2 General relations

We consider the case where 2-D turbulence is excited in a finite box of size $L$ by an external forcing. It is assumed to be a random quantity with statistical properties that are homogeneous in time and space. We assume also that correlation functions of the pumping force are isotropic. The main object of our investigation is the stationary (in the statistical sense) turbulent state caused by such forcing. To excite turbulence the forcing should be stronger than dissipation related both to the bottom friction and to the viscosity at the pumping scale. That implies that the characteristic velocity gradient of the fluctuations produced by the forcing should be much larger than the flow damping at the pumping scale. The velocity gradient is estimated as $\unicode[STIX]{x1D716}^{1/3}k_{f}^{2/3}$ , where $\unicode[STIX]{x1D716}$ is the energy flux (energy production rate per unit mass) and $k_{f}$ is the absolute value of the characteristic wavevector of the pumping force. Thus we arrive at the inequalities

(2.1) $$\begin{eqnarray}\unicode[STIX]{x1D716}^{1/3}k_{f}^{2/3}\gg \unicode[STIX]{x1D6FC},~\unicode[STIX]{x1D708}k_{f}^{2}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D6FC}$ is the bottom friction coefficient and $\unicode[STIX]{x1D708}$ is the kinematic viscosity coefficient, therefore $\unicode[STIX]{x1D708}k_{f}^{2}$ is the viscous damping rate at the pumping scale $k_{f}^{-1}$ . In simulations, hyperviscosity is often used. In the case the inequalities (2.1) are still obligatory for exciting turbulence, where $\unicode[STIX]{x1D708}k_{f}^{2}$ has to be replaced by the hyperviscous damping rate at the pumping scale  $k_{f}^{-1}$ .

If the inequalities (2.1) are satisfied then turbulence is excited in the box and random pulsations of different scales are formed due to nonlinear hydrodynamic interaction. The pumped energy flows to larger scales whereas the pumped enstrophy flows to smaller scales (Kraichnan Reference Kraichnan1967; Leith Reference Leith1968; Batchelor Reference Batchelor1969). Thus two cascades are formed: the energy cascade (inverse cascade) realized at scales larger than the forcing scale $k_{f}^{-1}$ and the enstrophy cascade realized at scales smaller than that scale. In an unbound 2-D system the inverse energy cascade is terminated by the bottom friction at the scale

(2.2) $$\begin{eqnarray}L_{\unicode[STIX]{x1D6FC}}=\unicode[STIX]{x1D716}^{1/2}\unicode[STIX]{x1D6FC}^{-3/2},\end{eqnarray}$$

where a balance between the energy flux $\unicode[STIX]{x1D716}$ and the bottom friction is achieved. The enstrophy cascade is terminated by viscosity (or hyperviscosity) (Boffetta & Ecke Reference Boffetta and Ecke2012).

In a finite box the above two-cascade picture is realized if the box size $L$ is larger than $L_{\unicode[STIX]{x1D6FC}}$ . Here we consider the opposite case $L<L_{\unicode[STIX]{x1D6FC}}$ . Then the energy, transferred by the nonlinearity to the box size $L$ by the inverse cascade, is accumulated there, giving rise to a mean (coherent) flow. We analyse the statistically stationary case where the mean flow is already formed and stabilized by the bottom friction. To describe the flow, we use the Reynolds decomposition, that is the flow velocity is presented as the sum $\boldsymbol{V}+\boldsymbol{v}$ where $\boldsymbol{V}$ is the velocity of the coherent flow and $\boldsymbol{v}$ represents velocity fluctuations on the background of the coherent flow. Let us stress that $\boldsymbol{V}$ is an average over time; it possesses a complicated spatial structure.

As numerical simulation and experiment show, the coherent flow contains some vortices separated by a hyperbolic flow. The characteristic velocity $V$ of the coherent motion can be estimated as $V\sim \sqrt{\unicode[STIX]{x1D716}/\unicode[STIX]{x1D6FC}}$ . This estimate is a consequence of the energy balance: in the stationary case the incoming energy rate $\unicode[STIX]{x1D716}$ is equal to the bottom friction rate. The characteristic mean vorticity in the hyperbolic region is estimated as $\unicode[STIX]{x1D6FA}\sim L^{-1}\sqrt{\unicode[STIX]{x1D716}/\unicode[STIX]{x1D6FC}}$ . However, inside the coherent vortices the mean vorticity $\unicode[STIX]{x1D6FA}$ is much higher than the estimate (Chertkov et al. Reference Chertkov, Connaughton, Kolokolov and Lebedev2007; Xia et al. Reference Xia, Shats and Falkovich2009; Laurie et al. Reference Laurie, Boffetta, Falkovich, Kolokolov and Lebedev2014). The maximal value of the mean vorticity $\unicode[STIX]{x1D6FA}$ is achieved in the viscous core of the vortex. The radius of the core can be estimated as $(\unicode[STIX]{x1D708}/\unicode[STIX]{x1D6FC})^{1/2}$ (Kolokolov & Lebedev Reference Kolokolov and Lebedev2016).

