Theory of energy spectra in superfluid He-4 counterflow turbulence

Following LABEL:DNS-He4,LP-2017,BKLPPS-2017, we describe the large scale turbulence in superfluid ⁴He by the gradually-damped versionHe4 of the coarse-grained Hall-Vinen HV -Bekarevich-KhalatnikovBK (HVBK) equations, generalized in LABEL:decoupling for the counterflow turbulence. In these equations the superfluid vorticity is assumed continuous, limiting their applicability to large scales with characteristic scale of turbulent fluctuations $R>\ell$. HVBK equations have a form of two Navier-Stokes equations for the turbulent velocity fluctuations ${\bm{u}}_{\text{n}}({\bm{r}},t)$ and ${\bm{u}}_{\text{s}}({\bm{r}},t)$:

			$\displaystyle\thinspace\frac{\partial\,{\bm{u}}_{\text{s}}}{\partial t}+[({\bm{u}}_{\text{s}}+{\bm{U}}_{\text{s}})\cdot{\bm{\nabla}}]{\bm{u}}_{\text{s}}-\frac{1}{\rho_{\text{s}}}{\bm{\nabla}}p_{\text{s}}=\nu_{\text{s}}\,\Delta{\bm{u}}_{\text{s}}+{\bm{f}}_{\text{ns}}\,,$
			$\displaystyle\thinspace\frac{\partial\,{\bm{u}}_{\text{n}}}{\partial t}+[({\bm{u}}_{\text{n}}+{\bm{U}}_{\text{n}})\cdot{\bm{\nabla}}]{\bm{u}}_{\text{n}}-\frac{1}{\rho_{\text{n}}}{\bm{\nabla}}p_{\text{n}}=\nu_{\text{n}}\,\Delta{\bm{u}}_{\text{n}}-\frac{\rho_{\text{s}}}{\rho_{\text{n}}}{\bm{f}}_{\text{ns}}\,,$
			$\displaystyle\thinspace p_{\text{n}}=\frac{\rho_{\text{n}}}{\rho}[p+\frac{\rho_{\text{s}}}{2}|{\bm{u}}_{\text{s}}-{\bm{u}}_{\text{n}}|^{2}]\,,p_{\text{s}}=\frac{\rho_{\text{s}}}{\rho}[p-\frac{\rho_{\text{n}}}{2}|{\bm{u}}_{\text{s}}-{\bm{u}}_{\text{n}}|^{2}]\,,$
	$\displaystyle{\bm{f}}_{\text{ns}}$	$\displaystyle\simeq$	$\displaystyle\Omega_{\text{s}}\,({\bm{u}}_{\text{n}}-{\bm{u}}_{\text{s}})\,,\quad\Omega_{\text{s}}=\alpha(T)\kappa{\mathcal{L}}\,,$		(4b)

coupled by the mutual friction force ${\bm{f}}_{\text{ns}}$ in the form (4b), suggested in LABEL:LNV. It involves the turbulent velocity fluctuations of the normal- and superfluid components, the temperature dependent dimensionless dissipative mutual friction parameter $\alpha(T)$ and the superfluid vorticity $\kappa{\mathcal{L}}$, defined by the vortex line density ${\mathcal{L}},\,\ell={\mathcal{L}}^{-1/2}$.

Other parameters entering Eqs. (4) include the pressures $p_{\text{n}}$, $p_{\text{s}}$ of the normal and the superfluid components, the total density $\rho\equiv\rho_{\text{s}}+\rho_{\text{n}}$ of ⁴He and the kinematic viscosity of normal fluid component $\nu_{\text{n}}=\eta/\rho_{\text{n}}$ with $\eta$ being the dynamical viscosityDB98 of normal ⁴He component. The dissipative term with the Vinen’s effective superfluid viscosity $\nu_{\text{s}}$ Vinen was added in LABEL:He4 to account for the energy dissipation at the intervortex scale $\ell$ due to vortex reconnections, the energy transfer to Kelvin waves and similar effects.

Generally speaking, Eqs. (4) involve also contributions of a reactive (dimensionless) mutual friction parameter $\alpha^{\prime}$, that renormalizes the nonlinear terms. For example, in Eq. (4) $({\bm{u}}_{\text{s}}\cdot{\bm{\nabla}}){\bm{u}}_{\text{s}}\Rightarrow(1-\alpha^{\prime})({\bm{u}}_{\text{s}}\cdot{\bm{\nabla}}){\bm{u}}_{\text{s}}$. However, in the studied range of temperatures $|\alpha^{\prime}|\lesssim 0.02\ll 1$ and this renormalization can be ignored. For similar reasons we neglected all other $\alpha^{\prime}$-related terms in Eqs. (4).

Our theory is based on the stationary balance equations for the 1D energy spectra $E_{\text{n}}(k)$ and $E_{\text{s}}(k)$ of the normal and superfluid components, defined by Eq. (1).

