Freeze two model temperatures, change infrared transfer and release time. Follow every energy route, move the accounting boundary and compare the model with real atmospheric evidence.
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Stored energy increases when absorbed input exceeds outgoing energy. A change in infrared escape can start that imbalance before temperatures have changed.
Sunlight and the initial temperatures stay fixed. Infrared escape falls. What changes first?
Incoming power now exceeds outgoing power. Finite heat capacity makes temperature respond over time.
The EPIC image shows Earth’s sunlit disk using combined red, green and blue filtered exposures. It is real remote-sensing imagery, not a thermal-radiation map. The separate calculation uses a global average: sunlight intercepted by a disk is distributed over four times that area on a sphere.
Some incoming sunlight is reflected. The rest supplies energy to the model surface. The surface emits thermal infrared; part goes directly to space and part is absorbed by the model atmosphere. That atmosphere emits both upward and downward.
Keep initial temperatures and sunlight fixed while increasing the model’s infrared absorptivity. Less energy immediately escapes. The surface and atmosphere do not jump to new temperatures; finite heat capacities make them respond over time.
A transfer from surface to atmosphere is a loss for one and a gain for the other. Enclose both and those internal entries cancel. They still occur physically; they simply do not add or remove energy from the combined system.
As the reservoirs warm in this example, their thermal emission changes. The imbalance shrinks. Equilibrium allows sunlight, infrared and internal transfer to continue while both reservoir budgets balance.
With more heat capacity per area, the same initial perturbation changes temperature more slowly. The fixed-feedback equilibrium is the same. Heat capacity is storage per kelvin, not an extra energy source.
The surface is a thermal blackbody. One isothermal atmospheric layer absorbs a fraction ε of surface infrared and emits εσTₐ⁴ in each direction. Solar and thermal channels are separate. Atmospheric shortwave absorption is omitted after aggregate planetary reflection. The layer is a calculation, not a solid lid or an observed emission altitude.
Q=S₀/4, A=(1−α)Q, Eₛ=σTₛ⁴ and Eₐ=σTₐ⁴. OLR=(1−ε)Eₛ+εEₐ. Surface net is A+εEₐ−Eₛ−F꜀; atmosphere net is εEₛ−2εEₐ+F꜀. Their sum is A−OLR. W/m² is power per area; integrate over seconds to obtain J/m².
S₀=1361 W/m² and α=0.30 are rounded reference choices. σ=5.670374419×10⁻⁸ W m⁻² K⁻⁴ is the displayed CODATA value. The chosen ε values 0.75 and 0.85, surface capacities 2×10⁸ and 4×10⁷, and atmospheric capacity 10⁷ J m⁻² K⁻¹ are illustrative. They are not fitted CO₂ concentrations, actual ocean depths or calibrated climate response times.
The optional closure limits Tₛ−Tₐ to 39.2 K: the dry lapse-rate scale 9.8 K/km times an author-chosen 4 km separation. On an active boundary, F꜀=(CₐRₛ−CₛRₐ)/(Cₛ+Cₐ) must be nonnegative; both temperatures then change at (A−OLR)/(Cₛ+Cₐ). A negative required transfer releases the cap. This is not resolved wind, moist convection or the original Manabe model.
The model uses RK 4 with event splitting for the dry cap. An initial over-threshold contrast transfers J=(Tₛ−Tₐ−D)/(1/Cₛ+1/Cₐ) upward, conserving CₛTₛ+CₐTₐ. Daily steps reproduce the selected independent reference histories. Integrated net input is checked against the change in reservoir energy.
At ε=0.75, Tₛ≈267.0064 K and Tₐ=250 K, the total incoming and outgoing power can match while the surface gains about 116.10 W/m² and the atmosphere loses the same amount. Zero total storage change alone does not establish thermal equilibrium.
A cooler atmosphere can emit toward a warmer surface. For Tₛ>Tₐ, the net longwave exchange εσ(Tₛ⁴−Tₐ⁴) is upward. Both directions belong in the ledger. Internal downward emission must not be counted again as new sunlight.
At ε=0, a separate no-longwave-atmosphere reference has T≈254.578 K and no relevant atmospheric temperature or convection. It is not a real atmosphere-removal forecast. At ε=1, direct surface escape is zero, but the atmospheric emission to space continues.
The calculated dry-adjusted temperature difference for ε 0.75→0.85 is about 4.4645 K. No ε-to-ppm relationship is supplied. It cannot be labeled the warming from doubling real CO₂. Real spectroscopy, abundance records and attribution analyses have distinct quantities and methods.
Choose a boundary before adding arrows. An internal transfer changes the parts without supplying new energy to the whole.
A sustained power imbalance accumulates energy. Capacity controls the temperature response per joule.
Atmospheric concentration, radiation and temperature are different quantities. Linking them quantitatively requires appropriate observations and physical analysis.
Mark surface, atmosphere and space. Add an outline around surface plus atmosphere. Sort radiation routes separately from upward nonradiative energy transfer.
