Unit 8: PDE Modelling in Key AIMS Domains

Published

12/06/2026

A short, intuition-first tour of the reef-scale physics and biogeochemistry that powers the capstone — written for people fluent in PDEs and ML but new to reef science. The goal is to read the equations in Unit 9 not as a list of symbols but as a familiar set of mechanisms in a new domain: temperature and nutrients transported by flow, smeared by turbulence, exchanged with the atmosphere at the surface and with the deeper shelf at the bottom.

8.1 Why reef modelling? Why PDEs?

The Australian Institute of Marine Science monitors and models the Great Barrier Reef — the largest coral ecosystem on Earth, and one of the most thermally and chemically sensitive. A typical reef sits in 20–100 m of water on a shallow continental shelf, with tidal currents, river plumes, episodic storms, and a daily solar cycle all pushing on it at once. The practical questions AIMS cares about read like a list of PDE problems in disguise:

Thermal stress and coral bleaching. Corals expel their symbiotic algae when temperatures stay above \sim 30\,°\mathrm{C} for days to weeks. Predicting which reefs heat up and how deep the warm layer reaches needs a column thermal model.
Nutrient supply and water quality. Phytoplankton blooms, dissolved oxygen, and coral physiology all depend on vertical and horizontal transport of nitrate, phosphate, and particulate matter — same advection-diffusion machinery, with reaction terms layered on top.
Sediment, light, and turbidity. Suspended sediment from rivers and resuspension events attenuates the light corals need; the transport equations are identical in form to the heat equation.
Larval dispersal and connectivity. Where coral larvae land after a few-day pelagic phase governs whether a damaged reef can recover. That’s a passive-tracer transport problem on the shelf hydrodynamics.

We model these with PDEs because the physical processes — advection by currents, diffusion by turbulence, surface fluxes, radiative absorption, biological uptake — are local and continuous. PDEs are the natural language for something at each point in space changes in response to its neighbours and to external forcings. The same mathematics describes heat in a metal rod, pollutant in a river, and nitrate in a lagoon; switching to a reef site is mostly a matter of choosing which terms to keep and which to drop.

This unit is about which terms reef-scale modellers keep, and why. The capstone focuses on the thermal column problem (a single sensor near a reef, over a few weeks) but the same ideas transfer directly to nutrients and tracers — same equation, different source term and boundary conditions.

✏️ Section exercise — write the turbidity model

The section claims sediment transport is “identical in form to the heat equation”. Earn that claim: write down the 1-D column PDE for suspended-sediment concentration C(z, t) near a reef after a resuspension event. Start from the temperature equation and decide, term by term, what changes: which transport terms survive unchanged, what plays the role of the body source, what new sinking term appears (sediment falls at a settling velocity w_s), and what the natural top and bottom boundary conditions become (no flux through the surface; erosion/deposition at the bed). Then say in one sentence how light attenuation for the corals would be computed from your C(z, t).

💡 Hint

Settling is just advection at velocity w_s relative to the water — so it enters exactly like w, with a sign. The top BC is the subtle part: ‘no flux’ means the total flux (diffusive + settling) vanishes, which makes it a Robin condition. For the light question, Beer–Lambert with a C-dependent attenuation coefficient.

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8.2 The column idealisation

Reef waters are three-dimensional, with tides, wind-driven currents, river plumes, and lagoon flushing all operating at once. Modelling that in full takes a regional ocean solver. But many AIMS-style questions only require one slice:

What does temperature (or nitrate, or sediment) look like, as a function of depth, at a single mooring location, over a few weeks?

That’s the column idealisation. We treat the water column as a stack of horizontal sheets at fixed (x, y), with T = T(z, t) (or C = C(z, t) for a tracer) and nothing depending on (x, y) inside the column. Horizontal currents still exist — we don’t pretend the water is at rest — but their effect on the column is summarised through one quantity: the vertical velocity w(z, t) those horizontal flows induce via continuity.