To derive these equations, Eqs. (4) were Fourier transformed, multiplied by the complex conjugates of the corresponding velocities and properly averaged. The pressure terms were eliminated using the incompressibility conditions. Finally, the energy balance equations have a form:

$$\frac{d\varepsilon_{j}}{dk}=\Omega_{j}\big{[}E_{\text{ns}}(k)-E_{j}(k)\big{]}-2\,\nu_{j}k^{2}E_{j}(k)\ .$$

(5)

Here we use subscript “_j” to denote either superfluid or normal fluid $j\in\{\mbox{s, n}\}$ and define $\Omega_{\text{n}}=\Omega_{\text{s}}\rho_{\text{s}}/\rho_{\text{n}}$. The normal-fluid-superfluid cross-correlation function $E_{\text{ns}}(k)$ is defined similarly to Eq. (1):

$$E_{\text{ns}}=\frac{1}{2V}\int\left\langle{\bm{u}}_{\text{n}}({\bm{r}},t)\cdot{\bm{u}}_{\text{s}}({\bm{r}},t)\right\rangle d{\bm{r}}=\int\limits_{0}^{\infty}E_{\text{ns}}(k)\,dk\ .$$

(6)

The energy transfer term $d\varepsilon_{j}/dk$ in Eq. (5) originates from the nonlinear terms in the HVBK Eqs. (4) and has the same formLP-95 ; LP-2 ; BL as in classical turbulence:

	$\displaystyle\frac{d\varepsilon_{j}(k)}{dk}$	$\displaystyle=$	$\displaystyle 2\,\mbox{Re}\Big{\{}\int V^{\xi\beta\gamma}({\bm{k}},{\bm{q}},{\bm{p}})\,F_{j}^{\xi\beta\gamma}({\bm{k}},{\bm{q}},{\bm{p}})$		(7a)
			$\displaystyle\times\delta({\bm{k}}+{\bm{q}}+{\bm{p}})\frac{d^{3}q\,d^{3}p}{(2\pi)^{6}}\,\Big{\}}\,,$
	$\displaystyle V^{\xi\beta\gamma}({\bm{k}},{\bm{q}},{\bm{p}})$	$\displaystyle=$	$\displaystyle i\Big{(}\delta_{\xi\xi^{\prime}}-\frac{k^{\xi}k^{\xi^{\prime}}}{k^{2}}\Big{)}$		(7b)
			$\displaystyle\times\Big{(}k^{\beta}\delta_{\xi^{\prime}\gamma}+k^{\gamma}\delta_{\xi^{\prime}\beta}\Big{)}\ .$

Here $F_{j}^{\xi\beta\gamma}({\bm{k}},{\bm{q}},{\bm{p}})$ is the simultaneous triple-correlation function of turbulent (normal or superfluid) velocity fluctuations in the ${\bm{k}}$-representation, that we will not specify here and $V^{\xi\beta\gamma}({\bm{k}},{\bm{q}},{\bm{p}})$ is the interaction vertex in the HVBK (as well as in the Navier-Stokes) equations. Importantly, the right-hand-side of Eq. (7a) conserves the total turbulent kinetic energy (i.e. the integral of $E_{j}(k)$ over entire ${\bm{k}}$-space) and therefore can be written in the divergent form as ${d\varepsilon_{j}}/{dk}$.

One of the main problems in the theory of hydrodynamic turbulence is to find the triple-correlation function $F^{\xi\beta\gamma}({\bm{k}},{\bm{q}},{\bm{p}})$, which determines the energy flux $\varepsilon(k)$ in Eqs. (7). The simplest way is to directly model $\varepsilon(k)$ using dimensional reasoning to connect $\varepsilon(k)$ and the energy spectrum $E(k)$ with the same wavenumber $k$:

$$\varepsilon(k)=Ck^{5/2}E^{3/2}(k)\,,$$

(8)

Here $C$ is the phenomenological constant with the value $C\approx 0.5$, corresponding to the fully developed turbulence of classical fluid CK ; YeungZhou . The algebraic closure (8) is basedFri mainly on the Kolmogorov-1941 (K41) assumption of the universality of turbulent statistics in the limit of large Reynolds numbers and on the Richardson-1922 step-by-step cascade picture of the energy transfer towards large $k$. The energy-cascade picture combined with the K41 idea that in this case $\varepsilon(k)$ is the only relevant physical parameter determining the level of turbulent excitations and their statistics, lead to Eq. (8).

