This in turn requires a definition of the instability amplitude. Previous experience has found that for a finite range of conditions relatively far from the critical point, the Landau series for analogous problems appear to converge. For the tropical cyclone problem, the mean flow parameters are relatively far from critical, and we find that for most of the region outside the RMW, we usually need to treat these expansions as asymptotic, keeping terms out to a 4 at most.
Within the RMW the series appear to converge. It may be possible to improve the convergence properties through a judicious choice of inner product and eigenfunction normalization. Figures 8a—d show a vertical cross section through the nonlinear solution corresponding to the eigenfunction shown in Figs. This solution shows that the updraft regions become narrower and stronger than the downdraft regions.
Similarly the along-roll wind is reduced in relatively narrower bands below the updrafts since the upward branch of the overturning circulation transports the lower-momentum near-surface air upward.
Because the updrafts are stronger than the downdrafts, the negative along-roll perturbations are stronger and narrower than the positive ones. The cross-roll velocities converge near the bottom of the updrafts and diverge below the downdrafts. In this case as well, the negative across-roll perturbation winds are stronger than the positive perturbations.
This asymmetry enhances the asymmetry in the along-roll velocity. This pattern tends to strengthen the roll circulation and to enhance the vertical velocity. The circulation asymmetries increase with increasingly unstable stratification. When nonlinear effects are included these lateral roll translation speeds are reduced, most strongly at larger radii.
In neutral stratification the phase velocities are larger than for unstable stratification but still less than what the linear analysis would estimate. The analysis presented here has important similarities to and significant differences from previous studies of BL rolls. Faller , Lilly , Brown , and Etling all examined the linear stability of the neutrally stratified Ekman layer and found that the normal modes agreed well with both the laboratory measurements and the observed characteristics of midlatitude BL rolls.
Faller and Kaylor first considered the nonlinear stability of the Ekman layer by time-integrating the Ekman layer equivalents to 13 — 16 seeded with the normal mode shape and searching for steady state solutions.
The equilibrium solutions had reasonable circulation magnitudes. Brown attempted a simple nonlinear stability calculation in which the first Landau coefficient was estimated from a kinetic energy equation based on only the overturning component 2D of the roll fluxes calculated from the normal mode shape Stuart Because the normal mode shape is assumed throughout the nonlinear evolution, there is no asymmetry in the equilibrium solution.
In contrast the expansion solution as presented here and applied to the Ekman layer is in excellent agreement with the fully nonlinear solution Foster , a. Ekman layer stability in unstable stratification was studied by Brown , Etling , and Asai and Nakasiji The linear analyses were all consistent with each other. Brown applied the same kinetic energy approach ignoring the roll-induced heat fluxes to estimate the nonlinear amplitude of the overturning part of the roll flow, which results in a symmetric circulation.
Foster , a found that unstable stratification has a profound effect on the nonlinear Ekman layer solutions and significantly increases the asymmetry between the ascending and descending branches of the roll circulation. He also found that baroclinic shear in the BL strongly affects both the wavelength and orientation of the rolls and that simple superposition of roll, stratification and thermal wind effects will underestimate their combined effects on the momentum transfer across the midlatitude BL Foster and Levy A canonical stability problem closely related to the Ekman layer considers the unbounded flow above a rotating disc e.
This flow is frequently studied in the laboratory because it is easy to establish, it includes flow curvature and is susceptible to cross-flow instabilities discussed below that are good analogs to flow instabilities over airfoils. The mean suction flow above the disc is a similarity solution to the Navier—Stokes equations in cylindrical coordinates Greenspan A scaling argument similar to that described here also reduces the rotating disc problem to a set of locally Cartesian stability problems at each radius.
Analogous to the inapplicability of Ekman BL instabilities to the hurricane BL roll problem, Malik and Orszag showed that applying the Orr—Sommerfeld equation to the rotating disc results in the wrong answer. The recent rotating disc literature is primarily concerned with the linear stability problems of calculating the neutral stability surface, the effects of nonparallel flow and the difference between stationary and traveling modes e. Faller proposed that an Ekman layer instability might be responsible for the formation of rain bands in hurricanes because the inflow angles are similar.
