On the Instabilities of Tropical Cyclones Generated by Cloud Resolving Models

SERIES A DYANAMIC METEOROLOGY Tellus AND OCEANOGRAPHY PUBLISHED BY THE INTERNATIONAL METEOROLOGICAL INSTITUTE IN STOCKHOLM

On the instabilities of tropical cyclones generated by cloud resolving models

By DAVID A. SCHECTERÃ, NorthWest Research Associates, Boulder, CO, USA (Manuscript received 30 May 2018; in final form 29 August 2018)

ABSTRACT An approximate method is developed for finding and analysing the main instability modes of a tropical cyclone whose basic state is obtained from a cloud resolving numerical simulation. The method is based on a linearised model of the perturbation dynamics that distinctly incorporates the overturning secondary circulation of the vortex, spatially inhomogeneous eddy diffusivities, and diabatic forcing associated with disturbances of moist convection. Although a general formula is provided for the latter, only parameterisations of diabatic forcing proportional to the local vertical velocity perturbation and modulated by local cloudiness of the basic state are implemented herein. The instability analysis is primarily illustrated for a mature tropical cyclone representative of a category 4 hurricane. For eddy diffusivities consistent with the fairly conventional configuration of the simulation that generates the basic state, perturbation growth is dominated by a low azimuthal wavenumber instability having greatest asymmetric kinetic energy density in the lower tropospheric region of the inner core of the vortex. The characteristics of the instability mode are inadequately explained by nondivergent 2D dynamics. Moreover, the growth rate and modal structure are sensitive to reasonable variations of the diabatic forcing. A second instability analysis is conducted for a mature tropical cyclone generated under conditions of much weaker horizontal diffusion. In this case, the linear model predicts a relatively fast high-wavenumber instability that is insensitive to the parameterisation of diabatic forcing. The prediction is in very good quantitative agreement with a previously published analysis of how the instability develops in a cloud resolving model on the way to creating mesovortices slightly inward of the central part of the eyewall.

Keywords: tropical cyclones, instabilities, numerical modeling

1. Introduction of perturbation growth may involve the mutual amplifi- Satellite and radar images of mature tropical cyclones cation of counter-propagating vortex Rossby waves or a commonly reveal deformed eyewalls and mesovortices destabilising wave-critical layer interaction. Depending along the periphery of the eye. There has been longstand- on specifics, the instability may generate robust arrays ing interest in understanding how such features develop of mesovortices or engender transient turbulence that and whether the process appreciably affects the temporal thoroughly redistributes inner core vorticity into a cen- trend of vortex intensity. One plausible explanation for tralised monopole (Schubert et al., 1999; Kossin and the emergence of prominent waves and mesovortices Schubert, 2001). The latter transformation may appre- involves an instability of the local circular shear flow. ciably deepen the central pressure deficit while diminish- Although such an explanation is prevalent in the ing the maximum azimuthally averaged wind speed literature, there has been limited progress in advancing (ibid). Adding simplified parameterisations of diabatic an instability theory for realistically modelled trop- forcing (moist convection) to a nondivergent 2D model ical cyclones. or a shallow-water system generally modifies the devel- Basic insights have been gained through the study of idealised two-dimensional (2D) models. Such models opment of an instability and the coinciding change of show that a vorticity annulus similar to that on the vortex intensity. Details depend on the parameterisation, inward edge of an eyewall is usually unstable. The onset and published results on the topic (Rozoff et al., 2009; Hendricks et al., 2014; Lahaye and Zeitlin, 2016)await rigorous comparison to more realistic theories and ÃCorresponding author. e-mail: [email protected] numerical simulations. Tellus A: 2018. # 2018 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1 Citation: Tellus A: 2018, 70, 1445379, http://dx.doi.org/10.1080/16000870.2018.1111111 2 D. A. SCHECTER

