Turbulence on Hyperbolic Plane: The Fate of Inverse Cascade

Abstract

We describe ideal incompressible hydrodynamics on the hyperbolic plane which is an infinite surface of constant negative curvature. We derive equations of motion, general symmetries and conservation laws, and then consider turbulence with the energy density linearly increasing with time due to action of small-scale forcing. In a flat space, such energy growth is due to an inverse cascade, which builds a constant part of the velocity autocorrelation function proportional to time and expanding in scales, while the moments of the velocity difference saturate during a time depending on the distance. For the curved space, we analyze the long-time long-distance scaling limit, that lives in a degenerate conical geometry, and find that the energy-containing mode linearly growing with time is not constant in space. The shape of the velocity correlation function indicates that the energy builds up in vortical rings of arbitrary diameter but of width comparable to the curvature radius of the hyperbolic plane. The energy current across scales does not increase linearly with the scale, as in a flat space, but reaches a maximum around the curvature radius. That means that the energy flux through scales decreases at larger scales so that the energy is transferred in a non-cascade way, that is the inverse cascade spills over to all larger scales where the energy pumped into the system is cumulated in the rings. The time-saturated part of the spectral density of velocity fluctuations contains a finite energy per unit area, unlike in the flat space where the time-saturated spectrum behaves as $\,k^{-5/3}$.

Appendices

Appendix 1: Some Notions of Riemannian Geometry

Let $M$ be a $d$-dimensional oriented manifold equipped with a Riemannian metric $\,g_{ij}{\mathrm {d}}x^i{\mathrm {d}}x^j$.  Such a metric may be used to lower the indices of the contravariant tensor fields,  e.g. $v_i=g_{ij}v^j$.  One has $\,v^j=g^{ij}v_i$,  where $\,(g^{ij})\,$ is the matrix inverse to $\,(g_{ij})$.  The metric together with the orientation induces the volume form $\,\chi =\sqrt{g}\,{\mathrm {d}}x^1\wedge \cdots \wedge {\mathrm {d}}x^d \equiv \sqrt{g}\,{\mathrm {d}}x$,  where $\,\sqrt{g}\,$ is the shorthand notation for $\,\sqrt{\mathrm{det}(g_{ij})}$.  A Riemannian metric induces the Levi-Civita connection acting on tensor fields by

$$\begin{aligned}&\nabla _it_{k_1\ldots k_r}^{j_1\ldots j_p}=\,\partial _it_{k_1\ldots k_r}^{j_1\ldots j_p} \,+\,\Gamma _{il}^{j_1}t_{k_1\ldots k_r}^{lj_2\ldots j_p} + \cdots +\Gamma _{il}^{j_p} t_{k_1\ldots k_r}^{j_1\ldots j_{p-1}l} \,-\,\Gamma _{ik_1}^lt_{lk_2\ldots k_r}^{j_1\ldots j_p}- \cdots -\Gamma _{ik_r}^lt_{k_1\ldots k_{r-1}l}^{j_1\ldots j_p}, \nonumber \\ \end{aligned}$$

(14.1)

where $\,\Gamma ^k_{ij}\,$ are given by the Christoffel symbols

$$\begin{aligned} \Gamma ^k_{ij}\equiv \{ \begin{array}{c} {}_k\\ {}^{ij} \end{array}\}={_1\over ^2}g^{kn}(\partial _i g_{jn}+\partial _j g_{in}-\partial _n g_{ij})=\, \{\begin{array}{c} {}_k\\ {}^{ji} \end{array}\}. \end{aligned}$$

(14.2)

For the Levi-Civita connection, the covariant derivatives of the tensors $g_{ij}$ and $g^{ij}$ vanish so that $\nabla _i$ commutes with the raising and lowering of indices. The Riemann curvature tensor is defined by the formula

$$\begin{aligned} (\nabla _k\nabla _l-\nabla _l\nabla _k)t^i\,=\,R^i_{\,jkl}t^j \end{aligned}$$

(14.3)

and the Ricci tensor by

$$\begin{aligned} ({\mathrm {Ric}})_{jl}\,=\,R^i_{\,jil}\,=\,({\mathrm {Ric}})_{lj}. \end{aligned}$$

(14.4)

The scalar curvature is $S\,=\,g^{jl}({\mathrm {Ric}})_{jl}$. The totally antisymmetric tensors may be identified with the differential forms by the correspondence

$$\begin{aligned} (\eta _{k_1\ldots k_r})\quad \leftrightarrow \quad \eta ={_1\over ^{r!}}\,\eta _{k_1\ldots k_r}{\mathrm {d}}x^{k_1} \wedge \cdots \wedge {\mathrm {d}}x^{k_r}. \end{aligned}$$

(14.5)

The exterior derivative in terms of this identification is given by the formula

$$\begin{aligned} ({\mathrm {d}}\eta )_{k_0k_1\ldots k_r}&= \partial _{k_0}\eta _{k_1\ldots k_r} -\partial _{k_1}\eta _{k_0k_2\ldots k_r}\ \cdots \ (-1)^r\partial _{k_r} \eta _{k_0\ldots k_{r-1}}\nonumber \\&= \nabla _{k_0}\eta _{k_1\ldots k_r}-\nabla _{k_1}\eta _{k_0k_2\ldots k_r}\ \cdots \ (-1)^r\nabla _{k_r}\eta _{k_0\ldots k_{r-1}}. \end{aligned}$$

(14.6)

The Hodge star on $r$-forms is defined by

$$\begin{aligned} (*\eta )_{i_1\ldots i_{d-r}}\,=\,{_1\over ^{r!}}\, \eta ^{j_1\ldots j_r}\,\epsilon _{j_1\ldots j_ri_1\ldots i_{d-r}}\,\sqrt{g}. \end{aligned}$$

(14.7)

