# Mechanisms driving ocean stratification and mixed layer deepening

Éric Zeltz

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## To cite this version:

Éric Zeltz. Mechanisms driving ocean stratification and mixed layer deepening. Explora: Environment and Resource, 2024, 1 (1), pp.4578. 10.36922/eer.4578. hal-04857066

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Submitted on 27 Dec 2024

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ORIGINAL RESEARCH ARTICLE

# Mechanisms driving ocean stratification and mixed layer deepening

Éric Zeltz*

Independent Scholar, La Motte en Champsaur, Hautes-Alpes, France

## Abstract

This article aims to better understand the links between climate change and the recently observed significant increases in ocean stratification and the deepening of the mixed layer. Using a time series of ocean stratification data, we investigate potential "signals" of interactions with other climatic phenomena. Our findings reveal that the observed temporal patterns in stratification are primarily influenced by the El Niño-Southern Oscillation. However, this influence alone does not account for the overall increase in stratification or the deepening of the mixed layer observed since the 1960s. Instead, we demonstrate that these two phenomena are largely driven by the warming of the upper oceanic stratum (UOS). To incorporate these findings, we adapted a climate model developed in 2023, enabling it to account for the identified drivers. The enhanced model generates projections for five key global climate parameters: atmospheric temperature at the ocean level, UOS temperature, ocean cloudiness, ocean stratification, and mixed layer deepening. Our mathematical analysis of the model indicates that the relationships underlying these variables impose a clear growth trend on each of the five parameters, which eventually stabilizes toward finite limits. This stabilization results in values that remain within reasonable bounds, minimizing their impact on the planet and the global ecosystem. Under a comparable scenario, for example, the model predicts an increase of approximately 1°C in global atmospheric temperature by 2100 – significantly lower than the 2°C increase projected by the Intergovernmental Panel on Climate Change (IPCC).

**Keywords:** Climate modeling; Ocean stratification; Mixed layer; El Niño-Southern Oscillation; Natural climate regulation

## 1. Introduction

Recent studies indicate that ocean stratification and vertical stability have increased significantly over the past 50 years.<sup>1-3</sup> The main drivers of this phenomenon, as identified in these studies, include:

(i) Surface warming due to global warming: The increase in heat at the ocean's surface intensifies density differences between successive water layers, thereby strengthening thermal barriers between the upper ocean layer and deeper waters.<sup>1-3</sup>

(ii) Melting polar ice: Warming-induced ice melt releases substantial amounts of freshwater, which is lighter than saltwater. This freshwater remains on the top water layer, further warming and reinforcing the thermal barrier with deeper layers of the oceans.<sup>2</sup>

<footnote>*Corresponding author:</footnote>

<footnote>Éric Zeltz
(ericzeltz@wanadoo.fr)</footnote>

<footnote>**Citation:** Zeltz É. Mechanisms driving ocean stratification and mixed layer deepening. *Explora Environ Resour.* 2024;1(1):4578. doi: 10.36922/eer.4578</footnote>

<footnote>**Received:** August 19, 2024</footnote>

<footnote>**Accepted:** October 31, 2024</footnote>

<footnote>**Published Online:** December 5, 2024</footnote>

<footnote>**Copyright:** © 2024 Author(s). This is an Open-Access article distributed under the terms of the Creative Commons Attribution License, permitting distribution, and reproduction in any medium, provided the original work is properly cited.</footnote>

<footnote>**Publisher's Note:** AccScience Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.</footnote>
(iii) Disruption of precipitation and river runoff: Climate change has altered precipitation patterns and increased river discharge, leading to the formation of a less salty surface layer. This layer becomes more isolated from deeper waters, with salinity changes playing an important role primarily at local scales compared to the global impact of surface warming.<sup>1-6</sup>

Building on temporal data on global stratification provided by Li *et al.*¹ and analyzed using a “signal” detection method,⁷,⁸ this study examines the temporal evolution of ocean stratification. Subsequently, we adopt a global climate model⁷,⁸ to simulate the evolution of global averages for key climate parameters: atmospheric temperature at ocean level, upper oceanic stratum (UOS) temperature, oceanic cloud cover, the deepening of the mixed layer, and oceanic stratification.

This work provides three significant outcomes:

(i) Theoretical insight: A deeper understanding of the evolution of ocean stratification and mixed-layer deepening.

(ii) Forecast capability: A low-cost model capable of simulating critical climate parameters with precision.

(iii) Policy implications: The simulations indicate that all five parameters exhibit limited growth, stabilizing at levels by the end of the century that are relatively benign for the global ecosystem. This challenges current projections by the Intergovernmental Panel on Climate Change (IPCC) and suggests that the potential impacts of greenhouse gas emissions may be less severe than anticipated.

The structure of this article is as follows:

(i) Section 2: Description of the studied entities, data sources, and analysis methods (Sections 2.1 and 2.2)

(ii) Section 3: Implementation of the analysis and quantitative evaluation of heat transfer mechanisms in the UOS due to climate change (Sections 3.1–3.4). Section 3.5 details the mathematical model based on these mechanisms. Section 3.6 presents simulation results, including the strong negative feedback effect of oceanic cloud cover on atmospheric temperatures. Section 3.7 mathematically demonstrates this feedback on a theoretical level. Section 3.8 highlights contradictions with IPCC perspectives. Section 3.9 discusses the advantages and limitations of the model compared to those used by major climate institutes. Section 3.10 compares our approach to Hasselmann’s theory, which also employs differences in atmospheric and oceanic memory along with Markov chains.

(iii) The conclusion (Section 4) summarizes the key results and emphasizes the need for further verification of findings that contradict current IPCC projections. If

these results prove accurate, they could significantly alter the guidance provided to policymakers on addressing climate change.