Fung expanded on Faller and considered the global linear stability problem of the entire hurricane BL. She assumed a stationary, symmetric vortex in neutral stratification, and a mean flow profile that was a three-piecewise linear approximation of an Ekman layer under a vortex, thus reducing the order of the stability problem from eight to six.
This singular perturbation reduced the number of boundary conditions that could be satisfied hence the horizontal component of the roll flow was assumed to be free slip at the surface. Note that Brown and Lee found that the gross characteristics of the full Ekman layer linear stability problem are retained under similar singular perturbations.
The pure Ekman stability problem even for the singular perturbation would have obtained local rolls on the 2—4-km scale range similar to the previous Ekman layer studies. However, the global stability analysis found structures of 20—50 km with shapes quite reminiscent of hurricane rainbands. Even so, recent studies suggest that hurricane rainbands are not associated with BL instabilities Chen and Yau ; Wang Interestingly Gall et al. This question should be revisited with a better BL representation and stability analysis.
Normal mode instabilities are not the only possible perturbations of the mean flow. As shown, for example, by Farrell ; Butler and Farrell , Trefethen et al. Foster b examined this possibility for the Ekman layer and found small-scale transient perturbations whose structure resemble the near-surface streaks seen in large-eddy simulations Drobinski and Foster and in observations of the midlatitude BL Drobinski et al.
Foster b also showed that the same mechanism could transfer instability energy into unstable normal modes and activate them much more rapidly than their initial exponential growth rates would imply. We speculate that a similar mechanism could be present in the hurricane BL to speed the formation of the hurricane BL rolls in the presence of intense turbulence and other processes.
It is also possible that smaller-scale, transient streaky structures might exist near the surface; however it is unlikely that the existing observations would be capable of resolving them. An important conclusion of this nonlinear analysis is that the hurricane BL should be considered a nearly ideal environment for roll formation even though prior to the observations discussed in the introduction, the existence of rolls in hurricane BL flow was not suspected.
There are two related lines of reasoning that we discuss below. We may first appeal to analogy with related instabilities. The governing equations in neutral stratification, 13 — 16 , ignoring the Coriolis and curvature terms, describe cross-flow instabilities.
While the curvature and Coriolis terms affect the instability, they are relatively weak. A cross-flow instability is a consequence of a 3D BL profile in which the lateral component has an inflection point. Canonical examples of cross-flow instabilities are pure Ekman BLs and the flow above a rotating disc. Engineering examples include laminar or turbulent flow over an airfoil.
The instability aligns nearly in the downstream direction and, in these standard examples, tends to reach a nonlinear equilibrium state with an embedded secondary circulation in a modified mean flow. Laminar Ekman instabilities have been shown to be a useful paradigm for turbulent planetary BL roll vortices when a constant or variable eddy viscosity is used to parameterize the smaller-scale turbulent eddies. Since these canonical instabilities are usually studied in neutral stratification we should consider how unstable stratification will affect hurricane BL cross-flow instabilities.
Ordinary BL rolls usually form in conditions of slightly to moderately unstable stratification. Unlike pure shear instabilities, pure convection does not have a preferred instability shape because there are too many degrees of freedom in the instability problem Chandrasekhar Because the dynamical instability has already found a mechanism to relieve the gradients that led to the instability in the system, we suppose that sheared convection initially reinforces the purely dynamical mode.
The primary effects of unstable stratification are to align the rolls closer to the downstream direction, increase the asymmetry between updrafts and downdrafts, and strengthen the circulation.
It is routinely observed that midlatitude BL rolls become more convective in character including larger aspect ratios as either the flow slows down or the convective forcing increases Etling and Brown ; Atkinson and Zhang ; Young et al. For strong convection and weak shear the roll instability mechanism can break down.
Similar patterns can be seen in SAR images of the sea surface underneath cold air outbreaks. As a result, the character of the roll instability remains essentially dynamical and the convective contribution moderately reinforces the rolls rather than establishing a convective BL.
Thus the shear instability mechanism is paramount and remains a good predictor of the roll characteristics. This is a different situation than the typical midlatitude roll—containing BL where the dynamical mode is relatively weaker than in the hurricane BL. The observations suggest that rolls do not form in the hurricane rainbands or at least are not the dominant structures.