Figure 1: Schematic of the 1-D thermistor column. Five sensors at fixed depths z_1, \ldots, z_5 sample temperature T(z_k, t) while the column evolves under vertical advection w(z, t) (green), turbulent diffusion \kappa(z, t) (purple), three surface fluxes (red — Q_{\text{SW}} penetrating, Q_{\text{LW}}, Q_{\text{lat}} + Q_{\text{sens}} non-penetrating), penetrating shortwave decaying through the water (orange), and a fixed deep reservoir at z = -H. The right-hand panels collect the governing PDE, its boundary conditions, and the externally prescribed forcings.

How the column equation falls out of 3-D

Starting from the 3-D advection–diffusion equation for a passive scalar T(\mathbf{x}, t),

\frac{\partial T}{\partial t} \;+\; \mathbf{u}\!\cdot\!\nabla T \;=\; \nabla\!\cdot\!\bigl(\kappa\,\nabla T\bigr) \;+\; \mathcal{S},

assume horizontal homogeneity (\partial_x T = \partial_y T = 0 inside the column) and split the velocity into its horizontal and vertical parts \mathbf{u} = (u_h, v_h, w). The horizontal advection u_h\,\partial_x T + v_h\,\partial_y T vanishes by the homogeneity assumption; what survives is the 1-D equation in Figure 1,

\boxed{\;\; \frac{\partial T}{\partial t} \;=\; \underbrace{-\, w(z, t)\,\frac{\partial T}{\partial z}}_{\text{vertical advection}} \;+\; \underbrace{\frac{\partial}{\partial z}\!\left(\kappa(z, t)\,\frac{\partial T}{\partial z}\right)}_{\text{Reynolds-averaged turbulent diffusion}} \;+\; \underbrace{\mathcal{S}(z, t)}_{\text{penetrating shortwave (Beer–Lambert)}} \;\;} \tag{1}

closed by the two boundary conditions

\rho\, c_p\, \kappa(0, t)\,\frac{\partial T}{\partial z}\bigg|_{z = 0} \;=\; Q_{\text{np}}(t), \qquad T(-H, t) \;=\; T_{\text{deep}}. \tag{2}

The Neumann condition at the surface is the bulk air–sea heat-flux relation: with z measured upward, \kappa\,\partial_z T is the downward diffusive heat flux entering the ocean, set equal to the non-penetrating heat input Q_{\text{np}} (positive = heat into the ocean; longwave + sensible + latent combined). The Dirichlet condition at z = -H pins the bottom to a much-slower-varying deep reservoir. \rho_0 \approx 1025\,\mathrm{kg/m^3} and c_p \approx 3990\,\mathrm{J\,kg^{-1}\,K^{-1}} are the seawater density and heat capacity — they’re the only “where do these numbers come from” constants in Equation 2.

When the column idealisation breaks

If you care about reef-front gradients (river plumes meeting clear water), lagoon-vs-open-water exchange, or near-coast bathymetric steering, the column idealisation fails because the horizontal physics matters. The capstone problem — a mooring on the shelf, away from the immediate reef break and away from a freshwater plume — is exactly the regime where it works. Outside it, you need the 2-D shallow-water solver of Unit 9 coupled to the column, or a full 3-D model.

The win is dimensionality: a 1D PDE in (z, t) instead of a 3D system in (x, y, z, t). The cost is that you have to prescribe or derive the vertical velocity rather than solve for it; the horizontal flow that drives it lives in a separate, lower-cost 2D model (the shallow water solver in Unit 9).

✏️ Section exercise — units police

Check every term of Equation 1 and Equation 2 dimensionally — the fastest way to internalise what each symbol is:

Confirm all four terms of Equation 1 have units of K/s, given [w] = \text{m/s}, [\kappa] = \text{m}^2/\text{s}.
What must the units of \mathcal{S}(z, t) be, and how does dividing Q_{\text{SW}} (W/m²) by \rho c_p deliver them? (Track the watts: \text{W} = \text{J/s}.)
In the surface BC, show both sides have units K·m/s, and compute the equilibrium surface gradient \partial_z T implied by a net cooling Q_{\text{np}} = -200\,\text{W/m}^2 through a mixed-layer \kappa = 10^{-3}\,\text{m}^2/\text{s}. Is the answer (in K per metre) big or small compared to a typical thermocline gradient of ~0.3 K/m?

