Transformed eddy-PV flux and positive synoptic eddy feedback onto low-frequency flow

Abstract

Interaction between synoptic eddy and low-frequency flow (SELF) has been the subject of many studies. In this study, we further examine the interaction by introducing a transformed eddy-potential-vorticity (TEPV) flux that is obtained from eddy-potential-vorticity flux through a quasi-geostrophic potential-vorticity inversion. The main advantage of using the TEPV flux is that it combines the effects of the eddy-vorticity and heat fluxes into the net acceleration of the low-frequency flow in such a way that the TEPV flux tends to be analogous to the eddy-vorticity fluxes in the barotropic framework. We show that the anomalous TEPV fluxes are preferentially directed to the left-hand side of the low-frequency flow in all vertical levels throughout the troposphere for monthly flow anomalies and for climate modes such as the Arctic Oscillation (AO). Furthermore, this left-hand preference of the TEPV flux direction is a convenient three-dimensional indicator of the positive reinforcement of the low-frequency flow by net eddy-induced acceleration. By projecting the eddy-induced net accelerations onto the low-frequency flow anomalies, we estimate the eddy-induced growth rates for the low frequency flow anomalies. This positive eddy-induced 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 low-frequency atmospheric flow by balancing some of the strong damping effect of surface friction.

Appendix: 3-D Laplacian inversion for the PV equation with eddy-PV forcing

To examine the contributions of transient synoptic eddy-PV forcing to the low-frequency flow by combining the eddy-vorticity and heat feedback, we adopt the approach outlined by Lau and Holopainen (1984). To obtain the eddy-induced geopotential tendency, we need to solve the simplified 3-D quasi-geostrophic PV equation

$$ \left[ {\nabla^{2} + f^{2} {\frac{\partial }{\partial p}}\left( {\frac{1}{\sigma }}{\frac{\partial }{\partial p}} \right)} \right]\left( {\frac{1}{f}{\frac{{\partial \overline{\Upphi }^{a} }}{\partial t}}} \right)_{\text{ed}} = D^{\text{PV}} , $$

(18)

where the eddy-PV 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

$$ D^{\text{PV}} = - \nabla \cdot \overline{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V}^{\prime}\zeta^{\prime}}} + f{\frac{\partial }{\partial p}}\left( {{\frac{{\nabla \cdot \overline{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V}^{\prime}\theta^{\prime}}} }}{{\tilde{\bar{S}}}}}} \right), $$

(19)

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 thermo-dynamical equation

$$ - \frac{p}{R}\left( {{\frac{{p_{0} }}{p}}} \right)^{{{R \mathord{\left/ {\vphantom {R {C_{\text{P}} }}} \right. \kern-\nulldelimiterspace} {C_{\text{P}} }}}} {\frac{\partial }{\partial p}}\left( {{\frac{{\partial \overline{\Upphi }^{a} }}{\partial t}}} \right)_{\text{ed}} = - \nabla \cdot \overline{{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {V}^{\prime}\theta^{\prime}}} ,\, p \, = { 1,}000{\text{ and 1}}00{\text{ mb}}, $$

(20)

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 eddy-PV forcing $D^{\text{PV}}$ at all levels, the eddy-heat 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 wave-numbers) 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 2-D 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 eddy-induced tendency and its gradient vector or the so-called 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 thermo-dynamical 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 eddy-PV flux and stream function tendency.