More realistic modeling of $\varepsilon(k)$ can be reached in the framework of integral closures, widely used in analytic theories of classical turbulenceFri , for example, so-called Eddy-damped quasinormal Markovian (EDQNM) closure or Kraichnan’s Direct Interaction Approximation Kra1 ; Kra2 . These closures are based on the representation of third-order velocity correlation function $F^{\xi\beta\gamma}_{\text{s}}$ in Eq. (7a) as a product of the vertex $V$, Eq. (7b), two second-order correlations $E(k)$, and the response (Green’s) functions $G(k)\sim\Gamma(k)-$the typical relaxation frequencies at scale $k$. Keeping in mind the uncontrolled character of integral closures, L’vov, Nazarenko and Rudenko (LNR) suggested in LABEL:LNR a simplified version of EDQNM closure with the same level of justification for isotropic turbulence. LNR replaced a volume element [$d^{3}q\,d^{3}p\,\delta^{3}({\bm{k}}+{\bm{q}}+{\bm{p}})$] in Eq. (7a), involving 3-dimensional vectors ${\bm{k}}$, ${\bm{q}}$, and ${\bm{p}}$, by its isotropic version [$q^{2}dq\,p^{2}dp\ \delta(k+q+p)/(k^{2}+q^{2}+p^{2})$], involving only one-dimensional vectors $k$, $q$, and $p$ varying over the interval $(-\infty,+\infty)$. In addition, they replaced the interaction amplitude $V^{\xi\beta\gamma}({\bm{k}},{\bm{q}},{\bm{p}})$, Eq. (7b) by its scalar version $(ik)$. The resulting LNR closure can be written as follows:

	$\displaystyle\frac{d\varepsilon(k)}{dk}=\frac{A_{1}\,k}{2\pi^{2}}\int_{-\infty}^{\infty}\frac{dq\,dp\,\delta(k+q+p)}{2\pi\,(k^{2}+q^{2}+p^{2})}$		(9a)
	$\displaystyle\thinspace\times\frac{k^{3}\,E(|q|)E(|p|)+q^{3}\,E(|k|)E(|p|)+p^{3}\,E(|q|)E(|k|)}{\Gamma(|k|)+\Gamma(|q|)+\Gamma(|p|)}\ .$
Here $A_{1}$ is a dimensionless parameter of the order of unity.

The integral closure (9a) may be related to the algebraic closure (8) by assuming that the integral (9a) converges (i.e. the main contribution to it comes from the wavenumbers of similar scales $q\sim p\sim k$, so-called locality of interaction) and estimating $\Gamma(k)$ as $\sqrt{k^{3}E(k)}$ and $d\varepsilon(k)/dk$ as $\varepsilon(k)/k$.

The LNR model (9a) satisfies all the general closure requirements: i) it conserves energy, $\int\frac{d\varepsilon(k)}{dk}dk=0$ for any $E(k)$; ii) $\frac{d\varepsilon(k)}{dk}=0$ for the thermodynamic equilibrium spectrum $E(k)\propto k^{2}$ and for the cascade K41 spectrum $E(k)\propto|k|^{-5/3}$. Importantly, the integrand in Eq. (9a) has the correct asymptotic behavior in the limits of small and large $q/k$, as required by the sweeping-free Belinicher-L’vov representation BL . This means that the model (9a) adequately reflects the contributions of the extended-interaction triads and thus can be used for the analysis of the non-local energy transfer, which become importantBKLPPS-2017 when the scaling exponent $m$ approaches $m=3$.

The mutual friction terms in Eqs. (4) cause additional energy dissipation at all scales. Therefore, we can expect that the energy spectra may be steeper than K41 and not scale-invariant.

To generalize the closure (9a) for such a situation, assume that the energy spectrum has a form $E(k)=E_{0}/k^{m}$ with a scaling exponent $m\geqslant 5/3$. As $m\to 3$, the main contribution to the integral (9a) comes from the distant interactions with wavevectors of different scales. In particular, the $\delta$-function in the integral dictates that for $q\ll k$, $p\approx-k$ and the integral (9a) may be approximated as:

	$\displaystyle\frac{d\varepsilon(k)}{dk}$	$\displaystyle\approx$	$\displaystyle\frac{A_{1}\,E_{0}}{8\pi^{3}k\Gamma(|k|)}\int_{-k}^{k}\frac{dq}{q^{m}}$		(9c)
			$\displaystyle\times\big{[}k^{3}\,E(|k+q|)-(k+q)^{3}E(k)\big{]}$
			$\displaystyle\thinspace\approx\frac{A_{1}\,E_{0}}{8\pi^{3}k\Gamma(|k|)}\,\frac{d^{2}}{dq^{2}}\big{[}k^{3}\,E(|k+q|)-(k+q)^{3}E(k)\big{]}_{q\to 0}$
			$\displaystyle\times\int\limits_{0}^{\infty}q^{2-m}dq\simeq\frac{A_{1}\big{[}kE(k)\big{]}^{3/2}}{8\pi^{3}(3-m)}\ .$

Dimensionally, Eq. (9c) coincides with the K41 algebraic closure (8), but contains additional prefactor $1/(3-m)$. This may be interpreted as $m$-dependent parameter $C$ in front of Eq. (8), which diverges as $m\to 3$. The physical reason for such a dependence is simple: as $m$ increases, more and more extended triads that involve $k$- and $(q\ll k)$-modes contribute to the energy influx in the $k$-mode. As a result, the energy flux grows $\propto 1/(3-m)$, according to Eq. (9c). Moreover, when $m\geqslant 3$ the integral (9c) formally diverges, meaning that the leading contribution to the flux at $k$-mode comes not from comparable $(q\sim k)$-modes (as assumed in the Richardson-Kolmogorov step-by-step cascade picture of the energy flux), but directly from the largest, energy-containing modes in the turbulent flow.

