ApV

matrix component that multiplies $\tilde{v}$ to compute $A_{p}$ .

Description

Complex valued property with dimensions $(j, k l)$ and no units.

Discussion

These are the row 1, column 2 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,

ApV \equiv \frac{l ω - i k f_{0}}{2 ω K}

in the manuscript. In code this is computed with,

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

There are no $k^{2} + l^{2} > 0, j = 0$ wave solutions for a rigid lid,

ApV(:,:,1) = 0;

The inertial solutions occupy the $k^{2} + l^{2} = 0$ portion of the matrix,

ApV(1,1,:) = -sqrt(-1)/2;