Potential-enstrophy lengthscale for the turbulent/nonturbulent interface in stratified flow

We study properties of the turbulent/nonturbulent interface (TNTI) between two layers of stratified fluids through direct numerical simulations (DNSs). Zero mean shear forcing creates moderate turbulence in one of the layers with the Taylor microscale Reynolds numbers in the mixed region of ${\text{Re}}_{\lambda }=35,44$. We focus on the similarities and differences of the properties of stratified TNTIs due to two distinct types of forcing: (a) convection due to a boundary heat source and (b) agitation resembling a vertically oscillating grid experiment. Similarly to other stratified flows, the small scale dynamics of the TNTI in the present DNSs differ from what would be expected in comparable yet unstratified TNTIs. The interface cannot be indeed uniquely identified by the commonly used vorticity $\omega$. Instead, the potential enstrophy ${\mathrm{\Pi }}^2$ is shown to be the most appropriate marker in these flow cases. It is emphasized that the Kolmogorov lengthscale ${\eta }_K\sim \sqrt{\nu /\omega }$ is not representative of the small scale dynamics of the interface. Hence, an alternative lengthscale, ${\eta }_{\mathrm{\Pi }}$, is defined, in analogy to the Kolmogorov scale, based on the potential enstrophy, ${\eta }_{\mathrm{\Pi }}={({\nu }^3/{\mathrm{\Pi }}_{*})}^{1/6}$, being ${\mathrm{\Pi }}_{*}=|g/{\rho }_0\mathrm{\Pi }|$. The conditionally averaged profiles of potential enstrophy ${\mathrm{\Pi }}^2$, enstrophy ${\omega }^2$, and turbulent kinetic energy dissipation $\epsilon$ of the two distinctly different turbulence forcing cases collapsed when scaled by ${\eta }_{\mathrm{\Pi }}$ at different time instants in each simulation. This implies not only the self-similarity of the small scale statistics of the TNTI in either of the two cases, but also the similarity between the statistics of the two different turbulent flows in the proximity of TNTI.

Figure 1

Oscillating grid DNS case G: (a) Three-dimensional (3D) representation of the grid within the domain at $t/t^{*}=3.8$; the colored fields represent the fluid vertical velocity. (b) Spatial distribution of $u_{\mathrm{RMS}}^{\prime }$ and (c) longitudinal integral lengthscale as function of the distance from the grid, $d$.

Figure 2

Vertical profiles of case C (upper row), and case G (lower row) over time: (a, e) buoyancy; (b, f) turbulent kinetic energy; (c, g) diffusive flux; (d, h) turbulent flux.

Figure 3

Case C (upper row) and G (lower row): vertical cross section of (a–d) normalized ${\mathrm{\Pi }}^2$, (b–e) normalized ${\omega }^2$, and (c–f) turbulent volume fraction as function of ${\mathrm{\Pi }}^2$. Black lines mark threshold (solid), $1/5$ of threshold (dashed), $1/10$ of threshold (dotted).

Figure 4

Vertical conditional profiles of (a, b) ${\mathrm{\Pi }}^2$, (c, d) ${\omega }^2$, (e, f) $k$, (g, h) $\epsilon$ scaled with ${\eta }_K$ (left column) and ${\eta }_{\mathrm{\Pi }}$ (right column). G, black curves; C, red curves.

Figure 5

Vertical conditional profiles of (a, b) ${\text{Re}}_{\epsilon }$, (c, d) $\overline{w^{\prime }{b}^{\prime }}$ scaled with ${\eta }_K$ (left column) and ${\eta }_{\mathrm{\Pi }}$ (right column). C, red curves; G, black curves.

Figure 6

Alignment of $\mathit{\omega }$ with the TNTI normal $\mathbf{n}$. C, red curves; G, black curves. (a) Vertical profiles of $\langle cos\theta \rangle$. PDFs of $cos\theta$ at different interval height: (b) $-20<z_I/{\eta }_{\mathrm{\Pi }}<-15$, (c) $-10<z_I/{\eta }_{\mathrm{\Pi }}<-5$, and (d) $-2.5<z_I/{\eta }_{\mathrm{\Pi }}<2.5$.