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).

Installation

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

Running

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

./eval_lambda l m q_min q_max n_q log_q rossby filename

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, 2003a)

m (integer scalar)

Azimuthal order \(m\)

rossby (logical scalar)

Rossby family flag

q (real array)

Spin parameter \(q\)

lambda (real array)

Eigenvalue \(\lambda\)