Mathematical Theory of Radiative Feedbacks

Overview

This tutorial explains the ways to calculate radiative feedbacks, the math behind the calculations, and their respective advantages and limitations.

After this tutorial, you will know:

  1. what radiative kernels are;

  2. the calculations of radiative feedback with radiative kernels; and

  3. the advantages, limitations, assumptions and uncertainties of the radiative kernel method.

Prerequisites

The following concepts will be referred to or explicitly used in this tutorial.

Concepts

Importance

Notes

Calculus

Necessary

we love math

  • Time to learn: 24-7 :)

Methods to Calculate Raditaive Feedback

Typically radiative feedback are calculated with global climate models.

  • Partial Radiative Perturbation (PRP) method (Wetherald and Manabe 1988)

  • Cloud Radiative Forcing (CRF) approach (Cess et al. 1990, 1996)

  • Online Feedback Suppression Approach (Hall and Manabe 1999, Schneider et al. 1999)

  • Radiative Kernel Methods (Soden et al. 2008)

Evaluates partial derivatives of model TOA

Calculating Feedbacks with Radiative Kernels

As of when this cookbook section is written (2024), there are a couple radiative kernels available in the community.

We will discuss the radiative kernel method in details based off of kernels discussed in Soden et al. 2008.

Planck Feedback: Temperature Kernel

Water Vapor and Lapse Rate Feedback: Water Vapor Kernel

Albedo Feedback: Surface Albedo Kernel

Cloud Feedback: Cloud Kernel…?

Clouds are generally assumed to be the largest source of uncertainty in the global climate response to radiative forcing. Cloud formation and lifetime are dictated by complex processes occurring at micro and macro-scale, and their radiative properties depend on the cloud type and optical properties. The nonlinear radiative effects of clouds are thus not suitable for direct calculation by kernels.

Several methods have evolved to estimate cloud feedbacks. Soden and Held (2006) computed the cloud feedback as the residual difference between the effective climate sensitivity and all other feedbacks. Soden et al. (2008) calculated cloud feedback as a sum of the change in cloud radiative forcing and the difference between the full-sky and clear-sky kernels. However, Held and Shell (2012) have pointed out that the decomposition of radiative feedbacks into the mechanisms discussed herein is rather arbitrary and proposed the use of tropospheric relative humidity

Adjustment vs. residual method

Relative Humidity Framework Kernels

Summary

Yay you made it!

What’s next?

Now that you have learned all the foundations about the energy balance, the feedback-forcing framework, how to calculate and the reasons behind the calculations of feedbacks with the use of radiative kernels, let’s get crackin! (is this old school?)

On this page