Additionally, persistent residuals are also observed in the annual mean over continental regions, which remain unaffected by computational specifics (Figs. <ref>(b) and <ref>(b)). These residuals are primarily due to numerical errors in the $\omega_p$ output from ModelE3's diagnostic code, particularly pronounced in regions with intense surface slopes (Fig. <ref>). This problem arises from deficiencies in the diagnostic code, necessitating modifications to more effectively address the impacts of surface slopes on $\omega_p$ computations. With the current state of the diagnostic code, $\omega_p$ does not meet the quality standards necessary for reliable MSE budget analysis over continental regions. This indicates a need to bypass its use in MSE budget calculations in these areas.

One important supplementary implication from this study is the potential advantages of using an advective form equation (Eqs. <ref> or <ref>) over a flux-divergence form (Eq. <ref>) for computing the column MSE budget. Although the advective form does not align with the model's actual transport processes and relies on potentially contaminated $\omega_p$, the errors resulting from the advective form are considerably smaller than those from inaccuracies in reconstructing flux divergence from model outputs.

In models where the prognostic temperature variable is an entropic quantity, the MSE budget does not close if MSE is treated in the same manner as other conserved tracers. This fact necessitates a reevaluation of what constitutes an accurate computation of the MSE budget. We distinguish between two types of computations: 1) numerically consistent computation, and 2) physically consistent computation. Both approaches can be considered accurate, but only the physically consistent approach enables proper closure of the budget.

In the numerically consistent computation, MSE is initially calculated and then input into the advection scheme used for other conserved tracers. Although this approach ensures that the computation of MSE transport aligns with that of other tracers within the model, it does not achieve complete closure of the MSE budget. Moreover, this approach increases computational demands, which is generally undesirable.

Our preferred approach, the physically consistent computation, is predicated on the assumption that the MSE budget equation (Eqs. <ref>, <ref>, or <ref>) ought to remain valid in discretized models, or equivalently, that there exists an advection scheme ensuring its complete closure. Based on this assumption, we compute the dynamical tendency of column MSE by using the process increment method, which calculates the difference in the values of $\langle\widetilde{h} \rangle +z_sp_s$ before and after the dynamical integration scheme is applied. This method facilitates the accurate computation of column MSE flux divergence, as detailed in Section <ref>. Additionally, we calculate the column-integrated vertical MSE advection indirectly as a residual, given the column-integrated horizontal advection, the tendency from column mass changes, and the column MSE flux divergence computed using the process increment method. This approach effectively bypasses the use of $\omega_p$, which is often contaminated with numerical errors. The process increment approach is adaptable and can be easily implemented across various models with differing structures, thereby potentially enhancing inter-model comparisons. Additionally, it is applicable for computing the column DSE flux divergence, as elaborated in <ref>.

The process increment method for calculating column MSE flux divergence and the indirect computation of vertical MSE advection enable precise MSE budget analysis over all regions, including continental areas, as shown in Figs. <ref>, <ref>, and <ref>. This level of precision is unattainable through direct computations using standard model outputs, regardless of the meticulousness of the computational details.

Most critically, Fig. <ref> highlights the significant impact of vertical interpolation into the $p$-coordinate system, showing that such interpolation can introduce errors substantial enough to reverse the sign of $-\overline{\kiab{\omega_p\partial h/\partial p}}$ over the oceanic ITCZ. Additionally, daily averaging exacerbates these errors. This underscores the sensitivity of vertical MSE advection computations to vertical coordinate systems and temporal resolutions, necessitating careful consideration in MSE budget analysis.

In the $p$-coordinate system, our results show negative $-\overline{\kiab{\omega_p\partial h/\partial p}}$ over the Western Pacific and parts of the Indian Ocean, consistent with previous studies that suggest top-heavy $\omega_p$ profiles in these regions lead to MSE export, while bottom-heavy profiles in the Eastern Pacific lead to MSE import. This view, encapsulated by the concept of GMS, has been foundational in many theories of tropical climate dynamics. However, when using the model’s native vertical coordinates, the sign of $-\overline{\kiab{\omega_p\partial h/\partial p}}$ can be flipped, particularly over the warm pool regions, indicating a potential energy import via vertical circulations from the subtropics into the tropics. These findings suggest that vertical interpolation errors, particularly when transitioning from the model’s native vertical levels to pressure levels, are significant enough to alter the computed GMS. Therefore, the conventional beliefs about vertical MSE advection and GMS may require reevaluation in light of these sensitivities.

