Article Type:
Review Article

Centrifugal Waves in Tornado-Like Vortices: Kelvin’s Solutions and Their Applications to Multiple-Vortex Development and Vortex Breakdown

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

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Online Publication:
13 Sep 2021
Print Publication:
01 Oct 2021
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Review Articles in Monthly Weather Review
DOI:
https://doi.org/10.1175/MWR-D-20-0426.1
Page(s):
3173–3216
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Abstract

About 140 years ago, Lord Kelvin derived the equations describing waves that travel along the axis of concentrated vortices such as tornadoes. Although Kelvin’s vortex waves, also known as centrifugal waves, feature prominently in the engineering and fluid dynamics literature, they have not attracted as much attention in the field of atmospheric science. To remedy this circumstance, Kelvin’s elegant derivation is retraced, and slightly generalized, to obtain solutions for a hierarchy of vortex flows that model basic features of tornado-like vortices. This treatment seeks to draw attention to the important work that Lord Kelvin did in this field, and reveal the remarkably rich structure and dynamics of these waves. Kelvin’s solutions help explain the vortex breakdown phenomenon routinely observed in modeled tornadoes, and it is shown that his work is compatible with the widely used criticality condition put forth by Benjamin in 1962. Moreover, it is demonstrated that Kelvin’s treatment, with the slight generalization, includes unstable wave solutions that have been invoked to explain some aspects of the formation of multiple-vortex tornadoes. The analysis of the unstable solutions also forms the basis for determining whether, for example, an axisymmetric or a spiral vortex breakdown occurs. Kelvin’s work thus helps explain some of the visible features of tornado-like vortices.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

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

Abstract

About 140 years ago, Lord Kelvin derived the equations describing waves that travel along the axis of concentrated vortices such as tornadoes. Although Kelvin’s vortex waves, also known as centrifugal waves, feature prominently in the engineering and fluid dynamics literature, they have not attracted as much attention in the field of atmospheric science. To remedy this circumstance, Kelvin’s elegant derivation is retraced, and slightly generalized, to obtain solutions for a hierarchy of vortex flows that model basic features of tornado-like vortices. This treatment seeks to draw attention to the important work that Lord Kelvin did in this field, and reveal the remarkably rich structure and dynamics of these waves. Kelvin’s solutions help explain the vortex breakdown phenomenon routinely observed in modeled tornadoes, and it is shown that his work is compatible with the widely used criticality condition put forth by Benjamin in 1962. Moreover, it is demonstrated that Kelvin’s treatment, with the slight generalization, includes unstable wave solutions that have been invoked to explain some aspects of the formation of multiple-vortex tornadoes. The analysis of the unstable solutions also forms the basis for determining whether, for example, an axisymmetric or a spiral vortex breakdown occurs. Kelvin’s work thus helps explain some of the visible features of tornado-like vortices.

© 2021 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Johannes M. L. Dahl, johannes.dahl@ttu.edu
Keywords: Atmosphere; Instability; Kelvin waves; Stability; Stationary waves; Wave properties; Waves, atmospheric; Oscillations; Tornadoes; Vortices

1. Introduction

Most atmospheric scientists are familiar with the type of inertia-gravity wave that is trapped along coastlines or the equator, known as a Kelvin wave (e.g., Thomson 1879; Gill 1982, chapter 10; Wang 2003). These waves play a prominent role in tropical dynamics including El Niño–Southern Oscillation (e.g., Battisti 1988). Perhaps less familiar within the atmospheric science community is the work by Lord Kelvin on vortex waves (Thomson 1880), in which he laid out the general strategy for obtaining a wide variety of solutions of waves traveling within infinitely extended, concentrated cylindrical vortices resembling the flow in tornadoes or dust devils some distance away from the lower boundary. These waves are usually referred to as “Kelvin waves” in the fluid physics literature, and they are also known as vortex waves, centrifugal waves, or inertial waves (Lugt 1989). Such waves play an important role in the vortex breakdown phenomenon as well as in the development of multiple-vortex tornadoes (e.g., Lewellen 1993; Rotunno 2013). Examples of these waves within tornadoes are shown in Fig. 1. Here so-called bending waves (to be introduced in section 3) are visible, which are often well pronounced during the rope stage of the tornado (Fig. 1b). The effects of vortex waves are also seen in Fig. 2, where Fig. 2a shows a multivortex structure that results from unstable growth of spiral modes (section 3), and Fig. 2b displays what appears to be a vortex breakdown. This phenomenon often occurs in vortices with nonzero axial velocity and it will be reviewed in section 8.

Fig. 1.
Fig. 1.

Bending waves within a tornado: (a) El Reno, OK, on 30 Apr 1978 (photo courtesy of NOAA/NSSL via James Murnan); (b) large-amplitude bending wave during the rope stage of a Tornado near Waldo, KS, on 28 May 2019 (video capture, courtesy of Tom Smetana).

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

Fig. 2.
Fig. 2.

(a) A multiple-vortex tornado near Katie, OK, on 9 May 2016 (video capture, courtesy of Twisting Fury). (b) Possible vortex breakdown, suggested by the abrupt widening of the vortex core near the ground of a tornado near Seymour, TX, on 1 May 2019 (photo courtesy of Jason Weingart).

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

The need for this review arises from the observation that, despite the ubiquity of Kelvin vortex waves in tornado-like vortices, a detailed introduction to the structure and behavior of these waves appears to be absent from the atmospheric science literature. Although this topic is introduced in some fluid physics texts (e.g., Thomson 1880; Lamb 1932, section 158; Chandrasekhar 1961, chapter VII, especially p. 284; Drazin and Reid 1981, p. 75; Saffman 1992, chapters 11 and 12; Lim and Redekopp 1998; Fritts et al. 1998; Rossi 2000; Batchelor 2002, p. 559; Alekseenko et al. 2007, chapter 4), in a given presentation either only limited, or rather advanced, aspects of vortex waves are discussed. As a consequence, there is a large gap between the knowledge base offered in mainstream texts about atmospheric dynamics (e.g., Gill 1982; Holton and Hakim 2013; Markowski and Richardson 2010) and the advanced peer-reviewed literature on the remarkably rich Kelvin vortex wave dynamics. Thus, the present paper seeks to

  • narrow the gap between the atmospheric-dynamics literature and the advanced fluid-physics literature by providing a detailed introduction to these waves. Except where the mathematical steps are readily available in mainstream texts, the complete mathematical development is included in this paper;

  • retrace Kelvin’s original approach, and point to possible applications of his contributions to tornado-like flows, including multiple-vortex development and vortex breakdown.

To achieve this, an almost trivial generalization of Kelvin’s equations is included by allowing for a piecewise constant axial flow, which admits a wide class of unstable solutions not considered in Kelvin’s original work. These solutions have been arrived at previously, but along with a review of these topics, it is demonstrated that the results follow from Kelvin’s approach. In the context of stability analyses, Kelvin’s (slightly modified) approach presented in this review has been superseded by much more advanced analysis techniques, but these are still based on analyses such as those presented here, and will be touched upon in section 8.