💡 Hint

Track the watt: \text{W} = \text{J/s}, and \rho c_p \approx 4.09\times10^6 J m⁻³ K⁻¹ is the conversion factor between heat flux and ‘temperature flux’. Part 3 is one division: \partial_z T = Q/(\rho c_p \kappa) — then compare the number against 0.3 K/m.

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8.3 Advection and diffusion in fluids

Two transport mechanisms move heat around the column.

Advection — temperature riding along with a moving fluid

If a parcel of water is rising at speed w > 0, it carries its temperature with it. A fixed sensor at depth z then sees the temperature of the parcel that used to be at depth z - w \Delta t. That’s advection:

\left.\frac{\partial T}{\partial t}\right|_{\text{advection}} = -w\,\frac{\partial T}{\partial z}.

The minus sign is the same minus sign you see in the linear advection equation u_t + cu_x = 0 from Unit 6: rightward motion at fixed location feels like the field shifted leftward in space.

In ocean terms: upwelling (w > 0) brings cold deep water up past your sensor; downwelling (w < 0) brings warm surface water down. Either way, the parcel doesn’t change temperature on the way — advection just relocates it.

Where w(z, t) actually comes from on the GBR. Three contributions, each with its own timescale:

Tides drive a regular \sim 12.4-h vertical sloshing (the dominant semidiurnal M_2 tide, locally up to a few metres on the GBR mid-shelf). On a single column this shows up as a sinusoidal w(z, t) component that the thermistor data sees clearly but the recovered storm source must be separated from.
Wind-driven Ekman pumping during storms — the intermittent, days-long envelope our capstone treats as the unknown \tau(t) (Unit 9).
Mean shelf circulation — a slow, near-constant background rise/fall.

The capstone deliberately keeps the tidal contribution as part of the known prescribed w(z, t) (read off from a tide table plus the local barotropic mode), so the inversion focuses on the storm-driven piece. Real operational systems perform an equivalent “de-tiding” of the column data before inversion, analogous to the harmonic-fit removal Unit 1 §1.1 describes for the surface gauges.

Diffusion — turbulent stirring as a Fickian closure

Real ocean turbulence is chaotic on scales below ~1 m. We don’t resolve it. What we do notice is that, averaged over a few minutes, turbulence mixes the column the way molecular diffusion mixes a gas — gradient-down, with an effective diffusivity \kappa many orders of magnitude larger than the molecular value (\kappa_{\text{molecular}} \sim 1.4\times 10^{-7}\,\text{m}^2/\text{s} vs \kappa_{\text{turbulent}} \sim 10^{-5} to 10^{-3}\,\text{m}^2/\text{s}):

\left.\frac{\partial T}{\partial t}\right|_{\text{diffusion}} = \frac{\partial}{\partial z}\!\left(\kappa(z, t)\,\frac{\partial T}{\partial z}\right).

This is the Reynolds closure applied to temperature: split T = \bar T + T' into mean and fluctuation, average the advection equation, and identify the unresolved correlation \overline{w' T'} as -\kappa\,\partial_z \bar T by analogy with molecular diffusion. The “closure” is the choice of \kappa.

How \kappa(z, t) is actually set is the central design choice in any column model. Three standard parameterisations:

K-profile / KPP closure (Large, McWilliams & Doney 1994) — the operational default. Splits the column into a boundary layer (depth diagnosed from a bulk Richardson criterion) where \kappa has a smooth cubic profile, and an interior with a small background diffusivity plus shear-instability and internal-wave terms.
Mellor–Yamada level-2.5 (Mellor & Yamada 1982) — a two-equation second-moment closure that carries turbulent kinetic energy and a length scale as prognostic variables. Heavier and slower than KPP but more honest about transients.
Pacanowski–Philander (Pacanowski & Philander 1981) — a simple Richardson-number-dependent diffusivity. Cheap, used in many GCMs for the interior, but blind to surface-driven boundary-layer dynamics.

For the capstone we use a prescribed \kappa(z, t) — the column inherits it from a parent regional model or a simple analytical profile that mimics KPP — rather than running a closure internally. The recovery question in Unit 9 is about inferring effective \kappa from sparse mooring data, which is exactly what KPP and Mellor–Yamada try to do prognostically.

