UAm

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

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Description

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

Discussion

These are the row 1, 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,

\[\textrm{UAm} \equiv \frac{k \omega + i l f_0}{\omega K}\]

in the manuscript. In code this is computed with,

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

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

UAm(:,:,1) = 0;

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

UAm(1,1,:) = 1;