5. Hydrodynamic modes

In order to find the hydrodynamic modes, or decay rates [37], we need the Fourier transform of the linear system (4), which yields an algebraic system of

equations in terms of the variables k ! and ω: The hydrodynamic modes are obtained by calculating its eigenvalues λ ¼ iω, given by the roots of the characteristic equation <sup>p</sup>ð Þ¼ <sup>λ</sup> pLð Þ<sup>λ</sup> <sup>p</sup>Tð Þ¼ <sup>λ</sup> <sup>0</sup>, where pLð Þ<sup>λ</sup> and <sup>p</sup>Tð Þ<sup>λ</sup> are the characteristic polynomials of fifth and second order in λ of the matrices N<sup>L</sup> and NT, respectively. These roots are calculated below.