Thus, by accounting for the possible scale-dependent nonlocal energy transfer over scales, we generalize the standard K41-closure (8) by including the $k$-dependence of the prefactor $C$:

	$$\varepsilon(k)=C(k)k^{5/2}E^{3/2}(k)\,,\ C(k)=\frac{4\,C}{3\,[3-m(k)]}\ .$$		(10a)
For convenience, the prefactor $C(k)$ is chosen to reproduce the Kolmogorov constant $C$ for the K41 scaling exponent $m(k)=5/3$. As follows from above arguments, the function of $m(k)$ in (10a) should be understood as a local scaling exponent of $E(k$):
	$$m(k)=\frac{d\ln E(k)}{d\ln(k)}\ ,$$		(10b)

making the new closure a self-consistent differential closure.

The general form of the cross-correlation functiondecoupling $E_{\text{ns}}$ reads as:

	$\displaystyle E_{\text{ns}}(k)$	$\displaystyle=$	$\displaystyle D(k)E_{\text{ns}}^{(0)}(k)\,,\ D(k)=\frac{\arctan[\xi(k)]}{\xi(k)}\,,~~~~~~~~$		(11a)
	$\displaystyle\ \xi(k)$	$\displaystyle=$	$\displaystyle\frac{k}{k_{\times}}\,,\quad k_{\times}=\frac{\Gamma(k)}{U_{\text{ns}}}\ .$		(11b)
Here $D(k)$ is the $U_{\text{ns}}$-dependent decoupling function, and $E_{\text{ns}}^{(0)}(k)$ has the formLNS ; He4 :
	$\displaystyle E_{\text{ns}}^{(0)}(k)$	$\displaystyle=$	$\displaystyle\frac{\Omega_{\text{ns}}\big{[}\rho_{\text{n}}E_{\text{n}}(k)+\rho_{\text{s}}E_{\text{s}}(k)\big{]}}{\Gamma(k)\,\rho}\,,$		(11c)
	$\displaystyle\Gamma(k)$	$\displaystyle=$	$\displaystyle\Omega_{\text{ns}}+\gamma_{\text{s}}(k)+\gamma_{\text{n}}(k)+(\nu_{\text{s}}+\nu_{\text{n}})\,k^{2}\,,$		(11d)
	$\displaystyle\Omega_{\text{ns}}$	$\displaystyle=$	$\displaystyle\Omega_{\text{n}}+\Omega_{\text{s}}=\alpha\kappa{\mathcal{L}}\frac{\rho}{\rho_{\text{n}}}\,,\ \gamma_{j}(k)=C_{\gamma}\sqrt{k^{3}E_{j}(k)}\ .~~~$

Here $C_{\gamma}$ is a phenomenological parameter, the same for both components. However $E_{\text{n}}(k)$ and $E_{\text{s}}(k)$ in Eq. (11c) are not the energy spectra at $U_{\text{ns}}=0$, but the $U_{\text{ns}}$-dependent energy spectra, found self-consistently by solving Eqs. (5) with $E_{\text{ns}}(k)$ given by Eqs. (11).

The Eqs. (5) are coupled via cross-correlation function $E_{\text{ns}}$, which depends on the energy spectra of both components. To find the leading contributions to $E_{\text{ns}}$, we recall that at $k$ close to $k_{0}$ the velocities are well correlateddecoupling , meaning that $E_{\text{n}}(k)\simeq E_{\text{s}}(k)$, while for $k\gg k_{0}$ they are almost decorrelated and $E_{\text{ns}}(k)$ is negligible compared to $E_{\text{n}}$ and $E_{\text{s}}$. Therefore, without loss of accuracy we can replace $E_{\text{n}}(k)$ by $E_{\text{s}}(k)$ in Eqs. (11) for $E_{\text{ns}}$ that enters into balance equation (5) for the superfluid and $E_{\text{s}}(k)$ by $E_{\text{n}}(k)$ for $E_{\text{ns}}$ that enters into normal-fluid balance equation.