It satisfies $\,*^2=(-1)^{(d-r)r}$.  The $L^2$ scalar product of $r$-forms is given by

$$\begin{aligned} (\eta ,\xi )\ =\,\int \eta \wedge *\xi \ =\ {_1\over ^{r!}}\int \eta ^{i_1\ldots i_r}\,\xi _{i_1\ldots i_r}\,\chi \ =\ (\xi ,\eta )\ =\ (*\eta ,*\xi ). \end{aligned}$$

(14.8)

The (formal) adjoint of the exterior derivative $d$ mapping $r$-forms to $(r+1)$-forms is the operator

$$\begin{aligned} {\mathrm {d}}^\dagger \,=\,(-1)^{r+1}*^{-1}{\mathrm {d}}\,* \end{aligned}$$

(14.9)

mapping $(r+1)$-forms to $r$-forms.  In terms of the tensor components,

$$\begin{aligned} ({\mathrm {d}}^\dagger \eta )_{i_1\ldots i_r}\,=\,-\nabla _{i}\eta ^i_{\ i_1\ldots i_r}. \end{aligned}$$

(14.10)

The (negative) Laplace–Beltrami operator is

$$\begin{aligned} \Delta \,=\,-({\mathrm {d}}^\dagger {\mathrm {d}}+{\mathrm {d}}{\mathrm {d}}^\dagger ). \end{aligned}$$

(14.11)

In the action on functions, it gives

$$\begin{aligned} \Delta \psi \ =\ \frac{_1}{^{\sqrt{g}}}\,\partial _i\,g^{ij}\sqrt{g}\,\partial _j\psi . \end{aligned}$$

(14.12)

The Laplace–Beltrami operator act also on vector fields $\,v\,$ by the formula

$$\begin{aligned} (\Delta v)^{\flat }\,=\,\Delta v^{\flat }, \end{aligned}$$

(14.13)

where $\,v^{\flat }\,$ is the 1-form represented by $(v_i)$ associated to the vector field represented by $\,(v^i)$.  We also need the Lie derivative of tensors w.r.t. vector fields $\,u\,$ given by the formulae

$$\begin{aligned} (\mathcal{L}_uv)^i&= [u,v]^i\,=\,u^j\partial _jv^i-v^j\partial _ju^i\,=\, u^j\nabla _jv^i-v^j\nabla _ju^i,\end{aligned}$$

(14.14)

$$\begin{aligned} (\mathcal{L}_u\eta )_i&= (\partial _iu^j)\eta _j+u^j(\partial _j\eta _i) \,=\,(\nabla _iu^j)\eta _j+u^j(\nabla _j\eta _i), \end{aligned}$$

(14.15)

in the action on the vector fields and 1-forms, respectively, and by the Leibniz rule on higher tensors. In particular, the vector fields $\,u\,$ are called Killing vectors if $\,\mathcal{L}_u\,$ annihilate the metric tensor $\,g$:

$$\begin{aligned} (\mathcal{L}_ug)_{ij}=(\nabla _iu^k)g_{kj}+(\nabla _ju^k)g_{ik}\,=\,\nabla _iu_j+ \nabla _ju_i\,=\,0, \end{aligned}$$

(14.16)

We also consider divergenceless vector fields $\,v\,$ defined by the condition $\,\mathcal{L}_v\chi =0$. In coordinates, the condition is equivalent to the conditions $\,\partial _k(v^k\sqrt{g})=0\,$ or $\,\nabla _kv^k=\sqrt{g}^{-1} \partial _k(v^k\sqrt{g})=0$.  Yet another form of this condition is the requirement that $\,{\mathrm {d}}^\dagger v^\flat =0$.  Locally, every 1-form satisfying such condition can be written via $(d-2)$-form $\,\psi \,$:

$$\begin{aligned} v^\flat \,=\,*{\mathrm {d}}\psi \end{aligned}$$

(14.17)

Appendix 2: Variation Principle for Euler Equation

It will be convenient to perform the extremization over all diffeomorphisms imposing the volume-preservation constraint by adding to the action a term

$$\begin{aligned} S_{Lm}(\Phi ,\lambda )\ =\ \int \lambda \,(\Phi ^*_t\chi -\chi )\,{\mathrm {d}}t \end{aligned}$$

(15.1)

with a Lagrange multiplier $\,\lambda (t,x)$.  For the variation of $\,S_g\,$ at volume-preserving $\,\Phi$,  we obtain³:

$$\begin{aligned} \delta S_g(\Phi )&= {_1\over ^2}\delta \int g_{ij}(y) v^i(t,y)v^j(t,y)\chi (y){\mathrm {d}}t \nonumber \\&= {_1\over ^2}\delta \int g_{ij}(\Phi (t,x))\partial _t\Phi ^i(t,x)\partial _t\Phi ^j(t,x)\chi (x){\mathrm {d}}t \nonumber \\&= {_1\over ^2}\int \partial _k g_{ij}(\Phi (t,x))\delta \Phi ^k(t,x)\partial _t\Phi ^i(t,x)\partial _t\Phi ^j(t,x)\chi (x){\mathrm {d}}t \nonumber \\&+\int g_{ij}(\Phi (t,x))\partial _t\Phi ^i(t,x)\partial _t\delta \Phi ^j(t,x)\chi (x){\mathrm {d}}t \nonumber \\&= \int \delta \Phi ^j(t,x)\bigg [{_1\over ^2}\partial _j g_{ik}(y)v^i(t,y)v^k(t,y)-\partial _k g_{ij}(y)v^i(t,y)v^k(t,y) \nonumber \\&-g_{ij}(y)\left( \partial _t v^i(t,y)+v^k(t,y)\partial _k v^i(t,y)\right) \bigg ]\chi (y){\mathrm {d}}t \nonumber \\&= -\int g_{ij}u^j\bigg [\partial _t v^i+v^k\partial _k v^i+{_1\over ^2}g^{in}(\partial _k g_{ln}+\partial _l g_{kn}-\partial _n g_{lk})v^k v^l\bigg ]\chi {\mathrm {d}}t \nonumber \\&= -\int g_{ij}\,u^j\,(\partial _tv^i+v^k\nabla _kv^i)\,\chi \,{\mathrm {d}}t, \end{aligned}$$

