# Emergent Scale Invariance and Climate Sensitivity

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## Abstract

**:**

## 1. Introduction

## 2. Linear Response Models and Scale-Dependent Sensitivity

#### 2.1. The 1-Box Energy Balance Model

#### 2.2. Generalizations of the 1-Box Model

#### 2.3. Scale-Invariant Models

## 3. Materials and Methods

#### 3.1. Data

#### 3.2. Parameter Estimation

## 4. Results

## 5. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ECS | Equilibrium climate sensitivity |

ESM | Earth system model |

IPCC | Intergovernmental Panel on Climate Change |

GMST | Global mean surface temperature |

CERES | Clouds and Earth’s Radiant Energy System |

CMIP5 | Coupled Model Intercomparison Project Phase 5 |

ENSO | El Niño Southern Oscillation |

PSD | Power Spectral Density |

LRD | Long-range dependence |

fGn | fractional Gaussian noise |

Probability density function | |

INLA | Integrated nested Laplace approximation |

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**Figure 1.**(

**a**) The red curve is the adjusted forcing for the NorESM1-M model provided by Forster et al. [15]. The black curve is the forcing data of Hansen et al. [16] modified so that its 17-year moving average equals the 17-year moving average of the red curve. (

**b**) The black curve is the global surface temperature in the historical run of the NorESM1-M model, and the blue curve is the response to the modified Hansen forcing (the black curve in (

**a**)) for the model given by Equation (5). Parameters are estimated as $\beta =0.67$ and $\mu =7.8\times {10}^{-3}$ y${}^{-1}$.

**Figure 2.**(

**a**) The sloping lines are double-logarithmic plots of the scale dependent sensitivity $R\left(f\right)$ for each ESM in the ensemble. The different slopes correspond to different $\beta $-estimates. The horizontal lines indicate the ECS of the ESMs obtained from the Gregory plots and reported in [1], and the black dots indicate for which frequency f we have $R\left(f\right)=\mathrm{ECS}$ for each model. (

**b**) Correlation (over the ensemble of ESMs) between the scale-dependent sensitivity $R\left(f\right)$ and the Gregory estimate of the ECS. The correlation coefficient is plotted as a function of the frequency f.

**Figure 3.**(

**a**) The letters (see the legend inserted in panel (

**b**)) show the Gregory estimate of ECS versus R(f)m evaluated at $f={10}^{-3}$ y${}^{-1}$ for each model in the ensemble. The contour plot shows the conditional probability density function (PDF) $p\left(\mathrm{ECS}\right|R)$. The vertical black line is $R=2.9$ K, which is obtained from the parameters $\beta =0.66$ and $\mu =11.9\times {10}^{-3}$ y${}^{-1}$ estimated from the instrumental temperature record using the Hansen forcing. The thick, black curve is the estimated PDF of $R\left(f\right)$. (

**b**) The full curve shows the PDF for ECS computed from Equation (14), where $p\left(R\right)$ is the PDF shown in (

**a**). The histogram is the distribution of ECS in the model ensemble, and the dotted curve is a Gaussian fit to the histogram.

**Figure 4.**(

**a**) The points in the scatter plot are the same as (the letters) in Figure 3a. The line is the least-square fit of the model $\mathrm{ECS}=aR+b$. The vertical lines correspond to the R-values estimated from the instrumental temperature record for the different modified Hansen forcing time series. The horizontal lines show how these R-values are mapped to ECS-values by the linear model. (

**b**) As in (

**a**), but prior to the analysis the volcanic forcing is reduced to half of its original values. (

**c**) As in (

**b**), but using a linear model $\mathrm{ECS}=aR$ with zero intercept to map R-values to ECS-values.

**Table 1.**Estimated quantities for the Earth system model (ESM) in the ensemble. The columns for equilibrium climate sensitivity (ECS) and ${F}_{2\times {\mathrm{CO}}_{2}}$ are obtained from Gregory plots for 4 × CO${}_{2}$ runs and taken from [1]. The columns denoted $\beta $, $\mu $ and $\sigma $ show the estimates of the parameters in the model given by Equation (2) with response function given by Equation (5), obtained from historical runs of the ESMs and the modified Hansen forcing for each model. The last column displays the scale-dependent sensitivity $R\left(f\right)$ obtained from Equation (6) using the values of ${F}_{2\times {\mathrm{CO}}_{2}}$, $\beta $ and $\mu $ that are listed in the columns to the left, and evaluated at $f={10}^{-3}$ y${}^{-1}$.

Model | ECS (K) | ${\mathit{F}}_{2\times {\mathbf{CO}}_{2}}$ (W/m${}^{2}$) | $\mathit{\beta}$ | $\mathit{\mu}$ (${10}^{-3}$ y) | $\mathit{\sigma}$ (W/m${}^{2}$) | R (K) * |
---|---|---|---|---|---|---|

