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Heat Equation for Atmospheric Dynamics

Table of Contents

1. Overview

J. David Neelin's Climate Change and Climate Modeling give the following description of the equation governing atmospheric thermodynamics (it's the heat equation plus a term describing how an air parcel's volume increases as it elevates):

\begin{equation} c_{p}\frac{\mathrm{D}T}{\mathrm{D}t}-\frac{1}{\rho}\frac{\mathrm{D}p}{\mathrm{D}t} = Q \end{equation}

where \(c_{p}\) is the heat capacity of air at constant pressure, and \(Q\) is the heating term. For us, the heating term has several contributions

\begin{equation} Q = Q_{\text{solar}} + Q_{\text{IR}} + Q_{\text{convection}} + Q_{\text{mixing}}. \end{equation}

Beyond this, I'm relatively stumped, to be completely honest. This can also be found in, e.g., Vallis's Atmospheric and Oceanic Fluid Dynamics, chapter 1 §6. It seems Vallis discusses radiative transfer in chapter 18, but it's…a bit of work to get to some expression for \(Q\) out of it.

2. References

  • J. David Neelin,
    Climate Change and Climate Modeling.
    CUP, 2011; see especially §3.3.2 for the heat equation for air.

Last Updated 2022-04-13 Wed 09:48.