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## Abstract

A formula for the computation of a solenoidal term, for example, **k** · (**
∇
**

*ψ*×

**ϒ) or ∂(**

**∇***ψ,*ϒ)/∂(

*x, y*) in Jacobian form, from three or more noncollinear stations or grid points is presented. The formula is based on a geometric interpretation of the solenoid as the ratio of an elementary area (expressed as a line integral) in

*ψ*–ϒ dependent-variable or phase space to the corresponding area in the

*x–y*plane. The Arakawa finite-difference Jacobian on a rectangular grid in physical space is shown to be a linear combination of such ratios. Thus the Arakawa method is a line-integral method. The new interpretation readily provides the forms of the Jacobian at boundary points needed to maintain the integral constraints in a closed domain. Elementary properties of Jacobians ensure that the Arakawa Jacobian can be used on a regular mesh in any general orthogonal curvilinear coordinate system, thus permitting the use of “stretched” grids that have a lesser density of grid points away from boundaries and also away from the axis if the flow is axisymmetric. The proper forms of the advective term on the axis of an axisymmetric flow and at a pole also are deduced using the line-integral approach.

## Abstract

A formula for the computation of a solenoidal term, for example, **k** · (**
∇
**

*ψ*×

**ϒ) or ∂(**

**∇***ψ,*ϒ)/∂(

*x, y*) in Jacobian form, from three or more noncollinear stations or grid points is presented. The formula is based on a geometric interpretation of the solenoid as the ratio of an elementary area (expressed as a line integral) in

*ψ*–ϒ dependent-variable or phase space to the corresponding area in the

*x–y*plane. The Arakawa finite-difference Jacobian on a rectangular grid in physical space is shown to be a linear combination of such ratios. Thus the Arakawa method is a line-integral method. The new interpretation readily provides the forms of the Jacobian at boundary points needed to maintain the integral constraints in a closed domain. Elementary properties of Jacobians ensure that the Arakawa Jacobian can be used on a regular mesh in any general orthogonal curvilinear coordinate system, thus permitting the use of “stretched” grids that have a lesser density of grid points away from boundaries and also away from the axis if the flow is axisymmetric. The proper forms of the advective term on the axis of an axisymmetric flow and at a pole also are deduced using the line-integral approach.

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## Abstract

A new technique for computing the wet-bulb potential temperature of a parcel and its temperature after pseudoadiabatic ascent or descent to a new pressure level is presented. It is based on inverting Bolton’s most accurate formula for equivalent potential temperature *θ _{E}
* to obtain the adiabatic wet-bulb temperature

*T*on a given pseudoadiabat at a given pressure by an iterative technique. It is found that

_{w}*T*is a linear function of equivalent temperature raised to the −1/

_{w}*κ*(i.e., −3.504) power, where

_{d}*κ*is the Poisson constant for dry air, in a significant region of a thermodynamic diagram. Consequently, Bolton’s formula is raised to the −1/

_{d}*κ*power prior to the solving. A good “initial-guess” formula for

_{d}*T*is devised. In the pressure range 100 ≤

_{w}*p*≤ 1050 mb, this guess is within 0.34 K of the converged solution for wet-bulb potential temperatures

*θ*≤ 40°C. Just one iteration reduces this relative error to less than 0.002 K for −20° ≤

_{w}*θ*≤ 40°C. The upper bound on the overall error in the computed

_{w}*T*after one iteration is 0.2 K owing to an inherent uncertainty in Bolton’s formula. With a few changes, the method also works for finding the temperature on water- or ice-saturation reversible adiabats.

_{w}The new technique is far more accurate and efficient than the Wobus method, which, although little known, is widely used in a software package. It is shown that, although the Wobus function, on which the Wobus method is based, is supposedly only a function of temperature, it has in fact a slight pressure dependence, which results in errors of up to 1.2 K in the temperature of a lifted parcel. This intrinsic inaccuracy makes the Wobus method far inferior to a new algorithm presented herein.

## Abstract

A new technique for computing the wet-bulb potential temperature of a parcel and its temperature after pseudoadiabatic ascent or descent to a new pressure level is presented. It is based on inverting Bolton’s most accurate formula for equivalent potential temperature *θ _{E}
* to obtain the adiabatic wet-bulb temperature

*T*on a given pseudoadiabat at a given pressure by an iterative technique. It is found that

_{w}*T*is a linear function of equivalent temperature raised to the −1/

_{w}*κ*(i.e., −3.504) power, where

_{d}*κ*is the Poisson constant for dry air, in a significant region of a thermodynamic diagram. Consequently, Bolton’s formula is raised to the −1/

_{d}*κ*power prior to the solving. A good “initial-guess” formula for

_{d}*T*is devised. In the pressure range 100 ≤

_{w}*p*≤ 1050 mb, this guess is within 0.34 K of the converged solution for wet-bulb potential temperatures

*θ*≤ 40°C. Just one iteration reduces this relative error to less than 0.002 K for −20° ≤

_{w}*θ*≤ 40°C. The upper bound on the overall error in the computed

_{w}*T*after one iteration is 0.2 K owing to an inherent uncertainty in Bolton’s formula. With a few changes, the method also works for finding the temperature on water- or ice-saturation reversible adiabats.