Additional insights have been gained from the study of a realistic instability analysis and remains an open issue. three-dimensional (3D) stratified vortices whose basic A separate challenge pertinent to analysing instabilities is states do not possess secondary circulations. The domin- to move beyond the conventional but questionable simpli- ant modes of instability often resemble their 2D counter- fication of using constant eddy diffusivities. parts but differ in quantitative details [Nolan and Section 2 of this paper presents a somewhat distinct Montgomery, 2002 (NM02)]. The qualitative similarities linear model of the perturbation dynamics that includes can extend beyond wave growth to nonlinear mesovortex tuneable formulas for diabatic forcing and subgrid turbu- formation and potential vorticity mixing (Hendricks and lent transport with inhomogeneous eddy diffusivities. The Schubert, 2010). On the other hand, adding vertical struc- parameterisation of diabatic forcing does not provide a ture to the vortex introduces the possibility of baroclinic definitive closure of the thermodynamic equation, but instability (Kwon and Frank, 2005). Moreover, instability facilitates assessment of how an instability mode might mechanisms involving the interactions of vortex Rossby change with plausible variation in the treatment of cloud waves and inertia-gravity waves become potentially rele- processes. Section 3 outlines a numerical method for find- vant in the parameter regime of a major hurricane ing the main instability modes of a tropical cyclone and (Schecter and Montgomery, 2003, 2004; Hodyss and the second-order response of symmetric fields to the Nolan, 2008; Menelaou et al., 2016; Schecter and growth of an asymmetric mode. Section 4 describes the Menelaou, 2017). basic state of a mature tropical cyclone generated by an The final step toward a realistic perturbation theory is axisymmetric model with explicit cloud microphysics. to generalise a 3D model to incorporate moisture and Section 5 analyzes the 3D instability of the aforemen- secondary circulation into the basic state of the vortex. tioned system and illustrates sensitivities to the represen- The inclusion of cloud coverage alone has the effect of tations of diabatic forcing and subgrid turbulence in the substantially reducing static stability (Durran and perturbation equations. Results of the analysis are com- Klemp, 1982). In principle, such reduction can alter the pared to those of an ostensibly analogous 2D (baro- structure and growth rate of the linear eigenmode asso- tropic) model. Section 6 presents an additional instability ciated with an instability (Schecter and Montgomery, analysis for one of the tropical cyclones examined in 2007 (SM07); Menelaou et al., 2016). The importance of NS14. The adequacy of the analysis is evaluated by direct the secondary circulation to the prevailing mechanism comparison to the initial stage of perturbation growth of perturbation growth is presently unclear. Although simulated (in NS14) with a three-dimensional cloud secondary circulations are known to significantly resolving model. Section 7 summarises our main findings. influence the inner core instabilities of tornado-like The appendices provide some technical details excluded vortices (Rotunno, 1978;Gall,1983;Walkoand from the main text. Gall, 1984;Nolan,2013), tropical cyclones are distinct atmospheric systems. 2. The perturbation equations Needless to say, cloud coverage in a mature tropical cyclone is largely linked to the secondary circulation. The present study is based on a compressible nonhydro- Therefore, including one without the other in a model static model of a tropical cyclone. The equations of could yield misleading results. Naylor and Schecter (2014) motion are expressed in a cylindrical coordinate system (NS14) recently examined the consequences of having whose central axis corresponds to that of the vortex. both. They found only subtle differences between perturb- The radial, azimuthal and vertical coordinates are ation growth in realistically simulated (moist convective) respectively denoted by r, u and z.Asusual,timeis tropical cyclones and the instabilities of analogous dry denoted by t. The prognostic fluid variables are the (nonconvective) vortices. However, there is no firm rea- radial velocity u, the azimuthal velocity v, the vertical son to believe that the results of NS14 are general. A velocity w, the density potential temperature hq and the more extensive investigation is necessary. total density q. Tendency equations for the mixing ratios NM02 contains the underpinnings of an appropriate of water vapour and hydrometeors are not explicitly linear model for investigating perturbation dynamics in a considered. The influence of cloud processes on the moist convective tropical cyclone. The NM02 model perturbation dynamics is parameterized as explained in accommodates the incorporation of an adequately Section 2.2. resolved boundary layer and the complete overturning secondary circulation of the basic state, but does not 2.1. Basic formulation of the model close the book on the thermodynamics. Proper parameterisation of the perturbation to diabatic forcing as a The nonlinear equations of motion governing the tropical function of the prognostic fluid variables is necessary for cyclone are given by INSTABILITIES OF TROPICAL CYCLONES 3