The remainder of the paper is structured as follows. In the next section a brief history of vortex waves will be offered and the relevance of these waves will be described, and section 3 offers an overview of the classification of vortex waves. In section 4 the governing equations describing infinitely extended columnar vortices will be presented following Kelvin’s approach. Subsequently, consecutively more refined scenarios are introduced that follow directly from Kelvin’s equations. Starting with the scenario of vanishing base-state axial flow, a cylindrical domain bounded by rigid walls will be considered (section 5) to gain an intuition for the structure and dynamics of these waves. Thereafter the Rankine vortex will be discussed in section 6, and two scenarios that allow for unstable wave growth will be considered in section 7, i.e., a Rankine vortex with upward motion in its core as well as a two-celled vortex with descending motion in its irrotational core, and rising motion outside of it. The unstable waves in the latter scenario provide a rudimentary model for multiple-vortex formation in tornado-like vortices. These ideas will then be applied to vortex breakdown in section 8. Limitations of Kelvin’s approach and its applicability to tornadoes will be addressed in section 9. Finally, section 10 offers concluding remarks and possible directions for future investigations.

2. A brief history of Kelvin vortex waves

Around the mid-1800s, Sir William Thomson, who was granted the title Lord Kelvin in 1892, pursued the idea of describing the previously discovered atoms in terms of microscopic knotted vortex rings, called “vortex atoms” (Thomson 1867; Fabre et al. 2006). The medium in which these vortex rings were thought to exist was the hypothesized all-pervading, homogeneous perfect fluid known as the aether. This idea was fueled by Hermann von Helmholtz’s discovery of the laws of vortex motion (von Helmholtz 1858), specifically his second law, which implies that these vortex rings could persist forever in perfect homogeneous (and thus barotropic) fluids. Falconer (2019) offers a detailed summary of Kelvin’s vortex atom theory. Kelvin was particularly interested in the vibrational modes of these vortex rings, postulating that different vibrational modes could account for the different atomic spectra that had previously been discovered. As a first step, Kelvin formulated the equations describing wave motions within an infinitely long, cylindrical vortex. This effort led to the paper entitled “Vibrations of a columnar vortex” (Thomson 1880), which is the basis of this review. Although the idea of vortex atoms did not survive past the 1890s, it does bear an intriguing resemblance to string theory, and Kelvin vortex waves still do play an important role in fundamental physics, e.g., in the dynamics of quantum vortices in superfluids (e.g., Fonda et al. 2014).

Kelvin’s work entered the field of aeronautical engineering following the discovery that lift-generating devices produce a pair of trailing vortices, which pose a hazard to aircraft that encounter these vortices, which consequently reduce airspace capacity (Hallock and Holzäpfel 2018). The stability of the wake vortices is directly related to the unstable growth of bending Kelvin modes (the different Kelvin modes will be introduced in the next section). This instability is known as “cooperative instability” and is sometimes visibly manifest as a contortion of aircraft contrails and the formation of contrail lobes (Lewellen and Lewellen 2001; Wu et al. 2006, p. 499; Schultz and Hancock 2016).

In the late 1950s, another phenomenon was discovered by aerodynamicists, i.e., the vortex breakdown. Peckham and Atkinson (1957) are generally credited for first documenting this phenomenon during the investigation of lift-generating vortices produced at the leading edge of ogival delta wings. At large angles of attack, the observation included a disintegration of the vortex structure, which appeared to “bell out before disappearing—as though the core was becoming more diffuse” (Peckham and Atkinson 1957, p. 5), indicating turbulence and an undesirable drop of lift. (The reader may skip ahead to Fig. 31 for an example of such a breakdown in a tornado.)1 On the other hand, vortex breakdown was found to have beneficial effects in some applications such as stabilizing the flame in combustion chambers of gas turbines (e.g., Escudier 1988; Spencer and McGuirk 2008). Vortex breakdown has thus garnered considerable interest in the engineering and fluid physics community, and soon after its official discovery, Squire (1960) and Benjamin (1962) explained the vortex breakdown phenomenon as an axisymmetric analog of the hydraulic jump, which is related to the inability of wave perturbations, in this case axisymmetric Kelvin vortex waves, to propagate upstream (Squire 1960; Benjamin 1962). Among other vortex profiles, these studies also considered the Rankine vortex, and the vortex breakdown criterion discovered in these studies directly follows from Kelvin’s analysis. Vortex breakdown and its relation to Kelvin vortex waves will be reviewed in section 8.

Aside from the application to engineering problems, the roles of Kelvin vortex waves have been studied in the general context of transition to turbulence in rotating flows (e.g., Hopfinger et al. 1982). Importantly, these experiments revealed the existence of solitary-like vortex waves (Maxworthy et al. 1985). In the atmospheric science literature, Kelvin’s work does not appear to be mentioned frequently. One exception is Fultz (1959), who studied oscillations in convective motion. He observed vortex waves in his rotating tank experiments and pointed to a textbook by Bjerknes et al. (1933), where vortex waves are introduced. Apparently, however, Bjerknes et al. (1933) had been unaware of Kelvin’s treatment, prompting Fultz to note that “its significance in meteorological dynamics had to be rediscovered.” Kelvin vortex waves feature prominently in the studies by Andreassen et al. (1998), Fritts et al. (1998) and Fritts and Alexander (2003), which are concerned with turbulence generation as a result of shear instability as well as in breaking internal gravity waves (Fritts and Alexander 2003). These authors refer to vortex waves as “twist waves” (Arendt et al. 1997). Kieu (2016) extended Kelvin’s solutions with the goal of describing waves in the inner core region of tropical cyclones.

In the field of tornado research, centrifugal waves attracted some attention in the 1970s and 1980s when research using tornado vortex chambers flourished (Ward 1972; Church et al. 1977; Rotunno 1979; Snow 1982; Church and Snow 1993). However, the focus of these analyses was the determination of how the flow parameters, most notably the swirl ratio (Davies-Jones 1973), led to different tornado structures including vortex breakdown and multiple vortices. The importance of centrifugal waves is mentioned in these studies in the context of vortex breakdown (Church et al. 1977; Rotunno 1979; Snow 1982; Fiedler and Rotunno 1986), but the structure and dynamics of these waves are not analyzed further. Nolan and Farrell (1999) analyzed these waves in detail, and they observed downward propagating axisymmetric waves in their numerically simulated tornado-like vortex. Their analysis suggests that these waves are less likely to be observed in flows with low swirl ratios, consistent with such flows being supercritical (section 8), but their linear analysis did not fully explain the behavior of these waves. More recently, Nolan (2012) revisited this phenomenon by studying the linear instability of fully nonlinear flows.

