How much heat must escape through the floor?

Summary

A fluid heated throughout its interior can carry much of that heat upwards. How little can still escape through its cold floor? This candidate supplies stronger mathematical lower bounds on that bottom heat loss in an idealized infinite-Prandtl model.

With rigid, no-slip plates the guaranteed fraction scales as (R log R)^(−1/3); with stress-free plates it scales as R^(−1/2). Here R measures internal heating, and the horizontal periods are held fixed. The result concerns classical solutions, not arbitrary turbulent weak solutions or every physical mantle model.

The central advance is analytical. The flow calculations illustrate the gap between the bound and computed solutions; they do not prove that the bound can be attained.

Summary for specialists

For uniform internal heating in a horizontally periodic slab in dimension two or three, with T = 0 on both plates and infinite Prandtl number, the candidate derives

\[ \liminf_{\tau\to\infty}\frac1\tau\int_0^\tau F_B(t)\,dt \geq \mathcal B[(R\mathcal T/2)\Phi]. \]

The functional is

\[ \mathcal B[m]=\frac{\int_0^1 z e^{-\int_0^z m(s)ds}\,dz}{\int_0^1 e^{-\int_0^z m(s)ds}\,dz}. \]

The near-wall estimates yield orders (Λ log Λ)^(−1/3) for no-slip plates and Λ^(−1/2) for stress-free plates, with Λ = R𝒯/2 and the stated large-Λ thresholds. A drift-independent eventual temperature ceiling depends on the enlarged horizontal-cell volume and eliminates 𝒯. This produces the advertised orders at fixed horizontal periods, not uniformly over arbitrary joint large-domain limits.

Technical account

Nonnegative temperature gives J(z) ≤ m(z)Θ(z), where J is the averaged convective flux and Θ the averaged temperature. The heat balance becomes a differential inequality for Θ. Multiplication by an integrating factor and the two zero boundary values give the ratio-of-integrals lower bound.

The admissible m comes from the Stokes response. The horizontal kernel has zero integral, permitting subtraction of the temperature midrange and a factor of one half. For no-slip walls, the wall function behaves as z² log(1/z); its integral creates the cube-root logarithmic scale. Stress-free walls instead give a linear wall function and a square-root scale.

The revised stress-free proof removes the low-frequency cusp before applying the radial Sobolev estimate. The exact functional asymptotics concern the lower-bound functional, not optimality among solutions of the PDE. The banner's arrows are a heat-flow schematic, not a computed velocity field; its exponent comparison refers to fixed-domain large-R orders.

Evidence, assurance and limitations

The supplied review has been actioned. Sampled Carlson maxima and plotted prefactors are labelled estimates, not continuum upper enclosures. All eight saved snapshots now have 384 horizontal coordinates matching their padded physical fields, with the original 256 coordinates retained separately; temperature and velocity arrays are unchanged.

One finite-window no-slip profile at R = 10⁶ passes the old grid test but exceeds the Chen triangle near the upper wall. Its boundary-slope ratio is approximately 1.000119051132. The revised code checks slopes and off-grid extrema of the spectral interpolant. This cooling-window effect does not refute the stationary or infinite-time hypothesis.

The supporting suite passes 194 checks in normal and optimized Python. Profile controls and a corrupted-snapshot rejection check supplement kernel, Stokes-map and small steady/DNS comparisons. These are internal checks, partly on stored data. They are not a fresh integration of all runs, formal verification, unaffiliated reproduction or authenticated external peer review.

The stored trajectories are finite-window illustrations: spatial resolution and heat-budget closure do not establish temporal or statistical convergence. Step changes restart with SBDF1. Roll scans establish best-found values and non-detection within their sampled ranges, not branch disappearance, global optimality or asymptotic exclusion. No rigorous numerical prefactor, logarithm removal, finite-Prandtl theorem or saturating solution is claimed.

Relationship to earlier work

Arslan and Rojas obtained earlier unconditional leading orders R^(−2/3) and R^(−40/29). The present claim improves these powers under the fixed-domain assumptions. Chen and colleagues already obtained a related one-third rate conditional on a temperature-profile hypothesis; this is not the first appearance of that exponent.

Minimum principles, logarithmic Stokes estimates and drift-independent temperature bounds have precedents. The proposed contribution is their assembly into this internally heated bottom-flux bound. Sondak and colleagues and Wen and colleagues also constrain the novelty of the roll calculations: competing optima and three-tenths scaling are established antecedents. The source search is bounded, not a priority certificate.

Who should care, and why

Why the problem matters

Internally generated heat is important in planetary interiors. The idealized model asks a narrower but useful question: what restrictions follow from the equations before choosing a particular flow? A stronger lower bound rules out more extreme concentration of heat loss through the top. Translating it to physical predictions requires assumptions absent here, including finite-Prandtl effects and realistic geometry and material properties.

How to inspect or reproduce the recorded checks

Start with AI_INDEX.md, REVIEW_RESPONSE_V21.md, the manuscript's Theorem 2.1 and temperature ceiling, and REPLAY_ENVIRONMENT.json. Verify the release checksums and internal manifest. In an isolated environment matching the recorded replay dependencies, run:

python code/test_publication.py

That command runs the arithmetic/profile checks in both normal and optimized Python and verifies rejection of a malformed snapshot. The broader reproduce.sh regenerates derived checks, tables, figures and the manuscript. Neither command promises a fresh replay of every simulation and parameter sweep; those separate scripts and their stored inputs are mapped in DEPENDENCIES.md.

The most valuable next projects

Independent scrutiny of the Stokes-kernel and temperature-ceiling arguments is the first priority. A validated uniform numerical prefactor would turn estimated plots into certified numerical evaluations. Longer-window and time-step sensitivity would strengthen empirical claims if those are pursued. Tracking roll extrema by continuation would be more informative than asserting disappearance from isolated scans. Attainment and logarithm removal remain separate research problems.

What is in the evidence package

The versioned package contains the 25-page manuscript and LaTeX/Markdown sources, supporting scripts, stored simulation and roll data, all eight repaired snapshots, current replay logs, input hashes, revision response, source-comparison records, licences and an AI index. Original prose and data are CC0; original code is MIT; the citation style retains its upstream licence. Media explains the result and supplies no additional scientific evidence.

@misc{internallyheatedconvectionfluxbounds2026,
  title        = {How much heat must escape through the floor?},
  author       = {Anonymous},
  year         = {2026},
  doi          = {10.5281/zenodo.23172948},
  url          = {https://doi.org/10.5281/zenodo.23172948},
  version      = {2.1.0-candidate},
  howpublished = {Zenodo},
  note         = {Unrefereed; internally replayed evidence package. Press page: https://evidencepress.org/releases/internally-heated-convection-flux-bounds/}
}