3 Coherent vortex

Here we examine the flow inside the coherent vortex. We attach the origin of our reference system to the vortex centre, which is determined as the point of maximum vorticity. This definition corresponds to the procedures used in the works by Chertkov et al. (Reference Chertkov, Connaughton, Kolokolov and Lebedev2007), Xia et al. (Reference Xia, Shats and Falkovich2009) and Laurie et al. (Reference Laurie, Boffetta, Falkovich, Kolokolov and Lebedev2014) to establish the mean vortex profile. The position of the vortex centre fluctuates: in the laboratory experiments it fluctuates near a fixed position determined by the cell geometry. For the periodic set-up (used in the numerical simulations) the vortex centre can shift significantly from its initial position, and only the average relative position of the vortices is fixed. The reference system is not inertial, and the velocity of the vortex centre is subtracted from the flow velocity in the system. However, the flow vorticity in the reference system coincides with that in the laboratory reference system.

As was established experimentally and numerically (Chertkov et al. Reference Chertkov, Connaughton, Kolokolov and Lebedev2007; Xia et al. Reference Xia, Shats and Falkovich2009; Laurie et al. Reference Laurie, Boffetta, Falkovich, Kolokolov and Lebedev2014), in the chosen reference system the mean flow possesses axial symmetry. Such flow can be characterized by the polar velocity $U$ , which depends on the distance $r$ from the vortex centre. Then the mean vorticity is calculated as $\unicode[STIX]{x1D6FA}=\unicode[STIX]{x2202}_{r}U+U/r$ . To obtain an equation for the profile $U(r)$ , one has to use the complete Navier–Stokes equation. Assuming that the average pumping force is zero, one finds the Reynolds equation after averaging (Monin & Yaglom Reference Monin, Yaglom and Limley1971). Outside the viscous core where the viscous term is irrelevant we arrive at

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}U=-\left(\unicode[STIX]{x2202}_{r}+\frac{2}{r}\right)\langle uv\rangle ,\end{eqnarray}$$

where $v$ and $u$ are the radial and polar components of the velocity fluctuations, and angular brackets mean averaging over time.

To analyse the flow fluctuations inside the vortex, it is convenient to start from the equation for the fluctuating vorticity  $\unicode[STIX]{x1D71B}$ ,

(3.2) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D71B}+(U/r)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D71B}+v\unicode[STIX]{x2202}_{r}\unicode[STIX]{x1D6FA}+\unicode[STIX]{x1D735}(\boldsymbol{v}\unicode[STIX]{x1D71B}-\langle \boldsymbol{v}\unicode[STIX]{x1D71B}\rangle )=\unicode[STIX]{x1D719}-\hat{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D71B},\end{eqnarray}$$

which is obtained from the same Navier–Stokes equation. Here $\unicode[STIX]{x1D711}$ is the polar angle, $\unicode[STIX]{x1D719}$ is the curl of the pumping force, $\boldsymbol{v}$ is the fluctuating velocity, and the operator $\hat{\unicode[STIX]{x1D6E4}}$ represents dissipation including some terms. Among the terms are the bottom friction $\unicode[STIX]{x1D6FC}$ and the viscosity term, $-\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}$ . For the case of hyperviscosity the last contribution to $\hat{\unicode[STIX]{x1D6E4}}$ is replaced by $(-1)^{p+1}\unicode[STIX]{x1D708}_{p}(\unicode[STIX]{x1D6FB}^{2})^{p}$ where $p$ is an integer. An additional contribution to $\hat{\unicode[STIX]{x1D6E4}}$ is related to the nonlinear interaction of the fluctuations. Though the interaction is weak, it could be larger than $\unicode[STIX]{x1D6FC}$ and $-\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}$ because of the smallness of the contributions.