(15.2)

where $\,u^j(t,\Phi (t,x))=\delta \Phi ^j(t,x)\,$ and $\,\nabla _k\,$ is the covariant derivative with respect to the Levi-Civita connection.  On the other hand, the variation of $\,S_{Lm}\,$ is given by

$$\begin{aligned} \delta S_{Lm}(\Phi ,\lambda )&= \ \delta \int \lambda (t,x)\left( \sqrt{g(\Phi (t,x))} \,\frac{{\partial (\Phi (t,x))}}{{\partial (x)}}\,-\,\sqrt{g(x)}\right) {\mathrm {d}}x\,{\mathrm {d}}t \nonumber \\&= \int (\delta \lambda )(t,x)\left( \sqrt{g(\Phi (t,x))}\, \frac{{\partial (\Phi (x))}}{{\partial (x)}}\,-\,\sqrt{g(x)}\right) {\mathrm {d}}x \,{\mathrm {d}}t \nonumber \\&+\int \lambda (t,x)\,\Big (\frac{_1}{^2}g^{ij}(\Phi (t,x)) (\partial _kg_{ji})(\Phi (t,x))\,(\delta \Phi ^k)(t,x)\,\nonumber \\&+\,\big [\big (\frac{_{\partial \Phi (t,x)}}{^{\partial x}}\big )^{-1} \big ]^j_k\,(\partial _j\delta \Phi ^k)(t,x)\Big )\, \sqrt{g(\Phi (t,x))}\, \frac{{\partial (\Phi (t,x))}}{{\partial (x)}}\,{\mathrm {d}}x\,{\mathrm {d}}t \nonumber \\&= \int (\delta \lambda )(t,x)\left( \sqrt{g(\Phi (t,x))}\, \frac{{\partial (\Phi (x))}}{{\partial (x)}}\,-\,\sqrt{g(x)}\right) {\mathrm {d}}x \,{\mathrm {d}}t \nonumber \\&+\int p(t,y)\left( \frac{_1}{^2}g^{ij}(y)(\partial _kg_{ij}(y)\,u^k(t,y)\, +\,(\partial _ku^k)(t,y)\right) \sqrt{g(y)}\,{\mathrm {d}}y\,{\mathrm {d}}t,\nonumber \\ \end{aligned}$$

(15.3)

where we have introduced the pressure by the formula

$$\begin{aligned} p(t,\Phi (t,x))=\lambda (t,x). \end{aligned}$$

(15.4)

Integrating by parts in the last term, we finally obtain

$$\begin{aligned} \delta S_{Lm}(\Phi ,\lambda )&= \int (\delta \lambda )(t,x)\left( \sqrt{g(\Phi (t,x))}\, \frac{{\partial (\Phi (x))}}{{\partial (x)}}\,-\,\sqrt{g(x)}\right) {\mathrm {d}}x \,{\mathrm {d}}t \nonumber \\&-\int \,u^k(t,y)\,(\partial _kp)(t,y)\,\sqrt{g(y)}\,{\mathrm {d}}y\,{\mathrm {d}}t. \end{aligned}$$

(15.5)

Equating to zero the variation $\,\delta S(\Phi ,\lambda )\,$ for $\,S(\Phi ,\lambda )=S_g(\Phi )+S_{Lm}(\Phi ,\lambda )$,  we obtain the (generalized) Euler equation (2.2) together with the volume-preserving condition on $\,\Phi (t)\,$ which, in terms of the velocity is the incompressibility condition (2.3).  On noncompact manifolds, these should be accompanied by the decay conditions at infinity that assure that the integrations by parts above may be performed and eliminate the solutions $\,v^i\,=\,g^{ij}\partial _jf\,$ with $\,\Delta f=g^{ij}\nabla _i\partial _jf=0\,$ and arbitrary $t$-dependence (solving also the unforced Navier–Stokes equations on Einstein manifolds, see [10, 19]).

Appendix 3: Velocity Covariance in Terms of Stream Functions

${{\mathbb H}}_{_R}\,$ is a contractible space. On the other hand, due to incompressibility of velocity, the equal-time 2-point function $\,\langle (*v^\flat )(\mathcal{X}_1) \otimes (*v^\flat (\mathcal{X}_2)\rangle$ is a closed 1-form in its dependence on each of the two points. We may then obtain a stream-function correlator $\,\langle \psi (\mathcal{X}_1)\,\psi (\mathcal{X}_2)\rangle \,$ satisfying Eq. (6.26) by integrating $\,\langle (*v^\flat )(\mathcal{X}_1) \otimes (*v^\flat (\mathcal{X}_2)\rangle \,$ in each variable from a fixed point of $\,{{\mathbb H}}_{_R}\,$ to $\,\mathcal{X}_1\,$ and $\,\mathcal{X}_2$,  respectively. Function $\,\langle \psi (\mathcal{X}_1)\,\psi (\mathcal{X}_2)\rangle \,$ obtained this way is symmetric and of positive type, but it depends on the choice of the initial point of integration. As a result, the $\,SL(2,{{\mathbb R}})$-covariance of the velocity 2-point function does not imply the $\,SL(2,{{\mathbb R}})\,$ invariance of $\,\langle \psi (\mathcal{X}_1)\, \psi (\mathcal{X}_2)\rangle$.  However, for $\,\gamma \in SL(2,{{\mathbb R}})$,  the function

$$\begin{aligned} \langle \psi (\gamma \mathcal{X}_1)\psi (\gamma \mathcal{X}_2)\,\rangle \,-\, \langle \psi (\mathcal{X}_1)\,\psi (\mathcal{X}_2)\rangle \ \equiv \ f_\gamma (\mathcal{X}_1,\mathcal{X}_2) \end{aligned}$$

