The wave-vortex transformation

The total transformations to and from wave-vortex space are defined so that

\[\label{linear-transformations} \left[\begin{array}{c} \hat{A}_+ \\ \hat{A}_- \\\hat{A}_0 \end{array}\right] = \underbrace{T_\omega^{-1} \cdot S^{-1} \cdot D}_{\mathcal{L}} \cdot \left[\begin{array}{c}u \\v \\ \eta \end{array}\right] \textrm{ and } \left[\begin{array}{c} u \\ v \\ \eta \\ w \\ p \end{array} \right] = \underbrace{D^{-1} \cdot S \cdot T_\omega}_{\mathcal{L}^{-1}} \left[\begin{array}{c} \hat{A}_+ \\ \hat{A}_- \\\hat{A}_0 \end{array}\right]\]

where it is understood that the physical variables are functions of \((x,y,z,t)\) and wave-vortex coefficients are indexed with \(jkl\) and taken to be at some reference time \(t_0\).

The full linear transformation from physical variables to wave-vortex space is \(\mathcal{L} = T_\omega^{-1} \cdot S^{-1} \cdot D\cdot\), with inverse \(\mathcal{L}^{-1} = D^{-1} \cdot S \cdot T_\omega\). The three parts to this transformation are,

• \(D : (u,v,\eta) \to (\hat{u},\hat{v},\hat{\eta})\) discrete transforms of the physical variables,
• \(S^{-1} : (\hat{u},\hat{v},\hat{\eta}) \to (A_+,A_-,A_0)\) transform to wave-vortex amplitude and,
• \(T_\omega^{-1} : \left(A_+(t),A_-(t),A_0(t)\right) \to \left(A_+(t_0),A_-(t_0),A_0(t_0)\right)\) an unwinding of wave phases to \(t_0\).

Discrete transformations

The discrete transformations \(D\) include two Fourier transformations (one in the \(x\) direction, the other in \(y\)) followed by the projection onto the vertical modes with the \(\mathcal{F}\) and \(\mathcal{G}\) matrices. The discrete transforms \(D\) and its inverse is therefore,

\[D = \left[\begin{array}{ccc} \mathcal{F} & 0 & 0 \\ 0 & \mathcal{F} & 0 \\ 0 & 0 & \mathcal{G} \end{array}\right] \cdot \mathcal{DFT}_y \cdot \mathcal{DFT}_x, \qquad D^{-1} = \mathcal{DFT}_x^{-1} \cdot \mathcal{DFT}_y^{-1} \cdot \left[\begin{array}{ccccc} \mathcal{F}^{-1} & 0 & 0 & 0 & 0 \\ 0 & \mathcal{F}^{-1} & 0 & 0 & 0 \\ 0 & 0 & \mathcal{G}^{-1} & 0 & 0 \\ 0 & 0 & 0 & \mathcal{G}^{-1} & 0 \\ 0 & 0 & 0 & 0 & \mathcal{F}^{-1} \end{array}\right]\]

There are thus two basic operations,

\[\begin{align} \hat{u}^{klj} =& \mathcal{F} \left[\mathcal{DFT}_y \left[\mathcal{DFT}_x \left[ u(x,y,z,t) \right] \right] \right] \\ \hat{\eta}^{klj} =& \mathcal{G} \left[\mathcal{DFT}_y \left[\mathcal{DFT}_x \left[ \eta(x,y,z,t) \right] \right] \right] \end{align}{}\]

which, in code, are performed with,

    u_hat = wvt.transformFromSpatialDomainWithF(u);
    w_hat = wvt.transformFromSpatialDomainWithG(w);

The two inverse transformations of \(D^{-1}\) are,

\[\begin{align} u(x,y,z,t) =& \mathcal{DFT}_x^{-1} \left[\mathcal{DFT}_y^{-1} \left[ \mathcal{F}^{-1} \left[ \hat{u}^{klj} \right] \right] \right] \\ \eta(x,y,z,t) =& \mathcal{DFT}_x^{-1} \left[\mathcal{DFT}_y^{-1} \left[ \mathcal{G}^{-1} \left[ \hat{\eta}^{klj} \right] \right] \right] \end{align}{}\]

and implemented in code with

    u = wvt.transformToSpatialDomainWithF(u_hat);
    w = wvt.transformToSpatialDomainWithG(w_hat);

