Quantum turbulence is the name given to the turbulent flow – the chaotic motion of a fluid at high flow rates – of quantum fluids, such as superfluids. The idea that a form of turbulence might be possible in a superfluid via the quantized vortex lines was first suggested by Richard Feynman. The dynamics of quantum fluids are governed by quantum mechanics, rather than classical physics which govern classical (ordinary) fluids. Some examples of quantum fluids include superfluid helium (4He and Cooper pairs of 3He), Bose–Einstein condensates (BECs), polariton condensates, and nuclear pasta theorized to exist inside neutron stars. Quantum fluids exist at temperatures below the critical temperature T c {\displaystyle T_{\text{c}}} at which Bose-Einstein condensation takes place.
General properties of superfluids
The turbulence of quantum fluids has been studied primarily in two quantum fluids: liquid helium and atomic condensates. Experimental observations have been made in the two stable isotopes of helium, the common 4He and the rare 3He. The latter isotope has two phases, named the A-phase and the B-phase. The A-phase is strongly anisotropic, and although it has very interesting hydrodynamic properties, turbulence experiments have been performed almost exclusively in the B-phase. Helium liquidizes at a temperature of approximately 4 K. At this temperature, the fluid behaves like a classical fluid with extraordinarily small viscosity, referred to as helium I. After further cooling, helium I undergoes Bose–Einstein condensation into a superfluid, referred to as helium II. The critical temperature T c {\displaystyle T_{\text{c}}} for Bose–Einstein condensation of helium is 2.17 K (at the saturated vapour pressure), while only approximately a few millikelvin for 3He–B. Although in atomic condensates there is not as much experimental evidence for turbulence as in helium, experiments have been performed with rubidium, sodium, caesium, lithium and other elements. The critical temperature for these systems is of the order of micro-Kelvin. There are two fundamental properties of quantum fluids that distinguish them from classical fluids: superfluidity and quantized circulation.
Superfluidity Superfluidity arises as a consequence of the dispersion relation of elementary excitations, and fluids that exhibit this behaviour flow without viscosity. This is a vital property for quantum turbulence as viscosity in classical fluids causes dissipation of kinetic energy into heat, damping out motion of the fluid. Landau predicted that if a superfluid flows faster than a certain critical velocity v c {\displaystyle v_{\text{c}}} (or alternatively an object moves faster than v c {\displaystyle v_{\text{c}}} in a static fluid) thermal excitations (rotons) are emitted as it becomes energetically favourable to generate quasiparticles, resulting in the fluid no longer exhibiting superfluid properties. For helium II, this critical velocity is v c ≈ 60 m/s {\displaystyle v_{\text{c}}\approx 60{\text{m/s}}} .
Quantized circulation The property of quantized circulation arises as a consequence of the existence and uniqueness of a complex macroscopic wavefunction Ψ {\displaystyle \Psi } , which affects the vorticity (local rotation) in a very profound way, making it crucial for quantum turbulence. The velocity and density of the fluid can be recovered from the wavefunction Ψ ( x , t ) {\displaystyle \Psi (\mathbf {x} ,t)} by writing it in polar form Ψ ( x , t ) = | Ψ ( x , t ) | e i ϕ ( x , t ) {\displaystyle \Psi (\mathbf {x} ,t)=|\Psi (\mathbf {x} ,t)|e^{i\phi (\mathbf {x} ,t)}} , where | Ψ | {\displaystyle |\Psi |} is the magnitude of Ψ {\displaystyle \Psi } and ϕ {\displaystyle \phi } is the phase. The velocity of the fluid is then v ( x , t ) = ( ℏ / m ) ∇ ϕ {\displaystyle \mathbf {v} (\mathbf {x} ,t)=(\hbar /m)\nabla \phi } , and the number density is n ( x , t ) = | Ψ | 2 {\displaystyle n(\mathbf {x} ,t)=\vert \Psi \vert ^{2}} . The mass density is related to the number density by ρ ( x , t ) = m n {\displaystyle \rho (\mathbf {x} ,t)=mn} , where m {\displaystyle m} is the mass of one boson. The circulation Γ {\displaystyle \Gamma } is defined to be the line integral along a simple closed path C {\displaystyle C} within the fluid
Γ = ∮ C v ⋅ d r {\displaystyle \Gamma =\oint _{C}\mathbf {v} \cdot \mathbf {dr} }
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