This study presents a straightforward computational framework for accurately computing the column MSE budget, underscoring the critical role of precision in MSE budget analysis. Accurate computation is vital as it could potentially challenge and revise traditional views on vertical MSE advection and GMS, which are fundamental to our understanding of tropical climate dynamics. However, achieving this accuracy requires access to models' in-line computation codes and a deep understanding of the complex details inherent in each model. Given the vast array of models available for analysis, this task can be overwhelming for individual model users. Therefore, we strongly recommend that each modeling institute provide all necessary terms to comprehensively close the column MSE budget. Making these terms readily available will enable more reliable and thorough MSE budget analysis, thereby enhancing our understanding of climate dynamics and contributing to more robust climate modeling and forecasting.

This paper establishes the foundations for the upcoming second part of our two-part series. Using the column MSE budget terms acurately derived as outlined in this paper, the next installment provides a comprehensive assessment of each term, including those that are often neglected in conventional MSE budget approximations, examining their significance and broader implications. Additionally, we evaluate the impact of postprocessing procedures on MSE budget analysis via a detailed comparison of budget terms derived through our new methodology against those derived from postprocessed data. The impact on vertical MSE advection is thoroughly examined, highlighting how accurate MSE budget computations can significantly affect the understanding of tropical climate dynamics.  Through this rigorous analysis, we aspire to provide deeper insights and enhance understanding in the field of MSE budget computations.

§ COLUMN INTEGRATION OF FLUX-DIVERGENCE FORM

We obtain Eq. <ref> from Eq. <ref> as follows:

\begin{eqnarray}
\int_{z_s}^{z_t}\kicb{ \frac{\partial}{\partial t}\kirb{\rho h} + \nabla\cdot\kirb{\rho h\mathbf{U}} } \ud z& = &  \int_{z_s}^{z_t}\kicb{ \frac{\partial}{\partial t}\kirb{\rho h} + \nabla_z\cdot\kirb{\rho h\mathbf{v}} + \frac{\partial}{\partial z} \kirb{\rho h w}} \ud z \label{eq:flux_div1}\\
& = & \frac{\partial}{\partial t}\int_{z_s}^{z_t}\rho h \ud z + \nabla_z\cdot\int_{z_s}^{z_t}\rho h\mathbf{v}\ud z \nonumber \label{eq:flux_div2}\\
&+& \rho_sh_s\kirb{\frac{\partial z_s}{\partial t} + \mathbf{v}_s\cdot\nabla_z z_s -w_s}\;,
\end{eqnarray}

where the subscript $s$ represents a surface values, and $\nabla_z$ denotes the two-dimensional horizontal gradient operator. From Eq. <ref> to Eq. <ref>, we apply the Leibniz integral rule with the assumption that $\rho=0$ at the top of the atmosphere. Since we have $w_s\equiv Dz_s/Dt = \partial z_s/\partial t + \mathbf{v}_s\cdot\nabla_z z_s$ (generally, $\partial z_s/\partial t =0$), the last term of Eq. <ref> vanishes. Consequently, we obtain

\begin{equation}
\int_{z_s}^{z_t}\kicb{ \frac{\partial}{\partial t}\kirb{\rho h} + \nabla\cdot\kirb{\rho h\mathbf{U}} } \ud z = \frac{\partial \kiab{h}}{\partial t} + \nabla_z\cdot\kiab{h\mathbf{v}}\;.
\end{equation}

§ STRESS TENSOR AND TENSOR ALGEBRA

The stress on a fluid particle, defined as force per unit area, requires nine components for complete specification and is represented as a second-order tensor. For example, the viscous stress tensor in Eq. <ref> is expressed as:

\begin{equation}
\kitens{\tau} \equiv
\begin{bmatrix}
\tau_{11} & \tau_{12} & \tau_{13} \\
\tau_{21} & \tau_{22} & \tau_{23} \\
\tau_{31} & \tau_{32} & \tau_{33}
\end{bmatrix} \;,
\end{equation}

where $\tau_{ij}$ represents the $j$-th component of the force acting on a surface, with its outward normal vector pointing in the $i$-direction. It is essential to note that this stress tensor is symmetric (i.e., $\tau_{ij}=\tau_{ji}$) to avoid the scenario of infinite angular acceleration in an infinitesimal fluid particle.