_{w}The new technique is far more accurate and efficient than the Wobus method, which, although little known, is widely used in a software package. It is shown that, although the Wobus function, on which the Wobus method is based, is supposedly only a function of temperature, it has in fact a slight pressure dependence, which results in errors of up to 1.2 K in the temperature of a lifted parcel. This intrinsic inaccuracy makes the Wobus method far inferior to a new algorithm presented herein.

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## Abstract

Several new formulas for pseudoadiabatic equivalent potential temperature (EPT) are devised and compared to previous ones. The maximum errors of all the formulas are determined from calculations on a dense grid of points in the region of a thermodynamic diagram defined by wet-bulb potential temperature ≤32°C (EPT ≤ 400 K) and pressure between 100 and 1050 mb. One of the new formulas has an accuracy of 0.015 K in the specified region. Finding the imitation first law of thermodynamics that they satisfy approximately reveals how the formulas work.

## Abstract

Several new formulas for pseudoadiabatic equivalent potential temperature (EPT) are devised and compared to previous ones. The maximum errors of all the formulas are determined from calculations on a dense grid of points in the region of a thermodynamic diagram defined by wet-bulb potential temperature ≤32°C (EPT ≤ 400 K) and pressure between 100 and 1050 mb. One of the new formulas has an accuracy of 0.015 K in the specified region. Finding the imitation first law of thermodynamics that they satisfy approximately reveals how the formulas work.

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## Abstract

Given wind data from three noncollinear observing stations, divergence and vorticity can be computed very efficiently by fitting a linear velocity field to the observed wind components. The four wind gradients and the four kinematic quantities (divergence, vorticity, and stretching and shearing deformation) can be expressed as simple algebraic functions of the station coordinates and the observed wind components. Computation of all eight quantities requires only 31 arithmetic operations. All the methods for computing divergence from three stations (linear fitting, Bellamy's graphical method, the line-integral method, and the linear vector point function method) are shown to be equivalent. The fitting method is extended to a six-station network, using a quadratic velocity field to fit the data. Apart from the inversion of a 6×6 matrix, which needs to be performed only once for a fixed network geometry, the solution is again simple. It is shown that the 6×6 matrix is singular when the stations all lie on a conic section.

For an *n*-sided (*n*>3) polygon of stations, simple formulas for divergence and the other quantities are obtained from the generalized, piecewise-linear, Bellamy, and line-integral methods, which again are found to be equivalent. Also presented are least-squares solutions for both the linear and quadratic fitting methods for networks with more than three and six stations, respectively. Apart from the elements of the square matrices and column vectors being summations instead of individual-station quantities, the least-squares solutions have the same form as the exact ones. Simplified versions of the formulas are presented for networks configured in the form of regular polygons with an interior station at the center included to eliminate the singularity of the quadratic method. In this special case, the 6×6 matrix can be inverted analytically. For such networks, all the methods give the same value of mean divergence, which is independent of the central observation. Tests involving observations from two particular networks of an analytical wind field show that the computed divergence is a much better estimate of the mean divergence over the network than of the divergence at the centroid. The general simplicity of the analytical formulas for the derived quantities permits analysis of errors due to random wind observing errors. Truncation errors also are discussed.

## Abstract

Given wind data from three noncollinear observing stations, divergence and vorticity can be computed very efficiently by fitting a linear velocity field to the observed wind components. The four wind gradients and the four kinematic quantities (divergence, vorticity, and stretching and shearing deformation) can be expressed as simple algebraic functions of the station coordinates and the observed wind components. Computation of all eight quantities requires only 31 arithmetic operations. All the methods for computing divergence from three stations (linear fitting, Bellamy's graphical method, the line-integral method, and the linear vector point function method) are shown to be equivalent. The fitting method is extended to a six-station network, using a quadratic velocity field to fit the data. Apart from the inversion of a 6×6 matrix, which needs to be performed only once for a fixed network geometry, the solution is again simple. It is shown that the 6×6 matrix is singular when the stations all lie on a conic section.

