Importance of the continuous spectrum in the excitation of sheared surface gravity waves

Abstract

The initial value problem is solved for the excitation of long surface gravity waves in a continuously sheared flow. This reveals the presence of a continuous spectrum along side the standard normal modes of gravity wave propagation. An analytical similarity solution for the evolution of the free surface displacement from the continuous spectrum is found for the impulse response to surface excitation. It is demonstrated that the continuous spectrum contribution can be a significant fraction of the surface response, with the amplitude of the continuous spectrum exceeding that of the upstream gravity wave mode for Froude numbers of order unity. The Landau damped mode description of the continuous spectrum is found to provide a link between methods using dispersion relations for phase speeds within the range of the velocity profile, and the variable-shear profiles that do not admit normal modes in this range.

Figures (4)

• Figure 1: (a) Dispersion relation for the unsheared and parabolic profiles. For both profiles up- and downstream modes are shown, together with the Landau damped mode from the parabolic profile. For the parabolic profile, the range of the velocity is shown in gray. (b) Weights of the normal modes ($\alpha_+, \alpha_-$) and the continuous spectrum ($\alpha_\mathrm{cts}$) for both profiles. The unsheared profile has $\alpha_\pm = 0.5$ and $\alpha_\mathrm{cts} = 0$. The values of the weights for each profile sum to unity $\alpha_+ + \alpha_- + \alpha_\mathrm{cts} = 1$.
• Figure 2: (a) Space-time evolution of the continuous spectrum component of the surface displacement, $\eta_\mathrm{cts}(x,t)$, for the parabolic profile with delta function forcing and $F = 1.5$. (b) The self-similar form for the continuous spectrum evolution of the free surface, $\beta(x/t;F)$, as a function of $F$. The vertical dashed lines in (a,b) correspond to the position and speed, respectively, of the Landau damped mode.
• Figure 3: Possible branch cuts of the function $\mathcal{H}(c)$ for $F = 1.5$. Colors denote the argument of the complex function $\mathcal{H}(c)$, and brightness denotes the amplitude. (a) Branch cut corresponding to the principle branches of the logarithm and square root functions in $\mathcal{D}(c)$ of equation (\ref{['eq:D_parabolic']}). The cut corresponds with $\mathrm{Re}(c)$ in the range of the velocity profile. (b) Alternate branch cut for $\mathcal{D}(c)$ with cuts corresponding to the negative imaginary axis for the logarithm and square root functions. Crosses denote the locations of poles, and the dashed lines denote the branch cuts of $\mathcal{H}(c)$. The damped Landau pole is revealed by the choice of cut in (b).
• Figure 4: (a) Normal mode phase speeds for the kinked piecewise $U(z)$ profile (solid colored lines). Plotted additionally are the unsheared gravity wave modes, and the long wave vorticity mode which is independent of $F$. (b) Weights of each mode of the kinked profile obtained from the residues with colors matching the modes in panel (a).