In this study, we employ four distinct tensor operations to manipulate the wind vector and the viscous stress tensor. These operations are the contraction of tensors ($\cdot$), the double contraction of tensors ($:$), the divergence of a tensor ($\nabla\cdot$), and the gradient of a vector ($\nabla$). We provide explicit definitions for each operation, as different authors may use varying conventions.

First, we define the contraction of two tensors of arbitrary order, $\kitens{A}$ and $\kitens{B}$, as follows:

\begin{equation}
\kitens{A}\cdot\kitens{B} = \sum_k A_{\ldots k}B_{k\ldots}\;.
\end{equation}

In this operation, the last index of the first tensor ($\kitens{A}$) is equated to the first index of the second tensor ($\kitens{B}$), and a summation is performed over this shared index. When applied to two vectors (or first-order tensors with one index), this operation simplifies to $\sum A_kB_k$, which represents the dot product of the vectors. Notably, in Einstein notation, the summation symbol is often omitted.

Second, utilizing this index convention, we define the divergence of the viscous stress tensor in Cartesian coordinates. The $j$-component of this divergence is expressed as:

\begin{equation}
\left( \nabla\cdot\kitens{\tau} \right)_j = \sum_{i=1}^3\nabla_i\tau_{ij}=\sum_{i=1}^3\frac{\partial \tau_{ij}}{\partial x_i} \;.
\end{equation}

Some authors define this as $\partial\tau_{ij}/\partial x_j$, but both formulations are equivalent due to the symmetry of the stress tensor, where $\tau_{ij}=\tau_{ji}$.

Third, similarly to the contraction, we define the double contraction of two tensors of arbitrary order as follows:

\begin{equation}\label{eq:double_contraction}
\kitens{A}:\kitens{B} = \sum_j\sum_k A_{\ldots jk}B_{jk\ldots}\;.
\end{equation}

When applied to two second-order tensors (or matrices), this becomes $\sum\sum A_{ij}B_{ij}$, which is a scalar and called the Frobenius inner product.

Finally, we define the gradient of a vector $\mathbf{U}$ in Cartesian coordinates, whose $ij$-component is expressed as:

\begin{equation}
\left( \nabla \mathbf{U} \right)_{ij} = \frac{\partial U_j}{\partial x_i}\;.
\end{equation}

With the aforementioned definitions, we can derive the following tensor identity:

\begin{equation}\label{eq:tens_id}
\mathbf{U}\cdot \left( \nabla\cdot\kitens{\tau} \right) = \nabla \cdot \left( \kitens{\tau}\cdot \mathbf{U}   \right) -\kitens{\tau}:\nabla\mathbf{U}\;.
\end{equation}

This can be proven as follows:

\begin{eqnarray}
\mathbf{U}\cdot \left( \nabla\cdot\kitens{\tau} \right) & =& \sum_j U_j \left( \nabla\cdot\kitens{\tau}  \right)_j \\
& =& \sum_j  U_j \sum_i \frac{\partial \tau_{ij}}{\partial x_i}  \\
& =& \sum_i\sum_j \left\{ \frac{\partial (\tau_{ij} U_j) }{\partial x_i } -\tau_{ij}\frac{\partial U_j}{\partial x_i}   \right\} \\
& =& \sum_i\frac{\partial}{\partial x_i}  \sum_j \tau_{ij}U_j  - \sum_i\sum_j \tau_{ij}\frac{\partial U_j}{\partial x_i} \\
& =& \sum_i \frac{\partial}{\partial x_i} \left( \Bar{\Bar{\tau}}\cdot \mathbf{U}  \right)_i - \sum_i\sum_j\tau_{ij} \left( \nabla \mathbf{U} \right)_{ij} \\
& =& \nabla \cdot \left( \kitens{\tau}\cdot \mathbf{U}   \right) -\kitens{\tau}:\nabla\mathbf{U}\;.
\end{eqnarray}

