On the realism of quasi steady-state hurricanes

Transcription

1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215) On the realism of quasi steady-state hurricanes John Persing a, Michael T. Montgomery a, Roger K. Smith b and James C. McWilliams c a Dept. of Meteorology, Naval Postgraduate School, Monterey, CA, USA b Meteorological Institute, Ludwig Maximilians University of Munich, Munich, Germany c Department of Atmospheric and Oceanic Sciences, University of California at Los Angeles, Los Angeles, CA, USA Correspondence to: John Persing, Department of Meteorology, Naval Postgraduate School, Monterey, CA, USA. jjpersin@nps.edu We show that sources of cyclonic relative angular momentum must be considered when studying the canonical quasi steady-state hurricane problem. Steadiness is assessed here using several metrics including: the traditional one, the magnitude of the maximum cyclonic wind; the magnitude of the maximum anticyclonic wind; and the magnitude of certain integral quantities. Consistent with recent literature, we were able to construct a simulation with prolonged, quasi-steady characteristics only by including artificial sources of angular momentum that are numerical expedients with no real-life analogue. Copyright c 215 Royal Meteorological Society Key Words: Hurricanes; tropical cyclones; typhoons; steady-state Received today s date Citation: Introduction Tropical cyclones are intense, warm-cored, cyclonicallyrotating, convectively-driven vortices that form over the tropical oceans. A large fraction of these storms form from pre-existing disturbances in the Inter-Tropical Convergence Zone or monsoon trough, typically within 15 deg latitude of the equator. If conditions are favourable, intensification may be rapid and storms may last for a week or more. Storms that become particularly intense may remain intense (though not maintaining the same peak intensity or radius of maximum wind) for many days (e.g., Hurricane Ivan 24). However, storms eventually weaken, sometimes due to their northwestward propagation out of the tropics and over cooler sea-surface temperatures, sometimes because of increasing vertical wind shear in their environment, or sometimes because of their encounter with land or with rugged topography associated with island terrain. Certainly, from an observational perspective, one could justifiably regard tropical cyclones as transient vortices. Notwithstanding the flow complexities of individual tropical cyclones, our understanding of these vortices has been influenced strongly by the simple, axisymmetric, steady-state hurricane model described in a pioneering study by Emanuel (1986; henceforth E86). For almost three decades now, this model has served to underpin many ideas about how tropical cyclones function and even how they intensify from a small, but finite amplitude cyclonic vortex a few hundred kilometres in diameter (Rotunno and Emanuel 1987; Emanuel 23; Persing and Montgomery 23; Emanuel 24; Smith et al. 28; Bryan and Rotunno 29a,b; Bister et al. 211; Montgomery and Smith 214). As an example of its far-reaching influences, the E86 theory is still used as a basis for deriving updates to the so-called a priori potential intensity (PI) theory (Bryan and Rotunno 29b; Garner 215) as well as for estimates of the impact of tropical cyclone intensity and structure change due to global change scenarios (Camargo et al. 214; Emanuel 1988). Several recent numerical studies have described longlived simulations of tropical cyclones in axisymmetric (Hakim 211; Chavas and Emanuel 214; Frisius 215) and three-dimensional (3D) geometries (Brown and Hakim 213; Camargo et al. 214; Khairoutdinov and Emanuel 213; Zhou et al. 214). These studies may present candidate models for steady-state tropical cyclones if 1) the simulation is more-or-less steady (admitting some measure of temporal variability) throughout the domain, not just for a point metric like maximum wind speed, 2) the sources and sinks of conserved quantities such as mass, water mass, and absolute angular momentum are accounted for, and 3) that the resulting structure of the storm and computed sources and sinks are realistically represented. Copyright c 215 Royal Meteorological Society [Version: 211/5/18 v1.2]

2 2 John Persing, Michael T. Montgomery, Roger K. Smith and James C. McWilliams A recent study by Smith et al. (214) has questioned the existence of a realistic globally quasi steady-state tropical cyclone. By global quasi steady state we mean that the macro-scale flow does not vary systematically with time. Among other things, echoing Anthes (1982), the Smith et al. study showed that if such a state were to exist, then a source of cyclonic relative angular momentum (RAM) would be necessary to maintain the vortex against the frictional loss of angular momentum at the sea surface. It showed also that while a supply of cyclonic RAM is a necessary condition for a globally steady state cyclone, it is not sufficient. It would be necessary also that, above the boundary layer and outside regions where turbulent diffusion is significant, the spin up function S vanish, where S is the dot product of the absolute vertical vorticity of the azimuthal mean vortex and the gradient of the diabtic heating rate divided by the potential vorticity of the azimuthal mean vortex (see Section 3.2 of Smith et al. 214). It would seem to be a logical consequence that, without a steady source of cyclonic RAM, tropical cyclones must be globally transient, a deduction that accords with observations. In the E86 steady-state hurricane model, cyclonic RAM is assumed to be steadily replenished at large radii in the upper troposphere, but is such an assumption realistic? Even if it were, one might ask: how long does it take to attain this state and does the steady-state vortex predicted by the model bear any relationship to typical observed structures on realistic forecast time scales of a few days? In other models where the vortex is deemed to attain a quasi-steady state, the question arises: what is the source of RAM needed to support this state? To investigate these questions, we describe a series of cloud-permitting numerical simulations with increasing degrees of complexity. For the few cases which appear to attain a quasi-steady state, we