Abstract

A series of numerical experiments on two-dimensional decaying turbulence is performed for a barotropic fluid on a rotating sphere. Numerical calculations have confirmed two important asymptotic features: emergence of the banded structure of zonal flows and their extreme latitudinal inhomogeneities in which kinetic energy is accumulated into the easterly circumpolar jets. The banded structure of zonal flows is established relatively early on in the initial stage. Later, after extended periods of time integration, only the circumpolar jets are intensified gradually, while there is no further evolution in the banded structure in the low and midlatitudes. Wave activity flux analysis illustrates that the initial vortices in the low and midlatitudes propagate poleward as Rossby waves and converge to produce easterly circumpolar flows. In association with this convergence, accumulation of the mean zonal component of kinetic energy proceeds. The tendency for the accumulation becomes strong as the rotation rate is increased, and nearly all of the kinetic energy is concentrated to the circumpolar flows in cases of rapid rotation.

A theoretical model is constructed under the assumption that a circumpolar jet emerges around the latitude where the local Rhines scale is equal to the distance from the Pole, and that initial vortices at the lower latitudes contribute to the generation of the jets. The model describes the mean zonal component of kinetic energy and the averaged speed and width of the circumpolar jets as functions of the rotation rate, which agree satisfactorily with the numerical results.

1. Introduction

Banded structures and associated zonal flows of planetary atmospheres, typically observed in Jovian planets, are one of the prominent planetary-scale fluid dynamical phenomena and have attracted the interests of many researchers. Some of the studies have sought their generation and maintenance mechanism in deep convective motion (e.g., Busse 1983; Heimpel et al. 2005), while others have extensively discussed the dynamical properties of two-dimensional fluid motions in a surface thin layer under the influence of rotation and stratification in relation to the emergence and maintenance of banded structures (e.g., Rhines 1975; Williams 1978; Cho and Polvani 1996; Li and Montgomery 1996, 1997; Nozawa and Yoden 1997a, b; Huang and Robinson 1998; Taniguchi et al. 2002).

Among such attempts, the two-dimensional decaying turbulence on a rotating sphere is one of the basic problems of the most idealistic setups. Yoden and Yamada (1993) first performed numerical experiments of decaying turbulence on a rotating sphere with full spherical geometry. They investigated the statistical tendency for the development of a flow field from a random set of initial states and found that a banded structure of zonal flows emerges and that there was a tendency for circumpolar flows to be easterly jets. However, especially in cases when the rotation rate is large, their initial fields are in a state of “wave turbulence”; that is, the initial kinetic energy spectrum is concentrated around the total wavenumber that is lower than the characteristic “Rhines wavenumber,” at which the magnitudes of the linear and nonlinear terms are on the same order. Later, by using a higher-resolution model, Ishioka et al. (1999) and Yoden et al. (1999) performed time integration from initial states with kinetic energy spectra concentrated around the higher total wavenumbers and confirmed the emergence of the banded structure and the circumpolar easterly jets. They also found a decreasing trend in the width of the circumpolar jet and the bands with increase in rotation rate (for a more detailed review of these studies, see also Hayashi et al. 2007).

Since these numerical observations were only qualitative, Takehiro et al. (2007) conducted a quantitative investigation on the strength and width of the easterly circumpolar jets and, by performing a series of experiments with longer time integration and larger rotation rates than the previous studies, discovered asymptotic behaviors in rapidly rotating cases. The integration periods of the preceding studies seemed to have been insufficient to fully develop the calculated flow fields, which were still developing at the end of the computations. Numerical calculations in Takehiro et al. (2007) revealed that the banded structure of zonal flows is extremely inhomogeneous and that most of the kinetic energy is accumulated inside the easterly circumpolar jets. It was argued that the averaged speed and width of the circumpolar jet are proportional to ¼ power and −¼ power of the rotation rate, respectively, in the asymptotic limits of large rotation rates.

In this paper, we extend the study of Takehiro et al. (2007) and discuss not only the asymptotic features of the circumpolar jets but also the development of the structures by consulting the wave quantities. In section 2, the model and setup of the experiment are presented again for the readers’ convenience, and the numerical results are examined more specifically in section 3. To clarify the origin of easterly circumpolar jets, section 4 presents an analysis of wave activity performed from the viewpoint of Rossby wave propagation by using the ensemble average with 50 members. In section 5, a theoretical model of kinetic energy accumulation is constructed and is compared with the numerical results. Section 6 summarizes our conclusions.

