Energy and enstrophy spectra and fluxes for the inertial-dissipation range of two-dimensional turbulence

In this paper, using Pao's conjecture [Y.-H. Pao, Phys. Fluids 8, 1063 (1965)], we derive expressions for the spectra and fluxes of kinetic energy and enstrophy for two-dimensional (2D) forced turbulence that extend beyond the inertial range. In these expressions, the fluxes and the spectra contain additional factors of the exponential form. To validate these model predictions, we perform numerical simulations of 2D turbulence with external force applied at wavenumber band $(k_f,k_f+1)$ in the intermediate range. The numerical results match with the model predictions, except for the energy and enstrophy fluxes for $k<k_f$, where the fluxes exhibit significant fluctuations. We show that these fluctuations arise due to the unsteady nature of the flow at small wavenumbers. For $k<k_f$, the shell-to-shell energy transfers computed using numerical data show forward energy transfers among the neighboring shells, but backward energy transfers among other shells.

Figure 1

(a) Plot of total energy $E$ vs $t$ for ${2048}^2$ grid (blue solid curve) and ${8192}^2$ grid (dashed red curve). (b) Plot of $E_u\left(k\right)$ vs $k$ at various times for the ${8192}^2$ grid simulation. These plots indicate approximate steady states for the simulation runs.

Figure 2

Kinetic energy spectra for single time frames each of 2D forced turbulence on the (a) ${2048}^2$ grid and (b) ${8192}^2$ grid. The forcing wavenumbers are indicated by dashed vertical lines. The plots exhibit numerical results (solid red curves), model predictions [Eqs. (25) and (32)] (dashed black curves), and Kraichnan [22]'s predictions (solid black lines). The model predictions are in general agreement with numerical results. Fitting parameters are given in Tables 1 and 2.

Figure 3

Plots of kinetic energy flux, ${\mathrm{\Pi }}_u\left(k\right)$ (left panel), and enstrophy fluxes, ${\mathrm{\Pi }}_{\omega }\left(k\right)$ (right panel): (a, b) for a single time frame of ${2048}^2$ run, (c, d) time averaged ${\mathrm{\Pi }}_u\left(k\right)$ and ${\mathrm{\Pi }}_{\omega }\left(k\right)$ over 35 frames of the ${2048}^2$ run, and (e, f) time averaged ${\mathrm{\Pi }}_u\left(k\right)$ and ${\mathrm{\Pi }}_{\omega }\left(k\right)$ over 35 frames of the ${8192}^2$ run. The numerical results (red solid lines) match with the model predictions [Eqs. (30) and (33)] (black dashed lines) in the $k>k_f$ regime. The discrepancies in the $k<k_f$ regime are possibly due to the unsteady nature of the flow. Note that the plots are in log-log scale because of the huge range of scales of ${\mathrm{\Pi }}_u\left(k\right)$ and ${\mathrm{\Pi }}_{\omega }\left(k\right)$.

Figure 4

For 2D simulation on the ${2048}^2$ grid, plots of $|-D\left(k\right)-d{\mathrm{\Pi }}_u\left(k\right)/dk|$ and $|\partial E(k,t)/\partial t|$. Significant measures of $|\partial E(k,t)/\partial t|$ indicate the unsteady nature of the flow.

Figure 5

For 2D simulation on the ${8192}^2$ grid, plots of the shell-to-shell energy transfers $T_{u,n}^{u,m}$ vs $n-m$ in the inertial range of the $k<k_f$ regime (a) for a single time frame, (b) for time-averaged data with 35 frames, and (c, d) density plots of shell-to-shell energy transfers $T_{u,n}^{u,m}$ corresponding to (a) and (b), respectively. The $x$ and $y$ axes represent the receiver and giver shells, respectively. The energy transfers are local and forward for neighboring shells, but nonlocal and backward for distant shells.