For an *n*-sided (*n*>3) polygon of stations, simple formulas for divergence and the other quantities are obtained from the generalized, piecewise-linear, Bellamy, and line-integral methods, which again are found to be equivalent. Also presented are least-squares solutions for both the linear and quadratic fitting methods for networks with more than three and six stations, respectively. Apart from the elements of the square matrices and column vectors being summations instead of individual-station quantities, the least-squares solutions have the same form as the exact ones. Simplified versions of the formulas are presented for networks configured in the form of regular polygons with an interior station at the center included to eliminate the singularity of the quadratic method. In this special case, the 6×6 matrix can be inverted analytically. For such networks, all the methods give the same value of mean divergence, which is independent of the central observation. Tests involving observations from two particular networks of an analytical wind field show that the computed divergence is a much better estimate of the mean divergence over the network than of the divergence at the centroid. The general simplicity of the analytical formulas for the derived quantities permits analysis of errors due to random wind observing errors. Truncation errors also are discussed.

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## Abstract

A nonlinear formula for updraft motion in supercell storms is derived from Petterssen's formula for the motion of systems and the vertical equation of motion, and tested on form-preserving disturbances. At each level, continuous propagation of an updraft maximum is determined largely by the horizontal gradient of the nonhydrostatic vertical pressure-gradient force (NHVPGF) at the updraft center. The NHVPGF is deduced from the formal solution of the Poisson equation for nonhydrostatic pressure in anelastic flow subject to homogeneous Neumann boundary conditions at the ground and top boundary. Recourse also is made to published fields of partitioned vertical pressure-gradient force. Updraft motion is partitioned into parts forced by horizontal gradients of hydrostatic pressure, linear interaction between the environmental shear and updraft, and nonlinear dynamical effects.

The dynamics of supercell storms for nearly straight and highly curved hodographs are found to be different. Nonlinear rotationally induced propagation is important during storm splitting in fairly unidirectional shear where the vortex pair, formed at midlevels by lifting of environmental vortex tubes, straddles the initial updraft. After the initial storm splits into severe right-moving (SR) and left-moving (SL) supercells, anomalous motion is maintained by the distribution of the NHVPGF. For the SR storm, the NHVPGF is upward below the cyclonic vortex on the right side of the updraft and downward on the left side. The anticyclonic vortex on the left side of this storm migrates to the downdraft and so does not affect updraft propagation. For a storm in shear that turns markedly clockwise with height, the cyclonic vortex is nearly coincident with the updraft, while the anticyclonic vortex is located in the downdraft so that the horizontal gradient of nonlinear NHVPGF at the updraft center associated with rotationally induced propagation is relatively small. Linear shear-induced propagation now becomes the dominant mechanism. At each level, propagation off the hodograph to the concave side increases with updraft width. Once the propagation has been deduced, tilting of storm-relative environmental streamwise vorticity explains the origins of overall updraft rotation in all cases.

## Abstract

A nonlinear formula for updraft motion in supercell storms is derived from Petterssen's formula for the motion of systems and the vertical equation of motion, and tested on form-preserving disturbances. At each level, continuous propagation of an updraft maximum is determined largely by the horizontal gradient of the nonhydrostatic vertical pressure-gradient force (NHVPGF) at the updraft center. The NHVPGF is deduced from the formal solution of the Poisson equation for nonhydrostatic pressure in anelastic flow subject to homogeneous Neumann boundary conditions at the ground and top boundary. Recourse also is made to published fields of partitioned vertical pressure-gradient force. Updraft motion is partitioned into parts forced by horizontal gradients of hydrostatic pressure, linear interaction between the environmental shear and updraft, and nonlinear dynamical effects.

The dynamics of supercell storms for nearly straight and highly curved hodographs are found to be different. Nonlinear rotationally induced propagation is important during storm splitting in fairly unidirectional shear where the vortex pair, formed at midlevels by lifting of environmental vortex tubes, straddles the initial updraft. After the initial storm splits into severe right-moving (SR) and left-moving (SL) supercells, anomalous motion is maintained by the distribution of the NHVPGF. For the SR storm, the NHVPGF is upward below the cyclonic vortex on the right side of the updraft and downward on the left side. The anticyclonic vortex on the left side of this storm migrates to the downdraft and so does not affect updraft propagation. For a storm in shear that turns markedly clockwise with height, the cyclonic vortex is nearly coincident with the updraft, while the anticyclonic vortex is located in the downdraft so that the horizontal gradient of nonlinear NHVPGF at the updraft center associated with rotationally induced propagation is relatively small. Linear shear-induced propagation now becomes the dominant mechanism. At each level, propagation off the hodograph to the concave side increases with updraft width. Once the propagation has been deduced, tilting of storm-relative environmental streamwise vorticity explains the origins of overall updraft rotation in all cases.