2. Model and setup for numerical experiments

The model and experiment setups are basically the same as those of Takehiro et al. (2007). Two-dimensional barotropic flow on a rotating sphere is governed by the vorticity equation for the vertical component of vorticity in the rotating frame of reference:

 
formula

where ψ(λ, μ, t) is the streamfunction whose sign follows the geophysical convention, ζ(λ, μ, t) is the vertical component of vorticity (= ∇2ψ), t is time, λ is longitude, and Ω is the rotation rate of the sphere. The nonlinear term J( f, g) ≡ (∂λf ) (∂μg) − (∂μf ) (∂λg) is the horizontal Jacobian and μ = sinϕ with ϕ representing latitude. Here ν2p is a coefficient of the hyperviscosity, which we employ for the viscous dissipation occurring in a higher wavenumber range than in the case of the normal viscosity. Note that “2” in the parentheses of the hyperviscosity term ensures the conservation of angular momentum. In Eq. (1), length, velocity, and time are nondimensionalized with the radius of the sphere a, the characteristic velocity amplitude of the initial state U0 (or square root of the mean kinetic energy 2), and the advection time scale a/U0, respectively. For numerical calculation, a spectral model of Eq. (1) is constituted by expanding each physical variable into a series of spherical harmonic functions, and by truncating higher wavenumber components whose total wavenumbers are larger than N (Ishioka et al. 2000):

 
formula

The nonlinear Jacobian term is computed with the spectral transform method, in which the nonlinear term is evaluated on physical space and transformed to spectral space each time. The core of the numerical model used in this study is available as “SPMODEL” of GFD Dennou Club, whose technical coding features are described in Takehiro et al. (2006).

For the numerical time integration, the linear terms are calculated analytically by transforming the variable with exponential factors, and the remaining nonlinear Jacobian term is dealt with by the fourth-order Runge–Kutta method. The number of grid points on the physical space is 512 and 256 points in longitude and latitude, respectively, while the truncated total wavenumber N is set to 170 in order to avoid aliasing error in the nonlinear term. The parameters of the artificial hyperviscosity are selected as p = 8 and ν2p = 10−34.

The initial velocity fields that we adopted in the simulation have the same spectral form of

 
formula

as those in Yoden et al. (1999), in which the kinetic energy spectrum is defined as E(n, t) = (1/2)n(n + 1)Σnm=−n|ψ̃mn(t)|2. Note that n0 is the peak wavenumber of the kinetic energy spectrum, which is set to 50 here, and γ gives the width of the spectrum and takes the value of 100. We generated 25 initial velocity fields by using random numbers for amplitude and phase of each coefficient of the spherical harmonics expansion, while retaining the above kinetic energy spectral form. The coefficient A is tuned to satisfy the condition that the mean kinetic energy be unity, that is, = ΣNn=2E(n, t = 0) = 1.

Studies preceding Takehiro et al. (2007) considered only the cases of Ω ≤ 400 (Ishioka et al. 1999; Yoden et al. 1999, 2002; Hayashi et al. 2000). In the present study where the parameter space was expanded toward the rapid rotation side following Takehiro et al. (2007), we adopted Ω = 0, 50, 100, 200, 400, 1000, 4000, and 10 000. For each rotation rate, we first performed time integration from 25 initial states. Then we analyzed the ensemble of the final flow fields, which was constructed from the 25 final fields from these runs and another 25 fields generated by reflecting the former 25 final fields with respect to the equator, resulting in 50 members in total. This procedure was performed to preserve the statistical reflection symmetry of the ensemble, taking into account the fact that the equation of motion is invariant to reflection with respect to the equator.

Time integrations were performed until the zonal flows are fully developed and their values of kinetic energy are saturated. The values of time increment and integration period are shown in Table 1. Note that the time developments were computed until t = 5 in the previous studies (Ishioka et al. 1999; Yoden et al. 1999, 2002; Hayashi et al. 2000). As is demonstrated in Takehiro et al. (2007) and will be shown later in this paper, the flow fields in previous studies were still developing at the end of the computation; thus, the final state of the mean zonal flows, especially those of the circumpolar jets, could not be obtained from those earlier results.

Table 1.

Values of time increment Δt, length of time integration tfinal, and characteristic wavenumber nβ = for each value of the rotation rate Ω.

Values of time increment Δt, length of time integration tfinal, and characteristic wavenumber nβ =  for each value of the rotation rate Ω.
Values of time increment Δt, length of time integration tfinal, and characteristic wavenumber nβ =  for each value of the rotation rate Ω.

3. The results of individual numerical calculations

First, we will examine the results of individual developments of the fields. Many of the characteristics presented here have already been described by Takehiro et al. (2007), and hence we will only present a brief summary for the convenience of readers. In addition to these, we emphasize here the inhomogeneous feature of the decaying turbulence on a rotating sphere, as is shown in the comparison of the zonal-mean zonal-flow profiles between t = 5 and final state (Fig. 2, right panels).