(16.1)

is annihilated by the product of exterior derivatives $\,d(1)d(2)\,$ in $\,\mathcal{X}_1\,$ and $\,\mathcal{X}_2$.  It has then to be of the form $\,f^1_\gamma (\mathcal{X}_1)+f^2_\gamma (\mathcal{X}_2)\,$ where, by symmetry, we may take $\,f^1_\gamma (\mathcal{X})=f^2_\gamma (\mathcal{X})=\frac{1}{2}\,f_\gamma (\mathcal{X},\mathcal{X})$.  Definition (16.1) implies the cocycle condition

$$\begin{aligned} f^1_{\gamma _1\gamma _2}(\mathcal{X})\,=\,f^1_{\gamma _1}(\gamma _2\mathcal{X}) +f^1_{\gamma _2}(\mathcal{X}). \end{aligned}$$

(16.2)

Taking, in particular, $\,\mathcal{X}=\mathcal{X}_0\,$ corresponding to $\,w=i\,$ and $\,\gamma _2\in SO(2,{{\mathbb R}})$,  which is the stabilizer subgroup of $\,\mathcal{X}_0$,  we obtain

$$\begin{aligned} f^1_{\gamma _1\gamma _2}(\mathcal{X}_0)\,=\,f^1_{\gamma _1}(\mathcal{X}_0) +f^1_{\gamma _2}(\mathcal{X}_0). \end{aligned}$$

(16.3)

Taking also $\,\gamma _1\in SO(2,{{\mathbb R}})$,  we infer that $\,f^1_\gamma (\mathcal{X}_0)\,$ is an additive function of $\,\gamma \in SO(2,{{\mathbb R}})$,  so it must be equal to zero when restricted to such $\gamma$.  Consequently, for general $\,\gamma _1$,  one has $\,f^1_{\gamma _1\gamma _2}(\mathcal{X}_0)\,=\,f^1_{\gamma _1}(\mathcal{X}_0)$,  so that $\,f^1_\gamma (\mathcal{X}_0)\,$ defines a function $\,g(\mathcal{X})\,$ on $\,{{\mathbb H}}_{_R}\cong SL(2,{{\mathbb R}})/SO(2,{{\mathbb R}})\,$ (vanishing at $\,\mathcal{X}=\mathcal{X}_0$)  and it determines $\,f^1_\gamma (\mathcal{X})\,$ by the formula

$$\begin{aligned} f^1_{\gamma _1}(\gamma _2\mathcal{X}_0)\,=\,g(\gamma _1\gamma _2\mathcal{X}_0)- g(\gamma _2\mathcal{X}_0) \end{aligned}$$

(16.4)

following from the cocycle condition (16.2). This, in fact, provides a general solution of this condition if we consider arbitrary functions $\,g(\mathcal{X})\,$ on $\,{{\mathbb H}}_{_R}$.  Note that the symmetric function $\,\Psi (\mathcal{X}_1,\mathcal{X}_2)= \big \langle \,\psi (\mathcal{X}_1)\,\psi (\mathcal{X}_2)\big \rangle -g(\mathcal{X}_1)-g(\mathcal{X}_2)\,$ satisfies the relation $\,\Psi (\gamma \mathcal{X}_1,\gamma \mathcal{X}_2)=\Psi (\mathcal{X}_1,\mathcal{X}_2)$,  hence it depends only on the hyperbolic distance $\,\delta _{12}$.  It defines the same velocity correlators if used in formula (6.26) instead of $\,\big \langle \psi (\mathcal{X}_1)\, \psi (\mathcal{X}_2)\big \rangle$. If $\,\Psi '\,$ is another such functions, then

$$\begin{aligned} \Psi '(\mathcal{X}_1,\mathcal{X}_2)\,=\,\Psi (\mathcal{X}_1,\mathcal{X}_2)\,+\,g^1(\mathcal{X}_1)+g^1(\mathcal{X}_2). \end{aligned}$$

(16.5)

for some function $\,g^1(\mathcal{X})$.  The $SL(2,{{\mathbb R}})$-covariance of $\,\Psi \,$ and $\,\Psi '\,$ implies then that

$$\begin{aligned} g^1(\gamma \mathcal{X}_1)-g^1(\mathcal{X}_1)\,=\,-\,g^1(\gamma \mathcal{X}_2)+g^1(\mathcal{X}_2)\,=\,c_\gamma , \end{aligned}$$

(16.6)

but for each $\,\gamma \,$ there is $\,\mathcal{X}_1\,$ such that $\,\gamma \mathcal{X}_1=\mathcal{X}_1\,$ and we infer that $\,c_\gamma =0\,$ so that function $\,g^1\,$ is constant.

Appendix 4: Forcing Covariance on $\,{{\mathbb H}}_{_{R/\lambda }}$

Equation (7.2) implies the formulae:

$$\begin{aligned} \lambda ^{-2}C^{rr}(\mathcal{X}_1',\mathcal{X}_2')&= \frac{_{\sqrt{1+(R/\lambda )^2r_1^{-2}} \,\sqrt{1+(R/\lambda )^2r_2^{-2}}}}{^{R^2}}\,\,\partial _{\varphi _1}\partial _{\varphi _2}\mathcal{C}(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }})\nonumber \\&= -\,\frac{_{\sqrt{1+(R/\lambda )^2r_1^{-2}}\,\sqrt{1+(R/\lambda )^2 r_2^{-2}}}}{^{R^2}}\,\Big (\frac{_{r_1r_2 \cos (\varphi _1-\varphi _2)}}{^{(R/\lambda )^2}} \,\mathcal{C}'(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }}) \nonumber \\&+\,\frac{_{r_1^2r_2^2\sin ^2(\varphi _1-\varphi _2)}}{^{(R/\lambda )^4}}\, \mathcal{C}''(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }})\Big ). \end{aligned}$$

