Angular Momentum Transports by Moving Spiral Waves

K. C. Chow Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

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Kwing L. Chan Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, China

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Abstract

Theoretical results based on the shallow water equations show that outgoing spiral waves generated by a nonaxisymmetric vortex can effectively remove angular momentum from the vortex, and the removal rate is sensitive to the strength of the vortex. When the model is applied to tropical cyclones, the results suggest that radiation of spiral waves may play a noticeable role in the total angular momentum budget of the core and affect the intensification of a tropical cyclone.

An Eliassen–Palm relation is derived for the shallow water equations. It shows that the total angular momentum carried by shallow water waves does not change during their propagation through an axisymmetric mean flow.

Corresponding author address: K. C. Chow, Dept. of Physics and Material Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong, China. Email: apkchow@cityu.edu.hk

Abstract

Theoretical results based on the shallow water equations show that outgoing spiral waves generated by a nonaxisymmetric vortex can effectively remove angular momentum from the vortex, and the removal rate is sensitive to the strength of the vortex. When the model is applied to tropical cyclones, the results suggest that radiation of spiral waves may play a noticeable role in the total angular momentum budget of the core and affect the intensification of a tropical cyclone.

An Eliassen–Palm relation is derived for the shallow water equations. It shows that the total angular momentum carried by shallow water waves does not change during their propagation through an axisymmetric mean flow.

Corresponding author address: K. C. Chow, Dept. of Physics and Material Science, City University of Hong Kong, Tat Chee Avenue, Kowloon Tong, Hong Kong, China. Email: apkchow@cityu.edu.hk

1. Introduction

It is well known that the angular momentum transport in a tropical cyclone plays a key role in determining the cyclone's intensity. Angular momentum budget analysis of tropical cyclones, such as Holland (1983), has shown that the horizontal eddy flux is the main transport mechanism in the outer region (6°–10° latitude radius) of the cyclone, while, in the innermost region (0°–2° latitude radius), the contribution by the mean flow is dominant. Although such a role played by the horizontal eddies has been recognized, the generation mechanism of eddies, as well as the factors affecting the transport of angular momentum by these eddies, have not been clarified. Atmospheric waves propagating horizontally in the outer region account for one kind of these eddies, and the eddy processes associated with waves can be more readily analyzed. Transport of angular momentum by swirl waves in tornadolike vortices has been investigated by Chimonas and Hauser (1997). They considered free swirl waves propagating in a static mean atmosphere and found that swirl waves are very effective in damping out the core vortex that excites the waves.

Elliptical distributions of vorticity are commonly found in cores of tropical cyclones. In Chow et al. (2002, hereafter CCL), the spiral wave field generated by a rotating elliptical vortex has been determined in the framework of the shallow water model. In this paper, the role played by these waves in transporting angular momentum will be investigated.

Shallow water equations are widely used in studies of geophysical fluid dynamics. In the appendix of this paper, an Eliassen–Palm relation will be derived for this system of equations. It is shown that the total angular momentum carried by spiral shallow water waves is conserved during propagation through an axisymmetric mean flow. Based on this property, angular momentum transport to the outer region by waves excited at the core of a tropical cyclone can be estimated.

2. Horizontal velocities of a linear spiral shallow water wave

In CCL, a shallow water wave is expressed in terms of a perturbation in the geopotential. In order to compute the angular momentum carried by the wave, we need to first find the perturbations of the velocity components associated with the wave. The basic governing equations used in the present analysis are the f-plane shallow water equations in cylindrical coordinates. If the wave field of the geopotential perturbation is given, the corresponding perturbations in the horizontal velocity components can be obtained from the momentum equations.

