Formulas for Parcel Velocity and Vorticity in a Rotating Cartesian Coordinate System

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

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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.

Corresponding author address: Dr. Robert Davies-Jones, Doggetts Farm, New Street, Stradbroke, Eye, Suffolk IP21 5JG, United Kingdom. E-mail: bobdj1066@yahoo.com

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.

Corresponding author address: Dr. Robert Davies-Jones, Doggetts Farm, New Street, Stradbroke, Eye, Suffolk IP21 5JG, United Kingdom. E-mail: bobdj1066@yahoo.com

1. Introduction

Understanding of atmospheric vortices, such as tornadoes [see reviews by Davies-Jones et al. (2001) and Davies-Jones (2015)], lee vortices (Smolarkiewicz and Rotunno 1989; Davies-Jones 2000), and larger-scale cyclones (Lackmann 2011, 101–102) often involves determining the mechanisms by which air parcels obtain large vorticities. One approach to investigating tornadogenesis is to use a “bare-bones computer model” that forms a tornado (Davies-Jones 2008). The results are easy to interpret, but a loss of realism naturally comes with the simplifying assumptions. More realistic three-dimensional models of supercell storms produce tornadoes in favorable environments, but the origins of these simulated vortices are difficult to decipher, because these models are complex and use Eulerian coordinates, whereas the laws governing vorticity are Lagrangian in nature (Salmon 1988, p. 226). The modeler usually resorts to computing how circulation evolves around a material circuit drawn around the near-ground vorticity maximum and traced back to an arbitrary “initial time” using computed backward parcel trajectories (Rotunno and Klemp 1985; Davies-Jones and Brooks 1993; Adlerman et al. 1999; Markowski et al. 2012). This method allows the analyst to determine the barotropic (i.e., the initial) circulation and the change in circulation around the circuit, which is the baroclinic circulation plus circulation generated by frictional torque (when significant). Computational restraints generally limit the analysis to only one circuit. A diagnostic method using millions of forward trajectories has been developed recently (Dahl et al. 2014), but the resulting evaluations of barotropic and nonbarotropic vorticity (the latter computed as the residual vorticity) are along individual trajectories, which are spaced irregularly in Eulerian coordinates. The barotropic and baroclinic vorticity cannot be presented easily as fields by this method.

This paper develops formulas for the velocity and vorticity of an individual parcel in a general flow with friction and diabatic heating that can be used to generate the baroclinic- and barotropic vorticity fields if auxiliary equations are added to the model and integrated forward in time alongside the model equations. The formulas are relatively simple if the flow is isentropic and inviscid. We use a rectangular Cartesian coordinate system that is rotating with Earth rather than the inertial systems used in previous work and assume that all fields are continuous and differentiable. The Lagrangian coordinates and initial velocities appear in the formulas as dependent variables. We find a matrix integrating factor for integration of the Euler equations of motion and a propagator for integration of the vector vorticity equation. We then show how the formulas can be obtained variationally from Hamilton’s principle of least action. We check the formulas by showing that the barotropic part of the vorticity is equivalent to Cauchy’s formula (Dutton 1976, p. 385) and by using them to derive Ertel’s potential vorticity theorem, the helicity-conservation theorem, and the circulation theorems.

