According to Taylor et al. (2020), the continuum energy budget is defined as K+I+PK≡12∂π∂sv2I=cpΘvΠ−∂π∂s1ρp+ptopϕtopP=πsϕ \begin{align*} K + I + P \\ K \equiv \frac{1}{2} \pder{\pi}{s} \boldsymbol{v}^2 \\ I = c_p \Theta_v \Pi - \pder{\pi}{s}\frac{1}{\rho}p + p_{\textrm{top}} \phi_{\textrm{top}} \\ P = \frac{\pi}{s} \phi \end{align*} K+I+PK≡21∂s∂πv2I=cpΘvΠ−∂s∂πρ1p+ptopϕtopP=sπϕ
K+I+PK≡12∂π∂sv2I=cpΘvΠ−∂π∂s1ρp+ptopϕtopP=πsϕ \begin{align*} K + I + P \\ K \equiv \frac{1}{2} \pder{\pi}{s} \boldsymbol{v}^2 \\ I = c_p \Theta_v \Pi - \pder{\pi}{s}\frac{1}{\rho}p + p_{\textrm{top}} \phi_{\textrm{top}} \\ P = \frac{\pi}{s} \phi \end{align*} K+I+PK≡21∂s∂πv2I=cpΘvΠ−∂s∂πρ1p+ptopϕtopP=sπϕ