Shear Flow and Kelvin-Helmholtz Instability in Superfluids

The first realization of instabilities in the shear flow between two superfluids is examined. The interface separating the $A$ and $B$ phases of superfluid ${}^{\mathrm{3}}\mathrm{H}\mathrm{e}$ is magnetically stabilized. With uniform rotation we create a state with discontinuous tangential velocities at the interface, supported by the difference in quantized vorticity in the two phases. This state remains stable and nondissipative to high relative velocities, but finally undergoes an instability when an interfacial mode is excited and some vortices cross the phase boundary. The measured properties of the instability are consistent with the classic Kelvin-Helmholtz theory when modified for two-fluid hydrodynamics.

Figure 1

Stabilization of the first-order ${}^{\mathrm{3}}\mathrm{H}\mathrm{e}\mathrm{\text{−}}A–{}^{\mathrm{3}}\mathrm{H}\mathrm{e}\mathrm{\text{−}}B$ phase boundary. At pressures $p\ge 21\mathrm{\text{bars}}$, the $A$ phase extends to lower temperatures in external magnetic field (see phase diagram on the right). The sample (length 11 cm, radius $R=0.3\mathrm{c}\mathrm{m}$) is first cooled to the $A$ phase. On further cooling, the $A\to B$ transition happens in the coldest place at the bottom. The $AB$ boundary then starts to move up and rises to a height $z$ where the barrier field $H_{\mathrm{b}}(z)$ equals the value of the thermodynamic $A\to B$ transition $H_{AB}(T,p)$. The $A$ phase in the top section remains in a metastable supercooled state [9]. Ultimately, the boundary disappears when $H_{AB}(T,p)>[H_{\mathrm{b}}(z){]}_{\max }$. For $p=29.0\mathrm{\text{bars}}$ and $H_{\mathrm{b}}(z)$ as shown in the figure with a current of 4 A in the barrier solenoid, the stable $AB$ boundary exists below 2.07 mK down to 1.33 mK. The NMR spectrometers operate in homogeneous static magnetic fields of 10 and 35 mT, chosen for best measuring sensitivity of the single-vortex signal.

Figure 2

The two configurations of vortex lines with the $AB$ boundary in rotation. (a) $\Omega <{\Omega }_{\mathrm{c}}$: Vortex lines are formed in ${}^{\mathrm{3}}\mathrm{H}\mathrm{e}\mathrm{\text{−}}A$ while ${}^{\mathrm{3}}\mathrm{H}\mathrm{e}\mathrm{\text{−}}B$ remains vortex-free. Near the $AB$ boundary the $A$-phase vortices bend parallel to the interface and form a vortex sheet between the sliding superfluids. The radial distributions of the normal and superfluid velocities far from the $AB$ interface are depicted below, with $v_{\mathrm{n}A}=v_{\mathrm{n}B}=\Omega r$, while $v_{\mathrm{s}B}=0$. (b) $\Omega \ge {\Omega }_{\mathrm{c}}$: Vortices are observed to appear in the $B$ phase in events of a few lines at a time. They form a central cluster in the $B$-phase section. (c) A hydrodynamically stable state with respect to externally imposed perturbations in $\Omega$, $T$, or $H$ exists at the $AB$ interface for $\Omega \ge {\Omega }_{\mathrm{c}}$. A topologically stable configuration for the vortex-line intersection with the $AB$ interface is suggested in Ref. 14: The doubly quantized $A$-phase vortex terminates at the $AB$ interface in two point singularities, known as boojums. These, in turn, are the end points of two singly quantized $B$-phase vortices.

Figure 3

Instability events during slowly increasing rotation ($d\Omega /dt=2.5\times {10}^{-4}\mathrm{r}\mathrm{a}\mathrm{d}/{\mathrm{s}}^2$). NMR absorption signals as a function of $\Omega$ are recorded (shifted arbitrarily on the vertical scale). The vertical jumps mark events in which $\delta N$ new vortex lines enter the $B$ phase section of the sample. The height of each jump is proportional to $\delta N$, a small stochastic number. When averaged over a large interval of $\Omega$, the number of $B$-phase vortex lines grows linearly with $\Omega$. This is demonstrated also by the dashed line: the instability occurs independently of $\Omega$ at constant critical drive, i.e., $u_{\mathrm{c}}=|v_{\mathrm{s}B}-v_{\mathrm{n}}{|}_{r=R}={\Omega }_{\mathrm{c}}R=\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}$. The three topmost signal traces were recorded at the same temperature; i.e., measurements at fixed $T$ yield the same ${\Omega }_{\mathrm{c}}$ (if small variations in $v_{\mathrm{c}A}$ are accounted for [13]). The two bottom traces at different $T$ illustrate that ${\Omega }_{\mathrm{c}}$ depends on temperature. Here $T_{AB}(H=0)=0.785T_{\mathrm{c}}$ and $H_{\mathrm{b}}$ as in Fig. 1.

Figure 4

Critical velocity ${\Omega }_{\mathrm{c}}$ for the first appearance of $B$-phase vortex lines in the presence of the $AB$ boundary as a function of temperature, while $\Omega$ is slowly increased ($\dot{\Omega }=5\times {10}^{-4}\mathrm{r}\mathrm{a}\mathrm{d}/{\mathrm{s}}^2$). The current in the barrier solenoid is constant but different for the three sets of data. It controls the magnetic force $F$ which is largely responsible for the shape of the curves: As a function of decreasing temperature the curves start at $T_{AB}(H_{AB}=35\mathrm{m}\mathrm{T})$ and end at the temperature at which $H_{AB}$ equals the maximum value of the barrier field $H_{\mathrm{b}}$ at the given current. The steep slopes at both ends of the curves are caused by the rapidly changing value of $\nabla H_{\mathrm{b}}$. The solid curves represent Eq. (2) if one sets ${\Omega }_{\mathrm{c}}=|v_{\mathrm{s}B}-v_{\mathrm{n}}|/R$ and $v_{\mathrm{s}A}=v_{\mathrm{n}}$. No fitting parameters are used. The values for $\sigma (T)$, ${\chi }_A-{\chi }_B(T,H)$, and ${\rho }_{\mathrm{s}B}(T,H)$ are obtained from accepted references [8, 18, 19, 20], while $H_{\mathrm{b}}$ and $\nabla H_{\mathrm{b}}$ apply for the profile $H_{\mathrm{b}}(z)$ at the location of the $AB$ boundary: $H_{AB}(T)=H_{\mathrm{b}}(z)$.