Pencil, spreadsheet, simulation. Optional. It belongs to no milestone and is not required. Best taken after Spreadsheet discipline and Sanity checks and orders of magnitude, and before you build anything large. About four hours.
A large model that returns a number has not explained anything. It has produced a number, and the number is only as trustworthy as your ability to say why it came out that way. If you cannot, you cannot tell a finding from a bug, and neither can your committee.
The cure is old and has three names. The one Jones was taught as a PhD student is analytical → numerical → computational. The plainer ones are crawl, walk, run, and the one this lab uses: pencil, spreadsheet, simulation.
The rungs do different jobs, and the job each one does is the reason you need all three. The pencil shows the mechanism the big model hides. The big model tests whether the pencil’s simplifications matter. Neither answers the other’s question.
None.
From Jones’s own carbon research. Two equations: atmospheric carbon C, a reservoir R that the atmosphere drains into, emissions E, and pre-industrial level C₀.
dC/dt = E − k(C − R)
dR/dt = γ·k(C − R) − μ(R − C₀)
Pencil. Set both derivatives to zero. The long-run equilibrium is
C_eq = C₀ + E(1/k + γ/μ)
and the emissions that hold a target concentration forever are
E_hold = (C_target − C₀) / (1/k + γ/μ)
That is two resistances in series. Read it and the mechanism is in front of you: CO₂ falls whenever E < k(C − R), and the slower of the two drains sets how much you can emit. Each simpler model is this one with a parameter switched off. Set γ = 0 and the reservoir never fills, so C relaxes toward C₀ like a one-box model. Set μ = 0 and nothing leaves the reservoir, a two-box model with no equilibrium at all.
Spreadsheet. Step the equations one year at a time. Fit k, γ and μ to the Mauna Loa record by grid search. Then hold E constant, run it long, and check the number it settles at against C_eq. If they disagree, one of them is wrong, and you find out which now.
Simulation. Cross-check against published climate models, FaIR and Hector. The simple model says where the balance point is and why. The big models say whether it survives more physics: temperature feedbacks, ocean layers, a saturating land sink.
Do this on your own dissertation problem. It works for optimization as well as dynamics; most of this lab’s problems are optimization, so there are examples of both below.
1. Pencil. Write the smallest version of your problem that still contains the thing you care about, and solve it by hand. Stop when you have a result in symbols: a closed form, a threshold, a sign, or a comparative static (which way the answer moves when one input rises).
2. Spreadsheet. Build that same small model numerically, in a spreadsheet or notebook. Put real numbers in it, from data where you can. Then choose one case where the pencil result applies exactly and show the two agree to rounding.
3. Simulation. Take your full model and set it to the degenerate case: switch off the features the pencil model lacks (one node, one period, one technology, no uncertainty, whatever applies). It must reproduce the spreadsheet. If your full model cannot be reduced that far, say which feature cannot be turned off and reduce it as far as it will go.
4. Write down what each rung showed. One sentence each: what the pencil made obvious that the simulation hides, and what the simulation revealed that the pencil got wrong or could not see. If the answer to the second is “nothing,” your full model may not be earning its cost.
If you have not built your full model yet, do rungs 1 and 2 now and write rung 3 as the check you will run when it exists. That is a pass. The hierarchy is most useful before the big model, not after.
A one-page PDF: the pencil derivation, the spreadsheet or notebook link, the degenerate-case comparison table (pencil / spreadsheet / simulation, same inputs, three outputs), and the two sentences from step 4.
If you do it, do it as one week’s arithmetic entry. Submit it to that week’s WA assignment in Canvas, the same one you would have submitted anyway, with the one-page PDF attached and, in the text box, the pencil result in one line and whether the three rungs agreed.
It does not count toward a milestone and nothing else needs submitting. It is here because it is how this lab checks a model, including one an AI wrote for you: if the full model, switched down to the simple case, does not reproduce the hand result, something is wrong, and you now know where to look.