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## Abstract

This paper investigates whether the descending rain curtain associated with the hook echo of a supercell can instigate a tornado through a purely barotropic mechanism. A simple numerical model of a mesocyclone is constructed in order to rule out other tornadogenesis mechanisms in the simulations. The flow is axisymmetric and Boussinesq with constant eddy viscosity in a neutrally stratified environment. The domain is closed to avoid artificial decoupling of a vortex from the storm-scale circulation. In the principal simulation, the initial condition is a balanced, slowly decaying, Beltrami flow describing an updraft that is rotating cyclonically at midlevels around a low pressure center surrounded by a concentric downdraft that revolves cyclonically but has anticyclonic vorticity. The boundary conditions are no slip on the tangential wind and free slip on the radial or vertical wind to accommodate this initial condition and to allow strong interaction of a vortex with the ground.

Precipitation is released through the top above the updraft and falls to the ground near the updraft–downdraft interface in an annular curtain. The downdraft enhancement induced by the precipitation drag upsets the balance of the Beltrami flow. The downdraft and its outflow toward the axis increase low-level convergence beneath the updraft and transport air with moderately high angular momentum downward and inward where it is entrained and stretched by the updraft. The resulting tornado has a corner region with an intense axial jet and low pressure capped by a vortex breakdown and a transition to a broader vortex aloft (a tornado cyclone). A clear slot of subsiding air with anticyclonic vorticity surrounds the vortex. The vertical kinetic energy of the entire circulation declines dramatically prior to tornado formation.

## Abstract

This paper investigates whether the descending rain curtain associated with the hook echo of a supercell can instigate a tornado through a purely barotropic mechanism. A simple numerical model of a mesocyclone is constructed in order to rule out other tornadogenesis mechanisms in the simulations. The flow is axisymmetric and Boussinesq with constant eddy viscosity in a neutrally stratified environment. The domain is closed to avoid artificial decoupling of a vortex from the storm-scale circulation. In the principal simulation, the initial condition is a balanced, slowly decaying, Beltrami flow describing an updraft that is rotating cyclonically at midlevels around a low pressure center surrounded by a concentric downdraft that revolves cyclonically but has anticyclonic vorticity. The boundary conditions are no slip on the tangential wind and free slip on the radial or vertical wind to accommodate this initial condition and to allow strong interaction of a vortex with the ground.

Precipitation is released through the top above the updraft and falls to the ground near the updraft–downdraft interface in an annular curtain. The downdraft enhancement induced by the precipitation drag upsets the balance of the Beltrami flow. The downdraft and its outflow toward the axis increase low-level convergence beneath the updraft and transport air with moderately high angular momentum downward and inward where it is entrained and stretched by the updraft. The resulting tornado has a corner region with an intense axial jet and low pressure capped by a vortex breakdown and a transition to a broader vortex aloft (a tornado cyclone). A clear slot of subsiding air with anticyclonic vorticity surrounds the vortex. The vertical kinetic energy of the entire circulation declines dramatically prior to tornado formation.

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## Abstract

An expression is obtained for effective buoyancy per unit volume in inviscid anelastic flow with surrounding horizontal density variations. Effective buoyancy is defined here as the statically forced part of the locally nonhydrostatic, upward pressure-gradient force because this is the part of the net vertical force that is both absolute (i.e., independent of an arbitrary base state) and dependent solely on variations in the specific weight of the air–hydrometeor mixture. For the case where the domain is the half space *z* > 0 above flat ground, effective buoyancy *β*(**x**) at a given field point **x** is given by *β*(**x**) = ^{∞}
_{−∞}
^{∞}
_{−∞}
^{∞}
_{−∞}
*g*
^{2}
_{
H
}
*ρ*
_{
T
}(**x̂**)]/(4*π*|**x** − **x̂**|) *dx̂*
*dŷ*
*dẑ,* where *gρ*
_{
T
} is the total weight of air and hydrometeors per unit volume, the **x̂** ≡ (*x̂,*
*ŷ,*
*ẑ*) are source points, ^{2}
_{
H
}
**x̂**, and ^{2}
_{
H
}
*ρ*
_{
T
}(*x̂,*
*ŷ,* −*ẑ*) ≡ −^{2}
_{
H
}
*ρ*
_{
T
}(*x̂,*
*ŷ,*
*ẑ*).