(17.1)

$$\begin{aligned} \lambda ^{-2}C^{r\varphi }(\mathcal{X}_1',\mathcal{X}_2')&= -\,\frac{_{\sqrt{1+(R/\lambda )^2r_1^{-2}}\,\sqrt{1+(R/\lambda )^2r_2^{-2}} }}{^{R^2}}\,\partial _{\varphi _1}\partial _{r_2}\mathcal{C}(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }}) \nonumber \\&= -\,\frac{_{\lambda ^2r_1\sin (\varphi _1-\varphi _2)}}{^{R^4}}\, \mathcal{C}'(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }}) \nonumber \\&-\,\frac{_{\lambda ^4r_1^2r_2\sin (\varphi _1-\varphi _2)\Big ( \sqrt{1+(R/\lambda )^2r_1^{-2}}-\sqrt{1+(R/\lambda )^2r_2^{-2}} \,\cos (\varphi _1-\varphi _2)\Big )}}{^{R^4\, \sqrt{1+(R/\lambda )^2r_2^{-2}}}}\, \mathcal{C}''(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }}).\nonumber \\ \end{aligned}$$

(17.2)

$$\begin{aligned}&\lambda ^{-2}C^{\varphi \varphi }(\mathcal{X}_1',\mathcal{X}_2')\ = \frac{_{ \sqrt{1+(R/\lambda )^2r_1^{-2}}\,\sqrt{1+(R/\lambda )^2r_2^{-2}}}}{^{R^2}}\, \partial _{r_1}\partial _{r_2}\mathcal{C}(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }})\nonumber \\&\quad =\frac{_{\lambda ^2\Big (1-\sqrt{1+(R/\lambda )^2r_1^{-2}}\, \sqrt{1+(R/\lambda )^2r_2^{-2}}\, \cos (\varphi _1-\varphi _2)\Big )}}{^{R^4}}\, \mathcal{C}'(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }}) \nonumber \\&\qquad +\, \frac{_{\lambda ^4r_1r_2\Big (\sqrt{1+(R/\lambda )^2r_2^{-2}}- \sqrt{1+(R/\lambda )^2r_1^{-2}}\,\cos (\varphi _1-\varphi _2)\Big ) \Big (\sqrt{1+(R/\lambda )^2r_1^{-2}} -\sqrt{1+(R/\lambda )^2r_2^{-2}}\,\cos (\varphi _1-\varphi _2)\Big )}}{^{R^6}} \nonumber \\&\qquad \cdot \,\mathcal{C}''(\cosh \frac{_{\delta _{1'2'}}}{^{R/\lambda }}). \end{aligned}$$

(17.3)

The asymptotic behavior (12.5), (12.6) and (12.7) follow from these expressions in a straightforward way.

Appendix 5: Rescaled Velocity 2-Point Functions on $\,{{\mathbb H}}_{_{R}}$

Assuming the form (12.27) of the 2-point correlator of the stream function on ${{\mathbb H}}_{_{R}}$,  the rescaled version of the velocity 2-point functions given by Eq. (6.26) may be written in the form:

$$\begin{aligned}&\lambda ^{\frac{4}{3}-2}F^{rr}_{{{\mathbb H}}_{_R},\mathcal{C}}(\lambda ^{\frac{2}{3}}t; \lambda r_1,\varphi _1; \lambda r_2,\varphi _2)\ =\ \lambda ^{-\frac{2}{3}} \frac{_{\sqrt{1+(R/\lambda )^2r_1^{-2}}\,\sqrt{1+(R/\lambda )^2r_2^{-2}}}}{^{R^2}}\Bigg [x\,\Psi '(\lambda ^{\frac{2}{3}}t,x) \nonumber \\&\quad +\ x^2\Psi ''(\lambda ^{\frac{2}{3}}t,x)\, -\,\frac{_{r_1r_2\,\sqrt{1+(R/\lambda )^2r_1^{-2}}\, \sqrt{1+(R/\lambda )^2r_2^{-2}}}}{^{(R/\lambda )^2}} \Big (\Psi '(\lambda ^{\frac{2}{3}}t,x)\, +\,2x\,\Psi ''(\lambda ^{\frac{2}{3}}t,x)\Big ) \nonumber \\&\quad +\,\left( 1+\frac{_{r_1^2+r_2^2}}{^{(R/\lambda )^2}} \right) \Psi ''(\lambda ^{\frac{2}{3}}t,x)\Bigg ],\end{aligned}$$

(18.1)

$$\begin{aligned}&\lambda ^{\frac{4}{3}-1}F^{r\varphi }_{{{\mathbb H}}_{_R},\mathcal{C}} (\lambda ^{\frac{2}{3}}t;\lambda r_1, \varphi _1;\lambda r_2,\varphi _2)\ =\ -\lambda ^{-\frac{2}{3}}\,\frac{_{\sqrt{1+(R/\lambda )^2r_1^{-2}} \,\sqrt{1+(R/\lambda )^2r_2^{-2}}\,r_1\sin (\varphi _1-\varphi _2)}}{^{R^2(R/\lambda )^2}} \nonumber \\&\quad \times \,\Bigg [\Psi '(\lambda ^{\frac{2}{3}}t,x)\, +\,x\,\Psi ''(\lambda ^{\frac{2}{3}}t,x)\, -\,\frac{_{r_1\,\sqrt{1+(R/\lambda )^2r_1^{-2}}}}{^{r_2\, \sqrt{1+(R/\lambda )^2r_2^{-2}}}}\, \Psi ''(\lambda ^{\frac{2}{3}}t,x)\Bigg ],\end{aligned}$$

