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2.4. Vorticity - or potential flow?

With the growing and decaying isentropic perturbations, and the isocurvature mode, we have accounted for three of the expected five linear modes. The remaining two degrees of freedom were lost when we took the divergence of the Euler equation, thereby annihilating any transverse (rotational) contribution to v. We consider them now.

Theorem: Any differentiable vector field v(x) may be written as a sum of longitudinal (curl-free) and transverse (divergence-free) parts, v|| and vperp, respectively:

Equation 2.22 (2.22)

The proof follows by construction, by solving nabla . v|| = theta and nabla × vperp = bold omega where theta ident nabla . v and bold omega ident nabla × v. In a flat Euclidean space, solutions are given by

Equation 2.23 (2.23)

Equation 2.24 (2.24)

Note that this decomposition is not unique; we may always add to v|| a curl-free solution of nabla . v|| = 0 and to vperp a divergence-free solution of nabla × vperp = 0 (e.g., constant vectors). With suitable boundary conditions (e.g., integ v|| d3x = 0 when integrated over all space) this freedom can be eliminated. The variables theta and bold omega are called the (comoving) expansion scalar and vorticity vector, respectively.

In our preceding discussion of perturbation evolution we have implicitly considered only v||. The remaining two degrees of freedom correspond to the components of vperp (the transversality condition nabla . vperp = 0 removes one degree of freedom from this 3-vector field). Fortunately, we can get a simple nonlinear equation for vperp - actually, for its curl, bold omega - by taking the curl of the Euler equation:

Equation 2.25 (2.25)

where we have assumed an ideal monatomic gas in writing the second form. The term arising from entropy gradients is called the baroclinic term. It is very important for the dynamics of the Earth's atmosphere and oceans (Pedolsky 1987).

An important general result follows from eq. (2.25), the Kelvin Circulation Theorem: If bold omega = 0 everywhere initially, then bold omega remains zero (even in the nonlinear regime) if the baroclinic term vanishes. (We are assuming that other torques such as magnetic ones vanish too.) The reason for the importance of this result in cosmology is that many models assume irrotational, isentropic initial conditions. With adiabatic evolution, it follows that bold omega = 0. Such a flow is also called potential flow because the velocity field may then be obtained from a velocity potential: v = v|| = -nabla Phiv.

Nonadiabatic processes (heating and cooling) and oblique shock waves can generate vorticity. In a collisionless fluid, if the fluid velocity is defined as the mass-weighted average of all the mass elements at a point, this averaging behaves like entropy production in regions where trajectories intersect, and so vorticity can be generated in the mean (fluid) velocity field. Vorticity also arises from isocurvature initial conditions. Equation (2.21) implies deltaS propto nabla2S for long wavelengths in the linear regime, giving a baroclinic torque proportional to nabla deltaS × nablaS propto nabla(nabla2S) × nablaS, which is nonzero in general (though it appears only in second-order perturbation theory).

For most structure formation models, vorticity generation is quite small until shocks form (or trajectories intersect, for collisionless dark matter). In this case, one may obtain the velocity potential from the line integral of the velocity field:

Equation 2.26 (2.26)

Taking the path to be radial with the observer in the middle allows one to reconstruct the velocity potential, and therefore the transverse velocity components, from the radial component. This idea underlies the potential flow reconstruction method, POTENT (Bertschinger & Dekel 1989). If the (smoothed) density fluctuations are sufficiently small for linear theory to be valid, we can estimate the density fluctuation field from an additional divergence. If pressure is unimportant, so k << kJ and delta propto delta+(tau), the linearized continuity equation gives

Equation 2.27 (2.27)

For a wide range of cosmological models, d lndelta+ / d ln a ident f (Omega) approx Omega0.6 depends primarily on the mass density parameter and weakly on other cosmological parameters (Peebles 1980, Lahav et al. 1991). Thus, combining measurements of v (radial components from galaxy redshifts and distances) and independent measurements of delta (from the galaxy density field plus an assumption about how dark matter is distributed relative to galaxies) allows estimation of Omega (Dekel et al. 1993). A review of the POTENT techniques and results is given by Dekel (1994).

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