Use the ε=.75 dry-equilibrium cards: absorbed solar 238.18; surface emission 359.60; atmospheric absorption 269.70; direct escape 89.90; atmospheric emission 148.28 in each direction; upward dry transfer 26.85 W/m².
Surface gains 238.18+148.28 and loses 359.60+26.85. Atmosphere gains 269.70+26.85 and loses 148.28+148.28. Small residuals arise from rounding. Do not count absorbed surface emission as an additional surface loss.
Cross out paired internal debits and credits. The combined model gains absorbed sunlight and loses the two radiation routes to space. Crossing out an internal pair does not switch off its physical transfer.
For ε=.85 at those same temperatures, direct escape is 53.94 and atmospheric upward emission 168.05 W/m². Absorbed input remains 238.18. The gain is about 16.19 W/m², before temperatures change.
Separately, a constant 1 W/m² sustained for one 365-day year adds 31,536,000 J/m². Dividing by the combined baseline capacity 2.1×10⁸ J m⁻² K⁻¹ gives 0.150171 K. Label this constant-rate arithmetic; it is not the varying-rate model history or an Earth forecast.
Can the same energy transfer disappear from one ledger and remain real?
Use calculated model cards. No heating or sealed jars. This paper investigation demonstrates accounting; it is not a measurement of the real atmosphere.
Incoming solar is mostly shorter wavelength; Earth mainly emits longer-wave thermal infrared The Sun is much hotter than Earth; sunlight includes infrared too.
Outgoing radiation falls and stored-energy rate becomes positive The selected model gains about 16.19 W/m² before temperature changes.
Both reservoir budgets balance while transfers continue Check the two residuals as well as the combined budget.
Cancels between internal loss and gain The surface loses exactly what the atmosphere gains.
Approaches the same equilibrium more slowly Capacity changes storage per kelvin and the time response.
Yes; both directions matter, and net exchange remains upward here For Tₛ>Tₐ, εσ(Tₛ⁴−Tₐ⁴)>0 upward.
Its cover changes air exchange and other processes together A cover changes several heat-transfer routes; appropriate atmospheric evidence is needed.
No, there is no supplied ε-to-ppm relationship or complete climate response This is a mechanism model with author-selected parameters.
Cannarsa et al., DOI 10.1063/5.0136673, author manuscript v 2 equations 1.3/1.6. The radiative terms match the declared simplifications; this lesson’s dry cap is a separate authored closure.
Cannarsa et al. · Two-layer energy balanceNASA SORCE reference 1361 W/m²; newer TSIS-1 solar-minimum estimate 1361.6±0.3. The lesson holds a rounded value fixed.
NASA · Solar irradiance scienceNIST 2022 CODATA displayed decimal, SI kelvin and W m⁻² K⁻⁴.
NIST · Fundamental physical constantsNASA account distinguishes incoming/reflected/thermal radiation, convection and evaporation. Its historical imbalance figures are not used as current measurements.
NASA · Earth’s energy budgetRounded α=.30 is an Earth-like reference, not a measured constant for every place/year.
NASA · Earth’s albedoNOAA/NWS adiabatic glossary. The 4 km effective separation is our choice, not an observed emission height.
NOAA/NWS · Adiabatic processesManabe and Strickler 1964, Journal of Atmospheric Sciences 21,361–385. This lesson does not reproduce that multi-level model or its figures.
Manabe & Strickler · Convective adjustmentNASA DSCOVR/EPIC, acquired July 6,2015; combined RGB exposures. Original PNG unchanged, no third-party credit present.
NASA · EPIC Earth imageEducational/informational media guidance; no endorsement implied and no blanket CC license asserted.
NASA · Images and mediaLan,Tans,Thoning, version 2026-08, DOI 10.15138/9 N 0 H-ZH 07. Original annual file pinned with header/hashes, 1979–2025. Dry-air micromol/mol, provider uncertainty not relabeled 95%.
NOAA GML · Global CO₂ dataSource header creation 2026-08-05 10:21:45, timezone unspecified. Values may be revised; retrieved 7 September 2026.
NOAA GML · Annual mean source fileNOAA explanation of calibrated nondispersive infrared CO₂ analysis. Absorption is not mirror reflection.
NOAA GML · Measuring CO₂Feldman et al.2015 DOI 10.1038/nature 14240. Clear-sky surface forcing analysis at two sites over 2000–2010; not global all-sky top-of-atmosphere flux.
Feldman et al. · Surface radiative forcingKramer et al.2021 DOI 10.1029/2020 GL 091585. Radiative kernels applied to satellite observations; not a direct CO₂-only measurement.
Kramer et al. · Global radiative forcingGovernment information public domain unless annotated; acknowledgment retained. Brytalearn’s plotted rendering is not an official NOAA chart.
NOAA GML · Terms and acknowledgmentIndependent subject review is pending.
Read the sources and model assumptions
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