determine the source of RAM and assess the realism of this source. An outline of the paper is as follows. Section 2 reviews briefly the angular momentum requirements needed to support a quasi-steady hurricane on an f-plane. Section 3 presents the numerical model and its configuration. The basic evolutionary characteristics of the cyclonic and anticyclonic portions of the model hurricanes are presented in Section 4. The angular momentum budget for these simulations is analysed in Section 5. A pathological case of a sustained hurricane is illustrated in Section 6. Section 7 touches briefly on the far-field energetics of the simulated storms. A discussion of our findings and their relationship to recent work is given in Section 8, where an assessment of some but not all of these published steady tropical cyclone solutions is given also. The conclusions are given in Section Angular momentum fundamentals Before presenting our series of long-time (more than a week or two) cloud-permitting numerical experiments and analyses, it proves useful to review the angular momentum requirement needed to sustain a steady-state hurricane. Although most of this material was presented by Smith et al. (214), some repetition may both help the reader and facilitate our discussion of the key angular momentum issue examined in this study. We consider a vertically-aligned, swirling flow on an f-plane expressed in cylindrical coordinates (r, λ, z) with corresponding azimuthally-averaged velocity components (u, v, w) and departures therefrom denoted by primes. The azimuthal-mean absolute angular momentum M per unit mass about the vortex axis is defined by the equation: M = rv fr2, (1) where f is the Coriolis parameter. Making the anelastic approximation ρ ρ(z), Smith et al. showed that the flux form of the equation for M is: where M t + 1 rum + 1 ρwm r r ρ z E = 1 r ru M r 1 ρ = E +D, (2) ρw M z. (3) represents the momentum fluxes associated with asymmetric (eddy) processes, and D = 1 [ r 3 ( v ) ] K r + 1 [ ] M ρk z, (4) r r r r ρ z z represents the unresolved horizontal and vertical diffusive processes. Here, the brackets <... > denote an azimuthal average and K r and K z are horizontal and vertical eddy diffusivities, respectively (Note that for simplicity azimuthal variations in eddy diffusivity have been ignored). Integrating the steady-state version of (2) over a control volume extending from the axis to r = R and from the surface to z = H, and using the boundary conditions that u = at r =, and w = at z = and z = H, gives R [ρrum + ρru M ] r=r dz = [ ] M ρk z rdr + z z= [ ρk r r 3 ( v r r) ] dz. r=r (5) After vertically integrating the steady-state version of the continuity equation ru r + 1 ρwr = (6) ρ z from z = to z = H, radially integrating this equation from r = to r = R, and again using the boundary conditions that u = at r =, and w = at z = and z = H, one obtains It follows trivially that [ρru] r=r dz =. (7) [ ρru( 1 ] 2 fr2 ) dz =, (8) r=r showing that, in a steady state, the vertically-integrated radial influx of planetary angular momentum density must Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

3 On the realism of quasi steady-state hurricanes 3 vanish at any radius. The vanishing of this integral is independent of the rate at which the azimuthally-averaged, low-level radial velocity decays with radius 1. For sufficiently large radius R, the relative angular momentum will be small compared to the planetary angular momentum, i.e., M(R,z) 1/2fR 2. This result, together with (8), implies that [ρrum] r=r dz [ ρru( 1 ] 2 fr2 ) dz =. (9) r=r Inserting (9) into (5) then yields the steady-state angular momentum balance of the vortex: R [ ] M ρk z rdr + z z= [ ρru M ] r=r dz = [ ρk r r 3 ( v r r) ] dz r=r (1) valid for sufficiently large radius R and for either axisymmetric or three-dimensional hurricane flows. From Equation (1), it is evident that the planetary angular momentum has completely dropped out of the atmospheric angular momentum balance of the vortex 2. In other words, the Earth s rotation cannot supply net angular momentum to maintain a steady vortex. The only way that the Earth s rotation can enter the angular momentum balance of the vortex is indirectly via the boundary conditions where the swirling flow component, itself, must exhibit special characteristics as discussed below. According to Eq. (1), the only possible sources of M available to maintain a hurricane in a steady state could be the radial diffusion of cyclonic angular velocity 3 at r = R at heights where the flow is anticyclonic (the second term on the right of Eq. (1)), the vertical diffusion at the ocean surface at radii where the flow is anticyclonic (contained in the first term on the right of Eq. (1)), a positive radial eddy flux of tangential momentum at r = R (through the term [ ρru M ] r=r ), or vertical diffusion and/or frictional torque of relative angular momentum arising at the upper boundary of the model (term not written here) 4. That the Earth s rotation cannot supply a net radial angular momentum flux to the steady vortex seems to be an under-appreciated result in the hurricane and geophysical fluid dynamics communities. The underlying assumption of many with whom we have spoken seems to be that the Earth s rotation provides an unlimited supply of cyclonic angular momentum. If this assumption was correct, then a plausible explanation of the dynamical elements needed for a steady-state hurricane would be trivial indeed! 1 This observation sharpens the conclusions of Smith et al. (214; their footnote on p4) 2 The vanishing of the integrated influx of planetary angular momentum into a hurricane is the analogue of the vanishing Coriolis torque across a meridional surface when considering the angular momentum balance of the middle-latitude westerly jet (see, e.g., Lindzen 199). 3 Note that the quantity that is radially diffused in the tangential momentum equation is the angular velocity, v/r, and not the angular momentum. 