1. „Noise“ – an integral part of climate dynamics and analysis
Hans von Storch12, 陈学恩2 (Chen X.) and 唐声全12 (Tang S)
1Institute of Coastal Research, HZG Research Center, Germany 2Ocean University of China, Qingdao
20 min in Olomouc
30-45 min in Boguszów-Gorce (10 September Seminar of Applied Mathematics)
2 - 8 September IAMG 2018, Olomouc, Czech Republic

2. Quasi-realistic climate models
process-based, i.e., with state variables  and a variety of processes Pk.
maximum number of state variables and processes, as long as the system can still be integrated.
goal is complexity and detailed realism (numbers), not simplicity and understanding (knowledge). After having run the model, statistical and conceptual models are used to extract knowledge from the numbers.
serve as substitute reality, allowing for „numerical experiments“ and realistic simulation of the complex interaction of many processes.
have nonlinear components all over the place e.g., advective terms in equations of motion, or changes of phases (condensation and formation of precipitation)
are chaotic in the sense that a small initial disturbances causes „divergence“ at a later time ...

3. Noise-generating simulations
Variability in a 20-year simulation in the hierarchy of ocean models of the South China Sea with different resolutions
Atmospheric variability in a 1000 year coupled atmosphere-ocean simulation
Both simulations are “forced” with periodic (annual) conditions – no forcing on time scales longer than a year, and shorter that a month

4. Three ocean models are integrated with prescribed atmospheric forcing, which has no „weather“ (only monthly means are used), and is strictly periodic, with all Januaries identical, all Februaries identical etc.
One model is almost global, with a grid resolution of about 1o, a second embedded in the first covering most of SE Asia with a grid resolution of 0.2o, and a third one with an even smaller region, the South China Sea, and a grid resolution of 0.04o.
The first model is hardly describing macroturbulent eddy dynamics, but the other two models become better and better in doing so.
All three models generate inter-annual variability (e.g., summer-to-summer differences), as well as variability within seasons – the more so the better the macro-turbulence is covered.
The regions of the three-layer nested simulation, which includes an (almost) global model, a West-Pacific (WP) model and a South China Sea (SCS) model.
The spatial distributions of logarithm of the mean intra-seasonal variance of sea surface height in the SCS simulated in summer by tthe global model (left), WP model (middle) and the SCS model (right)).
Tang S., H. von Storch, and Chen X., 2018: “Noise” in climatologically driven ocean models with different grid resolution, submitted

5. Simulation with a coupled atmosphere-ocean general circulation model for 1000 years. The model is subject to periodic (annual) forcing of solar radiation, but otherwise no variable forcing is acting. (No CO2 increase, no volcanoes, no variable solar activity).
The model simulates the full range of thermo-dynamical variables, such as air temperature, sea surface temperature, currents and winds etc.
Even if no forcing on time scales longer than 1 year (the annual cycle) is present, the system generates significant and persistent anomalies.
The system generates “noise”, i.e., variability unprovoked by external factors.
1000 years
Zorita, 2003

6. We have smoke without fire.
Thus, these models of (components of) the climate system generate variability on all time scales, without being provoked for doing so.
We have smoke without fire.

7. Klaus Hasselmann‘s (1976) concept of the Stochastic climate model.
Before the introduction of that concept, and again more and more so in recent times, people were convinced that all remarkable developments in climate must be related to one or more external reasons.
He demonstrated that there is “smoke without fire” in climate variability, – low-frequency variability is generated as the integrated response to short-term fluctuations (as for instance macro-turbulent atmospheric flow at mid-latitudes).
Thus, a new task for climate analysis emerged, namely the question if an “anomaly” is just part of the internally generated low-frequency variability of caused by specific external causes (“detection and attribution”).
Dynamical systems are given by a differential equations system
(*) x‘ = D(x,f)
With the state variables x and the forcing f, and the dynamics D. Let us assume that x is made up of two components y and z, which fluctuate on long and on short time scaes: x = y + z,
After linearization of (*) we find for the slow component
y’ = d y + w
With d < 0, as x is stationary, and w represents independent short term variability (noise). After discretization we arrive at
Yt+1 =  yt + wt
If w is Gaussian, y is governed by an autoregressive (uni- or multivariate) process, with a ”red spectrum”.
Hasselmann, K., 1976: Stochastic climate models. Part I. Theory. Tellus 28,

8. The Mikolajewicz- case of the working of the stochastic climate model
Numerical experiment with global ocean model: standard simulation with steady forcing (wind, heat and fresh water fluxes)
plus random zero-mean precipitation overlaid.
Mikolajewicz, U. and E. Maier-Reimer, 1990: Internal secular variability in an OGCM. Climate Dyn. 4,
forcing
response

