Corrigendum

Johannes M. L. Dahl Department of Geosciences, Texas Tech University, Lubbock, Texas

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Jannick Fischer Karlsruher Institut für Technologie, Germany

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© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Johannes Dahl, johannes.dahl@ttu.edu

© 2025 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Johannes Dahl, johannes.dahl@ttu.edu

This corrigendum concerns two equations in Dahl and Fischer’s (2023) derivation: one erroneously omitted term in their Eq. (13), which, however, will be shown not to affect the downstream mathematical development, and one typo in Eq. (30).

1. Equation (13)

In the following, the correct form of the time integral of the momentum equation, Eq. (13) in Dahl and Fischer (2023), is presented, and it is demonstrated that an additional term that appears in that equation is irrotational and hence does not contribute to the vorticity integral, Dahl and Fischer’s Eq. (14). The momentum equation is given by
dvdt=f,
where v is the velocity vector and f is the net force acting on the fluid parcel. Using the same notation as in Dahl and Fischer (2023), this may be written as
dvdt=ddt(uαgα)=duαdtgα+uαdgαdt=f.
The term on the rhs involving duα/dt describes the time rate of change of the coordinate velocity uα, which includes the effect of the force as well as the effect of the time-dependent coordinate basis vectors. The second term, involving dgα/dt, is the geometric term that corrects for the effect of the time-dependent coordinate basis. In the present case, the basis vectors change because the material volume, including the coordinate system materially attached to it, is deformed by the flow.1 To obtain an expression for the time evolution of gα, we first recognize that
gα=ξα.
Now, for any parcel property Φ, we may write Φ(t) = Φ[r(t), t], where r is the parcel’s location. Using the chain and product rules, we find
(dΦdt)=(Φt+vΦ)=ddt(Φ)+(v)Φ.
With Φ = ξα and α/dt = 0, we see that
dgαdt=ddt(ξα)=(v)ξα.
The rhs is equivalent to the 3D vector frontogenesis function [vector frontogenesis in 2D is discussed by Keyser et al. (1988)], which connects the evolution of the coordinate curves to the flow field: Like isentropes in the usual frontogenesis case, material coordinate curves are deformed and rotated by the flow; the contravariant basis vectors correspond to the gradients of the isentropes. This implies that the geometric term G may be written as
G=uαdgαdt
=uα(v)ξα
=(v)gαuα
=(v)v.
With this, the momentum equation becomes
δuβδt=duβdtgβ[(v)v]=fβ.
This equation is equivalent to Eq. (12) in Dahl and Fischer (2023). When integrating this equation with respect to time, however, the geometric term needs to be carried along, so Dahl and Fischer’s Eq. (13) should read
uβ(t)=uβ(t0)+t0tdtfβ(t)+t0tdtgβ(t){[v(t)]v(t)}.
When this equation is inserted into the expression for the vorticity,
ωγ(t)=ϵαβγG(t)uβ(t)ξα,
one obtains
ωγ(t)=ϵαβγG(t)uβ(t0)ξα+t0tdtϵαβγG(t)fβ(t)ξα+t0tdtϵαβγG(t)ξα{gβ(t)[v(t)v(t)]}.
This equation corresponds to Eq. (14) in Dahl and Fischer (2023), but it includes the geometric term (the last term on the rhs). To show that this term is zero, we use Eq. (23) from Dahl and Fischer (2023) and find that
t0tdtϵαβγG(t)ξα{gβ(t)[v(t)v(t)]}=t0tdtρ(t)ρ(t)ϵαβγG(t)Gβ(t)ξα
=t0tdtρ(t)ρ(t)[×G(t)]γ
=t0tdtρ(t)ρ(t)gγ(t)[×G(t)].
If ∇ × G = 0, this integral vanishes. To see that this is indeed the case, we first apply the product rule and see that
G=(v)v=12(vv)=12V2,
where V=vv is the velocity magnitude. Then,2
×G=12×V20.

2. Equation (30)

Equation (30) in Dahl and Fischer (2023) is written in symbolic form, so there should be no index on the lhs. The correct equation is given by
ω(t)=rξ(t)[ρ(t)ρ(t0)ω0+t0tdtρ(t)ρ(t)ξr(t)τ(t)].

Footnotes

1
In many applications, the coordinate basis is constant in time but nonuniform in space, such that the geometric term is given by (e.g., Simmonds 1994, p. 57)
uαdgαdt=uαuβdgαdξβ=uαuβΓβγαgγ
where uβgα/∂ξβ is the time rate of change of gα due to “advection” [see also appendix A of Dahl and Fischer (2023)] and Γβγα is the Christoffel symbol.
2

Note that (v)v=^(uj/xi)uj is not equal to the advection term (v)v=^ui(uj/xi), whose curl is generally not zero.

Acknowledgments.

Exchanges with Drs. Bob Davies-Jones and Giorgio Bornia led to a more general and simplified treatment.

REFERENCES

  • Dahl, J. M. L., and J. Fischer, 2023: On the origins of vorticity in a simulated tornado-like vortex. J. Atmos. Sci., 80, 13611380, https://doi.org/10.1175/JAS-D-22-0145.1.

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  • Keyser, D., M. J. Reeder, and R. J. Reed, 1988: A generalization of Petterssen’s frontogenesis function and its relation to the forcing of vertical motion. Mon. Wea. Rev., 116, 762781, https://doi.org/10.1175/1520-0493(1988)116<0762:AGOPFF>2.0.CO;2.

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  • Simmonds, J. G., 1994: A Brief on Tensor Analysis. Springer, 112 pp.

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  • Dahl, J. M. L., and J. Fischer, 2023: On the origins of vorticity in a simulated tornado-like vortex. J. Atmos. Sci., 80, 13611380, https://doi.org/10.1175/JAS-D-22-0145.1.

    • Search Google Scholar
    • Export Citation
  • Keyser, D., M. J. Reeder, and R. J. Reed, 1988: A generalization of Petterssen’s frontogenesis function and its relation to the forcing of vertical motion. Mon. Wea. Rev., 116, 762781, https://doi.org/10.1175/1520-0493(1988)116<0762:AGOPFF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Simmonds, J. G., 1994: A Brief on Tensor Analysis. Springer, 112 pp.

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