Abstract
Interaction between synoptic eddy and lowfrequency flow (SELF) has been the subject of many studies. In this study, we further examine the interaction by introducing a transformed eddypotentialvorticity (TEPV) flux that is obtained from eddypotentialvorticity flux through a quasigeostrophic potentialvorticity inversion. The main advantage of using the TEPV flux is that it combines the effects of the eddyvorticity and heat fluxes into the net acceleration of the lowfrequency flow in such a way that the TEPV flux tends to be analogous to the eddyvorticity fluxes in the barotropic framework. We show that the anomalous TEPV fluxes are preferentially directed to the lefthand side of the lowfrequency flow in all vertical levels throughout the troposphere for monthly flow anomalies and for climate modes such as the Arctic Oscillation (AO). Furthermore, this lefthand preference of the TEPV flux direction is a convenient threedimensional indicator of the positive reinforcement of the lowfrequency flow by net eddyinduced acceleration. By projecting the eddyinduced net accelerations onto the lowfrequency flow anomalies, we estimate the eddyinduced growth rates for the low frequency flow anomalies. This positive eddyinduced growth rate is larger (smaller) in the lower (upper) troposphere. The stronger positive eddy feedback in the lower troposphere may play an important role in maintaining an equivalent barotropic structure of the lowfrequency atmospheric flow by balancing some of the strong damping effect of surface friction.
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Acknowledgments
This work is jointly supported by National Science foundation (NSF) grants ATM 0652145 and ATM 0650552 and NSF of China (NSFC) grants 40705021 and 40805028, the Meteorological Special Project (GYHY200806005), and the National Science and Technology Support Program of China (2007BAC29B03, GYHY200906015). J.S. Kug is supported by KORDI (PM55290, PP00720) and the Korea Meteorological Administration Research and Development Program under Grant CATER_20102209. The authors especially express their thanks to A. F. Z. Levine for his careful proofreading and revising.
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An erratum to this article can be found at http://dx.doi.org/10.1007/s0038201111221
Appendix: 3D Laplacian inversion for the PV equation with eddyPV forcing
Appendix: 3D Laplacian inversion for the PV equation with eddyPV forcing
To examine the contributions of transient synoptic eddyPV forcing to the lowfrequency flow by combining the eddyvorticity and heat feedback, we adopt the approach outlined by Lau and Holopainen (1984). To obtain the eddyinduced geopotential tendency, we need to solve the simplified 3D quasigeostrophic PV equation
where the eddyPV forcing is \( D^{\text{PV}} =  \nabla \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {F}^{\text{PV}} . \) In this study, \( D^{\text{PV}} \) are not estimated by the explicit function of q in Eq. 1, but alternatively by approximate scheme as follows
where \( \tilde{\bar{S}} =  \partial \theta /\partial p \), taken as the extratropical hemispheric average by using a climatological monthly mean θ.
Following the approach of Lau and Holopainen (1984), the upper and lower boundary conditions are set at 1,000 and 100 mb, respectively, such that \( (\partial \overline{\Upphi }^{a} /\partial t)_{\text{ed}} \) is specified by the following thermodynamical equation
where \( p_{0} = 1,000\;{\text{mb,}} \) R is the gas constant, C _{p} the heat capacity at constant pressure.
Equations 18–20 are solved in a procedure as follows:

1.
This equation is first rewritten in terms of spectral coefficients based on the Fourier extension in the longitudinal direction. The number of zonal harmonic components is truncated to 71 (= 144/2 − 1), where 144 is the number of zonal grid points in data. Then, the eddyPV forcing \( D^{\text{PV}} \) at all levels, the eddyheat forcing \(  \nabla \cdot \overline{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V}^{\prime}\theta^{\prime}}} \) at the upper and lower boundaries, and the associated geopotential tendency \( (\partial \overline{\Upphi }^{a} /\partial t)_{\text{ed}} \) are transformed into Fourier coefficients from grid point data.

2.
The 143 sets of equations (one for the zonal mean, and two for each of the first 71 wavenumbers) are formed, where every set consists of a separable elliptic equation with latitude and pressure as independent variables, and the boundary conditions (20) at the upper and lower boundaries. For each set of equations, the finite difference scheme is used for discretization in latitude and pressure.

3.
The horizontal domain for the inversion is northern hemisphere, where the 2D Laplacian operator is expressed in spherical coordinates and the dimensionality in the latitudinal direction is specified as 37 extending from 0°N to 90°N. The boundary conditions at the equator and north pole are assumed to be \( (\partial \overline{\Upphi }^{a} /\partial t)_{\text{ed}} = 0 \) and \( \partial (\partial \overline{\Upphi }^{a} /\partial t)_{\text{ed}} /\partial y = 0, \) respectively. The vertical domain covers the entire troposphere spanning from 1,000 to 100 hPa with 12 degrees of freedom at standard pressure levels. The Coriolis parameter f is allowed to vary with latitude and the static stability parameter σ is the function of pressure only.
The inverted fields of eddyinduced tendency and its gradient vector or the socalled irrotational flux are unique under these set boundary conditions. The result is insensitive to the horizontal boundary conditions and inversion in the global or hemispheric domain is the same for the QG inversion. The inversion, however, does depend how to set vertical boundary condition. We chose to adopt the reasonable approach primarily following the calculation procedure designed by Lau and Holopainen (1984) by using the same vertical boundary condition derived from a simplified thermodynamical equation.

4.
The solved geopotential tendency is finally assembled by summation of the solutions for individual harmonic components for each month in the selected 30 years. Based on the geopotential tendency dataset, we can further estimate the transformed eddyPV flux and stream function tendency.
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Ren, HL., Jin, FF., Kug, JS. et al. Transformed eddyPV flux and positive synoptic eddy feedback onto lowfrequency flow. Clim Dyn 36, 2357–2370 (2011). https://doi.org/10.1007/s0038201009130
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DOI: https://doi.org/10.1007/s0038201009130
Keywords
 Synoptic eddy and lowfrequency flow feedback
 Transformed eddypotentialvorticity flux
 Acceleration of low frequency flow
 Eddyvorticity flux
 Eddyheat flux
 Lefthand preference
 Eddyinduced growth rate