4 Universal interval

Further we consider the region outside the vortex core where the coherent velocity gradient is large enough,

(4.1) $$\begin{eqnarray}U/r\gg \unicode[STIX]{x1D716}^{1/3}k_{f}^{2/3}.\end{eqnarray}$$

In this case fluctuations in the interval of scales between the pumping scale $k_{f}^{-1}$ and the radius $r$ are strongly suppressed by the coherent flow. The inequality (4.1) means that the mean velocity gradient $U/r$ is larger than the gradient of the velocity fluctuations in the region at all scales larger than $k_{f}^{-1}$ . Therefore the passive regime is realized there, that is, the self-interaction of the velocity fluctuations is weak. The interval of scales outside the vortex core where the inequality (4.1) is satisfied will be called hereafter the universal interval of scales.

Moreover, the passive regime is realized for scales smaller than the pumping scale $k_{f}^{-1}$ . Indeed, in the direct cascade the velocity gradients can be estimated as $\unicode[STIX]{x1D716}^{1/3}k_{f}^{2/3}$ , up to logarithmic factors that depend weakly on scale; see Kraichnan (Reference Kraichnan1971, Reference Kraichnan1975) and Falkovich & Lebedev (Reference Falkovich and Lebedev1994a ,Reference Falkovich and Lebedev b , Reference Falkovich and Lebedev2011). Therefore the inequality (4.1) means domination of the coherent velocity gradient in the interval of scales where the direct cascade would be realized. The passive regime can be consistently analysed. Then one neglects the nonlinear term in (3.2), retaining a linear equation for the vorticity fluctuation  $\unicode[STIX]{x1D71B}$ . The equation enables one to express $\unicode[STIX]{x1D71B}$ in terms of the pumping $\unicode[STIX]{x1D719}$ and then to calculate correlation functions of $\unicode[STIX]{x1D71B}$ via the correlation functions of  $\unicode[STIX]{x1D719}$ .

Further we focus on the case where the pumping $\unicode[STIX]{x1D719}$ is short-correlated in time and has Gaussian statistics. Direct calculations (Kolokolov & Lebedev Reference Kolokolov and Lebedev2016) show that in this case

(4.2) $$\begin{eqnarray}\langle uv\rangle =\unicode[STIX]{x1D716}/\unicode[STIX]{x1D6F4},\end{eqnarray}$$

where $\unicode[STIX]{x1D6F4}$ is the local shear rate of the coherent flow:

(4.3) $$\begin{eqnarray}\unicode[STIX]{x1D6F4}=r\unicode[STIX]{x2202}_{r}(U/r)=\unicode[STIX]{x2202}_{r}U-U/r.\end{eqnarray}$$

Expression (4.2) is derived for the condition $\unicode[STIX]{x1D6F4}\gg \unicode[STIX]{x1D6E4}_{f}$ , where $\unicode[STIX]{x1D6E4}_{f}$ is the damping of the velocity fluctuations at the pumping scale. The validity of the condition is guaranteed by the inequalities (2.1) and (4.1). Some additional condition $\unicode[STIX]{x1D708}k_{f}^{2}\gg \unicode[STIX]{x1D6FC}$ is needed for validity of the expression (4.2); the inequality is assumed to be satisfied in our scheme. (Note that the inequality is satisfied in numerical simulations (Laurie et al. Reference Laurie, Boffetta, Falkovich, Kolokolov and Lebedev2014).) The opposite case needs some additional analysis that is beyond the scope of our work.

Substituting expression (4.2) into (3.1), one finds a solution

(4.4a,b ) $$\begin{eqnarray}U=\sqrt{3\unicode[STIX]{x1D716}/\unicode[STIX]{x1D6FC}},\quad \unicode[STIX]{x1D6F4}=-U/r,\end{eqnarray}$$

for the mean profile. Thus we arrive at the flat profile of the polar velocity found in Laurie et al. (Reference Laurie, Boffetta, Falkovich, Kolokolov and Lebedev2014) and confirmed analytically in Kolokolov & Lebedev (Reference Kolokolov and Lebedev2016). It is characteristic of the universal region.