## 2. Materials and methods

### 2.1. Schematic description of ocean stratification

Ocean stratification refers to the formation of distinct layers within the ocean, characterized by variations in water density. The least dense layers are found at the surface, while the densest layers reside at greater depth. These density differences arise primarily due to variations in temperature and/or salinity. Specifically, water becomes less dense as it warms or loses salinity. The upper layer of the ocean, which is more exposed to solar radiation than the deeper layers and frequently influenced by precipitation or melting ice, constitutes a relatively stable stratum. This stability resists disruption from external forces such as strong winds or ocean currents originating from other regions. Beneath this upper layer lies a highly stable transition zone, marked by a strong negative temperature gradient (thermocline) and a strong positive density gradient (pycnocline). For the purposes of this study, these two gradients are treated as equivalent. This transition zone acts as a barrier, separating the upper ocean from the deeper layers, which are characterized by higher density and colder temperatures. Figure 1 illustrates the vertical structure of the world’s oceans, highlighting the relative positions of the mixed layer, pycnocline, and deep ocean.

### 2.2. Method for analyzing the time series

To analyze the time series of stratification data for the UOS, we used a data processing method initially introduced in Zeltz⁹ and subsequently implemented in Zeltz⁷, Zeltz⁸, and Zeltz.⁹ The detailed steps of this method are outlined in Zeltz.⁸

We began by verifying whether the derived series – constructed from the differences in stratification between consecutive semesters – could be modeled as “white noise,”

<figure><img src="image_3.png" /><figcaption>**Figure 1.** An idealized schematic of the vertical structure of the world's oceans
Abbreviation: UOS: Upper oceanic stratum.</figcaption></figure>
as defined in time series theory.¹⁰ From this derived series, we generated a binary series composed of 1s or 0s using the following conventions:

* 1: Indicates positive difference, that is, an increase in stratification compared to the previous semester.

* 0: Indicates no increase in stratification.

Next, we examined whether the frequencies of 0s and 1s were approximately equal to 0.5, suggesting that increases and decreases in stratification are equiprobable. This observation, combined with the finding that stratification differences can be modeled as white noise, supports the hypothesis that the binary series (0s and 1s) can be modeled as a Markov process.

We considered three potential scenarios for the Markov process:

(i) Markov-0 binomial model:⁷⁻⁹ Each value is independent of the previous one, resulting in a binomial distribution governed by parameters *n* (number of elements) and *p* = 0.

(ii) Markov-1 lengthening case: Each value depends on the previous one such that if the previous value is 0 (or 1), the next value is more likely to remain 0 (or 1).

(iii) Markov-1 alternating case: Each value depends on the previous one such that if the previous value is 0 (or 1), the next value is more likely to be 1 (or 0).

To distinguish among these scenarios, we analyzed the average lengths of successive runs of 1s (or 0s) and compared them to the theoretical average length of 1.94 expected under the Markov-0 binomial model. The classification criteria were as follows:

* Less than 1.80: Markov-1 alternating case

* Between 1.80 and 2.10: Markov-0 binomial model

* Greater than 2.10: Markov-1 lengthening case.

If necessary, we confirmed the calculation by calculating the probability under a binomial distribution. When the binary series is classified as Markov-1 alternating or Markov-1 lengthening, this classification implies an interaction with another entity that exhibits a similar type of signal. In such cases, it is necessary to identify a potential interacting entity and justify this choice using additional arguments, which, in our study, are primarily climatological.

As demonstrated in Zeltz⁷, Zeltz⁸, and Zeltz,⁹ this method of time series analysis enables the identification of interactions that are undetectable by traditional approaches. It provides valuable insights into the type of alternation within the series and its potential dependence on another series. However, this approach also results in a significant loss of information, as the original series is reduced to a derived binary series composed solely of 0s and 1s.

This reduction can lead to cases where the type of alternation in the binary series indicates an interaction with a specific entity, even when there is no strong correlation in the original time series. Conversely, a robust correlation between the original series may not correspond to a shared alternation pattern in the binary series. These two types of information are fundamentally different and complementary:

(i) Quantitative variations: Insights into the overall dynamics driving the changes in the series, which help formulate or refute hypotheses about its underlying "engine."

(ii) Qualitative alternation patterns: Understanding the factors influencing the rhythm of the alternation between increases and decreases in the phenomenon.

An analogy can help illustrate the distinction between these two types of information. Consider a musician using an electronic synthesizer: They can lengthen the cumulative duration of the ascending and descending phases of the sound power (analogous to alternation patterns) or adjust the average sound power using the potentiometer on their amplifier (analogous to quantitative variations). These are two distinct processes: the average sound power is directly influenced by the potentiometer, while the alternation speed is governed by the musician's rhythmic choices.

# 3. Results and discussion

## 3.1. Description of data used for global stratification (upper 0 – 2000 m) and upper layer stratification (upper 0 – 200 m)

We used stratification data provided by Li et al.,¹ initially published in 2020 and subsequently updated by the authors. The dataset is publicly accessible at the following link: https://pan.cstcloud.cn/web/share.html?hash=E0zjDQOeRfs

The dataset primarily relies on data from the Institute of Atmospheric Physics at Chinese Academy of Sciences (China), covering the period from 1955 to the present. It is available with a horizontal resolution of 1° × 1° and includes 41 vertical levels for the upper 0 – 2000 m of the ocean. The researchers applied several quality assurance techniques, including instrumental bias correction, an advanced gap-filling algorithm for reconstructing temperature and salinity changes, and validation against recent Argo data (available at https://argo.ucsd.edu). In addition, the data include 95% confidence intervals calculated using a Monte Carlo approach that accounts for all sources of error. Figure 2 presents the global monthly anomalies of stratification for upper 0 – 2000 m, updated by the authors through the end of 2023 and calculated relative to the 1981 – 2010 reference period.
<figure><img src="image_4.png" /><figcaption>**Figure 2.** Global monthly anomalies for ocean stratification in the upper 0 – 2000 m, expressed in $N^2$
Source: https://pan.cstcloud.cn/web/share.html?hash=E0zjDQOeRfs</figcaption></figure>

According to this dataset, stratification in the upper 0 – 2000 m increased by approximately 6% (95% confidence interval: 5.4 – 6.6) over the period 1955 – 2023, equating to an average increase of approximately 0.8%/decade.

