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Long-term change of the atmospheric energy cycles and weather disturbances

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Abstract

Weather disturbances are the manifestation of mean atmospheric energy cascading into eddies, thus identifying atmospheric energy structure is of fundamental importance to understand the weather variability in a changing climate. The question is whether our observational data can lead to a consistent diagnosis on the energy conversion characteristics. Here we investigate the atmospheric energy cascades by a simple framework of Lorenz energy cycle, and analyze the energy distribution in mean and eddy fields as forms of potential and kinetic energy. It is found that even the widely utilized independent reanalysis datasets, NCEP-DOE AMIP-II Reanalysis (NCEP2) and ERA-Interim (ERA-INT), draw different conclusions on the change of weather variability measured by eddy-related kinetic energy. NCEP2 shows an increased mean-to-eddy energy conversion and enhanced eddy activity due to efficient baroclinic energy cascade, but ERA-INT shows relatively constant energy cascading structure between the 1980s and the 2000s. The source of discrepancy mainly originates from the uncertainties in hydrological variables in the mid-troposphere. Therefore, much efforts should be made to improve mid-tropospheric observations for more reliable diagnosis of the weather disturbances as a consequence of man-made greenhouse effect.

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Acknowledgements

This study was supported by RP-Grant 2015 of Ewha Womans University, Korea and by “Development of cloud algorithms” project, funded by ETRI, which is a subproject of “Development of Geostationary Meteorological Satellite Ground Segment (NMSC-2016-01)” program funded by NMSC of KMA. W. Kim acknowledges the support from the APEC Climate Center. Y.-S. Choi is supported by Jet Propulsion Laboratory, California Institute of Technology.

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Correspondence to Yong-Sang Choi.

Appendices

Appendix 1: governing equations

We start from a set of primitive equations for non-divergent hydrostatic dry air:

$$\begin{gathered} \frac{{Du}}{{Dt}} - \frac{{\tan \varphi }}{a}uv - fv = - \frac{1}{{a\cos \varphi }}\frac{{\partial \Phi }}{{\partial \lambda }} + F_{\lambda } \hfill \\ \frac{{Dv}}{{Dt}} - \frac{{\tan \varphi }}{a}u^{2} + fu = - \frac{1}{a}\frac{{\partial \Phi }}{{\partial \varphi }} + F_{\varphi } \hfill \\ 0 = - \frac{{\partial \Phi }}{{\partial p}} - \alpha \hfill \\ \frac{{D\theta }}{{Dt}} = \left( {\frac{\partial }{{\partial t}} + U \cdot \nabla } \right)\theta = \left( {\frac{\partial }{{\partial t}} + \frac{u}{{a\cos \varphi }}\frac{\partial }{{\partial \lambda }} + \frac{v}{a}\frac{\partial }{{\partial \varphi }} + \omega \frac{\partial }{{\partial p}}} \right)\theta = \left( {\frac{{p_{0} }}{p}} \right)^{\kappa } \frac{Q}{{C_{p} }} \hfill \\ \nabla \cdot U = \frac{1}{{a\cos \varphi }}\left( {\frac{{\partial u}}{{\partial \lambda }} + \frac{{\partial v\cos \varphi }}{{\partial \varphi }}} \right) + \frac{{\partial \omega }}{{\partial p}} = 0 \hfill \\ \end{gathered}$$

where \(\lambda\), \(\varphi\), \(p\), and \(t\) is longitude, latitude, pressure, and time, respectively; \(U=\left(u,v,w\right)\) is 3-dimensional wind velocity, \(\theta\) is potential temperature (\(={\left({p}_{0}/p\right)}^{\kappa }T\)), \(\Phi\) is geopotential height, \(\alpha\) is specific volume; \(a\) is the radius of the earth, \(f\) is the Coriolis parameter, \(F=\left({F}_{\lambda },{F}_{\varphi },0\right)\) is friction, \(Q\) is diabatic heating, \({p}_{0}\) is reference pressure (1000 hPa). All other notations follow general form.

Appendix 2: time-averaged energy terms

Monthly means of available potential energy and kinetic energy are expressed as follows:

$$\begin{aligned} P_{M} &= \frac{{C_{p} }}{2}\left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \left( {\bar{\theta } - \left\langle {\bar{\theta }} \right\rangle } \right)^{2} \hfill \\ P_{E} & = \frac{{C_{p} }}{2}\left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \overline{{\theta ^{{\prime 2}} }} = \frac{{C_{p} }}{2}\gamma \overline{{T^{{\prime 2}} }} \hfill \\ K_{M} & = \frac{{\bar{u}^{2} + \bar{v}^{2} }}{2} \hfill \\ K_{E} &= \frac{{\overline{{u^{{\prime 2}} }} + \overline{{v^{{\prime 2}} }} }}{2} \hfill \\ \end{aligned}$$

where \(\left\langle X \right\rangle\) denotes global averages over a given pressure level and \(\gamma =-\kappa /p{{\left( {{p}_{0}}/p \right)}^{\kappa }}{{\left( d\left\langle {\bar{\theta }} \right\rangle /dp \right)}^{-1}}\) is the static stability of the dry atmosphere. P and K denote available potential energy and kinetic energy, while subscripts M and E indicate mean and eddy energy, respectively. It can be inferred from the equation that the available potential energy is a function of temperature deviation from its surrounding, so that relatively warmer (colder) air can potentially induce air motion. Due to strong hydrostatic constraints kinetic energy is solely expressed by the horizontal winds.

