# Absolute and Relative Vorticity in Geophysical Fluid Dynamics

### Key Takeaways

• Vorticity is a microscopic measurement that indicates the spin and rotation of a fluid.

• The relative vorticity can be generalized as the vorticity observed in the rotating frame.

• In geophysical fluid dynamics, the relative vorticity is produced by the airflow through the curved path and wind shear. Three-dimensional rotation, otherwise called vorticity, depicts the behavior of ocean circulation as well as weather systems

On Earth’s surface, energy, momentum, and moisture are redistributed through large-scale fluid fluctuations. Three-dimensional rotation observed in the atmosphere is responsible for the transfer of energy, momentum, and moisture from one point to another. The three-dimensional rotation, otherwise called vorticity, depicts the behavior of ocean circulation as well as weather systems. When describing the vorticity in the atmosphere, absolute vorticity and relative vorticity are two terms that need to be understood and clearly explained.

## What Is Vorticity?

Vorticity is a microscopic measurement that indicates the spin and rotation of a fluid. Vorticity describes the vector representation of local rotation in fluids. In Earth’s systems, vorticity is expressed as the net magnitude of change in wind components.

Generally, vorticity is defined as the curl of the velocity. The wind components denoted using u, v, and w, along the orthogonal Cartesian axes x, y, and z, describe the complete rotation or spin experienced by a fluid parcel as: The unit vectors in the x, y, z coordinates are given by i, j, and k, respectively.

## Representing Rotational Dynamics of the Atmosphere Using Vorticity Components

Vorticity is an inevitable quantity when discussing atmospheric or meteorological models. The rotational dynamics associated with the hydrosphere, troposphere, and atmosphere are described using the parameter vorticity.

Vorticity is a quantity with horizontal and vertical components. The orientation of the horizontal vorticity vector along with the horizontal velocity vector influences the updraft rotation, especially during thunderstorms. Similarly, while focusing on the atmospheric circulation study, the vertical component of vorticity is considered, as it is related to divergence, vertical motion in the atmosphere, and horizontal vorticity.

## Geophysical Fluid Dynamics and Vorticity

In geophysical fluid dynamics, the vertical and horizontal components of vorticity are of great importance. To describe the low levels of the atmosphere, the horizontal component is utilized, whereas the vertical component plays a significant role whenever there are speed or direction changes in the wind.

The vorticity can be expressed in the inertial reference frame using the term absolute vorticity: Relative to the rotation of the earth, the vorticity is described using the following equation: While discussing the motion of fluid in Earth’s system, absolute vorticity and relative vorticity are two terms that are alternately used.

## Absolute Vorticity

The vertical component of vorticity is of great importance in large-scale dynamics. When spinning about the z axis, fluids resolve into vertical components of vorticity given by the equation: In the Earth-reference frame, the total vertical vorticity experienced by a fluid or fluid parcel is called absolute vorticity. The mathematical representation of absolute vorticity (ζA) can be given as: ζR is the relative vorticity and f is the Coriolis parameter.

### Coriolis Parameter

The Coriolis parameter is the quantity that accounts for the background rotation of the Earth. The Coriolis parameter is the vorticity imparted to the atmospheric air by the Earth’s surface. When describing Earth’s vorticity, the Coriolis parameter ‘f’ is a function of latitude. In the Northern Hemisphere, the value of ‘f’ is positive, whereas in the Southern Hemisphere, it is negative.

## Relative Vorticity

The relative vorticity can be generalized as the vorticity observed in the rotating frame. To determine the absolute vorticity in an inertial frame, an understanding of relative vorticity and the Coriolis parameter is required. The equation of absolute vorticity can be rewritten as: Comparing the last two equations of absolute vorticity, it can be concluded that relativity vorticity is equal to: In geophysical fluid dynamics, the relative vorticity is produced by the airflow through the curved path and wind shear. The relative vorticity can be positive or negative.

Relative vorticity due to wind shear or shear vorticity - When there are horizontal differences in the wind speed, shear vorticity arises.

Relative vorticity due to curved wind flow - When the air or fluid parcels acquire vorticity due to curved flow, it is called curvature vorticity.

## Measuring Relative Vorticity

Relative vorticity is measured with respect to the ground, which is in contrast to the absolute vorticity measurement, where the rotation of the Earth is taken as an additional component. Typically, the relative vorticity is expressed in rotation per second. Since rotation is a dimensionless quantity, the relative vorticity is expressed per second.

Cadence’s CFD software can help you determine the relative vorticity associated with fluid flow and assist you in modeling the vorticity in fluids. Cadence offers advanced solvers to solve vorticity-related problems in fluid dynamics.