Forest-Flow: Multi-Disciplinary Transport Dynamics

• Forest-Flow is a multidisciplinary framework that defines transport processes in heterogeneous, forest-like environments spanning atmospheric, cosmological, granular, and data-driven systems.
• It employs techniques such as explicit canopy drag, conditional emulation, and flow matching to accurately capture turbulence, coherent structures, and dynamic obstacle interactions.
• The approach informs practical applications from improved weather prediction and hydrological management to enhanced machine learning for high-fidelity generative modeling.

Forest-Flow encompasses a spectrum of models, methodologies, and observational phenomena spanning atmospheric science, cosmology, granular physics, machine learning, and materials science. In all contexts, “forest flow” refers to the dynamics and consequences of transport processes (fluid, moisture, particles, or information) through systems characterized by forest-like heterogeneity, whether physical canopies, pillar arrays, or high-dimensional statistical structures.

1. Atmospheric Boundary-Layer Flow Over Forest Canopies

Forested canopies significantly modulate neutral boundary-layer flow via enhanced drag and turbulence, especially over complex terrain such as ridges. Two principal parameterization strategies exist in numerical weather prediction (NWP):

• Roughness-Length Method: Enhances the surface roughness length $z_0$ and zero-plane displacement $d$ to approximate forest drag. Above $z > d + z_0$, mean wind follows the log-law, but the approach lacks representation of turbulence below $d$ and fails to generate realistic turbulence under the canopy (Tolladay et al., 2021).
• Explicit Canopy-Drag Model: Treats the canopy as a porous layer with plant area density $a$ and drag coefficient $C_d$, introducing a momentum sink $\mathbf{F}_c = - C_d a |\mathbf{u}| \mathbf{u}$ directly into the equations for all grid cells below the canopy top. This canopy drag augments turbulence kinetic energy (TKE) dissipation and captures the vertical structure and magnitude of turbulence more accurately.

Wind-tunnel-calibrated LES using the WRF model show explicit canopy drag (WRF-C) yields lower RMSE for mean wind speed, realistic turbulence statistics, and proper wake depth and recirculation following ridge flow separation. In contrast, roughness-only approaches (WRF-R) systematically over-predict mean winds and under-represent turbulence. Grid resolution $\Delta x < 0.07L$ and vertical layering $\Delta z \approx 0.1 h_c$ are necessary for resolving canopy-induced flow features, especially for tall canopies or highly turbulent regimes.

2. Turbulence and Coherent Structures in Forest-Like Canopies

Large-eddy simulation (LES) frameworks for turbulent canopy flows must address subgrid-scale (SGS) energy transfer affected by vortex stretching and coherent structure generation (Bhuiyan et al., 2020). Three SGS closure paradigms are prominent:

• SGS-d ("Deardorff" TKE-based): Evolves SGS TKE $k$, balancing production $P$, dissipation $\epsilon$, and transport $T$. Eddy viscosity is proportional to $C_s \Delta k^{1/2}$.
• SGS-s (Lagrangian dynamic Smagorinsky): Utilizes Lagrangian averaging of Germano identity, dynamically adjusting $C_s$ to capture the backscatter associated with coherent intermittency.
• SGS-w (Vortex-Stretching/Scale-Adaptive): Locally adjusts $\nu_\mathrm{sgs}$ in proportion to enstrophy production, accounting for rapid changes near canopy tops where vortex stretching dominates.

Simulation results show SGS-w resolves approximately 18% more TKE compared to classical models, accurately producing hairpin-type eddy structures and sweep–ejection events dominating transport. The choice of canopy representation (immersed-porous vs. immersed-solid obstacles) also influences intermittency and grid requirements.

3. Cosmological Lyman-α Forest Clustering and ForestFlow Emulator

“ForestFlow” specifically designates a surrogate-model framework for predictive cosmological modeling of the Lyman-α forest clustering from linear to nonlinear scales (Chaves-Montero et al., 9 Sep 2024). The central elements are:

• 3D Flux Power-Spectrum Model: Utilizes a bias expansion $\delta_F = b_\delta \delta + b_\eta \eta$ combined with a nonlinear distortion kernel $D_\mathrm{NL}(k, \mu)$ parameterized by eight physical parameters ($b_\delta$, $b_\eta$, $q_1$, $q_2$, $k_v$, $a_v$, $b_v$, $k_p$).
• Conditional Normalizing Flow Emulator: Trains on high-resolution hydro simulations, mapping six-dimensional physical summaries (linear spectrum amplitude/slope, mean flux, thermal broadening, $\gamma$, and pressure smoothing scale) to the eight model parameters, achieving percent-level accuracy across redshift $z=2$ to $4.5$ and varied cosmologies.
• Integration with Boltzmann Solvers: Emulator outputs are combined with CLASS/CAMB for linear $P_\mathrm{lin}(k)$, enabling prediction of $P_\mathrm{3D}$, $P_\mathrm{1D}$, correlation functions, and cross-spectra over a broad range in $k$.