Fig. 2.

Examples of the fields for various values of Ω using Fig. 1 as the initial state. (left) Relative vorticity at the final states. Orthographic projection from the equator is used. (right) The profiles of relative angular momentum. Solid and broken lines represent the profiles in the final states and at t = 5, respectively.

Fig. 2.

Examples of the fields for various values of Ω using Fig. 1 as the initial state. (left) Relative vorticity at the final states. Orthographic projection from the equator is used. (right) The profiles of relative angular momentum. Solid and broken lines represent the profiles in the final states and at t = 5, respectively.

Figure 1 shows an example of the initial states. The kinetic energy spectrum is concentrated in the wavenumber zone located at the total wavenumber of 50 with width of about 20 wavenumbers. Small vortices corresponding to this spectrum are observed uniformly over the sphere. The amplitude of the mean zonal flow of the initial state is less than unity and the number of the bands—that is, the number of maxima of easterlies and westerlies is also about 50—corresponding to the kinetic energy spectrum.

Fig. 1.

An example of the initial states.

Fig. 1.

An example of the initial states.

The left panels in Fig. 2 show the vorticity fields of the final states when integration is started from the initial state given in Fig. 1 using various values of Ω (the final time of each run is listed in Table 1). When Ω = 0 (not shown here; see Takehiro et al. 2007), the initial small vortices merge with each other into a few larger vortices surviving to the final state. When the system is rotating, a longitudinally elongated structure becomes predominant. As the rotation rate is increased, the longitudinal elongation becomes stronger and the width of the zonal band becomes narrower. However, in the case of Ω = 10 000, trains of small vortices whose aspect ratio is of the order of unity are found in the banded structure. Strong polar vortices also emerge when Ω ≥ 400, although they are barely seen in the figures.

The right panels in Fig. 2 show the profiles of zonally averaged relative angular momentum at t = 5 and at the final stage. As already mentioned in the previous studies (Yoden et al. 1999; Ishioka et al. 1999; Hayashi et al. 2000), a banded structure of zonal flows in the low and midlatitudes and circumpolar easterly jets in high latitudes emerge simultaneously at relatively early stages. A noteworthy point is that their later evolutions follow completely different paths.

The banded structure in the low and midlatitudes are seldom developed further; the position and amplitude of each band does not change greatly, although a latitudinally uniform westerly acceleration with small amplitude does occur. On the other hand, the circumpolar easterly jets continue to grow into the late stage of the long integration periods, and become predominant over the banded structure in the low and midlatitudes in the final state. The small but latitudinally uniform westerly bias observed in the low and midlatitudes compensates for the accumulation of easterly momentum in the polar caps.

It should be emphasized that, in the decaying turbulence on a rotating sphere, the banded structure of the zonal flows as a whole is extremely inhomogeneous in the latitudinal direction; there are weak and slightly westerly biased banded zonal flows in the low and midlatitude and intense easterly circumpolar jets. This tendency is enhanced as the rotation rate is increased. As seen in Figs. 2b–d, the profiles of angular momentum of the circumpolar jets become sharp without accompanying changes in their amplitudes as the rotation rate is increased, while banded structures in the low and midlatitudes become narrower and weaker.

The appearance of this extremely inhomogeneous feature in the evolution of zonal-mean zonal-flow profile was not clearly described in the previous studies. The inhomogeneous feature and slow development of the circumpolar jets have been recognized in earlier studies such as Yoden and Yamada (1993), Yoden et al. (1999), and Hayashi et al. (2000). However, what was presented in their studies does not actually correspond to the present inhomogeneous feature of the banded structure, but to the robust appearance of the circumpolar easterly jets and the ensemble-averaging reduction of the amplitudes of randomly distributed zonal-mean flows in the low and midlatitudes (see also Fig. 3). Moreover, since the final states of time integration of these studies were t = 5, at most, and the rotation rates utilized there were not so large, the zonal flows were still evolving and the accumulation of easterly momentum around the poles had not been completed. At t = 5 for Ω = 400, indicated by the broken line in Fig. 2b that corresponds to the final states of the largest rotation rate of Ishioka et al. (1999), Yoden et al. (1999), and Hayashi et al. (2000), the amplitudes of banded zonal flows in the low and midlatitudes are comparable to those of the circumpolar jets. However, as the integration proceeds further, the circumpolar jets continue to grow, while the other zonal flows in the low and midlatitudes are only slightly accelerated in the westerly direction. In the final state, after sufficiently long time integration, at t = 20 for Ω = 400 indicated by the solid line of Fig. 2b, the easterly circumpolar jets dominate the zonal flows in the low and midlatitudes. The time period required to achieve the final state decreases with increasing Ω. Figure 2d indicates that t = 5 can be regarded as the final state for the case of Ω = 10 000. It must be noted that, for the case of Ω = 10 000, the initial spectral energy peak is located at n = 50, and as is indicated by the characteristic wavenumber nβ (Nozawa and Yoden 1997a) listed in Table 1, most of the domains except for the polar caps are regarded to be wavy like the case of Ω = 400 of Yoden and Yamada (1993). However, we believe that, as in the case of Ω = 400, if we start from an initial field having a higher wavenumber energy peak, we will still have the similar inhomogeneous feature of the banded zonal flows.