Because the vortex chamber experiments reproduced the observed transition of a single tornado vortex into multiple vortices, investigations into the stability of the tornado followed (e.g., Rotunno 1978; Walko and Gall 1984; Nolan 2012). This instability is a manifestation of unstable centrifugal waves, and will be discussed in section 7. With the exception of Nolan’s analyses (Nolan and Farrell 1999; Nolan 2012), these centrifugal waves apparently have not played important roles in interpreting tornadic flows since the 1980s, and Snow (1982) at the end of his review on tornado dynamics, added that “features that should be investigated…include the nature of the wavelike features often seen on the walls of condensation funnels” (Snow 1982, p. 963). It is fair to say that his recommendation has not materialized, and consideration of centrifugal waves in the context of tornadoes appears to be scarce. Aside from Nolan and Farrell (1999), some examples include Lugt (1989) and Trapp (2000), who discuss vortex breakdown in tornadoes and mesocyclones, respectively, and in this context briefly mention centrifugal waves; Bluestein et al. (2003) observed oscillations of a tornado’s intensity and as a possible explanation, centrifugal waves were invoked; in their textbook, Markowski and Richardson (2010) introduce the stability criterion for centrifugal oscillations, which correspond to an axisymmetric Kelvin mode.

This brief overview reveals that, while the relevance of vortex waves in tornadoes is generally acknowledged, the association with Kelvin’s work is not widely appreciated, and the structure and behavior of these waves has attracted somewhat limited attention in the atmospheric science community. To start the discussion of vortex wave dynamics, the basic types and naming conventions of these waves are introduced in the next section. The main distinction between the different wave types is the azimuthal wavenumber m, which determines whether a wave is manifest as, e.g., axisymmetric or spiral perturbation. The azimuthal wavenumber is dimensionless and must be an integer to guarantee continuity of the wave perturbation. The vortex column serves as waveguide, and for infinitely extended vortex columns, each wave type (determined by m) may generally attain an arbitrary axial wavenumber k.

3. Classification of Kelvin vortex waves

a. Axisymmetric modes, m = 0

If m = 0, the perturbation is axisymmetric, resulting in a widening and narrowing of the vortex (e.g., in a Rankine vortex the radius of maximum winds extends and contracts). The axial wavenumber must be nonzero for axisymmetric waves to exist. A commonly used name of this mode goes back to Lord Rayleigh (Strutt 1902, p. 444). When discussing photographs of a liquid jet undergoing axisymmetric oscillations before breaking up into drops (as happens to a water stream emanating from a faucet), he noted: “…I have often been embarrassed for want of an appropriate word to describe the condition in question. But a few days ago, during a biological discussion, I found that there is a recognised, if not very pleasant, word. The cylindrical jet may be said to become varicose, and varicosity goes on increasing with time, until eventually it leads to absolute disruption.” Chandrasekhar (1961, p. 515), after citing this passage from Rayleigh, adds: “In recent times, ‘sausage instability’ has been used to describe the same condition; but this is also not a very ‘pleasant’ description, and varicose instability would seem preferable.” The designations “sausage” or “sausaging” as well as “varicose” modes have been adopted widely to describe axisymmetric vortex waves, though in this paper these waves will be referred to mainly as axisymmetric waves. This wave can only propagate in the axial direction and it plays an important role in the onset of vortex breakdown (section 8).

b. Spiral modes, |m| > 0

Waves with azimuthal wavenumbers |m| ≥ 1 are referred to as helical or spiral modes. If |m| = 1, the waves displace the vortex axis and are called “bending” modes. For all other choices of m the vortex axis remains centered. An example of how an azimuthal wave perturbation is related to the vortex structure is shown in Fig. 3, demonstrating that a wavenumber of two corresponds to an elliptic deformation of the vortex core. The sign of the azimuthal wavenumber determines the handedness of the spiral. As shown in Fig. 4, m > 0 implies left-handedness and m < 0 implies right-handedness.2

Fig. 3.
Fig. 3.

Demonstration of how the wavenumber along a straight axis is related to the azimuthal wavenumber m of a perturbed circle for m = 2. In this example the disturbance is proportional to sin(mθ) and θ increases in a counterclockwise sense starting from the 3 o’clock position. The initially circular surface is deformed into an ellipse.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

Fig. 4.
Fig. 4.

(a) Schematic of a left-handed spiral (m = +1). Shown are also horizontal cross sections at two different altitudes, with the red circle showing the displaced vortex core [~cos(φ)], and the black dashed circle representing the mean core location. The phase φ = kz + mθ has been added to the periphery of the circles to demonstrate how the sign of the wavenumber determines the handedness of the helix (which is indicated by the curved arrow at the top of each panel). (b) As in (a), but for a right-handed spiral (m = −1).

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

If the slope of the phase lines in the (θ, z) plane wrapping around the cylindrical vortex is taken to be dh/dθ, where h is the local height of the phase line and θ is the azimuthal angle, this slope is identical to the pitch of the wave, given by −m/k (Fabre et al. 2006). Defined this way, pitch is proportional to the distance by which the helix advances during one revolution (i.e., to the axial wavelength of the helix). Saffman (1992) and Alekseenko et al. (2007) use slightly different definitions of pitch, which only apply to |m| = 1 but are proportional to each other as well as to the definition used here. It follows that, e.g., left-handed spirals have a negative pitch and axisymmetric waves have zero pitch. For a given axial (vertical) wavenumber, the pitch becomes larger in magnitude as the number of spirals winding around the vortex (i.e., m) is increased.

An azimuthal wavenumber of m = 2 implies a double helix or double spiral structure. Waves with k = 0 and m ≠ 0 are sometimes called “fluted” modes (Maxworthy 1988), which are 2D waves that only propagate in azimuthal direction. Hopfinger et al. (1982), Hopfinger and Browand (1982), and Maxworthy et al. (1985) coined the term “kink wave” for helical solitary waves (m = ±1) they observed in their experiments. These waves resembled the soliton solution by Hasimoto (1972).3

The azimuthal angular phase speed of spiral modes is given by ω/m, where ω is the wave frequency. Assuming that the angular velocity of the vortex Ω is positive, implying cyclonic rotation, then if the azimuthal phase speed is larger than the angular velocity of the vortex, the waves are said to be “cograde,” meaning these waves propagate downstream relative to the motion within the vortex. If the angular speed of the wave is less than Ω, but the waves still move in the same sense as the azimuthal vortex flow, the wave is said to be “retrogade.” These waves propagate upstream relative to the vortex flow, but are advected downstream. Finally, the waves are “countergrade” if they move in the opposite sense than the vortex, now also for a stationary observer (Fabre et al. 2006). From these basic definitions and conventions the rich structure of vortex waves is already becoming apparent. The next section introduces the governing equations describing the dynamics of these waves.

4. Governing equations

One goal of this paper is to reintroduce Kelvin’s original mathematical treatment, which is summarized in Fig. 5. Starting with the linearized, inviscid, incompressible equations of motion in cylindrical coordinates, the existence of a base state flow is assumed, and the equations are linearized about this base state. Subsequently, a normal-mode solution is inserted, and the resulting simplified equation set is solved for the radial and azimuthal flow components. One then specifies the base state flow for a given vortex configuration, uses the mass continuity equation, and arrives at a linear ordinary differential equation (ODE) whose solution governs the radial wave structure. Application of the boundary conditions finally leads to the complete flow field as well as the dispersion relation, from which the wave frequencies and speeds, as well as the existence and growth rates of unstable modes, may be inferred.

Fig. 5.
Fig. 5.