2. Previous work

The quest for these formulas starts with the fact that a velocity field (or any other vector field) u and its curl ζ can be represented locally by
e1
e2
(Truesdell 1954) where ϕ, ψ, and χ are called Clebsch potentials (Lamb 1932; Serrin 1959) or Monge potentials (Truesdell 1954; Aris 1962). These solutions predict that flows in which all the vortex tubes are closed have no helicity (Bretherton 1970; Salmon 1988, p. 241), because
e3
(Here, Ξ is the fluid volume, Σ is its bounding surface, and n is the outward unit normal to the surface.) Thus, they fail to represent all flows. For instance, they exclude the Beltrami flows in closed domains (e.g., Davies-Jones 2008). Consequently, the representation (1) is generally local rather than global. We can make the formulas global by adding more terms such that
e4
e5
According to Salmon (1988, 236–237), two is the minimum value of N for representing either a general homentropic (uniform entropy) or a general isentropic flow.
For further elaboration, we need the following definitions. The Eulerian coordinates are (x, y, z, t), where t is the current time and xxi + yj + zk is the position vector in terms of eastward, northward, and upward unit vectors i, j, and k. The corresponding Lagrangian coordinates are (X, Y, Z, τ), where τ is the symbol for time in the Lagrangian system, and XXi + Yj + Zk is the initial position vector of a parcel at the initial time τ0. The coordinates are measured from the origin of a tangent plane, which is rotating with the earth. The trajectory of a parcel is denoted by
e6
where the tilde denotes quantities at time . The current and initial velocities of a parcel are u ≡ (u, υ, w) and U ≡ (U, V, W), respectively. The gradient operator in Lagrangian coordinates is
e7
and it is
e8
at an intermediate time in space. The material derivative in the rotating frame is D/Dt in the Eulerian framework and ∂/∂τ in the Lagrangian one. Thus, ∂x/∂τ = u = ux + υy + wz, ∂X/∂τ = ∂U/∂τ = 0, and . The potential energy of a parcel is gz, where g is the gravitational acceleration. The thermodynamic properties of a parcel are its entropy S; its specific volume α; its density ρ = 1/α; its pressure p; its enthalpy cpT, where T is its temperature and cp is the specific heat at constant pressure; and its internal energy EcυT, where cυ is the specific heat at constant volume (Salmon 1988, p. 47). By the ideal gas law, = RT, where R = cpcυ is the gas constant for air. The internal energy is related to α and S by
e9
Consequently,
e10
Ertel’s potential vorticity is αωS. It is conserved following a parcel in inviscid isentropic flow.
In inertial reference frames, Serrin (1959), Dutton (1976), Mobbs (1981), Epifanio and Durran (2002), and Davies-Jones (2000, 2006, hereafter DJ06) found integrals of the vector vorticity equation for inviscid isentropic flow and decomposed the vorticity ω into a barotropic part ωBT and a baroclinic part ωBC. Serrin generalized Weber’s transformed equations of motion to isentropic flow by adding an additional term. Transformation back to Eulerian coordinates results in the N = 4 formula for parcel velocity:
e11
(Mobbs 1981), where Λ is the parcel’s cumulative temperature or the integral over time of its temperature T from the initial time τ0 to the current time τ, Ψ is defined by DΨ/DtcpT + gzu u/2, Ψ = 0 initially. By definition, Λ = 0 initially, and DΛ/Dt = T. The corresponding vorticity formula is
e12
where the last term is the baroclinic vorticity, and the remaining terms on the right are the barotropic vorticity. Alternatively, the barotropic vorticity is expressed by Cauchy’s formula, which dates back to 1815 (Dutton 1976, p. 385). In general, there is a frictional part to vorticity. For nonisentropic flows with friction, Epifanio and Durran (2002) and DJ06 derived more complicated formulas involving propagators, also known as state-transition matrices.
Using variational analysis in the Eulerian framework with constraints of entropy and mass conservation but without the Lin constraints, Sasaki (2014) obtained formulas of the types of equations (1) and (2) for inviscid isentropic flow: namely,
e13
e14
where ϕ and Λ (α and β in Sasaki’s notation) are Lagrange multipliers for mass and entropy conservation, and S is entropy. The rotational velocity is not unique because it contains an arbitrary irrotational part that can be exchanged with the potential term. For example, an equally valid decomposition of the wind is
e15
Since N = 1, these formulas must apply only to a subset of solutions of the perfect-fluid equations (Salmon 1988, p. 234). The vorticity in (14) is, in fact, the baroclinic vorticity in isentropic flow ωBC (Dutton 1976). The formulas exclude homentropic flows with barotropic (i.e., initial) vorticity, all flows with nonzero potential vorticity, and some flows with nonzero helicity (see above). The barotropic vorticity is missing because it depends only on the initial and current states of the flow, which is when the variations vanish, not on the flow configurations at intermediate times, which is when variations are allowed.
The Lin constraints are
e16
where a ≡ (a, b, c) is a unique parcel label or identifier. Justification for the Lin constraints is provided in section 7. Classically, the initial position vector (X, Y, Z) is used to identify each parcel, but other conserved quantities can serve as labels provided that the labeling is unique. Salmon (1988, 1998, 327–329) included the Lin constraints, with (X, Y, S) as the labels and obtained the following compact formulas with N = 2:
e17
e18
Even though the variations are still assumed to vanish at the beginning and end times, the initial velocity field enters the analysis as the Lagrange multipliers of the Lin constraints (see section 7). Thus, (18) allows barotropic vorticity and nonzero potential vorticity. However, the physical significance of the apparently nonmeteorological Lagrange multiplier A associated with the X label constraint is unclear, and there are generally insufficient degrees of freedom to enter a parcel’s 3D initial position when N = 2. Consequently, the formula does not seem useful for following the evolution of a parcel’s vorticity.

Dellar (2011) used variational analysis in the Hamiltonian framework to derive equations of motion for generalized β planes that incorporate the vertical and horizontal components of the rotation vector and their changes with latitude. Since they are derived from Hamilton’s principle, the equations conserve energy, angular momentum, and potential vorticity.

