Theory: Structure Functions in Turbulence

This document explains the mathematical foundation of structure functions and how they're used to characterize turbulent flows.

Definition
Longitudinal vs Transverse
Order
Dimensional Variants
Curved Geometry: Structure Functions on a Sphere
Kolmogorov Theory
References

Definition

A structure function measures the statistical properties of velocity differences in a turbulent flow. Formally, the nth-order structure function in 1D is:

\[S_n(\Delta x) = \left\langle |\Delta u(\Delta x)|^n \right\rangle\]

where:

\[\Delta u(\Delta x) = u(x + \Delta x) - u(x)\]
is the velocity increment over separation $\Delta x$
The angle brackets $\langle \cdot \rangle$ denote ensemble/time averaging
\[n\]
is the order (typically 2, 3, or higher)

Physical Interpretation

Structure functions quantify how velocity changes as you move through a turbulent medium:

$S_2$ (Second-order): Measures the variance of velocity increments; related to the energy cascade
$S_3$ (Third-order): Directly related to the energy dissipation rate; satisfies the famous "-4/5 law" in Kolmogorov-scale turbulence
Higher orders: Capture tail statistics and intermittency; used to detect coherent structures

Longitudinal vs Transverse

For a point pair separated by vector $\vec{r}$, there are two natural projections of velocity differences:

Longitudinal Structure Function

The velocity difference along the separation vector:

\[S^L_n(r) = \left\langle ([\Delta \vec{u}(\vec{r})] \cdot \hat{r})^n \right\rangle\]

where $\hat{r} = \vec{r} / |\vec{r}|$ is the unit vector in the direction of separation.

Intuition: Measures how velocity changes in the direction you're moving.

Transverse Structure Function

The velocity difference perpendicular to the separation vector:

\[S^T_n(r) = \left\langle ([\Delta \vec{u}(\vec{r})] \times \hat{r})^2 \right\rangle^{n/2}\]

Intuition: Measures how velocity changes perpendicular to your direction of motion.

In most real turbulent flows (e.g., stratified atmosphere, rotating fluids), longitudinal and transverse structure functions differ. The ratio $S^L_n / S^T_n$ quantifies flow anisotropy — deviations from isotropy.

Order

Second-Order: Energy Spectrum

The second-order structure function is most commonly used:

\[S_2(\Delta x) = \langle (\Delta u)^2 \rangle\]

In isotropic turbulence, this relates to the energy spectrum $E(k)$ via:

\[S_2(r) = \int_0^{k_r} E(k) \sin(kr) / (kr) dk\]

Third-Order: Energy Dissipation

The third-order longitudinal structure function has special significance:

\[S_3^L(r) = -\frac{4}{5} \varepsilon r\]

This "-4/5 law" (Kolmogorov, 1941; Landau & Lifshitz, 1987) directly relates the third-order SF to the energy dissipation rate $\varepsilon$, provided you're in the inertial range of turbulence.

Higher Orders: Intermittency

Higher-order SFs are sensitive to intermittency — the non-Gaussian nature of velocity increments in real turbulence. The scaling exponents $\zeta_n$ defined by:

\[S_n(r) \sim r^{\zeta_n}\]

deviate from linear scaling ($\zeta_n \neq n/3$) in intermittent flows. These anomalous exponents are studied in multifractal analysis of turbulence.

Dimensional Variants

StructureFunctions.jl supports calculations in multiple dimensions:

2D Flows

For 2D velocity fields $(u_x, u_y)$ in horizontal planes:

Compute longitudinal/transverse SFs in the $(x, y)$ plane
Study anisotropy between meridional and zonal components
Common in meteorology (weather systems, jet streams)

3D Flows

For full 3D velocity fields $(u_x, u_y, u_z)$:

Compute SFs along any separation vector
Study isotropy/anisotropy in all directions
Standard in direct numerical simulations (DNS)

Projected / Vector Variants

Full vector SF: Uses complete velocity difference vector
Projected SF: Projects onto specific directions (e.g., vertical, meridional)
Flexibility for analyzing anisotropic flows

Curved Geometry: Structure Functions on a Sphere

Everything above assumes flat space: $\hat{r} = \vec{r}/|\vec{r}|$ is a single vector, and $\Delta\vec{u}$ is a plain difference of two velocity vectors. On a sphere neither holds. The two points have different tangent planes, so "the separation direction" is two different vectors, and subtracting velocities that live in different tangent spaces is not defined.