We assume that the magnitudes of all the perturbed quantities are small compared with those of the basic flow, and an arbitrary variable x can be decomposed as the sum of a steady axisymmetric basic state and a time-dependent nonaxisymmetric perturbation field; that is, x(r, λ, t) = x(r) + x′(r, λ, t), where r and λ are the radial and azimuthal coordinates, respectively. By neglecting high-order terms that involve products of perturbations, the horizontal velocity perturbations can be related to the geopotential height perturbation by the linearized momentum equations as
i1520-0469-60-16-2004-e1
where ϕ′ = gh′, ζ = f + Ω + ∂υ/∂r, κ2 = (2Ω + f)ζ, and d/dt = ∂/∂t + V · ∇ = ∂/∂t + (υ/r)(∂/∂λ). In the above equations, u and υ are the radial and azimuthal velocities; ϕ is the geopotential; h is the height; f is a constant Coriolis parameter; and Ω = υ/r, ζ, and κ are the local angular speed, the absolute vorticity, and the local inertia frequency of the mean flow, respectively.
Equation (1) can be written in the form of small oscillations forced by ϕ′:
i1520-0469-60-16-2004-e2
For a spiral wave with m arms, the geopotential per-turbation can be written in the form ϕ′ = ϕ̂ cosφ, where ϕ̂ is a function of r, and φ = + krωt (m and k are wavenumbers in azimuthal and radial directions; ω is the wave frequency). The corresponding perturbations in the horizontal velocities can be derived from (2).
With the solutions of u′ and υ′ expressed in the form u′ = C cosφ + D sinφ and υ′ = E cosφ + F sinφ, where the coefficients C, D, E, and F are functions of r only, Eq. (2) gives
i1520-0469-60-16-2004-eq1
where ν = m(Ω − Ωp)/κ is a dimensionless frequency, with Ωp = ω/m being the pattern frequency. Assuming that the inertia frequency does not vanish (κ ≠ 0) and that the rotational speed of the wave is faster than that of the local azimuthal mean flow in the wave propagation region so that ν2 ≠ 1, we have
i1520-0469-60-16-2004-e3
with C = νκϕ̂k, D = νκ(dϕ̂/dr) + (m/r)(2Ω + f)ϕ̂, E = ζ(/dr) + (mνκ/r)ϕ̂, and F = −ζϕ̂k.

3. Angular momentum loss of an elliptical vortex

As in Chimonas and Hauser (1997), the radial current of angular momentum at radius r is given by
i1520-0469-60-16-2004-eq2
where (·) = ½π 2π0 (·) represents taking the azimuthal mean; ρ is the constant density of the fluid. In the present analysis, the configuration of the domain can be considered to be composed of two concentric cylindrical regions: the outer and inner regions. The basic flow is axisymmetric in the outer region. The waves traveling in the outer region with radial current of angular momentum I are generated by processes in the inner region.
In the appendix, an Eliassen–Palm relation for the shallow water equations is derived in a way similar to that of Andrews and McIntyre (1978). This relation shows that for quasi-steady shallow water waves propagating in a region with axisymmetric mean flow and constant density, the radial current of angular momentum is constant. Therefore, if the waves come from an inner region (wave-generation region) with total angular momentum L, the rate of angular momentum loss in the inner region due to the emitted waves is
i1520-0469-60-16-2004-eq3
Since the objective of this paper is not to explain the creation of the vorticity in the inner region, we are going to ignore any other angular momentum source/sink. The fractional decrease of the angular momentum of the inner region in a time period corresponding to one full rotation of the wave pattern (Δt = 2πp) can be estimated as
i1520-0469-60-16-2004-e4

In the following analysis, we estimate the rate of decrease of angular momentum for an elliptical vortex that generates spiral shallow water waves by the Lighthill radiation mechanism as discussed in CCL.

The shape of the elliptical vortex is taken to be a small deviation of a circle with radius a; it has semimajor/minor axes equal to a(1 ± ϵ), and the positive parameter ϵ is ≤1. The vortex rotates with the angular velocity Ωp = ω/2. The spiral wave solution obtained by CCL is a far field solution, applicable to waves in the outer region.