**Figure 3** presents the global monthly anomalies of stratification of the upper 0 – 200 m layer, derived from the same dataset.¹

This data indicates that stratification in the upper 0 – 200 m layer increased by approximately 7% (95% confidence interval: 6.2 – 7.5) over the same period, or nearly 1%/decade.

Although the absolute values differ, the variations in stratification between the upper 0 – 200 m layer and the upper 0 – 2000 m layer are remarkably similar. This similarity can be attributed to the fact that temperature changes driving stratification are more pronounced at the ocean surface than at greater depths. Consequently, stratification in the upper 0 – 200 m layer predominantly determines the overall stratification signal.

Li *et al.*¹, p.1 highlight this dynamic, stating: “Most of the increase (~71%) occurred in the upper 200 m of the ocean and resulted largely (>90%) from temperature changes, although salinity changes play an important role locally.”

Our study primarily focuses on the relationships between stratification in the UOS (approximately 0 – 200 m), the thermal energy it contains, and atmospheric temperature. The UOS, the upper 0 – 200 m layer, and the mixed layer (located above the pycnocline) are closely related. For simplicity, we treat these three entities as equivalent throughout this work. This approach allows us to leverage specific results derived from studies on the upper 0–200 m layer and the mixed layer in Zeltz⁷, Zeltz⁸ to refine the global climate model Z.2, developed in Zeltz⁸, as an improvement over the earlier Z.1 model described in Zeltz.⁷

## 3.2. Interpretation of stratification data from the UOS

Table 1 summarizes the average lengths of sequences of 0s and 1s per semester for the UOS.

<figure><img src="image_5.png" /><figcaption>**Figure 3.** Global monthly anomalies for ocean stratification in the upper 0 – 200 m, expressed in $N^2$
Source: https://pan.cstcloud.cn/web/share.html?hash=E0zjDQOeRfs</figcaption></figure>

<table><caption>**Table 1.** Average lengths of sequences of 1s and 0s per semester for the upper oceanic stratum (0 – 200 m)</caption><thead><tr><th>Per semester and for UOS (0 – 200 m)</th><th>Chains of</th><th>1</th><th>0</th></tr></thead><tbody><tr><td>Period 1955 – 2023</td><td>Average</td><td>2.74</td><td>2.37</td></tr><tr><td>137 values (0 or 1)</td><td>Number of chains</td><td>27</td><td>27</td></tr></tbody></table>

The data exhibit a pronounced Markov-1 lengthening behavior. When modeled using a binomial distribution with parameters *n* = 137 and *P* = 0.5, such a pattern is highly improbable (<1% likelihood).

To explain the observed Markov-1 lengthening behavior in the semiannual UOS stratification data, we provide a detailed analysis below.

### 3.2.1. Seasonal alternation

Sallée *et al.*² emphasize the summer-winter seasonal alternation affecting UOS stratification. This alternation is evident not only in UOS stratification but also in the pycnocline, which separates the deep ocean from the seasonal mixed layer, and in the deepening of the mixed layer itself.

* In temperate and cold regions, the 0 – 200 m layer exhibits a seasonal thermocline during summer, reinforced by processes like ice melting.¹¹ This thermocline dissipates during winter.¹²

* Summer corresponds to August–October in the northern hemisphere and January–March in the southern hemisphere, while the reverse applies to winter.

An important factor is the asymmetry in land-sea distribution between hemispheres: 80% of the southern hemisphere is covered by oceans, compared to only 60% in the northern hemisphere. This imbalance prevents full compensation of southern hemisphere stratification by northern hemisphere stratification. If seasonal alternation was the only influencing factor, we would expect a Markov-1 alternating signal. However, the data reveal Markov-1 lengthening behavior, indicating the presence of additional factors.
### 3.2.2. Influence of the intertropical convergence zone (ITCZ)

The ITCZ is one such factor.

* As illustrated in Figure 4, the ITCZ shifts seasonally, moving further south during the austral summer and further north during the boreal summer.

* The ITCZ brings rainy seasons and monsoons over vast oceanic regions, cooling the UOS and reducing stratification resulting from warming.

The marine influence of the ITCZ is greater in the southern hemisphere during the austral summer, counteracting stratification increases intemperate zones of that hemisphere. A similar effect occurs in the northern hemisphere during the boreal summer. This phenomenon compensates for much of the summer increase in stratification in the temperate zones of the hemisphere where the ITCZ is temporarily positioned. Without additional influences, the signal would likely resemble a binomial Markov-0 type. However, the observed data still strongly exhibit Markov-1 lengthening behavior, indicating another significant influence.

### 3.2.3. Role of the El Niño-Southern Oscillation

The El Niño-La Niña provides the most plausible explanation for the observed Markov-1 lengthening behavior. Analysis of the Oceanic Niño Index (ONI) provided by the National Oceanic and Atmospheric Administration (NOAA) (https://origin.cpc.ncep.noaa.gov/products/analysis_monitoring/ensostuff/ONI_v5.php), supports this hypothesis. The ONI, calculated over a large geographical region (5°N – 5°S, 170°W – 120°W) and subjected to strict standardization criteria, is among the most reliable indices for studying ENSO phenomena, as supported by Trenberth.¹³ Figure 5 illustrates its evolution over the period 1950 – 2016.

