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## Abstract

The area effectively covered by dual-Doppler radars is presented as a function of spatioal resolution and accuracy of horizontal velocity measurement. Implications of this relationship for the spacing of a dual-Doppler network tornadic storm research are discussed.

## Abstract

The area effectively covered by dual-Doppler radars is presented as a function of spatioal resolution and accuracy of horizontal velocity measurement. Implications of this relationship for the spacing of a dual-Doppler network tornadic storm research are discussed.

## 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.

## 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.

## 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.

## Abstract

A formula is derived for the rate of change of circulation around an updraft perimeter at a constant elevation. This quantity depends on the continuous propagation of points on the edge, so an expression for local propagation of the edge is obtained from Petterssen's formula for the motion of an isopleth and the vertical equation of motion. On the edge of an updraft in inviscid anelastic flow, the local propagation velocity along the outward normal is equal to the local nonhydrostatic vertical pressure-gradient force (NHVPGF) divided by the magnitude of the local vertical-velocity gradient. Circulation around an updraft perimeter increases at a rate equal to the line integral around the edge of vertical vorticity times the outward propagation velocity. Formulas are also found for the propagation of an updraft's centroid at a given height and for the acceleration of an updraft's vertical helicity. All of the formulas are tested on exact Beltrami-flow solutions of the governing equations.

The relevance of two paradigms of supercell dynamics to local edge propagation and circulation growth of updrafts is evaluated by decomposing the NHVPGF into linearly and nonlinearly forced parts and examining results of supercell simulations in different types of shear. Propagation across the shear and rate of increase of circulation depend mostly on the nonlinear part of the NHVPGF (as in the vertical-wind-shear paradigm) for updrafts in nearly unidirectional shear and on the linear part (as in the helicity paradigm) for updrafts in shear that turns markedly with height.

## Abstract

A formula is derived for the rate of change of circulation around an updraft perimeter at a constant elevation. This quantity depends on the continuous propagation of points on the edge, so an expression for local propagation of the edge is obtained from Petterssen's formula for the motion of an isopleth and the vertical equation of motion. On the edge of an updraft in inviscid anelastic flow, the local propagation velocity along the outward normal is equal to the local nonhydrostatic vertical pressure-gradient force (NHVPGF) divided by the magnitude of the local vertical-velocity gradient. Circulation around an updraft perimeter increases at a rate equal to the line integral around the edge of vertical vorticity times the outward propagation velocity. Formulas are also found for the propagation of an updraft's centroid at a given height and for the acceleration of an updraft's vertical helicity. All of the formulas are tested on exact Beltrami-flow solutions of the governing equations.

The relevance of two paradigms of supercell dynamics to local edge propagation and circulation growth of updrafts is evaluated by decomposing the NHVPGF into linearly and nonlinearly forced parts and examining results of supercell simulations in different types of shear. Propagation across the shear and rate of increase of circulation depend mostly on the nonlinear part of the NHVPGF (as in the vertical-wind-shear paradigm) for updrafts in nearly unidirectional shear and on the linear part (as in the helicity paradigm) for updrafts in shear that turns markedly with height.

## Abstract

In a prior paper, insights into tornadogenesis in supercell storms were gained by discovering analytical formulas for vorticity variations along streamlines in idealized, steady, frictionless, isentropic inflows of dry air imported from a horizontally uniform environment. This work is simplified and extended to the evolution of parcel vorticity in unsteady, nonisentropic flows by integrating the vorticity equation using nonorthogonal Lagrangian coordinates. The covariant basis vectors

### Significance Statement

Air parcels rising in a tornado spin rapidly about their direction of motion. Theory herein describes the processes that can produce this streamwise spin in supercells. Cyclonic updraft rotation originates from strong low-level environmental storm-relative winds that turn clockwise with height. Parcels flowing into the updraft have initially large streamwise spins that are amplified by streamwise stretching. Rain curtains falling through the cyclonic updraft cause other parcels to descend and turn leftward. Buoyancy and frictional torques give them horizontal spin. Even if these spins are transverse to the flow initially, they are turned streamwise by secondary flow that develops in left-hand bends. As the parcels reach the ground and converge into the tornado, streamwise stretching greatly magnifies their streamwise spins.

## Abstract

In a prior paper, insights into tornadogenesis in supercell storms were gained by discovering analytical formulas for vorticity variations along streamlines in idealized, steady, frictionless, isentropic inflows of dry air imported from a horizontally uniform environment. This work is simplified and extended to the evolution of parcel vorticity in unsteady, nonisentropic flows by integrating the vorticity equation using nonorthogonal Lagrangian coordinates. The covariant basis vectors

### Significance Statement

Air parcels rising in a tornado spin rapidly about their direction of motion. Theory herein describes the processes that can produce this streamwise spin in supercells. Cyclonic updraft rotation originates from strong low-level environmental storm-relative winds that turn clockwise with height. Parcels flowing into the updraft have initially large streamwise spins that are amplified by streamwise stretching. Rain curtains falling through the cyclonic updraft cause other parcels to descend and turn leftward. Buoyancy and frictional torques give them horizontal spin. Even if these spins are transverse to the flow initially, they are turned streamwise by secondary flow that develops in left-hand bends. As the parcels reach the ground and converge into the tornado, streamwise stretching greatly magnifies their streamwise spins.

## Abstract

Abstract not available.

## Abstract

Abstract not available.

## 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*).

## 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.