4 The specific source terms ofm that arise at the upper boundary using the chosen numerical model are written out explicitly in Section 5 and in turn analysed there and in Section 6. Specific contributions to the vortex-scale M-balance (1) will be discussed in Sections 5 and The Model and Experiments 3.1. Model core The base experiments are performed using the numerical model CM1, version 16, a non-hydrostatic and fully compressible cloud model (Bryan and Fritsch 22) 5. Over the sea, as is the case here, the terrain-following coordinate system simplifies to regular Cartesian coordinates. An attractive feature of this model is that it has options for being run in either axisymmetric (AX) or three-dimensional (3D) configurations. Another attractive feature is the way that moisture is treated, with all water species accounted for in the density potential temperature θ ρ (see below) as it appears in the pressure gradient force. There are predictive equations for the three components of the velocity vector u, water vapour q v, suspended liquid q l and cloud moisture q c mixing ratios, perturbation Exner function π = (p /p ) R/cp, and perturbation potential temperature θ, where perturbation quantities are defined relative to a prescribed hydrostatic basic state. Here c p is the specific heat of dry air at constant pressure p, R is the gas law constant for dry air, and p = 1 5 Pa is a reference pressure. For simplicity, ice microphysical processes are neglected. The reference sounding is a nearly moist-neutral sounding generated from the axisymmetric Rotunno- Emanuel (1987) model. Near-neutral (very low CAPE) soundings have served as a prototype in idealized studies since E86 demonstrated that ambient CAPE is unnecessary for maintaning a mature tropical cyclone and since many numerical simulations (e.g. Rotunno and Emanuel 1987; Montgomery et al. 29) are able to simulate intensification with very little environmental CAPE. Persing and Montgomery (25) presented a suite of axisymmetric simulations where approximately the same maximum intensity was found for a wide range of environmental soundings with varying CAPE. On the numerical side, the advection terms are calculated in flux form. The pressure gradient force per unit mass takes the form F P = c p θ ρ π, where θρ is the density potential temperature defined by: θ ρ = θ(1+q v /ǫ)/(1+ q v +q c +q l ), where ǫ = R/R v and R v is the gas constant for water vapour. The vertical momentum equation includes a buoyancy force per unit mass B = g(θ ρ θ ρ )/(θ ρ ), where θ ρ (z) is the basic state profile of θ ρ, and g is the gravitational acceleration. The tendencies of θ and π are calculated using the mass- and energy-conserving equations derived by Bryan and Fritsch (22). For simplicity, dissipative heating is not included (cf. Bister and Emanuel 1998) but its omission should not alter any of the conclusions or interpretations herein. The calculations are carried out on an f-plane with the Coriolis parameter f = s 1, corresponding to 2 N. 5 For a complete description of the three-dimensional model and variable definitions see the technical document The governing equations for CM1, available for download at and also available from G. Bryan. For a complete description of the axisymmetric version of CM1, see the paper by Bryan and Rotunno (29a). Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

4 4 John Persing, Michael T. Montgomery, Roger K. Smith and James C. McWilliams 3.2. Radiation physics The effects of radiation are represented variously in the simulations presented below. Several simulations are carried out with no radiative effects, and several others are carried out using the simple Newtonian cooling scheme (Rotunno and Emanuel 1987; RE87), that is capped at 2 K per day. This simple scheme prevents the environmental sounding from drifting away from the known initial sounding of θ(z). Finally, several simulations are carried out with a more-complete radiation scheme, described in the CM1 documentation as the NASA-Goddard longwave and shortwave scheme (Chou and Suarez 1994, 1999; abbreviated NASA-G where necessary); this scheme is credited to the Advanced Regional Prediction System of the Center for Analysis and Prediction of Storms at the University of Oklahoma Boundary conditions The lateral boundary condition in the AX version of the model is a free-slip, rigid wall. At the top boundary, a Rayleigh damping layer of the form Q/ t = αq is added above heights of 2 km to mitigate the reflection of internal gravity waves. HereQis one of{u,v,w,θ }, where θ = θ θ(z) is the perturbation of potential temperature from the initial basic sounding, and α is defined by { [ ( )] z < 2 km α(z) = 1 z 2 km A 1 cos π 25 2 km z 2 km Here we take constant A to have the default value of 3 s (following the similar numerical setup of Persing et al. 213). It will be shown that the Rayleigh damping of the horizontal winds in the upper-troposphere and lower-stratosphere has non-trivial consequences for the angular momentum balance of the putative steady-state storms. Hitherto, the effect of Rayleigh damping has not been investigated quantitatively in near-cloud resolving simulations. Since we desire solutions that avoid the artificial source of RAM at the lateral boundary as discussed in the Introduction, lateral damping is applied in this study in only one case (EX-6) below Air-sea interaction All simulations have a fixed sea surface temperature of K (26.15 C). The parameters determining the exchange of heat and momentum at the air-sea interface are set as follows. The surface exchange coefficients of heat and momentum are taken to be constant in both space and time. The moist enthalpy transfer coefficient C k is set equal to This value is close to the mean value ( ) derived from the Coupled Boundary Layers/Air-Sea Transfer (CBLAST) experiment (Fig. 6 of Black et al. 27; Fig. 4 of Zhang et al., 27), a recent laboratory study (Fig. 1 of Haus et al. 21) near and slightly above marginal hurricane wind speeds, and an energy and momentum budget analysis of the lowertropospheric eyewall region at major hurricane wind speeds (Bell et al. 212). The drag coefficient is set to be twice the enthalpy exchange coefficient C D = 2 C k = , and is close to the estimated mean value of C D = from observations derived from CBLAST for major hurricane wind speeds by (Bell et al. 212) Turbulence parameterization Subgrid-scale turbulence is represented by choosing option iturb=3 in the model, which is designed for problems that do not resolve any part of the turbulent Kolmogorov inertial