(18.2)

$$\begin{aligned}&\lambda ^{\frac{4}{3}}F^{\varphi \varphi }_{{{\mathbb H}}_{_R}, \mathcal{C}}(\lambda ^{\frac{2}{3}}t;\lambda r_1,\varphi _1;\lambda r_2,\varphi _2)\ =\ \lambda ^{-\frac{2}{3}}\,\frac{_{\sqrt{1+(R/\lambda )^2r_1^{-2}}\, \sqrt{1+(R/\lambda )^2r_2^{-2}}}}{^{r_1r_2\,R^2}} \nonumber \\&\quad \times \,\Bigg [ \Big (x\,+\,\frac{_{r_1r_2\,(1-(1+(R/\lambda )^2r_1^{-2})(1+(R/\lambda )^2 r_2^{-2}))}}{^{(R/\lambda )^2\,\sqrt{1+(R/\lambda )^2r_1^{-2}}\, \sqrt{1+(R/\lambda )^2r_2^{-2}}}}\Big )\,\Psi '(\lambda ^{\frac{2}{3}}t,x) \nonumber \\&\quad +\,\Big (x-\frac{_{r_1\,\sqrt{1+(R/\lambda )^2r_1^{-2}}}}{^{r_2\,\sqrt{1+(R/\lambda )^2r_2^{-2}}}}\Big ) \Big (x-\frac{_{r_2\,\sqrt{1+(R/\lambda )^2r_2^{-2}}}}{^{r_1\, \sqrt{1+(R/\lambda )^2r_1^{-2}}}}\Big )\, \Psi ''(\lambda ^{\frac{2}{3}}t,x)\Bigg ], \end{aligned}$$

(18.3)

where $\,x\,$ is given by the right hand side of Eq. (12.28) with $\,R\,$ replaced by $\,R/\lambda$.  The above expressions should be compared to Eqs. (12.16), (12.17) and (12.18) for the velocity correlators on $\,{{\mathbb H}}_{_0}$.

Appendix 6: Harmonic Analysis on $\,{{\mathbb H}}_{_R}\,$ in the $\,R\rightarrow 0\,$ limit

To study the harmonic analysis on $\,{{\mathbb H}}_{_R}\,$ in the limit $\,R\rightarrow 0$,  we rewrite it using the coordinates $\,(r,\varphi )$.  The harmonic analysis in space $\,L^2({{\mathbb H}}_{_R})\,$ is realized by the formulae that combine the Gel’fand-Graev transformation $\,\psi \mapsto \psi ^{GG}\,$ from functions on the hyperboloid $\,{{\mathbb H}}_{_R}\,$ to functions on the cone $\,{{\mathbb H}}_{_0}\,$

$$\begin{aligned} \psi ^{GG}(Y)\ =\ \int \limits _{{{\mathbb H}}_{_R}}\,\delta (Y^1X^1+Y^2X^2-Y^3X^3+R)\, \,\psi (X)\,\,\chi (X) \end{aligned}$$

(19.1)

with the Fourier transform in the angular direction and Mellin transform in the radial one [23].  In the coordinates $\,(r,\varphi )\,$ it is given by the relations:

$$\begin{aligned} \psi (r,\varphi )&= \frac{_1}{^{2\pi R}}\sum \limits _{m=-\infty }^\infty \int \limits _0^\infty \Big (\int \limits _{-\pi }^\pi {\mathrm {d}}\vartheta \int \limits _0^\infty s^{-\frac{1}{2}-i\sigma }\,\mathrm{e}^{-im\vartheta } \,\delta \big (sr\cos (\vartheta -\varphi )-s\sqrt{R^2+r^2}+R\big )\,\,ds\Big ) \nonumber \\&\times \ a_m(\sigma )\,\,\sigma \,\tanh (\pi \sigma )\,{\mathrm {d}} \sigma \nonumber \\&= \frac{_1}{^{2\pi R^2}}\!\sum \limits _{m=-\infty }^\infty \!\!\mathrm{e}^{-im\varphi } \int \limits _0^\infty \! \Big (\int \limits _{-\pi }^\pi \mathrm{e}^{-im\vartheta }\,\big [\frac{_{\sqrt{R^2+r^2}}}{^R}-\frac{_r}{^R}\cos \vartheta \big ]^{i\sigma -\frac{1}{2}}\,{\mathrm {d}}\vartheta \Big ) a_m(\sigma )\,\sigma \,\tanh (\pi \sigma )\,{\mathrm {d}}\sigma \nonumber \\&= \frac{_1}{^{R^2}}\sum \limits _{m=-\infty }^\infty \mathrm{e}^{-im\varphi } \int \limits _0^\infty \mathcal{P}^{-\frac{1}{2}+i\sigma }_{m0}(\frac{_{\sqrt{R^2+r^2}}}{^R})\,\,a_m(\sigma )\,\sigma \,\tanh (\pi \sigma )\,{\mathrm {d}}\sigma , \end{aligned}$$

(19.2)

$$\begin{aligned} a_m(\sigma )&= \frac{_R}{^{4\pi ^2}}\int \limits _{-\pi }^\pi \mathrm{e}^{im\vartheta }\,{\mathrm {d}}\vartheta \int \limits _0^\infty s^{-\frac{1}{2}+i\sigma }ds \nonumber \\&\quad \times \, \int \delta \big (sr\cos (\vartheta -\varphi )- s\sqrt{R^2+r^2}+R\big )\,\psi (r,\varphi ) \,\frac{_{Rr}}{^{\sqrt{R^2+r^2}}}\,{\mathrm {d}}r\wedge {\mathrm {d}}\varphi \nonumber \\&= \frac{_1}{^{4\pi ^2}} \int \mathrm{e}^{im\varphi }\Big (\int \limits _{-\pi }^\pi \mathrm{e}^{im\theta } \big [\frac{_{\sqrt{R^2+r^2}}}{^R}-\frac{_r}{^R} \cos \vartheta \big ]^{-\frac{1}{2}-i\sigma }\,d\theta \Big )\, \psi (r,\varphi )\,\frac{_{Rr}}{^{\sqrt{R^2+r^2}}}\,{\mathrm {d}}r\wedge {\mathrm {d}}\varphi \nonumber \\&= \frac{_1}{^{2\pi }}\int \mathrm{e}^{im\varphi }\,\,\mathcal{P}^{-\frac{1}{2}-i\sigma }_{m0} (\frac{_{\sqrt{R^2+r^2}}}{^R})\,\, \psi (r,\varphi )\,\frac{_{Rr}}{^{\sqrt{R^2+r^2}}}\,{\mathrm {d}}r\wedge {\mathrm {d}}\varphi ,\end{aligned}$$

(19.3)

$$\begin{aligned}&\int |\psi (r,\varphi )|^2\,\,\frac{_{Rr}}{^{\sqrt{R^2+r^2}}} \,{\mathrm {d}}r\wedge {\mathrm {d}}\varphi \ =\ \frac{_{2\pi }}{^{R^2}}\sum \limits _{m=-\infty }^\infty \int \limits _0^\infty |a_m(\sigma )|^2\,\,\sigma \,\tanh \pi \sigma \,{\mathrm {d}}\sigma , \end{aligned}$$

(19.4)

which reproduce Eqs. (6.9), (6.15) and (6.16) upon setting $\,\sigma =Rk$.

