1. Introduction
In the context of the two-layer quasigeostrophic (QG) model, it has been known for almost five decades that Ekman pumping, if present only at the lower boundary, can destabilize baroclinic waves. For example, Holopainen (1961) performed a linear stability analysis of the two-layer model and found that the lower boundary Ekman pumping broadens the marginally stable curve of the inviscid flow so as to destabilize both longer and shorter zonal waves. Essentially the same result was found by Pedlosky (1983), and also by Weng and Barcilon (1991), who used a linear Eady-like model (Eady 1949). Pedlosky (1983) further showed, with a weakly nonlinear analysis, that while the destabilized wave can grow initially, as the wave changes the mean flow the wave eventually decays. The final result is zero wave amplitude with an altered mean flow.
Nonlinear numerical model calculations also found that lower-layer Ekman damping can energize baroclinic waves. In their study of QG turbulence with a doubly periodic two-layer model, Hua and Haidvogel (1986) found that lower-layer Ekman pumping acts as a source of energy for the baroclinic waves. This finding was supported by Rivière et al. (2004), who used a primitive equation model to study effect of bottom friction on baroclinic eddies in an oceanic jet. Using a two-layer QG model, Lee (2010) showed that surface Ekman pumping acting directly on the eddies can increase eddy potential enstrophy. Thompson and Young (2007) found that for β less than a critical value, the addition of bottom friction produces a heat flux that is weaker than the inviscid prediction by Held and Larichev (1996) and Lapeyre and Held (2003). However, for β larger than a critical value, the inclusion of bottom friction results in a heat flux that is stronger than the inviscid prediction. Although linear destabilization may be relevant for these nonlinear results, to distinguish between linear and nonlinear influences the nonlinear behavior will be referred to as “dissipative energization” and the linear instability as “dissipative destabilization.”
The above nonlinear results suggest for the atmosphere that surface Ekman pumping may play a nontrivial role for the equilibration process of midlatitude baroclinic waves. In spite of the potentially important role of surface Ekman pumping, the physical process by which Ekman pumping can energize baroclinic waves is not well understood. Dissipative energization of baroclinic waves, found in various numerical models, has often been attributed to the barotropic governor mechanism (James and Gray 1986; James 1987). While this may indeed be the case, the mechanism to be presented in this study differs from the barotropic governor mechanism, for which surface friction influences the eddies through the horizontal shear of the zonal mean flow.
In this study, the results of Lee (2010) are further analyzed from the viewpoint of the Lorenz energy cycle (Lorenz 1955) to help us better understand the workings of the dissipative energization.
2. Model
The model is driven toward a prescribed thickness field ψ̂e ≡ (ψe1 − ψe2)/2, which is analogous to the radiative equilibrium temperature field. For simplicity, it is assumed that ue2 ≡ −∂ψe2/∂y = 0 everywhere. The coefficients κT and κM are the thickness and Ekman damping rates, respectively. To emphasize its analogous role in the atmosphere, the thickness damping will be referred to hereafter as thermal damping. The term ν∇6ψj represents the enstrophy cascade toward subgrid scales, and the value of ν determines the cascade rate.
3. Energetics
The Ekman pumping can contribute to EKE growth if the term EKEEk in (8.3) is positive. It can be seen, after integrating by parts, that EKEEk is positive if 
Similarly, Ekman pumping can contribute toward EAPE growth if EAPEEk > 0, that is, if 
Figure 1 illustrates schema for the mechanism by which EKE and EAPE can grow through the Ekman pumping. Here, it is assumed that |ψ1| > |ψ2|. For the EAPE, η in this two-layer model is equivalent to temperature in a continuous model, with upward displacement (η > 0) corresponding to cold air and downward displacement (η < 0) to warm air. The solid curve in Fig. 1 denotes an initial η field, and the dashed curve indicates η after being subjected to Ekman pumping. If ψ1 and ψ2 are out of phase, the surface Ekman pumping can help convert the EAPE to EKE because it reduces |η| (Fig. 1a). Similarly, if ψ1 and ψ2 are in phase, the surface Ekman pumping can generate EAPE, since it enhances |η| (Fig. 1b).