range. This option requires the external specification of the horizontal mixing length l h = 7 m and the vertical mixing length far above the surface layer l = 5 m, which for simplicity are assumed constant in both space and time (Smagorinsky 1963; Lilly 1962). Near the bottom boundary the vertical mixing length l v = l v (z;l ) approaches zero to model a logarithmic surface layer 6. The subgrid scale scheme follows Smagorinsky (1963) and Lilly (1962), except that different eddy viscosities must be used for the horizontal and vertical directions. The flow-dependent momentum diffusivities in the horizontal and vertical directions are specified as follows: K m,h = l 2 h S h andk m,v = l 2 vs v 1 Ri/Pr, where the m subscript refers to momentum, and the second subscript h or v refer to the horizontal and vertical directions, S h and S v denote the terms found in the total deformation, S, that involve the horizontal and vertical flow components, Ri = N 2 m/s 2 is the moist Richardson number, N 2 m is the moist Brunt-Väisälä frequency, and Pr is the Prandtl number (set to unity in this option) (see Bryan and Fritsch 22 for complete definitions of these parameters). In this scheme, the vertical eddy diffusivity is proportionally reduced in regions with positive moist Richardson number and the heat and momentum diffusivities are taken to be identical,k h = K m. Whenever Ri exceeds unity, the vertical momentum and vertical heat diffusivities are set to zero Precipitation physics Rainfall is represented here by one of two schemes. (1) The simple scheme of Rotunno and Emanuel (1987; RE87) is used for some simulations. This scheme has no ice and represents both cloud and rain by a single variable q l where any such condensate with a value above a small threshold is assumed to fall at a fixed 7 m s 1 fall speed. (2) A more-complete scheme used in other simulations is the NASA-Goddard Cumulus Ensemble Model (Tao and Simpson 1993; NASA-G). This includes separate predictive variables for liquid cloud, rain, ice cloud, snow, and hail and is described by the CM1 documentation (distributed with the model as described in the footnote above) as being matched to the available NASA-Goddard radiation scheme. For simplicity, we do not employ ice physics in this study Initial conditions The same initial cyclonic vortex is used for all simulations. The initial radial and vertical velocity are set to zero. The initial tangential velocity is taken to be in gradient wind balance with a maximum cyclonic velocity of 13 m s 1 at the surface at a 1 km radius from the centre of circulation. The tangential velocity varies smoothly in space and tends 6 In Smith and Montgomery (213) the existence of a traditional loglayer in the inner-core region of a hurricane vortex was questioned on both theoretical and observational grounds. Smith and Montgomery (213) refute the stance of Kepert (212) that hurricane models which do not have a log layer implementation should not be used. Nevertheless, we accept the current modification as an expedient for incorporating a vanishing vertical mixing length as the height approaches zero. We reserve further discussions for a later study. Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

5 On the realism of quasi steady-state hurricanes 5 to zero at large radii: it vanishes beyond 4 km radius and above z=2 km. As noted above, the initial sounding is the near-neutral sounding of Rotunno and Emanuel (1987) and is the same as employed in Persing et al. (213) Model geometries Since one of the main objectives of this study is to identify the sensitivity of long-time sustained tropical cyclones to artificial sources of RAM, the AX configuration presents several advantages compared with the 3D configuration for examining this issue: (1) a simplifying absence of resolved eddy effects which permits readier comparison to the extant theories for steady state TCs (a la E86); (2) reduced data size permitting very long simulations, and (3) the storm maintains a fixed centre above the origin without possibility for vortex tilt. For these reasons, most of the numerical simulations presented here employ the AX geometry with a uniform horizontal grid spacing of 3 km and an outer boundary at a radius of 9 km (see Table 1). In the AX experiments, this large outer radius permits the simulated storm to have a size in which the influence of the outer boundary is minimized. In the vertical, there is a stretched grid with 4 grid points between the surface and model top of 25 km; the first grid level is at 25 m altitude and there is finer grid spacing near the surface The experiments As pointed out in the Introduction, without a steady source of RAM, tropical cyclones must be globally transient phenomena. To investigate this inference, we examine a series of cloud-permitting numerical simulations with increasing degrees of complexity. These experiments are listed in Table 1. As summarized above, we have attempted to configure the CM1 model with parameters based on the latest observational guidance. In this sense our experiments are believed to be as realistic as possible within an idealized framework. All experiments use version 16 of CM1, except for EX-2. The experiments are as follows: (1) Simulation EX-1 is an AX simulation using both the RE87 simple-rain approximation and the RE87 expedient for radiation via Newtonian relaxation. (2) Simulation EX-2 is the same as EX-1, but uses version 14 of CM1. This version was the one used by Smith et al. (214) and does not have a strict logarithmic surface layer. This aim of this experiment is to test the robustness of the findings to the representation of the surface layer 7. (3) Simulation EX-3 has an AX configuration and uses the NASA-Goddard microphysics scheme, but without radiation. (4) Simulation EX-4 is the same as EX-3, but includes the NASA-Goddard radiation scheme. (5) EX-5 is the principal 3D simulation discussed here. This simulation uses the stretched grid mesh described in Persing et al. (213), the inner region of which has a fixed horizontal grid spacing of 3 km. The computational domain is larger in extent than 7 CM1 version 14 differs from version 16 in several small ways. Of interest here, the vertical mixing length l v = 5 m is constant and replaces the proposed