatial distribution of the superflow around the moving obstacle just below
vc is examined. The particle density at the tip of the obstacle decreases as
V0 increases to
Vc0, and at the critical strength, a vortex dipole is suddenly formed and dragged by the moving obstacle, indicating the onset of vortex pinning. The minimum
vc exhibits power-law scaling with the obstacle size $\ensuremath{\sigma}$ as ${v}_{c}\ensuremath{\sim}{\ensuremath{\sigma}}^{\ensuremath{-}\ensuremath{\gamma}}$ with $\ensuremath{\gamma}\ensuremath{\approx}1/2$.
We study theoretically the critical velocity
Uc for quantum vortex generation by a thin plate-shaped obstacle moving through a uniform two-dimensional Bose-Einstein condensate. Our results based on the Gross-Pitaevskii theory reveal that the critical velocity monotonically decreases with increasing the plate size
L. In the limit of large
L, the critical velocity is asymptotic to ${L}^{\ensuremath{-}1/2}$ predicted by the potential flow theory for an incompressible ideal fluid with a phenomenological length correction. As
L decreases, however, the incompressible analysis breaks down quantitatively. By performing a perturbative analysis to incorporate compressibility into the potential flow theory, we successfully reproduced the numerical results analytically over a wide parameter range.
We report systematic validation of the Davis Law of Critical Velocity governing vortex nucleation in two-dimensional Gross-Pitaevskii dynamics:
C=Kτ⇒β=csvc=f(Keff)(species-independent, geometry-indexed) Through GPU-accelerated Gross-Pitaevskii simulations across three atomic species (⁸⁷Rb, ²³Na, ⁷Li) spanning a 12:1 mass ratio, we demonstrate that the dimensionless critical velocity β = v_c/c_s is species-independent—a universal geometric invariant within each curvature regime, determined solely by obstacle geometry. Key Results Universal Critical Velocity: Hard obstacles (tanh-wall): β_hard = 0.425 ± 0.000 across all three species Semi-hard obstacles (Gaussian, V₀ = 50μ): β_soft = 0.486 ± 0.000 across all species Excellent agreement with Frisch-Pomeau-Rica result (β ≈ 0.4) Perfect Species Independence: The coefficient of variation is 0.0% for both obstacle types across a 12:1 mass ratio, definitively proving that β depends only on obstacle geometry, not atomic properties. Davis Law Support: The relation C = τ/K from the Davis Field Equations correctly predicts that obstacle curvature controls the critical velocity. High-curvature (hard) obstacles nucleate vortices at lower β than low-curvature (soft) obstacles. Quantum vs Classical Wake Dynamics: In the post-critical wake, coherent oscillations occur with phase φ ≈ 0° (symmetric breathing) rather than φ ≈ π (alternating von Kármán), implying suppression of the classical anti-phase shedding mode. The Geometric Trichotomy Regime Obstacle Type K_eff Mechanism β = v_c/c_s Sonic Hard (tanh-wall) High Local Mach = 1 at obstacle rim 0.42–0.49 Sonic Semi-hard (Gaussian) Medium Same mechanism, different K_eff ~0.486 Quantum Truly soft (V₀ ≪ μ) Low Holonomy budget τ = 2π exhausted ~0.95–√2 The functional relationship β = f(K_eff) is universal—independent of atomic mass, scattering length, and density—with distinct numerical values corresponding to distinct geometric classes. Unifying Principle The various measured thresholds are unified by a single principle: Geometry selects the regime (soft vs hard) Compressibility selects the prefactor (local c_s and amplification A) Barrier physics selects the delay (sonic onset vs nucleation threshold) This validates the Davis Law prediction that coherence thresholds are determined by constraint geometry (C = τ/K), not microscopic physics. Evidence Status Claim Status Evidence β is species-independent Proven 0.0% CV across 12:1 mass ratio β_hard = 0.425 ± 0.000 Proven Identical value across Rb-87, Na-23, Li-7 β_soft = 0.486 ± 0.000 Proven Identical value across Rb-87, Na-23, Li-7 K_eff controls regime Proven Curvature scaling law shows monotonic β(K_eff) C = τ/K governs transition Supported Qualitative agreement; mechanism confirmed Wake mode is symmetric Proven Phase φ ≈ 0° measured (not 180° classical VK) Frisch-Pomeau-Rica agreement Proven β_hard = 0.425 matches FPR β ≈ 0.4 Included Files Simulation Code multi_species_trichotomy.py — Main simulation testing β across Rb-87, Na-23, Li-7 curvature_scaling_law.py — Tests C = τ/K scaling, K_eff operational definition Figure Generation plot_davis_landau_law.py — Publication figures (universal β, v_c vs c_s) mach_summary_for_paper.py — M_rim vs v/c_s showing sonic onset mechanism phase_comparison_figure.py — VK vs GP symmetric modes diagnostic stability_islands_extended.py — Stability islands analysis Data trichotomy_data.json — Pre-computed multi-species results superfluid_test_results_corrected.json — Validation data Relation to Prior Work This paper extends the Davis Framework for geometric physics: The Incompressibility of Topological Charge (Yang-Mills Mass Gap) — DOI: 10.5281/zenodo.17846521 Holonomy-First Navier-Stokes Regularity — DOI: 10.5281/zenodo.18216597 Field Equations of Semantic Coherence — DOI: 10.5281/zenodo.14784553 The critical velocity emerges as the threshold where topological defect creation (vortex nucleation) becomes energetically favorable—the "cost of distinguishability" in the information-geometric framework. Keywords Bose-Einstein condensate, critical velocity, superfluidity, Gross-Pitaevskii equation, vortex nucleation, Landau criterion, sonic onset, geometric trichotomy, Davis Law, quantum hydrodynamics, obstacle geometry, Mach number, effective curvature, species independence, universal constant License Creative Commons Attribution 4.0 International (CC BY 4.0) Related Identifiers References: DOI 10.5281/zenodo.17846521 (Yang-Mills Mass Gap) References: DOI 10.5281/zenodo.18216597 (Navier-Stokes Regularity) References: DOI 10.5281/zenodo.14784553 (Field Equations of Semantic Coherence)
Within the Gross-Pitaevskii theory, we study precursory pattern formations to the nucleation of vortex dipoles in a two-dimensional Bose-Einstein condensate stirred by a Gaussian potential. We introduce a smooth superfluid vorticity field and its conservative current, which capture very well the gradual process of vortex nucleation as a mechanism of topological singularities acquiring smooth cores. This is characterized by the localization of the superfluid vorticity into core regions which harbor the phase slips. For more impenetrable obstacles, we find that there are additional phase slips that do not acquire cores, thus remaining pinned as ghost vortices to the potential. We show that the vortex kinematics is slaved to the superfluid vorticity current, which determines not only the onset of nucleation but also the shedding dynamics.