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A windowed Fourier transform (WFT) [@gomes99fourier] is a method that brings out the “local" spectral elements of a function in the neighborhood of a given position or time. It differs from the normal Fourier transform by including a Gaussian function centered at the position ($x$) or time ($t$) of interest. The spatial WFT $F(k,x)$ of a function $f(x)$ is defined as: $$\begin{aligned}
\label{eq:FTkx0}
F(k,x)= \int^{\infty}_{-\infty} dy\, f(y)w(y-x;D)e^{-iky},\end{aligned}$$ where $w(y-x;D)=\exp (-(y-x)^2/D^2)/\left(\sqrt{\pi}D\right)$ is a Gaussian window function of width $D$. With the filtering of the window, the transformed function $F(k,x)$ constitutes a local Fourier transform of $f(x)$, capturing features that vary on length scales much smaller than $D$. For instance, given a function, $f(x)=f_{q}(x) \exp (iqx)$, with wavevector $q$ and slowly varying amplitude $f_{q}(x)$, the transformed function is $F(k,x)\approx f_{q}(x) \exp (-(k-q)^2(D/2)^2)$ : a Gaussian in $k$-space, centered at $k=q$ with width $2/D$, and the peak height is the local amplitude, $f_{q}(x)$.

The WFT is able to resolve locally (at a given $x$ or $t$) the Fourier components in Eq. \[eq:psi0\] as peaks in the resulting wavevector (or frequency) spectrum, in which peak position and height indicate the wavevector and amplitude of each component. Specifically, for a background condensate flow, $\Psi_{\rm 