Global Stratification Trends Diagnosed Using the Center of Mass of the Ocean

Fabien Roquet; Gustav Holmquist; Jakob Gunnarsson

doi:10.16993/tellus.4121

Global Stratification Trends Diagnosed Using the Center of Mass of the Ocean

Tellus

Volume 79 (2026): Issue 1

By: Fabien Roquet , Gustav Holmquist and Jakob Gunnarsson

Open Access

|Apr 2026

Figures & Tables

Spatial distribution of the time-mean **(a)** *D_COM* (in m), and **(b)** mean-N² (in $s^{- 2}$ ) computed for the upper 2000 m of each horizontal grid points at location deeper than 2000 m. Estimated using the ECCOv4r4 for the period 1992–2017 (see Method section). The thick black contour corresponds to the corresponding global mean value, highlighting important local differences between the spatial distributions of the two indices. Regions shallower than 2000 m were masked (white).

Table 1

Short correspondence table for a given control volume (V in z-coordinates, $V_{p}$ in p-coordinates). In all cases, $D_{COM} = \frac{1}{g} ⟨ Π_{0} - Π ⟩$ (with the appropriate averaging operator). Notation: S is Absolute Salinity, $Θ$ is Conservative Temperature, $ρ$ is in-situ density, $ρ_{0}$ is a constant reference density, $ν = 1 / ρ$ is specific volume, $σ_{θ}$ is potential density anomaly (referenced to the surface), $ϕ$ is PEA, b is buoyancy defined as $b = (ρ_{0} - ρ) g / ρ_{0}$ , z is height (positive upward) and p is pressure; $z_{v} = ⟨ z ⟩$ and $p_{v} = {⟨ p ⟩}_{p}$ are the corresponding reference levels; angle brackets $⟨ \cdot ⟩$ denote a volume average over V, and ${⟨ \cdot ⟩}_{p}$ a pressure-coordinate volume average over $V_{p}$ ; $h_{d}^{b, z_{v}}$ and $h_{d}^{p_{v}}$ are dynamic enthalpies as defined in the table, and the subscript 0 denotes the fully mixed reference state obtained by replacing $S, Θ$ by their corresponding volume means over the same control volume.

	POT. DENSITY APPROX.	BOUSSINESQ APPROX.	COMPRESSIBLE FLUID
“Vertical” coordinate	Height z	Height z	Pressure p
Infinitesimal volume	$dV = d x d y d z$	$dV = d x d y d z$	$d V_{p} = d x d y d p$
“Volume” average	$⟨ \cdot ⟩ = \frac{1}{V} ∭_{V} (\cdot) 𝑑 V$	$⟨ \cdot ⟩ = \frac{1}{V} ∭_{V} (\cdot) 𝑑 V$	${⟨ \cdot ⟩}_{p} = \frac{1}{V_{p}} ∭_{V_{p}} (\cdot) 𝑑 V_{p}$
Reference level	$z = 0 m$	$z_{v} = ⟨ z ⟩$	$p_{v} = {⟨ p ⟩}_{p}$
Effective PE (local) $Π$	$\frac{σ_{θ} g z}{ρ_{0}}$	$h_{d}^{b, z_{v}} = \int_{z}^{z_{v}} b (S, Θ, z^{’}) 𝑑 z^{’}$	$h_{d}^{p_{v}} = \int_{p_{v}}^{p} ν (S, Θ, p^{’}) 𝑑 p^{’}$
Fully-mixed PE (local) $Π_{0}$	$\frac{⟨ σ_{θ} ⟩ g z}{ρ_{0}}$	$h_{d, 0}^{b, z_{v}} = h_{d}^{b, z_{v}} (⟨ S ⟩, ⟨ Θ ⟩, z)$	$h_{d, 0}^{p_{v}} = h_{d}^{p_{v}} ({⟨ S ⟩}_{p}, {⟨ Θ ⟩}_{p}, p)$
Depth anomaly $D_{COM}$	$⟨ \frac{⟨ σ_{θ} ⟩ z - σ_{θ} z}{ρ_{0}} ⟩ = \frac{ϕ}{ρ_{0} g}$	$\frac{1}{g} ⟨ h_{d, 0}^{b, z_{v}} - h_{d}^{b, z_{v}} ⟩$	$\frac{1}{g} {⟨ h_{d, 0}^{p_{v}} - h_{d}^{p_{v}} ⟩}_{p}$
Mean- $N^{𝟐}$	$- \frac{g}{ρ_{0}} ⟨ \frac{\partial σ_{θ}}{\partial z} ⟩$	$⟨ - \frac{g}{ρ_{0}} (\frac{\partial ρ}{\partial z} - \frac{\partial ρ}{\partial z} \|_{S, Θ} ⟩$	${⟨ - \frac{g}{ρ} (\frac{\partial ρ}{\partial z} - {\frac{\partial ρ}{\partial z} \|}_{S, Θ}) ⟩}_{p}$

Time series of **(a)** global *D_COM* (in cm), and **(b)** global mean-N² (in $10^{- 6} s^{- 2}$ ). Plots include monthly mean (blue) and the associated linear regression line (red) with an envelope representing $\pm$ the standard error (shaded orange). The annual mean is superimposed (orange). Corresponding variations relative to the mean are indicated on the right axis.