near-surface log-layer implementation of version 16. See also footnote 6. in Persing et al. (213): 6 km in each horizontal direction for the fine mesh region, with a total domain size of 3 3 km. The outer boundary of this experiment is open. In EX-5 we use the NASA- Goddard microphysics scheme, but radiation is turned off. This combination serves as a 3D analogue for the simulation EX-3. (6) Simulation EX-6 is an AX experiment designed to highlight the influence of the upper and lateral damping layers used in contemporary studies reviewed in the Introduction. A discussion and interpretation of this simulation is given in section Synopsis of cyclonic and anticyclonic flow evolution The time evolution of the domain maximum and minimum tangential velocity for simulations EX-1 through EX-5 is shown in Figure 1. The maximum tangential velocity is used as a metric for the intensity of the cyclonic vortex core. For the 3D simulation (EX-5), the maximum and minimum of the azimuthally-averaged tangential velocity are plotted. In each of the simulations, the vortex attains a maximum tangential velocity exceeding 6 m s 1 within 2 days. After this time, the intensities in EX-1, EX-2, and EX-3 weaken significantly, while that in EX-4 attains a quasisteady value within the range 45-6 m s 1. The vortex in the 3D simulation weakens comparatively slowly. In this case, as time proceeds, the vortex centre wanders within the inner domain so that a part of the storm circulation moves into the outer domain where the representation of convective processes is less accurate on the coarse (variable) outer grid mesh. When this happens, the calculations are suspended. In EX-1, EX-3, and EX-4, a sustained quasi-steady anticyclone is found with an intensity exceeding 4 m s 1. In EX-1 and EX-3 this peak intensity occurs after the intensity of the cyclone core has weakened below tropical storm strength (< 2 m s 1 ). These results highlight the non-uniform changes in global vortex structure that occur over the course of extended integrations. EX-4 has a particularly strong upper-level anticyclone, with a tangential wind minimum of -7 m s 1, noticeably greater in magnitude than the intensity of cyclone core (between 45 and 6 m s 1 ). Table I summarizes key features of each of the simulations, as well as peak and final cyclonic and anticyclonic intensities. Figure 2 shows the time evolution of the mass integral of kinetic energy (KE), partitioned into cyclonic (v > ) and anticyclonic (v < ) portions as in Smith et al. (214). The domain of integration is chosen to be a circular cylinder large enough to contain the principal cyclonic and anticyclonic tangential wind regions. EX-1 and EX-3 slowly expands to fill out to 6 km, in contrast to EX-4, where environmental convection at 4 km radius leads to a v > signature that limits the extent of the anticyclone. This is seen for example, in the Hovmöller diagram shown later in Figure 3. In EX-5, a 289 km radius was chosen for the cylinder of integration (Fig. 2b), which is contained within the 6 6 km domain in Cartesian geometry and allows for some modest vortex wander near the center of the domain. In all simulations, the anticyclonic KE subsequently becomes at least several times larger than the cyclonic KE, except in EX-2, where the central cyclone is so rapidly diminished so neither a cyclone or anticyclone signal is shown (Fig. 2b). These calculations affirm the novel Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

6 6 John Persing, Michael T. Montgomery, Roger K. Smith and James C. McWilliams Table I. Prolonged characteristics of simulations Intensity Cycl./Antic. Simulation Geometry CM1 Microphysics Radiation Peak Cycl. Final t final Behavior Ver. (m s 1 ) (m s 1 ) (days) EX-1 AX 16 RE87 RE87 8 / / decayed EX-2 AX 14 RE87 RE87 65 / / -1 1 decayed EX-3 AX 16 NASA-G none 8 / -5 1 / decayed EX-4 AX 16 NASA-G NASA-G 8 / / quasi-steady EX-5 3D 16 NASA-G none 75 / / decaying EX-6 AX 16 NASA-G NASA-G 8 / -3 7 / quasi-steady NOTE: EX-6 has a 15 km radial domain with Rayleigh damping at both the top and lateral boundaries quantitative analyses of Smith et al. (214) who found in their canonical integration that the KE of the outflow anticyclone was comparable to or greater than that of the cyclonic flow during the mature and decaying phases of the storm lifecycle. Extreme cyclonic and anticyclonic winds for EX-4, shown in Figure 1d, presents quasi-steady intensities of approximately equal magnitude. The integrated cyclonic and anticyclone kinetic energy for EX-4, shown in Figure 2d, presents an interesting comparison with the anticylonic measure now an order of magnitude larger than cyclonic kinetic energy. 5. Angular momentum budget for CM1 experiments The absolute angular momentum budget for the extended simulations derived in section 2 are summarized in Table II. For the budget we apply a time-dependent form of Equation (5) to the simulations described in section 3. In addition, we include Rayleigh damping terms near the upper and lateral boundaries. The angular momentum budget integrated over a circular cylinder of radius R and height H = z top, the top of the numerical model, is as follows + = + R R R [ρr u M + ρru M ] r=r dz ρ M rdrdz t C D ρ v sfc c sfc r 2 dr [ ρ K r r 3 v /r r R α ρ v r 2 drdz ] dz r=r α ρ v r 2 drdz (11) where the first term on the left hand side is the volume integral of both the radial mean and eddy flux ofm and the second term on the left hand side is the volume-integrated tendency of M. On the right hand side are: the integrals of the vertical diffusion of angular momentum across the bottom boundary due to surface drag using the surface momentum transfer in the model; the horizontal diffusion of angular velocity across the lateral boundary; the integral of Rayleigh damping in the model stratosphere; and finally a Rayleigh damping contribution confined to a layer adjacent to the outer lateral boundary, which is used only in EX-6 8. The additional lateral Rayleigh damping function used in EX-6 is given by the formula: α (r) = { [ ( 1 A 1 cos )] r < 14 km r 14 km π r 14 km km (12) which applies velocity damping on all three velocity components to a state of rest in a radial band 14 < r 15 km; the potential temperature perturbation θ is similarly damped. The surface