Reconstructing the subsurface ocean decadal variability using surface nudging in a perfect model framework

Abstract

Initialising the ocean internal variability for decadal predictability studies is a new area of research and a variety of ad hoc methods are currently proposed. In this study, we explore how nudging with sea surface temperature (SST) and salinity (SSS) can reconstruct the three-dimensional variability of the ocean in a perfect model framework. This approach builds on the hypothesis that oceanic processes themselves will transport the surface information into the ocean interior as seen in ocean-only simulations. Five nudged simulations are designed to reconstruct a 150 years “target” simulation, defined as a portion of a long control simulation. The nudged simulations differ by the variables restored to, SST or SST + SSS, and by the area where the nudging is applied. The strength of the heat flux feedback is diagnosed from observations and the restoring coefficients for SSS use the same time-scale. We observed that this choice prevents spurious convection at high latitudes and near sea-ice border when nudging both SST and SSS. In the tropics, nudging the SST is enough to reconstruct the tropical atmosphere circulation and the associated dynamical and thermodynamical impacts on the underlying ocean. In the tropical Pacific Ocean, the profiles for temperature show a significant correlation from the surface down to 2,000 m, due to dynamical adjustment of the isopycnals. At mid-to-high latitudes, SSS nudging is required to reconstruct both the temperature and the salinity below the seasonal thermocline. This is particularly true in the North Atlantic where adding SSS nudging enables to reconstruct the deep convection regions of the target. By initiating a previously documented 20-year cycle of the model, the SST + SSS nudging is also able to reproduce most of the AMOC variations, a key source of decadal predictability. Reconstruction at depth does not significantly improve with amount of time spent nudging and the efficiency of the surface nudging rather depends on the period/events considered. The joint SST + SSS nudging applied everywhere is the most efficient approach. It ensures that the right water masses are formed at the right surface density, the subsequent circulation, subduction and deep convection further transporting them at depth. The results of this study underline the potential key role of SSS for decadal predictability and further make the case for sustained large-scale observations of this field.

Appendix: Statistical methods

The influence of the nudging method is assessed by quantifying the agreement in terms of temporal variations between the target and the nudged simulations with the help of statistical skill scores.

The first score is the correlation coefficient R (Eq. (1), significance estimated with a Student t test) to quantify the temporal agreement between the target and the nudged simulations.

$$R = \frac{1}{T - 1}\frac{{\mathop \sum \nolimits_{t = 1}^{T} \left( {\left( {X\left( t \right) - \bar{X}} \right)\left( {Y\left( t \right) - \bar{Y}} \right)} \right)}}{{\sigma_{X} \sigma_{Y} }}$$

X and Y are the same variable, X is the nudged simulation and Y the target; t represents the time, T is the length of the period, the overbar denotes a time average over which this quantity is computed (usually the length of the nudged simulations) and $\sigma_{X}$ and σ _Y are the standard deviation of X and Y respectively. The Pearson correlation coefficient (commonly used correlation coefficient) is used for Gaussian variables, like temperature, salinity and AMOC index. For non-Gaussian variables (precipitation and mixed-layer depth), a Spearman’s rank correlation coefficient is used.

To compare the amplitude of the simulated variability with the target, a Fisher F test on the ratio of variance is performed (Von Storch and Zwiers (2002)). For non-Gaussian variables (such as precipitation), the Fisher F-test is applied on the logarithm of the strictly positive values of the time series.

Estimating the significance of a correlation on 150 years of an oceanic variable as a function of depth has to deal with a low number of degrees of freedom of the time series, due to the presence of potentially dominating low frequency (multi-decadal and more). An iterative detrending procedure (see Boer (2004)) has therefore been used to remove this low frequency: a linear trend is first removed from the time series at the first iteration, and the amount of the total variance of the detrended time series that this trend represents is evaluated. If it represents more than 5 % of the variance of the detrended time series, a spline of growing order (second, third…) is iteratively removed until the variance of the lastly removed trend represents less than 5 % of the variance of the detrended time series. With this procedure, the highest order of spline removed was six. We computed the autocorrelation function of a set of randomly chosen detrended time series, and checked that this detrending procedure retains the decadal variability. It is important to note here that this detrending procedure allows estimating correlation coefficients on time series with more than ten degrees of freedom, which is much more relevant in terms of statistics than on time series presenting important trends and having not more than five degrees of freedom. However, the decadal variability of a temperature time series at more than one thousand meters in depth can be quite small compared to the amplitude of the trends. This limitation has been kept in mind when interpreting the results.

The statistical significance is computed by taking into account the serial correlation of the time series in the estimation of the number of degrees of freedom (Bretherton et al. 1999). Nevertheless, and despite the detrending procedure, some significant correlations between the free run and the target still appear. Because no causal relationship behind these correlations is assumed, we have highlighted the correlation coefficients between the nudged simulations and the target that are not significantly different (test of the difference of correlation based on the Fisher transform at the 99 % level) in absolute value from the corresponding correlation coefficient in absolute value between the target and the free run. Correlation coefficients failing this last test have to be interpreted carefully.

A bootstrap method using the 1,000 years of the control simulation was also used to estimate the significance of the correlation and gave the same estimation of significance than the method described above.

In this study, a variable or a phenomenon is qualified as “reconstructed” or “reproduced” when a certain level of agreement between a given nudged simulation and the target, and as computed with these statistical tools, has been shown.