Distribution of **(a)** mean ( $\pm 2 S E$ ), **(b)** trends ( $\pm 2 S E$ ), **(c)** normalized trends ( $\pm 2 S E$ ), and **(d)** relative trend ( $\pm 2 S E$ ) of stratification indices as a function the maximum depth H of the considered ocean volume. In the four panels, the depth anomaly *D_COM* is shown in red and the mean-N² in blue. Trends that are not statistically significant are dashed (based on p-value $> 10 %$ for a Kendall’s tau test (Kendall, 1938). The mean depth $⟨ H ⟩$ of the ocean volume shallower than H is used to normalize trends in **(c)**. The depth of global mean mixed layer $= 78 m$ is indicated with the vertical dashed black line.

Distribution of **(a)** *D_COM* and mean-N² partial trends ( $\pm 2 S E$ ), and **(b)** *D_COM* partial trends per unit depth as a function of depth H. Partial trends are obtained on the full volume of the ocean, when the properties deeper than H have been time-averaged. Horizontal dashed red lines in (a) show *D_COM* that are 25%, 50%, 75%, and 90% of the global trend. The dashed blue line indicates an exponential fit of partial trends per unit depth in (b). The global-mean mixed layer depth is indicated with a dashed black line.

Spatial patterns of the 0–2000 m stratification trends, based on **(a)** *D_COM*, **(b)** $D_{COM, Θ}$ , and **(c)** *D_COM,S*. Linear trends are computed for the 1992–2017 period. Values are given in mm per year, where positive values indicate enhanced stratification and negative values indicate weakened stratification. A 0.5 mm yr^–1 contour interval is used. The stippling in (a–c) indicates regions that are not statistically significant at 90% confidence. On average, the depth anomaly is increasing at an average rate of 0.36 mm yr^–1, but local trend values can be more than 10 times larger in certain areas.

Zonally averaged trends of local depth anomaly for the upper 2000 m water column *D_COM*, and the respective contribution from temperature $D_{COM, Θ}$ , and salinity *D_COM,S*. The residual *D_COM* minus $D_{COM, Θ} + D_{COM, S}$ is negligible at all latitudes. Trends are scaled by the total area of open ocean (i.e., deeper than $2000 m$ ) per unit latitude.

Table 2

Mean, trend, relative trend, and SE of the relative trend are tabulated for the two stratification indices (depth anomaly and mean-N²), considering ocean volumes for different depth ranges.

	GLOBAL MEAN	0–207 m	0–513 m	0–958 m	0–1993 m	FULL DEPTH
D_COM	Mean (cm)	3.52	10.29	19.08	35.32	68.87
	Trend (mm y^-1)	0.01	0.13	0.28	0.39	0.66
	Relative Trend (% year^-1)	0.03	0.13	0.15	0.11	0.10
	SE of the relative trend (% year^-1)	0.023	0.009	0.005	0.003	0.002
mean-N²	Mean (10^-5 s^-2)	9.08	4.93	2.86	1.52	0.75
	Trend (10^-8 s^-2 year^-1)	3.57	4.93	2.86	1.52	0.75
	Relative Trend (% year^-1)	0.04	0.10	0.10	0.09	0.09
	SE of the relative trend (% year^-1)	0.019	0.013	0.011	0.010	0.009

Table 3

Spatial correlations between various local trends.

	D_COM	mean-N²
Buoyancy anomaly (0–2000 m)	0.857	0.573
Heat anomaly (0–2000 m)	0.462	0.192
Salt anomaly (0–2000 m)	–0.314	–0.314
Surface buoyancy	0.451	0.691
Surface temperature	0.384	0.408
Surface salinity	-0.188	-0.441

Time series of global *D_COM* computed using the potential density approximation (see Table 1). As in Figure 2, the linear regression line (red) is shown with an envelope representing ± the standard error (shaded orange), and the annual mean is superimposed (orange). Corresponding variations relative to the mean are indicated on the right axis.

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DOI: https://doi.org/10.16993/tellus.4121 | Journal eISSN: 3035-9554

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Language: English

Page range: 20 - 34

Submitted on: Oct 15, 2025

Accepted on: Apr 1, 2026

Published on: Apr 20, 2026

Published by: Stockholm University Press

In partnership with: Paradigm Publishing Services

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© 2026 Fabien Roquet, Gustav Holmquist, Jakob Gunnarsson, published by Stockholm University Press
This work is licensed under the Creative Commons Attribution 4.0 License.

Volume 79 (2026): Issue 1

Global Stratification Trends Diagnosed Using the Center of Mass of the Ocean

Figures & Tables

Figure 1

Table 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Table 2

Table 3

Figure A.1

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