Fig. 3.

Zonal-mean flow profiles obtained from the average of 50 ensemble members (reproduction of Takehiro et al. 2007). (a) Solid, long-dashed, short-dashed, and dashed–dotted lines denote the cases of Ω = 0, 50, 100, and 200, respectively. (b) Solid, long-dashed, short-dashed, and dashed–dotted lines denote the cases of Ω = 400, 1000, 4000, and 10 000, respectively.

Fig. 3.

Zonal-mean flow profiles obtained from the average of 50 ensemble members (reproduction of Takehiro et al. 2007). (a) Solid, long-dashed, short-dashed, and dashed–dotted lines denote the cases of Ω = 0, 50, 100, and 200, respectively. (b) Solid, long-dashed, short-dashed, and dashed–dotted lines denote the cases of Ω = 400, 1000, 4000, and 10 000, respectively.

4. Results of the ensemble-mean fields

As for the development of the circumpolar easterly jets, it has been suggested that the easterly momentum transport from the low and midlatitudes associated with Rossby waves contributes to its maintenance (Hayashi et al. 2000, 2007). The problem, however, is that in order to illustrate the contribution of wave activity to the development of circumpolar zonal-mean flow, a single time series of a run from a single initial state produces quite noisy wave-quantity profiles. Ensemble averaging may reduce such fluctuations caused by phase configurations of particular disturbances that appear in individual runs.

Before getting into the profiles of wave properties, let us briefly summarize the properties of ensemble-averaged zonal-mean zonal flows demonstrated in Takehiro et al. (2007). Figure 3 shows the ensemble average of the profiles of mean zonal flows over 50 members at the final point of numerical integration for each rotation rate. At Ω = 50, the circumpolar easterly is located at the lower latitude and there is a circumpolar westerly flow near the pole. As the rotation rate is increased, the circumpolar easterly jet emerges with narrower and stronger features, and with its axis shifts poleward. Note that the banded structure in the low and midlatitudes, which is observed in the individual distributions (Fig. 2), is now invisible in the ensemble-mean fields; it is smoothed out by the averaging process since their amplitude is weak and moreover the location and direction of mean zonal flow are different among the ensemble members. Only slight westerly flow biases can be observed in the low and midlatitudes. Although the magnitude of the westerly bias decreases with increasing Ω, there does exist a westerly bias even for Ω = 10 000 to compensate for the accumulation of the easterly angular momentum associated with the circumpolar jets.

The trend in change in the width and averaged velocity of the easterly circumpolar jet with increasing Ω is also shown in Fig. 4. The strength of the jet is defined as the area-mean easterly velocity over the circumpolar jet area, ∫π/2π/2−θ |u| cosϕ dϕ/(1 − cosθ), | where the jet is assumed to extend from the pole to the latitude π/2 − θ (θ is the colatitude, where the pole is indicated by θ = 0, of the jet boundary). The equator-side boundary of the jet is defined as the latitude at which the cumulative integral of kinetic energy associated with the zonal-mean component cosϕ dϕ changes its slope significantly. This definition of jet width is different from that of Takehiro et al. (2007) where the equator-side boundary is given as the latitude at which jet velocity is equal to the one-tenth of that at the jet’s peak. The definition of Takehiro et al. (2007) tends to yield a smaller width.

Fig. 4.

Dependence of width (asterisk) and strength (circle) of the ensemble-averaged circumpolar easterly jet on rotation rate Ω. Dotted and broken lines are the width and strength estimated by the theoretical model described in section 5, respectively.

Fig. 4.

Dependence of width (asterisk) and strength (circle) of the ensemble-averaged circumpolar easterly jet on rotation rate Ω. Dotted and broken lines are the width and strength estimated by the theoretical model described in section 5, respectively.

