Could the Deep Ocean Be Warming Because Its Layers Are Moving?

The deep ocean warming problem has a complication…water moves.

If I measure temperature at one fixed depth and it gets warmer, that doesn’t necessarily mean the water sitting there gained that heat locally.

Whole layers of water can rise and sink. If a warmer layer moves into the depth I’m measuring, my thermometer records warming even if that layer itself has not warmed very much.

That’s the idea behind isopycnal heave. And, after testing it against the observed abyssal warming acceleration, heave officially has my attention. Where I can track the density layers reliably, heave reproduces a surprisingly large part of the observed pattern.

There is an important caveat: the historical abyssal observations are sparse enough that I can only make that calculation robustly for about 21.7% of the total positive warming signal in the 4,000 - 6,000 dbar (4 -6 kdbar) range.

So the result is: Heave is plausible where testable, although I can’t yet say that it explains the dominant global abyssal warming signal.

First: what is isopycnal heave?

The ocean isn’t just one homogeneous tank of water.

Temperature and salinity together determine seawater density, so the ocean forms layers with different densities.

Oceanographers can follow surfaces of equal density called isopycnals.

For this test, I used sigma-4, which describes seawater density referenced to a pressure of 4 kdbar. That makes it useful specifically in the deep and abyssal ocean.

I like the stacked blankets on a bed analogy. If you have a fixed horizontal line through a stack of blankets on a bed, and the bed moved, the blanket crossing the line at any given time changes.

Nothing happened to the blanket itself, the stack just moved.

The same thing can happen at a fixed depth in the ocean. If a warmer density layer moves downward through 4,500 meters, for example, a measurement at that depth can get warmer just because different water now occupies that depth.

That’s the mechanism I wanted to quantify.

The observational data

I used the same validated observational dataset behind my reproduction of the Johnson [1 abyssal warming calculation.

The profile dataset combines:

• shipboard CTD observations from the World Ocean Database;

• Deep Argo profiles;

• Conservative Temperature;

• Absolute Salinity;

• profile date and geographic position;

• Johnson's geographic bins;

• and the pressure levels needed to follow the deep water column.

The combined validated profile cache contains 1,124,918 profiles. Deep Argo is already included, so the observational limitation I eventually hit isn’t because the modern deep float data were omitted.

For the warming field itself, I used the fixed pressure Conservative Temperature acceleration that I had already reproduced from the Johnson workflow. Johnson's calculation fits individual profiles as a quadratic function of time, centered on 2005, and defines acceleration as twice the quadratic coefficient.

The main heave test focused on 4-6 kdbar.

Step 1: Calculate the density of the water

For every usable temperature-salinity observation, I calculated sigma-4 using TEOS-10:

σ₄ = σ₄(SA, CT)

where:

• SA is Absolute Salinity;

• CT is Conservative Temperature.

This gives us a way to ask…what is happening to this same density layer through time?

That difference is the core of the test.

Step 2: Define where each density layer was in 2005

For each geographic bin and each target pressure between 4-6 kdbar, I constructed a reference temperature and salinity profile for 2005.

From that profile I calculated the sigma-4 value occupying each target pressure.

So if a particular density surface sat near 4,500 dbar in 2005, I could then search later and earlier profiles for that same density surface and see where it moved.

Step 3: Track the density surface through time

For every profile with enough temperature and salinity coverage, I found the pressure where that particular sigma-4 surface crossed the profile.

That gave me a time series:

p_σ(t)

which means: Where was this density surface at this time?

I then fitted its pressure through time with a quadratic model:

p_σ(τ) = c₀ + c₁τ + c₂τ²

with:

τ = year − 2005

The density-surface pressure acceleration is:

a_pσ = 2c₂

That tells me whether the layer's vertical movement itself is changing through time.

Step 4: Calculate how much temperature change that movement should produce

This is where the heave test becomes physical instead of just statistical.

Suppose a density surface moves downward by some amount:

Δp_σ(t) = p_σ(t) − p_σ(2005)

I used the actual 2005 vertical Conservative Temperature profile to ask:

If that layer moved by this amount, what temperature change should appear at the original fixed pressure?

The finite-displacement calculation was:

CT_heave(p₀, t) = CT_ref(p₀ − Δp_σ(t)) − CT_ref(p₀)

Then I fitted that predicted heave temperature through time with the same quadratic form:

CT_heave(τ) = h₀ + h₁τ + h₂τ²

giving:

a_heave = 2h₂

Now I had two directly comparable quantities:

Observed warming acceleration at fixed pressure

versus

Warming acceleration predicted purely from vertical movement of the density surfaces.