In the strongly convective hurricane rainbands we expect that diabatic heating and cooling due to condensation and evaporation induce intense, local vertical motions that presumably disrupt the roll circulation. The second argument for hurricane rolls comes from the theory itself. The locally uniform relative to the roll time and length scales basic-state hurricane BL flow is ideal for the generation and maintenance of rolls throughout the majority of the BL where hurricane BL shear and surface buoyancy fluxes dominate i.
The nonlinear roll solution is strong and robust and results in large local enhancements in the surface winds and fluxes that are consistent with the observations. Noting that the contour plot of a 0 is broadly peaked we can consider relaxing the restriction of considering only the fastest growing instability.
Any of the nearby unstable modes could reach a nonlinear equilibrium if perturbed sufficiently to compete with the fastest growing mode. The stability analysis presented here includes the high-order nonlinear advective terms; however, the mean flow profile of Kepert that was used to define the mean BL profiles only includes a linearized mean radial advection and omits the mean vertical advection. Kepert and Wang used a numerical model that included all of the advection terms in order to investigate their importance.
For the same conditions and at the same radius as those assumed in this stability analyses presented here, the boundary layer mean flow profiles in Kepert and Wang are very similar in shape to those of Kepert The most noticeable effect is an increase in the mean vertical shear.
Because of the similarity in shape, we expect that the basic instability characteristics wavenumber, orientation, and normal mode shape would be very little changed if the analysis was performed around this basic state.
However, the strength of the nonlinear instability ought to increase because there is greater shear in the nonlinear profiles and consequently greater potential for transfer of kinetic energy into the perturbation. To test this assumption, we fit a smoothed curve through the mean flow profiles shown in Fig. As expected, the wavelength, orientation and overall normal mode shape were little changed.
However, the nonlinearly predicted roll magnitudes were significantly stronger. This suggests that the relatively low predicted roll strength in the model presented in section 7 is largely due to the use of an idealized mean flow profile that underestimates the mean vertical shear in the hurricane BL. Based on this, we expect that incorporating observed mean flow profiles into this model would result in nonlinear solutions with strengths comparable to the observations.
It is interesting to consider how a roll circulation such as that shown in Fig. Depending on the fall rate and the magnitude of the cross-roll winds, very near the surface it might be swept toward the lower surface wind branch of the roll circulation and warmer temperatures. While it is difficult at this early stage in hurricane boundary layer roll research to place it in context with overall hurricane theory, some comments are in order. There have been many recent investigations into the processes by which hurricanes intensify that have been summarized in Camp and Montgomery Emanuel , describes hurricanes as atmospheric heat engines that convert heat acquired from the ocean into the kinetic energy of the large-scale flow.
Boundary layer inflow transports this heat energy into the eyewall where it ascends and releases latent heat in the deep convective clouds. This is the energy source that drives the hurricane flow. The high level outflow radiates excess heat to space and the cycle is assumed to close as the cooled air descends and reenters the boundary layer. Based on this Carnot cycle model, the maximum intensity of the storm, expressed in terms of the maximum possible wind speed, may be derived Emanuel , , A key parameter in this theory is the ratio of the effective bulk transfer coefficients for momentum and enthalpy, neither of which is well characterized at hurricane wind speeds.
The drag coefficient arises because the loss of kinetic energy to friction is an important part of the energy cycle. The ratio of bulk transfer coefficients is often assumed to be one, even though numerical models have shown strong sensitivity to this ratio Emanuel Simple axisymmetric models of hurricanes have generally confirmed this prediction of maximum hurricane intensity Emanuel , although these models often exclude important processes that can significantly alter the hurricane dynamics and intensity.
These include vertical wind shear, convective asymmetry, spiral arm bands, ocean spray, dissipational heating, wind-induced ocean cooling, and vortex Rossby waves see Persing and Montgomery , hereinafter PM03 , and its included references for a detailed discussion. Furthermore, the same simple models can obtain hurricane strengths that greatly exceed the theoretical maximum intensity when the model resolution is increased to kilometer scale compared to the standard km resolution PM Analysis of these results and of very high resolution 3D models e.
PM03 also quote observational evidence that the low-level entropy in the eye can exceed that of the eyewall, which sets up the conditions for this process to occur in actual hurricanes. Hence, it appears that kilometer-scale or perhaps even subkilometer-scale processes omitted in the derivation of the maximum potential intensity theory that alter the transport of heat in the boundary layer or that act as sources of heat can potentially have a significant effect on the predicted strength of hurricanes.
