Near-inertial waves and geostrophic turbulence

Wind-forced near-inertial waves form a high-energy wave component in the upper ocean. The weakly dispersive nature of these large horizontal and small vertical scale waves make them suitable candidates for energetic interactions with mesoscale balanced flows. We take advantage of an idealized two-vertical-mode system obtained by projecting the hydrostatic Boussinesq equations onto the barotropic and a single high baroclinic mode to examine wave-balanced flow interactions. Our detailed analysis using results of freely evolving numerical simulations demonstrate how the well established two-mode quasigeostrophic turbulence phenomenology changes in the presence of high-energy near-inertial waves. In the absence of waves, the barotropic flow, which contains most of the balanced energy, undergoes an inverse energy cascade resulting in the formation of large-scale coherent vortices. In contrast, high-energy near-inertial waves transfer energy to the barotropic flow, facilitating a forward energy cascade of the balanced flow. The balanced flow in turn assists in the forward energy cascade of the wave field, which transforms the wave field from low-frequency near-inertial waves to high-frequency inertia-gravity waves. Given that the idealized model we employ is two-dimensional, the forward energy cascade of wave and balanced flow is an unexpected and intriguing feature.

Figure 1

Schematic showing the three components of the model, Eqs. (6). The barotropic mode (denoted by $T$) is balanced with no linear dynamics while the baroclinic mode consists of inertia-gravity waves (denoted by $W$) and a balanced component (denoted by $G$). In the absence of waves, the two-mode QG Eqs. (12) describes the interactions between the $G$ and the $T$ modes. In two-mode QG turbulence, the barotropic flow exhibits an inverse energy cascade (shown by small leftward pointing black arrows in the $T$ box), while the baroclinic balanced flow exhibits a forward energy cascade (shown by small rightward pointing blue arrows in the $G$ box). (Notice that the wavenumbers, $k$, increases towards right, as shown by the long horizontal black arrow at the bottom). Overall, the baroclinc flow loses energy to the barotropic flow in two-mode QG turbulence, as indicated by the long blue arrow connecting the $T$ and $G$ boxes. In this study we will examine how high baroclinic mode waves (indicated by the red box above), which are initially NIWs, can modify QG turbulence phenomenology.

Figure 2

(a) Barotropic vorticity and (b) baroclinic PV at $t=5000$ based on a simulation of Eqs. (6) with initial conditions $E_T={\text{Ro}}^2, E_G={\text{Ro}}^3, E_W=0$. (b) Energy change (from initial time) of $T, G$, and $W$ modes.

Figure 3

Results of a simulation with $\text{Bu}={\text{Ro}}^2$ and $\text{Ro}=0.1$ with initial conditions: $E_T={\text{Ro}}^2, E_G=0$, and $E_W=1$. (a)–(d) ${\zeta }_T$, (e)–(h) wave speed, $\sqrt{u_W^2+v_W^2}$ (i)–(l) total energy spectra of $T, W$, and $G$, and (m)–(p) frequency spectra of $u_W$. The four columns above correspond to four different times indicated above the first row. The dashed vertical lines in frequency spectra correspond to frequencies: $\omega =1$, 2, 3, and 4. The wavenumbers corresponding to these frequencies were obtained based on the dispersion relationship $\omega \left(k\right)=\sqrt{1+\text{Bu}{k}^2}$ and are marked by dashed vertical lines in the energy spectra.

Figure 4

Plotted above is the time series of the variable $R$ defined in Eq. (13). Observe that $R$ is negative throughout, indicating that waves are positively correlated with anticyclonic regions in ${\zeta }_T$, i.e., regions where ${\zeta }_T<0$.

Figure 5

(a) WPE:WKE ratio and the root-mean-square pressure gradient term. (b) $e_W(k,t)$ showing the fraction of the total wave energy contained in wavenumbers lower than $k$; see Eq. (14).

Figure 6

Change in energy of $T, W$, and $G$ modes, and WPE and WKE. Panel (a) shows a short duration of phase 1 while panel (b) shows the entire duration of the dynamics.

Figure 7

Time series of the energy exchange terms given in Eq. (16). Panels (a) and (b) show the barotropic mode's budget based on Eq. (16a); panels (c) and (d) show the wave's budget based on Eq. (16b); and panels (e) and (f) shows the baroclinic balanced flow's budget based on Eq. (16c). Note that in panel (c) above, the time series of $E_{WTW}$ and $\mathrm{\Delta }{E}_W$ overlap, while $E_{WTG}$ and $D_W$ are negligible in magnitude and lie on top of each other.

Figure 8

Spectral fluxes computed at $t=2000$ based on Eq. (18). The fluxes were time-averaged over a window $\mathrm{\Delta }t=25$ to remove high-frequency fluctuations.

Figure 9

Frequency spectra of three wavenumbers. The dashed vertical lines on each plot indicate the linear wave frequency corresponding to that specific wavenumber based on the dispersion relationship: $\omega \left(k\right)=\sqrt{1+\text{Bu}{k}^2}$.

Figure 10

(a) Frequency spectrum of $u_T$ before (black) and after (red) the running time averaging based on Eq. (19) was performed. (b) Evolution of the slow (red) and unaveraged (black) barotropic flow energy.

Figure 11

$\text{Bu}=2{\text{Ro}}^2$ with initial conditions: $E_T={\text{Ro}}^2, E_G=0$, and homogeneous wave field with $E_W=1$.

Figure 12

$\text{Bu}={\text{Ro}}^2/2$ with initial conditions: $E_T={\text{Ro}}^2, E_G=0$, and homogeneous wave field with $E_W=1$.

Figure 13

Schematic showing energy flow pathways between the three modes based on our simulations. The barotropic flow receives energy from waves via the $E_{WTW}$ term. Although the baroclinic balanced flow gains energy from both waves and the barotropic mode by $E_{GTW}=E_{WTG}+E_{TGW}$, the magnitude of this energy gain is comparable with the amount of energy the baroclinic balanced mode directly transfers to the barotropic mode via the $E_{GTG}$ term. As a result, the baroclinic balanced mode's energy remains relatively small at all times. Importantly, all three modes exhibit a forward energy cascade, resulting in the generation of small-scale features in the balanced and wave fields. The small horizontal arrows pointing right indicates this, wavenumber $k$ increasing from left to right. Concomitant with the generation of small-scale features, the waves' forward energy cascade transforms the wave field from NIWs at early times to a high-frequency inertia-gravity wave field at later times.