2 v oun oubun oun oub o u v u fv c h o P D (1a) inXbun wb n vn wn t pd q r u ot ¼À or À À oz þ b À oz ¼À Ár þ r þ À þ o o (3a) uv Pb Pb Pb ~ o o cpd hqn lhqb hqn q Dun tv v v fu cpd hq uP=r Dv (1b) À or À or h þ q n þ ¼À Ár À r À À þ qb b otw v w g cpd hqozP Dw (1c) ovn ovn ubvn ovn ovb ¼À Ár À À þ o gbun ub o inXbvn wb o o wn othq v hq Sh Dh (1d) t ¼À À r À r À À z À z ¼À Ár þ þ inl Pbhqb ~ o q div vq S ; (1e) Pbhqn q Dvn t q À r þ q n þ ¼À ðÞþ b in which v is the three-dimensional velocity vector, f is (3b) the (constant) Coriolis parameter, g is the gravitational own owb own owbwn g~ un ub inXbwn hqn acceleration, and cpd is the specific heat of dry air at con- ot ¼À or À or À À oz þ hqba o (3c) stant pressure. The Exner function satisfies the relation Pb Pb ~ lhqb hqn qn Dwn R À oz h þ q þ Rd d qb b c p cpd Rdhqq vd oh oh oh oh P ; (1f) qn qb u qb w u qn pa ¼ pa o o n o n b o t ¼À r À z À r (3d) ohqn 5 inX h w S D~ in which p is total pressure, pa 10 hPa, Rd is the gas b qn b o hn hn À À z þ þ constant of dry air, and c is the specific heat of dry air o o o o vd qn 1 rqbun inqbvn qbwn 1 rubqn at constant volume. Each Da represents a tendency (of ot ¼Àr or À r À oz À r or o (3e) field-a) induced by surface fluxes and unresolved turbu- wbqn ~ inXbqn o Sqn; lence within the vortex. Sh represents the tendency of hq À À z þ induced by cloud processes, radiative transfer and dissipa- in which Xb vb=r; n 2Xb f , g or rvb =r f , b þ b ð Þ þ tive heating. Sq is the density tendency attributable to g~ r; z cpdhqbaozPb and l cpdRd=cvd . To stay within ð ÞÀ mass changes of water content. Standard notations have the realm of standard practice, the arbitrary function hqba o 1o been used for the gradient operator ^r r u^ rÀ u is equated to hqb rB; z , in which rB is the outer boundary 1r þ 1 þ ð Þ ^zoz and the divergence div h rÀ or rhr rÀ ouhu radius of the model. To prevent artificial trapping of ð Þ ð Þþ þ ~ ozhz of the vector field h hr; hu; hz . The symbol ox is radiated waves in a finite domain, we have let Dun ðo Þ ~ used interchangeably with o in this paper to denote a Dun cun and likewise for all other D-functions. Similarly, x À ~ partial derivative with respect to any variable x. we have let Sqn Sqn cqn. In the preceding definitions, À A generic field F may be written as follows: the terms proportional to c r; z represent sponge-damp- ð Þ F Fb r; z F r; u; z; t , in which the subscript b ing near rB and near the upper vertical boundary of the ð Þþ 0ð Þ denotes the component associated with a suitably defined model. By design, the positive function c is negligible basic state of the vortex. The preceding decomposition inside the tropical cyclone. To simplify matters, the may be applied to both the fluid variables u; v; w; parameterisations utilised for this study restrict Dan and f hq; q; P and the forcing functions Du; Dv; Dw; San (for all applicable a) to be linear functions of the g f Dh; Sh; Sq in the nonlinear model [Equations (1a)–(1f)]. wavenumber-n components of the prognostic fluid varia- g The result is a perturbation equation for each prognostic bles. It follows that Equations (3a)–(3e) constitute an fluid variable of the form autonomous linear system. Note that the reality condition