Empirical validation demonstrates $2$–$3\%$ precision on $P_\mathrm{3D}$ and $P_\mathrm{1D}$, robustness to changes in ionization history, neutrino masses, and curvature, and generalizability to out-of-training cosmologies.

4. Data Generation with Forest Flow in Machine Learning

Forest Flow (FF) is a flow-matching based generative model for tabular data, employing XGBoost regressors in place of neural score networks (Akazan et al., 20 Oct 2024). Key aspects are:

• Flow Matching ODE: Learns a velocity field $f^\theta_t$ for ODE integration from Gaussian noise to real data, minimizing $$J(\theta) = E_{t, x_0, x_1} \,\|f^\theta_t(x_t) - (x_1 - x_0)\|^2$$ with temporal transitions $x_t = (1-t)x_0 + t x_1$.
• Treatment of Categorical Variables: Original FF used one-hot continuous encoding, leading to rounding errors and degraded fidelity for large categorical variables.
• HS³F Extension: Generates features sequentially, conditioning each on previously generated columns. Categorical features are directly modeled via XGBoost classifiers with multinomial sampling. Incorporation of higher-order Runge–Kutta solvers (RK4) increases robustness and speed, achieving up to $27\times$ generation speedup for categorical-rich tables and superior data fidelity (Wasserstein, F1 metrics).

HS³F demonstrates enhanced resilience to initial-noise mismatches, higher sample diversity, and pragmatic acceleration on heterogeneous datasets.

5. Granular Flow Through Forests of Obstacles

Experimental and theoretical studies of downslope granular flow through pillar forests illuminate collective effects of obstacle-induced drag (Texier et al., 2023). The depth-averaged $\mu(I)$ rheology, augmented with pillar drag, predicts the discharge rate:

$$\tilde Q(h, \theta) = \frac{2 I_0}{5} \frac{\tan\theta - \tan\theta_1 - \beta \chi h D}{\tan\theta_2 + \beta \chi h D - \tan\theta} \sqrt{\phi \cos\theta} \bigg(\frac{h}{d}\bigg)^{5/2}$$

where $\chi$ is pillar density, $\beta$ quasi-static drag coefficient, $C_d$ inertial drag coefficient, and other parameters fit directly to observed data. High pillar density induces sub-linear $Q(h)$ scaling and, at extremes, apparent flow arrest, highlighting the non-trivial granular-resistance structuring by obstacle forests.

6. Forest Flow in Hydrology: Moisture Regimes and Transpiration

Forest transpiration modulates atmospheric moisture convergence and runoff, with dichotomous behavior contingent on column humidity (Makarieva et al., 2022):

• Wet Regime ($W > W^*$): Increased transpiration ($T \uparrow$) enhances moisture convergence ($C \uparrow$), rainfall, and soil moisture.
• Dry Regime ($W < W^*$): Increased transpiration diminishes moisture convergence and runoff due to reduced atmospheric import capacity.

This critical transition is defined by $k_P(W^*) = k_E(W^*)$, where $k_P$ and $k_E$ are empirical sensitivities of precipitation and evapotranspiration to humidity. Quantitative analysis (e.g., Amazon) attests that deforestation leads to dramatic reductions in runoff in the wet regime, but may paradoxically increase runoff in the dry regime. Management strategies rely on tracking $W$, recycling ratio $E/P$, runoff ratio $R/P$, and identifying minimum points on $C(W)$ curves to anticipate regime shifts.

7. Forest Dislocations and Plastic Flow in Phase-Field Models

In the context of crystal plasticity, forest dislocations modeled as perforations in phase-field domains act as frictional obstacles (Dondl et al., 2017):

• The gradient-flow evolution for Peierls–Nabarro energy with forest dislocations yields a stored-energy term in the $\Gamma$-limit but manifests as a one-sided dry friction law rather than a driving force.
• Sub- and super-solution constructions for the fractional Allen–Cahn equation demonstrate the kinetic implications: interface motion is pinned unless external loading exceeds the friction threshold $A$, resulting in hysteretic, Bauschinger-type response under load reversal.
• This mechanism preserves the possibility of plastic slip under rate-dependent evolution, distinguishing these models from purely elastic, non-slipping scenarios.

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Forest-Flow, across its subfields, characterizes the rich transport and dynamical phenomena induced by spatially complex, forest-structured environments. Explicit canopy drag models, conditional flow-based emulators, featurewise generative pipelines, granular drag laws, and phase-field friction thresholds each reflect the multiscale, multifaceted impacts of forests—real or abstract—on physical and statistical flows.