Given that the direct impact of the Ekman pumping on the TEE is always dissipative (i.e., TEEEk < 0), the observation that dissipative energization can occur implies that there must be an indirect influence through which Ekman pumping can increase TEE. From the Lorenz energy cycle, which in Fig. 2 flows in a counterclockwise direction, waves can grow via this indirect effect only if EAPEEk > 0. This is because a positive EAPEEk can increase |η|, which in turn can promote an energy conversion (BC) from ZAPE to EAPE. This process is indicated schematically with the black arrows in Fig. 2 and will be referred to as the BCEk growth.
If EKEEk is positive (Fig. 1a), which results in an increase in |υi|, then BC can also increase in response. However, because an increase in |υi| also enhances BT, compared with the BCEk growth, this is an inefficient route toward dissipative energization. Holopainen (1961) provided a physical explanation for linear dissipative destabilization in terms of his version of EKEEk being positive. However, the above analysis for EAPEEk suggests an alternative interpretation.
The barotropic governor mechanism of James and Gray (1986) and the self-maintaining jet mechanism of Robinson (2006) are also included in Fig. 2. For the latter mechanism, as discussed in Robinson (2006), the eddy momentum flux convergence at the jet center enhances the vertical shear of the zonal-mean zonal wind beyond that of the radiative equilibrium state. Although not explicitly stated in that paper, in order for this to result in wave growth, surface friction must be stronger than radiative damping (Lee 2010). Energetically, this means that changes in the ZKE, with the help of surface Ekman pumping, can increase ZAPE. As such, there are at least three different ways that surface Ekman pumping can energize baroclinic eddies through their impact on the ZAPE to EAPE energy conversion.
4. Test of the BCEk growth
a. An overview from statistically steady states
There are two noteworthy features from the simulations summarized in Fig. 3. First, for all nonzero values of β, EAPEEk first increases with κM, peaking either at κM = 0.20 (for β = 0.15) or at κM = 0.30 (for β = 0.25 and 0.30). Within the range 0.05 < κM < 0.2, the fractional increase in EAPEEk is greater as β increases; the fractional increase is 58% for β = 0.15, 101% for β = 0.25, and 117% for β = 0.30. Second, for β = 0.25 and 0.30, where the fractional increase in EAPEEk is large, BC and EAPE also increase with κM. The above two features indicate that there is a range of optimal values of β and κM where the dissipative energization is more effective. While a direct comparison is impossible because of the differences in forcing, the dependency of BC on κM is consistent with the finding of Thompson and Young (2007). They found that for sufficiently large values of β, northward heat flux intensifies as their surface friction increases.
The dependency on κM is also consistent with the above interpretation that condition (11) favors a sufficiently small δϕ and thus the dissipative energization. Figures 3e–h show that δϕ monotonically increases with κM for all β, implying a weakening of the dissipative energization with κM. However, because the amplitude of EAPEEk is not only inversely proportional to δϕ but also proportional to κM, there would be optimal, intermediate values of κM where maximum EAPEEk can be produced. As discussed above, Figs. 3a–d indeed show this behavior.
While this analysis provides a coherent interpretation for the dependency of EAPEEk and δϕ on κM, the question still remains as to why δϕ increases monotonically with κM. This increase in δϕ is consistent with (∂Q2/∂y)(y = 0) becoming increasingly negative with κM (see Figs. 3e–h) while (∂Q1/∂y)(y = 0) remains essentially constant. [This can be inferred by the fact that [(∂Q2/∂y)/(∂Q1/∂y)](y = 0) (see Figs. 3e–h) closely follows ∂Q2/∂y.] The only isolated exceptions occur for (β, κM) = (0.0, 0.05) and (0.0, 0.1). Because this simultaneous increase in (|∂Q2/∂y|/|∂Q1/∂y|)(y = 0) and δϕ with κM is consistent with condition (11), we interpret this increase in δϕ as arising from (∂Q2/∂y)(y = 0) becoming increasingly negative with κM. This behavior can in turn be attributed to the Ekman damping effect on the zonal mean flow; as the damping strengthens, −∂2U2/∂y2, which is positive, would become smaller at the jet center. Because ∂Q2/∂y ≡ β − ∂2U2/∂y2 − (U1 − U2)/2 the above change in −∂2U2/∂y2 would allow (∂Q2/∂y)(y = 0) to become increasingly negative as κM increases.
