8. Bonus: Climate Sensitivity and Feedbacks

8. Bonus: Climate Sensitivity and Feedbacks#

Tutorials at the 2026 paleoCAMP | June 15–June 29, 2026

Jiang Zhu
jiangzhu@ucar.edu
Climate & Global Dynamics Laboratory
NSF National Center for Atmospheric Research

Learning Objectives#

Understand equilibrium climate sensitivity (ECS) and effective radiative forcing (ERF)
Compare ECS and the individual radiative feedbacks (Planck, lapse-rate, water-vapor, albedo, and cloud) across CMIP6 models
Estimate ECS and ERF from a CESM2 abrupt-4xCO2 simulation using the Gregory method

Time to learn: 15 minutes

Load Python packages

from datetime import timedelta

import xarray as xr
import numpy as np
import pandas as pd

import matplotlib.pyplot as plt

import warnings
warnings.filterwarnings("ignore")

Example 1: Look at Climate sensitivity and feedbacks in CMIP6 models with pre-computed values from Mark Zelinka#

https://github.com/mzelinka/cmip56_forcing_feedback_ecs
Zelinka, M. D., Myers, T. A., McCoy, D. T., Po-Chedley, S., Caldwell, P. M., Ceppi, P., et al. (2020). Causes of higher climate sensitivity in CMIP6 models. Geophysical Research Letters, 47, e2019GL085782. https://doi.org/10.1029/2019GL085782

url = "https://raw.githubusercontent.com/mzelinka/cmip56_forcing_feedback_ecs/master/CMIP6_ECS_ERF_fbks.txt"

df = pd.read_csv(
    url, sep=r'\s+', skiprows=10, skipfooter=4, engine='python',
    names=["Model", "Variant", "ECS", "ERF2x", "PL", "PL*", "LR",
           "LR*", "WV", "RH", "ALB", "CLD", "SWCLD",
           "LWCLD", "NET", "ERR"]
)

df = df[~df['Model'].str.startswith('-')].reset_index(drop=True)
df = df.drop(columns=["PL*", "LR*"])