wind speed is denoted by c sfc. Quantities involving c sfc in EX-2 use the first grid level (z = 25 m) data, while those for version 16 use the surface (log-) layer extrapolation scheme to estimate the 1-m wind M budget for the 3D simulation We begin by calculating the M-budget for the 3D simulation, EX-5. This case provides an illustration of the spin-up and subsequent spin-down of a model hurricane of realistic duration. For this vortex, Figure 1e shows a weakening trend in peak cyclonic intensity and a strengthening trend in peak anticyclonic intensity beyond 7 days. Figure 4a shows a negative trend of the volumeintegrated, mass-weighted M-tendency integral (red curve). This negative tendency in M tracks the surface drag term (black curve). The surface drag term increases in magnitude even beyond the attainment of peak cyclonic intensity (at 7 days). This increase is accompanied by a broadening of the surface tangential wind profile (not shown). The residual between the drag term and the tendency term is accounted for primarily by the term involving the vertical integral of the mean radial flux ofm (Fig. 4a, green curve). As the developing anticyclone begins to impinge on the damping layer, the Rayleigh damping term (green dotted curve, Fig. 4b) contributes to a positive tendency in volumeintegrated M from the time of peak cyclonic intensity onwards. The eddy term (blue curve, Figure 4b) contributes also a positive tendency to the volume-integrated M and is of similar magnitude to the Rayleigh damping term. However, both the eddy and Rayleigh damping terms are two orders of magnitude smaller than the surface drag term and are unable to counter the sink ofm associated with the 8 In reference to version 16 of CM1, selection of parameters to suppress Rayleigh damping at the top boundary ( irdamp= ) and with no-slip top and bottom boundary condition ( bcuturbu=3, required for bulk aerodynamic surface formula) leads to an erroneous stress term at the top boundary. In the results presented here, this error has been corrected. Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

7 On the realism of quasi steady-state hurricanes 7 Figure 1. Maximum (solid) and minimum (dotted) tangential wind from the (a) EX-1, (b) EX-2, (c) EX-3, (d) EX-4, and (e) EX-5 simulations. Note the time ranges on the x-axis are different for each simulation. drag term. As in all other experiments considered here, the radial diffusion term (Figure 4b, black dotted curve, with multiplier) is many orders of magnitude smaller than the other terms calculated. Figure 2. Mass-integral kinetic energy within a circular cylinder partitioned into cyclonic (solid) and anticyclonic (dotted) portions, normalized by the initial cyclonic kinetic energy. (a) From EX-1 inside 6 km radius. (b) From EX-2 inside 15 km radius. (c) From EX-3 inside 6 km radius. (d) From EX-4 inside 42 km radius. (e) From EX-5 inside km radius. Note that different ranges of time are shown on the x-axis and ranges of value on they-axis. Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

8 8 John Persing, Michael T. Montgomery, Roger K. Smith and James C. McWilliams Figure 3. Mean tangential winds at z = 12 km height in EX-4, plotted radius versus time. The contour interval is 1 m s 1, with blue contours negative, yellow contour is zero, and red contours positive. Figure 5. Terms of the angular momentum budget (Eq. 11) computed for EX-4 inside an integration cylinder of radius R = 41 km. Two panels are used for clarity. The terms represented are, refering to corresponding terms of Table 2: 1) Tendency red solid, 2) MeanFlux green solid, 3) Surface black solid, 4) UpperRayl., top boundary Rayleigh damping, green dotted, 5) UpBndyStr., the upper boundary stress term, with a multiplier of 15 for clarity, blue dotted, and 6) HorizDiff, black dotted. integrated M (see Table II). As a result, the vortex in each case ultimately decays. EX-2 is a rather special case in that, with the complete collapse of the cyclone/anticyclone pair, a convective signal is likely being reported. Table II summarizes quantitatively the role that the artificial Rayleigh damping term provides as a source for cyclonic torque in offsetting the spin-down torque due to surface friction. Unlike all of the experiments discussed above, experiment EX-4 (Fig. 1d) shows a prolonged and quasi-steady state after 1 days. In this state, the M -tendency term and the mean radial influx of M (Fig. 5a, red and green curves, respectively), have the largest magnitudes. However, these two terms average out with little bias (see column for EX4 in Table II: the means of these quantities being nearly two orders of magnitude smaller than the standard deviation across the quasi-steady averaging period.) Because of the large temporal variability shown in the M -budget calculations and the coarse (six-hourly) temporal sampling of the output, a statistical interpretation of the budget results is presented. Overall, an approximate balance is found Figure 4. Terms of the angular momentum budget (Eq. 11) computed for between the spin-down tendency from the surface drag EX-5 inside an integration cylinder of radius R = 2845 km. Two panels are used for clarity. The terms represented are, refering to corresponding term (black curve) and the spin-up tendency due Rayleigh terms of Table 2: 1) Surface black solid, 2) Tendency red solid, damping in the model stratosphere (green dotted curve, 3) MeanFlux green solid, 4) UpperRayl., top boundary Rayleigh Fig. 5b; and column for EX-4 in Table II). The spin-up damping, green dotted, 5) HorizDiff, horizontal diffusion, with a torque represented by the upper-level Rayleigh damping multiplier of 15 for clarity, black dotted, and 6) EddyFlux, blue solid. is the numerical device used to control the reflection of gravity waves off of the rigid top boundary. In this case, the damping layer acts upon the outflow anticyclone, which 5.2. M budget for the AX simulations abuts the stratosphere and reduces the strength of the upper The behaviour of the M budget in experiments EX-1 anticyclone. Of course, there is no real-life analogue for and EX-3 are broadly similar to that of EX-5 in the this Rayleigh damping of momentum, which is introduced sense that the surface drag term dominates the sources of merely as a numerical expedient. Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

