Solve the BVPs for the Poisson pdes

\[ \epsilon_{\delta \eta \alpha} \frac{\partial \omega_{\alpha}}{\partial x_{\eta}}=\frac{\partial^{2} v_{\eta}}{\partial x_{\delta} \partial x_{\eta}}-\frac{\partial^{2} v_{\delta}}{\partial x_{\eta} \partial x_{\eta}} \]

or in symbolic notation

\[ abla \times \boldsymbol{\omega}=abla(abla \cdot \mathbf{v})-\Delta \mathbf{v} \]

for the disturbance velocity for a doubly periodic flow in a cubic domain \(\mathcal{D}\) in Cartesian coordinates. The domain is defined by

\[ \mathcal{D} \equiv\left\{\left(x_{1}, x_{2}, x_{3}\right): 0 \leq x_{1} \leq 2 \pi L_{1},-L_{2} \leq x_{2} \leq L_{2},-\pi L_{3} \leq x_{3} \leq \pi L_{3}\right\} \]

with \(x_{1}\) and \(x_{3}\) as periodic directions. Velocity is the sum of the disturbance field \(\mathbf{v}(\mathbf{x})\) periodic with respect to \(x_{1}\) and \(x_{3}\) and a basic/mean field

\[ V_{\alpha}\left(x_{1}, x_{2}, x_{3}\right)=\delta_{\alpha, 1} V_{1}\left(x_{2}\right)+v_{\alpha}\left(x_{1}, x_{2}, x_{3}\right) \]

The basic field \(V_{1}\left(x_{2}\right)\) is assumed known, for instance, constructed analytically using the Gaussian error function. Assume an incompressible fluid.

(4.3.1) Fourier transform the Poisson pde to set up the BVP for the odes governing the complex-valued Fourier modes \(\tilde{v}_{\alpha}^{k, m}\left(x_{2}, t\right)\), where \(k\) and \(m\) are the wavenumbers corresponding to \(x_{1}\) and \(x_{3}\).

(4.3.2) Solve the BVP for the Fourier modes.