The left-hand side of the inequality (4.1) diminishes as $r$ grows. Therefore it is broken at some $r\sim R_{u}$ . Substituting expression (4.4) into (4.1), one obtains

(4.5) $$\begin{eqnarray}R_{u}=L_{\unicode[STIX]{x1D6FC}}^{1/3}k_{f}^{-2/3}=\unicode[STIX]{x1D716}^{1/6}\unicode[STIX]{x1D6FC}^{-1/2}k_{f}^{-2/3}.\end{eqnarray}$$

The scale $R_{u}$ determines the boundary of the region where flow fluctuations are passive. It could be treated as the vortex size if $R_{u}$ is less than the box size, as in numerical simulations (Laurie et al. Reference Laurie, Boffetta, Falkovich, Kolokolov and Lebedev2014) where the universal region is well separated from the outer region, which is not completely passive. The case $R_{u}\gtrsim L$ is characteristic of the numerical simulations (Chertkov et al. Reference Chertkov, Connaughton, Kolokolov and Lebedev2007; Frishman, Laurie & Falkovich Reference Frishman, Laurie and Falkovich2016) and the experiment (Xia et al. Reference Xia, Shats and Falkovich2009); in this case the passive regime is realized everywhere in the box. Even in this case the equations determining the global structure of the condensate are essentially nonlinear. Solutions of this equations can undergo instabilities and bifurcations of various types as the friction parameter $\unicode[STIX]{x1D6FC}$ and the geometry of the system vary. For example, it is found in Frishman et al. (Reference Frishman, Laurie and Falkovich2016) that diminishing $\unicode[STIX]{x1D6FC}$ in the rectangular geometry leads to a rise of the new vortex. Note that the global structure of the mean flow is a subject of special investigation that is beyond the scope of our article.

5 Vorticity fluctuations

Since the flow fluctuations inside the universal region are passive we can use the linearized version of (3.2),

(5.1) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D71B}+(U/r)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D711}}\unicode[STIX]{x1D71B}+v\unicode[STIX]{x2202}_{r}\unicode[STIX]{x1D6FA}+\hat{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D71B}=\unicode[STIX]{x1D719}.\end{eqnarray}$$

Since the pumping is assumed to be short-correlated in time, its statistics is determined by the pair correlation function

(5.2) $$\begin{eqnarray}\langle \unicode[STIX]{x1D719}(t,\boldsymbol{k})\unicode[STIX]{x1D719}(t^{\prime },\boldsymbol{k}^{\prime })\rangle =2(2\unicode[STIX]{x03C0})^{2}\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\unicode[STIX]{x1D6FF}(t-t^{\prime })k^{2}\unicode[STIX]{x1D712}(\boldsymbol{k})\end{eqnarray}$$

for the space Fourier transform of $\unicode[STIX]{x1D719}$ . The function $\unicode[STIX]{x1D712}(\boldsymbol{k})$ has a profile with the characteristic pumping wavevector $k_{f}$ and is normalized:

(5.3) $$\begin{eqnarray}\int \frac{\text{d}^{2}\boldsymbol{k}}{(2\unicode[STIX]{x03C0})^{2}}\unicode[STIX]{x1D712}(\boldsymbol{k})=1.\end{eqnarray}$$

Then $\unicode[STIX]{x1D716}$ is the energy (per unit mass, per unit time) pumped into the system, i.e. the energy flux.

We analyse the fluctuations near a radius $r=r_{0}$ with scales much smaller than the radius. Then the shear approximation for the mean velocity can be used. We pass to the reference system rotating with the angular velocity $\unicode[STIX]{x1D6FA}(r_{0})$ and expand all terms in (5.1) in $x_{r}=r-r_{0}$ and $x_{\unicode[STIX]{x1D711}}=r_{0}\unicode[STIX]{x1D711}$ . We assume that the parameter $(k_{f}r)^{-1}$ is small. Then the term $v\unicode[STIX]{x2202}_{r}\unicode[STIX]{x1D6FA}$ in (5.1) can be discarded and we end up with the following equation:

(5.4) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D71B}+\unicode[STIX]{x1D6F4}x_{r}\unicode[STIX]{x2202}_{2}\unicode[STIX]{x1D71B}+\hat{\unicode[STIX]{x1D6E4}}\unicode[STIX]{x1D71B}=\unicode[STIX]{x1D719}.\end{eqnarray}$$

Here $\unicode[STIX]{x1D6F4}$ is the local shear rate (4.4) in the universal region $\unicode[STIX]{x1D6F4}\propto r^{-1}$ . Let us rewrite the evolution equation (5.4) for the spatial Fourier components of the vorticity $\unicode[STIX]{x1D71B}_{\boldsymbol{k}}$ :