Figures 5 and 6 show the time development of zonally averaged wave quantities defined as linear Rossby waves for Ω = 400 and Ω = 10 000, respectively. Note that these figures are equatorially symmetric because of the ensemble-mean process utilized, but we have kept both hemispheres to enable us to clearly recognize the contrast of behaviors between the polar caps and the other regions. The plotted quantities are wave activity (pseudo–angular momentum associated with Rossby waves) cosϕ/(2β̂), the latitudinal component of wave activity flux cosϕ, its divergence, and wave activity dissipation rate. Here u = −∂ϕψ and υ = ∂λψ/cosϕ are the longitudinal and latitudinal components of velocity, respectively; and •′ = • − denote zonal mean and its deviation from the mean, respectively; and β̂ ≡ ∂ϕ(2Ω sinϕ + ) is the latitudinal derivative of zonally averaged potential vorticity. Initially, as is apparent in Fig. 5a, the distribution of wave activity is latitudinally uniform except for the polar regions where β̂ is small. This is because the ensemble-mean amplitude of is uniform over the sphere, and the behavior of β̂ is roughly proportional to 2Ω cosϕ.

Fig. 5.

Time development of zonal-mean wave quantities averaged over 50 ensemble members from t = 0 to 10 for Ω = 400. (a) Latitudinal component of wave activity flux, (b) wave activity, (c) wave activity flux divergence, and (d) wave activity dissipation. Note that the tone interval for (d) is 4 times that of (c).

Fig. 5.

Time development of zonal-mean wave quantities averaged over 50 ensemble members from t = 0 to 10 for Ω = 400. (a) Latitudinal component of wave activity flux, (b) wave activity, (c) wave activity flux divergence, and (d) wave activity dissipation. Note that the tone interval for (d) is 4 times that of (c).

Fig. 6.

Same as in Fig. 5, but for Ω = 10 000. Note that the tone interval for (a) is 1/25 that in Fig. 5, and the tone interval for (d) is now the same as that of (c).

Fig. 6.

Same as in Fig. 5, but for Ω = 10 000. Note that the tone interval for (a) is 1/25 that in Fig. 5, and the tone interval for (d) is now the same as that of (c).

When Ω = 400, the decrease in wave activity in the low and midlatitudes is mostly caused by dissipation. Note that the tone interval of wave flux divergence shown in Fig. 5b is ¼ that of wave activity dissipation shown in Fig. 5d. The magnitude of wave activity that is latitudinally uniform in the low and midlatitudes decreases also uniformly in the latitudinal direction and rapidly during the early stage of the integration up to around t = 1. The smooth distribution of decrease in magnitude is the result of taking the ensemble average; the distribution of each realization is significantly noisier. Even with the 50-ensemble average, wave activity convergence is still noisy, as shown in Fig. 5c.

An interesting feature of wave activity dissipation for Ω = 400 is that it occurs not at the very beginning of the integration but reaches its maximum at around t = 0.5 (Fig. 5d). That is the time when energy upcascade and enstrophy downcascade occur, and the banded structure of zonal flows begins to be established (not shown here). It seems that the development of the banded structure of zonal flows is caused locally by the Rhines effect on a local β plane. The duration during which dissipation is active corresponds to the evolving period of the banded structure of zonal flows in the low and midlatitudes. At around t = 1, the development of the banded structure of zonal flows in the low and midlatitudes is almost completed (not shown here), and most of the initial wave activity of these latitudes has ceased. The decrease in wave activity in the low and midlatitudes in the next stage of integration is caused by the wave activity flux. The wave activity flux is poleward as indicated in Fig. 5b. The development of poleward flux requires a bit of time from the initial state. It becomes evident at around t = 1, that is, after the end of the intense wave activity dissipation period described above, and lasts for a fairly long time. Wave activity convergence occurs continuously at the high latitudes (Fig. 5c).

The characteristic features of this poleward wave activity flux and its convergence can be recognized more clearly in the case of Ω = 10 000. With increasing Ω, the progress of the field development becomes faster. Now, for Ω = 10 000, the poleward flux becomes evident after around t = 0.2 (Fig. 6b). The magnitude of flux continuously increases and reaches its maximum at around t = 1 at latitudes of around 75°. Then the maximum value decreases and the location of maximum moves toward lower latitudes. Figure 6a shows that accumulation of wave activity in the high latitudes occurs simultaneously with the establishment of the poleward wave activity flux. Wave activity increases around the latitudes where the flux converges (Fig. 6c), but tends to decrease soon after. Correspondingly, wave activity dissipation is quite active in the polar regions (Fig. 6d). This indicates that, as the Rossby waves reach those latitudes, they are quickly stirred and dissipated, and the easterly momentum associated with them is transformed into the mean zonal flow to enhance the easterly circumpolar jet there. As the easterly circumpolar jet grows, the latitude of the maximum convergence of wave activity flux shifts equatorward. This tendency indicates that the critical-level absorption of Rossby waves occurs in those latitudes.