For example, at hurricane wind speeds the heat generated by turbulent dissipation in the boundary layer is no longer trivial and ought to be included as a source of heat Bister and Emanuel ; Businger and Businger MBMDB have shown that hurricane boundary layer rolls significantly enhance the surface stress.
The question remains of what effect they have on the fluxes of heat and water vapor. In particular, what is the ratio of roll-induced enthalpy flux to that for momentum? Future hurricane roll model development and observational studies ought to investigate the roll effect on air—sea exchange and on the transport of heat and water vapor across the boundary layer.
A simple similarity boundary layer model connecting the gradient-level flow with the air—sea exchange for the hurricane boundary layer including rolls could then be developed along the lines of the midlatitude oceanic boundary layer model described in Brown and Liu Incorporation of such a model into maximum potential intensity theories could be used to address the question of whether or not the common presence of rolls in the hurricane boundary layer is important to predictions of hurricane intensity.
We have presented a theory that correctly predicts the existence of equilibrium roll solutions in the hurricane boundary layer over the ocean inside and outside the radius of maximum winds. The predicted wavelengths and orientation angles agree with the observations, and, in particular, the aspect ratio is in excellent agreement. We cannot directly compare the perturbation magnitudes to observations; however, we find that the predictions are at least of the correct order of magnitude.
However, we also find that the nonlinear amplitudes increase with increasing shear in the boundary layer. Applying the same theory to a more realistic mean flow profile yields roll strengths comparable to the observations.
The theory also predicts features of the roll circulation that have not yet been documented and that can be looked for in future field campaigns. These features include the asymmetry in perturbation winds between updraft and downdraft branches of the circulation, the warmer near-surface air below the updrafts and the dependence of the roll drift velocity with radius from the storm center. Hurricane BL flows are much more intense and have much higher turbulent fluxes than ordinary BL flows and present a major challenge to hurricane BL theory, modeling and parameterization.
The fluxes of momentum across the hurricane BL containing rolls are substantially larger and different in character than those predicted by the standard downgradient diffusive parameterizations of turbulence. A direct calculation of the roll-induced momentum fluxes from this theory shows that the nonlocal contribution in the mid-BL is of comparable magnitude to that due to the mean flow in the absence of rolls.
Attempting to include these effects in a standard boundary layer parameterization is unlikely to succeed since the roll-induced fluxes are inherently nongradient Zilitinkevich et al. This point is convincingly made by Glendening who simulated strong midlatitude roll vortices in an LES. However, it was found impossible to describe this flux using any reasonable gradient diffusion turbulence parameterization. Hence there is currently a wide gulf between recent observations of rolls in the hurricane BL, which show a large increase in surface stress due to rolls MBMDB , and numerical models of hurricane BL flow.
The strong sensitivity of numerical model simulations to boundary layer parameterization is discussed in Braun and Tao The theory presented here, suitably modified to ingest observed mean conditions and to include the feedback between the rolls and the surface stress, is well suited for parameterizing the roll contributions to the hurricane BL mean flow and fluxes. Hence it should be possible to introduce roll effects into hurricane BL parameterizations and thus, assuming that the numerical models are sensitive to changes in the air—sea interaction induced by rolls, improve the performance of numerical hurricane simulations.
We thank J. Businger, S. Businger, and K. Katsaros for bringing this problem to our attention and for many helpful discussions and comments on this paper.
Barnard, G. Young, P. Black, J. Patoux, R. Brown, M. Montgomery, and anonymous referees provided insightful comments that significantly improved the presentation of the results. Kepert also provided useful comments and the mean flow profiles corresponding to Fig. We thank P. Citation: Journal of the Atmospheric Sciences 62, 8; Sign in Sign up.
Advanced Search Help. Journal of the Atmospheric Sciences. Sections Abstract 1. Introduction 2. Equations and flow decomposition 3. Undisturbed basic-state flow 4. Scales and nondimensional parameters 5. Perturbation equations 6. Linear stability 7. Nonlinear analysis 8. Comparison with similar stability problems 9. Discussion Export References. Wyngaard , J. Izumi , and E. Ferziger , and P. Carlotti , R. Newsom , R. Banta , R. Foster , and J. Tuttle , and P.