(5.5) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D71B}(\boldsymbol{k})-\unicode[STIX]{x1D6F4}k_{\unicode[STIX]{x1D711}}\unicode[STIX]{x2202}\unicode[STIX]{x1D71B}(\boldsymbol{k})/\unicode[STIX]{x2202}k_{r}+\unicode[STIX]{x1D6E4}(k)\unicode[STIX]{x1D71B}(\boldsymbol{k})=\unicode[STIX]{x1D719}(t,\boldsymbol{k}),\end{eqnarray}$$

where the components $k_{r},k_{\unicode[STIX]{x1D711}}$ of the wavevector $\boldsymbol{k}$ correspond to the variables $x_{r},x_{\unicode[STIX]{x1D711}}$ . Solving the evolution equation (5.5), one obtains a formal solution

(5.6) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D71B}(t,\boldsymbol{k}) & = & \displaystyle \int ^{t}\text{d}\unicode[STIX]{x1D70F}\,\unicode[STIX]{x1D719}[\unicode[STIX]{x1D70F},k_{r}+\unicode[STIX]{x1D6F4}(t-\unicode[STIX]{x1D70F})k_{\unicode[STIX]{x1D711}},k_{\unicode[STIX]{x1D711}}]\nonumber\\ \displaystyle & & \displaystyle \times \exp \left\{-\int _{\unicode[STIX]{x1D70F}}^{t}\text{d}\unicode[STIX]{x1D70F}^{\prime }\,\unicode[STIX]{x1D6E4}\left[\sqrt{(k_{r}+\unicode[STIX]{x1D6F4}(t-\unicode[STIX]{x1D70F}^{\prime })k_{\unicode[STIX]{x1D711}})^{2}+k_{\unicode[STIX]{x1D711}}^{2}}\,\right]\right\}.\end{eqnarray}$$

Now we can find the simultaneous pair vorticity correlation function for the Fourier transform from (5.2):

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D71B}(t,\boldsymbol{k})\unicode[STIX]{x1D71B}(t,\boldsymbol{k}^{\prime })\rangle =2(2\unicode[STIX]{x03C0})^{2}\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\int _{0}^{\infty }\text{d}\unicode[STIX]{x1D70F}\,q^{2}\unicode[STIX]{x1D712}(\boldsymbol{q})\exp \left[-2\int _{0}^{\unicode[STIX]{x1D70F}}\text{d}\unicode[STIX]{x1D70F}^{\prime }\,\unicode[STIX]{x1D6E4}(\boldsymbol{q}^{\prime })\right].\quad & \displaystyle\end{eqnarray}$$

Here

(5.8) $$\begin{eqnarray}\boldsymbol{q}=(k_{r}+\unicode[STIX]{x1D6F4}\unicode[STIX]{x1D70F}k_{\unicode[STIX]{x1D711}},k_{\unicode[STIX]{x1D711}}),\end{eqnarray}$$

and $\boldsymbol{q}^{\prime }$ differs from $\boldsymbol{q}$ by a substitution $\unicode[STIX]{x1D70F}\rightarrow \unicode[STIX]{x1D70F}^{\prime }$ . The factor $\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })$ in the expression (5.7) reflects space homogeneity that is not destroyed by a shear flow.

Let us consider scales larger than $k_{f}^{-1}$ , that is wavevectors $k\ll k_{f}$ . With this condition the main contribution to the integral (5.7) is gained from times $\unicode[STIX]{x1D70F}\sim k_{f}/(\unicode[STIX]{x1D6F4}k_{\unicode[STIX]{x1D711}})$ . In the case $|k_{\unicode[STIX]{x1D711}}|\gg \unicode[STIX]{x1D6E4}_{f}k_{f}/\unicode[STIX]{x1D6F4}$ the dissipation is irrelevant and the last exponential factor in (5.7) can be replaced by unity. Here, as above, $\unicode[STIX]{x1D6E4}_{f}$ is the flow damping at the pumping scale. Passing then to the integration over the wavevector (5.8), one obtains

(5.9) $$\begin{eqnarray}\displaystyle & \displaystyle \langle \unicode[STIX]{x1D71B}(t,\boldsymbol{k})\unicode[STIX]{x1D71B}(t,\boldsymbol{k}^{\prime })\rangle =\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\frac{2(2\unicode[STIX]{x03C0})^{2}\unicode[STIX]{x1D716}q_{f}}{\unicode[STIX]{x1D6F4}|k_{\unicode[STIX]{x1D711}}|}, & \displaystyle\end{eqnarray}$$