b. Transient evolutions
To test if BCEk growth can occur, it is necessary to examine the transient evolution of the energetics. We also ask how BCEk would differ between the regime where TEE increases with κM and that where TEE decreases with κM. Based on the results summarized in Fig. 3, we choose three cases: (β, κM) = (0.25, 0.1), (0.25, 0.2), and (0.25, 0.5). The first case is where TEE increases with κM; the second case is where the TEE maximum occurs; the third case is where TEE decreases with κM. We choose the cases with β = 0.25 rather than β = 0.30 because most of the finite-amplitude states with β = 0.30 arise from subcritical instability (the arrows in Figs. 3a–d indicate the linear stability boundary). That is, the finite-amplitude states occur only when the model is initialized with finite-amplitude eddies (Lee and Held 1991). For this subcritical region, therefore, it is impossible to examine the initial transient evolution from the normal mode form.
For each of the three cases, the initial transient evolution is shown in Figs. 4a–c. For κM = 0.1 and 0.2, the initial spinup stage is followed by growth of the most unstable normal mode. (The spinup stage is not shown.) As can be inferred by the constant EAPE growth rates, the normal mode growth continues until day 100 for κM = 0.1, and until day 550 for κM = 0.2. In the latter case, the linear growth stage is followed by a brief period (between days 550 and 750) of higher growth rates. Lee and Held (1991) interpreted this behavior in terms of a nonlinear growth (γ > 0) in the weakly nonlinear amplitude equation ∂A/∂t = αA + γ|A|2A (Pedlosky 1970), where A is the wave amplitude. The finding in Lee (2010) suggests that this nonlinear growth may be due to the eddy-driven baroclinicity (“self-maintaining jet” mechanism; Robinson 2006). While this nonlinear growth is likely to foster the dissipative energization (Lee 2010), as will be explained below, the dissipative energization still occurs for κM = 0.1 where the nonlinear growth does not occur.
During the nonlinear growth period, the disturbance attains a finite amplitude, while sharply deviating from its normal mode form. The latter feature can be detected from the occurrence of a large change in the meridional scale and the vertical phase tilt of the wave field (see Figs. 4g,h). As can be seen, the deviation from normal mode form is characterized by a decrease in the meridional scale of the upper-layer eddy streamfunction Ly1, an increase in the meridional scale of the lower-layer eddy streamfunction Ly2, and a decrease in the vertical phase tilt δϕ. (The meridional eddy scales Ly1 and Ly2 were defined as the distance between the two points, on either side of the jet center, where the eddy streamfunction first changes sign.) The smallness in δϕ implies that—as illustrated by Fig. 1—the EAPE can grow more effectively by the Ekman pumping. (In Figs. 4g–i, 1 − EAPEEk is displayed to highlight the fact that the increase in EAPEEk is associated with the decrease in δϕ.) In fact, EAPEEk, while smaller than BC initially, rapidly increases afterward; during the equilibrium stage EAPEEk is about twice as large as BC. This time evolution also demonstrates that the Ekman pumping–driven growth of EAPE is a nonlinear process and cannot be explained by the linear theory. As κM is increased from 0.2 to 0.5 (the initial state is day 1000 of the κM = 0.2 case), δϕ rapidly increases (Fig. 4i), but EAPEEk still dominates over BC.
Having demonstrated that EAPEEk is the main contributor to the nonlinear growth of the eddy energy, we now test the hypothesis that the direct effect of the Ekman pumping can further enhance BC, and thus EAPE. To perform this test, we choose for the initial flow the model state at a time when EAPEEk attains a large value and integrate the model forward in time with κM = 0.0 in the eddy potential vorticity equation. In this integration, the zonal mean flow is still subject to the same Ekman damping so that the effect of the Ekman pumping on the eddies can be isolated from that on the zonal mean flow, as in the barotropic governor mechanism of James and Gray (1986) and James (1987). Although this model setting is unphysical, the initial behavior of the eddy energy can be used to test our hypothesis. For each of the three cases, with the model state indicated by the thick arrow (in Figs. 4a–c) as the initial state, the test run was performed and the resulting energetics over the next 50 model days are shown in Figs. 4d–f. For κM = 0.5, to test the sensitivity to the initial state, an additional calculation was performed using the model state indicated by the thin arrow (Fig. 4c) as the initial state. The result (not shown) is very similar to that shown in Fig. 4f.
