Active versus Passive Scalar Turbulence

Active and passive scalars transported by an incompressible two-dimensional conductive fluid are investigated. It is shown that a passive scalar displays a direct cascade towards the small scales while the active magnetic potential builds up large-scale structures in an inverse cascade process. Correlations between scalar input and particle trajectories are found to be responsible for those dramatic differences as well as for the behavior of dissipative anomalies.

Figure 1

Power spectra of active and passive scalar variances $E_a(k)=\pi k|\widehat{a}(\mathit{k},t){|}^2$ and $E_c(k)=\pi k|\widehat{c}(\mathit{k},t){|}^2$. The fluctuations of $a$ and $c$ injected at the forcing wavelength $k_f$ flow towards smaller and larger wave numbers, respectively. At $k<k_f$ we observe power-law behaviors $E_a(k)\sim k^{-2.0\pm 0.1}$ and $E_c(k)\sim k^{0.7\pm 0.1}$, while at $k>k_f$ we find $E_a(k)\sim k^{-3.6\pm 0.1}$ and $E_c(k)\sim k^{-1.4\pm 0.1}$ (for a discussion about the spectra of two- and three-dimensional magnetohydrodynamics, see, e.g., Refs. 3, 4, 5, 6, 7, 8). In the lower left corner, the fluxes of scalar variance ${\Pi }_{a,c}$ out of wave number $k$. Negative values indicate an inverse cascade. In the upper right corner, the total scalar variance $e_{a,c}(t)=\int E_{a,c}(k,t)dk$. The active variance $e_a(t)$ grows linearly in time, whereas $e_c(t)$ fluctuates around a finite value (see text). The rate of active to passive scalar dissipation is ${ϵ}_a/{ϵ}_c\simeq 0.005$. The data result from the numerical integration of Eqs. (1, 2, 3) by a dealiased pseudospectral parallel code, on a doubly periodic box of size $2\pi$ and resolution ${4096}^2$. The forcing terms $f_a$ and $f_c$ are homogeneous independent Gaussian processes with zero mean and correlation $⟨{\widehat{f}}_i(\mathit{k},t){\widehat{f}}_j({\mathit{k}}^{\prime },t^{\prime })⟩=[F_0/(2\pi k_f)]{\delta }_{ij}\delta (\mathit{k}+{\mathit{k}}^{\prime })\delta (k-k_f)\delta (t-t^{\prime })$ where $i,j=a,c$. The coefficients $\kappa$ and $\nu$ are chosen to obtain a dissipative length scale of the order of the smallest resolved scales. All fields are set to zero at $t=0$, and time is defined in units of eddy-turnover time $\tau =l_f/v_{\mathrm{r}\mathrm{m}\mathrm{s}}$ where $l_f=2\pi /k_f$.

Figure 2

PDF’s of active (solid line) and passive (dashed line) scalar fields normalized by their standard deviation. The active scalar PDF is indistinguishable from a Gaussian (dotted line).

Figure 3

Time runs from bottom to top. First column: time evolution of the active scalar field resulting from the numerical integration of Eqs. (1, 3). Second column: backward evolution of the particle propagator according to Eq. (4). Third column: time evolution of the passive scalar field in the same flow.

Figure 4

Top: ${\varphi }_{a,c}(s)=\int d\mathit{y}{f}_{a,c}(\mathit{y},s)p(\mathit{y},s\mid \mathit{x},t)$. The two graphs have the same scale on the vertical axis. Here, $t=32$. Bottom: time integrals ${\int }_0^s{\varphi }_a(s^{\prime })d{s}^{\prime }$ (upper curve) and ${\int }_0^s{\varphi }_c(s^{\prime })d{s}^{\prime }$ (lower curve). Note the different scale on the vertical axis. Recall that ${\int }_0^t{\varphi }_a(s^{\prime })d{s}^{\prime }=a(\mathit{x},t)$ (and similarly for $c$).