df

	Model	Variant	ECS	ERF2x	PL	LR	WV	RH	ALB	CLD	SWCLD	LWCLD	NET	ERR
0	ACCESS-ESM1-5	r1i1p1f1	3.88	2.83	-3.27	-0.30	1.70	0.02	0.46	0.56	0.43	0.13	-0.73	0.12
1	AWI-CM-1-1-MR	r1i1p1f1	3.16	3.63	-3.23	-0.41	1.77	0.03	0.46	0.22	-0.31	0.53	-1.15	0.04
2	BCC-CSM2-MR	r1i1p1f1	3.02	3.11	-3.31	-0.36	1.67	-0.07	0.43	0.51	0.16	0.35	-1.03	0.04
3	BCC-ESM1	r1i1p1f1	3.25	3.01	-3.29	-0.35	1.66	-0.05	0.47	0.52	0.02	0.50	-0.92	0.07
4	CAMS-CSM1-0	r1i1p1f1	2.29	4.17	-3.31	-0.62	2.01	0.05	0.26	-0.36	-0.72	0.36	-1.82	0.19
5	CESM2	r1i1p1f1	5.15	3.27	-3.34	-0.53	1.87	0.00	0.40	0.96	0.79	0.17	-0.63	0.01
6	CESM2-FV2	r1i1p1f1	5.16	2.86	-3.30	-0.51	1.74	-0.07	0.43	1.06	0.95	0.11	-0.55	0.03
7	CESM2-WACCM	r1i1p1f1	4.68	3.30	-3.34	-0.54	1.86	-0.01	0.40	1.17	1.05	0.12	-0.71	-0.25
8	CESM2-WACCM-FV2	r1i1p1f1	4.80	2.92	-3.31	-0.52	1.75	-0.09	0.41	1.13	0.97	0.15	-0.61	-0.06
9	CIESM	r1i1p1f1	5.63	3.86	-3.25	-0.35	1.60	-0.10	0.29	0.81	0.68	0.13	-0.69	0.21
10	CMCC-CM2-SR5	r1i1p1f1	3.55	3.79	-3.32	-0.47	1.82	-0.01	0.34	0.52	0.48	0.04	-1.07	0.04
11	CMCC-ESM2	r1i1p1f1	3.58	3.75	-3.31	-0.47	1.84	0.02	0.37	0.50	0.45	0.05	-1.05	0.03
12	CNRM-CM6-1	r1i1p1f2	4.90	3.64	-3.29	-0.42	1.75	-0.02	0.53	0.55	-0.02	0.57	-0.74	0.13
13	CNRM-CM6-1-HR	r1i1p1f2	4.33	3.96	-3.33	-0.56	1.92	0.02	0.40	0.54	-0.10	0.64	-0.92	0.12
14	CNRM-ESM2-1	r1i1p1f2	4.79	2.97	-3.29	-0.47	1.83	0.01	0.51	0.56	0.03	0.53	-0.62	0.24
15	CanESM5	r1i1p1f1	5.64	3.68	-3.33	-0.61	1.97	0.07	0.47	0.80	-0.02	0.82	-0.65	0.03
16	E3SM-1-0	r1i1p1f1	5.31	3.33	-3.33	-0.54	1.86	0.01	0.34	0.94	0.75	0.18	-0.63	0.11
17	EC-Earth3	r8i1p1f1	4.26	3.22	-3.24	-0.16	1.68	0.11	0.60	0.31	0.05	0.26	-0.76	0.05
18	EC-Earth3-AerChem	r1i1p1f1	3.87	3.63	-3.24	-0.27	1.74	0.08	0.59	0.20	-0.09	0.28	-0.94	0.06
19	EC-Earth3-CC	r1i1p1f1	4.23	3.25	-3.23	-0.25	1.72	0.09	0.63	0.29	0.06	0.23	-0.77	0.07
20	EC-Earth3-Veg	r1i1p1f1	4.33	3.37	-3.26	-0.20	1.73	0.11	0.60	0.29	0.02	0.27	-0.78	0.07
21	EC-Earth3-Veg-LR	r1i1p1f1	4.23	3.25	-3.26	-0.29	1.72	0.05	0.56	0.41	0.12	0.28	-0.77	0.09
22	FGOALS-f3-L	r1i1p1f1	2.98	4.17	-3.20	-0.37	1.65	-0.05	0.40	-0.01	-0.21	0.20	-1.40	0.14
23	FGOALS-g3	r1i1p1f1	2.87	3.58	-3.29	-0.37	1.90	0.14	0.57	-0.01	-0.73	0.72	-1.25	-0.06
24	GFDL-CM4	r1i1p1f1	3.89	3.19	-3.24	-0.53	1.84	0.03	0.55	0.56	0.03	0.53	-0.82	-0.00
25	GFDL-ESM4	r1i1p1f1	2.65	3.77	-3.31	-0.68	1.94	-0.01	0.37	0.44	-0.15	0.59	-1.42	-0.19
26	GISS-E2-1-G	r1i1p1f1	2.71	3.95	-3.26	-1.03	2.19	-0.01	0.27	0.00	-0.63	0.64	-1.46	0.37
27	GISS-E2-1-H	r1i1p1f1	3.12	3.53	-3.22	-0.55	1.91	0.07	0.48	-0.03	-0.53	0.50	-1.13	0.27
28	GISS-E2-2-G	r1i1p1f1	2.43	3.65	-3.24	-0.84	2.11	0.03	0.39	-0.09	-0.88	0.79	-1.51	0.17
29	GISS-E2-2-H	r1i1p1f1	2.68	3.42	-3.19	-0.43	1.73	0.00	0.46	-0.04	-0.62	0.58	-1.28	0.20
30	HadGEM3-GC31-LL	r1i1p1f3	5.55	3.49	-3.27	-0.41	1.68	-0.07	0.41	0.79	0.98	-0.19	-0.63	0.18
31	HadGEM3-GC31-MM	r1i1p1f3	5.44	3.57	-3.29	-0.44	1.69	-0.08	0.35	0.87	1.03	-0.17	-0.66	0.17
32	IITM-ESM	r1i1p1f1	2.37	4.51	-3.28	-0.74	1.99	0.02	0.29	-0.08	-0.69	0.62	-1.91	-0.09
33	INM-CM4-8	r1i1p1f1	1.83	2.70	-3.29	-0.09	1.61	0.08	0.40	-0.13	-0.19	0.06	-1.48	0.02
34	INM-CM5-0	r1i1p1f1	1.92	2.92	-3.29	-0.11	1.58	0.03	0.45	-0.11	-0.11	-0.00	-1.52	-0.03
35	IPSL-CM5A2-INCA	r1i1p1f1	3.82	3.07	-3.31	-0.73	2.06	0.04	0.36	0.98	0.50	0.48	-0.81	-0.17
36	IPSL-CM6A-LR	r1i1p1f1	4.70	3.58	-3.28	-0.50	1.96	0.10	0.48	0.38	0.14	0.23	-0.76	0.21
37	IPSL-CM6A-LR-INCA	r1i1p1f1	4.13	3.15	-3.28	-0.52	1.93	0.07	0.49	0.43	0.19	0.24	-0.76	0.19
38	KACE-1-0-G	r1i1p1f1	4.75	3.41	-3.30	-0.51	1.76	-0.07	0.36	0.80	0.98	-0.19	-0.72	0.18
39	MIROC-ES2L	r1i1p1f2	2.66	4.11	-3.31	-0.70	1.98	-0.01	0.44	-0.02	-0.35	0.33	-1.54	0.07
40	MIROC6	r1i1p1f1	2.60	3.65	-3.34	-0.64	2.03	0.05	0.51	0.12	-0.13	0.26	-1.40	-0.10
41	MPI-ESM-1-2-HAM	r1i1p1f1	2.95	4.16	-3.16	-0.66	1.92	0.04	0.39	-0.23	-0.62	0.39	-1.41	0.33
42	MPI-ESM1-2-HR	r1i1p1f1	2.98	3.65	-3.20	-0.53	1.83	0.03	0.42	0.20	-0.41	0.61	-1.22	0.06
43	MPI-ESM1-2-LR	r1i1p1f1	3.03	4.22	-3.16	-0.76	1.97	0.02	0.39	0.12	-0.68	0.79	-1.39	0.05
44	MRI-ESM2-0	r1i1p1f1	3.13	3.43	-3.27	-0.48	1.71	-0.08	0.60	0.38	0.12	0.26	-1.10	-0.04
45	NESM3	r1i1p1f1	4.76	3.73	-3.15	-0.55	1.83	0.02	0.54	0.38	-0.15	0.53	-0.78	0.17
46	NorCPM1	r1i1p1f1	3.03	3.34	-3.23	-0.18	1.56	-0.00	0.50	0.23	-0.01	0.24	-1.10	0.01
47	NorESM2-LM	r1i1p1f1	2.56	3.43	-3.35	-0.67	2.03	0.03	0.40	0.36	0.21	0.15	-1.34	-0.10
48	NorESM2-MM	r1i1p1f1	2.49	3.73	-3.36	-0.75	2.11	0.06	0.34	0.43	0.30	0.14	-1.50	-0.27
49	SAM0-UNICON	r1i1p1f1	3.72	3.89	-3.33	-0.58	1.77	-0.13	0.42	0.69	0.89	-0.20	-1.04	-0.01
50	TaiESM1	r1i1p1f1	4.36	3.85	-3.31	-0.43	1.77	-0.02	0.43	0.64	0.55	0.09	-0.88	0.02
51	UKESM1-0-LL	r1i1p1f2	5.36	3.61	-3.26	-0.35	1.60	-0.10	0.53	0.81	0.93	-0.12	-0.67	-0.00