For κM = 0.1, it can be seen that BC strengthens briefly, but it rapidly weakens during the next 10–15 days. The brief increase in BC can be understood as being due to the strengthening of the eddy meridional wind υj, as evidenced by the rapid increase in the EKE, which is due to the zero EKEEk. (As can be seen from Fig. 4a, EKEEk is the main sink of EKE when κM ≠ 0.) In the face of this rapid EKE increase, the subsequent weakening in BC implies either that |η| is becoming small or that the correlation between υj and η has declined. Comparing the EAPE between Figs. 4a and 4d, it can be seen that |η| of the test run is clearly smaller than that of the control run, even during the first five days when BC undergoes a strengthening. Because this decline in |η| is due to EAPEEk being zero, we are led to conclude that the decline in BC (between days 3 and 18) is due to the absence of EAPEEk and its impact on BC (the black arrows in Fig. 2). The thermal damping is not found to play an important role here because the change in the thermal damping during the transient stage is small. In Fig. 4d, it is also interesting to observe that the initial EKE gain is followed by a slight reduction of EKE (between days 14 and 20). This EKE decline is consistent with the preceding reduction in C(EAPE, EKE), which closely follows BC. Beyond day 25, the EKE continues to increase, but again this is due to the zero EKEEk. This long-time behavior is unphysical and therefore not meaningful.
As the value of κM increases (cf. Figs. 4d and 4e, and Figs. 4e and 4f), the initial period of BC strengthening becomes longer, and the subsequent decline of BC becomes smaller. In accordance with the earlier interpretation, this lengthening of the initial period is consistent with the more rapid EKE increase, and the smaller BC reduction is consistent with the lesser decline of |η|. This finding indicates that the dissipative energization occurs for all values of κM. However, for κM = 0.5, BC never drops below the initial value, indicating that in the regime where TEE decreases with κM, BC is more strongly influenced by EKEEk than by EAPEEk.
5. Conclusions
In this study, we investigated how nonlinear dissipative energization of baroclinic waves occurs in a two-layer model where Ekman damping is applied only to the lower layer. Because the total eddy energy of this system is always damped by the Ekman pumping, the nonlinear growth must arise from an enhanced interaction via the Ekman pumping between the eddies and the zonal mean flow. It is found that this growth involves the following process:
If the phase difference between the upper- and lower-layer eddies is less than one quarter wavelength, then the lower-layer Ekman pumping can act to produce EAPE (EAPEEk > 0) while damping the EKE.
This EAPE production in turn increases the baroclinic energy conversion (BC) from the ZAPE to the EAPE.
The relationships between the eddy scale and δϕ, and between δϕ and EAPEEk, as discussed earlier, imply that eddy scale and EAPEEk may also be related to each other. For the three nonzero β cases considered in this study, as can be seen by comparing the eddy scales shown in Figs. 3f–h with the corresponding EAPEEk values in Figs. 3b–d, there is a hint that Ly1 tends to be relatively small when EAPEEk is relatively large. Lee (2010) showed that the reduction in δϕ (Figs. 4g,h) coincides with jetward movements of upper-layer critical lines and confinement of the upper-layer eddy PV flux toward the jet center. The interpretation of this behavior was that this confinement of the PV flux, in the region where ∂Q1/∂y is also large (∂Q1/∂y is maximum at the jet center), favors the occurrence of the inequality in (11). This interpretation can explain the apparent inverse relationship between Ly1 and EAPEEk. One implication of this conclusion is that in the presence of surface Ekman pumping, there is a selective generation of small scales in the upper layer. It would be interesting to investigate whether this process can help explain the finding of Rivière et al. (2004) that bottom friction results in significant horizontal scale selection.