9 On the realism of quasi steady-state hurricanes 9 Table II. M-budget summary Simulation EX-1 EX-2 EX-3 EX-4 EX-5 EX-6 Analysis interval (d) 2 < t < 9 4 < t < 8 4 < t < 12 1 < t < 3 1 < t < 3 1 < t < 3 Intensity during interval Weakened Weakening Weakening Quasi-steady Weakening Quasi-steady R (km) Tendency ρ M rdrdz t Mean (kg m 2 s 2 ) Std. Dev (kg m 2 s 2 ) Mean flux ρr u M dz Mean (kg m 2 s 2 ) Std. Dev (kg m 2 s 2 ) Eddy flux ρru M dz Mean (kg m 2 s 2 ) Std. Dev (kg m 2 s 2 ) Surface C D ρ v sfc c sfc r 2 dr Mean (kg m 2 s 2 ) Std. Dev (kg m 2 s 2 ) Horizontal diffusion ρ K r r 3 v /r dz r Mean (kg m 2 s 2 ) Std. Dev (kg m 2 s 2 ) Upper Rayleigh damping α ρ v r 2 drdz Mean (kg m 2 s 2 ) Std. Dev (kg m 2 s 2 ) Lateral Rayleigh damping α ρ v r 2 drdz Mean (kg m 2 s 2 ) Std. Dev (kg m 2 s 2 ) In the 3D simulation (EX-5) the Rayleigh damping term is weaker than in EX-4 because the anticyclone is shallower and weaker than that in EX Evolution of environmental sounding The vertical structure of the mean tropical atmosphere results from a balance between the supply of sensible and latent heat from the ocean surface, the infrared radiative cooling of the atmosphere to space and the vertical transport of energy by various forms of convection (e.g., Riehl 1954; Goody and Yung 1989; Brown and Bretherton 1997). Most current numerical models of tropical cyclone behaviour do not have the spatial resolution at large radii to represent the convective processes involved in this balance. In particular, in axisymmetric models, the convection at large radii is especially unrealistic as it takes the form of annular rings encircling the storm. In all models that cannot represent explicit convection in the far field, one approach is to adopt a simple relaxation scheme. As noted in Smith et al. (214), this choice serves as a simple expedient for circumventing an explicit representation of the radiativeconvective equilibrium process, which operates to maintain the ambient tropical sounding over realistic forecast time scales of several days. Traditionally, a relaxation scheme is used to avoid an excessive stabilization of the inner-core region over integration times of many days and possible changes in the ambient sounding as warm air is advected through the open boundary of the flow domain at large radius. For the foregoing reasons, in their idealized tropical cyclone simulations, Rotunno and Emanuel (1987) relaxed the temperature profile to a convectively-neutral profile determined on the basis of their model physics. Some of our experiments employ this relaxation method, while some do not. In those that do not, it is important to know how much the ambient profiles of temperature and moisture drift away from the initial profile during the course of the integration. A drift of the profiles may represent an increasing energy supply to the storm in the form of convective available potential energy (CAPE). Where there is relaxation (EX-1 and EX-2, see Table 1), by construction there is no drift of the ambient temperature and moisture profile. In EX-3 and EX-5, where there is no relaxation and no radiation, we find there is little drift in the soundings. An example of such a environmental sounding is shown in Figure 6a using experiment EX-3. The changes in temperature are everywhere less than 2 K. However, in the case of EX-4, which includes the NASA- Goddard radiation scheme, there is a larger departure of the final sounding from the initial sounding, in which the troposphere is cooler and drier (see Figure 6b). In the case of EX-4 the domain-averaged CAPE is found to vary between 4-9 J kg 1 K 1, within the range of realistic values (Smith and Montgomery 212). 7. The pathological case: EX-4 The long-lived vortex that maintains hurricane-intensity in EX-4 has a structure that distinguishes it from the decaying simulations and one that is not consistent with known observations of hurricanes. The intense anticyclone (7 m s 1 Fig. 1d) in this simulation extends from the surface to tropopause and has a very large horizontal extent, reaching to approximately 4 km radius (Fig. 7a). The radius of maximum tangential wind shows variability between 12 and 3 km; the persistent eyewall convection is found to lie within the same radial range. At smaller radii, within the simulated eye, a secondary maximum of tangential winds of more than 3 m s 1 is found at 1 km height, a feature with no known real-life analogue. Large annular cells of cyclonic/anticyclonic winds exist in the far-field environment (r > 5 km) and exhibit a quasi-steady character there (e.g, the standing pattern of positive/negative tangential winds shown in Figure 3). Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

10 1 John Persing, Michael T. Montgomery, Roger K. Smith and James C. McWilliams (a) (b) Figure 6. Domain-average vertical soundings at initialization (solid) and at 25 days (dotted) for a) EX-3 (top) and b) EX-4 (bottom). The left panels show temperaure (T, right curve) and dew point temperature (T D, left curve). The right panels show potential temperature (θ, left curve) and equivalent potential temperature (θ e, right curve). As demonstrated in Section 2, the long-term maintenance of the vortex in EX-4 can be achieved only by supplying mean cyclonic RAM to offset the loss of RAM by the surface frictional torque at radii where the surface wind is cyclonic. Since this simulation has frictionless, rigid walls, the angular momentum loss must be offset by either surface boundary torques or by the Rayleigh damping found in the model stratosphere. (Recall, that the lateral Rayleigh damping is disabled in all experiments, except EX-6.) When describing a physically plausible pathway to sustaining a hurricane (though not necessarily realistic as observed in the real tropical atmosphere), Smith et al. (214) anticipated the descent of the outflow anticyclone to the surface and noted prior studies that had constructed such solutions. The important role of the surface anticyclone in EX- 4 is its ability to draw cyclonic RAM from the lower boundary, thereby opposing, in part, the loss of RAM beneath the cyclone. The RAM source associated with the surface anticylonic winds