A two-scale Stefan problem arising in a model for tree sap exudation
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Bibliographic record
Abstract
The study of tree sap exudation, in which a (leafless) tree generates elevated stem pressure in response to repeated daily freeze–thaw cycles, gives rise to an interesting multiscale problem involving heat and multiphase liquid/gas transport. The pressure generation mechanism is a cellular-level process that is governed by differential equations for sap transport through porous cell membranes, phase change, heat transport, and generation of osmotic pressure. By assuming a periodic cellular structure based on an appropriate reference cell, we derive an homogenized heat equation governing the global temperature on the scale of the tree stem, with all the remaining physics relegated to equations defined on the reference cell. We derive a corresponding strong formulation of the limit problem and use it to design an efficient numerical solution algorithm. Numerical simulations are then performed to validate the results and draw conclusions regarding the phenomenon of sap exudation, which is of great importance in trees such as sugar maple and a few other related species. The particular form of our homogenized temperature equation is obtained using periodic homogenization techniques with two-scale convergence, which we investigate theoretically in the context of a simpler two-phase Stefan-type problem corresponding to a periodic array of melting cylindrical ice bars with a constant thermal diffusion coefficient. For this reduced model, we prove results on existence, uniqueness and convergence of the two-scale limit solution in the weak form, clearly identifying the missing pieces required to extend the proofs to the fully nonlinear sap exudation model. Numerical simulations of the reduced equations are then compared with results from the complete sap exudation model.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it