Phenomenology of nonrelativistic parity-violating hydrodynamics in 2+1 dimensions

Parity-violating fluids in two spatial dimensions can appear in a variety of contexts such as liquid crystal films, anyon fluids, and quantum Hall fluids. Nonetheless, the consequences of parity violation on the solutions to the equations of motion are largely unexplored. In this paper, we explore phenomenological consequences of parity violation through simple, illustrative examples. Although incompressible velocity fields are essentially unchanged by parity violation, we discuss examples where parity violation plays a role at boundaries, or in the dynamics of temperature. We then discuss types of compressible flows which only exist in a parity-violating fluid, including sound waves, and solitons in the dissipationless limit. We conclude with a discussion of some curious features in Rayleigh-Bénard convection of a parity-violating fluid.

Figure 1

An example of Eq. (23) for $\kappa =\tilde{\kappa },n=1$, and $L_x=L_y$. Here we are plotting $T-T_0$, and we plot the dimensionless $(T-T_0)/max(T-T_0)$.

Figure 2

The setup of flow on an inclined plane.

Figure 3

The nine most unstable modes of the parity-violating waveguide. (a) $k=4$ with varying $a$. (b) $a=1$ with varying $k$.

Figure 4

The (a) real and (b) imaginary parts of the most unstable mode $\omega \left(k\right)$, for various values of $a$.

Figure 5

(a) $v_{\mathrm{shake}}/v_{\mathrm{drive}}$ with fixed $\tilde{\nu }/\nu =0.1$. (b) $v_{\mathrm{shake}}/v_{\mathrm{drive}}$ with fixed ${(ch/\nu )}^2=60$.

Figure 6

A plot of the function $V\left(f\right)$ for various choices of $C$.

Figure 7

The seven most unstable modes in the Rayleigh-Bénard problem for select values of $R,G,F$, and $k$. So that all data fits in the same plot, we have had to choose the same value $Q=100$ for all data, but we have observed that the qualitative types of behavior depicted above occur for other values of $Q$.

Figure 8

Plots of the most unstable mode in the Rayleigh-Bénard problem for a variety of parameters. Here are the parameters which are held fixed in each plot: (a) $R=2400,Q=10,G=0$; (b) $R=2400,Q=10,F=0$; (c) $R=2400,Q=10,F=100$; (d) $R=500,Q=10,F=100$; (e) $R=500,Q=10,G=0.6$; (f) $R=500,Q=100,F=100$.

Figure 9

Plots of the most unstable eigenvector in the Rayleigh-Bénard problem for fixed $k=4,R=2400,Q=10$. Rows from top to bottom are as follows: $F=G=0$; $F=50,G=0$; $F=0,G=0.4$; $F=20,G=0.2$.