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Analytical solution for the vertical structure eigenfunction in diagnosis of the low frequency wave-type perturbations in primitive dynamics equation

This study presents an analytical solution to the problem of determining the vertical modal structure of low-frequency atmospheric oscillations. This analytical solution is critically compared with the numerical solution of this problem known in the literature, and good agreement for vertical normal modes with the wave number less than 9 is found. Applicability of the analytical solution for diagnosing the atmospheric climate variability and the quality of its simulation by contemporary GCM is briefly discussed.


Introduction
Using a classical approach, a special diagnostic technique to analyze the long-wave-type eigensolutions of governing primitive equations, which characterize free, unforced, temporal changes in the atmospheric dynamics, simulated by GCM. These results have demonstrated that at special conditions, the generated eigen-oscillations influence the climate statistics by changing the time-averaged basic state of the atmospheric circulation [3].
Within this general eigenvalue problem, the problems of determination of the horizontal normal modes and of the vertical normal modes, respectively, are intrinsically coupled. So, in general case, this problem does not allow for separation of variables and has to be studied numerically. The horizontal normal modes can be specified as a sum of associated Legendre polynomials or, what sometimes is preferred, as a sum of Hough functions being the eigenfunctions of Laplace's tidal equations on a sphere. However, in this short note we would like to remind the reader on existence of an exact analytical solution by Monin and Gavrilin to a very similar problem of determination of vertical normal modes in the quasi-geostrophic atmospheric dynamics. This analytical approach has a potential of testing and improving the diagnostic technique applied to atmospheric data and GCM outputs.

Diagnostic Problem Formulation
We use the primitive equations written in the log-pressure coordinates. These equations are linearized with respect to the perturbations superimposed on a basic state. The basic state corresponds to a real horizontal & vertical distribution of meteorological parameters, averaged over time. To study the eigensolutions of the primitive equations and to recognize the individual atmospheric wave modes, by following the approach of Tanaka, this system was re-formulated in terms of the Galerkin projection onto orthogonal basis functions. As the result, the original set of governing equations is split into two sub-systems, corresponding to the horizontal normal modes and the vertical normal modes, respectively.

Analytical Solution for the Vertical Normal Mode Structure
The thermodynamic energy balance equation, taken under adiabatic approximation, gives an ordinary differential equation describing the vertical structure of oscillation [1].
with p as the pressure, G m is the vertical structure function depending on height (pressure) and on the vertical modal index m;  is the static stability parameter variable with height as dp dT p c where T0(p) is the basic state temperature. The boundary condition follows from the energy balance at the top and the bottom (Earth surface) levels in the atmosphere, as in [1].
It is taken both at the surface level p = p0 and at the top level p = pH. To solve this problem, we apply an exact analytical solution of by Monin and Gavrilin for a similar differential operator and similar boundary conditions. In terms of (3.  Table 1).
The input T0(p) and γ(p) data at pressure levels similar as in Tanaka and Kung [1] have been used from the paper just cited; see 1 st -3 rd columns of Table 1.

Results and Analysis
The analytical solution (3.4) is in good agreement with the numerical results (Figures 1 to 4). Note, that no indication on the tropopause is apparent in the global mean state shown in Table 1, because of insufficient model resolution in the stratosphere. The wavenumber m = 0 corresponds to the barotropic mode, whereas the modes with m ≥1 are regarded as the baroclinic modes, with the number of nodes given by m. In numerical computations, the vertical structure of the higher baroclinic modes may depend on the selection of vertical levels. Since these modes have an aliasing problem in the stratosphere, these issues are addressed in the subsequent computations.
In particular, the baroclinic mode with m = 2 has a single maximum at 150 hPa. For m = 3 this maximum is shifting down to 180 hPa, but a secondary maximum appears at 100 hpa.  The comparison of the analytical solution with the numerical eigenfunction profiles shows good agreement for m = 9, but for higher m-values the analytical solution does not match the numerical computations. However, depending on the considered problem the analytical solution can be easily used in various circumstances for the purpose of comparison with numerical calculations of the baroclinic modes.

Summary and Conclusuions
In this work, we deal with an analytical solution to the problem of the vertical eigenmodes determination. In distinction to previously obtained numerical solutions, our approach is based on the exact analytical solution by Monin [2] which greatly facilitates analysis and allows for easier diagnostics of different dynamical mechanisms, which can govern the waves vertical structure and their propagation with altitude. Comparison of our analytical results with the previously obtained numerical profiles show good correspondence and confirm the applicability and usefulness of the analytical solution.
Following the approach proposed by H. Tanaka, the present investigation deals with developing of diagnostic techniques to be applied to analysis of the wave-type modal structures of the atmospheric general circulation, by using the primitive governing equations. This study might be useful to illuminate such an important modern question as to what extent can the atmospheric dynamics be responsible for the low frequency atmospheric climate variability? Additionally, this diagnostic could assist in finding some deficiencies in the numerical solutions to the climate models, as far as it concerns the adequate reproduction of the low-frequency climate variability and climate statistics.
Finally, our approach opens an easier way to future diagnostics of nonlinear interactions between modes, of mechanisms responsible for the upscale energy fluxes, which contribute to the atmospheric mean climate state and its variations.