3. The velocity due to the earth’s rotation in a tangent plane

To find a formula for parcel velocity in a nontraditional f plane (i.e., a tangent plane in solid-body rotation with both a horizontal and vertical component) or a nontraditional β plane, we first need a formula for the planetary velocity (the velocity relative to the fixed stars of a point in the atmosphere that rotates with the earth). The planetary velocity is the vector
e19
where Ω ≡ Ω(cosϕj + sinϕk) is the earth’s angular velocity (with magnitude Ω), rx + ak is the position vector from the earth’s center, xxi + yj + zk is the position vector from the tangent plane’s origin (at r = ak), ϕ is latitude, and a is the earth’s radius. Note that uE is a vector potential for 2Ω because × uE = 2Ω. The derivative of (19) following a parcel is
e20
where D/Dt is the material derivative in the rotating frame of the earth. (Note that DuE/Dt ≠ 0, despite ∂uE/∂t = 0.) Thus, the material derivative of uE supplies one-half of the Coriolis acceleration in the derivation of the equations of motion.
On a nontraditional f plane, the earth’s angular velocity is the constant value Ω(0, cosϕ0, sinϕ0), where ϕ0 is the latitude at the origin. In terms of the Coriolis parameters f0 ≡ 2Ω sinϕ0 and κ0 ≡ 2Ω cosϕ0, the planetary velocity on the f plane is
e21
where R0 = a cosϕ0 is the distance from the earth’s axis to the origin. This velocity field is one of solid-body rotation with planetary vorticity (0, κ0, f0). It satisfies (19) and (20).
In Grimshaw’s (1975) nontraditional β-plane approximation, the horizontal component κ of the Coriolis parameter is constant (=κ0), and the vertical component f varies with latitude according to f = f0 + βy, where βκ0/a. In Cartesian geometry, uE must now have deformation and thus cannot be a field of solid-body rotation. Consequently, it cannot be obtained from (19) or as a solution of (20). It can only be determined up to the additive gradient of an indeterminate potential βχ(x, y) because a β plane is unphysical. For the analysis in section 6, it is convenient to define
e22
where Ff0y + βy2/2. The planetary velocity is uP = uβ + βχ(x, y). This choice of planetary velocity is permissible because it reduces to uf in the case β = 0 and yields the correct planetary vorticity 2Ω = (0, κ0, f). Dellar (2011) defined a vector potential for 2Ω [his (B4)] that, for Grimshaw’s β plane, is compatible with uP.
The absolute velocity of a parcel is
e23
and its initial value is
e24
where
e25
and is the parcel’s initial value of χ. The initial absolute vorticity is
e26

4. Useful matrices

In this section, we introduce some important matrices and investigate their properties. We can think of the Lagrangian coordinates X, Y, and Z either as curvilinear coordinates that are dragged by the flow through location space, the space with Eulerian coordinates (x, y, z) as a Cartesian system, or as Cartesian coordinates in label (X) space (Salmon 1998, p. 5). By the chain rule,
e27
where
e28
is the Jacobian matrix of the transformation ≡ x(X, τ) from Lagrangian to Eulerian coordinates. For consistency, we use square brackets for matrices throughout this paper. The columns of , ∂x/∂X, ∂x/∂Y, and ∂x/∂Z, which are tangent to the coordinate curves of X, Y, and Z, respectively, are the covariant basis vectors e1, e2, and e3 (Margenau and Murphy 1956, p. 193). This fact is expressed in the notation
e29
Hence, the elements of the column vector dX are the differentials of the contravariant coordinates. More generally, for a generic vector A with contravariant components A1, A2, and A3,
e30
Thus, the matrix (−1) operating on the column vector of contravariant (Cartesian) components of a vector yields its Cartesian (contravariant) components. The Jacobian of is the determinant of , det, which = e1 ⋅ (e2 × e3). The inverse of (27) is
e31
where −1 is the Jacobian matrix of the reverse transformation −1 ≡ X(x, t), and, by the chain rule,
e32
The determinants of and −1 are related to a parcel’s specific volume by the Lagrangian continuity equation and its reciprocal as follows:
e33
e34
where α0 is the parcel’s initial specific volume at time τ0, and e1, e2, and e3 are the contravariant basis vectors (Borisenko and Tarapov 1979, p. 25).
We can obtain a different expression for −1 by expanding (29) into the form
e35
taking the transpose of the matrix of cofactors of and dividing the result by the determinant of . The identity of Lagrange (Kreyszig 1972, p. 215) is useful here. For example, the cofactor
e36
We thus find that
e37
where the rows of −1, are the reciprocal or contravariant basis vectors, and e1 = e2 ×e3/[e1 · (e2 × e3)], etc. (Borisenko and Tarapov 1979, p. 25). From (37) and (32), it is apparent that e1 = X, e2 = Y, and e3 = Z. Substituting for the covariant basis vectors in (37) gives
e38
via (33). We can form the inverse of −1 similarly. This yields the alternative form for :
e39
The generic vector A also has the following expansion with respect to the contravariant basis e1, e2, and e3:
e40
where A1, A2, and A3 are its covariant components. Taking the dot product with the unit vectors i, j, and k gives us
e41
The square matrix in (41) we define as
e42
where from (6), superscript T denotes transpose, and, hereinafter, the arguments of (and below) will be shortened to (τ, τ0). Note that operating on the column vector of covariant components of a vector yields its Cartesian components, and −1 performs the reverse transformation. These matrices are important because −1 converts any gradient σ in location space to the corresponding gradient in label space, and transforms the gradient back to location space. This follows from the chain rule, whereby
e43
or
e44
The reverse transformation is
e45
where
e46
The columns of are the contravariant basis vectors, so
e47
which shows that in location space ∂σ/∂X, ∂σ/∂Y, and ∂σ/∂Z are the covariant components of σ.
We will also need to find a matrix operator such that
e48
e49
With the aid of the rule (47) applied to A and B, we obtain
e50
because e1 = e2 × e3/[ e1 · (e2 × e3)], etc. This becomes
e51
after use of (29) and (34). Thus,
e52
e53
from (38), (39), (33), and (34). It is shown in the appendix that is the propagator for a directed material element of area. Hence, is related to vortex-tube stretching.