9. Noise as nuisance: masking the signal
The 300 hPa geopotential height fields in the Northern Hemisphere: the mean January field, the January 1971 field, which is closer to the mean field than most others, and the January 1981 field, which deviates significantly from the mean field. Units: 10 m
Noise as nuisance: masking the signal

10. Noise as a concealing element
The presence of noise hampers the identification of forced changes in the models (and in the climate system)
Interpreting numerical experiments with dynamical models. (Example: Influence of the shape of ocean surface waves on atmospheric state.)
Identifying the effect of anthropogenic influences on regional climate: “Detection and attribution” (Examples: recently changing climate statistics and their relation to elevated greenhouse gases and changed land-use.)

11. Example: Does the shape of the ocean wave spectrum affect atmospheric weather phenomena?
A numerical experiment with a coupled regional atmosphere-ocean-waves model (ECAWOM).
Two ensembles of simulations were generated: (CTR) The sea surface roughness is obtained from the standard Charnock relation. (ESD) The impact of waves is explicitly accounted for in the calculation of the sea surface roughness using a wind-over- waves coupling theory.
It is found here that the largest differences between the two ensembles occurred synchroneously with high inherent model variability. An possibly existing impact of the sea state–dependent roughness on the atmospheric circulation could not be discriminated from the background variability. Thus, the alternative hypothesis that the shape of the wave spectrum matters for the atmospheric state, could not be accepted at a given risk.
Local SLP differences in hPa between the ensemble means of the ESD and the CTR simulations. Areas for which the local t statistics exceed the value for the 95% confidence limit are indicated in gray.
Weisse, R., H. Heyen and H. von Storch, 2000: Sensitivity of a regional atmospheric model to a sea state dependent roughness and the need of ensemble calculations. Mon. Wea. Rev. 128:

12. Detecting externally caused climate change
Detection: Determination if observed variations are within the limits of variability of a given climate regime. If this regime is the undisturbed, this is internal variability (of which ENSO, NAO etc. are part). If not, then there must be an external (mix of) cause(s) foreign to the considered regime.
Detection is achieved by using a statistical test. It is associated with rigorous evidence.
Null hypothesis: the recent change is within the range of “normal” stationary variations.
The Rybski-et al. approach
Temporal development of Ti(m,L) = Ti(m) – Ti-L(m) divided by the standard deviation of the m-year mean reconstructed temp record
The thresholds R = 2, 2.5 and 3σ are given as dashed lines; they are derived from temperature variations modelled as Gaussian long-memory processes fitted to various reconstructions of historical temperature.
The null hypothesis of the recent changes of global mean temperature being in the range of historical trends, is rejected with very small risks.
An external cause must be at work, but this analysis does not inform which this cause may be.
Rybski, D., A. Bunde, S. Havlin,and H. von Storch, 2006: Long-term persistence in climate and the detection problem. Geophys. Res. Lett. 33, L06718, doi: /2005GL025591

13. Detecting externally caused climate change and attributing plausible causes
Attribution: In case of a positive detection: Determination of a mix of plausible external forcing mechanisms that best “explains” the detected deviations
Attribution is the result of a plausibility argument. It identifies a set of possible causes, which efficiently describe a cause, which was earlier detected as inconsistent with internal variability alone.
The case of Tropical South American rainfall in fall, and the influence of greenhouse gases (GHGs) and land-use (LU).
The recent trends are attributed to the simultaneous influence of GHGs and LU.
Observed precipitation trends in August-October over the 1983–2012 time period where externally forced changes are detectable (at 5% level).
The ellipse displays the joint two-dimensional 90% uncertainty interval for the greenhouse gas (GHG) and land-use change (LU) when observed data are regressed onto two signals simultaneously during 1983–2012. The whiskers indicate 95th percentile uncertainty intervals for the two signals.
Barkhordarian, A., H. von Storch, A. Behrangi, P. C. Loikith, C. R. Mechoso, and J. Detzer, 2018: Simultaneous regional detection of land-use changes and elevated GHG levels: the case of springtime precipitation in tropical South America. Geophys Res. Letters, DOI: /2018GL078041

14. Conclusion
„Noise“ is ubiquitous in the climate system - on all scales - due to myriads of nonlinear mechanisms It is not relevant, if this variability is “real” noise or if it is just impossible to discriminate the variability from “real noise”. Note: “Real” noise is a –most useful- mathematical construct.
The presence of noise changes the dynamics of the system – the mean circulation would be different if there were no storms; the storms were different if there were not convective cells …
Noise hinders the identification of externally forced signals (and the utility of forecasts), and the attribution of anomalies to specific causes.