When the Lake Rings — Reading the Results

Figure 5.1 · Observed Ensemble

Fifty-One Storms Walk Into a Bar

Let's start with the most honest thing this thesis does: it looks at fifty-one real storms, one by one, and asks what happens to the lake underneath each one. Not just the average. All of them, together, on the same plot. The result is what scientists lovingly call a spaghetti plot — and it earns the name.

Figure 5.1 — Observed Near-Inertial Velocity, 51 Storm Events

Figure 5.1: Observed spaghetti plot of 51 near-inertial events at LLO Western Mooring

Each gray line is one storm's current at 10 m depth, phase-aligned so every storm begins pointing east. The black line is the average of all 51. The red lines mark ±1 standard deviation. Time runs left to right in units of the inertial period T_f ≈ 16.8 hours.

Every gray line starts somewhere between −0.6 and +0.6 metres per second. At time zero, they have all been phase-aligned — rotated mathematically so that every storm begins with the current pointing east. Think of it as making all fifty-one pendulum swings start from the same spot, even though in real life they started in all sorts of directions. That alignment is the trick that lets us average them meaningfully.

The chorus analogy. Imagine fifty-one singers, each humming the same note but starting at a different moment in the phrase. If you hit play on all of them simultaneously and rotate each recording so the phrase starts together — that's phase alignment. Now you can actually hear the melody instead of chaos.

The plot reveals three distinct acts, visible right there in the data before any theory is applied.

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Act I: The Ringing

t < 5 T_f (~3.5 days)

All 51 currents move together. The lake behaves like a single bell just struck. The spread is small.

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Act II: The Unravelling

5 – 15 T_f

The average fades. Individual storms diverge, each following its own subsequent wind. The spaghetti spreads.

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Act III: The Hum

t > 20 T_f (~14 days)

The average reaches zero. But the spread stabilises — the lake keeps oscillating at a steady background level, forever.

Two numbers come directly out of this picture with no theory at all: the damping timescale r⁻¹ ≈ 7 days, read from how fast the black line decays, and the quasi-stationary standard deviation σ∞ ≈ 0.17 m/s, read from how wide the red lines settle. These two numbers will do almost everything in the rest of the thesis.

Key observational result

The lake's memory of any one storm lasts about seven days. After that, it settles into a permanent, random background oscillation whose typical speed is 0.17 metres per second. That background hum never disappears as long as wind keeps blowing.

Figures 5.2 & 5.3 · Model Validation

Can a Random Number Generator Predict an Ocean?

Here is where the audacity of the whole project becomes visible. We took two numbers from the data — how energetically the wind fluctuates (Q_τ), and how fast the lake forgets (r) — fed them into a model that uses actual random numbers to simulate wind, and asked: does it reproduce the fifty-one storm ensemble without cheating?

The dice game analogy. You've never seen a casino, but someone tells you two things: how often a die lands on six, and how much the house takes from each bet. From just those two numbers, you can predict exactly how much money the average gambler will have after an hour — and how spread-out those amounts will be. No casino floor required. Same idea here: two measurable wind statistics, and the physics gives you everything else.

Figure 5.2 — 500 Simulated Storms

Figure 5.2: Simulated 500-member ensemble

Five hundred independent simulated currents, each driven by a different randomly generated wind sequence but using the same two parameters. The spread pattern, decay rate, and long-run plateau all match the observations without any tuning.

Figure 5.3 — Theory vs. 500 Simulations

Figure 5.3: Analytical predictions vs Monte Carlo statistics

The dashed lines are closed-form predictions from Itô calculus — no simulation at all, just algebra. The solid lines are the actual average and spread from 500 computer runs. Correlation ρ = 0.999. They are, for all practical purposes, identical.

Figure 5.3 is the most important validation plot. It shows the analytical prediction — derived purely from equations on a page — sitting exactly on top of simulation results. The correlation is 0.999. The error is about 3 millimetres per second. That is not a coincidence; it is what happens when the physics is right.

