In addition to the vertical mode expansion of the oceans described in the other page, we will perform vertical mode expansion using meteorological data.

 

 Internal Link: Vertical mode expansion of the tropical Pacific Ocean. 

 

 Atmospheric and oceanic phenomena are deeply interconnected at various spatio-temporal scales, and many common theories are used in their study. The vertical mode expansion is one of them. Vertical mode expansion means solving an eigenvalue problem, called the Strum-Liouville problem, for the vertical structure of the atmosphere (or ocean). The physical interpretation is to consider a phenomenon as a superposition of simple individual phenomena called modes. The eigenvalue decomposition yields a single forward pressure mode and a myriad of tilting pressure modes.

 

In this case, we will use Python (see above link for eigenvalue and eigenvector formulas). The data set used is the global mean data for 37 observed altitudes from 1000 hPa to 1.0 hPa in pressure altitude (downloadable from the following link).

  

 File Link: Vertical Atmospheric Dataset 

 

The columns are, from left to right: layer number, pressure height (hPa), height (m), and buoyancy frequency (1/s).

  

I have also uploaded a reference program.

 

 File Link: Vertical Mode Expansion (Atmospheric Version)

 

The tasks to be performed in the program flow as shown below: 

 1. Reading the data

 2. Create matrix equation from the vertical structure of the atmosphere based on the buoyancy frequency

 3. Solve the determinant and generate the vertical profile data from the eigenvectors.

  

For the matrix calculation, we use Numpy's linalg.eig function. This function implements the _geev routine of LAPACK. This routine can compute the eigenvalues and eigenvectors of large matrices (including asymmetric matrices). We also use argsort, a Numpy library, to sort the output eigenvectors (2D arrays) in descending order according to their eigenvalues.

The figure below shows the vertical profiles of the tilting pressure modes (1st to 3rd) among the output eigenvectors