Mathematics

The energy budget equations are as follows:

\begin{align*} \frac{\partial A_Z}{\partial t} &= C_K - C_A + BA_Z + \Delta G_Z \\ \frac{\partial A_E}{\partial t} &= C_A - C_E + BA_E + \Delta G_E \\ \frac{\partial K_Z}{\partial t} &= C_K - C_Z + BK_Z - \Delta R_Z \\ \frac{\partial K_E}{\partial t} &= C_E - C_K + BK_E - \Delta R_E \end{align*}

In these equations, available potential energy (APE) is divided into zonal (\(A_Z\)) and eddy (\(A_E\)) components, as is kinetic energy (\(K_Z\) and \(K_E\), respectively). The transformations between these forms of energy are denoted by \(C\), with subscripts \(Z\) and \(E\) for conversions between zonal and eddy forms, and \(A\) and \(K\) indicating conversions between APE and kinetic energy, respectively. Thus, \(C_A\) represents the conversion between \(A_Z\) and \(A_E\), \(C_E\) denotes the conversion from \(A_E\) to \(K_E\), \(C_K\) signifies the transformation from \(K_E\) to \(K_Z\), and \(C_Z\) describes the conversion from \(A_Z\) to \(K_Z\). The residual terms are defined as:

\begin{align*} \Delta R_Z &= B \Phi_Z - D_Z + \epsilon_{KZ} \\ \Delta R_E &= B \Phi_E - D_E + \epsilon_{KE} \\ \Delta G_Z &= G_Z + \epsilon_{GZ} \\ \Delta G_E &= G_E + \epsilon_{GE} \end{align*}

Where APE generation and dissipation of kinetic energy are indicated by \(G\) and \(D\), with \(G_Z\) and \(G_E\) marking the generation of \(A_Z\) and \(A_E\), and \(D_Z\) and \(D_E\) representing the dissipation of \(K_Z\) and \(K_E\), respectively.

Firstly, we define the zonal mean of a variable \(X\), between longitudes \(\lambda_{1}\) and \(\lambda_{2}\):

\[\begin{align*} [X]_\lambda &= \frac{1}{\lambda_2 - \lambda_1} \int_{\lambda_2}^{\lambda_1} X d\lambda \end{align*}\]

The eddy component of this variable is its deviation from the zonal mean:

\[\begin{align*} (X)_\lambda &= X - [X]_\lambda \end{align*}\]

The domain mean of the variable \(X\), defined over the computational domain bounded by longitudes \(\lambda_1\) and \(\lambda_2\), and latitudes \(\varphi_1\) and \(\varphi_2\), is given by:

\[\begin{align*} [X]_{\lambda\phi} &= \left(\frac{1}{\lambda_2 - \lambda_1}\right) \left(\frac{1}{\sin\phi_2 - \sin\phi_1}\right) \int_{\lambda_2}^{\lambda_1} X \cos\phi d\lambda d\phi \end{align*}\]

Similarly, we define the deviation of the zonal mean from the domain mean:

\[\begin{align*} ([X]_\lambda)_\phi &= [X]_\lambda - [X]_{\lambda\phi} \end{align*}\]

From the definitions above, the four energy components used in the LEC computation are defined as follows:

\[\begin{split}\begin{align*} A_Z &= \int_{p_t}^{p_b} \frac{([(T)_\lambda ])_\phi^{2}]_{\lambda \phi}} {2[\sigma]_{\lambda \phi}} dp \\ A_E &= \int_{p_t}^{p_b} \frac{[(T)_\lambda^{2}]_{\lambda \phi}]} {2[\sigma]_{\lambda \phi}} dp \\ K_Z &= \int_{p_t}^{p_b} \frac{[[u]_\lambda^2 + [v]_\lambda^2]_{\lambda \phi}}{2g} dp \\ K_E &= \int_{p_t}^{p_b} \frac{[(u)_\lambda^2 + (v)_\lambda^2]_{\lambda \phi}}{2g} dp \end{align*}\end{split}\]

where \(p\) is the atmospheric pressure, with subscripts \(b\) and \(t\) denoting the lower (base) and upper (top) pressure boundaries of the atmosphere, respectively. \(T\) represents temperature, \(g\) is the acceleration due to gravity, and \(u\) and \(v\) are the zonal and meridional wind components, respectively. The static stability parameter \(\sigma\) is defined as:

\[\begin{align*} \sigma &= \left[\frac{gT}{c_p}-\frac{pg}{R}\frac{\partial T}{\partial p}\right]_{\lambda \phi} \end{align*}\]

where \(c_p\) is the specific heat at constant pressure, and \(R\) is the ideal gas constant for dry air.