What this means

The model was never shown the current data during calibration. The wind parameter came from the wind record. The damping came from the decay curve. Then the model looked at the currents cold — and matched them to four significant figures. That is a prediction, not a fit.

Figure 5.4 · Amplitude & Phase Dynamics

The Clock That Won't Lose Its Mind

A rotating current has two properties, the same two a clock hand has: how far it reaches (amplitude) and which direction it points (phase). The model tracks both, and the phase result is the surprising one.

The spinning top analogy. When you spin a top, its spin gradually slows (amplitude decays). And its axis slowly wanders — it precesses, tracing a wobbly cone. Does that wobble grow without bound until the top falls over? Or does something keep it from going completely crazy? For Lake Superior's currents, something does.

Figure 5.4 — Amplitude A(t) and Phase θ(t)

Figure 5.4: Back-rotated amplitude and phase for 500-member ensemble

Top panel: amplitude of the oscillation, decaying from 0.25 m/s toward the plateau at σ∞ = 0.168 m/s. Bottom panel: the direction of oscillation across 500 simulations. Phase spread grows but then freezes at about ±70° after roughly 15 inertial periods — bounded phase diffusion under linear damping.

The phase spread grows for a while, then stops growing. It saturates at about ±70°. The reason is the same friction that damps the amplitude: the lake forgets old wind events after about seven days, which means it also forgets old directional nudges. Randomness cannot accumulate indefinitely because friction keeps erasing it.

The physical insight

Damping doesn't just slow the oscillation down. It caps how confused the direction can get. This is why Green et al. (2025) found that the lake's current direction drifted by only a few hours over 23 days of continuous random wind. The lake has its own internal compass, maintained by friction and Earth's rotation.

Figure 5.5 · Predictability Horizon

The Four-Day Wall

One of the cleanest results in the thesis, with a clean honest number: 4.1 days. That is how long useful information about a storm-excited current persists before random wind noise buries it completely.

The shouting across a crowded room analogy. Someone shouts your name across a noisy party. Right after the shout, you can hear it clearly — signal beats noise. But the noise keeps accumulating while the sound of the shout fades. There is a specific moment when background noise becomes louder than the echo. After that, no amount of careful listening helps. For Lake Superior currents, that moment arrives at t* ≈ 4.1 days.

Figure 5.5 — Predictability Horizon

Figure 5.5: Predictability crossing time t* = 4.1 days

Blue line: the decaying ensemble mean — the "signal" of the initial storm. Orange dashed: the growing uncertainty ±σ(t). Where they cross is t* ≈ 4.1 days. Before that crossing, knowing the initial current is useful. After it, the best forecast is: "somewhere around ±0.17 m/s, direction uncertain."

The formula for this crossing time is:

t* = (1/2r) · ln(1 + u₀²/σ∞²)

Doubling the storm's initial strength only adds about 2.4 more days. The predictability horizon grows only logarithmically with storm intensity — you cannot buy much extra forecast skill just by having a stronger storm. The lake's forgetting rate dominates everything.

Practical meaning

For anyone forecasting lake currents — for water quality, ship navigation, or sediment transport — four days is the honest limit. Beyond that, the distribution is your best forecast. Not the specific trajectory. And the distribution, as the next section shows, can be predicted perfectly well.

Figure 5.6 · Wind Forcing Statistics

Why the Wind Talks in Bell Curves (Once You Listen Carefully)

The whole model rests on one foundational assumption: that the random kicks the wind gives the lake follow a Gaussian (bell-curve) distribution. This has to be checked. And checking it requires fifteen years of ten-minute wind records from Buoy 45006.

Figure 5.6 — Wind Speed vs. Wind Stress Increments

Figure 5.6: Histograms of wind speed and wind stress increments

Top panel: how much wind speed changes every ten minutes. Sharp peak, fat tails — a Laplace distribution, where extreme gusts happen far more often than a bell curve predicts. Bottom panel: how much wind stress changes every ten minutes. Now the distribution fits a clean bell curve. Excess kurtosis ≈ 0. The model assumption holds.