The first step is to estimate the current of angular momentum of the spiral wave. From (3), and using the equalities sin2λ = cos2λ = 1/2 and sinλ cosλ = 0, we have
i1520-0469-60-16-2004-e5
The spiral wave solution given by Eq. (9) of CCL is
i1520-0469-60-16-2004-eq4
where ζ0 is the maximum vorticity in the elliptical region, and c = gh0 is the radial wave speed of the spiral wave, with h0 being a constant reference height. Parameter A describes the vorticity distribution; A = 1 corresponds to the case where the vorticity is constant throughout the elliptical region. This equation implies that
i1520-0469-60-16-2004-e6
From (5) and (6) and the phase speed relation k = ω/c, the radial current of angular momentum I at radius r transported by the spiral wave can be computed as
i1520-0469-60-16-2004-e7
In the outer region, the rotational speed of the spiral pattern is generally greater than the mean azimuthal wind speed, and the amplitude of the Doppler-shifted frequency ωmΩ (m > 0) is larger than the local inertia frequency κ (as in the case of moving spiral bands discussed in CCL); the current of angular momentum is positive. It means that trailing spiral waves always export angular momentum.

The second step is to estimate the total angular momentum L of the wave-generation region. Following CCL, we model this core region as a rotating elliptical vortex.

The vorticity distribution is taken to concentrate in an elliptical region and is monotonic decreasing from the center. The value of vorticity ζ is equal to a constant ζ0 from r = 0 to r = a0(1 + ϵ cos2λ), and decreases linearly to a minimum value ζm when the mean distance (the amplitude factor of r) increases from a0 to a. As ϵ is assumed to be small, we can get a good estimate of L by approximating the region with a circular disk of radius a and constant vorticity ζ0. This circular region has the same area as the elliptical region. By choosing the maximum vorticity ζ0 as the constant vorticity, this approximation yields a value larger than the total angular momentum of the elliptical vortex, and gives a lower bound for estimating the percentage decrease of angular momentum in (4). Since the velocity υ within this constant vortical region is given by υ = 0/2, the total angular momentum is then
i1520-0469-60-16-2004-e8
In deriving (8), the depth h inside the integral is taken to be a constant equal to the value at the outer boundary of the inner region. In the circular flow, h actually decreases inward. As before, this approximation also produces a larger value for the total angular momentum L and a lower bound for estimating the percentage decrease of angular momentum. Substituting (7) and (8) into (4), one obtains
i1520-0469-60-16-2004-e9
Since (ωmΩ)2κ2ω2, we can again get a lower-limit estimate for the loss if we replace this factor in the above expression by ω2. Using ω = mΩp, we get
i1520-0469-60-16-2004-e10

From (10), we can see that the loss of angular momentum per period of pattern revolution is mainly determined by three parameters:

  1. the number of arms m,

  2. the rotational speed of the pattern Ωp, and

  3. the central vorticity ζ0.

Therefore, spiral waves with more arms can spin down the central region more efficiently, and the rate is a rather steep function of m. The second and third parameters are closely related; the rotational speed of the pattern is basically proportional to the vorticity of the vortex. For example, in a Kirchoff vortex, which is an elliptical region of constant vorticity ζ0 surrounded by irrotational fluid, the rotational speed of the ellipse is exactly ζ0/4. Thus, the rate of angular momentum loss is approximately proportional to the fourth power of the central vorticity.

In the case of a tropical cyclone, the magnitude of the vorticity in the core region actually reflects its intensity. If the result of the above analysis is applied to a tropical cyclone, it can be inferred that the loss of angular momentum from the core to the spiral waves increases sharply when the tropical cyclone intensifies. This loss mechanism may play a role in the cyclone intensification process, especially in the late stages.

To produce some numbers as estimates, we deduce the parametric values from the simulated tropical cyclone model of CCL and apply them to (10). By setting m = 2, a = 32.75 km, ϵ = 0.145, A = 0.517, c = 62 m s−1, ζ0 = 55 × 10−4 s−1, and Ωp = 2π/75 min, we get
i1520-0469-60-16-2004-eq5
In each revolution of the spiral pattern, the angular momentum of the central vortex can be reduced by about 13%. This estimate, though very crude, suggests that angular momentum transport by spiral waves could reach a rather high magnitude.

4. Summary and discussion

In this paper, we consider the situation that spiral waves are generated in a compact inner region and propagate to an outer region where the far-field solution for a wave is applicable. If the medium can be considered as nondissipative, the angular momentum carried by the waves is conserved during propagation. This is demonstrated for the shallow water waves through the derivation of the Eliassen–Palm relation in the appendix. Modeling the inner region as a concentrated elliptical distribution of vorticity, CCL have derived an analytical expression for the emitted spiral wave in terms of the geopotential height. Here, we derive the expressions for the wave velocity components from which the angular momentum current can be computed. This current is found to be highly sensitive to the number of arms in the spiral wave and to the spin rate of the core vortex.

