Triad resonant instability of horizontally periodic internal modes

Theory is developed to predict the growth and structure of “sibling” waves developing through triadic resonant instability of a vertically confined mode-1 internal “parent” wave in uniform stratification including the influence of background rotation. For a sufficiently hydrostatic parent wave, two branches for growth of sibling waves are dominant. The branch with largest growth rate corresponds to sibling waves having frequencies much larger than that of the parent; the other branch corresponds to sibling waves having frequencies close to half the frequency of the parent. Numerical simulations show that sibling waves corresponding to the subharmonic branch appear in practice. In the absence of rotation, the sibling waves corresponding to this branch are predicted to have near-constant growth rate as their horizontal wave number increases. With rotation, however, the growth rate peaks at moderate wave number. In all cases, as confirmed by numerical simulations, the e-folding time for the growth of the sibling waves can be thousands of buoyancy periods for parent waves having amplitudes typical of realistic oceanic internal modes. In nonuniform stratification, the parent wave self-interacts immediately to force superharmonics. Nonetheless, numerical simulations with symmetric top-hat stratification show that triadic resonant instability eventually emerges. Such emergence is not evident in simulations with stratification more representative of the ocean. The results suggest a reconsideration of the efficacy of parametric subharmonic instability in leading to the breakdown of low-mode internal tides in the ocean.

Figure 1

Values of the sibling wave number $(k_{+},m_{+})$ with component normalized respectively by the parent wave number $(k_0,m_0)$ shown in four cases for a mode-1 wave, with $f=0$ and values of $k_{0}H$ as indicated. Pairs of sibling waves both with vertical wavelength smaller than $H$ are allowed in the light gray-shaded regions. The nondimensional growth rate, ${\sigma }_0$ is indicated by color with scale indicated in (a). Sibling waves with the largest growth rate having $k_{+}>k_0$ in the case $k_{0}H=3$, and by extension to cases with $k_{0}H=1,0.5$, and 0.2 are denoted by “branch 1”; sibling waves with the highest growth rate appearing for $k_{0}H=1,0.5$, and 0.2 are denoted “branch 2.”

Figure 2

Relative wave number ratio (blue), frequency (red), and the nondimensional growth rate ${\sigma }_0$ (thick black) computed for four different parent waves with $k_{0}H$ and $f/N_0$ specified as indicated. Solid (dotted) lines give the predicted values corresponding to branch 1 (branch 2) as indicated in Fig. 1. Open circles correspond to the relative wave number ratio (blue), frequency (red), and nondimensional growth rate (black) measured in corresponding numerical simulations with $A_{\xi }=0.01H$. Note that the ordinate in (a) and (b) is on a linear scale, whereas the ordinate in (c) and (d) is on a log scale. In all cases $k_{+}/k_0$ is plotted on a log scale, as indicated in the bottom plots.

Figure 3

Total horizontal velocity (left) and mode-filtered horizontal velocity (right) from a simulation with two horizontal wavelengths of a mode-1 internal mode with $k_{0}H=1$ and $A_{\xi }=0.02H$ shown at nondimensional times (a) $N_{0}t=3000$, (b) 5000, and (c) 7000. The values of velocity are represented by colors as indicated to the upper right in each plot. In all cases, the fields are shown in a frame of reference moving with the phase speed of the wave so that the phase is fixed.

Figure 4

(a) Vertical time series at $x=0$ and (b) horizontal time series at half-depth of $\left\{\tilde{u}\right\}\equiv \text{sgn}\left(\tilde{u}\right)ln\left(\right|\tilde{u}\left|\right)$ shown over nondimensional times $4000\le N_{0}t\le 5000$. In both cases the mode-filtered horizontal velocity $\tilde{u}$, is fixed in a frame of reference moving with the phase speed of the wave.

Figure 5

Log plot of the normalized disturbance kinetic energy plotted versus time. The best-fit line to the kinetic energy curve with slope ${\sigma }_k$ (dashed black line) is determined over times $4500\le N_{0}t\le 5500$.

Figure 6

(a) Log of the magnitude of the spectrum of horizontal velocity determined from horizontal time series at $z_0=-0.25H$ spanning 20 periods of the parent mode starting at $Nt=3500$, when the development of superharmonics becomes evident. (b) Log of the corresponding bispectrum for values of $B>0$. In both plots, negative frequencies correspond to leftward-propagating waves.

Figure 7

Snapshots of the total (left) and mode-filtered (right) horizontal velocity shown at times (a) $N_{0}t=100$ and (b) $N_{0}t=3500$ from a simulation initialized with a mode having $k_{0}H=1$ and $A_{\xi }=0.02H$ in symmetric top-hat stratification with $N^2\simeq N_0^2$ in the middle half of the domain and $N^2=0$ otherwise.

Figure 8

Mode-filtered energy versus time for the simulation shown in Fig. 7, with the total energy (black) being the sum of energy associated with disturbances having frequency less than ${\omega }_0$ (red) and greater than ${\omega }_0$ (blue). The best-fit line through the red curve for $3000\le N_{0}t\le 3500$ shows exponential growth of subharmonics with rate ${\sigma }_k$ indicated.

Figure 9

As in Fig. 6, but for the simulation with top-hat stratification shown in Fig. 7, showing (a) the spectrum of horizontal velocity and (b) the corresponding bispectrum computed for 20 periods of the parent mode starting at $Nt=3200$. The spectrum, $\widehat{u}$, is determined from a horizontal time series at $z_0=-0.325H$.

Figure 10

Results from simulations with (a) shifted top-hat stratification and (b) exponential stratification showing, as in right plots of Fig. 7, mode-filtered snapshots at $N_{0}t=30000$ (left) and, as in Fig. 8, the evolution of the mode-filtered energy partitioned into low and high frequency disturbances (right) The structure of the stratification is superimposed in white on left plots. In both simulations $A_0=0.002H$, $k=0.2H$.