Stuart , and W. Vachon , W. Liu , and P. Katsaros , and J. Snell , and Z. Businger , F. Marks , P. Vickery , and T. Schmid , and D. Moser , and M. Trefethen , S. Reddy , and T. Kristovich , M. Hjelmfelt , and R. Gryanik , V. Lykossov , and D. View in gallery Typical boundary layer mean flow profile.
View in gallery Mean flow parameters and linear instability characteristics as functions of radius and stratification for the generic hurricane described in the text. View in gallery a Hurricane BL depth squares , predicted roll wavelength triangles , and aspect ratio circles. View in gallery First Landau coefficient a 1 corresponding to the conditions in Fig. View in gallery Contour plot of nonlinear instability corresponding to Fig. View raw image Typical boundary layer mean flow profile.
View raw image Mean flow parameters and linear instability characteristics as functions of radius and stratification for the generic hurricane described in the text. View raw image a Hurricane BL depth squares , predicted roll wavelength triangles , and aspect ratio circles.
View raw image First Landau coefficient a 1 corresponding to the conditions in Fig. View raw image Contour plot of nonlinear instability corresponding to Fig. Chart I. Tracks of Centers of Anticyclones, December, Author: P. Next Article. Editorial Type: Article. Ralph C. Foster 1. Article History. Download PDF. Full access.
Introduction The first indication that hurricane boundary layers BLs might have coherent structures was given in Wurman and Winslow who found evidence of intense subkilometer-scale horizontal roll vortices roughly aligned with the mean azimuthal wind in high-resolution Doppler radar wind retrievals in the BL of Hurricane Fran during landfall. The length and time scales characterizing a hurricane are much larger than those of its BL flow, so we seek a basic-state hurricane BL flow that is quasi-steady and look for local perturbations in different regions of the storm.
We simplify by assuming the basic-state flow is azimuthally symmetric. The equations governing the basic-state mean flow are found by neglecting perturbation product terms, calculating the mean and assuming BL flow, that is,. Undisturbed basic-state flow The equations governing the basic-state hurricane BL mean flow are. Our solution to 1 — 2 closely follows the neutrally stratified, azimuthally symmetric one described by Kepert , who omitted the vertical advection terms and linearized both the horizontal advection, the radial curvature terms, and the flux lower boundary condition to arrive at a simplified system of equations for the boundary layer mean flow.
The major differences from Kepert are 1 we patch a stratification-dependent surface layer following Monin—Obukhov MO similarity below the BL solution; 2 we allow variable sea surface roughness Large and Pond ; Powell et al. Elliassen and Elliassen and Lystad explored the Ekman layer dynamics under a cyclostrophically balanced, axisymmetric vortex.
Of primary interest in their work were the spindown processes in such vortices. They showed that the decay is exponential when a no-slip lower boundary condition is applied and algebraic for a flux lower boundary condition. They also found that the region of maximum updrafts is located off the axis of symmetry. Notably they showed that the characteristic vertical length scale for this problem is.
The natural velocity scale is the local gradient wind speed V g. Perturbation equations The equations that govern the perturbation flow are found by subtracting 1 — 4 from the full system.
They are. Hence, for rolls oriented near the azimuthal wind direction, the x axis points roughly in the direction of the storm center and the y axis is pointed roughly upwind.
This scaling and rotation results in the following nondimensional system of equations relevant for hurricane BL perturbations:. Nonlinear analysis We now seek an approximate solution to the nonlinear problem defined in 13 — Following Herbert , we stretch the eigenvalue to form a Landau equation:.
It is believed that this highly coherent structure, likely caused by the inflection-point instability, plays an important role in organizing turbulent transport. Large-eddy simulations are conducted to investigate the impact of wind shear characteristics, such as the shear strength and inflection-point level, on the roll structure in terms of its spectral characteristics and turbulence… Expand.
View PDF. Save to Library Save. Create Alert Alert. Share This Paper. Figures and Tables from this paper. Citation Type. Has PDF. Publication Type. More Filters. Previous theoretical and numerical studies only focused on the formation of roll vortices rolls under a stationary and axisymmetric hurricane.
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