The steps summarizing Kelvin’s (Thomson 1880) approach, including a more systematic treatment of the boundary conditions. Here u^, υ^, and w^ are the amplitudes of the radial, azimuthal, and axial velocity components, respectively.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

a. Linearization of the momentum and mass-continuity equations

The radial, azimuthal (or tangential), and axial (here, vertical) velocity components, respectively, are written as (u, υ, w). The azimuthal velocity component is also called swirl velocity. The equation for the radial velocity u is given by (e.g., Drazin and Reid 1981, p. 71):
∂u∂t+u ∂u∂r+υr∂u∂θ+w ∂u∂z−υ2r=−1ρ∂p∂r,
where r is the radial distance from the center, θ is the azimuth, and p is the pressure perturbation relative to a hydrostatic reference state. The density ρ is assumed to be constant in this treatment. The last term on the lhs is the centripetal acceleration, which appears when writing the equations of motion in cylindrical coordinates. For the azimuthal component υ we have
∂υ∂t+u ∂υ∂r+υr∂υ∂θ+w ∂υ∂z+uυr=−1ρr∂p∂θ,
where the last term on the lhs describes the effect of angular-momentum conservation, which like the centripetal acceleration, enters the equation when using cylindrical coordinates. The vertical momentum equation reads as
∂w∂t+u ∂w∂r+υr∂w∂θ+w ∂w∂z=−1ρ∂p∂z.
Since gravity does not appear in the equation, the implication is that the air is neutrally stratified (i.e., buoyancy is zero). Finally, the mass continuity equation for incompressible flows is given by
∇⋅v=∂u∂r+ur+1r∂υ∂θ+∂w∂z=0.
The u/r term arises from the axisymmetry of the coordinate system (i.e., whenever the radial velocity u is nonzero there is nonzero radial divergence).
For the base-state vortex flow V(r) to be a solution of the above equations, a state of cyclostrophic balance is implied. The azimuthal base-state velocity V(r) may have an arbitrary dependence on r. The axial base-state flow W is constant, and there is no radial base-state flow. Then,
u=u′⁡(r,θ,z,t),
υ=V⁡(r)+υ′⁡(r,θ,z,t),
w=W+w′⁡(r,θ,z,t),
p=P⁡(r)+p′⁡(r,θ,z,t),
where P(r) is the base-state pressure and the primed variables denote perturbations from the base state. Throughout the analysis, V is assumed to be positive, so the base-state vortex spins cyclonically. To linearize the governing equations, this decomposition is inserted into Eqs. (1)–(4). For the radial momentum equation, one finds
∂u′∂t+u′ ∂u′∂r+1r [V+υ′] ∂u′∂θ+[W+w′] ∂u′∂z−⁡(V+υ′)2r=−1ρ∂∂r⁡ (P+p′).
Neglecting the products of perturbation variables and thereby linearizing the equation, gives
∂u′∂t+Vr∂u′∂θ+W ∂u′∂z−2 Vr υ′=−1ρ∂p′∂r−1ρdPdr+V2r.
The last two terms on the rhs cancel because the base state is cyclostrophically balanced, resulting in
∂u′∂t+Vr∂u′∂θ+W ∂u′∂z−2 Vr υ′=−1ρ∂p′∂r.
The analogous procedure for the azimuthal velocity component gives
∂υ′∂t+Vr∂υ′∂θ+W ∂υ′∂z+[Vr+dVdr]u′=−1rρ∂p′∂θ.
The last term on the lhs represents radial angular momentum advection. The vertical momentum equation becomes, after linearization,
∂w′∂t+Vr∂w′∂θ+W ∂w′∂z=−1ρ∂p′∂z.
Finally, for the continuity equation one obtains
∂u′∂r+u′r+1r∂υ′∂θ+∂w′∂z=0.
Now a normal mode solution is assumed,
u′=u^⁡(r)ei⁡(kz+mθ−ωt),
υ′=υ^⁡(r)ei⁡(kz+mθ−ωt),
w′=w^⁡(r)ei⁡(kz+mθ−ωt),
p′=p^⁡(r)ei⁡(kz+mθ−ωt),
where the “hatted” variables are complex amplitudes that only depend on r, and ω = ωr + iωi is the complex phase speed.4 The axial wavenumber k is assumed to be real in this analysis. It is implied that only the real part of the solution has physical relevance (e.g., Markowski and Richardson 2010, chapter 6; Holton and Hakim 2013, chapter 5), which is sometimes written explicitly by adding the complex conjugate to the above normal modes (e.g., Fabre et al. 2006). Upon inserting these normal modes into the linearized governing equations, one finds that
iu^g+2 Vr υ^=1ρdp^dr,
iυ^g−[Vr+dVdr]u^=imρp^r,
w^g=kp^ρ,
du^dr+u^r+imυ^r+ikw^=0,
where
g=ω−Vr m−Wk
is the intrinsic or Doppler-shifted angular wave frequency (i.e., the frequency observed in a reference system following the local base-state flow).

b. Equations for the radial and azimuthal velocity

Kelvin’s approach is based on obtaining equations for the radial and azimuthal velocities as a function of w^. To find the equation for u^, the variables p^ and υ^ must be eliminated from Eqs. (19)–(21). The details of the required manipulations are presented in appendix A. The result is
u^=i gkd[g dw^dr−mw^r⁡(Vr+dVdr)],
where
d≡2 Vr⁡(Vr+dVdr) − g2.
To obtain the υ^ equation, one needs to eliminate p^ and u^ from Eqs. (19)–(21), which is also carried out in appendix A, resulting in
υ^=1kd{mr[⁡(Vr)2 − ⁡(dVdr)2− g2]w^+g[Vr+dVdr]dw^dr}.
Now one may consider an inner and an outer region, and in each region separately specify the base-state azimuthal wind profile V(r) and a constant base-state axial flow W. While the axial flow is uniform within each of the two regions, it may be different in each region. This simplifies Eqs. (24) and (26), which are subsequently inserted into the continuity equation, Eq. (22), to obtain an ordinary differential equation for w^⁡(r). Before applying this approach to different scenarios, we slightly deviate from Kelvin’s original presentation and introduce the general formulation of matching conditions along the free, perturbed boundary between the two regions.

c. Conditions at the free boundary

In the inner region, defined by r < R, variables will have the suffix 1. The variables pertaining to the outer region, r > R, will be marked with the suffix 2. The free boundary Rb between these two regions is perturbed by the waves, and its location given by
Rb⁡(θ,z,t)=R+r′⁡(θ,z,t)=R+r^ei⁡(kz+mθ−ωt).

1) Kinematic boundary condition

For the kinematic boundary condition, the displacement on each side of the boundary needs to be matched (Drazin and Reid 1981, p. 76). To achieve this, first the normal velocity at the boundary is calculated:
DRbDt=u′=∂r′∂t+υr∂r′∂θ+w ∂r′∂z,
and linearized using Eqs. (6) and (7):
u′≈∂r′∂t+Vr∂r′∂θ+W ∂r′∂z.
Inserting the expression for r′ [Eq. (27)] in Eq. (29) results in
u′=−iωr′+i Vr mr′+iWkr′=−igr′,
and solving for the boundary displacement gives
r′=iu′g.
The matching condition, r1′=r2′, is thus
u1′g1=u2′g2,
or equivalently,
u^1g1=u^2g2.
Next, the dynamic boundary condition is specified.