In this way we obtain decoupled equations for the fluxes $\varepsilon_{\text{n}}$ and $\varepsilon_{\text{s}}$

	$\displaystyle\frac{d\varepsilon_{j}}{dk}$	$\displaystyle=$	$\displaystyle E_{j}(k)\big{\{}\Omega[D_{j}(k)-1]-2\,\nu_{j}k^{2}\big{\}}\,,$		(12a)
with different decoupling functions for the normal-fluid and superfluid components:
	$\displaystyle D_{j}(k)$	$\displaystyle=$	$\displaystyle\frac{\Omega_{\text{ns}}\arctan[kU_{\text{ns}}/\Gamma_{j}(k)]}{kU_{\text{ns}}}\,,$		(12b)
and different values of full damping frequencies:
	$\displaystyle\Gamma_{j}(k)$	$\displaystyle=$	$\displaystyle\Omega_{\text{ns}}+2C_{\gamma}\sqrt{k^{3}E_{j}(k)}+(\nu_{\text{s}}+\nu_{\text{n}})k^{2}\ .$		(12c)

The balance Eqs. (12), being uncoupled for the normal-fluid and superfluid components, are already much simpler than the fully coupled balance equations (5) and (11).

We are now ready to make next step and to analyze the relative importance of different contributions to the damping frequencies $\Gamma_{j}(k)$, comparing the dissipation due to mutual friction $\Omega_{\text{ns}}$, the rate of the energy transfer between scales $C_{\gamma}\sqrt{k^{3}E(k)}$ and the viscous dissipation $(\nu_{\text{s}}+\nu_{\text{n}})k^{2}$. We start with $\Omega_{\text{ns}}=\alpha(T)\kappa{\mathcal{L}}\rho/\rho_{\text{n}}(T)$. In the intermediate range of temperatures $\alpha(T)\rho/\rho_{\text{n}}(T)\approx 0.5$, weakly depending on $T$, cf Tab. 1. Therefore we can peacefully take $\Omega_{\text{ns}}\approx 0.5\kappa{\mathcal{L}}$.

Next, the sum $(\nu_{\text{s}}+\nu_{\text{n}})$ in this temperature interval also weakly depends on $T$ and is very close to $0.5\,\kappa$. (cf. Tab. 1 and Fig. 5 in LABEL:He4 and LABEL:LSS for discussion on physical origin of the numerical value). Estimating the largest wavenumber of the inertial interval by the quantum cutoff $k_{\text{max}}\sim 1/\ell=\sqrt{{\mathcal{L}}}$, we find that $(\nu_{\text{s}}+\nu_{\text{n}})k_{\text{max}}^{2}\simeq 0.5\kappa{\mathcal{L}}\approx\Omega_{\text{ns}}$. Therefore, $(\nu_{\text{s}}+\nu_{\text{n}})k^{2}<\Omega_{\text{ns}}$ in the entire inertial interval, except for its large-$k$ end, where the accurate representation of the decoupling functions is not important. This allows us to neglect in most cases the viscous contributions in Eq. (12c) for $\Gamma_{j}(k)$.

To estimate the energy transfer terms, recall that in the classical K41 turbulence (with $E(k)\propto k^{-5/3}$) $\sqrt{k^{3}E(k)}$ grows as $k^{2/3}$. It remains larger than viscous terms in the entire inertial range and become compatible with $\nu k^{2}$ at the large-$k$ end of the inertial interval, at the Kolmogorov microscale. In our case, the spectra $E_{j}(k)$ decay with $k$ faster than the K41 spectrum, owing to the energy dissipation due to mutual friction. Therefore, in the inertial interval of counterflow turbulence $\sqrt{k^{3}E_{j}(k)}$ grows slower than in K41 turbulence and remains smaller than $\Omega_{\text{ns}}$ for all $k$. Moreover, recent estimate of $C_{\gamma}$ using Direct Numerical Simulations of the superfluid ⁴He turbulenceDNS-He4 gives $C_{\gamma}\ll 1$. Therefore, for these conditions we can neglect the energy transfer terms compared to $\Omega_{\text{ns}}$. Having all these in mind, we approximate $\Gamma_{j}$ as $\Omega_{\text{ns}}$ and, using Eq. (12b), get

$$D_{\text{n}}(k)=D_{\text{s}}(k)\equiv D(k)\,,$$

(13)

with $\xi(k)=kU_{\text{ns}}/\Omega_{\text{ns}}$.

Now, we combine Eqs. (10), (12a) and (13) to finalize an approximation for the balance equations

	$\displaystyle C(k)\,\frac{d}{dk}k^{5/2}E_{j}^{3/2}(k)$	$\displaystyle=$	$\displaystyle E_{j}(k)\big{\{}\Omega_{j}[D(k)-1]$		(14a)
			$\displaystyle-2\nu_{j}k^{2}\big{\}},~~~~~~$
	$\displaystyle D(k)=\frac{\arctan[\xi(k)]}{\xi(k)}\,,$		$\displaystyle\quad\xi(k)=\frac{kU_{\text{ns}}}{\Omega_{\text{ns}}}\ .$		(14b)

In deriving Eq. (14a) we neglected for simplicity the $k$-derivative of $m(k)$ in the expression for $C(k)$ with respect to $d[k^{5/3}E(k)]/dk$ since $m(k)$ varies between 5/3 to 3 in the entire range of $k$, while $k^{5/3}E(k)$ varies by many orders of magnitude.