Appendix 3: time-averaged conversion terms

Conversion of energy C(A, B) from A to B is calculated as

$$\begin{gathered} C\left( {P_{M} ,\,K_{M} } \right) = - \bar{\omega }\bar{\alpha } ~~ ; \quad C\left( {P_{E} ,\,K_{E} } \right) = - \overline{{\omega ^{\prime}\alpha ^{\prime}}} \hfill \\ C\left( {P_{M} ,\,P_{I} } \right) = C_{p} \left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \left( {\bar{\theta } - \left\langle {\bar{\theta }} \right\rangle } \right)\nabla \cdot \left( {\overline{{\theta ^{\prime}U^{\prime}}} } \right) \hfill \\ C\left( {P_{E} ,\,P_{I} } \right) = C_{p} \left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \overline{{\theta ^{\prime}U^{\prime}}} \cdot \nabla \left( {\bar{\theta } - \left\langle {\bar{\theta }} \right\rangle } \right) \hfill \\ C\left( {K_{M} ,\,K_{I} } \right) = \bar{u}\nabla \cdot \overline{{u^{\prime}U^{\prime}}} + \bar{v}\nabla \cdot \overline{{v^{\prime}U^{\prime}}} - \frac{{\tan \varphi }}{a}\left( {\bar{u}\overline{{u^{\prime}v^{\prime}}} - \bar{v}\overline{{u^{\prime}u^{\prime}}} } \right) \hfill \\ C\left( {K_{E} ,\,K_{I} } \right) = \overline{{u^{\prime}U^{\prime}}} \cdot \nabla \bar{u} + \overline{{v^{\prime}U^{\prime}}} \cdot \nabla \bar{v} + \frac{{\tan \varphi }}{a}\left( {\bar{u}\overline{{u^{\prime}v^{\prime}}} - \bar{v}\overline{{u^{\prime}u^{\prime}}} } \right) \hfill \\ \end{gathered}$$

The conversion of available potential energy to kinetic energy [positive C(P M , K M ) or C(P E , K E )] results when relatively warm air (positive \({\alpha }=\frac{RT}{p}\) rises (negative \(\omega\)). The transfer of mean energy into (from) eddy energy is split into two parts that incorporate interaction term whose appropriate time mean is zero. Here, the subscript I stands for interaction energy that links between the mean energy and the eddy energy, but it disappears after monthly averaging. A detailed discussion of the interaction energy can be found in Murakami (2011). Regarding the calculation of energy conversion between the eddy terms, C(P E , K E ), the \(\omega \bullet \alpha\) formula is utilized to focus on the thermodynamic aspect of energy conversion (Kim and Kim 2013).

Appendix 4: time-averaged generation, dissipation, and boundary flux terms

Generation, dissipation, and boundary fluxes are as follows. In this study, however, generation terms were not calculated explicitly, but as a residual of fluxes. As the following terms are highly sensitive to reanalysis configuration, they are not extensively discussed in this research.

$$\begin{gathered} G\left( {P_{M} } \right) = \gamma \left( {\bar{T} - \left\langle {\bar{T}} \right\rangle } \right)\left( {\bar{Q} - \left\langle {\bar{Q}} \right\rangle } \right)~;\quad G\left( {P_{E} } \right) = \gamma \overline{{T^{\prime } Q^{\prime } }} \hfill \\ D\left( {K_{M} } \right) = - \bar{U} \cdot \bar{F}~~;\quad D\left( {K_{E} } \right) = - \overline{{U^{\prime } \cdot F^{\prime } }} \hfill \\ B\left( {P_{M} } \right) = C_{p} \left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \gamma \nabla \cdot \left[ {\frac{{\left( {\bar{\theta } - \left\langle {\bar{\theta }} \right\rangle } \right)^{2} - \left\langle {\bar{\theta }} \right\rangle ^{2} }}{2}\bar{U}} \right] \hfill \\ B\left( {P_{E} } \right) = C_{p} \left( {\frac{p}{{p_{0} }}} \right)^{{2\kappa }} \left( {\frac{{\overline{{\theta ^{{\prime 2}} }} }}{2}\bar{U} + \overline{{\frac{{\theta ^{{\prime 2}} }}{2}U^{\prime } }} } \right) \hfill \\ B\left( {K_{M} } \right) = \nabla \cdot \left[ {\left( {\frac{{\overline{u} ^{2} + \overline{v} ^{2} }}{2} + \bar{\Phi }} \right)\bar{U}} \right] \hfill \\ B\left( {K_{E} } \right) = \nabla \cdot \left( {\frac{{\overline{{u^{{\prime 2}} }} + \overline{{v^{{\prime 2}} }} }}{2}\bar{U}} \right) + \nabla \cdot \left[ {\overline{{\left( {\frac{{u^{{\prime 2}} + v^{{\prime 2}} }}{2} + \Phi ^{\prime } } \right)U^{\prime } }} } \right] \hfill \\ \end{gathered}$$

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Kim, W., Choi, YS. Long-term change of the atmospheric energy cycles and weather disturbances. Clim Dyn 49, 3605–3617 (2017). https://doi.org/10.1007/s00382-017-3533-0

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