Internal gravity waves in the energy and flux budget turbulence-closure theory for shear-free stably stratified flows

We have advanced the energy and flux budget turbulence closure theory that takes into account a two-way coupling between internal gravity waves (IGWs) and the shear-free stably stratified turbulence. This theory is based on the budget equation for the total (kinetic plus potential) energy of IGWs, the budget equations for the kinetic and potential energies of fluid turbulence, and turbulent fluxes of potential temperature for waves and fluid flow. The waves emitted at a certain level propagate upward, and the losses of wave energy cause the production of turbulence energy. We demonstrate that due to the nonlinear effects more intensive waves produce more strong turbulence, and this, in turn, results in strong damping of IGWs. As a result, the penetration length of more intensive waves is shorter than that of less intensive IGWs. The anisotropy of the turbulence produced by less intensive IGWs is stronger than that caused by more intensive waves. The low-amplitude IGWs produce turbulence consisting up to 90% of turbulent potential energy. This resembles the properties of the observed high-altitude tropospheric strongly anisotropic (nearly two-dimensional) turbulence.

Figure 1

The anisotropy parameter $A_z$ versus the wave Richardson number ${\mathrm{Ri}}_{{}_{\mathrm{W}}}$ for different values of the parameter $C_{\theta }$: $1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 2

The ratio of the turbulent potential energy, $E_P$, to the total turbulent energy $E=E_K+E_P$ versus the wave Richardson number ${\mathrm{Ri}}_{{}_{\mathrm{W}}}$ for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 3

The nondimensional turbulent kinetic energy $E_K/{\left({\ell }_0{N}_0\right)}^2$ versus ${\mathrm{Ri}}_{{}_{\mathrm{W}}}$ for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 4

The turbulent Prandtl number ${\mathrm{Pr}}_{{}_T}$ versus the wave Richardson number ${\mathrm{Ri}}_{{}_{\mathrm{W}}}$ for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 5

The nondimensional squared potential temperature flux $F_z^2/E_K{E}_{\theta }$ versus the wave Richardson number ${\mathrm{Ri}}_{{}_{\mathrm{W}}}$ for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 6

The vertical profile of parameter the wave Richardson number ${\mathrm{Ri}}_{{}_{\mathrm{W}}}$ for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 7

The vertical profile of the normalized turbulent kinetic energy, $E_K/{\left({\ell }_0{N}_0\right)}^2$, for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 8

The vertical profile of the normalized squared potential temperature flux $F_z^2/E_K{E}_{\theta }$, for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 9

The vertical profile of the anisotropy parameter $A_z$, for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 10

The vertical profile of the turbulent Prandtl number, for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).

Figure 11

The vertical profile of the ratio $E_P/E$, for different values of the parameter $C_{\theta }=1/15$ (solid), 0.1 (dashed), and 0.217 (dashed-dotted).