Quick plot of ECS in CMIP6 models#

df['ECS'].hist(bins=11, figsize=(5, 2))

<Axes: >

_images/bc1d6aafd066271c720a03620be665aa63f5e84ddafdfe34a5ce67851e0780e5.png

Plot of feedback strengths in CMIP6 models#

df["LR+WV"] = df["LR"] + df["WV"]

fbk_cols = ["PL", "LR", "WV", "LR+WV", "ALB", "CLD"]
fbk_colors = ["royalblue", "lightgreen", "tomato", "gold", "skyblue", "orange"]

fig, ax = plt.subplots(figsize=(10, 5))
x = np.arange(len(fbk_cols))

# MMM bars
mmm = [df[c].mean() for c in fbk_cols]
ax.bar(x, mmm, color=fbk_colors, edgecolor="k", linewidth=0.7, zorder=2, width=0.6)

# Label mean values
for i, (col, val) in enumerate(zip(fbk_cols, mmm)):
    std = df[col].std()
    if val >= 0:
        ypos = df[col].max() + 0.05
        ax.text(i, ypos, f"{val:.2f}±{std:.2f}", ha="center", va="bottom", fontsize=9, fontweight="bold")
    else:
        ypos = df[col].min() - 0.05
        ax.text(i, ypos, f"{val:.2f}±{std:.2f}", ha="center", va="top", fontsize=9, fontweight="bold")