The two mechanisms compete. The Péclet number \mathrm{Pe} = wH / \kappa measures who wins on the column scale: \mathrm{Pe} \gg 1 is advection-dominated, \mathrm{Pe} \ll 1 is diffusion-dominated, and \mathrm{Pe} \sim 1 is the messy interesting case the capstone column lives in (see §8.6).

The same advection–diffusion equation governs any passive scalar: replace T with a nutrient concentration C, a sediment concentration, or a tracer dye, and the equation is unchanged. What changes is the source term — biological uptake for nutrients, gravitational sinking for sediment, photochemical decay for dyes — and the boundary conditions. The capstone focuses on the thermal case; the nutrient case has the same machinery throughout.

✏️ Section exercise — which mechanism does your sensor see?

A thought experiment with numbers. A sensor sits at z = -20 m in a 60 m column with \kappa = 10^{-4}\,\text{m}^2/\text{s} below the mixed layer and a thermocline gradient \partial_z T \approx 0.3 K/m at the sensor.

Steady upwelling switches on at w = 2 \times 10^{-5} m/s (about 1.7 m/day). What cooling rate (K/day) does the advection term produce at the sensor?
How large would the curvature \partial_{zz} T have to be for the diffusion term to match that cooling rate? Is that a plausible number for a smooth thermocline (where \partial_{zz}T \sim 0.3\,\text{K/m} / 10\,\text{m})?
Conclude: at this depth and these parameters, which mechanism must a days-long cooling of ~0.5 K/day be attributed to — and what would have to change for diffusion to compete?

💡 Hint

Each rate is a single multiplication: w\cdot\partial_z T for advection, \kappa\cdot\partial_{zz}T for diffusion — convert to K/day with ×86 400 so the numbers mean something. For part 3, remember §8.5: the κ below the mixed layer is closer to 10^{-5} than the 10^{-4} given — does that strengthen or weaken the conclusion?

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8.4 Surface energy budget

Heat enters and leaves the column through the air–sea interface, and the size of the surface flux determines whether reef corals experience thermal stress. For our purposes the budget splits into two parts:

Penetrating shortwave Q_{\text{SW}}. Sunlight passes through the surface and is absorbed inside the water, exponentially with depth — red light absorbed in the top few centimetres, blue–green light reaching tens of metres. This is a body source, not a boundary flux. The Beer–Lambert decay model splits insolation into two bands (one absorbed near the surface, one penetrating deeper):

\mathcal{S}(z, t) \;=\; \frac{Q_{\text{SW}}(t)}{\rho\,c_p}\, \Bigl[\, R\, k_1\, e^{k_1 z} \;+\; (1 - R)\, k_2\, e^{k_2 z}\, \Bigr], \qquad z \leq 0,

with R \approx 0.58 the red-band fraction, k_1 \approx 0.5\,\text{m}^{-1} (red, decays fast), k_2 \approx 1/14\,\text{m}^{-1} (blue-green, reaches the thermocline). The two-band model is the classical Paulson & Simpson (1977) parameterisation used by essentially every coastal column model.

Non-penetrating fluxes Q_{\text{np}}. Three contributions that all act at the surface and add up to a single boundary flux:

Q_{\text{np}}(t) \;=\; \underbrace{Q_{\text{LW}}(t)}_{\text{net longwave}} \;+\; \underbrace{Q_{\text{sens}}(t)}_{\text{sensible heat}} \;+\; \underbrace{Q_{\text{lat}}(t)}_{\text{latent heat (evaporation)}}.

Each term has a standard bulk formula that ties it to the near-surface atmosphere (Fairall et al. 2003 — COARE 3.0 is the reference):

Sensible: Q_{\text{sens}} = \rho_a\, c_{p,a}\, C_H\, U_{10}\,(T_a - T_s)
Latent: Q_{\text{lat}} = \rho_a\, L_v\, C_E\, U_{10}\,(q_a - q_s)
Longwave: Q_{\text{LW}} = \varepsilon\,\sigma_{\text{SB}}\,T_s^4 - Q_{\text{LW}}^{\downarrow}

with U_{10} the 10-m wind, T_a, q_a air temperature/humidity, T_s, q_s skin-temperature/saturated humidity, C_H, C_E exchange coefficients (\sim 1.2 \times 10^{-3}), and \sigma_{\text{SB}} = 5.67 \times 10^{-8}\,\text{W m}^{-2} \text{K}^{-4} the Stefan–Boltzmann constant. Cooling typically wins at night and on cloudy days; heating wins on still sunny afternoons. For our forward solves we treat Q_{\text{np}}(t) and Q_{\text{SW}}(t) as prescribed inputs — observed at the mooring or pulled from reanalysis — rather than computing them from atmospheric fields.

