Evaluating Tidal Eigenvalues

This appendix describes the eval_lambda executable, which evaluates the eigenvalue \(\lambda\) appearing in Laplace’s tidal equations (see the Rotation Effects section). This executable is used for the calculations presented in Townsend (2020).

Compiling

The eval_lambda executable is automatically compiled when GYRE is built, and installed in the $GYRE_DIR/bin directory (see the Installation chapter).

Running

Unlike other GYRE executables, the parameters for eval_lambda are supplied directly on the command line, with the syntax

This evaluates \(\lambda\) for harmonic degree \(\ell\) and azimuthal order \(m\) on a grid \(\{q_{1},q_{2},\ldots,q_{N}\}\) in the spin parameter, writing the results to the file filename. If the flag log_q has the value T then the grid is logarithmically spaced:

\[q_{i} = 10^{(1 - w_{i}) \log q_{\rm min} + w_{i} \log q_{\rm max}},\]

where

\[w_{i} \equiv \frac{i-1}{N-1}.\]

Alternatively, if log_q has the value F, then the grid is linearly spaced:

\[q_{i} = (1 - w_{i}) q_{\rm min} + w_{i} q_{\rm max}.\]

As a special case, when \(n_{q}=1\), \(q_{\rm min}\) and \(q_{\rm max}\) must match, and the single \(q\) point is set to equal them.

If the flag rossby has the value T, then the Rossby \(\lambda\) family is evaluated; otherwise, the gravito-acoustic family is evaluated.

The table below summarizes the mapping between the user-definable controls appearing in the expressions above, and the corresponding command-line parameters:

Symbol Parameter
\(\ell\) l
\(m\) m
\(q_{\rm min}\) q_min
\(q_{\rm max}\) q_max
\(N\) n_q

Interpreting Output

The output file created by eval_lambda is in GYRE’s HDF Format, with the following data:

l (integer scalar)
Harmonic degree \(\ell\)
k (integer scalar)
Meridional order \(k\) (see Townsend, 2003)
l (integer scalar)
Azimuthal order \(m\)
rossby (logical scalar)
Rossby family flag
q (real array)
Spin parameter \(q\)
lambda (real array)
Eigenvalue \(\lambda\)