(5.10) $$\begin{eqnarray}\displaystyle & \displaystyle q_{f}=\int _{0}^{\infty }\text{d}q_{1}\,q^{2}\unicode[STIX]{x1D712}(q). & \displaystyle\end{eqnarray}$$

Here, we replaced the lower integration limit $|k_{r}|$ in the integral (5.10) by zero, since the main contribution to the integral comes from $q\sim k_{f}\gg k_{r}$ . The wavevector $q_{f}$ is of the order of the inverse pumping length.

There can exist an interval of the wavevectors $r^{-1}<|k_{\unicode[STIX]{x1D711}}|<\unicode[STIX]{x1D6E4}_{f}k_{f}/\unicode[STIX]{x1D6F4}$ where the dissipation is relevant. Then the exponential factor in (5.7) related to dissipation is relevant. The factor can be estimated as $\exp (-\unicode[STIX]{x1D6E4}_{f}\unicode[STIX]{x1D70F})$ , where the time $\unicode[STIX]{x1D70F}$ is needed to enhance $k_{\unicode[STIX]{x1D711}}$ to $k_{f}$ , that is $\unicode[STIX]{x1D70F}\sim k_{f}/(\unicode[STIX]{x1D6F4}k_{\unicode[STIX]{x1D711}})$ . Thus we come to the suppression factor $\exp (-A)$ , $A\sim \unicode[STIX]{x1D6E4}_{f}k_{f}/(\unicode[STIX]{x1D6F4}|k_{\unicode[STIX]{x1D711}}|)$ . Therefore the vorticity correlations are strongly suppressed in the region of wavevectors $|k_{\unicode[STIX]{x1D711}}|<\unicode[STIX]{x1D6E4}_{f}k_{f}/\unicode[STIX]{x1D6F4}$ . In real space, that means decay $\exp (-B)$ at $|x_{\unicode[STIX]{x1D711}}|>\unicode[STIX]{x1D6F4}/(\unicode[STIX]{x1D6E4}_{f}k_{f})$ where $B\sim \unicode[STIX]{x1D6E4}_{f}^{1/2}k_{f}^{1/2}|x_{\unicode[STIX]{x1D711}}|^{1/2}\unicode[STIX]{x1D6F4}^{-1/2}$ .

6 Velocity correlation functions

Knowing the vorticity correlation function, one can calculate the velocity correlation functions using the relation

(6.1) $$\begin{eqnarray}v_{\unicode[STIX]{x1D6FC}}(\boldsymbol{k})=\text{i}\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FD}}\frac{k_{\unicode[STIX]{x1D6FD}}}{k^{2}}\unicode[STIX]{x1D71B}(\boldsymbol{k}),\end{eqnarray}$$

valid for the Fourier transforms. If $k_{f}\gg |k_{\unicode[STIX]{x1D711}}|\gg \unicode[STIX]{x1D6E4}_{f}k_{f}/\unicode[STIX]{x1D6F4}$ and $k_{f}\gg |k_{r}|$ , then one finds from (5.9), (6.1)

(6.2) $$\begin{eqnarray}\displaystyle & \displaystyle \langle v(\boldsymbol{k})v(\boldsymbol{k}^{\prime })\rangle =2(2\unicode[STIX]{x03C0})^{3}\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\frac{q_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}}\frac{|k_{\unicode[STIX]{x1D711}}|}{\boldsymbol{k}^{4}}, & \displaystyle\end{eqnarray}$$

(6.3) $$\begin{eqnarray}\displaystyle & \displaystyle \langle u(\boldsymbol{k})u(\boldsymbol{k}^{\prime })\rangle =2(2\unicode[STIX]{x03C0})^{3}\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\frac{q_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}}\frac{k_{r}^{2}}{\boldsymbol{k}^{4}|k_{\unicode[STIX]{x1D711}}|}, & \displaystyle\end{eqnarray}$$

(6.4) $$\begin{eqnarray}\displaystyle & \displaystyle \langle v(\boldsymbol{k})u(\boldsymbol{k}^{\prime })\rangle =-2(2\unicode[STIX]{x03C0})^{3}\unicode[STIX]{x1D6FF}(\boldsymbol{k}+\boldsymbol{k}^{\prime })\frac{q_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}}\frac{k_{r}k_{\unicode[STIX]{x1D711}}}{\boldsymbol{k}^{4}|k_{\unicode[STIX]{x1D711}}|}. & \displaystyle\end{eqnarray}$$

If $r^{-1}<|k_{\unicode[STIX]{x1D711}}|<\unicode[STIX]{x1D6E4}_{f}k_{f}/\unicode[STIX]{x1D6F4}$ then the expressions are strongly suppressed due to dissipation.

