UAp
matrix component that multiplies \(A_p\) to compute \(\tilde{u}\).
Description
Complex valued property with dimensions \((j,kl)\) and no units.
Discussion
These are the row 1, column 1 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,
\[\textrm{UAp} \equiv \frac{k \omega - i l f_0}{\omega K}\]in the manuscript. In code this is computed with,
alpha = atan2(L,K);
fOmega = f./omega;
UAp = (cos(alpha)-sqrt(-1)*fOmega.*sin(alpha));
There are no \(k^2+l^2>0, j=0\) wave solutions for a rigid lid,
UAp(:,:,1) = 0;
The inertial solutions occupy the \(k^2+l^2=0\) portion of the matrix,
UAp(1,1,:) = 1;