From (51) and (30), it is evident that α0∂(A, B)/∂(Y, Z), α0∂(A, B)/∂(Z, X), and α0∂(A, B)/∂(X, Y) are the contravariant components of αA × B in location space. Even though A × B, like vorticity, is an axial vector (or pseudovector) that transforms differently from a true vector (Springer 1962, p. 76), αA × B, like α times vorticity, is a true vector.

5. Parcel velocity formula via integration

We now seek an expression for the velocity of a parcel. The equations of motion, continuity, and entropy equations in a frame rotating with the earth are
e54
e55
e56
where Φ ≡ gzuE · uE/2 is the sum of gravitational and centrifugal potentials, FF1x + F2y + F3z is the friction force, is the rate of entropy production in the parcel, and J is the diabatic heating rate.

The steps to obtaining a velocity formula are (i) use the matrix −1 to convert (54) to Lagrangian coordinates, (ii) utilize −1 as an integrating factor, (iii) integrate over time, and (iv) multiply by the inverse matrix to convert the resulting formula back to Eulerian coordinates.

When we pre-multiply (54) by −1 and use (45), we get
e57
By (46),
e58
since ∂x/∂τ = u. We also have
e59
But
e60
and
e61
Adding (60) and (61) gives
e62
Similarly,
e63
Therefore,
e64
from (59), (62), (63), and (22). Inserting (58) and (64) into (57) then gives
e65
where uB ≡ (uB, υB, wB) ≡ u + uβ. We now integrate (65) over time from the initial time τ0 to the current time τ, and apply the initial conditions x = X, uB = UB at τ0, where UBU + Uβ. We get
e66
where
e67
After integration by parts and use of (56), this becomes
e68
where Λ is the cumulative temperature defined in section 2, and
e69
is the integrand of a Lagrangian integral (integral over time following a parcel). For frictionless isentropic flow (G ≡ 0) in a nonrotating system, (68) reduces to Serrin’s (1959) generalization to isentropic flow of Weber’s transformation of the equations of motion for homentropic flow.
Finally, we pre-multiply (68) by (τ, τ0). Note first that
e70
via (24). We thus obtain
e71
on the Grimshaw β plane where Ψ = ψβ(χχ0). On the nontraditional f plane, the potential term is simply −Ψ, where Ψ is now
e72

The velocity formula is implicit through the definition of Ψ. However, the implicitness affects only the irrotational part of the wind. If the normal velocity is known at the boundaries of the domain, we may determine Ψ at time τ as the solution of the elliptic partial differential equation obtained by substituting the velocity formula into the continuity equation [(55)] (Hunt and Hussain 1991).

From the curl of (71) or, alternatively, (68), we obtain the following explicit formula of the form (5) for the absolute vorticity ω of a parcel on a nontraditional β plane:
e73

