Criteria for Kelvin-Helmholtz Instability I am presently engaged on a project to model elements of the behavior of Jupiter's atmosphere. I have been given by my authority some readings on fluid dynamics to raise awareness for the code. However, I do not entirely perceive why Kelvin-Helmholtz instability is an instability in the least bit. This is a quote from one of my assigned readings on the topic.

I don't see how there is an imaginary growing component for each $|U_1-U_2| > 0$ since there are two possible solutions for $\omega_{1/2}$, one of which appears to yield the growing pertubation, the opposite sign doesn't. Any clarification of why $|U_1-U_2| > 0$ has a an imaginary growing component for the two solutions of $\omega_{1/2}$?
 A: 
I don't see how there is an imaginary growing component for each $\lvert U_{1} - U_{2} \rvert > 0$ since there are two possible solutions for $\omega_{1/2}$, one of which appears to yield the growing pertubation, the opposite sign doesn't...

I think this is a matter of nomenclature and choice.  That is, you define subscript 1 regions as those with higher flow speeds than subscript 2.  Further, in wave phenomena one often chooses to have the frequency express real and imaginary parts but the wavenumber is purely real.  This is mostly a choice, not necessarily a physical reality.  Regardless, if one chooses this set of criteria, then the imaginary part of the frequency is often related to the increase or decrease of the wave amplitude.  Whether a positive imaginary part corresponds to growth or damping is also a choice.  That is, you, the user, get to choose the sign convention so long as you stay consistent throughout.
For instance, you could choose to have the amplitude be given by $A \propto e^{-\gamma t}$, where $\gamma = \Im{\omega}$.  Or you could use positive sign instead, if you so wished.  Once you decide the sign, then the sign of $\gamma$ determines whether a mode will grow or damp.

Any clarification of why $\lvert U_{1} - U_{2} \rvert > 0$ has a an imaginary growing component for the two solutions of $\omega_{1/2}$?

That would imply that positive $\gamma$ corresponds to growth in the authors' chosen sign convention.
If you want to understand all of this from a fundamental level, I recommend looking at French [1971] and/or Whitham [1999].
References

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*French, A.P. (1971), Vibrations and Waves, New York, NY: W. W. Norton & Company, Inc.; ISBN:0-393-09936-9.

*Whitham, G.B. (1999), Linear and Nonlinear Waves, New York, NY: John Wiley & Sons, Inc.; ISBN:0-471-35942-4.