2) Dynamic boundary condition

For the dynamic boundary condition, continuity of pressure is enforced across the perturbed free boundary to prevent infinite pressure gradient accelerations. Including the contributions from the base state and its perturbation on each side:
P1⁡(Rb)+p1′⁡(Rb,z,θ,t)=P2⁡(Rb)+p2′⁡(Rb,z,θ,t).
Expanding pressure in a Taylor series around R approximately gives
[P1⁡(R)+dP1dr|r=Rr′]+[p1′⁡(R,z,θ,t)+∂p1′∂r|r=Rr′]
=[P2⁡(R)+dP2dr|r=Rr′]+[p2′⁡(R,z,θ,t)+∂p2′∂r|r=Rr′].
Retaining only the first-order terms yields the dynamic boundary condition:
[P1⁡(R)+dP1dr|r=Rr′]+p1′⁡(R,z,θ,t)=[P2⁡(R)+dP2dr|r=Rr′]+p2′⁡(R,z,θ,t).
The first term in the brackets on either side of this equation represents the base-state pressure at the unperturbed boundary, and the second term in the brackets describes the variation of the base-state pressure along the perturbed boundary. The third term is the perturbation pressure at the unperturbed boundary.
Since the dynamic boundary condition is also fulfilled when there are no perturbations, P1 = P2, and the condition reduces to (see also Gallaire and Chomaz 2003)
dP1dr|r=Rr′+p1′⁡(R,z,θ,t)=dP2dr|r=Rr′+p2′⁡(R,z,θ,t).
The displacement r′ may be eliminated using the kinematic boundary condition, Eq. (31), giving
i dP1dr|r=R u1′g1+p1′⁡(R,z,θ,t)=i dP2dr|r=R u1′g1+p2′⁡(R,z,θ,t),
or
p1′⁡(R,z,θ,t)=p2′⁡(R,z,θ,t)+i u1′g1[dP2dr−dP1dr]r=R.
Equivalently,
p^1⁡(R)=p^2⁡(R)+i u^1g1[dP2dr−dP1dr]r=R.
Instead of u^1/g1, one may also use u^2/g2, per Eq. (33). With this, all the tools needed to find the wave solutions are available. Kelvin remarked that “crowds of exceedingly interesting cases present themselves” (Thomson 1880, p. 157). Following Kelvin, the first scenario considered is the simplest one, with the goal of gaining an intuition for the structure and general behavior of the waves.

5. Solid-body rotation in a bounded domain

a. General solution and boundary condition

The case of solid body rotation in a bounded domain may be realized using a rotating, fluid-filled cylindrical vessel with rigid boundaries, which is a special case of the Couette flow (e.g., Kundu and Cohen 2008, p. 303). This scenario has been studied extensively in laboratory experiments (e.g., Chandrasekhar 1961 chapter VII; Fultz 1959). Here,
W1=W=0, and
V1⁡(r)=V⁡(r)=Ωr,
where Ω is the uniform angular velocity of the fluid, and the rigid cylindrical boundary is located at r = R. With this, dV/dr = V/r = Ω. These conditions are inserted in Eqs. (24) and (26). First, it is noted that d [Eq. (25)] becomes
d=4Ω2−⁡(ω−mΩ)2
and
g=ω−mΩ.
Inserting this in Eq. (24) gives
u^=i gk[4Ω2−g2][g dw^dr−2Ωmr w^].
For the azimuthal velocity, Eq. (26), one finds
υ^=gk[4Ω2−g2][2Ω dw^dr−mgr w^].
These expressions for the velocity components are substituted into the continuity equation, Eq. (22), giving
igk⁡(4Ω2−g2)[g d2w^dr2−2Ωm 1rdw^dr+2Ωmr2 w^]
+ igk⁡(4Ω2−g2)[g 1rdw^dr−2Ωm w^r2]+i mrgk⁡(4Ω2−g2)[2Ω dw^dr−mgr w^]
+ikw=0.
Several terms cancel and one obtains
d2w^dr2+1rdw^dr+[k2⁡(4Ω2g2−1) − m2r2]w^=0.
Defining
β2≡k2[4Ω2g2−1],
gives
d2w^dr2+1rdw^dr+[β2−m2r2]w^=0.
This equation can be brought into a more convenient form by putting r = x/β, so that d/dr = β(d/dx), which leads to
x2 d2w^dx2+x dw^dx+(x2−m2) w^=0.
This is Bessel’s ODE, which, along with the boundary conditions, specifies an eigenvalue problem with β2 as the eigenvalue, and with w^⁡(r), which describes the radial structure of the wave, as the eigenvector (or eigenfunction). To solve this equation, one uses Frobenius’s method, inserts a power-series expansion w^⁡(x)=xμ∑n=0∞anxn into Eq. (54), and works out expressions for μ and the coefficients an. The power series then takes the following form (e.g., Arfken et al. 2013, p. 352; Deal 2018):
Jm⁡(x)=⁡(x2)m∑j=0∞ ⁡(−1)jj! ⁡(m+j)!⁡(x2)2j.
Further analysis (Arfken et al. 2013, p. 336 and 667; Deal 2018) shows that for integer m the general solution of Eq. (54) is
w^⁡(r)=AJm⁡(βr)+BYm⁡(βr),
where A and B are arbitrary constants, and Jm(βr) and Ym(βr) are, respectively, the Bessel functions of the first and second kind, of order m. These are shown in Fig. 6. The Bessel function of the first kind of order m = 0, J0(x), (red curve in Fig. 6a) qualitatively behaves like a damped cosine wave, and J1(x) (dashed black curve in Fig. 6a) behaves similarly to a damped sine wave. Since Ym(βr) is negatively infinite at r = 0, this solution is rejected, so that
w^⁡(r)=AJm⁡(βr).
The eigenvalue β2 represents the squared radial wavenumber. Because the boundary is rigid, the kinematic boundary condition is simply that the normal velocity at the boundary should vanish, so u^⁡(R)=0.5 Since the flow is inviscid, the flow parallel to the boundary, (υ, w), is free-slip. Using the equation for the radial velocity, Eq. (46), and the solution, Eq. (57), the boundary condition becomes
u^⁡(R)=igk[4Ω2−g2][AgβJm′⁡(βR) −A 2ΩmR Jm⁡(βR)]=0.
Here the chain rule has been used,
dw^dr=A ddr Jm⁡(βr)=AβJm′⁡(βr),
where the prime denotes the derivative with respect to the argument (βr). Equation (58) then becomes
βRgJm′⁡(βR) − 2ΩmJm⁡(βR)=0,
which may be written as
Jm′⁡(βR)Jm⁡(βR)=2mΩβRg.
The goal is to find the β values for which this equation is fulfilled. One then solves Eq. (52) for the wave frequency ω (which is defined via g) and obtains the dispersion relation. The ratio of the Bessel functions behaves qualitatively like the (negative of the) tangent function, implying that this equation has a countably infinite set of solutions.
Fig. 6.
Fig. 6.