Acknowledgments
This study was supported by the National Science Foundation under Grant ATM-0647776. The author acknowledges valuable comments from Steven Feldstein and two anonymous reviewers.
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Schema of (a) EKE and (b) EAPE production by surface Ekman pumping. In both frames, the wavy solid curve indicates the initial interface between the upper and lower layers for an inviscid fluid, and the dashed curve indicates the interface after the influence of the Ekman pumping takes place. It can be seen in (a) that the Ekman pumping helps convert EAPE to EKE, while in (b) the Ekman pumping generates EAPE by raising the interface.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
Schema of (a) EKE and (b) EAPE production by surface Ekman pumping. In both frames, the wavy solid curve indicates the initial interface between the upper and lower layers for an inviscid fluid, and the dashed curve indicates the interface after the influence of the Ekman pumping takes place. It can be seen in (a) that the Ekman pumping helps convert EAPE to EKE, while in (b) the Ekman pumping generates EAPE by raising the interface.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
Schema of (a) EKE and (b) EAPE production by surface Ekman pumping. In both frames, the wavy solid curve indicates the initial interface between the upper and lower layers for an inviscid fluid, and the dashed curve indicates the interface after the influence of the Ekman pumping takes place. It can be seen in (a) that the Ekman pumping helps convert EAPE to EKE, while in (b) the Ekman pumping generates EAPE by raising the interface.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
Schema of the BCEk growth (black arrows) in the context of the Lorenz energy cycle. The direction of the arrows between the Ekman pumping and EAPE/EKE depends on δϕ. As a reference, the barotropic governor mechanism (James and Gray 1986; James 1987) is also shown, along with the self-maintaining jet mechanism (Robinson 2006). Following Fig. 1 of James (1987), the two black bars represent a damper that acts to weaken the energy conversion.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
Schema of the BCEk growth (black arrows) in the context of the Lorenz energy cycle. The direction of the arrows between the Ekman pumping and EAPE/EKE depends on δϕ. As a reference, the barotropic governor mechanism (James and Gray 1986; James 1987) is also shown, along with the self-maintaining jet mechanism (Robinson 2006). Following Fig. 1 of James (1987), the two black bars represent a damper that acts to weaken the energy conversion.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
Schema of the BCEk growth (black arrows) in the context of the Lorenz energy cycle. The direction of the arrows between the Ekman pumping and EAPE/EKE depends on δϕ. As a reference, the barotropic governor mechanism (James and Gray 1986; James 1987) is also shown, along with the self-maintaining jet mechanism (Robinson 2006). Following Fig. 1 of James (1987), the two black bars represent a damper that acts to weaken the energy conversion.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
(a)–(d) The statistically steady state (average over days 1001–2000) of EKE, EAPE, baroclinic energy conversion, barotropic energy conversion, EKEEk, EAPEEk, thermal damping, and C(EAPE, EKE) for (a) β = 0.0, (b) β = 0.15, (c) β = 0.25, and (d) β = 0.3. In each case, the arrow indicates the stability boundary (to an accuracy of κM = 0.1). For the subcritical cases, the initial state is the final state of the case whose κM value is smaller by 0.1. (e)–(i) Statistically steady-state values for δϕ, ∂Q2/∂y, (∂Q2/∂y)/(∂Q1/∂y), and Ly1 and Ly2. The legend in (a) applies to (b)–(d), and that in (e) applies to (f)–(i).
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
(a)–(d) The statistically steady state (average over days 1001–2000) of EKE, EAPE, baroclinic energy conversion, barotropic energy conversion, EKEEk, EAPEEk, thermal damping, and C(EAPE, EKE) for (a) β = 0.0, (b) β = 0.15, (c) β = 0.25, and (d) β = 0.3. In each case, the arrow indicates the stability boundary (to an accuracy of κM = 0.1). For the subcritical cases, the initial state is the final state of the case whose κM value is smaller by 0.1. (e)–(i) Statistically steady-state values for δϕ, ∂Q2/∂y, (∂Q2/∂y)/(∂Q1/∂y), and Ly1 and Ly2. The legend in (a) applies to (b)–(d), and that in (e) applies to (f)–(i).