Using data from the simulated cyclone of CCL, we obtain an example to estimate the efficiency of the wave process. For this particular case, the amount of angular momentum carried away by the emitted waves during each pattern revolution may reach 13% of the total angular momentum of the core vortex. The magnitude of this value may sound surprisingly large, but it is still much less than that given by Chimonas and Hauser (1997) for a similar problem (about 30%). At any rate, it is only a rough estimate. First, the parameters used to evaluate the fractional loss rate in Eq. (10) appear with large exponents, the errors in estimating the parameters from the numerical simulation can be significantly amplified. To extract the parameters of the idealized model from the data of the simulated cyclone (with full physics) does involve certain inexact procedures. Second, the theoretical results obtained in this paper are based on shallow water dynamics with many idealized approximations (including the low Froude number assumption made by CCL). This simplified model can, at most, provide a rough description of a realistic situation.

The loss estimate may also be biased by the particular example chosen. The parameters used to compute the loss rate are obtained at an instance when the simulated cyclone is in a supertyphoon stage. During this phase the vorticity in the core region is extremely high, and the pattern speed of the spiral pattern is also very high (period = 75 min). This may correspond to a relatively rare situation for real tropical cyclones. Because the pattern speed depends approximately linearly with the core vorticity, and the relative loss rate given by Eq. (10) is proportional to the fourth power of the vorticity, the rate will drop to about 5% if the core vorticity is reduced to 0.8 of its original value. Despite the uncertainties, the analytical formula does provide information on which parameters participate and how sensitive the wave transport depends on the parameters.

The model suggests the possibility that moving spiral bands may affect the total angular momentum balance at the core region of a tropical cyclone. From the shallow water model, we know that one important requirement for the generation of waves is the persistence of an elliptical core vortex. It has been discussed in CCL that observations of tropical cyclones do show close association of elliptical eyes and two-arm spiral bands. According to Kossin et al. (2000), elliptical eyes are usually observed during the nonintensifying stages. This indicates that the wave emission mechanism could have a role in limiting the maximal intensity of the cyclone.

Finally, it is necessary to remark that, though the loss of angular momentum due to spiral waves could reach high values at certain stage, analysis of the angular momentum budget in numerical simulations shows that the rate of intensity change of a tropical cyclone is generally determined by the approximate balance between the horizontal convergence of angular momentum carried by the mean flow and the dissipation by surface friction. However, since the contributions from these two processes almost cancel with each other, an additional mechanism may tip the balance and exert significant influence on the cyclone intensity. Besides wave emission, there are many other mechanisms that may influence the intensity of a cyclone. For example when a tropical cyclone blows over a mountain, the disrupted flow at the boundary layer can enhance momentum loss. Furthermore, angular momentum of a tropical cyclone may also change when it is traveling in an environment with vertical shear of horizontal wind. However, these are beyond the scope of the present study.

Acknowledgments

This work was partially supported by the research Grant WITI93/94.RC01 (R5326) of the Center of Coastal and Atmospheric Research (CCAR) at HKUST, and the RGC Grant HKUST6118/00P.

REFERENCES

  • Andrews, D. G., and M. E. McIntyre, 1978: Generalized Eliassen–Palm and Charney–Drazin theorems for waves on axisymmetric mean flows in compressible atmospheres. J. Atmos. Sci., 35 , 175185.

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  • Chimonas, G., and H. M. Hauser, 1997: The transfer of angular momentum from vortices to gravity swirl waves. J. Atmos. Sci., 54 , 17011711.

    • Search Google Scholar
    • Export Citation
  • Chow, K. C., K. L. Chan, and A. K. H. Lau, 2002: Generation of moving spiral bands in tropical cyclones. J. Atmos. Sci., 59 , 29302950.