GISS-E2-R | 2.1 | 3.8 | 0.49 | 13.8 | 0.07 | 3.3 |

HadGEM2-ES | 4.6 | 2.9 | 0.95 | 7.3 | 0.32 | 4.8 |

IPSL-CM5A-LR | 2.6 | 3.1 | 0.79 | 9.6 | 0.16 | 4.0 |

NorESM1-M | 2.8 | 3.1 | 0.67 | 7.8 | 0.12 | 2.7 |

Access1-0 | 3.8 | 3.0 | 0.68 | 7.3 | 0.10 | 2.6 |

Miroc-ESM | 4.7 | 4.3 | 0.73 | 6.6 | 0.12 | 3.9 |

Miroc5 | 2.7 | 4.1 | 0.78 | 4.0 | 0.21 | 3.2 |

CanESM2 | 3.7 | 4.1 | 0.59 | 17.5 | 0.15 | 4.8 |

CCSM4 | 2.9 | 3.8 | 0.49 | 14.9 | 0.12 | 3.5 |

CNRM-CM5 | 3.3 | 3.6 | 0.60 | 15.0 | 0.12 | 3.9 |

GFDL-CM3 | 4.0 | 3.0 | 0.62 | 19.0 | 0.14 | 4.0 |

GFDL-ESM2G | 2.4 | 3.1 | 0.72 | 5.8 | 0.15 | 2.5 |

CSIRO-MK3 | 4.1 | 2.6 | 0.82 | 8.9 | 0.17 | 3.4 |

BCC-CSM1-1M | 2.8 | 3.2 | 0.53 | 15.8 | 0.09 | 3.2 |

GFDL-ESM2m | 2.4 | 3.1 | 0.47 | 15.3 | 0.16 | 2.8 |

INM-CM4 | 2.1 | 3.0 | 0.82 | 1.6 | 0.12 | 1.6 |

MPI-ESM-LR | 3.6 | 4.1 | 0.78 | 7.6 | 0.16 | 4.5 |

MRI-CGCM3 | 2.6 | 3.2 | 0.58 | 9.9 | 0.10 | 2.6 |

**Table 2.**The parameter estimates for the instrumental temperature record, and the resulting values of R, for varying forcing data.

Forcing | $\mathit{\beta}$ | $\mathit{\mu}$ (${10}^{-3}$ y) | $\mathit{\sigma}$ (W/m${}^{2}$) | R (K) * |
---|---|---|---|---|

GISS-E2-R | 0.61 | 5.8 | 0.12 | 2.4 |

HadGEM2-ES | 0.87 | 5.3 | 0.20 | 3.3 |

IPSL-CM5A-LR | 0.64 | 6.4 | 0.12 | 2.2 |

NorESM1-M | 0.74 | 6.3 | 0.14 | 2.8 |

Access1-0 | 0.67 | 6.3 | 0.12 | 2.3 |

Miroc-ESM | 0.68 | 7.6 | 0.14 | 3.7 |

Miroc5 | 0.68 | 6.6 | 0.12 | 3.3 |

CanESM2 | 0.68 | 5.5 | 0.13 | 3.0 |

CCSM4 | 0.62 | 4.5 | 0.13 | 2.1 |

CNRM-CM5 | 0.69 | 8.2 | 0.13 | 3.3 |

GFDL-CM3 | 0.86 | 5.6 | 0.19 | 3.4 |

GFDL-ESM2G | 0.75 | 4.4 | 0.14 | 2.3 |

CSIRO-MK3 | 0.86 | 6.0 | 0.19 | 3.1 |

BCC-CSM1-1M | 0.70 | 3.9 | 0.13 | 2.0 |

GFDL-ESM2m | 0.73 | 4.9 | 0.14 | 2.4 |

INM-CM4 | 0.67 | 5.9 | 0.12 | 2.2 |

MPI-ESM-LR | 0.81 | 1.8 | 0.15 | 2.3 |

MRI-CGCM3 | 0.79 | 6.9 | 0.15 | 3.4 |

Ensemble Run | $\mathit{\beta}$ | $\mathit{\mu}$ (${10}^{-3}$ y) | $\mathit{\sigma}$ (W/m${}^{2}$) | R (K) * |
---|---|---|---|---|

1 | 0.82 | 13.0 | 0.17 | 3.5 |

2 | 0.91 | 8.0 | 0.23 | 3.9 |

3 | 0.80 | 19.4 | 0.16 | 4.1 |

4 | 0.88 | 13.6 | 0.23 | 4.4 |

5 | 0.73 | 21.8 | 0.14 | 3.6 |

6 | 0.89 | 9.9 | 0.22 | 3.8 |

7 | 0.82 | 16.9 | 0.17 | 4.1 |

8 | 0.87 | 9.5 | 0.19 | 3.4 |

9 | 0.86 | 11.6 | 0.19 | 3.8 |

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**MDPI and ACS Style**

Rypdal, M.; Fredriksen, H.-B.; Myrvoll-Nilsen, E.; Rypdal, K.; Sørbye, S.H.
Emergent Scale Invariance and Climate Sensitivity. *Climate* **2018**, *6*, 93.
https://doi.org/10.3390/cli6040093

**AMA Style**

Rypdal M, Fredriksen H-B, Myrvoll-Nilsen E, Rypdal K, Sørbye SH.
Emergent Scale Invariance and Climate Sensitivity. *Climate*. 2018; 6(4):93.
https://doi.org/10.3390/cli6040093

**Chicago/Turabian Style**

Rypdal, Martin, Hege-Beate Fredriksen, Eirik Myrvoll-Nilsen, Kristoffer Rypdal, and Sigrunn H. Sørbye.
2018. "Emergent Scale Invariance and Climate Sensitivity" *Climate* 6, no. 4: 93.
https://doi.org/10.3390/cli6040093