Universal Profile of the Vortex Condensate in Two-Dimensional Turbulence

An inverse turbulent cascade in a restricted two-dimensional periodic domain creates a condensate—a pair of coherent system-size vortices. We perform extensive numerical simulations of this system and carry out theoretical analysis based on momentum and energy exchanges between the turbulence and the vortices. We show that the vortices have a universal internal structure independent of the type of small-scale dissipation, small-scale forcing, and boundary conditions. The theory predicts not only the vortex inner region profile, but also the amplitude, which both perfectly agree with the numerical data.

Figure 1

Plot of the total vorticity normalized by $(\epsilon /\alpha L^2{)}^{1/2}$ during the condensate regime of simulation B.

Figure 2

Radial profile of the mean vorticity $\mathrm{\Omega }$ normalized by $(\epsilon /\alpha L^2{)}^{1/2}$. The straight black dashed line corresponds to the theoretically predicted radial profile $\sqrt{3}{(r/L)}^{-1}$ determined from Eq. (8).

Figure 3

Radial profile of the mean polar velocity $U$ in log-lin coordinates. The horizontal black line is $(\alpha /\epsilon {)}^{1/2}U=\sqrt{3}$ [Eq. (8)].

Figure 4

Third-order moments, $⟨u^3⟩$, $⟨u^{2}v⟩$, $⟨u{v}^2⟩$, and $⟨v^3⟩$ for simulation C: an additional smallness of all odd in $v$ moments.

Figure 5

Polar velocity fluctuations.

Figure 6

Radial velocity fluctuations.

Figure 7

Normalized radial pressure profile. The data are compared to the prediction (9) with $R/L=0.143$.