ApU

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

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Description

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

Discussion

These are the row 1, column 1 components of the inverse wave-vortex (S)orting matrix, referred to as \(S^{-1}\) 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 C5.

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

\[\textrm{ApU} \equiv \frac{k \omega + i l f_0}{2 \omega K}\]

in the manuscript. In code this is computed with,

alpha = atan2(L,K);
fOmega = f./omega;
ApU = (1/2)*(cos(alpha)+sqrt(-1)*fOmega.*sin(alpha));

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

ApU(:,:,1) = 0;

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

ApU(1,1,:) = 1/2;