§ DIRECT DERIVATION OF EQUATION <REF>

Here, we derive Eq. <ref> by applying the Leibniz integral rule to $\nabla_z\cdot\kiab{h\mathbf{v}}$, as follows:

\begin{eqnarray}
\nabla_z\cdot\kiab{h\mathbf{v}} &\equiv& \nabla_z\cdot\frac{1}{g}\int_0^{p_s}h\mathbf{v}\ud p \\
&= & \frac{1}{g}h_s\mathbf{v}_s \cdot \nabla_z p_s + \frac{1}{g}\int_0^{p_s}\mathbf{v}\cdot\nabla_p h \ud p + \frac{1}{g}\int_0^{p_s}h\nabla_p\cdot\mathbf{v}\ud p \;, \label{eq:c2}
\end{eqnarray}

where the symbols are defined in a manner consistent with those in <ref>. By applying the mass conservation equation in the $p$-coordinate system, $\nabla_p\cdot\mathbf{v}+\partial \omega_p/\partial p = 0$, we obtain

\begin{eqnarray}
\text{The last term of Eq.~\ref{eq:c2}} &= & -\frac{1}{g}\int_0^{p_s}h\frac{\partial \omega_p}{\partial p}\ud p\\
&=& -\frac{1}{g}h_s\omega_{p,s} + \frac{1}{g}\int_0^{p_s}\omega_p\frac{\partial h}{\partial p}\ud p \; , \label{eq:c4}
\end{eqnarray}

where $\omega_p = 0$ at the top of the atmosphere. By definition, we have $\omega_{p,s}\equiv Dp_s/Dt = \partial p_s/\partial t + \mathbf{v}_s\cdot\nabla_z p_s$. Consequently, by applying this to Eq. <ref>, along with Eq. <ref>, we derive

\begin{equation}
\nabla_z\cdot\kiab{h\mathbf{v}} = -\frac{1}{g}h_s\frac{\partial p_s}{\partial t} + \frac{1}{g}\int_0^{p_s}\mathbf{v}\cdot\nabla_p h \ud p + \frac{1}{g}\int_0^{p_s}\omega_p\frac{\partial h}{\partial p}\ud p \; .
\end{equation}

§ DRY STATIC ENERGY BUDGET AND ITS ACCURATE COMPUTATION

By incorporating $D\phi/Dt \equiv gw$ into Eq. <ref>, we derive the dry static energy (DSE) budget equation:

\begin{equation}
\frac{Ds}{Dt} = -\frac{1}{\rho}\nabla\cdot\kirb{\mathbf{R}+\mathbf{F}_t} +L_v\mathcal{C}+L_f\mathcal{F} + L_s\mathcal{D} +\epsilon+ \delta\,,
\end{equation}

where $s\equiv c_pT +\phi$ represents the DSE. Multiplying by $\rho$ and following a similar derivation to that in Eq. <ref>, we obtain:

\begin{equation}
\frac{\partial \kiab{s}}{\partial t} = -\nabla_z\cdot\kiab{s\mathbf{v}} +R + H +\kiab{L_v\mathcal{C}}+\kiab{L_f\mathcal{F}} + \kiab{L_s\mathcal{D}} +\kiab{\epsilon}+ \kiab{\delta}\;.
\end{equation}

Utilizing the relation, $\kiab{\epsilon} = \partial \kiab{R_dT}/\partial t + \kiab{\widetilde{\epsilon}}$, this equation can be rearranged into:

\begin{equation}
\frac{\partial }{\partial t} \kiab{s-R_dT}= -\nabla_z\cdot\kiab{s\mathbf{v}} +R + H +\kiab{L_v\mathcal{C}}+\kiab{L_f\mathcal{F}} + \kiab{L_s\mathcal{D}} +\kiab{\widetilde{\epsilon}}+ \kiab{\delta}\; ,
\end{equation}

where $s-R_dT = c_vT + \phi$. Furthermore, by employing the relation, $\kiab{\phi} = \kiab{R_dT}+ z_sp_s$, the equation can also be expressed as:

\begin{equation}\label{eq:column_dse3}
\frac{\partial }{\partial t}\kirb{ \kiab{c_pT} + z_sp_s}= -\nabla_z\cdot\kiab{s\mathbf{v}} +R + H +\kiab{L_v\mathcal{C}}+\kiab{L_f\mathcal{F}} + \kiab{L_s\mathcal{D}} +\kiab{\widetilde{\epsilon}}+ \kiab{\delta}\;.
\end{equation}

Note that the term $\widetilde{\epsilon}$ in the above equations requires special treatment and interpretation in hydrostatic models, as discussed in Section <ref>.

