Numerical simulations of internal wave generation by convection in water

Water's density maximum at $4{}^{\circ }\mathrm{C}$ makes it well suited to study internal gravity wave excitation by convection: an increasing temperature profile is unstable to convection below $4{}^{\circ }\mathrm{C}$, but stably stratified above $4{}^{\circ }\mathrm{C}$. We present numerical simulations of a waterlike fluid near its density maximum in a two-dimensional domain. We successfully model the damping of waves in the simulations using linear theory, provided we do not take the weak damping limit typically used in the literature. To isolate the physical mechanism exciting internal waves, we use the spectral code dedalus to run several simplified model simulations of our more detailed simulation. We use data from the full simulation as source terms in two simplified models of internal-wave excitation by convection: bulk excitation by convective Reynolds stresses, and interface forcing via the mechanical oscillator effect. We find excellent agreement between the waves generated in the full simulation and the simplified simulation implementing the bulk excitation mechanism. The interface forcing simulations overexcite high-frequency waves because they assume the excitation is by the “impulsive” penetration of plumes, which spreads energy to high frequencies. However, we find that the real excitation is instead by the “sweeping” motion of plumes parallel to the interface. Our results imply that the bulk excitation mechanism is a very accurate heuristic for internal-wave generation by convection.

Figure 1

Four simulation snapshots near $t=37400\mathrm{s}$. The bottom part of the domain (below the thick black line) shows the temperature field. Recall that below $4{}^{\circ }\mathrm{C}$, cold water (red) is less dense than hot water (blue). The thick black line is the $5{}^{\circ }\mathrm{C}$ isotherm and shows the boundary between the convective region (below) and the stably stratified region (above). The top part of the domain (above the thick black line) shows the vorticity field associated with IGWs.

Figure 2

(a) Spectrogram of the kinetic energy density. The white curve is the imaginary part of the buoyancy frequency corresponding to the convection frequency. The black curve is the real part of the buoyancy frequency in the stably stratified region. The data have been smoothed to improve statistics (see the text for more details). Parts (b) and (c) plot the kinetic energy density as a function of $z$ (dashed lines) and $\omega$ (dot-dashed), respectively; the vertical and horizontal lines in (a) are color-coded to correspond to these slices. The thin solid lines in (b) show the predicted linear damping rates near the convective stably stratified boundary at $z=0.22\mathrm{m}$. Within the convection region [(c), yellow curve], the energy is peaked near the convection frequency. However, these low-frequency waves are quickly attenuated, and the stably stratified region is dominated with waves with frequencies (1–$2)\times {10}^{-2}\mathrm{Hz}$ [(c), blue curve].

Figure 3

Wave flux spectrum at $z=0.22\mathrm{m}$ calculated using Eq. (6). We assume that the flux at each $\omega$ is dominated by a single $k_x$, which we determine by comparing the energy decay rate at $z=0.22\mathrm{m}$ to ${\ell }_d(\omega ,k_x)$.

Figure 4

Spectrograms of kinetic energy in modes with $k_x=2\pi /0.1{\mathrm{m}}^{-1}$ testing different linear damping formulas. The black line shows the horizontally and temporally averaged buoyancy frequency profile. (a) The spectrogram for the simulation result. (b) The spectrum from the simulation evaluated at $z=0.22\mathrm{m}$ multiplied by the damping factor [Eq. (10)], using the weak dissipation limit expression for ${\ell }_d^{-1}$ [Eq. (11)]. (c) The spectrum from the simulation evaluated at $z=0.22\mathrm{m}$ multiplied by the damping factor, using the full expression for ${\ell }_d^{-1}$ [Eq. (12)]. While the weak damping formula significantly overestimates the damping, the full damping formula accurately describes the damping in the simulation.

Figure 5

Snapshots of vertical velocity in the stably stratified region at three different times for the full simulation (first column), bulk excitation simulation (second column), and interface forcing simulations with a $5{}^{\circ }\mathrm{C}$ interface (third column) and an $8{}^{\circ }\mathrm{C}$ interface (fourth column). The vertical velocities are normalized by the rms vertical velocity at each height. Although there is quite good agreement between the full simulation and the bulk excitation simulation, the interface forcing simulations are not as accurate.

Figure 6

Spectrogram of vertical velocity squared in four different simulations. The black line shows the horizontally and temporally averaged buoyancy frequency profile. (a) Full simulation, (b) bulk excitation simulation, (c) interface forcing simulations with $5{}^{\circ }\mathrm{C}$ interface, and (d) interface forcing simulations with $8{}^{\circ }\mathrm{C}$ interface. The bulk excitation simulation agrees very well with the full simulation. The interface forcing simulations overestimate the excitation of high-frequency waves. This is because the interface forcing cannot detect sweeping motions along the interface, and thus the wave excitation is treated as an impulsive event that produces high-frequency waves.