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A localized vorticity distribution with nonzero hydrodynamic impulse induces $\mathbf{u}\sim [3\mathbf{n}(\mathbf{I}\cdot \mathbf{n})-\mathbf{I}]/(4\pi r^3)$. This curl-free exterior velocity can remain long-ranged even when the vorticity is localized. In a random ensemble it produces dipolar longitudinal and transverse correlation tails.

In an unbounded incompressible fluid with localized vorticity and no added harmonic background, velocity is reconstructed by $\mathbf{u}(\mathbf{x})=(4\pi {)}^{-1}\int \mathbf{\omega }(\mathbf{y})\times (\mathbf{x}-\mathbf{y})/|\mathbf{x}-\mathbf{y}{|}^3{d}^{3}y$. Its multipole expansion relates hydrodynamic impulse to a dipolar far-field velocity.

Hydrodynamic impulse per unit density is $\mathbf{I}=\frac{1}{2}\int \mathbf{x}\times \mathbf{\omega }dV$ for localized vorticity. In unforced all-space incompressible flow its derivative vanishes when far-field fluxes vanish. For a spherical all-space limit, $\mathbf{I}=(3/2){\lim }_R{\int }_{V_R}\mathbf{u}dV$; a dipolar velocity tail makes the distinction from a bare momentum integral important.

The Lighthill elongated-body theory approximation treats each cross-section of a slender swimmer as a locally two-dimensional potential flow problem. The transverse dimension is small compared with the length and the scale over which cross-sections vary. The displacement and slope are small enough to linearize the impermeability condition about a straight body. The Reynolds number is sufficiently large for lateral loading to be dominated by fluid inertia; viscosity, axial friction and higher-order end corrections are neglected at this order. This reactive approximation must be distinguished from the viscous resistive-force theory used for a flagellum.

In coordinates translating with the swimmer, the undisturbed fluid passes in the positive $x$ direction at speed $U$. Applying the impermeability condition to the moving centre plane gives the transverse fluid velocityThe local added mass $m(x)$ is defined so that the transverse fluid hydrodynamic impulse per unit axial length is $P_y=m(x)v_y$. A fixed body-coordinate interval receives an impulse flux $U{P}_y$ at its leading end and loses the corresponding flux at its trailing end. Its rate of change of impulse plus the net outward impulse flux is thereforeThis is the transverse force exerted by the body on the fluid. By Newton's third law, the transverse force exerted by the fluid on the body has the opposite sign:The distinction is essential here: the printed positive expression is labelled as the force on the fish, so that label and expression are inconsistent. For example, take a uniform cross-section and a rigid lateral displacement $h=H(t)$. The swimmer must accelerate fluid of added mass $m$, which reacts on it with force $-m{H}^{\prime \prime }$, not $+m{H}^{\prime \prime }$. Thus the positive expression is recovered exactly with the force-on-fluid convention; the physically consistent force on the fish is the negative expression. In the following parts, $F_y$ denotes $F_{\mathrm{body}}$ so that the beam balance uses a genuine external force on the fish.

A large spherical region contains $N\sim V_R/{\ell }^3$ energy-containing eddies. If their hydrodynamic impulses have zero mean, finite variance and sufficiently short-range correlations, the mean-square sum grows as $N$, rather than $N^2$. In the ideal independent case,The spherical impulse-momentum relation makes the integrated velocity proportional to this sum for isolated enclosed structures. In a full field, external dipolar velocities and boundary effects can change the coefficient; they do not alter the volume scaling under these random-impulse statistics. ThusIt is a random walk law for momentum fluctuations, not an extensive mean flow. Although the mean impulse is zero by isotropy, its variance need not vanish. The Saffman integral measures this large-volume momentum-fluctuation density. If every eddy had exactly zero impulse or correlations enforced systematic cancellation, this leading density could vanish and a different small-wavenumber spectrum could result.

Consider an unforced incompressible flow with localized vorticity and sufficient smoothness for integration by parts. Define the hydrodynamic impulse per unit density by $\mathbf{I}=\frac{1}{2}\int \mathbf{x}\times \mathbf{\omega }dV$, denoting the vector impulse by $\mathbf{I}$ to distinguish it from the later scalar $L$. The vorticity equation can be writtenThe supplied identity, integrated over all space, gives $\frac{1}{2}\int \mathbf{x}\times \nabla \times \mathbf{F}dV=\int \mathbf{F}dV$, because $\mathbf{F}$ is localized and its surface terms vanish. ThereforeThe second integral is a vanishing surface integral. The vector identity $\mathbf{u}\times \mathbf{\omega }=\nabla (|\mathbf{u}{|}^2/2)-(\mathbf{u}\cdot \nabla )\mathbf{u}$ and incompressibility reduce the first to surface terms proportional to the quadratic velocity. Those vanish for $\mathbf{u}=O(R^{-3})$. Thus

For a large spherical control volume define ${\mathbf{P}}_R={\int }_{V_R}\mathbf{u}dV$. The impulse relation is $\mathbf{I}=(3/2){\lim }_{R\to \infty }{\mathbf{P}}_R$ for this localized field. The factor is important: impulse is not simply the integral of velocity, since an $R^{-3}$ velocity tail has a nonvanishing contribution in the impulse integration by parts involving a factor of $R$.

Using pressure per unit density $\Pi$ and the Navier-Stokes equation givesThe advective flux is $O(R^{-4})$, the pressure force is $O(R^{-1})$, and the viscous term is $O(\nu R^{-2})$ under the regular far-field derivative decay. All vanish as $R\to \infty$. The localized eddy cannot change its total impulse without a net external momentum input. Pressure redistributes momentum internally but its resultant force on an increasingly distant control surface tends to zero. At finite $R$, boundary forces need not vanish exactly; the all-space limit is the invariant statement.

Write $u^2=⟨u_x^2⟩$ for the one-component variance and $Q_{LL}=u^{2}f(r)$. Isotropy gives $Q_{ij}=u^2[(f-g)n_i{n}_j+g{\delta }_{ij}]$, and incompressibility gives $g=f+r{f}^{\prime }/2$. Taking the trace yieldsTherefore, with a regular correlation at the origin,For $L>0$, the longitudinal velocity correlation has the tail $u^{2}f(r)\sim L/(4\pi r^3)$. Its physical source is the velocity induced far from a localized eddy with nonzero hydrodynamic impulse. The hydrodynamic Biot-Savart kernel gives the dipolar far fieldThe absence of a vorticity monopole and the surviving impulse dipole give the $r^{-3}$ power. Random eddy impulses thus produce long-range velocity correlations despite the vorticity itself being concentrated in localized structures. Incompressibility also explains how a distant pressure response can redistribute velocity without transporting local vorticity there.

There is no inconsistency with integrability of the trace correlation. At leading order, $g\sim -f/2$, soThe longitudinal and transverse dipolar tails cancel in the trace. The scalar Saffman integral can remain finite even though individual tensor components have long-range, direction-dependent tails.