PyTurbo_SF: Structure Functions Reference

This document provides the mathematical foundations and API reference for structure function computations in PyTurbo_SF.

Mathematical Foundations 

Structure functions measure statistical properties of turbulent field increments across separation distances. The general form is:

\[S_n(r) = \langle |\phi(\vec{x} + \vec{r}) - \phi(\vec{x})|^n \rangle_{\vec{x}}\]

where \(\phi\) represents the field variable, \(\vec{r}\) is the separation vector, and \(\langle \cdot \rangle_{\vec{x}}\) denotes spatial averaging.

1D Structure Functions 

For 1D data (time series), PyTurbo_SF supports:

Scalar Structure Function (fun='scalar')

\[S_n(r) = \langle |f(x + r) - f(x)|^n \rangle_{x}\]

Scalar-Scalar Structure Function (fun='scalar_scalar')

\[S_{n,k}(r) = \langle |f(x + r) - f(x)|^n \cdot |g(x + r) - g(x)|^k \rangle_{x}\]

Important

1D only supports scalar and scalar_scalar function types.

2D Structure Functions 

For 2D velocity fields \(\vec{u} = (u, v)\):

Longitudinal Structure Function (fun='longitudinal')

Projects velocity increments onto the separation vector:

\[D_{LL}(\mathbf{r}) = \left\langle \left[ \delta u_L \right]^n \right\rangle\]

where \(\delta u_L = [\mathbf{u}(\mathbf{x} + \mathbf{r}) - \mathbf{u}(\mathbf{x})] \cdot \hat{\mathbf{r}}\)

Transverse Structure Function (fun='transverse')

Uses the perpendicular velocity component:

\[D_{TT}(\mathbf{r}) = \left\langle \left[ \delta u_T \right]^n \right\rangle\]

where \(\delta u_T = |[\mathbf{u}(\mathbf{x} + \mathbf{r}) - \mathbf{u}(\mathbf{x})] \times \hat{\mathbf{r}}|\)

Default Velocity Structure Function (fun='default_vel')

Sum of individual velocity component structure functions:

\[S_n(r) = \langle |u(\vec{x} + \vec{r}) - u(\vec{x})|^n + |v(\vec{x} + \vec{r}) - v(\vec{x})|^n \rangle\]

Advective Structure Function (fun='advective')

Third-order structure function including stress tensor components for energy flux:

\[D_{adv}(\mathbf{r}) = \left\langle \delta u \cdot \delta \tau \right\rangle\]

Scalar Structure Function (fun='scalar')

\[D_{\phi\phi}(r) = \left\langle \left[ \phi(\mathbf{x} + \mathbf{r}) - \phi(\mathbf{x}) \right]^n \right\rangle\]

Scalar-Scalar Structure Function (fun='scalar_scalar')

\[D_{\phi\psi}(r) = \left\langle \delta\phi^n \cdot \delta\psi^k \right\rangle\]

Longitudinal-Transverse (fun='longitudinal_transverse')

\[D_{LT}(\mathbf{r}) = \left\langle \delta u_L^n \cdot \delta u_T^k \right\rangle\]

Longitudinal-Scalar (fun='longitudinal_scalar')

\[D_{L\phi}(\mathbf{r}) = \left\langle \delta u_L^n \cdot \delta\phi^k \right\rangle\]

Transverse-Scalar (fun='transverse_scalar')

\[D_{T\phi}(\mathbf{r}) = \left\langle \delta u_T^n \cdot \delta\phi^k \right\rangle\]

3D Structure Functions 

For 3D velocity fields \(\vec{u} = (u, v, w)\):

Longitudinal (fun='longitudinal')

Requires all three velocity components ['u', 'v', 'w'].