## Abstract

An expression is obtained for effective buoyancy per unit volume in inviscid anelastic flow with surrounding horizontal density variations. Effective buoyancy is defined here as the statically forced part of the locally nonhydrostatic, upward pressure-gradient force because this is the part of the net vertical force that is both absolute (i.e., independent of an arbitrary base state) and dependent solely on variations in the specific weight of the air–hydrometeor mixture. For the case where the domain is the half space *z* > 0 above flat ground, effective buoyancy *β*(**x**) at a given field point **x** is given by *β*(**x**) = ^{∞}
_{−∞}
^{∞}
_{−∞}
^{∞}
_{−∞}
*g*
^{2}
_{
H
}
*ρ*
_{
T
}(**x̂**)]/(4*π*|**x** − **x̂**|) *dx̂*
*dŷ*
*dẑ,* where *gρ*
_{
T
} is the total weight of air and hydrometeors per unit volume, the **x̂** ≡ (*x̂,*
*ŷ,*
*ẑ*) are source points, ^{2}
_{
H
}
**x̂**, and ^{2}
_{
H
}
*ρ*
_{
T
}(*x̂,*
*ŷ,* −*ẑ*) ≡ −^{2}
_{
H
}
*ρ*
_{
T
}(*x̂,*
*ŷ,*
*ẑ*).

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## Abstract

Formulas in an Eulerian framework are presented for the absolute velocity and vorticity of individual parcels in inviscid isentropic flow. The analysis is performed in a rectangular Cartesian rotating coordinate system. The dependent variables are the Lagrangian coordinates, initial velocities, cumulative temperature, entropy, and a potential. The formulas are obtained in two different ways. The first method is based on finding a matrix integrating factor for the Euler equations of motion and a propagator for the vector vorticity equation. The second method is a variational one. Hamilton’s principle of least action is used to minimize the fluid’s absolute kinetic energy minus its internal energy and potential energy subject to the Lin constraints and constraints of mass and entropy conservation. In the first method, the friction and diabatic heating terms in the governing equations are carried along in integrands so that the generalized formulas lead to Eckart’s circulation theorem. Using them to derive other circulation theorems, the helicity-conservation theorem, and Cauchy’s formula for the barotropic vorticity checks the formulas further.

The formulas are suitable for generating diagnostic fields of barotropic and baroclinic vorticity in models if some simple auxiliary equations are added to the model and integrated stably forward in time alongside the model equations.

## Abstract

Formulas in an Eulerian framework are presented for the absolute velocity and vorticity of individual parcels in inviscid isentropic flow. The analysis is performed in a rectangular Cartesian rotating coordinate system. The dependent variables are the Lagrangian coordinates, initial velocities, cumulative temperature, entropy, and a potential. The formulas are obtained in two different ways. The first method is based on finding a matrix integrating factor for the Euler equations of motion and a propagator for the vector vorticity equation. The second method is a variational one. Hamilton’s principle of least action is used to minimize the fluid’s absolute kinetic energy minus its internal energy and potential energy subject to the Lin constraints and constraints of mass and entropy conservation. In the first method, the friction and diabatic heating terms in the governing equations are carried along in integrands so that the generalized formulas lead to Eckart’s circulation theorem. Using them to derive other circulation theorems, the helicity-conservation theorem, and Cauchy’s formula for the barotropic vorticity checks the formulas further.

The formulas are suitable for generating diagnostic fields of barotropic and baroclinic vorticity in models if some simple auxiliary equations are added to the model and integrated stably forward in time alongside the model equations.

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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.

## 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.

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## Abstract

In Part I, a general integral of the 2D vorticity equation was obtained. This is a formal solution for the vorticity of a moving tube of air in a 2D unsteady stratified shear flow with friction. This formula is specialized here to various types of 2D flow. For steady inviscid flow, the integral reduces to an integral found by Moncrieff and Green if the flow is Boussinesq and to one obtained by Lilly if the flow is isentropic. For steady isentropic frictionless motion of clear air, several quantities that are invariant along streamlines are found. These invariants provide another way to obtain Lilly’s integral from the general integral.

## Abstract

In Part I, a general integral of the 2D vorticity equation was obtained. This is a formal solution for the vorticity of a moving tube of air in a 2D unsteady stratified shear flow with friction. This formula is specialized here to various types of 2D flow. For steady inviscid flow, the integral reduces to an integral found by Moncrieff and Green if the flow is Boussinesq and to one obtained by Lilly if the flow is isentropic. For steady isentropic frictionless motion of clear air, several quantities that are invariant along streamlines are found. These invariants provide another way to obtain Lilly’s integral from the general integral.