**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**(

(2.22) |

The proof follows by construction, by solving
^{.}
*v*_{||} =
and
×
*v*_{}
=
where
^{.}
** v** and
×

(2.23) |

(2.24) |

Note that this decomposition is not unique; we may always add to
*v*_{||} a curl-free solution of
^{.}
*v*_{||} = 0 and to
*v*_{}
a divergence-free solution of
×
*v*_{}
= 0 (e.g., constant vectors). With suitable boundary conditions (e.g.,
*v*_{||} d^{3}*x* = 0
when integrated over all space) this freedom can be eliminated.
The variables and
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
*v*_{}
(the transversality condition
^{.}
*v*_{}
= 0 removes one
degree of freedom from this 3-vector field). Fortunately, we can get
a simple nonlinear equation for
*v*_{}
- actually, for its curl,
- by taking
the curl of the Euler equation:

(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
= 0 everywhere
initially, then
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
= 0. Such a
flow is also
called potential flow because the velocity field may then be obtained
from a velocity potential:
** v** =

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
_{S}
^{2}*S*
for long wavelengths in the linear regime, giving a baroclinic torque
proportional to
_{S} ×
*S*
(^{2}*S*)
× *S*,
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:

(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* << *k*_{J} and
_{+}(), the
linearized continuity equation gives

(2.27) |

For a wide range of cosmological models,
*d* ln_{+}
/ *d* ln *a*
*f* ()
^{0.6}
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
(from the galaxy density field plus an
assumption about how dark matter is distributed relative to galaxies)
allows estimation of
(Dekel et al. 1993).
A review of the POTENT techniques and results is given by
Dekel (1994).