Over the 74-year period from 1950 to 2023, approximately 21 years were affected by El Niño events, 22 years by La Niña events, and 33 years were classified as “neutral” (neither El Niño nor La Niña). According to the ONI data, the average duration of each event is as follows:

* El Niño events: ~9 months (more precisely 8.65)

* La Niña events: ~9 months (more precisely 9.21)

* Neutral phases: ~6 months (more precisely 6.03).

<figure><img src="image_6.png" /><figcaption>Figure 4. Intertropical convergence zone (ITCZ) positions in January and July 2006</figcaption></figure>

Source: Mats Halldin, Public domain, via Wikimedia Commons.

<figure><img src="image_7.png" /><figcaption>Figure 5. Evolution of the Oceanic Niño Index (ONI), showing warm (red) and cold (blue) phases of sea surface temperature anomalies in the tropical Pacific Ocean. Image reproduced from Trenberth.¹³ Copyright © 2024 UCAR</figcaption></figure>
For example, during a semester marked by El Niño, stratification is likely to increase, and this trend may persist into the next semester due to the average 9-month duration of El Niño events. Similarly, during a La Niña phase, stratification tends to decrease, with a higher probability that this trend will continue into subsequent months. Given the high frequency of such events – covering 58.5% of the 1950–2023 period – this phenomenon provides a sufficient explanation for the pronounced Markov 1 lengthening behavior observed in the biannual stratification patterns of the UOS and the 0–2000 m ocean layer.

### 3.3. Mechanisms linked to ocean stratification and deepening of the mixed layer

The biannual alternation in ocean stratification, closely linked to El Niño-Southern Oscillation (ENSO) events, explains a significant portion of the observed signals presented in Table 1. However, ENSO alone does not account for the observed global increase in ocean stratification, which has risen by approximately 1%/decade since the 1960s.<sup>1,2</sup> For example, the correlation coefficient between the annual ONI index of ENSO and the annual averages of $N^2$ stratification anomalies from Zeltz¹ is 0.13, a value too low to indicate a significant correlation.

Conversely, a strong correlation (coefficient = 0.84) exists between semiannual stratification changes and thermal energy in the UOS for the period 1955–2023, using data from Zeltz¹ and NOAA (downloadable from the link https://www.climate.gov/media/13603). This finding suggests that additional heat from global warming is the primary driver of increased stratification. The underlying mechanisms are elaborated below.

Water stratification arises from differences in water density, which are directly linked to temperature gradients. Significant warming of the UOS – caused by atmospheric heat and solar radiation – disrupts the balance of densities, enhancing stratification. As the heat gradient intensifies, so does the density gradient, leading to increased stratification.

Without ENSO, stratification signals would closely mirror ocean heat data, exhibiting Markov-1 alternating behavior, as stratification strongly depends on thermal energy. However, ENSO's long cycles and extensive influence modify these patterns, lengthening alternation periods. In summary, ENSO lengthens alternation periods, while additional heat in the UOS increases stratification.

The increase in stratification directly impacts the thermocline, which increasingly acts as a barrier to heat transfer into the deep ocean. According to Sallée et al.,² the density contrast at the base of the UOS during summer has increased at a rate of 8.9 ± 2.7%/decade over the past

50 years. The strengthened vertical density gradients limit mixing between upper and deep ocean layers.

Parallel to this phenomenon, Sallée et al.<sup>2</sup> observed a consistent deepening of the summer surface layer at a rate of 2.9 ± 0.5%/decade, corresponding to 5 – 10 m/decade, depending on the region. The winter mixed layer also appears to be deepening on a global scale, though this finding is based on shorter time series and carries greater uncertainty. Sallée et al.<sup>2</sup> attribute this deepening primarily to the mechanical action of increasing ocean winds.

Young et al.<sup>14</sup> estimated a global wind speed increase of approximately 0.25%/year during the period 1991 – 2008, while Zheng et al.<sup>15</sup> reported a significant rate of 3.35 cm/s/year for the period 1988 – 2011. However, reservations regarding the reliability of these data have been raised by Zieger et al.<sup>16</sup> and Wohland et al.<sup>17</sup> Even assuming their accuracy, such wind speed increases appear insufficient to account for the observed deepening rates of 5 – 10 m/decade.

Given that the average wind speed at the ocean surface is approximately 7 m/s,<sup>18</sup> an increase of 3.35 cm/s/year corresponds to a cumulative increase of approximately 5% over a decade.<sup>18</sup> This increase is inadequate to drive sustained additional mixing at the bottom of the mixed layer sufficient to cause the pycnocline to descend by 5 – 10 m/decade.

Surface winds generate ocean currents at approximately 2% of their speed,<sup>19</sup> meaning a decadal increase of 3.35 cm/s in wind speed results in a surface current increase of less than 1 mm/s. This current increment diminishes as it propagates to lower layers, ultimately reaching the pycnocline. In addition, the Coriolis effect introduces a spiral flow (Ekman spiral phenomenon), which further dissipates the energy transferred to deeper layers.<sup>19</sup>

In specific cases, such as in the tropical Atlantic basin, a linear relationship is often observed between surface wind speed and mixed layer deepening.<sup>20,21</sup> Episodes of intense winds, such as tropical cyclones, are increasing in frequency and intensity due to climate change.<sup>22</sup> These extreme events significantly influence mixed layer deepening, especially on large spatial scales. However, under typical conditions, the increase in wind-driven mixing near the thermocline is minimal. Extreme storm events are the exception, with their wind power capable of deepening the mixed layer enough to contribute to global averages. Furthermore, these events are correlated with increases in sea surface temperature and thermal energy in the UOS.<sup>23,24</sup>

To model mixed layer deepening (Section 3.8), we propose a thermal energy threshold of 2 × 10<sup>7</sup> J entering the UOS. Beyond this threshold, wind contributions to
the depth of the mixed layer (approximately 3 cm) will be included. This threshold accounts for approximately 8% of the average monthly thermal energy received by the UOS and approximately 10% of the average monthly deepening. These values were chosen to reflect the minor role of wind in mixed layer deepening and to ensure the model aligns with the observations of Sallée *et al.*²

The remaining 90% of the observed deepening must be attributed to other mechanisms. It is unlikely that this can be explained by thermal expansion of water or the influx of freshwater from melting ice, as these processes caused sea level rises of only 3.1 ± 0.7 mm/year during 1993 – 2003.²⁵

Analogous phenomena observed in large freshwater systems, such as the Great American Lakes during summer heatwaves,²⁶⁻²⁸ suggest that the primary driver of mixed layer deepening is the additional thermal energy entering the layer.