The transported frame

The separation is the great-circle arc. With $\hat{p}, \hat{q}$ the unit position vectors, $c = \hat{p}\cdot\hat{q}$, $\vec{w} = \hat{p}\times\hat{q}$, $s = |\vec{w}| = \sin\sigma$:

\[\sigma = 2\,\mathrm{atan2}\!\left(|\hat{q}-\hat{p}|,\ |\hat{q}+\hat{p}|\right), \qquad r = R\sigma\]

\[\hat{t}_A = \frac{\hat{q} - c\,\hat{p}}{s}, \qquad \hat{t}_B = \frac{c\,\hat{q} - \hat{p}}{s}, \qquad \hat{m} = \frac{\vec{w}}{s}\]

\[\hat{t}_A\]

and $\hat{t}_B$ are the geodesic tangents at each endpoint, and $\hat{m}$ is the great-circle normal. Two facts make this cheap: $|\hat{q}-c\hat{p}|^2 = |c\hat{q}-\hat{p}|^2 = s^2$, so one reciprocal square root normalises both; and $\hat{m}$ is the same vector at both endpoints, so the transverse direction needs no transport at all.

The increments are then

\[\delta u_L = \vec{u}_B\cdot\hat{t}_B - \vec{u}_A\cdot\hat{t}_A, \qquad \delta u_T = (\vec{u}_B - \vec{u}_A)\cdot\hat{m}\]

Projecting each velocity onto its own geodesic frame before differencing is parallel transport, not an approximation: a geodesic parallel-transports its own tangent, and transport on $S^2$ is an orientation-preserving isometry, so the frame carried from $A$ arrives at $B$ rotated by exactly the difference of forward azimuths.

Why a flat frame is wrong, and by how much

Using one flat direction for both endpoints ignores the meridian convergence — the angle $\psi$ between the two local frames:

\[\psi \approx \frac{r}{R}\tan\varphi\]

separation	latitude	$\psi$
100 km	45°	0.9°
1000 km	45°	9°
1000 km	70°	25°

\[D_{LT}\]

vanishes identically under reflection symmetry, so this leaks $O(\psi)\,(D_{TT}-D_{LL})/2$ into a quantity whose true value is zero, and the third-order cascade diagnostics $D_{LLL}$, $D_{LTT}$ inherit $O(\psi)$. It is the same pathology as spurious B-modes from flat-sky projection in weak lensing. A direct measurement of the effect: for a solid-body rotation — which has no strain, so $D_{LL}$ must vanish at every separation — the transported frame gives $\Sigma D_{LL}/\Sigma S_2 \approx 5\times10^{-32}$, while a flat lon/lat frame puts 36% of the total signal into $D_{LL}$.

Guidance: below ~10 km a flat tangent plane is fine; beyond ~100 km, or anywhere poleward of about 60°, use a spherical metric.

Conventions and limits

Radial component. For a thin shell ($D = 3$, $u = (\text{east},\text{north},\text{up})$) the radial component is differenced as a scalar and never transported: the geodesic frame is tangent to the shell, hence orthogonal to $\hat{p}$ at both endpoints, so radial motion cannot leak into $\delta u_L$ or $\delta u_T$. Ambient-transporting the full 3-vector would leak radial into tangential at $O(r/R)$ — a first-order error.
Coordinates. A point on a shell is located by two numbers. x is (2, N) lon/lat while u may be (2, N) or (3, N); the shell radius belongs to the metric, not to each point.
Angle units follow the metric. Distances.Haversine is degrees, Distances.SphericalAngle is radians. Mixing them rescales every separation by ~57.
Handedness. $\hat{n} = \hat{z}\times\hat{r}$, so $(\hat{r},\hat{n},\hat{z})$ is right-handed.
Degenerate pairs are excluded. Coincident points have no direction, and antipodal points are joined by infinitely many great circles, so parallel transport is genuinely non-unique. Both are masked out rather than producing a NaN.
Precision. Sub-metre separations on Earth need Float64 input: at 45°, one Float32 ulp of latitude is already ~4 m on the ground.
Isotropy relations are planar. KHM.jl's transverse_incompressibility_residual ($D_{TT} = \mathrm{d}(rD_{LL})/\mathrm{d}r$, Lindborg 1999 eq. 53) is a planar isotropic relation and holds on the sphere only to $O((r/R)^2)$.

What the literature does

Published velocity-SF work is almost entirely regional ($\lesssim 500$ km), where $\psi$ is negligible, and works in a local tangent plane: Balwada, LaCasce & Speer (2016) define $\vec{r} = \Delta x\,\hat{i} + \Delta y\,\hat{j}$; FluidSF uses a great-circle distance for the scalar separation only and then rotates with a flat grid-space angle. The transported-frame convention adopted here is the one established in CMB polarization and weak-lensing shear $\xi_\pm$, which face the identical problem.

Kolmogorov Theory

The Inertial Range

Kolmogorov's 1941 theory predicts that at scales $\eta \ll r \ll L$ (the inertial range):

\[S_2(r) \sim \varepsilon^{2/3} r^{2/3}\]