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
(a)–(d) The statistically steady state (average over days 1001–2000) of EKE, EAPE, baroclinic energy conversion, barotropic energy conversion, EKEEk, EAPEEk, thermal damping, and C(EAPE, EKE) for (a) β = 0.0, (b) β = 0.15, (c) β = 0.25, and (d) β = 0.3. In each case, the arrow indicates the stability boundary (to an accuracy of κM = 0.1). For the subcritical cases, the initial state is the final state of the case whose κM value is smaller by 0.1. (e)–(i) Statistically steady-state values for δϕ, ∂Q2/∂y, (∂Q2/∂y)/(∂Q1/∂y), and Ly1 and Ly2. The legend in (a) applies to (b)–(d), and that in (e) applies to (f)–(i).
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
(a)–(c) The time evolution of EKE, EAPE, baroclinic energy conversion, barotropic energy conversion, EKEEk, EAPEEk, thermal damping, the instantaneous EAPE growth rate (the EKE growth rate is essentially identical to the EAPE growth and is thus not shown), and C(EAPE, EKE). [See the legend in (a) for (κT, β) = (1/30, 0.25)]. Evolution is shown for (a) κM = 0.1, (b) κM = 0.2, and (c) κM = 0.5. The initial condition for the κM = 0.5 case is the day 1000 solution of the κM = 0.2 run. (d)–(f) The first 50-day segment of the model run where the Ekman damping is set to zero in the eddy potential vorticity equation. The initial state is that of (d) day 600 of the run shown in (a),(e) day 800 of (b), and (f) day 40 of (c). These model days are indicated by the thick arrows in (a)–(c). The legend in (a) applies to (b) and (c) and that in (d) applies to (e) and (f). (g)–(i) The time evolution of Ly1, Ly2, δϕ, and 1 − EAPEEk for (a),(b), and (c), respectively.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
(a)–(c) The time evolution of EKE, EAPE, baroclinic energy conversion, barotropic energy conversion, EKEEk, EAPEEk, thermal damping, the instantaneous EAPE growth rate (the EKE growth rate is essentially identical to the EAPE growth and is thus not shown), and C(EAPE, EKE). [See the legend in (a) for (κT, β) = (1/30, 0.25)]. Evolution is shown for (a) κM = 0.1, (b) κM = 0.2, and (c) κM = 0.5. The initial condition for the κM = 0.5 case is the day 1000 solution of the κM = 0.2 run. (d)–(f) The first 50-day segment of the model run where the Ekman damping is set to zero in the eddy potential vorticity equation. The initial state is that of (d) day 600 of the run shown in (a),(e) day 800 of (b), and (f) day 40 of (c). These model days are indicated by the thick arrows in (a)–(c). The legend in (a) applies to (b) and (c) and that in (d) applies to (e) and (f). (g)–(i) The time evolution of Ly1, Ly2, δϕ, and 1 − EAPEEk for (a),(b), and (c), respectively.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
(a)–(c) The time evolution of EKE, EAPE, baroclinic energy conversion, barotropic energy conversion, EKEEk, EAPEEk, thermal damping, the instantaneous EAPE growth rate (the EKE growth rate is essentially identical to the EAPE growth and is thus not shown), and C(EAPE, EKE). [See the legend in (a) for (κT, β) = (1/30, 0.25)]. Evolution is shown for (a) κM = 0.1, (b) κM = 0.2, and (c) κM = 0.5. The initial condition for the κM = 0.5 case is the day 1000 solution of the κM = 0.2 run. (d)–(f) The first 50-day segment of the model run where the Ekman damping is set to zero in the eddy potential vorticity equation. The initial state is that of (d) day 600 of the run shown in (a),(e) day 800 of (b), and (f) day 40 of (c). These model days are indicated by the thick arrows in (a)–(c). The legend in (a) applies to (b) and (c) and that in (d) applies to (e) and (f). (g)–(i) The time evolution of Ly1, Ly2, δϕ, and 1 − EAPEEk for (a),(b), and (c), respectively.
Citation: Journal of the Atmospheric Sciences 67, 7; 10.1175/2010JAS3295.1