Excess thermal energy in the UOS resulting from global warming is dissipated through three main mechanisms:

(i) Heating the mixed layer.

(ii) Evaporation into the atmosphere as latent heat.

(iii) Conduction into the thermocline, which does not act as a perfectly isolated “lid” from the mixing layer.

The thermocline's upper layer, in direct contact with the mixing layer exposed to excess heat, absorbs part of this energy through conduction. Over time, this layer warms enough to integrate into the mixing layer. This process repeats incrementally, with successive layers of the thermocline warming and integrating, always through conduction. The deepest thermocline layer, while remaining the coldest, develops an accentuated temperature gradient relative to the deep ocean.

This gradient eventually results in the formation of a new layer at the boundary of the deep ocean and the thermocline. This layer absorbs some heat, warms slightly, detaches from the deep layer, and becomes part of the thermocline. If excess heat continues to penetrate the UOS, this process repeats, ultimately leading to the increase in the thickness of the mixed layer and the observed summer deepening of the thermocline, as reported by Sallée *et al.*²

For the remainder of this study, the dissipation of excess heat from the mixing layer will be analyzed in terms of ratios representing the contributions of these three mechanisms: conduction into the thermocline, heating of the mixture layer, and evaporation as latent heat into the atmosphere.

### 3.4. Estimation of ratios

According to Sallée *et al.*,² over the period 1970 – 2018, the thermocline depth increased by 2.9 ± 0.5%/decade,

equivalent to a deepening of 5 – 10 m/decade, depending on the region. For this analysis, we adopt the median value of the range, that is, 7.5 ± 1.3 m/decade, corresponding to 75 ± 13 cm/year. Furthermore, the current thermocline strength evaluation standard follows the Standard of Marine Survey, where the intensity threshold is 0.2 °C/m for shallow water (depths <200 m) and 0.05°C/m for deeper waters (depths >200 m).²⁹

For oceanic thermoclines, an intermediate value of 0.1°C/m is assumed. Consequently, for a 75 cm deepening, the temperature gradient corresponds to 0.075°C. The volume of water affected by this change is approximately:

$$V = 0.75 \times S \qquad (I)$$

where S (≈ 3.6 × 10¹⁴ m²) represents the total surface of the oceans. Thus, V ≈ 2.7 × 10¹⁴ m³.

The mass of seawater at the relevant temperatures is approximately 1027 kg/m³ (as per Copin-Montégut).³⁰ The thermal energy required to heat this volume V by 0.075°C is given by:

$$Q \approx 1027 \times C \times V \times 0.075 \qquad (II)$$

where C ≈ 4 × 10³ J/K/kg is the specific heat capacity of liquid water.³¹ Substituting values yields Q ≈ 1027 × 4 × 10³ × 2.7 × 10¹⁴ × 0.075 ≈ 8.32 × 10¹⁹ J.

Over the studied period, the increase in thermal energy in the UOS is approximately 3.3 × 10²² J/decade, according to Zeltz⁷, or 3.3 × 10²¹ J/year. This means around 2.5% of the additional energy contributes to thermocline deepening. Further, calculations address the distribution of the remaining 97.5%.

The observed thermal expansion due to ocean warming primarily affects the upper layer, with an increase attributed to this phenomenon estimated at 0.89 ± 0.05 mm/year for the 0 – 700 m layer between 1970 and 2015.³²,³³ This estimate assumes no trend below 2000 m depth before 1992 and incorporates the modest contribution from the 700 – 2000 m layer. According to Desbruyères *et al.*,³⁴ this contribution increased by 10% during 2006 – 2015, consistent with the deepening of the thermocline.

To calculate the corresponding temperature increase T[0-700], we use Equation III:

$$700 \times d_{\text{water}} \times T_{[0-700]} = 0.89 \times 10^{-3} \text{ m/year} \qquad (III)$$

where d<sub>water</sub> is the thermal expansion coefficient of water. Asper Thomas,³⁵ d<sub>water</sub> ≈ 2.6 × 10⁻⁴ m/°C. Solving for T[0-700]:

$$T_{[0-700]} = (0.89 \times 10^{-3})/(700 \times d_{\text{water}}) \qquad (IV)$$

which gives T[0-700] ≈ 0.00489°C/year. This result is consistent with the estimate of 0.005°C/year provided by Hasselmann.³⁴ For the 0–200 m mixing layer, the
temperature increase is faster, approximately 0.008°C/year, as shown in Desbruyères et al.³⁴

The thermal energy Q' responsible for the annual warming of the 0 – 200 m mixed layer is approximately:

$$Q' \approx 1027 \times C \times 200 \times 3.6 \times 10^{14} \times 0.008 \approx 2.36 \times 10^{21} J \quad (V)$$

This value represents approximately 71.5% of the additional thermal energy absorbed annually by the oceans. Half of the remaining energy (26%, or 0.26 × 3.3× 10²¹ ≈ 0.858 × 10²¹ J) dissipates as latent heat through evaporation, while the other half offsets the cooling caused by evaporation. This process accounts for the surface temperature rise of 0.008°C. Consequently, the total energy used for surface warming is approximately 71.5% + 13% ≈ 84.5%, while 13% escapes as latent heat to the atmosphere – approximately 5 times the energy dissipated into the thermocline.