Wave-vortex sorting

The second part of the transformation is the wave-vortex (S)orting which takes \((\hat{u},\hat{v},\hat{\eta})\) and solves for the amplitudes of the wave and vortex solutions \((A_+,A_-,A_0)\). This \(3 \times 3\) matrix operation is uniquely defined for each \(k,l,j\). Components of the \(S\) and \(S^{-1}\) transformations would typically be referred to as \(S_{11}\), \(S_{12}\), etc, but instead we use the slightly more informative notation where

\[\begin{bmatrix} A_p^{klj}(t) \\ A_m^{klj}(t) \\ A_0^{klj}(t) \end{bmatrix} = \underbrace{\begin{bmatrix} A_pU^{klj} & A_pV^{klj} & A_pN^{klj} \\ A_mU^{klj} & A_mV^{klj} & A_mN^{klj} \\ A_0U^{klj} & A_0V^{klj} & A_0N^{klj} \end{bmatrix}}_{\equiv S^{-1}} \begin{bmatrix} \hat{u}^{klj} \\ \hat{v}^{klj} \\ \hat{\eta}^{klj} \end{bmatrix}\]

with inverse,

\[\begin{bmatrix} \hat{u}^{klj} \\ \hat{v}^{klj} \\ \hat{\eta}^{klj} \\ \hat{w}^{klj} \end{bmatrix} = \underbrace{ \begin{bmatrix} UA_p^{klj} & UA_m^{klj} & UA_0^{klj} \\ VA_p^{klj} & VA_m^{klj} & VA_0^{klj} \\ NA_p^{klj} & NA_m^{klj} & NA_0^{klj} \\ WA_p^{klj} & WA_m^{klj} & 0 \end{bmatrix}}_{\equiv S} \begin{bmatrix} A_p^{klj}(t) \\ A_m^{klj}(t) \\ A_p^{klj}(t) \end{bmatrix}\]

In code for example, after computing \((\hat{u},\hat{v},\hat{\eta})\), \(A_p^{klj}(t)\) is found with

    Apt = wvt.ApU.*u_hat + wvt.ApV.*v_hat + wvt.ApN.*n_hat;

or, in reverse,

    u_hat = wvt.UAp.*wvt.Apt + wvt.UAm.*wvt.Amt + wvt.UA0.*wvt.A0t

To be slightly less abstract, note that in the rigid lid case, when \(k^2+l^2>0\), \(j>0\), \(S\) and its inverse are just the internal gravity wave and geostrophic solutions,

\[S = \left[\begin{array}{ccc} \frac{k\omega - il f_0}{\omega K} & \frac{k\omega + il f_0}{\omega K} & -i \frac{g}{f_0} l \\ \frac{l \omega + i k f_0}{\omega K } & \frac{l\omega - i k f_0}{\omega K } & i\frac{g}{f_0} k \\ - \frac{K h}{\omega} & \frac{K h}{\omega} & 1 \\ -iKh & -iKh & 0 \\ -\rho_0 g \frac{K h}{\omega} & \rho_0 g \frac{K h}{\omega} & \rho_0 g \end{array}\right], \qquad S^{-1} = \left[\begin{array}{ccc} \frac{k \omega + i l f_0}{2\omega K} & \frac{l \omega - i k f_0}{2\omega K} & - \frac{gK}{2\omega} \\ \frac{k \omega - i l f_0}{2\omega K} & \frac{l \omega + i k f_0}{2\omega K} & \frac{gK}{2\omega} \\ i \frac{l h f_0}{\omega^2}& - i \frac{k h f_0}{\omega^2} & \frac{ f_0^2}{ \omega^2} \end{array}\right]\]

Phase winding

The final step of the transformation is to wind the phases to reference time \(t_0\). This is done using the the phase winding operator \(T_\omega\)

\[T_\omega = \left[\begin{array}{ccc} e^{i\omega t} & 0 & 0 \\ 0 & e^{-i\omega t} & 0 \\ 0 & 0 & 1 \end{array}\right], \qquad T_\omega^{-1} = \left[\begin{array}{ccc} e^{-i\omega t} & 0 & 0 \\ 0 & e^{i\omega t} & 0 \\ 0 & 0 & 1 \end{array}\right].\]

with inverse unwinding operator \(T_\omega^{-1}\).