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
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
4. Useful matrices
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 α∇A × ∇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), α∇A × ∇B, like α times vorticity, is a true vector.
5. Parcel velocity formula via integration
The steps to obtaining a velocity formula are (i) use the matrix
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).
6. Vorticity integral via propagator
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 = −J∂X/∂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.
8. Conservation and circulation theorems
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.
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
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
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