It follows from the expressions (6.2), (6.4) that the averages $\langle v^{2}\rangle$ and $\langle u^{2}\rangle$ are determined by the infrared integrals. Therefore

(6.5) $$\begin{eqnarray}\displaystyle & \displaystyle \langle v^{2}\rangle ,\langle u^{2}\rangle \sim \frac{k_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}}r\quad \text{if}~\unicode[STIX]{x1D6E4}_{f}k_{f}r\ll \unicode[STIX]{x1D6F4}, & \displaystyle\end{eqnarray}$$

(6.6) $$\begin{eqnarray}\displaystyle & \displaystyle \langle v^{2}\rangle ,\langle u^{2}\rangle \sim \frac{\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6E4}_{f}}\quad \text{if}~\unicode[STIX]{x1D6E4}_{f}k_{f}r\gg \unicode[STIX]{x1D6F4}. & \displaystyle\end{eqnarray}$$

The average $\langle uv\rangle$ needs an additional analysis (Kolokolov & Lebedev Reference Kolokolov and Lebedev2016). This analysis shows that the quantity is determined by expression (4.2).

Note that we work in the reference system attached to the (moving) vortex centre. Therefore, comparing the expressions (6.5), (6.6) with the numerical simulations (Laurie et al. Reference Laurie, Boffetta, Falkovich, Kolokolov and Lebedev2014), one has to take into account the global soft mode of fluctuations related to translations of the vortex. As a result $\langle u^{2}\rangle$ , $\langle v^{2}\rangle$ extracted from the numerical simulations do not go to zero as $r\rightarrow 0$ . Note that the soft mode does not contribute to the structure function inside the vortex.

A special problem is calculation of $\langle u_{0}^{2}\rangle$ where $u_{0}$ is the zeroth angular harmonic of the fluctuating polar velocity. There is no advection term related to the average flow in the equation for $u_{0}$ . Therefore the quantity $\langle u_{0}^{2}\rangle$ is determined solely by the damping. Strictly speaking, the calculation of $\langle u_{0}^{2}\rangle$ is outside our shear approximation. However, our logic can be easily extended to this case to obtain

(6.7) $$\begin{eqnarray}\langle u_{0}^{2}\rangle \sim \frac{\unicode[STIX]{x1D716}}{k_{f}r\unicode[STIX]{x1D6E4}_{f}}.\end{eqnarray}$$

An explanation of this expression is based on the expression

(6.8) $$\begin{eqnarray}\langle u_{0}^{2}\rangle =\int \frac{\text{d}\unicode[STIX]{x1D711}}{2\unicode[STIX]{x03C0}}\langle u(\boldsymbol{r}_{1})u(\boldsymbol{r}_{2})\rangle ,\end{eqnarray}$$

where the points 1 and 2 are separated by the same distance $r$ from the vortex centre and $\unicode[STIX]{x1D711}$ is the angle between the vectors $\boldsymbol{r}_{1}$ and $\boldsymbol{r}_{2}$ . The factor $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D6E4}_{f}$ is the contribution to $\langle u_{0}^{2}\rangle$ caused by the pumping that is effective if $\unicode[STIX]{x1D711}\lesssim (k_{f}r)^{-1}$ . The contribution (6.7) should be taken into account besides (6.5); the latter is related to the sum of non-zero angular harmonics. In the case (6.6) the contribution (6.7) is small in comparison with one related to non-zero harmonics.

The average $\langle u_{0}^{2}\rangle$ was calculated previously in the paper by Falkovich (Reference Falkovich2016), where the contribution related to the pumping was ignored and the nonlinear effects were taken into account instead. This approach is correct outside the universal region, at $r>R_{u}$ where $R_{u}$ is determined by expression (4.5). At the border, where $r\sim R_{u}$ , our estimate (6.7) coincides with one obtained in Falkovich (Reference Falkovich2016).