Default Velocity (fun='default_vel')

Sum of all three velocity component structure functions:

\[S_n(r) = \langle |u|^n + |v|^n + |w|^n \rangle\]

Transverse (plane-specific)

fun='transverse_ij' — xy-plane, requires ['u', 'v']
fun='transverse_ik' — xz-plane, requires ['u', 'w']
fun='transverse_jk' — yz-plane, requires ['v', 'w']

Longitudinal-Transverse (plane-specific)

fun='longitudinal_transverse_ij'
fun='longitudinal_transverse_ik'
fun='longitudinal_transverse_jk'

Transverse-Scalar (plane-specific)

fun='transverse_ij_scalar'
fun='transverse_ik_scalar'
fun='transverse_jk_scalar'

Scalar types: scalar, scalar_scalar, longitudinal_scalar

Bessel Energy Flux Decomposition 

Warning

Energy flux decomposition is only available in 2D via get_energy_flux_2d.

Only fun='advective' and fun='scalar_scalar' are supported.

Core Formula 

The spectral energy flux \(\Pi(K)\) is computed from the angle-averaged advective structure function using a Bessel \(J_1\) transform:

\[\boxed{\Pi(K) = -\frac{K}{2} \int_0^{\infty} \tilde{D}_{LLL}(r) \, J_1(Kr) \, dr}\]

where:

\(\Pi(K)\) — Spectral energy flux at wavenumber \(K\)
\(\tilde{D}_{LLL}(r)\) — Angle-averaged advective structure function
\(J_1(Kr)\) — Bessel function of the first kind, order 1
\(K\) — Wavenumber magnitude

Physical Interpretation 

Sign of \(\Pi\)	Physical Meaning
\(\Pi(K) > 0\)	Forward cascade: energy flows to smaller scales
\(\Pi(K) < 0\)	Inverse cascade: energy flows to larger scales

The magnitude \(|\Pi(K)|\) represents the rate of energy transfer across wavenumber \(K\).

Derivation 

The formula derives from the Plancherel theorem relating spectral energy flux to real-space structure functions. For homogeneous isotropic turbulence:

\[\Pi(K) = -\frac{1}{2} \frac{\partial}{\partial K} \left[ K \int_0^{\infty} \tilde{D}_{LLL}(r) \, J_0(Kr) \, r \, dr \right]\]

Using the identity \(\frac{d}{dz}[z J_0(z)] = z J_1(z)\) and integration by parts yields the working formula.

Numerical Implementation 

The integral is discretized over radial bins:

\[\Pi(K) \approx -\frac{K}{2} \sum_{i=1}^{N_r} \tilde{D}_{LLL}(r_i) \, J_1(Kr_i) \, \Delta r_i\]

Conditional Structure Functions 

Conditional structure functions compute statistics conditioned on a secondary field, enabling scale-dependent analysis of turbulence-scalar interactions.

Mathematical Definition 

\[\boxed{D(r \,|\, C) = \left\langle \left[ \delta u \right]^n \, \middle| \, C(\mathbf{x}) \in [C_{\text{lo}}, C_{\text{hi}}) \right\rangle}\]

where:

\(C(\mathbf{x})\) — Conditioning variable (e.g., vorticity, temperature, strain rate)
\([C_{\text{lo}}, C_{\text{hi}})\) — Conditioning bin range (half-open interval)

Indicator Function Formulation 

Using indicator functions:

\[D(r \,|\, C) = \frac{\left\langle \left[ \delta u \right]^n \cdot \mathbb{I}_C(\mathbf{x}) \right\rangle}{\left\langle \mathbb{I}_C(\mathbf{x}) \right\rangle}\]

where:

\[\begin{split}\mathbb{I}_C(\mathbf{x}) = \begin{cases} 1 & \text{if } C(\mathbf{x}) \in [C_{\text{lo}}, C_{\text{hi}}) \\ 0 & \text{otherwise} \end{cases}\end{split}\]

Implementation 

Single Conditioning Bin:

conditioning_bins = [C_lo, C_hi]

Multiple Conditioning Bins:

conditioning_bins = np.linspace(C_min, C_max, N+1)  # N bins
# or
conditioning_bins = np.logspace(log10(C_min), log10(C_max), N+1)

Note

When using multiple bins, the output dataset includes an additional cond_bin dimension with coordinates at bin centers.