It should be noted that, in Fig. 6d for the case of Ω = 10 000, the wave activity dissipation at the early development stage in the low and midlatitudes, which is prominent in Fig. 5d for the case of Ω = 400, is not evident. The decrease of wave activity in the low and midlatitudes for Ω = 10 000 occurs mainly due to wave activity flux divergence. Dissipation is quite small in those latitudes compared to the magnitude of wave activity flux divergence. Note that the tone intervals of wave activity flux divergence and wave activity dissipation in Fig. 6 are the same. For Ω = 10 000, the initial energy spectral peak is not located at the wavenumber small enough for the field to be regarded as turbulent; this situation is similar to that of Yoden and Yamada (1993). The initial disturbances placed in the low and midlatitudes propagate poleward as Rossby waves. Actually, the profile of zonal flows in the low and midlatitudes shows little evolution from the initial profile for Ω = 10 000 (cf. Figs. 1c and 2c).

The evolutionary features of wave activity, wave activity flux, its convergence, and dissipation in the polar caps after the initial rapid decrease of wave activity due to enstrophy cascade and dissipation do not depend qualitatively on the value of Ω. Because the disturbances produced by local energy upcascade or the disturbances placed initially are large enough, they can propagate poleward as Rossby waves from the low and midlatitudes toward the polar caps, where they are absorbed to produce the circumpolar easterly jets. The quantitative difference is that, as the Ω decreases, the evolution of wave activity flux progresses more slowly, accumulation of easterly momentum toward the polar regions lasts longer, and its maximum appears at the lower latitude.

5. Theoretical model of kinetic energy accumulation

One of the important aspects found by Takehiro et al. (2007) is that the total kinetic energy of the system is asymptotically concentrated into the easterly circumpolar jets as Ω increases. For all cases of Ω, total kinetic energy, which is defined as the integral of kinetic energy over the whole sphere, Etotal ≡ ∫ dS|ψ|2/2, decreases only 10% during the integration period. The eddy component of total kinetic energy, Eeddy ≡ ∫ dS|ψ′|2/2, is dominant in the initial stage, but is gradually converted into the zonal-mean component, Ezonal ≡ ∫ dS||2/2. This tendency is consistent with the absorption of Rossby waves into the mean zonal flows inferred by the analysis of wave activity. At the final state, when the time variation of each component becomes small and kinetic energy conversion process is nearly completed, the ratio of the zonal-mean component to the eddy component increases as the rotation rate is increased. Especially when Ω ≥ 1000, nearly all of the kinetic energy is concentrated in the zonal-mean component as seen clearly in Fig. 7. Since the amplitude of the zonal flow in the low and midlatitudes is small for Ω ≥ 1000, the zonal-mean component of kinetic energy is attributed to the circumpolar jets.

Fig. 7.

Dependence of total (circle) and zonal-mean part (asterisk) of kinetic energy in the final state for the rotation rate Ω. Dotted line shows the estimate of zonal-mean part of kinetic energy described in section 5.

Fig. 7.

Dependence of total (circle) and zonal-mean part (asterisk) of kinetic energy in the final state for the rotation rate Ω. Dotted line shows the estimate of zonal-mean part of kinetic energy described in section 5.

These tendency of kinetic energy may be interpreted as follows: among the initial vortices distributed over the whole sphere, vortices in the polar region interact to each other without influence of the β effect and there is no reason for kinetic energy there to be biased to the zonal component. As a result, they continue to contribute to the disturbance component of kinetic energy. In contrast, the β effect dominates the vortices in the low and midlatitudes, which therefore behave as Rossby waves. These Rossby waves propagate poleward and are absorbed into the circumpolar easterly jets. Most of the energy existing in the low and midlatitudes is also carried out associated with these Rossby waves from these latitudes to the polar caps. As Rossby waves are absorbed into the circumpolar jets in the polar regions, the associated energy from the low and midlatitudes is also converted into the mean zonal component of kinetic energy. Thus, kinetic energy of the initial vortices existing in the latitudes lower than the latitudes of the circumpolar jets accumulates into the zonal-mean component in the final state.

The important simplification assumed here is that the initial disturbances placed inside of the final circumpolar easterly jet area do not contribute to its angular momentum accumulation, while all of the energy in the low and midlatitudes outside of the polar caps is converted into the circumpolar jets. The assumption is that circumpolar jets are intensified only through the propagation and absorption of Rossby waves from the low and midlatitudes. Disturbances in the polar caps may contribute to determining the detailed profile of the polar jets, but they are assumed to remain in existence and to not be absorbed into the zonal-mean flows. On the other hand, the energy remaining as the banded structure of zonal flows in the low and midlatitudes is assumed to be negligible compared to the energy carried out by Rossby waves.

