Vorticity of Fourier Expanded Velocity

I have been reading some papers which find all three components of the vorticity vector for a Fourier expanded (perturbation) velocity field i.e

$$\mathbf{u'}(x,y,z,t)=\int\mathbf{\hat{u}}(x,y,t)e^{ikz} dk,$$

allowing for the decoupling of modes with wave-number $$k$$.

I tried to derive the equations for the vorticity field:

$$\boldsymbol{\omega}=\left( \frac{\partial w'}{\partial y} - \frac{\partial v'}{\partial z} \right)\tilde{i} + \left( \frac{\partial u'}{\partial z} - \frac{\partial w'}{\partial x} \right)\tilde{j} + \left( \frac{\partial v'}{\partial x} - \frac{\partial u'}{\partial y} \right)\tilde{k},$$

where the derivatives are:

$$\frac{\partial}{\partial ([x,y])}\mathbf{u'}= \frac{\partial \mathbf{\hat{u}}}{\partial ([x,y])} e^{ikz}$$

and

$$\frac{\partial}{\partial z}\mathbf{u'}=i k \mathbf{\hat{u}}e^{ikz}$$

such that the vorticity field is found by

$$\boldsymbol{\omega}=(\frac{\partial \hat{w}}{\partial y} - ik\hat{v})e^{ikz} \tilde{i} + (ik\hat{u}- \frac{\partial \hat{w}}{\partial x}) e^{ikz} \tilde{j} + (\frac{\partial \hat{v}}{\partial x} - \frac{\partial \hat{u}}{\partial y})e^{ikz} \tilde{k}.$$

I have problems with the $$\tilde{i}$$ and $$\tilde{j}$$ components of this equation. For example, (from some trial and error), these papers plot the $$\tilde{i}$$th component as $$\omega_i=\frac{\partial \hat{w}}{\partial y}-|k||\hat{v}|$$.

I don't understand this - as far as I can understand the sensible thing to do is to either plot the real and imaginary components separately or take the magnitude. Is there a good reason to only take the magnitude of the imaginary part and add it to the real part?

• To make you understand, publish links to articles. – Alex Trounev Aug 21 at 12:43