Implement NonnegativeLeastSquares.jl enhanced Energy Transfers from Third-Order Structure Functions

[jbphyswx]

An Improved Methodology to Estimate Cross-Scale Kinetic Energy Transfers from Third-Order Structure Functions Using Regularized Least Squares https://doi.org/10.1175/JTECH-D-25-0025.1 indicates that we should be able to use this method of optimized energy transfer calculations.

See also Third-order structure functions for isotropic turbulence with bidirectional energy transfer https://doi.org/10.1017/jfm.2019.651, and Direct observational evidence of an oceanic dual kinetic energy cascade and its seasonality https://doi.org/10.1126/sciadv.abq2566

PS, it also uses the Hankel transformation stuff that we never ported here, check what's going on and get that built in here too

[jbphyswx]

Implement Dimension-Aware Transforms Between Structure Functions and Spectra

Description
Our package calculates Structure Functions (SFs) of all orders, but lacks the ability to invert them into Isotropic Power Spectra ($E(k)$). Because our data spans different dimensions (1D tracks, 2D surfaces, 3D volumes), we cannot use a standard 1D FFT for all inversions. When multi-dimensional fields are averaged into 1D isotropic spectra, the geometry of the required transform changes.

Theoretical Basis

1. The 2nd-Order SF is the Bridge: By the Wiener-Khinchin theorem, only the 2nd-order structure function ($S_2(r)$) can be directly transformed into the kinetic energy power spectrum ($E(k)$).
2. The 3rd-Order SF is the Flux: The 3rd-order structure function ($S_3(r)$) represents spectral energy transfer (e.g., Kolmogorov's 4/5ths law). It cannot be inverted into a standard power spectrum.

Dimensionality Constraints & Required Basis Functions
To accurately convert an isotropic spectrum $E(k_r)$ back to a physical $S_2(r)$ (and vice versa), the integration basis function must match the dimension the data was azimuthally or spherically averaged from:

• 1D Data (Straight-line tracks): No angular averaging.
  • Transform: Standard 1D FFT / Discrete Cosine Transform.
  • Basis Function: Cosine wave ($\cos(kr)$).
• 2D Data (Ocean/Atmosphere surfaces): Averaged over circular rings.
  • Transform: Discrete Hankel Transform.
  • Basis Function: Bessel function of the first kind, order zero ($J_0(kr)$).
• 3D Data (Volume fields): Averaged over spherical shells.
  • Transform: Spherical Bessel Transform.
  • Basis Function: Sinc function ($\frac{\sin(kr)}{kr}$).

Implementation Checklist

• Implement 1D Cosine inversion for $S_2(r) \leftrightarrow E(k_{1D})$
• Implement 2D Hankel inversion (Bessel $J_0$) for isotropic surface fields.
• Implement 3D Spherical Bessel inversion (Sinc) for isotropic volume fields.
• Add explicit errors if a user attempts an $E(k)$ inversion on $S_n$ where $n \neq 2$.
• (Optional) Add a dedicated method to extract the turbulent dissipation rate ($\epsilon$) directly from $S_3(r)$.

This actually I think can used the generalized Hankel Transform for any dimensions, but the rest of the package needs to be updated to accept any dimension data.

[jbphyswx]

Hankel Links:
https://github.com/pbeckman/FastHankelTransform.jl, https://doi.org/10.48550/arXiv.2411.09583 (this is also NUFHT - nonuniform fast hankel transform)
https://github.com/chrisbrahms/Hankel.jl for QDHT

https://github.com/marcobonici/FFTLog.jl, https://www.marcobonici.com/FFTLogDocs.jl/dev/ also has an old potentially useful impelmentation to study