(a) Bessel function of the first kind, Jm(x), of order m = 0 (red), m = 1 (dashed black), and m = 2 (dotted blue); (b) Bessel function of the second kind, Ym(x), of order m = 0 (red), m = 1 (dashed black), and m = 2 (dotted blue).

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

b. Axisymmetric modes in a bounded domain

For m = 0, one can find the solution of Eq. (61) right away. In this case the rhs is zero and use can be made of the fact that J0′⁡(βR)=−J1⁡(βR) (Arfken et al. 2013, p. 646), resembling the relationship d/dx(cosx) = −sinx. The fraction on the lhs of Eq. (61) becomes zero only where the numerator J1(βr) is zero [the zeros of the denominator, which make the equation singular, do not coincide with J1(βr) = 0]. It follows that the requirement for a solution is
J1⁡(βR)=0.
These roots are tabulated, the first three being βjR ≡ αj = 3.8317, 7.0156, 10.1735, j = 1, 2, 3. The different βj values correspond to the different discrete radial modes of the waves, which will be introduced in the next paragraph. Since βj may be interpreted as radial wavenumber, the root α0 = 0 is not considered here. Now Eq. (52) can be used to determine gj(k):
⁡(αj)2=R2k2[4Ω2gj2−1].
Solving for the wave frequency, noting that now gj(k) = ωj(k), yields
ωj⁡(k)=±2Ω⁡(αj/kR)2+1,
which is the desired dispersion relation for the axisymmetric Kelvin vortex waves in a fluid in solid-body rotation. The axial (here, vertical) phase speed is given by
cj=ω⁡(k)k=±2Ω⁡(αj/R)2+k2.
The dispersion relation for the first three radial modes is plotted in Fig. 7. The positive branches pertain to upward propagating waves, and the negative branches to downward propagating waves. Given the symmetry about the horizontal axis, only the positive branches of the phase and group speeds are shown. It follows that the lowest-order axisymmetric radial mode, often referred to as the fundamental mode, moves faster than the higher-order modes. Moreover, the phase and group speeds coincide for small k, and the longest axisymmetric waves of the fundamental radial mode propagate the fastest.
Fig. 7.
Fig. 7.

Dispersion relation for axisymmetric Kelvin vortex waves bounded by a cylinder. (a) Wave frequency as a function of axial wavenumber, (b) axial phase speed as a function of axial wavenumber, and (c) axial group speed as a function of axial wavenumber. The different colors pertain to the radial modes as shown in the legend. Here Ω = 1 s−1, R = 1 m, and W = 0 m s−1. The solid curves in (a) approach 2 s−1 for large k, and the dashed curves approach −2 s−1 for large k.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

To gain insight into the structure of the different modes in meridional [(r, z)] plane, one uses the equations for w^ [Eq. (57)] and u^ [Eq. (46)], and we put w1 ≡ A. Since the problem at hand is governed by a linear, homogeneous ODE, the amplitude w1 may be selected arbitrarily. Then, starting with the radial velocity equation, Eq. (46), the solution w^⁡(r)=w1Jm⁡(βr) is inserted:
u^=ik[4Ω2−g2][g2βw1Jm′⁡(βr) − 2Ωmgr w1Jm⁡(βr)].
Per Eq. (52) the denominator is equal to β2g2/k, and one may write
u^=iw1kβ2g[gβJm′⁡(βr) − 2Ωmr Jm⁡(βr)].
If m = 0 this equation becomes
u^=i w1kβ J0′⁡(βr).
Using again J0′⁡(βr)=−J1⁡(βr),
u^=−i w1kβ J1⁡(βr)=u1J1⁡(βr),
with the amplitude
u1≡−iw1 kβ.
This shows that aside from the phase shift between the u and w waves, the vertical motion has a different magnitude than the radial motion; for instance, for small k, u1 ≪ w1, and the perturbation flow is dominated by the vertical perturbation velocity (i.e., the perturbations are manifest mainly as axial jets; the reader may skip ahead to Figs. 17e,f for an example), and for large k, the perturbation flow is mainly in the radial direction.
The full equation for the perturbation vertical velocity is
w′=w1J0⁡(βr)ei⁡(kz−ωt),
or, taking the real part,
w′=w1J0⁡(βr) cos⁡(kz−ωt).
For the radial velocity using Eq. (69), one finds
u′=w1kβ J1⁡(βr) sin⁡(kz−ωt).
Assuming the vortex is bounded by a cylinder with radius R, the boundary condition, u^⁡(R)=0, is fulfilled only if βj = αj/R = 3.8317/R, 7.0156/R, 10.1735/R, …. Since β may be interpreted as a radial wavenumber, these solutions dictate the radial structure of the wave. For the smallest eigenvalue one obtains a single cell in the radial direction, which is the classic “sausage” mode, shown in Figs. 8a and 8b. For the third eigenvalue, the result is three circulation cells in the radial direction, because the radial velocity distribution now has three roots. This configuration is shown in Figs. 8c and 8d. The higher the order of the solution, the more cells appear in the radial direction. The magnitude of these cells slowly decays with increasing distance from the axis. These different solutions are the radial modes (Alekseenko et al. 2007). Fultz (1959) presents laboratory observations of the fundamental and the second radial modes of these waves. Theoretically, for each azimuthal wavenumber there are infinitely many radial solutions [in reality, the high-order modes tend to be dampened by viscous effects (Arendt et al. 1997; Fabre et al. 2006)]. Shapiro (2001a) arrived at solutions for nonlinear axisymmetric centrifugal waves with a similar radial structure as Kelvin vortex waves, although he assumed no radial boundary and the wave perturbations increased in magnitude with r.
Fig. 8.
Fig. 8.

Wave structure in the meridional plane of upward-propagating, axisymmetric Kelvin vortex waves bounded by a cylinder. (a) Perturbation flow field and (b) perturbation streamlines of the fundamental mode. The red lines indicate the instantaneous displacement of a material surface at an average location r = 0.8 m. (c),(d) As in (a) and (b), but for the third radial mode. The placements of the unperturbed material surfaces are at r = (3, 6, 9) m. Here Ω = 1 s−1, W = 0 m s−1, k = 1.5 m−1, and u1 = 1 m s−1. The rigid boundary is located at R = 1 m in (a) and (b) and at R = 10 m in (c) and (d), marked by the thick black line.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

