Hasselmann reloaded: internal variability in marginal seas

Ensembles of regional ocean model simulations show the generation of intrinsic variability to be tied to the annual cycle and the activation of tides. This sensitivity can be attributed to"memory"- the central element in the Stochastic Climate Model of Klaus Hasselmann.


In 1976, Klaus Hasselmann initiated a groundbreaking shift in climate science when he introduced the Stochastic Climate Model (SCM) to the field, a conceptual framework aimed at elucidating the origins of red spectra observed in the dynamics of both the atmosphere and the oceans. While this concept had somewhat faded from the limelight as climate science increasingly morphed into climate change science, it regained prominence within the climate science community when Klaus Hasselmann was awarded the Nobel Prize in Physics in 2021.

Historically, the primary use of the SCM centered on explaining why the spectra of atmospheric and oceanic dynamics were continuous and ''red'' rather than a series of peaks. Our present paper  has expanded the scope of the SCM's applicability, when we used it as a valuable tool for understanding the generation of intrinsic variability (often referred to as ''noise'') within the hydrodynamics of a marginal sea. Our analysis of multiple simulation ensembles using a regional ocean model has uncovered a relationship between the generation of intrinsic variability tied to the annual cycle and the activation of tides. This relationship can be attributed to a single parameter, and notably, this parameter corresponds to the central element in the SCM—referred to as the "memory."

Some Context and Details
The emergence of noise, or unforced internal variability, in global atmospheric simulations was first recognized by Chervin et al. [1974]. After another 30 years later, internal variability was recognized in ocean simulations with eddy-resolving models (e.g., Jochum and Murtugudde [2004], Penduff et al. [2018]). Their research showed that in the presence of mesoscale turbulence, oceanic signals cannot be entirely attributed to atmospheric variability. Noise in regional models was first described for the atmosphere by Ji and Vernekar [1997], and for regional oceans by chmann and Söderkvist [2016], Tang et al. [2019], Lin et al. [2022].

Already in the 1970s, researchers postulated that the variability in the climate system is made of a few spikes, in particular the annual cycle, some broader peaks, and a smooth background, as depicted in Figure 1.
Mitchell's climate spectrum
Figure 1: Mitchell*s climate spectrum from 1976
It was Klaus Hasselmann’s (1976) strike of a genius, who noticed the separation of time scales, short-term weather variability, and long-term climate variations, which led him to the Stochastic Climate Model, which in its simplest form is a first-order auto-regressive process of first order (cf., von Storch and Heimbach [2022]):

Xt+1 = α · Xt + εt                    (1)

with Xt a state variable, α a term
representing the memory of the system, and εt short-term white noise. This is a statistical summary of the development of a complex system, which features many non-linearities, but in this summary (1) there is no non-linearity at work, but merely a white noise forcing and a negative feedback representing the memory of the system.
Such an auto-regressive process (1) has a continuous red spectrum, and we apply it to study the formation of unforced variability in an ensemble of regional simulations of the hydrodynamics in the Bo Hai and Yellow Sea. In our simulation, we noticed that
  • when we initiate the simulations at shifted times, but otherwise consistent
    atmospheric data [Lin et al., 2022], or run the same code on different computers [Lin et al., 2023b], there is considerable variability across the members of the ensemble.
  • this variability is significantly reduced, when tides were activated in the simulations [Lin et al., 2022], and
  • this variability showed a marked annual cycle, with a maximum in summer and a minimum in winter [Lin et al., 2022].

The question is, what would be behind this sensitivity of noise generation. Normally, this is done with a process-based analysis, when the significance of a diversity of processes is screened. Here, we use a different approach, namely, if these changes may be related to changes in the memory of the system. This memory is quantified by the time lag when the autocorrelation function has decayed to a certain threshold value.

Technically, the five members of our system were averaged each day, to obtain an estimate of the ”signal”. The ”noise” was taken to be the standard deviation of the deviation of this signal. The second step was to derive Empirical Orthogonal Functions from joint noise ensembles, with active and passive tides. These EOFs are not only ordered according to the intensity of the variability but also by the associated spatial scales.

Fig from paper
Figure 2: From the article Lin et al. [2023a]: Redness of the spectra of variations on time scales spatial scales larger or equal to 30 km. The blue stars and the red crosses represent the slopes of the spectra from with-tide and no-tide ensembles, respectively.on

The memory of time series of the EOF coefficients showed a significant difference in the two ensembles, with considerably reduced memory when tides are active, the more so, the larger the spatial scales are – in consistency with (1) – see Figure 2.