Step 5: Follow the density layer itself

I also performed the complementary test.

Instead of measuring CT at a fixed pressure, I measured CT while following the same sigma-4 surface.

Again I fitted:

CT_σ4(τ) = d₀ + d₁τ + d₂τ²

with:

a_CT,σ4 = 2d₂

If the entire fixed-depth warming signal were produced by vertical heave, then following the density surface should remove most of that apparent warming.

If CT is still increasing while I follow the same density layer, something besides simple vertical heave is happening.

That remaining signal could involve changes in water-mass properties, along-isopycnal transport, overturning circulation, or another mechanism. It shouldn’t automatically be interpreted as local heat being generated there. The stability audit explicitly preserves that distinction.

Then I tried to break the result

The first version of this calculation looked encouraging, but some predicted heave values were unrealistically large.

So, I ran a separate stability audit. For a density-surface fit to survive, it needed things like:

• enough repeated profiles to support a quadratic fit;

• real residual degrees of freedom;

• adequate time coverage;

• a well-conditioned numerical fit;

• a modeled 2005 density surface reasonably close to where the reference profile said it should be;

• a trajectory that stayed inside the measured pressure range;

• no single observation dominating the result;

• stability when individual profiles were removed one at a time;

• and no physically extreme heave prediction.

The stable refits used centered/scaled least-squares calculations instead of relying on poorly conditioned raw polynomial fits. That changed the result.

What survived the stability audit

On the final stable common support in the 4-6 kdbar range:

Heave captured about 73.3% of the observed positive warming.

The volume-weighted relationship between predicted heave acceleration and observed acceleration had:

• weighted Pearson correlation: 0.69;

• weighted Spearman correlation: 0.65;

• regression slope: 0.95;

• sign agreement: 74.5%;

• difference between the observed and predicted positive-depth centroids: only about 48 dbar.

A slope near 1 is particularly interesting: over the stable subset, the magnitude of the heave prediction is in approximately the right range. Those numbers are why I think heave is physically plausible where I can actually test it.

There are still mismatches. The stable heave field retains substantial overshoot in some locations, and roughly a quarter of the supported signal has the wrong sign. This isn’t a perfect reconstruction.

Hitting the observational wall

The strict stability filtering left only 21.7% of the total positive, volume-weighted 4-6 kdbar warming signal on common support.

This means that among the portion of the observed positive warming signal where I have enough repeated observations to track the density surfaces robustly, heave reproduces about 73% of the positive warming.

I ran one more audit specifically to find out why so much of the signal disappeared.

The largest loss was insufficient observations. Overall, the missing signal split into approximately:

• 41.7% of the total positive signal lost to observational limitations;

• 36.5% lost to numerical/stability requirements;

• 0% attributed to a purely arbitrary analysis-choice category.

I tested whether relaxing one stability criterion could rescue enough data to justify another correction, and it couldn’t.

Even dropping the largest adjustable gate (high leverage) would only raise common support from about 21.7% to about 25.6%, and most of the apparently recovered signal would still fail another independent stability check.

Deep Argo helps, but the problem is historical coverage. A quadratic heave calculation needs repeated observations through time. Much of the deepest ocean doesn’t have enough old, repeated, full-depth profiles in the same places to track those density surfaces robustly.

But how much of the abyss are we actually observing?

That 21.7% number raised another question.

It tells me how much of the positive, volume-weighted acceleration signal falls on stable heave support.

But what fraction of the actual 4-6 kdbar ocean does the observed signal represent? I went back and calculated the physical-volume denominators.

Across the entire physically existing 4-6 kdbar ocean represented by the Johnson grid:

• 17.36% of the total volume has a finite observed acceleration estimate.

• 10.10% of the total physical volume has positive temperature acceleration.

• Within the volume where acceleration can actually be estimated, 58.14% has positive acceleration.

Then, I asked how much of that positively accelerating volume is covered by the stable heave calculation.

The answer was 22.61% by physical volume.

That’s interesting because the earlier signal-weighted result was 21.70%.

In other words, the stable-heave subset contains about 22.6% of the positively accelerating observed volume and 21.7% of the magnitude-weighted positive acceleration signal. Those two numbers are very close.

Although the observations still vary with geography and depth, it makes it less likely that the heave result is being driven by a tiny collection of unusually strong hot spots.

The stable cells look much more like a real slice of the measured field.

Positive acceleration is different than warming

There was one more check I needed to make. A positive temperature acceleration means the temperature trend is becoming more positive through time.