A useful intuition

The ocean is a heat capacitor whose plates are the air–sea interface (top) and a deep reservoir (bottom). The capacitor charges when net surface flux is positive and discharges to the deep when it isn’t. Diurnal forcing wiggles the charge state on a 24-hour timescale; storm events kick it on a 1–3 day timescale; seasonal forcing shifts it over months. Each regime lives at a different depth because the diffusive penetration depth scales as \sqrt{\kappa\, \tau}.

✏️ Section exercise — where does the sunlight actually go?

Plot the Paulson–Simpson body source. With Q_{\text{SW}} = 800\,\text{W/m}^2 (noon sun), R = 0.58, k_1 = 0.5\,\text{m}^{-1}, k_2 = 1/14\,\text{m}^{-1}, plot \rho c_p\,\mathcal{S}(z) over z \in [-50, 0] m on a log x-axis. Then integrate (analytically — the integrals are elementary exponentials) to answer:

What fraction of the incoming shortwave is absorbed in the top 1 m? The top 10 m?
What fraction passes below 30 m — the typical thermocline depth?
The capstone’s single-band simplification uses \zeta = 10 m. Compare its below-30 m fraction with the two-band model’s. In which depth range do the two models disagree most, and would a thermistor at -40 m care?

💡 Hint

No numerical integration needed: the fraction of light still travelling downward past depth z is F(z) = R\,e^{k_1 z} + (1-R)\,e^{k_2 z}, so ‘absorbed above z’ is 1 - F(z). Compare the two models at z = -1, -10, -30 m and the answer to the design question falls out.

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8.5 Stratification, mixing, and the deep boundary

Ocean temperature usually decreases with depth — warm sun-heated water on top, cold deep water below, with a sharp transition called the thermocline in between. This is stable stratification: buoyancy resists mixing because warm parcels want to rise.

The eddy diffusivity \kappa therefore isn’t constant. In the mixed layer (above the thermocline, vigorously stirred by wind and breaking surface waves), \kappa is large — \mathcal{O}(10^{-3}) m²/s. Below the thermocline, stratification suppresses mixing and \kappa collapses to a background \mathcal{O}(10^{-5}) m²/s. The transition between the two is parametrised by a Richardson number \mathrm{Ri} = N^2 / (\partial_z U)^2 comparing buoyancy frequency N to shear \partial_z U — stable when \mathrm{Ri} > 1/4.

At the bottom of the column we need another boundary condition. We pick a simple one — a fixed deep reservoir T(-H, t) = T_{\text{deep}} — and treat it as exact. Physically: below the modelled column lies a much larger volume of cold deep water whose temperature responds on timescales irrelevant to ours. Dirichlet is a fair model; zero-flux (an insulating floor) is an alternative if the deep ocean beneath you is essentially a wall.

✏️ Section exercise — find the mixing barrier

Compute the stratification profile the capstone’s initial condition implies. Using T_0(z) from Unit 9 §9.4 (the \tanh thermocline: T_{\text{surface}} = 28, T_{\text{deep}} = 18, z_t = -30 m, \delta_t = 5 m) and the buoyancy frequency N^2 = \alpha g\,\partial_z T with \alpha = 2\times10^{-4}/K:

Derive \partial_z T_0(z) in closed form and find where N^2 peaks and its peak value.
With a background shear \partial_z U = 0.01\,\text{s}^{-1}, plot the Richardson number \mathrm{Ri}(z) and mark the region where \mathrm{Ri} > 1/4.
Plot closure 3 from Unit 9 §9.5, \kappa = \kappa_b + \kappa_0/(1 + 5\mathrm{Ri})^2, through the column. By what factor is mixing suppressed at the thermocline relative to the surface — and what does that mean for how fast a surface heat anomaly can reach the deep sensors?