Restoring force and propagation mechanism of the axisymmetric mode

To gain a better understanding of the propagation mechanism of the axisymmetric wave, it helps to consider the restoring force responsible for the oscillation. Markowski and Richardson (2010, p. 49) show that in the inviscid limit the radial displacement of a ring of parcels is governed by the following linear equation:
D2rDt2+1r3Γ¯2dr r=0,
where Γ¯=Vr is proportional to the base-state circulation, which corresponds to the angular momentum. This pertains to the shortwave limit where the perturbation flow is mostly in the radial direction per Eq. (70). Equation (74) is a linear harmonic oscillation equation if ⁡(1/r3)⁡(Γ¯2/dr)>0. In this context, the squared oscillation frequency, ω2=(1/r3)⁡(dΓ¯2/dr), is referred to as the Rayleigh discriminant (e.g., Chandrasekhar 1961). The restoring forces are the radial pressure gradient force and the centrifugal force. The oscillation frequency ω is related to the rate at which the ambient pressure gradient changes in the radial direction, which is linked to Γ¯ via cyclostrophic balance. A ring displaced in the positive radial direction is losing azimuthal speed due to angular-momentum conservation. Now if the circulation increases outward such that ω2 > 0, the radially displaced ring will find itself in a stronger pressure gradient than required for cyclostrophic balance. The result is an inward radial acceleration (i.e., stability; e.g., Markowski and Richardson 2010, p. 49). If the Rayleigh discriminant is negative, the displacement is unstable, which results in radial accelerations leading to toroidal circulation cells (e.g., Kundu and Cohen 2008, p. 486). For common vortex profiles, such as the Rankine and Burgers–Rott profiles (e.g., Wood and White 2011), or the Lamb–Oseen vortex (e.g., Fabre et al. 2006), Γ¯ increases in the outward direction in the core, so axisymmetric displacements are stable there (and neutral where Γ¯=const as, e.g., in the periphery of the Rankine vortex).

To understand why the phase speed depends on the axial wavenumber, consider the variation of the width of the vortex (e.g., as measured by the radial displacement of a concentric material surface relative to the base state), which serves as a measure of vertical vorticity. The narrower the vortex, the larger the vertical vorticity due to angular-momentum conservation. As shown in Fig. 9, which depicts an upward propagating wave, the regions of horizontal convergence near the vortex axis are shifted upward by a quarter wavelength relative to the axial vorticity maxima, implying that the vorticity extrema are propagating upward via vortex stretching and compression (see also Shapiro 2001a; Fabre et al. 2006). The regions of horizontal divergence and convergence are associated with meridional vorticity, which arises from tilting of the axial vorticity into and out of the meridional plane by the axial gradients of azimuthal velocity. The periodic vertical motion induced by the meridional vorticity stretches/compresses the axial vorticity. This is associated with a twisting and untwisting of the vortex lines (Melander and Hussain 1994; Arendt et al. 1997). The longer the wave, the larger are the pressure perturbations that accompany the vertical vorticity extrema [see, e.g., Markowski and Richardson (2010, p. 27) for the relationship between vorticity and pressure] or alternatively, the larger the induced velocity magnitudes (e.g., Dahl 2020) associated with the meridional vorticity extrema. As a consequence, as the vertical wavelength increases there are stronger vertical pressure-gradient accelerations and w′ gradients, resulting in more vigorous stretching/compression of vertical vorticity and hence faster wave motion.

Fig. 9.
Fig. 9.

As in Fig. 8a, but for k = 1.0 m−1; the labels reveal the locations of extrema of axial vorticity in the vortex center as well as of axial-vorticity stretching. The blue contours represent the azimuthal velocity perturbations υ′ (positive: solid; negative: dashed), implying, e.g., a maximum of vertical perturbation vorticity near the vortex axis where υ′ > 0. Regions of radial convergence and divergence are indicated by the black arrows.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

c. Spiral modes in a bounded domain

With m ≠ 0, the wave frequency, using Eq. (52) with g = ω − mΩ, is given by
ωj⁡(k)=mΩ±2Ω⁡(αj/kR)2+1.
When changing from a left-handed to a right-handed mode, i.e., m → −m with k > 0, the sign of ωj changes, but the dispersion behavior remains the same. Changing the sign of k does not change the wave frequency. Because of this symmetry, we may restrict the analysis without loss of generality to the case of k > 0 and m > 0, which describes the left-handed spiral modes (e.g., Saffman 1992, p. 232; Alekseenko et al. 2007). The dispersion relation, with αj obtained via Eq. (61), of the first three radial modes of the bending wave is shown in Fig. 10. Also displayed (Fig. 10a) is the graphical solution of Eq. (61). The cograde modes are represented by solid lines and have a positive oscillation frequency and axial phase speed (i.e., they move upward), and vice versa for the retrograde modes (Figs. 10b,c). The axial phase speeds behave like k−1 for small k, and thus attain infinite values in the longwave limit (which results from the fact that the phase lines are almost parallel to the z axis). The group speeds, Fig. 10d, remain finite and are always slower than the phase speeds.
Fig. 10.
Fig. 10.

Dispersion relation of the cograde m = 1 spiral modes in a vortex bounded by cylindrical walls. (a) Graphical solution of Eq. (61) for k = 1.2 m−1; (b) wave frequency as a function of axial wavenumber for the cograde (solid) and retrograde (dashed) modes as a function of axial wavenumber, (c) axial phase speed of the cograde (solid) and retrograde (dashed) modes as a function of axial wavenumber, and (d) axial group speed of the cograde (solid) and retrograde (dashed) modes as a function of axial wavenumber. The legend of (c) also applies to (d). The different colors pertain to the radial modes as shown in the legend. Here Ω = 1 s−1, W = 0 m s−1, and R = 1 m.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

The different modes, like in the axisymmetric case, correspond to the radial structure. Insight into the horizontal structure of these waves may be obtained using the expressions for u^ and υ^ [Eqs. (46) and (47)], and inserting the solution for w(r) = w1J1(βr). This gives, using again Eq. (52):
u′⁡(r,z,θ,t)=−w1kgβ2[gβJ1′⁡(βr) − 2Ωmr J1⁡(βr)] sin⁡(kz+mθ−ωt),
and
υ′⁡(r,z,θ,t)=w1kgβ2[2ΩβJ1′⁡(βr) − mgr J1⁡(βr)] cos⁡(kz+mθ−ωt).
To obtain the total flow, V(r) = Ωr must be added to the last equation. These flow fields are plotted in Figs. 11 and 12 . For m = 1, the perturbation flow (Fig. 11a) features a dipole, and the total flow (Fig. 11b) exhibits a displacement of the vortex center. For higher-order radial modes, the perturbation flow becomes more structured in the radial direction, but the total flow is still characterized mainly by a shift of the vortex center (Figs. 11c,d). To demonstrate the richness of structure of the higher-order modes, Fig. 12 shows the m = 4 spiral mode for the fundamental (Figs. 12a,b) and third (Figs. 12c,d) radial modes.
Fig. 11.
Fig. 11.

Wave structure in the horizontal plane of a counterclockwise-propagating Kelvin vortex wave mode m = 1, bounded by a cylinder. (a) Perturbation streamlines and (b) total flow field of the fundamental mode, and (c) perturbation streamlines and (d) total flow field of the third radial mode. Here Ω = 1 s−1, R = 1 m, and k = 1.2 m−1. The amplitude w1 was set to 1 m s−1.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

Fig. 12.
Fig. 12.