To study the $\,R\rightarrow 0\,$ limit, we use the asymptotic expansion from [16] holding for large positive $\,x$:

$$\begin{aligned} \!\!&\mathcal{P}^{-\frac{1}{2}+i\sigma }_{m0}(x)=\frac{_{\Gamma (\frac{1}{2}+i\sigma )x^{-\frac{1}{2}}}}{^{\Gamma (\frac{1}{2}+i\sigma +m)}} \Big (\frac{_{2^{-\frac{1}{2}+i\sigma } \Gamma (i\sigma )}}{^{\pi ^{\frac{1}{2}} \Gamma (\frac{1}{2}+i\sigma -m)}} x^{i\sigma } + \frac{_{2^{-\frac{1}{2}-i\sigma } \Gamma (-i\sigma )}}{^{\pi ^{\frac{1}{2}} \Gamma (\frac{1}{2}-i\sigma -m)}} x^{-i\sigma }\Big ) \left( 1+\mathcal{O}(x^{-2})\right) .\nonumber \\ \end{aligned}$$

(19.5)

Substituting this to Eq. (19.2) and using the relation $\,\sigma \tanh (\pi \sigma ) = {|\Gamma (\frac{1}{2}+i\sigma )|^2} /{|\Gamma }$ ${(i\sigma )|^2}$,  we obtain

$$\begin{aligned} \psi (r,\varphi )&= \frac{_1}{^{R^2}}\sum \limits _{m=-\infty }^\infty \mathrm{e}^{-im\varphi } \int \limits _0^\infty \Big ( \frac{_{(-1)^m\,2^{-\frac{1}{2}+i\sigma }\, \Gamma (\frac{1}{2}-i\sigma )\,(\frac{1}{2}-i\sigma )\cdots (\frac{1}{2}-i\sigma +m-1)}}{^{\pi ^{\frac{1}{2}}\,\Gamma (-i\sigma )\,(\frac{1}{2}+i\sigma ) \cdots (\frac{1}{2}+i\sigma +m-1)}}\, (\frac{_{\sqrt{R^2+r^2}}}{^R})^{-\frac{1}{2}+i\sigma } \nonumber \\&+\,\frac{_{(-1)^m\,2^{-\frac{1}{2}-i\sigma }\, \Gamma (\frac{1}{2}+i\sigma )}}{^{\pi ^{\frac{1}{2}}\, \Gamma (i\sigma )}} \,(\frac{_{\sqrt{R^2+r^2}}}{^R})^{-\frac{1}{2}-i\sigma }\Big ) \,a_m(\sigma )\,\,{\mathrm {d}}\sigma \ \left( 1+\mathcal{O}(R^{2})\right) ,\nonumber \\&= \sum \limits _{m=-\infty }^\infty \mathrm{e}^{-im\varphi }\int \limits _{-\infty }^\infty r^{^{-\frac{1}{2}+i\sigma }}\, \,\tilde{a}_m(\sigma )\,\,{\mathrm {d}}\sigma \ \left( 1+\mathcal{O}(R^{2})\right) , \end{aligned}$$

(19.6)

where we have set

$$\begin{aligned} \tilde{a}_m(\sigma )\ =\ \frac{_{(-1)^m\,2^{-\frac{1}{2}+i\sigma }\, \Gamma (\frac{1}{2}-i\sigma )}}{^{\pi ^{\frac{1}{2}}\,\Gamma (-i\sigma )}}\,R^{-\frac{3}{2}-i\sigma }\, \cdot {\left\{ \begin{array}{ll} \frac{_{(\frac{1}{2}-i\sigma )\,\cdots \, (\frac{1}{2}-i\sigma +m-1)}}{^{(\frac{1}{2}+i\sigma )\,\cdots \,(\frac{1}{2}+i\sigma +m-1)}} \,a_m(\sigma )\quad \ {\mathrm {if}} \quad \sigma >0,\\ a_m(-\sigma )\qquad \qquad \qquad \qquad {\mathrm {if}} \quad \sigma <0. \end{array}\right. }\qquad \end{aligned}$$

(19.7)

Hence, the limiting decomposition at $\,R=0\,$ of functions on the cone $\,{{\mathbb H}}_{_0}\,$ is given by the Mellin transform in $\,r\,$ combined with the Fourier transform in the angular variable $\,\varphi$:

$$\begin{aligned}&\psi (r,\varphi )\ =\ \sum \limits _{m=-\infty }^\infty \mathrm{e}^{-im\varphi } \int \limits _{-\infty }^\infty r^{-\frac{1}{2}+i\sigma }\,\tilde{a}_m(\sigma )\, {\mathrm {d}}\sigma \,\end{aligned}$$

(19.8)

$$\begin{aligned}&\tilde{a}_m(\sigma )\ =\ \frac{_1}{^{4\pi ^2}}\int \mathrm{e}^{im\varphi }\,r^{-\frac{1}{2}-i\sigma }\,\psi (r,\varphi )\,{\mathrm {d}}r \wedge {\mathrm {d}}\varphi , \end{aligned}$$