For facilitating comparison with previous studies, it is useful to express the DSE equation in an advective form within the $p$-coordinate system. By expanding the material derivative in the $p$-coordinate and using the relation from Eq <ref>, we derive the following:

\begin{equation}
\kirb{\frac{\partial s}{\partial t} }_p = - \mathbf{v}\cdot\nabla_p s - \omega_p\frac{\partial s}{\partial p} + g\frac{\partial}{\partial p}\kirb{R_v + F_{t,v}} + L_v\mathcal{C}+L_f\mathcal{F} + L_s\mathcal{D} +\epsilon+ \delta\; .
\end{equation}

Similar to Eq. <ref>, we can merge $\epsilon$ into the Eulerian tendency and horizontal advection as follows:

\begin{equation}
\kicb{\frac{\partial \kirb{c_pT} }{\partial t}}_p = - \mathbf{v}\cdot\nabla_p \kirb{c_pT} - \omega_p\frac{\partial s}{\partial p} + g\frac{\partial}{\partial p}\kirb{R_v + F_{t,v}} + L_v\mathcal{C}+L_f\mathcal{F} + L_s\mathcal{D} +\widetilde{\epsilon}_{eddy}+  \delta\; .
\end{equation}

Upon taking the vertical integration, we obtain:

\begin{equation}
\kiab{\kicb{\frac{\partial \kirb{c_pT} }{\partial t}}_p} = - \kiab{\mathbf{v}\cdot\nabla_p \kirb{c_pT}} - \kiab{\omega_p\frac{\partial s}{\partial p}} + R + H + \kiab{L_v\mathcal{C}}+\kiab{L_f\mathcal{F}} + \kiab{L_s\mathcal{D}} + \kiab{\delta} \; .
\end{equation}

Similar to Eq. <ref>, we can associate each term in Eq. <ref> with the corresponding integration schemes as follows:

\begin{equation}
\frac{\partial }{\partial t}\kirb{ \kiab{c_pT} + z_sp_s}=  \underbrace{-\nabla_z\cdot\kiab{s\mathbf{v}} +\kiab{\widetilde{\epsilon}}+ \kiab{\delta}}_{\text{Dynamics}} + \underbrace{H +\kiab{L_v\mathcal{C}}+\kiab{L_f\mathcal{F}} + \kiab{L_s\mathcal{D}} + R}_{\text{Turbulence, Cloud, Radiation}} \;.
\end{equation}

Therefore, the column MSE flux convergence at time level $n$, $-(\nabla_z\cdot\kiab{s\mathbf{v}})^n$, can be computed using the process increment method as follows:

\begin{equation}
-(\nabla_z\cdot\kiab{s\mathbf{v}})^n =  \frac{\kirb{\kiab{c_pT} + z_sp_s}^{n+1}_0 - \kirb{\kiab{c_pT} + z_sp_s}^n_3 }{\Delta t} -\kiab{\delta}^n  - \kiab{\widetilde{\epsilon}}^n \;.
\end{equation}

Finally, the column-integrated vertical DSE advection is calculated as a residual of the equation:

\begin{equation}
\kiab{ \omega_p\frac{\partial s}{\partial p}} = \nabla_z\cdot\kiab{s\mathbf{v}}+\frac{1}{g}s_s\frac{\partial p_s}{\partial t} - \kiab{\mathbf{v}\cdot\nabla_p s } \;,
\end{equation}

where $g^{-1}s_s\partial p_s/\partial t$ and $\kiab{\mathbf{v}\cdot\nabla_p s}$ are computed in the m-coordinate system.

§ OPEN RESEARCH SECTION

The ModelE3 simulation data used for making all figures in the study are available at <https://portal.nccs.nasa.gov/datashare/giss-publish/pub/tropicalclimate/entppe_4kuni_1yr_Q/>.

This work was primarily supported by the NASA Modeling, Analysis, and Prediction Program. K. Inoue was additionally supported by NSF Award 2303505. Resources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Center for Climate Simulation (NCCS) at Goddard Space Flight Center.

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More Information and Advice:

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