As in Fig. 11, but for m = 4.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

Propagation mechanism of the spiral mode

To gain an understanding of the propagation of these modes, the velocity and the pressure fields of the fundamental spiral mode (m = 1) are shown in Fig. 13. The fields are displayed on an unrolled cylindrical surface, i.e., in the (θ, z) plane and for a small axial wavenumber. Like in the axisymmetric case, positive axial vorticity perturbations are associated with a negative pressure perturbation, and negative axial vorticity perturbations are associated with a positive pressure perturbation, so p′ ~ −ζ′, where ζ′ is the axial vorticity perturbation (Fabre et al. 2006). It is apparent that there is axial stretching a quarter wavelength above the perturbation vorticity maxima for the cograde (upward propagating) mode (Fig. 13a). For the retrograde mode, (Fig. 13b), the maximum stretching occurs below the perturbation vorticity maxima. Similarly to the axisymmetric mode, these waves thus propagate due to stretching and compression of axial perturbation vorticity.

Fig. 13.
Fig. 13.

Perturbation pressure (shaded), velocity perturbations (vectors), and vertical stretching (dw′/dz; contoured, in s−1) on the (θ, z) plane. The pressure perturbations are proportional to the negative of the axial vorticity perturbations. The red arrows show the direction of the wave motion. Here m = 1, k = 0.24 s−1, Ω = 1 s−1, W = 0 m s−1, R = 1 m, r = 0.5 m, and w1 = 1 m s−1. (a) Cograde mode; (b) retrograde mode.

Citation: Monthly Weather Review 149, 10; 10.1175/MWR-D-20-0426.1

To summarize this section, each of the azimuthal modes (i.e., axisymmetric and spiral modes) have an infinite number of radial modes, which are characterized by an increasing number of circulation cells in the radial direction with increasing order of the radial mode. The perturbation flow of the fundamental mode has only one circulation cell in radial direction. Moreover, it was highlighted that the longwave limit of the axisymmetric fundamental mode moves the fastest. A better approximation of a tornado-like flow is a vortex not bounded by cylindrical walls, which will be considered next.

6. Rankine vortex

a. General solution and boundary conditions

For the Rankine vortex, one combines the solution for solid-body rotation of the previous section with an irrotational outer region. As mentioned in section 4c, the inner solution is designated by the suffix “1” and the outer solution is designated by the suffix “2.” For now, it is assumed that W1 = W2 = W, such that g1 = g2.

For the outer region, the irrotational azimuthal wind profile is given by (e.g., Kundu and Cohen 2008, p. 70):
V⁡(r)=ΩR2r,
where R may be interpreted as core radius. Inserting this into the equation for the radial velocity, Eq. (24) with d reducing to −g2 per Eq. (25), leads to
u^=i g−g2k[g dw^dr]=−ikdw^dr.
For the azimuthal velocity, Eq. (26), one finds
υ^=1−g2kmw^r⁡ (−g2)=mw^kr.
Like before, these expressions are inserted into the continuity equation, Eq. (22):
d2w^dr2+1rdw^dr−[k2+m2r2]=0.
This equation is similar to Eq. (53) and becomes the modified Bessel differential equation upon substituting r = x/k. The modified Bessel ODE equals the regular Bessel ODE if the argument is imaginary (Arfken et al. 2013, 680–683). If m is an integer, the solution is a linear combination of the Bessel functions encountered in the previous section but with imaginary argument (Arfken et al. 2013, 680–683). The solution of Eq. (81) becomes
w^⁡(r)=AIm⁡(kr)+BKm⁡(kr).
Here again A and B are arbitrary constants, and Im(kr) and Km(kr) are the modified Bessel functions of the first and second kind, respectively. These functions loosely resemble exponential functions, becoming infinite for large arguments in case of Im and for small arguments in case of Km, as shown in Fig. 14. In the present case, the Im solution is unphysical because the perturbations are required to be finite at radial infinity, so only the modified Bessel function of the second kind, Km, is retained. Then,
w^⁡(r)=BKm⁡(kr).
It follows that Kelvin vortex waves are primarily core waves that rapidly decay with increasing radius in the outer region. The last step is to match the boundary displacement and the total pressure at the boundary between the inner and outer regions, as discussed in section 4c. The kinematic condition, Eq. (33), with with g1 = g2 simply enforces continuity of the normal velocities on each side of the boundary. Using Eq. (67) for the inner side, one finds that
u^1⁡(R)=iA kgβ2[gβJm′⁡(βR) − 2 ΩmR Jm⁡(βR)].
The solution at the outer side of the boundary, per Eqs. (79) and (83), becomes
u^2⁡(R)=−i 1k kBKm′⁡(kR)=−iBKm′⁡(kR).
The kinematic boundary condition is thus
A kgβ2[gβJm′⁡(βR) − 2 ΩmR Jm⁡(βR)]=−BKm′⁡(kR).
Considering the dynamic boundary condition, the base-state pressure in the inner region is given by [appendix B, Eq. (B6)]
P1⁡(r)=P∞+ρ2 Ω2[r2−2R2],
so that at r = R,
dP1dr|r=R=ρΩ2R.
In the outer region [appendix B, Eq. (B6)],
P2⁡(r)=P∞−ρ2Ω2R4r2,
and at r = R,
dP2dr|r=R=ρΩ R4r3|r=R=ρΩ2R.
This is equal to Eq. (88) and in this case the dynamic boundary condition, Eq. (41), reduces to
p1′⁡(R)=p2′⁡(R).
From Eq. (21) it follows that p^=(ρg/k)w^, so the dynamic boundary condition is equivalent to equating w^ on each side of the boundary as was done by Kelvin (Thomson 1880). So,
g ρk AJm⁡(βR)=g ρk BKm⁡(kR),
or simply
AJm⁡(βR)=BKm⁡(kR).
Equations (86) and (93) form a linear system of equations, which has a nontrivial solution if the determinant of the coefficient matrix vanishes:
|kgβ2[gβJm′⁡(βR) − 2 ΩmR Jm⁡(βR)]Km′⁡(kR)Jm⁡(βR)−Km⁡(kR)|=0.
The requirement for a nontrivial solution is thus
kgβ2[gβJm′⁡(βR) − 2 ΩmR Jm⁡(βR)][−Km⁡(kR)] − Km′⁡(kR)Jm⁡(βR)=0.
Dividing through by Jm(βR)Km(kR) and some minor rearrangement gives the desired expression:
1βRJm′⁡(βR)Jm⁡(βR) − 2Ωmgβ2R2=−1kRKm′⁡(kR)Km⁡(kR).
This is the dispersion relation derived 140 years ago by Kelvin [his Eq. (50)], and it must be solved for β, which yields ω via Eq. (52). A few words about the stability of the waves described by this dispersion relation are in order. For unstable growth, the equation must be fulfilled for an imaginary wave frequency ωi, making g imaginary. However, the rhs of Eq. (96) is always real, so the imaginary part of the lhs must also be zero. Kelvin vortex waves in a Rankine vortex with uniform or vanishing axial base-state flow are thus stable. The solutions of this transcendental equation may be carried out graphically or numerically. Before proceeding with the numerical solutions, an important and in Kelvin’s words “curiously interesting” limit is considered.
Fig. 14.