It doesn’t automatically mean the water is already warming at the reference year.

The quadratic model I reproduced is:

CT(τ) = β₀ + β₁τ + β₂τ²

with:

τ = year − 2005

The instantaneous temperature trend at 2005 is therefore β₁, while the acceleration is:

a_CT = 2β₂

I separated every directly observed 4-6 kdbar cell into four groups according to the sign of its temperature trend at 2005 and the sign of its acceleration.

The result was:

• 43.02%: warming + accelerating

• 15.12%: cooling + accelerating

• 29.57%: warming + decelerating

• 12.29%: cooling + decelerating

That means 72.59% of the directly observed abyssal volume was warming at the 2005 reference year.

And 58.14% had positive temperature acceleration.

Most importantly, those two effects overlap substantially. Among the observed volume that was already warming, 59.27% was also accelerating. Looking at it from the other direction, among the volume with positive acceleration, 74.00% was already warming.

So, we now know: Within the directly observed 4-6 kdbar domain, more than half of the volume that was warming at 2005 was also accelerating in that warming. The simultaneously warming-and-accelerating category itself accounts for 43.02% of the observed volume, or about 7.47% of the entire physical 4-6 kdbar ocean volume.

This shows the behavior is common within the part we can actually measure.

Can we tell when the acceleration began?

If a large part of the observed abyss is both warming and accelerating, when did the acceleration change? One proposed date form ECDO theory [2] I wanted to test was 1973.

I now had a methodological problem. The quadratic model I’m currently using has constant acceleration. It can tell me that curvature exists, but it can’t tell me that the acceleration suddenly changed in a particular year.

To test a year like 1973, I would need a separate changepoint model. Before fitting one, I checked whether the observations were even capable of supporting it.

Guys, we don’t have enough data :(

In the 4-6 kdbar layer, only 1.95% of the presently observed volume has any pre-1973 observations at all. Even this tiny fraction that brackets 1973 has a large hole in the record. The volume-weighted median last pre-1973 observation falls in late 1972, while the median first observation afterward isn’t until roughly 1983. The median gap across the proposed transition is about 10.3 years.

So… I didn’t fit the changepoint. The existing abyssal observations cannot currently test the 1973 hypothesis at all.

This part of the project has been a useful reminder that there are two very different ways an idea can fail. One is: The observations contradict it. The other is: The observations aren’t good enough to test it yet.

The 1973 question currently falls into the second category.

The historical limitation also helps explain why the heave calculation lost so much support in the first place. Deep Argo gives us excellent recent measurements, but it can’t travel backward in time and provide the repeated abyssal observations that were never collected decades ago.

For the abyss, the limiting resource is increasingly clear: history.

We have much better measurements now than we had fifty years ago. But testing long-term changes in the deepest ocean requires measurements on both ends of that timeline.

Conclusions

This is what we’ve learned from investigating this mechanism:

• The observed abyssal acceleration isn’t confined to a few isolated cells. Within the directly observed 4–6 kdbar volume, positive acceleration occurs across 58.1% of that volume.

• Most of the observed abyss was already warming at the 2005 reference year. About 72.6% had a positive temperature trend.

• Warming and acceleration commonly occur together. About 43.0% of the observed volume is simultaneously warming and accelerating, and 59.3% of the warming volume is also accelerating.

• The stable heave subset does not consist of only the strongest acceleration hot spots. It represents 22.6% of the positively accelerating volume and 21.7% of the volume-weighted positive acceleration signal.

• We can’t yet determine if an inflection point exists around 1973 because the historical record is too sparse. That remains an untested timing hypothesis, not a rejected one.

So did heave explain the warming?

Here’s where I land: Heave is plausible where testable.

The signal itself looks increasingly difficult to dismiss as a handful of ephemeral measurements. What I still don’t know is how much of that signal comes from vertical movement of density layers, how much comes from changing water-mass properties and circulation, and whether another source contributes alongside them.

That brings me back to the next physical question:

Can changing Antarctic Bottom Water and deep-ocean overturning, coupled with heave, explain more of the signal that remains?

References

[1] Johnson, G. C. (2026). Observed multi-decadal acceleration of globally averaged abyssal ocean warming. Geophysical Research Letters, 53(14), e2026GL124104. https://doi.org/10.1029/2026GL124104

[2] R. B. Cunningham, Inversion: ECDO Theory-The Hidden Mechanism Driving Cataclysm, Cultural Tradition, and Climate. Atlanta, GA: Meridian Standard Press, 2026.