Steric sea level variability (1993–2010) in an ensemble of ocean reanalyses and objective analyses

Abstract

Quantifying the effect of the seawater density changes on sea level variability is of crucial importance for climate change studies, as the sea level cumulative rise can be regarded as both an important climate change indicator and a possible danger for human activities in coastal areas. In this work, as part of the Ocean Reanalysis Intercomparison Project, the global and regional steric sea level changes are estimated and compared from an ensemble of 16 ocean reanalyses and 4 objective analyses. These estimates are initially compared with a satellite-derived (altimetry minus gravimetry) dataset for a short period (2003–2010). The ensemble mean exhibits a significant high correlation at both global and regional scale, and the ensemble of ocean reanalyses outperforms that of objective analyses, in particular in the Southern Ocean. The reanalysis ensemble mean thus represents a valuable tool for further analyses, although large uncertainties remain for the inter-annual trends. Within the extended intercomparison period that spans the altimetry era (1993–2010), we find that the ensemble of reanalyses and objective analyses are in good agreement, and both detect a trend of the global steric sea level of 1.0 and 1.1 ± 0.05 mm/year, respectively. However, the spread among the products of the halosteric component trend exceeds the mean trend itself, questioning the reliability of its estimate. This is related to the scarcity of salinity observations before the Argo era. Furthermore, the impact of deep ocean layers is non-negligible on the steric sea level variability (22 and 12 % for the layers below 700 and 1500 m of depth, respectively), although the small deep ocean trends are not significant with respect to the products spread.

Appendix: Mathematical definitions

We briefly introduce in this Appendix some mathematical definitions that are used in the paper.

1.1 Signal-to-spread ratio

In order to evaluate how distinguishable is the climate signal of the reanalysis ensemble with respect to its uncertainty, we define the signal-to-spread ratio ($SSR$), or more generally, the signal-to-noise ratio of the ensemble for a generic parameter $p$ as

$$\begin{aligned} SSR = \frac{EM}{ES} = \frac{{<}p{>}}{ \sqrt{1/N \sum _{i=1}^{i=N} ( p_i - {<}p{>})^2 } } \end{aligned}$$

(5)

with $EM$ and $ES$ being the ensemble mean and spread, respectively, and N being the ensemble size, with

$$\begin{aligned} {<}p{>} = 1/N \sum _{i=1}^{i=N} p_i. \end{aligned}$$

(6)

Note that the ensemble spread is defined as the sample standard deviation. Values of $SSR$ smaller (greater) than 1 indicate that the discrepancy of reanalyses is greater (less) than their mean signal.

1.2 Annual and seasonal decomposition

It is also useful to decompose the steric sea level signal onto the seasonal (annual and semi-annual) and linear trend (inter-annual) components. To do this, we assume that every time-series of the variable $x$ be of the form:

$$\begin{aligned} x(t) = m t + c + A_a \cos \left( \frac{2\pi }{12} t - \varphi _a\right) + A_s \cos \left( \frac{2\pi }{6}t - \varphi _s\right) + \varepsilon (t) \end{aligned}$$

(7)

where $t$ is the time (in months), $m$ is the linear trend, $A_a$ and $\varphi _a$ are the annual amplitude and angular phase, respectively, and $A_s$ and $\varphi _s$ are the semi-annual amplitude and angular phase and $\varepsilon$ are the residuals. The decomposition is carried out by a least-squares fitting of Eq. (7), i.e. by minimizing the sum of $\varepsilon ^2 (t)$.

The seasonal signal $x_S$, introduced in the text in Sects. 3 and 4, is defined as the full signal minus the linear trend:

$$\begin{aligned} x_S(t) = x(t) - m t, \end{aligned}$$

(8)

namely it corresponds to the detrended signal. Conversely, the inter-annual signal $x_I$, is defined as the full signal to which the fitted seasonal signal is subtracted:

$$\begin{aligned} x_I(t) = x(t) - A_a \cos \left( \frac{2\pi }{12} t - \varphi _a\right) + A_s \cos \left( \frac{2\pi }{6}t - \varphi _s\right) . \end{aligned}$$

(9)

In the above definitions, only the complementary fitted signal is removed, while the residuals are always kept. Note that the time-series corresponding to the inter-annual and seasonal signals have the same length of the time-series of the full signal, implying that the same minimum values for testing the significance of the correlations apply.

1.3 Explained variance

The (percentage) explained variance of a component $y$ with respect to the (total) component $z$ is defined as

$$\begin{aligned} EV(y) = 100 \frac{VAR(z)-VAR(z-y)}{VAR(z)}, \end{aligned}$$

(10)

with $VAR(...)$ being the variance operator. When the explained variance of the inter-annual signal is introduced (e.g. in Fig. 11), it means that the explained variance is calculated on the timeseries, after removal of the seasonal signal, for both components $y$ and $z$.