VAm

matrix component that multiplies \(A_m\) to compute \(\tilde{v}\).

---

Description

Complex valued property with dimensions \((j,kl)\) and no units.

Discussion

These are the row 2, column 2 components of the wave-vortex (S)orting matrix, referred to as the \(S\) matrix in Early, et al. (2021). The primary internal gravity wave and geostrophic solutions that exist for \(k^2+l^2>0, j>0\) are summarized in equation C4.

For \(k^2+l^2>0, j>0\) this is written as,

\[\textrm{VAm} \equiv \frac{l \omega - i k f_0}{\omega K}\]

in the manuscript. In code this is computed with,

alpha = atan2(L,K);
fOmega = f./omega;
VAm = (sin(alpha)-sqrt(-1)*fOmega.*cos(alpha));

There are no \(k^2+l^2>0, j=0\) wave solutions for a rigid lid,

VAm(:,:,1) = 0;

The inertial solutions occupy the \(k^2+l^2=0\) portion of the matrix,

VAm(1,1,:) = -sqrt(-1);