It is worthwhile to characterize scales where the expressions (6.2)–(6.4) are correct by the velocity structure functions. One finds

(6.9) $$\begin{eqnarray}\displaystyle S_{11}(x_{r},x_{\unicode[STIX]{x1D711}}) & = & \displaystyle \langle [v(x_{r},x_{\unicode[STIX]{x1D711}})-v(0,0)]^{2}\rangle \nonumber\\ \displaystyle & = & \displaystyle \frac{2q_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}\unicode[STIX]{x03C0}}\int \text{d}^{2}\boldsymbol{k}\,\frac{|k_{\unicode[STIX]{x1D711}}|}{\boldsymbol{k}^{4}}(1-\text{e}^{\text{i}k_{r}x_{r}+\text{i}k_{\unicode[STIX]{x1D711}}x_{\unicode[STIX]{x1D711}}}),\end{eqnarray}$$

which is correct if $k_{f}^{-1}\ll |x_{r}$ , $x_{\unicode[STIX]{x1D711}}|\ll r$ , $k_{f}^{-1}\unicode[STIX]{x1D6F4}/\unicode[STIX]{x1D6E4}_{f}$ . Infrared divergence in the integral (6.9) can be regularized by substituting $k_{\unicode[STIX]{x1D711}}^{2}\rightarrow k_{\unicode[STIX]{x1D711}}^{2}+\unicode[STIX]{x1D707}^{2}$ , where $\unicode[STIX]{x1D707}\sim 1/r$ . The result of the integration can be expressed via the function

(6.10) $$\begin{eqnarray}{\mathcal{J}}(z)=\int _{0}^{\infty }\,\text{d}q\frac{\text{e}^{-z}}{q^{2}+\unicode[STIX]{x1D707}^{2}}\approx \frac{\unicode[STIX]{x03C0}}{2\unicode[STIX]{x1D707}}+z[\unicode[STIX]{x1D6E4}_{f}-1+\ln (\unicode[STIX]{x1D707}z)].\end{eqnarray}$$

Particularly, one finds

(6.11) $$\begin{eqnarray}S_{11}=\frac{2q_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}}\text{Re}\left[\frac{\unicode[STIX]{x03C0}}{2\unicode[STIX]{x1D707}}-{\mathcal{J}}+x_{r}\unicode[STIX]{x2202}_{1}{\mathcal{J}}\right],\end{eqnarray}$$

where ${\mathcal{J}}={\mathcal{J}}(x_{r}-\text{i}x_{\unicode[STIX]{x1D711}})$ , Calculating the expression (6.11), one finds

(6.12) $$\begin{eqnarray}S_{11}\approx \frac{2q_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}}\left[|x_{r}|+x_{\unicode[STIX]{x1D711}}\arctan \left(\frac{x_{\unicode[STIX]{x1D711}}}{|x_{r}|}\right)\right].\end{eqnarray}$$

Analogous expressions can be found for other components of the structure function:

(6.13) $$\begin{eqnarray}\displaystyle & \displaystyle S_{22}=\left\langle [u(x_{r},x_{\unicode[STIX]{x1D711}})-u(0,0)]^{2}\right\rangle \approx \frac{2q_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}}\left[x_{\unicode[STIX]{x1D711}}\arctan \left(\frac{x_{\unicode[STIX]{x1D711}}}{|x_{r}|}\right)-2|x_{r}|\ln \left(\unicode[STIX]{x1D707}\sqrt{x_{r}^{2}+x_{\unicode[STIX]{x1D711}}^{2}}\right)\right], & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and

(6.14) $$\begin{eqnarray}\displaystyle & \displaystyle S_{12}=\langle [v(x_{r},x_{\unicode[STIX]{x1D711}})-v(0,0)][u(x_{r},x_{\unicode[STIX]{x1D711}})-u(0,0)]\rangle \approx -\frac{2q_{f}\unicode[STIX]{x1D716}}{\unicode[STIX]{x1D6F4}}x_{r}\arctan \left(\frac{x_{\unicode[STIX]{x1D711}}}{|x_{r}|}\right).\qquad & \displaystyle\end{eqnarray}$$

In the region $|x_{r}|$ , $|x_{\unicode[STIX]{x1D711}}|\gg k_{f}^{-1}\unicode[STIX]{x1D6F4}/\unicode[STIX]{x1D6E4}_{f}$ the pair correlation functions are strongly suppressed and the structure functions are determined by the single-point averages.