Following this hypothesis, let us construct a theoretical model for finding the accumulation of the zonal-mean component of kinetic energy and the averaged velocity and width of the circumpolar jet as functions of the rotation rate. Let us suppose that the easterly circumpolar jet extends from the pole to the colatitude θ, where θ is the latitudinal width of the jet. First, we assume that in the polar region, where nonlinear interaction dominates, the jet spreads around the pole by the local Rhines length scale to the south. Using the local Rhines wavenumber, this can be expressed as

 
formula

which gives

 
formula

The local Rhines scale is evaluated around the center of the circumpolar jet. Here Up is the mean magnitude of the circumpolar jet velocity averaged over the jet extent, β(θ/2) is the β parameter at the colatitude θ/2, and a is the radius of the sphere, which is equal to unity in our experiments.

The second assumption is that the initial vortices existing outside of the circumpolar jet region contribute to the kinetic energy of the circumpolar jet, and this is almost equal to the zonal-mean component of kinetic energy in the final state Êzonal:

 
formula

where U0 is a typical velocity scale of the initial vortices, which is equal to 2 in our experiments because the initial state is given so that the averaged kinetic energy becomes unity. From Eq. (5), we obtain

 
formula

Equations (4) and (6) yield the dependences of Êzonal, Up, and θ on the rotation rate Ω.

The dotted line in Fig. 7 shows the estimated mean zonal component of kinetic energy Êzonal, which agrees fairly well with the results of numerical experiments. Note that since the total kinetic energy decreases by about 10% at the final stage, the estimate becomes larger than the calculated total kinetic energy when Ω is large. The broken and dotted lines in Fig. 4 are the estimated averaged velocity and width of circumpolar jets, respectively. Both of the lines explain well the dependences on the rotation rates.

Particularly, when the rotation rate is large, θ becomes small. Under the approximation of small θ, we obtain

 
formula

From Eq. (6), we can approximate U2p ∼ 2U20/θ2, and then we have

 
formula

which shows that the strength of the jets is proportional to the one-fourth power of the rotation rate. On the other hand, the width of the jet is calculated as

 
formula

The jet width is proportional to the minus one-fourth power of the rotation rate. We can also reproduce the asymptotic feature of circumpolar jets at the large rotation rate, which was previously obtained in Takehiro et al. (2007).

It may be worth mentioning here the roughness of the above argument. It may be argued that the dynamic transition from wavy to turbulence should occur at the equator-side boundary of the jet, and hence the evaluation should be made at θ rather than θ/2. It may also be argued that the velocity scale in estimating the Rhines wavenumber should be random velocity rather than the mean jet velocity. We are omitting these levels of differences. According to our calculation results, the velocity scale seems to be smaller than the mean jet velocity, but to be larger than the initially given random velocity. We have adopted Up simply because we have no other simple measure. Moreover, a similar θ dependence for the leading order of Up as seen in Eq. (4) can also be obtained from several different assumptions. For instance, based on considerations of the critical latitude absorption of Rossby waves, Up may be assumed to be equal to the phase velocity of Rossby waves whose wavenumber is on the order of the jet width (a quantitative argument is given in Hayashi et al. 2000). However, the issue here is whether the present choice of wavenumber may be justified. Based on considerations of the homogenization of potential vorticity in the polar jet, the latitudinal profile of zonal-mean zonal-wind u may be directly obtained. More simply, in order to compensate the variation of planetary vorticity by the relative vorticity associated with the established circumpolar jet, we have Up(2π/θ/2)2 ∼ 2Ω sin(θ/2) (note again that factors are rather arbitrary). However, now the problem is that, according to our numerical results, potential vorticity in the polar jet region is not so homogeneous, especially when Ω is large. As is indicated in Fig. 8, potential vorticity is extensively mixed over the circumpolar jet region for Ω = 400, while it is not as extensive for Ω = 10 000. Turbulent mixing in the polar caps is weak for large Ω cases. We suspect that the accumulation of momentum into the polar caps has, to a certain extent, the nature of critical latitude absorption, rather than the fully nonlinear Rhines effect. To develop a more sophisticated argument, it seems that we will need to reconstruct and refine the argument thoroughly from the beginning.

Fig. 8.

Zonal-mean potential vorticity gradient profile obtained from the average of 50 ensemble members. (a) Ω = 400 and (b) Ω = 10 000.

Fig. 8.

Zonal-mean potential vorticity gradient profile obtained from the average of 50 ensemble members. (a) Ω = 400 and (b) Ω = 10 000.