The speed camera analogy. Measure how fast cars drive past a camera. The distribution is skewed — a few people go wildly fast. Now measure the force the car exerts on the road instead. The squaring and averaging involved compresses the distribution toward a bell curve. Something similar happens with wind: speed has fat tails, but the force the wind exerts — wind stress — is computed by squaring the speed, compressing those fat tails back toward Gaussian.

The observational justification

The Gaussian wind model is not an assumption imposed on the data. It is what the data say when you look at the right variable. Wind stress increments at Buoy 45006 are empirically bell-shaped. The entire analytical framework depends on this, and the data deliver it.

Figure 5.7 · Extreme Event Return Periods

How Rare Is Dangerous? And Why the Naïve Answer Is Wrong by a Factor of Thirteen

How often does a near-inertial current in Lake Superior reach a speed dangerous enough to stir phosphorus-rich sediment off the lake floor? The answer has ecological consequences — storms that resuspend those sediments are effectively fertilising the lake, triggering algal responses that ripple through the food web for weeks.

Figure 5.7 — Return Period of Extreme Near-Inertial Velocities

Figure 5.7: Return period plot showing episodic correction

x-axis: current speed threshold. y-axis: average waiting time between events. Blue dashed: Gaussian model assuming wind never stops. Orange dotted: Laplace model, also continuous. Blue solid: Gaussian corrected for episodic storms (÷D). Black circles: what 15 years of observations actually show. The corrected model sits inside every error bar.

The broken vending machine analogy. A vending machine gives a candy bar once every 100 presses on average. Pressing continuously — 24 hours a day — you would expect one every day or two. But what if the machine is only turned on 7.5% of the time? The expected wait just got thirteen times longer. Not because the machine changed — because you forgot it was off most of the time.

Lake Superior storms work exactly like that. They arrive roughly once every 45 days and last about 3.4 days each. Active storm forcing occupies only 7.5% of the season. The corrected return period:

T^epis = T^cont / D

Speed threshold	Continuous Gaussian	Corrected (÷D)	Observed
0.30 m/s	9 days	~120 days	15 ± 3 days
0.40 m/s	40 days	~533 days	150 ± 40 days
0.50 m/s	232 days	~3,100 days (8.5 yr)	>5,479 days (never seen)

The central result

A current fast enough to seriously disturb the lake floor happens roughly once per eight to nine years, not once per eight months. The correction factor of thirteen is universal — it doesn't depend on speed threshold, damping rate, or noise level. Only on D, the storm duty cycle, which any wind record can tell you. This has never been calculated before for near-inertial oscillations in any enclosed basin.

Figure 5.8 · Annual Wind Variability

Is the Lake Getting Windier? (The Honest Answer)

Every result so far treated the wind as a constant statistical engine — same Q_τ every year. That deserves testing. Is Lake Superior's wind forcing actually the same from year to year?

Figure 5.8 — Annual Q Estimates, 2005–2019

Figure 5.8: Annual wind-stress spectral density estimates

Each orange dot is the wind energy spectral density for one stratified season, with error bars. The dotted horizontal line is the 15-year average. The dashed line is a linear trend. Year-to-year scatter is ten to twenty times larger than the measurement error — the variability is real, not noise. The trend is slight and statistically weak (r² = 0.04).

The coffee shop analogy. Count customers entering a coffee shop each day for three years. Some days are quiet, some are slammed. If the day-to-day scatter is ten times larger than your counting error, the variation is real — not you miscounting. Now fit a trend: is there one steadily more customer per week, year over year? That is exactly what this plot is asking about wind energy.

The answer: yes, Q_τ varies meaningfully — by as much as ±30% around the long-run average. This is real inter-annual variability driven by shifts in Great Lakes storm tracks. Some seasons are energetically wilder than others.

What this means for the future

As climate change shifts storm tracks, Q_τ will shift too. Because the return period formulas depend explicitly on Q_τ, any change in wind energy translates directly — and predictably — into a changed extreme event frequency. No new theory required. Update Q_τ from the latest wind records, plug it in, the corrected return period updates automatically. The framework is designed to age gracefully.