We also determined for all EOFs the memory in summer and winter in both ensembles, with and without tides (Figure 3), and found in both cases a marked reduction from summer to winter, which is consistent with the maximum of noise intensity in summer as opposed to a minimum in winter.

Figure 3: From the article Lin et al. [2023a]: Memories of the first 50 EOFs in summer and winter.
Squares are for no-tide ensembles; circles are for with-tide costumes.

So what?
To sum up, by adopting the Stochastic Climate Model,
we could relate changes in the overall hydrodynamics of a marginal sea to a system property, as opposed to pin-pointing a specific process - the latter we are trying to do in a separate companion paper.

This article is part of a series of papers, dealing with the characteristics of unforced (intrinsic, internal) variability in marginal seas. The main overall result is that unforced variability is a real issue in regional ocean models, a fact, that some reviewers and editors found difficult to accept. The claim that such noise would not emerge if tides are active, has been falsified - the phenomenon is indeed reduced but does not disappear.

A significant consequence is that when doing numerical experiments, say on the effect of including or modifying the representation of a process in a model, a separation of signal and noise is needed (as originally sketched by Chervin et al. [1974]), for instance by applying the technique of statistical hypothesis testing.

We had the privilege to be discretely motivated by Klaus and Susanne Hasselmann as well as Jinsong von Storch (Figure 4)

Figure 4: With spiritual supporters of our work, from right: Klaus and Susanne Hasselmann, Lin Lin, Jinsong and Hans von Storch.
Foto: Patriotische Gesellschaft Hamburg, Christian Augustin, 2022; with permission.

Our work was part of the long-year cooperation between the Helmholtz Zentrum hereon in Geesthacht and the Ocean University of China in Qingdao, which will be documented in 2024 by an anthology assembling many papers of this cooperation.


B. Büchmann and J. Söderkvist. Internal variability of a 3-d ocean model. Tellus A, 2016. doi: 10.3402/tellusa.v68.30417.
R. Chervin, W. Gates, and S. Schneider. The effect of the time averaging on the noise level of climatological statistics generated by atmospheric general circulation models. J. Atmos. Sci., 31:2216–2219, 1974.
K. Hasselmann. Stochastic climate models. Part I. Theory. Tellus, 28:473–485, 1976.
Y. Ji and A. D. Vernekar. Simulation of the Asian summer monsoons of 1987 and 1988 with a regional model nested in a global GCM. Journal of Climate, 10:1965–1979, 1997.
M. Jochum and R. Murtugudde. Internal variability of the tropical Pacific Ocean. Geophysical Research Letters, 31, 2004. doi: 10.1029/2004GL020488.
L. Lin, H. von Storch, D. Guo, S. Tang, P. Zheng, and X. Chen. The effect of tides on internal variability in the Bohai and Yellow Sea. Dynamics of Atmospheres and Oceans, 98:101301, 2022. doi: 10.1016/j.dynatmoce.2022.101301.
L. Lin, H. von Storch, and X. Chen. The Stochastic climate model helps reveal the role of memory in internal variability in the Bohai and Yellow Sea. Communications Earth & Environment, 2023a. doi: 10.1038/s43247-023-01018-7.
L. Lin, H. von Storch, and X. Chen. Seeding noise in ensembles of marginal sea
simulations – the case of Bohai and Yellow Sea. Advances in Computer and
Communication, 4:70–73, 2023b. doi: 10.26855/acc.2023.04.001.
J. Mitchell. An overview of climatic variability and its causal mechanisms. Quaternary Research, 6:481–493, 1976. doi: 10.1016/0033-5894(76)90021-1.
T. Penduff, G. Sérazin, S. Leroux, S. Close, J.-M. Molines, B. Barnier, L. Bessieres, L. Terray, and G. Maze. Chaotic variability of ocean heat content: Climate relevant features and observational implications. Oceanography, 2018. doi: 10.5670/oceanog.2018.210.
S. Tang, H. von Storch, X. Chen, and M. Zhang. “Noise” in climatologically driven ocean models with different grid resolutions. Oceanologia, 61(3):300–307, 2019. doi: 10.1016/j.oceano.2019.01.001.
H. von Storch and P. Heimbach. Klaus Hasselmann recipient of the Nobel Prize in Physics 2021. Oxford University Press Research Encylopedia of Climate Science, 2022. doi: 10.1093/acre-

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