The amount of additional water evaporated, E, can be calculated as:

$$E = (0.5 \times 0.858 \times 10^{21})/l \quad (VI)$$

where $l \approx 2470$ kJ/kg is the latent heat of water at the relevant temperatures.³¹ Substituting values yields $E \approx 1.73 \times 10^{14}$ kg, or approximately 168 km³ of seawater. This amount represents only a small fraction (≈0.035%) of the total oceanic evaporation rates, which is estimated at 480 km³ ± 10%.³⁶

Table S1 summarizes the primary results obtained in this section.

### 3.5. Modeling the deepening of the mixed layer

Table S2 summarizes the relationships used by the Z.3 model, which is employed to simulate the deepening of the mixed layer. Most of the relationships remain unchanged from the Z.2 model,⁸ which previously enabled simulations of changes in atmospheric temperature, the UOS, and oceanic cloud cover. Compared to the Z.2 model, only line 9 has been modified, and one new line (line 18) has been added.

Consequently, lines 1 – 8 and 10 – 17 are identical to those in the table,⁸ which outlines the relationships underlying the Z.2 model. Their detailed explanations, primarily derived from Zeltz,⁷,⁸ and supplementary information provided in Table S2, are discussed in Zeltz,⁷ except for lines $\delta_a$ and $\delta_b$, which are explained in Zeltz.⁸ The constants employed in the model have been calibrated using observational data, incorporating the ratios determined in Section 3.4.

The modifications and additions are described below:

* Line 9 (modified): The growth of stratification ($s_n$)

is strongly correlated with the increase in heat ($C_n$) absorbed by the UOS. This relationship occurs directly through heat penetration and indirectly through the addition of melted freshwater, which alters salinity.¹ This relationship is expressed as:

$$s_{n+1} = s_n + \mu(\theta_{n+1} - \theta_n) \quad (VII)$$

where $\mu$ is a constant estimated through calibration with data from Li et al.¹

* Line 18 (added): We define $L_n$ as the global average distance during month n between the base of the mixed layer and the sea surface. Estimating $L_n$ reliably is technically challenging due to the diverse criteria used (e.g., water pressure, temperature, salinity, or specific thresholds).³⁷ However, the change in depth, defined as:

$$S_n = L_n - L_{n-1} \quad (VIII)$$

On the other hand, for fairly obvious mathematical reasons, the depth $S_n$ of the layer defined by following Equation VIII depends much less on the techniques and criteria used.

$$S_{n+3} = S_{n+2} + \lambda(\theta_n - \theta_{n-1}) + L_{>Th} \quad (IX)$$

Here, $\theta_n - \theta_{n-1}$ reflects the interaction between UOS temperature variations and the deepening of the thermocline and mixing layer (Section 3.6). The 3-month lag between $S_{n+3}$ and $\theta_n - \theta_{n-1}$ accounts for the significant thermal inertia of ocean water.³¹,³⁸ The constant $\lambda$ is estimated through calibration with data from Sallée et al.² $L_{>Th}$ denotes additional deepening (L) caused by thermal energy inputs exceeding a certain threshold Th, driven by violent winds.

### 3.6. The Z.3 model: Confidence intervals and simulations

Using a sample of 250 simulations from the Z.3 model, we determined point estimates ($\mu_c$ and $\sigma_c$) for the means ($\mu$) and standard deviations ($\sigma$) of monthly changes in $S_n$, $s_n$, $t_n$, $\theta_n$, and $c_n$ over the period 1955 – 2095. After verifying normality, these estimates allowed us to calculate 95% confidence intervals. Table S3 summarizes the results.

Figures 6-9 illustrate Z.3 model simulations for:

* Deepening ($S_n$) in meters (Figure 6)

* Stratification ($s_n$) in percentage (Figure 7)

* Temperature anomalies ($t_n$ and $\theta_n$) in °C (Figure 8), and

* Oceanic cloudiness ($c_n$) in percentage (Figure 9).

Each figure includes the following components:

* A simulation curve of the variable,

* Two extreme curves delimiting the 95% confidence
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Z.3 model simulations for deepening</figcaption></figure>

Notes: Red curve: Simulation of anomalies of deepening S_n (in m). For the other curves: The 95% confidence interval is delimited by the upper curve and lower curve. The central blue curve is the theoretical curve (without taking into account natural variability) of the deepening anomalies.

<figure><img src="image_9.png" /><figcaption>**Figure 7.** Z.3 model simulations for stratification anomalies</figcaption></figure>

Notes: Red curve: Simulation of anomalies of stratification s_n (in %). For the other curves: the 95% confidence interval is delimited by the upper curve and lower curve. The central blue curve is the theoretical curve (without taking into account natural variability) of the stratification anomalies.

<figure><img src="image_10.png" /><figcaption>**Figure 8.** Z.3 model simulations for temperature anomalies</figcaption></figure>

Notes: Simulations of t_n (blue) and θ_n (orange) anomalies (in °C). For other curves: the 95% confidence interval is delimited by the upper curve and lower curve. The two central curves, which nearly overlap, are the theoretical curves (without taking into account natural variability) of t_n and θ_n.

<figure><img src="image_11.png" /><figcaption>**Figure 9:** Z.3 model simulations for oceanic cloudiness anomalies</figcaption></figure>

Notes: Light blue curve: Simulation of anomalies of oceanic cloudiness cl_n (in %). For the other curves: The 95% confidence interval is delimited by the upper curve and lower curve. The central blue curve is the theoretical curve (without taking into account natural variability) of the oceanic cloudiness anomalies.

intervals, and

* A “theoretical” central curve generated by the Z.3 model, neutralizing random coefficients to exclude natural variability.