6. Summary

As an extension of the study of Takehiro et al. (2007), we have performed numerical experiments of two-dimensional freely decaying barotropic turbulence on a rotating sphere for cases of larger rotation rates than those employed previously. Numerical time integrations were carried out for sufficiently long periods until the mean zonal component of kinetic energy is fully developed. The results have confirmed the emergence of easterly circumpolar jets and the banded structure of zonal flow in the low and midlatitudes. The intensity of the established zonal flows is extremely inhomogeneous with respect to the latitude: easterly circumpolar jets are quite strong, whereas zonal flows in the low and midlatitudes are relatively weak and unanimously westerly biased to compensate for the easterly angular momentum accumulation in the polar caps. As the rotation rate is increased, this inhomogeneity is intensified: easterly circumpolar jets are concentrated in the polar region and their velocity profile is sharpened and strengthened, while the amplitude of the zonal banded structure in the low and midlatitudes decreases in association with their narrowing.

The wave activity analysis clarified the evolution of zonal-mean flows and the formation of circumpolar jets from the perspective of Rossby waves. The development of the banded structure of zonal flows in the low and midlatitudes are associated with the large dissipation of wave activity, and hence can be regarded as caused by the local upcascade of energy and the Rhines effect of the local β plane. Then, the disturbances in the low and midlatitudinal region continuously propagate poleward as Rossby waves, and they are absorbed at the polar regions resulting in the enhancement of the easterly circumpolar jets there. Simultaneously, vortex motions and banded zonal-flow formation in the low and midlatitudes become inactive. As a result, total kinetic energy tends to be accumulated into the easterly circumpolar jets. In particular, nearly all of the kinetic energy is concentrated in the circumpolar jets when the rotation rate is large.

By applying the assumptions that the location of the circumpolar jets is determined by the latitude where the nonlinear term and the β term of the vorticity equation balances and that the kinetic energy of the initial vortices existing in the latitudinal region lower than the latitudes of the circumpolar jets contribute to that of the jets, we have succeeded in theoretically finding the dependence of the mean zonal component of kinetic energy and the averaged velocity and width of the jets on the rotation rate. These theoretical estimates agree fairly well with the numerical results. In particular, the asymptotic limit of large rotation rates for this theory shows that the strength and width of the jets are in proportion to one-fourth and minus one-fourth powers of the rotation rate, respectively. This asymptotic feature is consistent with the results by Takehiro et al. (2007).

Note that the inhomogeneity of banded structure of zonal flows inherent in the two-dimensional barotropic decaying turbulence on a rotating sphere, which is clearly shown in this study, does not appear in the framework of two-dimensional β plane whose governing vorticity equation is latitudinally homogeneous. Caution should be exercised when applying conclusions regarding banded structures of zonal flows obtained in β-plane cases to rotating sphere cases, or when applying them to the actual planetary atmospheres. This may lower the significance of the studies performed so far about the origin and maintenance of the zonal banded structure of the atmospheres of Jovian planets discussed within the framework of two-dimensional flows on a β plane. Latitudinal inhomogeneity of the β effect may be indispensable. On the other hand, we should also bear in mind the possibility that some other effects are involved in inhibiting the dominance of the inhomogeneous banded structure. If the amplitude of zonal banded flows in the low and midlatitudes is sufficiently large from the beginning, or is maintained by small-scale forcing, the zonal flows may prevent the propagation of the Rossby waves to higher latitudes and the subsequent accumulation of kinetic energy into the circumpolar jets. The forced turbulence cases on a rotating sphere exemplified by Nozawa and Yoden (1997a, b) and Huang and Robinson (1998) might be such cases. Continuously excited vortices might enhance in situ jets and cause homogeneous global banded structures.

Acknowledgments

All comments by reviewers are greatly appreciated by the authors. This work was supported by Grants-in-Aid for Scientific Research of the Japan Society for the Promotion of Science. Numerical computations were carried out on VPP5000 at the Astronomical Data Analysis Center of the National Astronomical Observatory, Japan, and on HPC2500 at the Academic Center for Computing and Media Studies of Kyoto University, Japan. The main part of the numerical model used in this study was constructed with a spectral transformation library “ISPACK” (http://www.gfd-dennou.org/library/ispack/) and a Fortran90 wrapper library “SPMODEL” (http://www.gfd-dennou.org/library/spmodel/); “gt4f90io” (http://www.gfd-dennou.org/library/gtool4/) and “NetCDF” (http://www.unidata.ucar.edu/packages/netcdf/) libraries were used for data IO routine of the model. Data analysis and visualization were done using the software products of the GFD Dennou Ruby project (http://ruby.gfd-dennou.org/).

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Footnotes

Corresponding author address: Shin-ichi Takehiro, Research Institute for Mathematical Sciences, Kyoto University, Sakyo-ku, Kyoto 606-8502, Japan. Email: takepiro@gfd-dennou.org

This article included in the Jets and Annular Structures in Geophysical Fluids (Jets) special collection.