6. Vorticity integral via propagator

An integral of the vector vorticity equation circumvents the complication caused by a β plane not having a unique planetary velocity. The vorticity equation is
e74
and the corresponding homogeneous equation, obtained by equating the left side to zero, is the barotropic vorticity equation.
Note that
e75
by continuity, the chain rule, (28), (32), (52), and (53). The vorticity equation now becomes
e76
Clearly, (τ, τ0) satisfies the homogeneous version of (76) (i.e., the barotropic vorticity equation) and is thus the propagator for the equation (DJ06). As well as satisfying its own homogeneous equation, the propagator has the following properties:
e77
where is the 3 × 3 unit matrix.
Pre-multiplying (76) by −1(τ, τ0) gives us
e78
after use of (49). Hence,
e79
Integrating (by parts in some places) over time from τ0 to τ yields the formula for the initial vorticity:
e80
where
e81
is the vector integrand of the Lagrangian integral [after use of (56)]. We then pre-multiply (80) by (τ, τ0) and utilize (48) to get
e82
where the first term on the right is the barotropic vorticity, the second term is the baroclinic vorticity if the flow is isentropic, and the last term includes the frictional vorticity and an alteration to the baroclinic vorticity owing to diabatic heating. Equation (82) is the same as (3.8b) in DJ06. It is demonstrated in section 8 that all the circulation theorems follow from (82). In inviscid isentropic flow, the integral term vanishes, leaving
e83
where the first and second terms on the right are the barotropic and baroclinic vorticity, respectively. Unlike the baroclinic and frictional vorticities, the barotropic vorticity is independent of the path that the parcel takes between X and x.
The barotropic vorticity in (83) may be written in several different forms. By (52),
e84
which is Cauchy’s formula. The inverse of (84), ω(τ0) = −1ωBT(τ), is (2) in Article 146 of Lamb (1932). Inserting (26) for the initial vorticity into (84) and using (48) produces
e85
We can relate the barotropic vorticity to the initial vorticity by expanding U, V, and W by the chain rule. For example,
e86
Substituting this and similar expressions for V × Y and W × Z into (85) gives us
e87
Inserting (85) into (82) gives us the general formula for the vorticity of a parcel on a Grimshaw β plane:
e88
where and H are given by (52) and (81). This is (73) again.

7. Velocity formula by calculus of variations

We now show how the isentropic frictionless version of velocity (71) can be derived from an Eulerian form of Hamilton’s principle of least action (Salmon 1988, 234–235). The action, a functional, is defined as the integral over volume and time of the Lagrangian density function L. The Lagrangian density is the absolute kinetic energy of the fluid minus its internal and potential energies (all per unit volume). Hamilton’s principle in particle dynamics states that the action is stationary with respect to small virtual displacements of the particles from their actual motion (Feynman et al. 1964). Because the Eulerian coordinates are independent variables here, the action is stationary with respect to small changes δX in the Lagrangian coordinates at each point rather than changes in the locations of individual particles (Salmon 1988, p. 234). The analysis has to account for the relationships between δX and the variations δu, δα, and δS in the other dependent variables. Solving DX/Dt = 0 for the velocity field yields u = −JX/∂t, so the variation in u is dependent on the Lagrangian coordinates as well as x and t (Salmon 1988, p. 234). The same is true for the density field because it is related to X via the continuity equation [(34)], and also for entropy, which is a function of X because it is conserved. However, the variations in α, S, and u can be considered independent of X if constraints are added to the Lagrangian density (Hildebrand 1965, 139–142). Therefore, we add constraints of mass and entropy conservation and the Lin constraints to L.

Thus, the variational problem is
e89
where Ξ is the 3D spatial domain with surface boundary Σ, the time integral is from the initial time t0 to the current time t, and
e90
Here, Ψ*, Λ*, U*, V*, and W* are the Lagrange multipliers. In the case of a nonrotating atmosphere, (89) and (90) reduce to Bretherton’s (1970) (10). We take the variations and use integration by parts where necessary in the manner prescribed by Hildebrand (1965, 135–136). After using (10) and the divergence theorem, we obtain
e91
where n is the unit outward normal on Σ, and we have omitted groups of terms that cancel owing to the constraints. Because the variations are arbitrary, we must have
e92
e93
e94
e95
The natural boundary conditions are that the normal velocity vanishes on Σ and that the variations δρ, δS, and δX vanish at the beginning and end times. We stipulate that Ψ* = 0 and Λ* = 0 initially. Then Λ* becomes identical to the cumulative temperature Λ, and (92) evaluated at the start gives us
e96
so U* is the initial absolute velocity Ua given by (24) and (25) with β = 0. Thus, (92) becomes
e97
By calculus of variations, we have found for a nontraditional f plane the inviscid isentropic version of (71), the velocity formula obtained by integration.

8. Conservation and circulation theorems

We now show that the formulas are fundamental to conservation of potential vorticity and the circulation theorems. Conservation of Ertel’s potential vorticity derives from the vorticity formula. For any scalar fields A, B, and Θ, we have
e98
with use of (33). Taking the scalar product of (88) with αΘ, where Θ is any conserved variable, and applying (98) to each term on the right gives
e99
when H ≡ 0. In inviscid isentropic flow, setting Θ = S eliminates the baroclinic term and proves potential vorticity conservation. We may also take the baroclinic term out of the equation by just considering the barotropic vorticity. Setting Θ equal to X, Y, and Z, in turn, gives us the conservation laws
e100
because X = e1, etc. Thus, the contravariant components in location space of αω(τ) are equal to the initial components of this vector and hence are conserved following the motion (Salmon 1998, p. 202; Dahl et al. 2014). The effects of vortex-tube stretching and tilting on a parcel are incorporated solely into the changing covariant basis vectors that are attached to the parcel. Incidentally, by (37), we may write (100) as J−1αωBT(τ) = α0ω(τ0), which is simply the inverse of Cauchy’s formula.

