# Integrals of the Vorticity Equation. Part I: General Three- and Two-Dimensional Flows

Robert Davies-Jones NOAA/National Severe Storms Laboratory, Norman, Oklahoma

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## Abstract

The integral of the vector vorticity equation for the vorticity of a moving parcel in 3D baroclinic flow with friction is cast in a new form. This integral of the vorticity equation applies to synoptic-scale or mesoscale flows and to deep compressible or shallow Boussinesq motions of perfectly clear or universally saturated air. The present integral is equivalent to that of Epifanio and Durran in the Boussinesq limit, but its simpler form reduces easily to Dutton’s integral when the flow is assumed to be isentropic and frictionless.

The integral for vorticity has the following physical interpretation. The vorticity of a parcel is composed of barotropic vorticity; baroclinic vorticity, which originates from solenoidal generation; and vorticity stemming from frictional generation. Its barotropic vorticity is the result of freezing into the fluid the w field (specific volume times vorticity) that is present at the initial time. Its baroclinic vorticity is the vector sum of contributions from small subintervals of time that partition the interval between initial and current times. In each subinterval, the baroclinic torque generates a small vector element of vorticity and hence w. The contribution to the current baroclinic vorticity is the result of freezing this element of w into the fluid immediately after its formation. The physical interpretation of vorticity owing to frictional generation is identical except the torque is frictional rather than solenoidal.

The baroclinic vorticity is decomposed into a part that would occur if the current entropy of the flow were conserved materially backward in time to the initial time and an adjustment term that accounts for production of entropy gradients in material coordinates during this interval. A method for computing all the vorticity parts in an Eulerian framework within a 3D numerical model is outlined.

The usefulness of the 3D vorticity integral is demonstrated further by deriving Eckart’s, Bjerknes’s, and Kelvin’s circulation theorems from it in relatively few steps, and by showing that the associated expression for potental vorticity is an integral of the potential vorticity equation and implies conservation of potential vorticity for isentropic frictionless motion of clear air (Ertel’s theorem). Last, a formula for the helicity density of a parcel is obtained from the vorticity integral and an expression for the parcel’s velocity, and is verified by proving that it is an integral of the equation for helicity density.

Corresponding author address: Dr. Robert Davies-Jones, NOAA/National Severe Storms Laboratory, 1313 Halley Circle, Norman, OK 73069-8493. Email: Bob.Davies-Jones@noaa.gov

## Abstract

The integral of the vector vorticity equation for the vorticity of a moving parcel in 3D baroclinic flow with friction is cast in a new form. This integral of the vorticity equation applies to synoptic-scale or mesoscale flows and to deep compressible or shallow Boussinesq motions of perfectly clear or universally saturated air. The present integral is equivalent to that of Epifanio and Durran in the Boussinesq limit, but its simpler form reduces easily to Dutton’s integral when the flow is assumed to be isentropic and frictionless.

The integral for vorticity has the following physical interpretation. The vorticity of a parcel is composed of barotropic vorticity; baroclinic vorticity, which originates from solenoidal generation; and vorticity stemming from frictional generation. Its barotropic vorticity is the result of freezing into the fluid the w field (specific volume times vorticity) that is present at the initial time. Its baroclinic vorticity is the vector sum of contributions from small subintervals of time that partition the interval between initial and current times. In each subinterval, the baroclinic torque generates a small vector element of vorticity and hence w. The contribution to the current baroclinic vorticity is the result of freezing this element of w into the fluid immediately after its formation. The physical interpretation of vorticity owing to frictional generation is identical except the torque is frictional rather than solenoidal.

The baroclinic vorticity is decomposed into a part that would occur if the current entropy of the flow were conserved materially backward in time to the initial time and an adjustment term that accounts for production of entropy gradients in material coordinates during this interval. A method for computing all the vorticity parts in an Eulerian framework within a 3D numerical model is outlined.

The usefulness of the 3D vorticity integral is demonstrated further by deriving Eckart’s, Bjerknes’s, and Kelvin’s circulation theorems from it in relatively few steps, and by showing that the associated expression for potental vorticity is an integral of the potential vorticity equation and implies conservation of potential vorticity for isentropic frictionless motion of clear air (Ertel’s theorem). Last, a formula for the helicity density of a parcel is obtained from the vorticity integral and an expression for the parcel’s velocity, and is verified by proving that it is an integral of the equation for helicity density.

Corresponding author address: Dr. Robert Davies-Jones, NOAA/National Severe Storms Laboratory, 1313 Halley Circle, Norman, OK 73069-8493. Email: Bob.Davies-Jones@noaa.gov

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