To analyze the energy balance Eq. (14) and to open a way to its numerical solution, we first rewrite it in the dimensionless form. To this end we introduce a dimensionless wavenumber $q$ and a dimensionless energy spectrum ${\mathcal{E}}(q)$:

$$q=k/k_{0}\,,\quad{\mathcal{E}}(q)=E(k)/E(k_{0})\ .$$

(15)

Here the minimal wave number is estimated as $k_{0}=2\pi/\Delta$, where $\Delta$ is outer scale of turbulence.

The resulting dimensionless equations for new dimensionless functions

	$$\Psi_{j}(q)=\sqrt{q^{5/3}{\mathcal{E}}_{j}(q)}\,,$$				(16a)
take the form
	$\displaystyle\frac{d\Psi_{j}(q)}{dq}$	$\displaystyle=$	$\displaystyle A_{j}\,\frac{D(qk_{0})-1}{q^{5/3}}-a_{j}\,q^{1/3}\,,~j\in\{s,n\}\,,~~~~$		(16b)
	$\displaystyle A_{j}$	$\displaystyle=$	$\displaystyle\frac{\Omega_{j}}{3C(k)\sqrt{k_{0}^{3}E_{j}(k_{0})}}\,,$		(16c)
	$\displaystyle\ a_{j}$	$\displaystyle=$	$\displaystyle\frac{2\nu_{j}\sqrt{k_{0}}}{3C(k)\sqrt{E_{j}(k_{0})}}\ .~~~~~$		(16d)

To clarify further the parameters in Eqs. (16), we define the dimensionless parameters that govern the counterflow superfluid turbulence: the turbulent Reynolds numbers, the efficiency of dissipation by mutual friction and the dimensionless counterflow velocity.

Similar to the classical hydrodynamic turbulence, the energy dissipation by viscous friction in the superfluid turbulence is governed by the Reynolds number. There are two such Reynolds numbers, ${\rm Re}_{\text{n}}$ and ${\rm Re}_{\text{s}}$:

	$${\rm Re\!}_{j}=\frac{u_{\scriptscriptstyle\rm{T}}}{k_{0}\nu_{j}}\,,\quad u_{\scriptscriptstyle\rm{T}}\simeq\sqrt{k_{0}E(k_{0})}\ .$$		(17a)
Here we ignore the presumably small difference between velocity fluctuations of the components at scale $k_{0}$ [i.e assume $E_{\text{s}}(k_{0})=E_{\text{n}}(k_{0})=E(k_{0})$] and estimate the root-mean-square (rms) turbulent fluctuations $u_{\scriptscriptstyle\rm{T}}$ as $\sqrt{k_{0}E(k_{0})}$. The ratio of the Reynolds numbers (17a) is defined by the viscosities ${\rm Re}_{\text{s}}/{\rm Re}_{\text{n}}=\nu_{\text{n}}/\nu_{\text{s}}$ and depends only on the temperature. Therefore, for a given temperature we are left with only one Reynolds number, say ${\rm Re}_{\text{n}}$.

There is one more mechanism of the energy dissipation in superfluid turbulence: the dissipation by mutual friction. This kind of dissipation is characterized by the frequency $\Omega_{\text{ns}}$, Eq. (11c). As was mentioned above, $\Omega_{\text{ns}}\approx 0.5\kappa{\mathcal{L}}$ in the entire temperature range. Then, the partial frequencies $\Omega_{\text{s}}$ and $\Omega_{\text{n}}$ that govern the energy dissipation by mutual friction in the superfluid and normal-fluid components with a given $\Omega_{\text{ns}}$, depend only on the densities $\rho_{\text{s}}$ and $\rho_{\text{n}}$, according to Eq. (11c). Therefore we can say that for a given temperature, the dissipation by mutual friction is governed by only one frequency $\Omega_{\text{ns}}\propto{\mathcal{L}}$.

As was shown in Refs. LNV ; BLPP-2012 ; BKLPPS-2017 , the rate of energy dissipation by mutual friction should be compared with the $k$-dependent rate of the energy transfer over the cascade, characterized by the turnover frequency of the eddies of scale $1/k$, $\gamma_{j}(k)$, Eq. (11c). This dictates a natural normalization of $\Omega_{\text{ns}}$ by $\gamma(k_{0})$, which can be estimated as $k_{0}u_{\scriptscriptstyle\rm{T}}$. In other words, we suggest to use

$$\widetilde{\Omega}_{\text{ns}}\equiv\Omega_{\text{ns}}/k_{0}u_{\scriptscriptstyle\rm{T}}\,,$$

(17b)

as the dimensionless parameter that characterizes the efficiency of dissipation by mutual friction .