The baroclinic vorticity term Λ × S is a solenoidal vector field and thus has its own vortex tubes. In isentropic flow, the baroclinic vortex lines are the intersections of the isentropic surfaces with the surfaces of constant Λ because ωBC is normal to both Λ and S (Davies-Jones 2000). The terms that compose the barotropic vorticity ωBT in (85) are all solenoidal and individual solutions of the barotropic vorticity equation. Therefore, these fields [or, more strictly, α times these fields—Salmon (1998, p. 199)] are frozen into the fluid. The vortex lines of the term U × X are the intersections of the surfaces of constant U and constant X, and similarly for the other terms in ωBT.

Helicity is conserved over a material volume Ξ of homentropic inviscid flow if the volume is made up of closed vortex tubes (zero vorticity flux on the bounding surface Σ). This follows from (71) and (87), which, for homentropic inviscid flow in an inertial frame, reduce to
e101
e102
where (ξ0, η0, ζ0) is the initial vorticity ω(τ0). The helicity density is
e103
and the helicity is therefore
e104
Hence, helicity is conserved if ωn = 0 on the bounding surface. The flow then has nonzero helicity only when the vortex tubes of u × x, υ × y, and w × z are interlinked (Moffatt and Tsinober 1992, p. 283).
Eckart’s (1960) circulation theorem follows immediately from the velocity (68). For the generic scalar fields A and B,
e105
Hence, applying the line-integral operator to (68) and using (105), (23), and (24) yields
e106
which is Eckart’s theorem for the circulation around a material curve C (Dutton 1976, p. 374). The barotropic circulation, obtained by setting the right side to zero, is constant, in agreement with Kelvin’s circulation theorem.
The circulation theorems can also be obtained from the vorticity (82). The element of vorticity flux is given by the scalar product of the vector element of material area and the vorticity vector. Note that a scalar product is equivalent to a 1 × 3 row vector times a 3 × 1 column vector. After integration over a finite material surface area and use of (A4), we obtain
e107
where dσ is a row vector, and ω and −1ω are column vectors. Inserting (82) and then (A4) into (107) yields
e108
Applying (48) at time gives us
e109
From (81), H is the sum of terms of the form By (109) and (A5),
e110
Thus, (108) becomes
e111
Stokes’ theorem then gives us
e112
which is (106) again.
In moist convection, dry entropy S is not conserved. It is better to use the −αp form of the pressure gradient term [see (54)], where α is the reciprocal of the density of the air and water system (e.g., Davies-Jones 2015). Then, (112) converts to
e113
For shallow convection,
e114
where b is the buoyancy force.

9. Concluding remarks

Equation (71), obtained by integrating the equations of motion, expresses the velocity of a parcel in a general flow with friction and diabatic heating on Grimshaw’s (1975) nontraditional β plane. The restriction of this formula to frictionless isentropic flow on a nontraditional f plane is obtained variationally from Hamilton’s principle of least action by utilizing the vital Lin constraints in an Eulerian framework.

Although the velocity formula is implicit, the rotational velocity is explicit. Thus, the curl of (71) provides the explicit (73) for a parcel’s absolute vorticity in general flow. This same formula is obtained by integrating the vector vorticity equation using the propagator defined in (52). The propagator method is more powerful than the variational one because it produces a vorticity formula for general flows on a β plane instead of just inviscid isentropic flows on an f plane.

The formulas pertain to individual parcels. If we knew all the trajectories x(X, τ), then we would have complete knowledge of the motion of the fluid. Simulations that use the Lagrangian description of fluid would provide this knowledge. However, keeping track of all the parcel trajectories is computationally expensive, and the trajectories may be chaotic, so almost all models use the mathematically simpler Eulerian description. The latter give us a subset of this knowledge: namely, the dependent variables (velocity, entropy, and specific volume) as functions of the Eulerian coordinates. The Eulerian method works with incomplete knowledge (the parcel paths are unknown), because the governing equations are independent of how the parcels are labeled (Salmon 1998, p. 337).