### 3.7. Asymptotic behavior of the sequences $(t_n)$, $(\theta_n)$, $(cl_n)$, $(s_n)$, and $(S_n)$

The theoretical curves of $t_n$, $\theta_n$, $cl_n$, $s_n$, and $S_n$, as shown in Figures 6-9, suggest that these sequences are strictly increasing and converge to finite limits.

If we denote the sequences in question by $(t_n^t)$, $(\theta_n^t)$, $(cl_n^t)$, $(s_n^t)$, and $(S_n^t)$, they satisfy the following proposition:

**Proposition:** The sequences $(t_n^t)$, $(\theta_n^t)$, $(cl_n^t)$, $(s_n^t)$, and $(S_n^t)$ are strictly increasing and convergent.

**Proof:**

From line 3 of Table S2, we observe the relationship:

$$ \theta_{n+3}^t = \theta_{n+2}^t + C_n / C_{dyn} \quad (X) $$

where $C_n$ represents the necessarily positive energy balance entering the UOS and leaving its surface or lower layers, and $C_{dyn}$ is a positive constant.

For any integer n, it follows that:

$$ \theta_{n+3}^t > \theta_{n+2}^t \quad (XI) $$
Hence, the sequence ($\theta_n^t$) is strictly increasing.

Similarly, from line 4 of Table S2, we have:

$$t_{n+1}^t = t_n^t + T_n/C_{dyn}^n \quad (XII)$$

where $T_n$ represents the necessarily positive energy balance entering the atmosphere and leaving it through the UOS or the top of the troposphere. Consequently:

$$t_{n+1}^t > t_n^t \quad (XIII)$$

Thus, the sequence ($t_n^t$) is strictly increasing.

Both sequences, ($\theta_n^t$) and ($t_n^t$), are further bounded above by the finite temperature of the Sun, which serves as the primary heat source warming the Earth. Therefore, these sequences are strictly increasing and convergent.

Next, consider the sequence ($cl_n^t$). Using the relation from line 6a:

$$cl_{n+1} = cl_n + \beta(\theta_{n+1} - \theta_n) \quad (XIV)$$

where $\beta>0$, and given that ($\theta_n^t$) is strictly increasing, it follows that ($cl_n^t$) is also strictly increasing. Since:

$$cl_{n+1}^t - cl_n^t = \beta(\theta_{n+1}^t - \theta_n^t) \quad (XV)$$

the sequence ($cl_n^t$) exhibits the same convergence behavior as ($\theta_n^t$) and is therefore convergent.

A similar argument applies to the sequences ($s_n^t$) and ($S_n^t$). These are derived respectively from the relationships in line 9 (Equation VII) and line 18 (Equation IX). These relationships demonstrate that ($s_n^t$) and ($S_n^t$) are strictly increasing and convergent.

Let $t^t$, $\theta^t$, $cl^t$, $s^t$, and $S^t$ denote the finite limits of the sequences ($t_n^t$), ($\theta_n^t$), ($cl_n^t$), ($s_n^t$), and ($S_n^t$), respectively. Their approximate values, calculated using the Z.3 model, rounded to the nearest hundredth, are as follows:

$$t^t \approx 0.93^\circ\text{C}, \theta^t \approx 0.87^\circ\text{C}, cl^t \approx 2.61\%, s^t \approx 17.40\%, \text{ and } S^t \approx 87.00 \text{ m.}$$

Except for minor fluctuations due to the Z.3 model accounting for natural variability, it follows from this proposition that the common tendency of the sequences ($t_n^t$), ($\theta_n^t$), ($cl_n^t$), ($s_n^t$), and ($S_n^t$) is continuous growth, capped by thresholds of the same order of magnitude as the finite limits described above. This is clearly demonstrated in the simulations presented in Figures 6-9 and corroborated by the means and 95% confidence intervals provided in Table S3, which were calculated from a sample of 250 simulations.

## 3.8. Comparison with Intergovernmental Panel on Climate Change forecasts

Figure 10 illustrates projections based on the Shared Socioeconomic Pathways (SSP) scenarios described in the contribution of Working Group I to the Sixth Assessment

<figure><img src="image_12.png" /><figcaption>Figure 10. Forecasts of global surface temperature evolution under different Shared Socioeconomic Pathways (SSP) scenarios</figcaption></figure>

Source: Masson-Delmotte et al. 38

Report of the IPCC (2021) (Masson-Delmotte et al. 38), the authors present projections by 2100 of global surface temperature in different scenarios of shared socioeconomic pathways (SSP): SSP1-1.9, SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5. For the intermediate scenario SSP2-4.5, Masson-Delmotte et al. 38 predict an increase of almost 2.5°C compared to the 1850-1900 with “very likely range”: (2.1°C; 3.5°C). Adding 0.85°C (observed increase in global surface temperature from 1850-1900 to 1995-2014) gives us an increase of 1.65°C (range [1.25°C; 2.65°C]) compared to 1995 – 2014. So around 2°C.

In the case of the Z.3 model, the constants used were calibrated by closely aligning simulations for the 1955-2023 period with historical observations. This alignment places the Z.3 model in an intermediate scenario, comparable to SSP2-4.5.

However, the simulations obtained by Z.3 predict an increase of around 1°C for the same period: 0.95°C with CI 95%: (0.59°C; 1.31°C). That is to say relating to significantly lower scenarios in terms of greenhouse gas emissions (SSP1-1.9 and SSP1-2.6). This difference can be attributed primarily to the Z.3 model’s incorporation of the strong negative feedback from the oceanic cloudiness, as highlighted by Zeltz⁸ and confirmed in the present work. In contrast, this feedback mechanism is not explicitly accounted for in the GCMs used in the IPCC reports. In addition, the asymptotic growth behavior toward a finite limit observed in the Z.3 model absent in the earlier Z.2 model is due to a revised consideration of stratification effects (line 9 of Table S2).