Typical Conditioning Variables 

Conditioning Variable	Application
Vorticity magnitude	Separate rotational from irrotational regions
Temperature	Stratification effects on turbulence
Strain rate	Strain-dominated vs. rotation-dominated regions
Scalar gradient	Frontal vs. well-mixed regions
Turbulent kinetic energy	High-energy vs. low-energy turbulence regions

Function Reference 

Important

1D supports only: scalar and scalar_scalar

Energy flux (Bessel) supports only: advective and scalar_scalar (2D only)

1D and 2D Function Types 

`fun`	`variables_names`	1D	2D bin	2D iso	2D flux
`scalar`	`['theta']`	✓	✓	✓
`scalar_scalar`	`['phi', 'psi']`	✓	✓	✓	✓
`default_vel`	`['u', 'v']` or `['u', 'v', 'w']`		✓	✓
`longitudinal`	`['u', 'v']`		✓	✓
`transverse`	`['u', 'v']`		✓	✓
`advective`	`['u', 'v', 'tau1', 'tau2']` or `['u', 'v', 'w', 'tau1', 'tau2', 'tau3']`		✓	✓	✓
`longitudinal_transverse`	`['u', 'v']`		✓	✓
`longitudinal_scalar`	`['u', 'v', 'theta']`		✓	✓
`transverse_scalar`	`['u', 'v', 'theta']`		✓	✓

3D Function Types 

`fun`	`variables_names`	3D bin	3D iso
`scalar`	`['theta']`	✓	✓
`scalar_scalar`	`['phi', 'psi']`	✓	✓
`default_vel`	`['u', 'v', 'w']`	✓	✓
`longitudinal`	`['u', 'v', 'w']`	✓	✓
`longitudinal_scalar`	`['u', 'v', 'w', 'theta']`	✓	✓
`transverse_ij`	`['u', 'v']` (xy-plane)	✓	✓
`transverse_ik`	`['u', 'w']` (xz-plane)	✓	✓
`transverse_jk`	`['v', 'w']` (yz-plane)	✓	✓
`longitudinal_transverse_ij`	`['u', 'v']`	✓	✓
`longitudinal_transverse_ik`	`['u', 'w']`	✓	✓
`longitudinal_transverse_jk`	`['v', 'w']`	✓	✓
`transverse_ij_scalar`	`['u', 'v', 'theta']`	✓	✓
`transverse_ik_scalar`	`['u', 'w', 'theta']`	✓	✓
`transverse_jk_scalar`	`['v', 'w', 'theta']`	✓	✓
`advective`	`['u', 'v', 'w', 'tau1','tau2','tau3']`	✓	✓
`pressure_work`	`['p', 'u', 'v', 'w']`	✓	✓

Note

3D Energy Flux (get_energy_flux_3d) is not implemented.

Methods Summary 

Method	Dim	Description	Output Coord
`bin_sf_1d`	1D	Binned SF for time series (scalar only)	`r`
`bin_sf_2d`	2D	Binned SF in Cartesian coordinates	`(dx, dy)`
`bin_sf_3d`	3D	Binned SF in Cartesian coordinates	`(dx, dy, dz)`
`get_isotropic_sf_2d`	2D	Radially-averaged SF with isotropy diagnostics	`r`
`get_isotropic_sf_3d`	3D	Spherically-averaged SF with isotropy diagnostics	`r`
`get_energy_flux_2d`	2D	Bessel J1 energy flux (advective/scalar_scalar)	`K`