Last important parameter of the problem is the counterflow velocity $U_{\text{ns}}$. It is natural to normalize it by the turbulent velocity $u_{\scriptscriptstyle\rm{T}}$, introducing a dimensionless velocity

$$\widetilde{U}_{\text{ns}}\equiv U_{\text{ns}}/u_{\scriptscriptstyle\rm{T}}\,,$$

(17c)

Using parameters (17) together with Eqs. (10) and (14b) we rewrite Eq. (16) as follows

	$\displaystyle 4C\,\frac{d\Psi_{j}}{dq}$	$\displaystyle=$	$\displaystyle-\Big{[}\frac{2}{3}+\frac{d\Psi_{j}}{dq}\,\frac{q}{\Psi_{j}}\Big{]}$
			$\displaystyle\times\Big{\{}\frac{\widetilde{\Omega}_{j}}{q^{5/3}}\Big{[}1-\frac{\arctan(q/q_{\times})}{q/q_{\times}}\Big{]}+\frac{2}{\mbox{Re}_{j}}\Big{\}}$
	$\displaystyle\widetilde{\Omega}_{\text{n}}=\widetilde{\Omega}_{\text{ns}}\frac{\rho_{\text{s}}}{\rho}\,,$		$\displaystyle\widetilde{\Omega}_{\text{s}}=\widetilde{\Omega}_{\text{ns}}\frac{\rho_{\text{n}}}{\rho}\,,\quad q_{\times}=\frac{\widetilde{\Omega}_{\text{ns}}}{\widetilde{U}_{\text{ns}}}\ .$		(18b)

These are the first-order ordinary differential equations for $\Psi_{j}(q)$. Aside from the temperature dependent parameter $\rho_{\text{n}}/\rho$, they include four dimensionless parameters that characterize the superfluid counterflow turbulence: $\widetilde{\Omega}_{\text{ns}}$, $\widetilde{U}_{\text{ns}}$ and Re_j.

To qualitatively analyze the energy spectra, we first neglect in Eqs. (18) the influence of the viscous dissipation. In Fig. 1 we show the energy spectra, obtained by solving Eq. (18) in a wide range of dimensionless parameters with Re${}_{\text{s,n}}\to\infty$. In all panels, the normal component spectra are shown by solid lines and the superfluid component spectra – by dashed lines.

Each row represent the spectra at one of four temperatures, from top to bottom: $T=1.65,1.85,1.95$ and $2.1$K.

The three columns show solutions for three different values of the dimensionless counterflow velocity (17c): $\widetilde{U}_{\text{ns}}=1$ (left), $\widetilde{U}_{\text{ns}}=4$ (middle) and $\widetilde{U}_{\text{ns}}=8$ (right).

Each of the panels contains the spectra for 5 values of the dimensionless frequency $\widetilde{\Omega}_{\text{ns}}$ from 0 (the horizontal gray lines) to $\tilde{\Omega}_{\text{ns}}=20$ (the lowest red lines), color-coded as described in the figure caption.

Comparing spectra shown in Fig. 1, we can make a set of observations:

i) The larger the counterflow velocity $\widetilde{U}_{\text{ns}}$, the stronger is the suppression of the energy spectra compared to the K41 prediction. This is an expected result: the normal-fluid and superfluid velocity fluctuations decorrelate faster with increasing $\widetilde{U}_{\text{ns}}$, leading to stronger energy dissipation by mutual friction and as a result– to a more prominent suppression of the energy spectra.

ii) In the absence of viscous dissipation, the dissipation by mutual friction defines the suppression of the spectra. The corresponding frequencies $\Omega_{j}$ [Eq. (18b)], are proportional to the other component’s densities: $\Omega_{\text{s}}\propto\rho_{\text{n}},\Omega_{\text{n}}\propto\rho_{\text{s}}$. Therefore, at low temperatures (two upper rows), when $\rho_{\text{n}}\ll\rho_{\text{s}}$, the normal-fluid component spectra are suppressed stronger than the superfluid spectra, while at high $T$ the relation is reversed (the lowest row).

iii) The competition between the velocity fluctuations coupling and the dissipation due to mutual friction leads to a complicated $q$-dependence of the spectra, described by Eq. (12a): the rate of energy dissipation is proportional to $\Omega_{j}[D_{j}(q)-1]$. The larger is $\Omega_{j}$ the stronger is the coupling, however, simultaneously $D_{j}(q)\to 1$. Which factor wins, depends on the scale: at large $q$ the dissipation wins and the spectra suppression is directly proportional to $\widetilde{\Omega}_{j}$. On the other hand, at small $q$, the spectra for larger $\widetilde{\Omega}_{j}$ are less suppressed, especially at weak counterflow velocity.