## 3.9. Strengths and limitations of the Z.3 model compared to models used by major climate institutes

The Z.3 model, developed primarily from ocean/ atmosphere energy balance equations, offers a tool that is both simple and quick to use, while still providing reliable results. Its accessibility makes it valuable for individuals with basic scientific knowledge and a standard microcomputer,
enabling them to perform simulations and gain a concrete understanding of the dynamics of climate change. This represents a significant improvement over the reliance on near-blind trust that non-specialists have traditionally placed in the results of more complex and effective models. Such advanced models, while powerful, are often inaccessible due to their dependence on sophisticated computing equipment, lengthy computation times, and a lack of transparency regarding the relationships and parameters employed.

However, the Z.3 model's simplicity also imposes inherent limitations. While it can simulate a few key parameters essential for understanding climate change, its rudimentary nature restricts its scope. Notably, it does not account for local changes in salinity caused by melting ice, precipitation anomalies, or increased river runoff driven by global warming, all of which affect stratification. Furthermore, the model, in its current form, cannot incorporate the potential to change the large-scale circulation of the global ocean, including its overturning circulation and horizontal flows (thermohaline circulation [THC]/meridional overturning circulation [MOC], commonly known as the "global conveyor belt").³⁹ These limitations present two interesting perspectives for future improvement of the Z.3 model, with undoubtedly many additional areas for refinement.

The Z.3 model, as it currently stands, integrates the mechanisms detailed in this article and previous works.⁷,⁸ It is capable of producing reliable simulations of five key climatic parameters – arguably the most critical for understanding the global climate – at a very low computational cost. However, it cannot replace the more sophisticated and comprehensive models employed by major climate institutes. Instead, it serves as a valuable complementary tool for climate research.

## 3.10. Comparison with the stochastic Hasselmann model

The stochastic Hasselmann model,⁴⁰ developed in 1976, remains in use today (e.g., Lin et al.⁴¹). Like the Z.3 model, it is based on the differing climatic “memories” of the oceans and the atmosphere and employs Markov chains. It is, therefore, worthwhile to highlight the key differences between the two approaches.

In Hasselmann's theory, short-term random noise (e.g., atmospheric weather) drives longer-term variations (e.g., red spectra at the ocean level). Mathematically, this is represented by an autoregressive process of the first order, where the next step, $y_{t+1}$, of the long variation depends on the previous state, $y_t$, scaled by the ocean's climatic memory, m, and disturbed by short-term variation $x_t$:

$$y_{t+1} = m y_t + x_t \qquad (XVI)$$

In contrast, the Z.3 model is guided by “signals” detected in the climate system. Specifically, it identifies Markov-1 alternating-type signals in atmospheric (monthly) and oceanic (quarterly) temperatures. The Z.2 model, developed in Zeltz,⁸ hypothesizes a direct interaction between these signals, where the difference in their periods is attributed to the vastly different climatic memories of the atmosphere and ocean.

Building on this interaction, the Z.3 model is fundamentally deterministic, incorporating random coefficients only to account for natural variability. This contrasts with Hasselmann's approach, which is more inherently stochastic. Despite these differences, both models underscore the importance of climate memory in understanding system dynamics.

## 4. Conclusion

This study demonstrates, through a detailed analysis of probabilistic signals present in relevant time series, that the ENSO exerts a significant influence on the biannual cycles of rise and fall in global average oceanic stratification. Conversely, the long-term increase in stratification and the deepening of the mixed layer r are predominantly explained by the warming of the UOS.

Our calculations indicate that the excess heat entering the UOS is dissipated as follows: out of a total of 40 units of additional heat, 34 units contribute to warming the UOS, five units are released as latent heat into the atmosphere, and one unit deepens the mixed layer by forming new lower strata, as described in Sections 3.3 and 3.4. This mechanism provides the primary explanation for the observed dual phenomenon of increasing stratification and deepening of the lower boundary layer. Additional factors, such as the intensification of storms (accounted for in the model) and the displacement of water masses from their regions of formation by ocean currents and eddies,⁴² also play a role. However, certain phenomena, particularly those related to changes in double diffusion driven by climate change – such as diffusive convection or salinization occurring in the expanding basins of the world's oceans⁴²⁻⁴⁵ – are not included in this work. This omission constitutes a significant limitation, adding to those noted in Section 3.9.

The Z.3 modeling, which builds on this and prior studies,⁷,⁸ incorporates five key climatic parameters: atmospheric temperature (t), UOS temperature (θ), oceanic cloudiness(cl), stratification (s), and the depth(S) of the mixed layer. The model reveals a common behavior across these parameters: growth toward an asymptotic threshold. The magnitude of variations projected for the
period 1955 – 2100 are as follows: for t, +1°C; for θ, +1°C; for cl, +3%; for s, +20%; and for S, +100 m.

While the mathematical demonstration provided by the model does not constitute absolute proof of these phenomena, it offers a compelling argument for their validity. If the model's projections are broadly accurate, these raise serious questions about the reliability of the scenarios envisaged by major climate institutes in recent IPCC reports. The robustness and validity of the results in this work, therefore, warrant thorough investigation, rather than dismissal. If these results reflect the climatic reality of future decades, many of the recommendations for policymakers provided by the IPCC would require significant reevaluation. Future work will address some of the limitations identified in this study, including the phenomena outlined above, to further refine the Z.3 model and its implications.

## Acknowledgments

None.

## Funding

None.

## Conflict of interest

The author declares no conflicts of interest.

## Author contributions

This is a single-authored article.

## Ethics approval and consent to participate

Not applicable.

## Consent for publication

Not applicable.

## Availability of data

All data and results obtained are free of any rights as long as the source is specified. In addition, the author can deliver the Z.3 model program by email to anyone who wishes it and whose expressed motivation suits the author.

## Further disclosure

Éric Zeltz is an independent scholar.

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