    • Search Google Scholar
    • Export Citation
  • Holland, G. J., 1983: Angular momentum transports in tropical cyclones. Quart. J. Roy. Meteor. Soc., 109 , 187209.

  • Kossin, J. P., W. H. Schubert, and M. T. Montgomery, 2000: Unstable interactions between a hurricane's primary eyewall and a secondary ring of enhanced vorticity. J. Atmos. Sci., 57 , 38933917.

    • Search Google Scholar
    • Export Citation

APPENDIX

Eliassen–Palm Relation for Shallow Water Equations

The Eliassen–Palm relation states that for quasi-steady waves without dissipation, the total angular momentum carried by the waves does not change during propagation through the mean flow. The derivation of this relation for the shallow water equations, here, is basically parallel to that by Andrews and McIntyre (1978), who derived the relation for the three dimensional compressible Euler equations.

Let ξ be the O(a) particle displacement vector associated with the wave motion. The length scale a is considered to be small. The displacement vector satisfies the following equations:
i1520-0469-60-16-2004-ea1
where Vl is the O(a) approximation to the Lagrangian velocity perturbation associated with the displacement, (·) represents the axisymmetric mean value, and (·)′ is the deviation from the mean value.
In cylindrical coordinates, the horizontal components of the perturbation can be expressed as
i1520-0469-60-16-2004-ea2
for u = 0 or as small as O(a2), and d/dt = ∂/∂t + (υ/r)∂/∂λ. In the fourth equation of (A2), the term ηυ/r does not appear due to a cancellation between the left- and right-hand sides. These metric terms come from the coordinate transformation.
From the azimuthal component of the momentum equation (1) we have
i1520-0469-60-16-2004-eqa1
By taking the azimuthal mean and using the relations ∂(·)/∂λ = 0, d(·)/dt = ∂(·)/∂t, and ′/dt = u′, we have
i1520-0469-60-16-2004-eqa2
Thus,
i1520-0469-60-16-2004-ea3
The left-hand side of (A3) is the Eliassen–Palm flux. In order to get the required relation, the second term on the right-hand side of (A3) is evaluated as follows:
i1520-0469-60-16-2004-ea4
To evaluate the right-hand side of (A4), the first step is to evaluate the term
i1520-0469-60-16-2004-eqaf
For this purpose, it can be shown that d(∇·ξ)/dt = ∇ · V′:
i1520-0469-60-16-2004-eqa3
So,
i1520-0469-60-16-2004-eqa4
Hence,
i1520-0469-60-16-2004-ea5
The second step is to determine the term in (A4). From the momentum equations of the shallow water equations, we have
i1520-0469-60-16-2004-eqa5
So, we get
i1520-0469-60-16-2004-ea6
Combining (A3) to (A6), we can write the Eliassen–Palm relation as
i1520-0469-60-16-2004-eqa6
For a quasi-steady wave with phase angle φ = + krωt, we have ∂/∂t = −(ω/m)∂/∂λ and = −(ω/m) = 0. Hence,
i1520-0469-60-16-2004-eqa7
Save
  • Andrews, D. G., and M. E. McIntyre, 1978: Generalized Eliassen–Palm and Charney–Drazin theorems for waves on axisymmetric mean flows in compressible atmospheres. J. Atmos. Sci., 35 , 175185.

    • Search Google Scholar
    • Export Citation
  • Chimonas, G., and H. M. Hauser, 1997: The transfer of angular momentum from vortices to gravity swirl waves. J. Atmos. Sci., 54 , 17011711.

    • Search Google Scholar
    • Export Citation
  • Chow, K. C., K. L. Chan, and A. K. H. Lau, 2002: Generation of moving spiral bands in tropical cyclones. J. Atmos. Sci., 59 , 29302950.

    • Search Google Scholar
    • Export Citation
  • Holland, G. J., 1983: Angular momentum transports in tropical cyclones. Quart. J. Roy. Meteor. Soc., 109 , 187209.

  • Kossin, J. P., W. H. Schubert, and M. T. Montgomery, 2000: Unstable interactions between a hurricane's primary eyewall and a secondary ring of enhanced vorticity. J. Atmos. Sci., 57 , 38933917.

    • Search Google Scholar
    • Export Citation
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