For physical understanding, however, this subset of knowledge is often insufficient. It does not tell us, for instance, how an intense vortex forms in a model simulation. To understand why the vortex is there, we need at least some trajectory information. In particular, we need to find out the paths of the parcels that end up in the vortex and to determine the deformations and the torques that these parcels experience along the way. We can do this by adding passive auxiliary equations to the model. If the diabatic heating and friction is weak, we might be able to diagnose the origin of a concentrated vortex by adding the equations DX/Dt = 0, DU/Dt = 0, DS/Dt = 0, and DΛ/Dt = T (with Λ = 0 initially) and marching them forward for a short while from an appropriate starting time using a stable, accurate, high-order, upstream-differencing scheme. The auxiliary equations cannot be integrated for too long, because the surfaces of constant X, etc., become increasingly convoluted. The relative strengths of the solenoids of the variable pairs, Λ and S, U and X, etc., at the time of vortex formation might determine the origin of rotation.

Acknowledgments

I am indebted to Dr. Qin Xu and the three anonymous reviewers for carefully reading the article, checking the mathematics, raising some interesting points, pointing out mistakes, and indicating where greater clarity was needed.

APPENDIX

The Propagator for a Material Surface Element

Following Dutton (1976, p. 386), we let the material surface be described by the equation x = x(λ, μ, τ) where λ and μ are parameters of the surface, and τ is time. The vector element of area on this surface is
ea1
when written as a row vector. At the initial time τ0, this area element is given by
ea2
By the chain rule for Jacobians (Margenau and Murphy 1956, p. 20),
ea3
Thus,
ea4
which shows that −1 must be the propagator of a material surface element.
At any intermediate time ,
ea5

REFERENCES

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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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    • Search Google Scholar
    • Export Citation
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  • Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.

  • Sasaki, Y., 2014: Entropic balance theory and variational field Lagrangian formalism: Tornadogenesis. J. Atmos. Sci., 71, 21042113, doi:10.1175/JAS-D-13-0211.1.

    • Search Google Scholar
    • Export Citation
  • Serrin, J., 1959: Mathematical principles of classical fluid mechanics. Fluid Dynamics I, C. Truesdell, Ed., Vol. VIII/1, Handbuch der Physik, Springer-Verlag, 125–263, doi:10.1007/978-3-642-45914-6_2.

  • Smolarkiewicz, P. K., and R. Rotunno, 1989: Low Froude number flow past three-dimensional obstacles. Part I: Baroclinically generated lee vortices. J. Atmos. Sci., 46, 11541164, doi:10.1175/1520-0469(1989)046<1154:LFNFPT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Springer, C. E., 1962: Tensor and Vector Analysis: with Applications to Differential Geometry. Ronald Press, 242 pp.

  • Truesdell, C., 1954: The Kinematics of Vorticity. Indiana University Press, 232 pp.

Save
  • Adlerman, E. J., K. K. Droegemeier, and R. P. Davies-Jones, 1999: A numerical simulation of cyclic mesocyclogenesis. J. Atmos. Sci., 56, 20452069, doi:10.1175/1520-0469(1999)056<2045:ANSOCM>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Aris, R., 1962: Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice Hall, 286 pp.

  • Borisenko, A. I., and I. E. Tarapov, 1979: Vector and Tensor Analysis with Applications. Dover, 257 pp.

  • Bretherton, F. P., 1970: A note on Hamilton’s principle for perfect fluids. J. Fluid Mech., 44, 1931, doi:10.1017/S0022112070001660.

    • Search Google Scholar
    • Export Citation
  • Dahl, J. M. L., M. D. Parker, and L. J. Wicker, 2014: Imported and storm-generated near-ground vertical vorticity in a simulated supercell. J. Atmos. Sci., 71, 30273051, doi:10.1175/JAS-D-13-0123.1.

    • Search Google Scholar
    • Export Citation
  • Davies-Jones, R. P., 2000: A Lagrangian model for baroclinic genesis of mesoscale vortices. Part I: Theory. J. Atmos. Sci., 57, 715736, doi:10.1175/1520-0469(2000)057<0715:ALMFBG>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Davies-Jones, R. P., 2006: Integrals of the vorticity equation. Part I: General three- and two-dimensional flows. J. Atmos. Sci., 63, 598610, doi:10.1175/JAS3646.1.

    • Search Google Scholar
    • Export Citation
  • Davies-Jones, R. P., 2008: Can a descending rain curtain in a supercell instigate tornadogenesis barotropically? J. Atmos. Sci., 65, 24692497, doi:10.1175/2007JAS2516.1.

    • Search Google Scholar
    • Export Citation
  • Davies-Jones, R. P., 2015: A review of supercell and tornado dynamics. Atmos. Res., 158–159, 274291, doi:10.1016/j.atmosres.2014.04.007.

    • Search Google Scholar
    • Export Citation
  • Davies-Jones, R. P., and H. E. Brooks, 1993: Mesocyclogenesis from a theoretical perspective. The Tornado: Its Structure, Dynamics, Prediction, and Hazards,Geophys. Monogr., Vol. 79, Amer. Geophys. Union, 105–114, doi:10.1029/GM079p0105.