The four conversion terms are defined as follows, integrating over the atmospheric column from the base (\(p_b\)) to the top (\(p_t\)) pressures:

\[\begin{split}\begin{aligned} &C_Z = \int_{p_t}^{p_b} - [\left([T]_\lambda)_\phi ([\omega]_\lambda\right)_\phi]_{\lambda\phi} \ \frac{R}{gp} \ dp \label{eq:CZ} \\ &C_E = \int_{p_t}^{p_b} - [(T)_\lambda (\omega)_\lambda]_{\lambda\phi} \ \frac{R}{gp} \ dp \label{eq:CE} \\ &C_A = \int_{p_t}^{p_b} - \left( \frac{1}{2a\sigma} \left[ (v)_\lambda (T)_\lambda \frac{\partial ([T]_\lambda)_\phi}{\partial \phi} \right]_{\lambda\phi} + \frac{1}{\sigma} \left[ (\omega)_\lambda (T)_\lambda \frac{\partial ([T]_\lambda)_\phi}{\partial p} \right]_{\lambda\phi} \right) dp \label{eq:CA} \\ &C_K = \int_{p_t}^{p_b} \frac{1}{g} \left(\left[ \frac{\cos\phi}{a} (u)_\lambda (v)_\lambda \frac{\partial}{\partial\phi} \left(\frac{[u]_\lambda}{\cos\phi}\right)\right]_{\lambda\phi} + \left[ \frac{(v)_\lambda^2}{a} \frac{\partial [v]_\lambda}{\partial\phi} \right]_{\lambda\phi} \right. \left. + \left[ \frac{\tan\phi}{a} (u)_\lambda^2 [v]_\lambda \right]_{\lambda\phi} + \left[ (\omega)_\lambda (u)_\lambda \frac{\partial [u]_\lambda}{\partial p} \right]_{\lambda\phi} + \left[ (\omega)_\lambda (v)_\lambda \frac{\partial [v]_\lambda}{\partial p} \right]_{\lambda\phi} \right) dp \label{eq:CK} \end{aligned}\end{split}\]

where \(a\) is the Earth’s radius and \(\omega\) is the vertical velocity in isobaric coordinates.

The APE generation and K dissipation terms are defined as:

\[\begin{split}\begin{align*} G_Z &= \int_{p_t}^{p_b} \frac{[([q]_\lambda)_\phi ([T]_\lambda)_\phi]_{\lambda \phi}}{c_p[\sigma]_{\lambda \phi}} dp \\ G_E &= \int_{p_t}^{p_b} \frac{[(q)_\lambda (T)_\lambda]_{\lambda \phi}}{c_p[\sigma]_{\lambda \phi}} dp \\ D_Z &= - \int_{p_t}^{p_b} \frac{1}{g} [[u]_\lambda [F_\lambda]_\lambda + [v]_\lambda [F_\phi]_\lambda]_{\lambda \phi} dp \\ D_E &= - \int_{p_t}^{p_b} \frac{1}{g} [(u)_\lambda (F_\lambda)_\lambda + (v)_\lambda (F_\phi)_\lambda]_{\lambda \phi} dp \end{align*}\end{split}\]

Here, \(F_{\lambda}\) and \(F_{\varphi}\) represent the zonal and meridional frictional components, respectively, and \(q\) is the diabatic heating term, computed as a residual from the thermodynamic equation:

\[\begin{align*} \frac{q}{c_p} &= \frac{\partial T}{\partial t} + \vec{V}_H \cdot \vec{\nabla}_p T - S_p\omega \end{align*}\]

where \(\vec{V}_H \cdot \vec{\nabla}_p T\) represents the horizontal advection of temperature and \(S_p\) approximates the static stability, given by:

\[\begin{align*} S_p &\equiv -\frac{T}{\theta}\frac{\partial \theta}{\partial p} \end{align*}\]

where \(\theta\) is the potential temperature.

The boundary terms are given by:

\[\begin{split}\begin{aligned} & \mathrm{BAZ}=c_1 \int_{p_1}^{p_2} \int_{\varphi_1}^{\varphi_2} \frac{1}{2[\sigma]_{\lambda_{\varphi}}}\left(2\left([T]_\lambda\right)_{\varphi}(T)_\lambda u+\left([T]_{\lambda_{\varphi}}\right)_{\varphi}^2 u\right)_{\lambda_1}^{\lambda_2} \nonumber \\ & \times d \varphi d p+c_2 \int_{p_1}^{p_2} \frac{1}{2[\sigma]_{\lambda \varphi}}\left(2\left[(v)_\lambda(T)_\lambda\right]_\lambda\left([T]_\lambda\right)_{\varphi} \cos \varphi \right. \left.+\left([T]_\lambda\right)_{\varphi}^2[v]_\lambda \cos \varphi\right)_{\varphi_1}^{\varphi_2} d p \nonumber \\ & -\frac{1}{2[\sigma]_{\lambda \varphi}}\left(\left[2(\omega)_\lambda(T)_\lambda\right]_\lambda\left([T]_\lambda\right)_{\varphi}+\left[[\omega]_\lambda\left([T]_\lambda\right)_{\varphi}^2\right]_{\lambda_{\varphi}}\right)_{p_1}^{p_2} \\ & \mathrm{BAE}=c_1 \int_{p_1}^{p_2} \int_{\varphi_1}^{\varphi_2} \frac{1}{2[\sigma]_{\lambda \varphi}}\left[u(T)_\lambda^2\right]_{\lambda_1}^{\lambda_2} d \varphi d p \nonumber \\ & +c_2 \int_{p_1}^{p_2} \frac{1}{2[\sigma]_{\lambda \varphi}}\left(\left[(T)_\lambda^2 v\right]_\lambda \cos \varphi\right)_{\varphi_1}^{^{\varphi_2}} d p \\ & -\left(\frac{\left[\omega(T)_\lambda^2\right]_{\lambda \varphi}}{2[\sigma]_{\lambda \varphi}}\right)_{p_1}^{p_2} \nonumber \\ & \mathrm{BKZ}=c_1 \int_{p_1}^{p_2} \int_{\varphi_1}^{\varphi_2} \frac{1}{2 g}\left(u\left[u^2+v^2-(u)_\lambda^2-(v)_\lambda^2\right]\right)_{\lambda_1}^{\lambda_2} \nonumber \\ & \times d \varphi d p+c_2 \int_{p_1}^{p_2} \frac{1}{2 g}\left(\left[v \cos \varphi \left[u^2+v^2\right.\right.\right. \left.\left.-(u)_\lambda^2-(v)_\lambda^2\right]\right]_{\varphi_1}^{\varphi_2} d p \\ & -\left(\frac{1}{2 g}\left[\omega\left[u^2+v^2-(u)_\lambda^2-(v)_\lambda^2\right]\right]_{\lambda \varphi}\right)_{p_1}^{p_2} \nonumber \\ & \mathrm{BKE}=c_1 \int_{p_1}^{p_2} \int_{\varphi_1}^{\varphi_2} \frac{1}{2 g}\left(u\left[(u)_\lambda^2+(v)_\lambda^2\right]\right)_{\lambda_1}^{\lambda_2} d \varphi d p \nonumber \\ & +c_2 \int_{p_1}^{p_2} \frac{1}{2 g}\left(\left[v \cos \varphi\left[(u)_\lambda^2+(v)_\lambda^2\right]\right]_\lambda\right)_{\varphi_1}^{\varphi_2} d p \\ & -\left(\frac{1}{2 g}\left[\omega\left[(u)_\lambda^2+(v)_\lambda^2\right]\right]_{\lambda \varphi}\right)_{p_1}^{p_2} \nonumber \end{aligned}\end{split}\]

where \(c_1=-\left[a\left(\lambda_2-\lambda_1\right)\left(\sin \varphi_2-\sin \varphi_1\right)\right]^{-1}, c_2=-\left[a\left(\sin \varphi_2-\sin \varphi_1\right)\right]^{-1}\).

Lastly, the terms \(B\Phi_Z\) and \(B\Phi_E\) are given by:

\[\begin{split}\begin{aligned} \mathrm{B} \Phi \mathrm{Z}= & c_1 \int_{p_1}^{p_2} \int_{\varphi_1}^{\varphi_2} \frac{1}{g}\left([v]_\lambda\left([\Phi]_\lambda\right)_{\varphi}\right)_{\lambda_1}^{\lambda_2} d \varphi d p \nonumber \\ & +c_2 \int_{p_1}^{p_2} \frac{1}{g}\left(\cos \varphi[v]_\lambda\left([\Phi]_\lambda\right)_{\varphi}\right)_{\varphi_1}^{\varphi_2} d p \\ & -\frac{1} {g}\left(\left[\left([\omega]_\lambda\right)_{\varphi}\left([\Phi]_\lambda\right)_{\varphi}\right]_{\lambda_{\varphi}}\right)_{p_1}^{p_2} \nonumber \\ \mathrm{~B} \Phi \mathrm{E}= & c_1 \int_{p_1}^{p_2} \int_{\varphi_1}^{\varphi_2} \frac{1}{g}\left((u)_\lambda(\Phi)_{\lambda_\lambda}\right)_{\lambda_1}^{\lambda_2} d \varphi d p \nonumber \\ & +c_2 \int_{p_1}^{p_2} \frac{1}{g}\left(\left[(v)_\lambda(\Phi)_{\lambda_\lambda}\right]_\lambda \cos \varphi\right)_{\varphi_1}^{\varphi_2} d p \\ & -\frac{1}{g}\left(\left[(\omega)_\lambda(\Phi)_\lambda\right]_{\lambda_{\varphi}}\right)_{p_1}^{p_2} \nonumber \end{aligned}\end{split}\]