Absolute vorticity

Absolute vorticity is the vorticity measured in an inertial frame. In a frame rotating with angular velocity

Ω

, it is

ω + 2 Ω

.

Table of contents
- Planetary vorticity Absolute vorticity
- Absolute-vorticity flux tensor Absolute vorticity
  - Radial vorticity conservation in a shearing sheet Absolute-vorticity flux tensor

Planetary vorticity ( $f = 2Ω sin ϕ$ )

The planetary contribution to absolute vorticity is supplied by the rotating frame. Its local vertical component is the Coriolis parameter

f = 2Ω sin ϕ

. On a beta plane, motion with northward velocity

v

changes this contribution at the rate

β v

.

Absolute-vorticity flux tensor ( $T_{ij} = u_{i} ω_{j} - ω_{i} u_{j}$ )

For a flow satisfying incompressibility, this antisymmetric tensor writes advection and vortex stretching in conservative form. With its first index as transport direction,

(\nabla \cdot T)_{j} = \partial_{i} T_{ij}

.

Radial vorticity conservation in a shearing sheet ( $\partial_{t} ω_{x} + \partial_{i} T_{i x} = 0$ )

When the only nonpotential force points radially, its curl has no radial component. The radial absolute vorticity therefore obeys a local flux conservation law, although vortex stretching can change it along a fluid trajectory.

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