# Individual model dots
for i, col in enumerate(fbk_cols):
    xjitter = np.random.uniform(-0.15, 0.15, len(df))
    ax.scatter(i + xjitter, df[col], color="k", s=15, alpha=0.5, zorder=3)

ax.axhline(0, color="k", linewidth=0.8)
ax.set_xticks(x)
ax.set_xticklabels(fbk_cols, fontsize=12)
ax.set_ylabel("Feedback (W m$^{-2}$ K$^{-1}$)", fontsize=12)
ax.set_title("CMIP6 Radiative Feedbacks\n(bars = MMM, dots = individual models)", fontsize=13)
ax.grid(axis="y", linestyle=":", alpha=0.5)

plt.tight_layout()
plt.show()

_images/a608eda830e0b1e9d118738db791edf03e71d12a5e94477c90c2f4b0f49ecc8e.png

Example 2: Use Gregory method to plot ECS and effective radiative forcing#

def load_ds(case, suffixes):
    files = [f"{archive}/{case}/atm/proc/tseries/month_1/{case}.cam.h0.{v}.{s}"
             for v in vnames for s in suffixes]

    ds = xr.open_mfdataset(
        files,
        combine='by_coords',
        data_vars='minimal',
        coords='minimal',
        compat='override',
        parallel=True
    )
    ds['time'] = ds.time.get_index('time') - timedelta(days=15)

    return ds.resample(time='YS').mean().compute()

archive = '/glade/campaign/collections/cmip/CMIP6/timeseries-cmip6'
vnames  = ['TS', 'FSNT', 'FLNT']

case1   = 'b.e21.B1850.f19_g17.CMIP6-piControl-2deg.001'
case2   = 'b.e21.BCO2x4.f19_g17.CMIP6-abrupt4xCO2-2deg.001'
suffix1 = ['030101-035012.nc', '035101-040012.nc', '040101-045012.nc']
suffix2 = ['000101-005012.nc', '005101-010012.nc', '010101-015012.nc']

ds_ctrl = load_ds(case1, suffix1)
ds_4co2 = load_ds(case2, suffix2)

Plot of the warming in 4xCO2#

fig, ax = plt.subplots(figsize=(4, 2))

ax.plot(np.arange(1, len(T_ctrl) + 1), T_ctrl, label='piControl')
ax.plot(np.arange(1, len(T_4co2) + 1), T_4co2, label='abrupt-4xCO2')
ax.set_xlabel('Year')
ax.set_ylabel('Global-mean TS (K)')
ax.legend(frameon=False)

<matplotlib.legend.Legend at 0x153cc3e51400>

_images/3edcc0263a0db5541f6db738e17ae1dbba20b050ab3d55b4c96a131e3c51b76b.png

Linear regression and make a plot#

slope, intercept = np.polyfit(dT, dR, 1)

ECS = -intercept / slope / 2
ERF =  intercept / 2

print(f"Slope (feedback): {slope:.3f} W m⁻² K⁻¹")
print(f"ERF (2xCO2):      {ERF:.3f} W m⁻²")
print(f"ECS:              {ECS:.3f} K")

Slope (feedback): -0.542 W m⁻² K⁻¹
ERF (2xCO2):      2.863 W m⁻²
ECS:              5.281 K

# Gregory plot
fig, ax = plt.subplots(figsize=(5, 3))

years = np.arange(len(dT))
sc = ax.scatter(dT.values, dR.values, c=years, cmap='plasma', s=20, zorder=3)
plt.colorbar(sc, ax=ax, label='Year')

# Regression line
dT_line = np.linspace(-1, 12, 100)
ax.plot(dT_line, slope * dT_line + intercept, 'k--', linewidth=1.5,
        label=f"slope = {slope:.2f} W m⁻² K⁻¹\nERF = {ERF:.2f} W m⁻²\nECS = {ECS:.2f} K")

ax.axhline(0, color='k', linewidth=0.8)
ax.axvline(0, color='k', linewidth=0.8)
ax.set_xlabel('ΔT (K)', fontsize=12)
ax.set_ylabel('ΔR (W m⁻²)', fontsize=12)
ax.set_title('Gregory Plot: CESM2 abrupt-4xCO2', fontsize=10)
ax.legend(fontsize=10, frameon=False)

<matplotlib.legend.Legend at 0x153c92fdb890>

_images/3f9f9a965696f2b2cdc5143b5a0a4499f24128b74afaa91594830fee5467dfef.png