💡 Hint

Differentiate the tanh profile: \partial_z \tanh((z-z_t)/\delta) = \mathrm{sech}^2(\cdot)/\delta. The Ri > 1/4 boundary solves $^2(x) = $ threshold, i.e. x = \operatorname{acosh}(1/\sqrt{\text{thr}}). For the consequence, compare crossing times L^2/\kappa through the suppressed band vs the mixed layer.

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8.6 Dimensionless regimes

A few dimensionless groups carry most of the physics. They tell you which terms dominate in a given problem before you solve anything.

Group	Definition	Interpretation
Péclet \mathrm{Pe}	wH/\kappa	advection vs. diffusion across the column
Diffusive time T_\kappa	H^2/\kappa	time for diffusion to mix the column
Richardson \mathrm{Ri}	N^2/(\partial_z U)^2	stability of stratification against mixing
Diurnal-to-diffusive	\tau_d / T_\kappa	how localised the daily cycle stays in depth
Storm-to-diffusive	\tau_{\text{storm}} / T_\kappa	how deep a storm signal penetrates

For the canonical mid-shelf column (Davies Reef in §9.1) with H = 100 m, \kappa_m = 10^{-3} m²/s, w_0 = 10^{-5} m/s:

\mathrm{Pe} \approx 1 — balanced regime; both transport mechanisms compete on equal footing.
T_\kappa \approx 116 days — a 30-day run only mixes a fraction of the column.
\tau_d / T_\kappa \sim 10^{-2} — the diurnal cycle is much faster than bulk mixing, so it lives in a thin near-surface layer.

These three numbers explain almost every qualitative behaviour you’ll see in the toy tasks in Unit 9 §9.11: why the diurnal warm layer is thin, why a 3-day storm leaves a fingerprint deeper than naive cooling would suggest, why SST cools slowly under upwelling, and why the steady state is reached only over many months.

✏️ Section exercise — three sites, three regimes, by the numbers

Do the §8.6 arithmetic for all three capstone sites (Unit 9 §9.1). Using each site’s depth (H_A = 15 m, H_B = 60 m, H_C = 100 m), with \kappa_m = 10^{-3}\,\text{m}^2/\text{s} and w_0 = 10^{-5}\,\text{m/s} for all three, compute per site: \mathrm{Pe} = w_0 H / \kappa_m, the diffusive time T_\kappa = H^2/\kappa_m in days, and the diffusive penetration depth \sqrt{\kappa_m \tau} for a 3-day storm. Check your numbers against the regime labels in the Unit 9 site table — do you reproduce “\mathrm{Pe} \ll 1 / \sim 1 / \gg 1”? One of the three labels deserves a caveat; identify which and explain (hint: how much of Site C’s column does a 30-day run actually mix?).

💡 Hint

Three formulas, three sites: \mathrm{Pe} = w_0H/\kappa, T_\kappa = H^2/\kappa, \sqrt{\kappa\tau} — a six-entry table you can fill in two minutes. The caveat hides in which κ you use: redo Site C’s Pe with the interior \kappa_b = 10^{-5} and watch the label change.

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8.7 Where this kind of column modelling appears in the literature

The 1-D thermistor-column problem of Figure 1 isn’t a workshop invention — it’s the standard way physical oceanographers reason about a single mooring’s worth of data. A few entry points:

Mixed-layer / column foundations. Price, Weller & Pinkel 1986 (“Diurnal cycling: Observations and models of the upper ocean response to diurnal heating, cooling, and wind mixing”) is the canonical paper for upper-ocean column response to surface forcing — the diurnal-warm-layer paragraph in §8.6 is a direct paraphrase. The KPP paper of Large, McWilliams & Doney 1994 is the closure most operational column models use today.
Great Barrier Reef specifically. Wolanski & Pickard 1985 and the long line of subsequent AIMS / James Cook University papers (e.g. Lambrechts et al. 2008; Hancock & Hughes 2017) describe the GBR’s lagoonal-vs-shelf thermal regimes — the observational backdrop the capstone synthesises.
Coral thermal stress as a column-modelling problem. Liu et al. 2014 (“Reef-scale thermal stress monitoring of coral ecosystems”) and the NOAA Coral Reef Watch operational product use SST and Degree-Heating-Weeks — proxies that miss the vertical distribution the column model resolves. The DHW shortcoming is the operational motivation for column modelling at AIMS.
Inverse problems on column data. Kunze 1985 (“Near-inertial wave propagation in geostrophic shear”) is the classical “what does this mooring tell me?” inverse-problem paper for internal-wave kinematics; modern variational data-assimilation work (e.g. ROMS 4D-Var at coastal scale) does the same thing at GCM cost.
PINN-style ocean-column work specifically. There’s a small but growing literature on physics-informed networks for ocean mixed-layer dynamics — e.g. column wind-mixing recovery (Bolton & Zanna 2019 is the precursor on subgrid closure; closer-to-PINN work has appeared in Ocean Modelling and JAMES through 2024–25). The AIMS-side AI4DA initiative at CSIRO is doing exactly this kind of column-model inverse problem with PINN-flavoured tooling.

For the workshop, the column model in Equation 1 is deliberately simple — KPP-style \kappa, two-band penetrating shortwave, bulk surface fluxes prescribed — so the focus stays on the inverse-problem mechanics rather than the closure menagerie. A realistic operational tool would substitute KPP for the prescribed \kappa(z, t) and replace prescribed w(z, t) with a coupled regional ocean model (ROMS / SHOC / SLIM / MITgcm-Coastal).

✏️ Section exercise — place the capstone in the literature

A matching drill that doubles as a reading list. For each capstone design choice below, name the literature entry above that justifies it (paper or product), and the operational shortcoming the capstone’s approach addresses:

The shape of the prescribed mixed-layer \kappa(z, t) profile.
The claim that a diurnal warm layer forms and erodes daily in the top few metres.
Using a vertical column of temperatures instead of satellite SST + Degree-Heating-Weeks for bleaching risk.
The whole enterprise of recovering an unknown driver from a single mooring’s record.

Then the honest-framing question: which item in the §8.7 list is closest to the capstone’s actual method, and what does the capstone simplify away relative to it?

💡 Hint

Match by mechanism, not author: closure shape → the KPP paper; diurnal layer → the 1986 observational classic; bleaching proxies → the Coral Reef Watch line; mooring inversion → the 1985 ‘what does this mooring tell me’ paper. The last question wants the §8.7 bullet that mentions PINN-flavoured tooling.

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8.8 From concepts to the capstone column

The capstone in Unit 9 is the column idealisation written down concretely:

One PDE: \partial_t T + w \partial_z T = \partial_z(\kappa \partial_z T) + \mathcal{S}.
Two boundary conditions: Neumann at the surface, Dirichlet at the deep reservoir.
Four prescribed forcings: w(z, t) (from a 2D shallow-water solver), \kappa(z, t) (closure), Q_{\text{np}}(t) (non-penetrating fluxes), Q_{\text{SW}}(t) (penetrating shortwave, sourced via Beer–Lambert).
A reference parameter set tuned for a tropical-reef site.

Everything else — the toy tasks, the storm scenario, the inverse problem — is choosing which parts of that setup to perturb. The “science” sits in §§8.2–8.6 above; the implementation sits in Unit 10.

✏️ Section exercise — reconstruct the spec from memory

Close this page (really). On a blank sheet, write down the complete capstone column problem: the PDE with all four terms, both boundary conditions, the initial condition, and the four prescribed forcings with their roles. Then reopen Unit 9 and grade yourself against §§9.3–9.5. The two items most commonly forgotten: where the penetrating shortwave enters (body source, not the surface BC), and which BC type sits at which end of the column. If you dropped either, reread §8.4 — that distinction is load-bearing for the whole inverse problem.

💡 Hint

Structure your sheet as: 4 PDE terms, 2 BCs (different types at the two ends), 1 IC, 4 forcings. The two classic mistakes are advertised in the exercise itself — so make sure your sheet takes an explicit position on where Q_{SW} enters and which BC type sits where.

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