API Reference 

bin_sf_1d 

bin_sf_1d(
    ds,                          # xarray.Dataset
    variables_names,             # ['scalar'] or ['phi', 'psi']
    order,                       # 2 or (1, 1)
    bins,                        # {'time': time_edges}
    bootsize=None,
    fun='scalar',                # 'scalar' or 'scalar_scalar' ONLY
    initial_nbootstrap=100,
    max_nbootstrap=1000,
    step_nbootstrap=100,
    convergence_eps=0.1,
    n_jobs=-1,
    backend='threading',
    conditioning_var=None,
    conditioning_bins=None,
    confidence_interval=0.95,
    seed=42
)

bin_sf_2d 

bin_sf_2d(
    ds,                          # xarray.Dataset with 2D fields
    variables_names,             # See function reference table
    order,                       # 2, 3, or (1, 1)
    bins,                        # {'x': x_edges, 'y': y_edges}
    bootsize=None,
    fun='longitudinal',
    initial_nbootstrap=100,
    max_nbootstrap=1000,
    step_nbootstrap=100,
    convergence_eps=0.1,
    n_jobs=-1,
    backend='threading',
    conditioning_var=None,
    conditioning_bins=None,
    seed=42
)

bin_sf_3d 

bin_sf_3d(
    ds,                          # xarray.Dataset with 3D fields
    variables_names,             # See function reference table
    order,
    bins,                        # {'x': x_edges, 'y': y_edges, 'z': z_edges}
    bootsize=None,
    fun='longitudinal',
    initial_nbootstrap=100,
    max_nbootstrap=1000,
    step_nbootstrap=100,
    convergence_eps=0.1,
    n_jobs=-1,
    backend='threading',
    conditioning_var=None,
    conditioning_bins=None,
    seed=42
)

get_isotropic_sf_2d 

get_isotropic_sf_2d(
    ds,
    variables_names,
    order=2.0,
    bins=None,                   # {'r': r_edges}
    bootsize=None,
    initial_nbootstrap=100,
    max_nbootstrap=1000,
    step_nbootstrap=100,
    fun='longitudinal',
    n_bins_theta=36,
    window_size_theta=None,
    window_size_r=None,
    convergence_eps=0.1,
    n_jobs=-1,
    backend='threading',
    conditioning_var=None,
    conditioning_bins=None,
    confidence_interval=0.95,
    seed=42
)

get_isotropic_sf_3d 

get_isotropic_sf_3d(
    ds,
    variables_names,
    order=2.0,
    bins=None,                   # {'r': r_edges}
    bootsize=None,
    initial_nbootstrap=100,
    max_nbootstrap=1000,
    step_nbootstrap=100,
    fun='longitudinal',
    n_bins_theta=36,
    n_bins_phi=18,
    window_size_theta=None,
    window_size_phi=None,
    window_size_r=None,
    convergence_eps=0.1,
    n_jobs=-1,
    backend='threading',
    conditioning_var=None,
    conditioning_bins=None,
    confidence_interval=0.95,
    seed=42
)

get_energy_flux_2d 

get_energy_flux_2d(
    ds,
    variables_names,             # ['u', 'v', 'tau1', 'tau2'] or ['u', 'v', 'w', 'tau1', 'tau2', 'tau3']
    order=3.0,
    wavenumbers=None,
    r_bins=None,
    bootsize=None,
    initial_nbootstrap=100,
    max_nbootstrap=1000,
    step_nbootstrap=100,
    fun='advective',             # 'advective' or 'scalar_scalar' ONLY
    n_bins_theta=36,
    n_r_bins=100,
    window_size_theta=None,
    window_size_k=None,
    convergence_eps=0.1,
    n_jobs=-1,
    backend='threading',
    conditioning_var=None,
    conditioning_bins=None,
    confidence_interval=0.95,
    seed=42
)