Note that the applicability range of HVBK [Eqs. (4)] is limited by $k<\pi/\ell=\pi\sqrt{{\mathcal{L}}}\simeq\pi\sqrt{2\Omega_{\text{ns}}/\kappa}$. However our analysis is performed for given values of $\widetilde{\Omega}_{\text{ns}}=\Omega_{\text{ns}}/k_{0}u_{\text{T}}$ with $k_{0}=1$ and an arbitrary value of $u_{\scriptscriptstyle\rm{T}}$. Therefore, there is no formal restriction on the range of $k$ in Figs. 1, (as well as in Figs. 2, 3, and Fig. 4), which was chosen as to expose all important features of the energy spectra.

To characterize in a compact form the energy spectra dependence on the flow parameters, $T$, $\widetilde{U}_{\text{ns}}$ and $\widetilde{\Omega}_{\text{ns}}$, we consider first a scaling exponent $m(q)$, Eq. (10b), in the beginning of the scaling interval $k=k_{0},q=1$. Using Eqs. (16a) and (18) we get for outer-scale exponent $m_{j}(q=1)$:

$\displaystyle m_{j}(1)=\frac{5}{3}+\frac{4}{3}\,\frac{\widetilde{\Omega}_{j}\big{[}1-q_{\times}\arctan(1/q_{\times})\big{]}}{4C+\widetilde{\Omega}_{j}\big{[}1-q_{\times}\arctan(1/q_{\times})\big{]}}\,.~~~~~$

(19)

The $\widetilde{\Omega}_{\text{ns}}$-dependencies of the normal-fluid outer-scale exponent $m_{n}(1)$ for different $\widetilde{U}_{\text{ns}}$ and $T$ are shown in Fig. 2. As expected, for $\widetilde{\Omega}_{\text{ns}}=0$ (no mutual friction damping) $m_{\text{n}}(1)=\frac{5}{3}$, the classical K41 value. In the limit $\widetilde{\Omega}_{\text{ns}}\to\infty$, $\displaystyle\lim_{q_{\times}\to\infty}[q_{\times}\arctan(1/q_{\times})]\propto 1/q_{\times}^{2}$ and $m_{\text{n}}(1)$ again tends to its K41 value $\frac{5}{3}$. In this limit, the normal and superfluid components are fully coupled and there is no energy dissipation by mutual friction. The resulting $\widetilde{\Omega}_{\text{ns}}$-dependence of $m_{n}(q)$ is non-monotonic with a maximum around $\widetilde{\Omega}_{\text{ns}}\sim 1$. As we saw before, the $m_{n}(1)$ is largest (i.e. the strongest suppression of normal-fluid energy spectra) at lowest $T=1.65\,$K (upper black lines), the smallest – at high $T=2.1\,$K (the lowest red lines).

Conversely, the superfluid exponents $m_{s}(1)$ (not shown) reflect strong suppression [larger $m_{s}(1)$] at high temperatures, while at low temperatures $m_{s}(1)$ are smaller.

To characterize the degree of deviation from the scale invariance, we introduce the mean scaling exponent over some $q$-interval from $q=1$ [with ${\mathcal{E}}_{j}(1)=1$] to a given value of $q$ :

$$\left\langle m_{j}\right\rangle_{q}=-\ln{\mathcal{E}}_{j}(q)\big{/}\ln q\ .$$

(20)

The value $\left\langle m_{j}\right\rangle_{q}$ should be $q$-independent and equal to the outer-scale exponent $m_{j}(1)=[d\ln{\mathcal{E}}_{j}(q)/d\ln q]_{q=1}$ for the scale-invariant spectra and vary otherwise. In Fig. 3 we compare $m_{j}(1)$ and $\left\langle m_{j}\right\rangle_{10}$ for $T=2.0$K and $\widetilde{U}_{\text{ns}}=4$. The outer-scale scaling exponent $m_{\text{n}}(1)$ is shown by solid line, $m_{\text{s}}(1)$ – by dashed line. The values of $\left\langle m_{\text{n}}\right\rangle_{10}$ and $\left\langle m_{\text{s}}\right\rangle_{10}$, calculated for a set of $\widetilde{\Omega}_{\text{ns}}$, are shown by full and empty circles, respectively. Clearly, the spectra coincide (at least within the first decade) for $\widetilde{\Omega}_{\text{ns}}\lesssim 3$, while for stronger $\widetilde{\Omega}_{\text{ns}}$ they vary significantly: the actual spectra are more suppressed than is suggested by $m_{j}(1)$ and $\left\langle m_{j}\right\rangle_{10}>m_{j}(1)$ for $\widetilde{\Omega}_{\text{ns}}>3$ at this temperature. The non-monotonic behavior is evident in all curves, although the maximum in the mean exponents $\left\langle m_{j}\right\rangle_{10}$ is broader and less prominent.