  • Davies-Jones, R. P., R. J. Trapp, and H. B. Bluestein, 2001: Tornadoes and tornadic storms. Severe Convective Storms,Meteor. Monogr., No. 50, Amer. Meteor. Soc., 167–221, doi:10.1175/0065-9401-28.50.167.

  • Dellar, P. J., 2011: Variations on a beta-plane: Derivation of non-traditional beta-plane equations from Hamilton’s principle on a sphere. J. Fluid Mech., 674, 174195, doi:10.1017/S0022112010006464.

    • Search Google Scholar
    • Export Citation
  • Dutton, J. A., 1976: The Ceaseless Wind: An Introduction to the Theory of Atmospheric Motion. Dover, 579 pp.

  • Eckart, C., 1960: Variation principles of hydrodynamics. Phys. Fluids, 3, 421427, doi:10.1063/1.1706053.

  • Epifanio, C. C., and D. R. Durran, 2002: Lee-vortex formation in free-slip stratified flow over ridges. Part II: Mechanism of vorticity and PV production in nonlinear viscous wakes. J. Atmos. Sci., 59, 11661181, doi:10.1175/1520-0469(2002)059<1166:LVFIFS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Feynman, R. P., R. B. Leighton, and M. Sands, 1964: The principle of least action. Mainly Electromagnetism and Matter, Vol. II, The Feynman Lectures on Physics, Addison–Wesley, 19-1–19-14.

  • Grimshaw, R. H. J., 1975: A note on the β-plane approximation. Tellus, 27A, 351357, doi:10.1111/j.2153-3490.1975.tb01685.x.

  • Hildebrand, F. B., 1965: Methods of Applied Mathematics. 2nd ed., Prentice-Hall, 362 pp.

  • Hunt, J. C. R., and F. Hussain, 1991: A note on velocity, vorticity and helicity of fluid elements. J. Fluid Mech., 229, 569587, doi:10.1017/S0022112091003178.

    • Search Google Scholar
    • Export Citation
  • Kreyszig, E., 1972: Advanced Engineering Mathematics. 3rd ed. Wiley, 866 pp.

  • Lackmann, G., 2011: Midlatitude Synoptic Meteorology: Dynamics, Analysis, and Forecasting. Amer. Meteor. Soc., 345 pp.

  • Lamb, H., 1932: Hydrodynamics. Dover, 738 pp.

  • Margenau, H., and G. M. Murphy, 1956: The Mathematics of Physics and Chemistry. 2nd ed. Van Nostrand, 604 pp.

  • Markowski, P., and Coauthors, 2012: The pretornadic phase of the Goshen County, Wyoming, supercell of 5 June 2009 intercepted by VORTEX2. Part II: Intensification of low-level rotation. Mon. Wea. Rev., 140, 29162938, doi:10.1175/MWR-D-11-00337.1.

    • Search Google Scholar
    • Export Citation
  • Mobbs, S. D., 1981: Some vorticity theorems and conservation laws for non-barotropic fluids. J. Fluid Mech., 108, 475483, doi:10.1017/S002211208100222X.

    • Search Google Scholar
    • Export Citation
  • Moffatt, H. K., and A. Tsinober, 1992: Helicity in laminar and turbulent flow. Annu. Rev. Fluid Mech., 24, 281312, doi:10.1146/annurev.fl.24.010192.001433.

    • Search Google Scholar
    • Export Citation
  • Rotunno, R., and J. B. Klemp, 1985: On the rotation and propagation of simulated supercell thunderstorms. J. Atmos. Sci., 42, 271292, doi:10.1175/1520-0469(1985)042<0271:OTRAPO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Salmon, R., 1988: Hamiltonian fluid mechanics. Annu. Rev. Fluid Mech., 20, 225256, doi:10.1146/annurev.fl.20.010188.001301.

  • Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.

  • Sasaki, Y., 2014: Entropic balance theory and variational field Lagrangian formalism: Tornadogenesis. J. Atmos. Sci., 71, 21042113, doi:10.1175/JAS-D-13-0211.1.

    • Search Google Scholar
    • Export Citation
  • Serrin, J., 1959: Mathematical principles of classical fluid mechanics. Fluid Dynamics I, C. Truesdell, Ed., Vol. VIII/1, Handbuch der Physik, Springer-Verlag, 125–263, doi:10.1007/978-3-642-45914-6_2.

  • Smolarkiewicz, P. K., and R. Rotunno, 1989: Low Froude number flow past three-dimensional obstacles. Part I: Baroclinically generated lee vortices. J. Atmos. Sci., 46, 11541164, doi:10.1175/1520-0469(1989)046<1154:LFNFPT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Springer, C. E., 1962: Tensor and Vector Analysis: with Applications to Differential Geometry. Ronald Press, 242 pp.

  • Truesdell, C., 1954: The Kinematics of Vorticity. Indiana University Press, 232 pp.

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