is shown in Figures 4 and 5: the surface stress of RAM is mostly negative, but occasionally positive during the quasi-steady period (after 15 days). Evidently, this term is not able to counter the loss of angular momentum associated with the drag term under the strong cyclonic wind region. However, a more-persistent spin-up torque is found in the top Rayleigh damping layer. When averaged between 1 < t < 3 d inside the 41 km circular cylinder, the surface drag torque is kg m 2 s 2 versus the torque from the Rayleigh damping layer of kg m 2 s 2. Thus the Rayleigh damping term, being of comparable magnitude to the surface drag torque and larger in magnitude than the net tendency and mean flux terms has an important role in offsetting the anticyclonic torque of drag Removal of the upper-level Rayleigh damping In light of the result from EX-4, an experiment was performed like EX-4, but without the top-boundary Rayleigh damping layer. In this case (not shown), the upper-level anticyclone extended up to the upper-most boundary (z = 25 km); above z > 2 km, the anticyclone extends out to 65 km radius. Otherwise, this case shows pathological structural similarities to EX-4, with a large radius of maximum winds (2 km) and the upper-level eye cyclone, which instead extends to the upper boundary. The simulation provides a long-lived cyclone, but steadiness is not achieved, even after 36 simulation days. There is a slow intensifying trend in cyclonic intensity, evidence for farfield evolution, and an integrated M tendency of sufficient magnitude to argue that steadiness is not achieved Revisiting Hakim s (211) results The oft cited study by Hakim (211) presents simulations of quasi-steady state hurricanes for prolonged periods of hundreds of days, begging the question: what is the source of RAM needed to maintain this state (Smith et al. 214). It turns out that Hakim (211) applies a damping layer Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)

11 On the realism of quasi steady-state hurricanes 11 Figure 7. Radius-height contours of the three components of the wind for EX-4 averaged from 125 to 165 d for a) the complete domain, and b) the innermost 75 km radius. Tangential windsv and radial windsuare shown with the contours ±{1,2,3,5,1,15,2,3,5} with cyclonic (v > ) values blue, anticyclonic (v < ) values black, outflow (u > ) values solid gray, and inflow (u < ) values dotted gray. Vertical winds w are shown with red contours with values {.1,.2,.3,.5, 1, 1.5, 2, 3, 5}. to momentum and potential temperature within 5 km of the model top (like EX-1 through EX-5 here), and to momentum within 1 km of the outer radius, which is 15 km in his simulations. The damping layer applied near this radius is reminiscent of that in the E86 model discussed in the Introduction. As noted in Smith et al. (214), inspection of Hakim s Figure 2 suggests that it is near the outer boundary where much of the cyclonic RAM is restored, typically between heights of 5 and 7 km where air parcels cross a large gradient in the M-surfaces as they descend in that region. To further test our hypothesis on the important role of a RAM source in supporting long-lived model hurricanes, we examine them budget for a simulation much like Hakim s. This experiment, EX-6 (see Table 1), replicates the mostpertinent settings of Hakim s simulation. In particular, it employs a relatively small radial domain of 15 km with Rayleigh damping near both the top and lateral boundaries. With this configuration, we can replicate Hakim s main result of a prolonged hurricane for hundreds of days (Figure 8). Shown in the lower panels of Fig. (8) is the angular momentum budget Eq. (11), except that an additional term for the Rayleigh damping near the lateral boundary has now been added. Comparing EX-6 and EX-4 (Table II), we see that the new lateral damping term replaces the role of the upper damping term as the principal source term of RAM, offsetting the loss of RAM due to surface friction. In retrospect, EX-4 can be viewed as a large-domain version of EX-6, since an analogue of EX-6 with lateral damping would present a damping zone more than twice the radius (4 km) of the extent of the anticyclone circulation. One could think of the lateral damping as representing processes at large radii (r > 15 km) that offset the loss of angular momentum due to friction in the Figure 8. a) Maximum (solid) and minimum (dotted) tangential wind from EX-6. Also, Terms of the angular momentum budget (Eq. 11) computed for EX-6 inside an integration cylinder of radius R = 1496 km, here two panels are used for clarity. The terms represented are: 1) Mean being the mean flux term in solid green, 2) Tend. being the tendency term in solid red, 3) Surf. Diff. being the vertical diffusion term at the bottom boundary in solid black, 4) Horiz. Diff. being the horizontal diffusion in dotted black, 5) Rayl. Top being the Rayleigh damping term at the top boundary in dotted green, and 6) Rayl. Lat. being the Rayleigh damping term at the lateral boundary in dotted red. A multiplier is applied to the horizontal diffusion term to improve the depiction. cyclone. However, the only realistic process to achieve this replenishment of M would be surface friction beneath a complimentary anticyclone. EX-4 does show an anticyclone with an intensity much greater than that found in EX-6 (Table I), but the anticyclonic circulation is most intense aloft, more readily able to access the artificial source of RAM in the stratosphere. The intensity of the cyclone is stronger in EX-6 than in EX-4, suggestive of the role that an artificial source of RAM closer to the cyclonic circulation might have on various measures of intensity. These experiments highlight a sensitivity of the simulations to the nature of angular momentum sources. EX-6 demonstrates the dominant role of artificial momentum sources in determining the structure of the steady-state. Compared to EX-4, the use of a stronger momentum damping term in the form of lateral damping (see Table II) produces a more realistic-looking anticyclone with intensity of just 3 m s 1. The dominant role of artificial momentum sources is futher illustrated by a further experiment, like EX-6, but Copyright c 215 Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1 14 (215)