# World Ocean Thermocline Weakening and Isothermal Layer Warming
## 3 Methods  
### Describing Mixed Layer Dynamics  
The connection between \(G\) (unit: \({}^{\circ}\)C m\({}^{-1}\)) and the ITL heat content \(H_{ITL}\) (unit: J m\({}^{-2}\)) can be found using the conservation of heat in the ITL [1; 3; 4; 5; 26]  
\[\begin{array}{l}H_{ITL}=\rho_{0}c_{p}hT_{I}\\ \frac{\partial T_{I}}{\partial t}=-\mathbf{v}\bullet\
abla T_{I}-\frac{Q}{ \rho_{0}c_{p}h}-\Lambda\frac{w_{e}}{h}\Delta T,\Delta T\equiv T_{I}(t, \mathbf{r})-T_{th}(t,\mathbf{r})\end{array} \tag{1}\]  
where \(\mathbf{v}\) (unit: m s\({}^{-1}\))is the bulk ITL horizontal velocity vector; \(h\) is the isothermal layer depth (unit: m); \(T_{I}\) is the ITL temperature (unit: \({}^{\circ}\)C); \(Q\) (unit: W m\({}^{-2}\)) is the net heat flux at the ocean surface (upward positive); \(w_{e}\) (unit: m s\({}^{-1}\)) is the entrainment velocity; \(\Lambda\) is the step function taking 0 if \(w_{e}<0\), and 1  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mean (m)** & **Standard Deviation (m)** & **Skewness** & **Kurtosis** \\ \hline
1970–1980 & 38.8 & 41.7 & 3.97 & 27.3 \\
1981–1990 & 55.0 & 54.6 & 3.22 & 21.0 \\
1991–2000 & 62.8 & 58.0 & 2.95 & 18.4 \\
2001–2010 & 66.3 & 61.5 & 2.82 & 17.0 \\
2011–2017 & 67.5 & 61.7 & 2.70 & 15.1 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Decadal variation of statistical characteristics of the global isothermal layer depth (\(h\)) in comparison to climatology.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Mean (\({}^{\circ}\)C/m)** & **Standard Deviation (\({}^{\circ}\)C/m)** & **Skewness** & **Kurtosis** \\ \hline
1970–1980 & 0.204 & 0.167 & 1.35 & 5.02 \\
1981–1990 & 0.162 & 0.150 & 1.77 & 6.61 \\
1991–2000 & 0.130 & 0.133 & 2.29 & 9.47 \\
2001–2010 & 0.124 & 0.132 & 2.35 & 9.69 \\
2011–2017 & 0.121 & 0.134 & 2.55 & 10.90 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Decadal variation of statistical characteristics of the global thermocline gradient (\(G\)) in comparison to climatology.  
otherwise due to the second law of the thermodynamics; \(\Delta T\) is the temperature difference between ITL and thermocline; \(T_{th}(t,\textbf{r})\) is the thermocline temperature (unit: \({}^{\circ}\)C). It is reasonable to assume that the temperature difference is proportional to the thermocline gradient, \(\Delta T=\gamma G\), with the proportionality \(\gamma\) (unit: m). Using conservation of mass in the ITL gives the \(H_{ITL}\) equation  
\[\frac{\partial H_{ITL}}{\partial t}=-\textbf{v}\bullet\
abla H_{ITL}-\rho_{0}c _{p}h\textbf{v}\bullet\
abla\hat{T}_{I}-\rho_{0}c_{p}\Big{(}T_{I}-\hat{T}_{I} \Big{)}w_{-h}-Q-\Lambda\rho_{0}c_{p}w_{c}\Big{[}\gamma G-\Big{(}T_{I}-\hat{T}_ {I}\Big{)}\Big{]} \tag{2}\]  
where \(\hat{T}_{I}\) is the temperature below the ITL; the last term in the right-hand side of (2) occurs only the mixed layer dynamics and thermodynamics are included. For the entrainment regime (\(w_{e}>0\), \(\Lambda=1\)), the thermocline gradient \(G\) is related negatively to \(\partial H_{ITL}\partial t\). Strong (weak) thermocline gradient causes the decrease (increase) in \(H_{ITL}\). Such an entrainment effect does not show up with the upper ocean heat content with fixed depth ranges commonly used in oceanography such as 0-300 m, 0-400 m, 0-700 m, 0-750 m, and 0-2000 m.