(19.9)

$$\begin{aligned}&\int |\psi (r,\varphi )|^2\,{\mathrm {d}}r\wedge {\mathrm {d}}\varphi \ =\ 4\pi ^2\int |\tilde{a}(\sigma )|^2\,{\mathrm {d}}\sigma . \end{aligned}$$

(19.10)

Action (11.21) of $\,\mathrm{Diff}S^1\,$ on $\,{{\mathbb H}}_{_0}\,$ induces the unitary action of that infinite-dimensional group in the space $\,L^2({{\mathbb H}}_{_0})\,$ of functions on $\,{{\mathbb H}}_{_0}\,$ square-integrable with respect to the volume measure $\,\chi _0$,  defined by the formula

$$\begin{aligned} \psi \,\mapsto \,D\psi ,\qquad (D\psi )(r,\varphi )\,=\,\psi (\frac{_r}{^{(D^{-1})'(\varphi )}},D^{-1}(\varphi )), \end{aligned}$$

(19.11)

Note that this action commutes with the Laplacian $\,\Delta =\partial _r r^2\partial _r$.  The functions $\,\mathrm{e}^{im\varphi }\,r^{-\frac{1}{2}+i\sigma }\,$ are eigenvectors of $\,\Delta \,$ corresponding to the eigenvalue $-(\frac{1}{4}+\sigma ^2)$.  The decomposition (19.8) realizes the decomposition of $\,L^2({{\mathbb H}}_{_0})\,$ into the spectral eigenspaces of $\,\Delta \,$ and, at the same time, the decomposition of the unitary representation of $\,\mathrm{Diff}S^1\,$ in $\,L^2({{\mathbb H}}_{_0})\,$ into the irreducible components:

(19.12)

Unlike for $\,R>0$,  where only $\,\sigma >0\,$ appeared in the decomposition (6.8),  now $\,\sigma >0\,$ and $\,\sigma <0\,$ give rise to inequivalent representations.

Appendix 7: Real Analyticity of the Spectrum

Spectrum $\,\mathcal{E}(k)\,$ is related to the stream function correlator $\,\Psi \,$ by the Fourier transform on $\,{{\mathbb H}}_{_R}\,$ (6.25) and Eq. (6.31) which imply the inverse transform relation

$$\begin{aligned} \frac{{8\mathcal{E}_{stat}(k)}}{{(1+(2Rk)^2)\,k\,\tanh (\pi Rk)}}\ =\, \int _1^\infty \overline{\mathcal{P}^{-\frac{1}{2}+iRk}_{00}(x)} \,\Psi (x)\,{\mathrm {d}}x, \end{aligned}$$

(20.1)

where the integral has to be interpreted in the appropriate sense  (by analytic continuation)  since, for large $\,x$, $\,|\mathcal{P}^{-\frac{1}{2}+i\sigma }_{00}(x)|=\mathcal{O}(x^{-\frac{1}{2}})$,  see Eqs. (19.5). Let us split the integration on the right hand side of (20.1) into the one from 1 to 2 and the rest.  The first integral gives an entire function of $\,k$,  as is easily seen from Eq. (6.11), and it contributes to $\,\mathcal{E}(k)\,$ a term $\,O(k^2)\,$ or less for small $\,k$.  For the other integral, we shall use the expression

$$\begin{aligned} \mathcal{P}^{-\frac{1}{2}+i\sigma }_{00}(x)\ =\ \frac{\Gamma (-i\sigma )}{\sqrt{\pi }\, \Gamma (\frac{1}{2}-i\sigma )}\,\zeta ^{-\frac{1}{2}-i\sigma }\,{}_2F_1 \left( \frac{_1}{^2},\frac{_1}{^2}+i\sigma ,1+i\sigma ;\zeta ^{-2}\right) \,+\ c.\,c. \qquad \end{aligned}$$

(20.2)

for $\,\zeta \equiv x+\sqrt{x^2-1}$.  Expanding the hypergeometric functions into a power series in $\,\zeta ^{-2}$,  we see that only the contribution

$$\begin{aligned} \int \limits _2^\infty \zeta ^{-\frac{1}{2}\mp iRk}\,\Psi (x)\,{\mathrm {d}}x \end{aligned}$$

(20.3)

of the leading term into the integral needs a special treatment.  Using the asymptotic expansion for $\,\Psi _{stat}\,$ rewritten in terms of variable $\,\zeta$,  we see that the integral (20.3) contributes few simple poles and a function analytic in $\,k\,$ in a neighborhood of the real axis. The pole at zero appears only if the power $\,x^{-\frac{1}{2}}\,$ occurs in the large $\,x\,$ expansion of $\,\Psi _{stat}$.  The other terms of the expansion of the hypergeometric functions contribute absolutely convergent integrals so that

$$\begin{aligned} \int \limits _1^\infty \zeta ^{-\frac{1}{2}\mp iRk}\, \left( {}_2F_1(\frac{_1}{^2},\frac{_1}{^2}\pm iRk,1\pm iRk;\zeta ^{-2})\,-\,1 \right) \,\Psi _{stat}(x)\,{\mathrm {d}}x\qquad \end{aligned}$$

(20.4)

is again analytic in $\,k\,$ in a neighborhood of the real axis. The prefactors $\,\pi ^{-\frac{1}{2}}\Gamma (\mp iRk)/$ $\Gamma (\frac{1}{2}\mp iRk)\,$ in Eq. (20.2) are meromorphic in $\,k\,$ in such a region with a simple pole at $\,k=0$.  Equation (20.1) implies than that the contribution of $\,\Psi _{stat}\,$ is analytic around $\,k\ge 0$. As for the contribution to $\,\Psi _0$,  a presence of the $\,x^{-\frac{2}{2}}\,$ term in its asymptotic expansion could leave us with a pole at $\,k=0\,$ of up to the second order (due to up to two derivatives over $\,k\,$ used to produced the logarithmic terms of $\,\Psi _0$),  but such pole would be incompatible with the finiteness of the energy density at finite times. Hence the contribution of $\,\Psi _0\,$ to the spectrum should also be analytic around $\,k\ge 0$.