Common Parameters 

Parameter	Description
`ds`	`xarray.Dataset` containing velocity/scalar fields
`variables_names`	List of variable names (see function reference tables)
`order`	Structure function order: `2`, `3`, or tuple `(1, 1)` for mixed
`bins`	Dict of bin edges: `{'x': [...], 'y': [...]}` or `{'r': [...]}`
`bootsize`	Bootstrap block size; should be `data_size / 2^n` for optimal results
`fun`	Structure function type (see tables above)
`initial_nbootstrap`	Initial bootstrap iterations (default: 100)
`max_nbootstrap`	Maximum bootstrap iterations (default: 1000)
`convergence_eps`	Convergence threshold for standard error (default: 0.1)
`n_jobs`	Parallel jobs: `-1` for all cores
`backend`	`'threading'`, `'multiprocessing'`, or `'loky'`
`conditioning_var`	Name of conditioning variable in dataset
`conditioning_bins`	`[lo, hi]` for single bin or array for multiple bins
`confidence_interval`	Confidence level (default: 0.95)
`seed`	Integer to ensure reproducibility at the initial phase of the bootstrapping

Output Variables 

All output datasets include:

Variable	Description
`sf` / `energy_flux`	Mean structure function or energy flux
`std_error`	Standard error from bootstrap
`ci_upper`, `ci_lower`	Confidence interval bounds
`point_counts`	Number of point pairs per bin
`converged`	Boolean convergence status
`n_bootstrap`	Bootstrap iterations used per bin
`error_isotropy`	Isotropy error (isotropic methods only)
`error_homogeneity`	Homogeneity error (isotropic methods only)
`mask_quality`	Combined quality mask

Usage Examples 

2D Longitudinal SF (Isotropic)

import numpy as np
from pyturbo_sf import get_isotropic_sf_2d

ds_sf = get_isotropic_sf_2d(
    ds,
    variables_names=['u', 'v'],
    order=2.0,
    bins={'r': np.logspace(2, 4, 30)},
    fun='longitudinal',
    bootsize={'y': 64, 'x': 64}
)

1D Scalar SF 

from pyturbo_sf import bin_sf_1d

ds_sf = bin_sf_1d(
    ds,
    variables_names=['temperature'],
    order=2,
    bins={'time': np.linspace(0, 3600, 50)},
    fun='scalar'
)

Energy Flux (2D)

from pyturbo_sf import get_energy_flux_2d

ds_flux = get_energy_flux_2d(
    ds,
    variables_names=['u', 'v', 'tau1', 'tau2'],
    order=3.0,
    fun='advective',
    bootsize={'y': 64, 'x': 64}
)

Conditional SF (Single Bin)

ds_sf = get_isotropic_sf_2d(
    ds,
    variables_names=['u', 'v'],
    fun='longitudinal',
    conditioning_var='vorticity',
    conditioning_bins=[0.001, 0.01]
)

Conditional SF (Multiple Bins)

import numpy as np

ds_sf = get_isotropic_sf_2d(
    ds,
    variables_names=['u', 'v'],
    fun='longitudinal',
    conditioning_var='vorticity',
    conditioning_bins=np.logspace(-4, -1, 6)  # 5 bins
)

# Access specific conditioning bin
ds_sf.sel(cond_bin=0.001, method='nearest')

3D Spherically-Averaged SF 

from pyturbo_sf import get_isotropic_sf_3d

ds_sf = get_isotropic_sf_3d(
    ds,
    variables_names=['u', 'v', 'w'],
    order=2.0,
    bins={'r': np.logspace(0, 3, 25)},
    fun='longitudinal',
    bootsize={'z': 32, 'y': 32, 'x': 32}
)

References 

Kolmogorov, A. N. (1941). The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers. Dokl. Akad. Nauk SSSR, 30, 301-305.
Lindborg, E. (1999). Can the atmospheric kinetic energy spectrum be explained by two-dimensional turbulence? J. Fluid Mech., 388, 259-288.
Cho, J. Y. N., & Lindborg, E. (2001). Horizontal velocity structure functions in the upper troposphere and lower stratosphere. J. Geophys. Res., 106(D10), 10223-10232.