Dynamics of saturated energy condensation in two-dimensional turbulence

In two-dimensional forced Navier-Stokes turbulence, energy cascades to the largest scales in the system to form a pair of coherent vortices known as the Bose condensate. We show, both numerically and analytically, that the energy condensation saturates and the system reaches a statistically stationary state. The time scale of saturation is inversely proportional to the viscosity and the saturation energy level is determined by both the viscosity and the force. We further show that, without sufficient resolution to resolve the small-scale enstrophy spectrum, numerical simulations can give a spurious result for the saturation energy level. We also find that the movement of the condensate is similar to the motion of an inertial particle with an effective drag force. Furthermore, we show that the profile of the saturated coherent vortices can be described by a Gaussian core with exponential wings.

Figure 1

Pseudocolor plot of vorticity showing Bose condensation. Red (the top-left vortex) and blue (the bottom-right vortex) represent positive and negative vorticity in physical space, respectively. The color scale is shown in the color bar on the right, which is chosen to make the fluctuation visible. The vorticity is normalized so that $\max \left|\omega \right|=1$.

Figure 2

Unnormalized energy evolution in simulations $\mathtt{a}\mathtt{16}$–$\mathtt{d}\mathtt{32}$. The thick red (lowermost), green, blue, and black (uppermost) curves are for different viscosity $\nu$ (see labels) while solid, dashed, and dotted styles denote different forcing wave number $k_{\mathrm{i}}$ (see legend). The green diamonds and blue squares are for simulations ${\mathtt{b}\mathtt{16}}^{\mathtt{f}}$ and $\mathtt{c}\mathtt{64}$${}^{\mathtt{b}}$, respectively. They correspond to single precision and underresolved simulations.

Figure 3

Energy spectrum in simulation $\mathtt{c}\mathtt{32}$. The thick solid curve is the energy spectrum $E_k$ computed at $t=1000$ with bin size $\Delta k=k_1$. The dashed line shows $k^{-5/3}$ while the dotted line is proportional to $k^{-3}$. The gray areas, which represent our three-scale model, are drawn in proper scales. The condensate at zone (i) contains a large amount of energy. The broken power law in zones (ii) and (iii) have slopes $-5/3$ and $-3$, respectively. See Sec. 3a for details of the model.

Figure 4

Normalized energy evolution in simulations $\mathtt{a}\mathtt{16}$–$\mathtt{d}\mathtt{32}$. The thick red, green, blue, and black curves (all under the “Simulations” label) are for different viscosity (see also labels in Fig. 2) while solid, dashed, and dotted styles denote different forcing wave number $k_{\mathrm{i}}$ (see legend). We normalize the energy and time using the results from our three-scale model. Almost all the curves collapse onto each other. The gray solid curve, green diamonds, and blue squares are the theoretical prediction ${\mathtt{b}\mathtt{16}}^{\mathtt{f}}$ and ${\mathtt{c}\mathtt{64}}^{\mathtt{b}}$, respectively.

Figure 5

Unfolded trajectory of the positive condensate vortex in simulation ${\mathtt{path}}^{\mathtt{100}}$. The computational domain is ${[0,2\pi )}^2$ (see gray box near the origin). We first unfold the trajectory into ${\mathbb{R}}^2$ and then shift the initial position to the origin.

Figure 6

The gray curves are the displacement square $\Delta r^2$ for all 16 ${\mathtt{path}}^{\mathtt{i}}$ runs. The thick solid curve shows their average. The dashed curve, which is indistinguishable from the solid curve, is the solution (35) with parameter $\xi =0.1$.

Figure 7

We apply coordinate transformations to shift the center of the positive condensate vortex to the origin in simulation $\mathtt{c}\mathtt{32}$. The noisy gray curves show the vorticity at several different snapshots along the $x$ axis (with $y=0$). The black solid curve is an average over 1000 such profiles. It consists of a sharp Gaussian core (dashed curve) and exponential wings (dotted curve), although for $\left|x\right|\gtrsim \pi /2$ it falls off superexponentially. The open circles show the reconstruction from the Fourier coefficients listed in Table .

Figure 8

The solid and dotted histogram are the energy spectra $E_k$ of the full ($\omega$) and the coherent ($\tilde{\omega }$) vorticity in simulation $\mathtt{c}\mathtt{32}$ at $t=10000$. The spectral bins have even width in log-scale. The valley and second peak right next to the condensate mode are physical. The valley corresponds to the $\left|\mathit{k}\right|=\sqrt{2}{k}_1$ and $2k_1$ modes, which have even $k_x+k_y$ (open circles in inset). The second peak corresponds to the $\left|\mathit{k}\right|=\sqrt{5}{k}_1$ modes, which are the first harmonics of the condensate (gray circles in inset).