o F F F F n FnÃ eliminates the need to explicitly solve for the tF0 ; (2) À ¼ ¼L þNL þB negative-n Fourier components. Here and elsewhere, F in which consists of terms linear in F0 and all other the superscript ‘ ’ denotes the complex conjugate of the L Ã perturbation fields, F represents nonlinear terms of dressed variable. NL higher order in the perturbation amplitude, and F The feedback of an asymmetric linear perturbation on B accounts for residual terms involving only basic state var- the mean vortex is essentially a second-order ‘eddy for- iables along with –g in the vertical velocity equation. cing’ of the symmetric (n 0) fluid variables. The full ¼ Ideally, the basic state is chosen to be sufficiently close to nonlinear equation of motion for a symmetric perturb- F equilibrium that the magnitude of is no greater than ation field (F0) is schematically given by B second-order in the perturbation amplitude. Neglecting o F0 F F F F F F ; (4) the relatively small terms and reduces the ot ¼L0 þNLA þNLS þB F B NL dynamics to otF0 . F L in which 0 is the right-hand side of the linear equation The azimuthal symmetry of the basic state facilitates L for F0 [see Equations (3a)–(3e)], and the leading order an azimuthal Fourier decomposition of the reduced contribution to F ( F ) is quadratic in the asym- inu NLA NLS system. Letting F0 n1 Fn r; z; t e for all F yields ¼ ¼À1 ð Þ metric (symmetric) component of the perturbation. The P 4 D. A. SCHECTER

primary quadratic part of the asymmetry term is conveni- facilitate discussion hereafter, Sh will be referred to as ently written as follows: diabatic forcing. Equation (6) indicates that Sh involves more than a term proportional to the dry-air heating F 1 F 2 m: (5a) rate. Nevertheless, in cloudy regions of a tropical cyc- NLA À m 1 NLA X¼ lone, the reasonable assumption that s_ d is of order The summands are given specifically by Lv sq_ =T suggests that s_ d will largely control the sign j = v j o o o of the sum in parentheses. Here, the symbol Lv=s has u umÃ umÃ vmvmÃ PmÃ R um wm cpd hqm been used to denote the latent heat of vapourisation/ NLAm ¼ or þ oz À r þ or o sublimation. ! vmumÃ cpd hqb mI ; (5b) To devise a parameterisation for S0 , one might first þ or þ r h consider an idealised cloudy vortex governed by reversible ov ov u v c hq P Ã v R u mÃ w mÃ m mÃ mI pd m m ; moist-adiabatic thermodynamics with ice-only or liquid- m m or m oz r r NLA ¼ þ þ þ only condensate. The diabatic forcing in such a system (5c) satisfies an equation of the form Sh vp_ , in which p_ is ¼ ow ow oP o! the material derivative of pressure p [SM07]. The coeffi- w R u mÃ w mÃ c h mÃ c h m m o m o pd qm o pd qb o cient of proportionality is given by N LA ¼ r þ z þ z þ z o s o u hq hq vmwmÃ v Hqt qv Hqv qt ; (7) mI ; (5d) ¼ ðÞÀ Ã op ! þ ðÞÃÀ op ! þ r sm;qt sm;qt oh oh v h h R qÃm qÃm I m qÃm in which qv is the saturation vapour mixing ratio with m um o wm o m ; (5e) Ã NLA ¼ r þ z þ r respect to ice or liquid. The step function H(x) is formally 1 orq u oq w defined to equal unity (zero) when x is positive (negative). q R m mÃ m mÃ ; (5f) m r or oz The subscripts on the partial derivatives with respect to NLA ¼ þ pressure indicate that the specific moist-entropy (sm) and in which Pm RdPb hqm=hqb qm=qb =cvd is the linear ½ð Þþð Þ total water mixing ratio (qt) are held constant. The sym- approximation of Pm in terms of prognostic fluid variables, s u bol hq (hq) represents the functional form of hq in terms 1 o2P 2 1 o2P 2 o2P and ! 2 hqm 2 q hqmqÃ . The 2 oh b 2 oq b m ohqoq b m of p, s and q under the assumption that the air is satu- ð q Þ j j þ ð Þ j j þð Þ m t operators R and I in Equations (5b)–(5f) respectively yield rated (unsaturated) and qv equals qv (qt). Appendix A Ã the real and imaginary parts of their operands. If every F0 is provides practical formulas for both partial derivatives initially subdominant to the asymmetric perturbation, that appear in Equation (7). F will be negligible for an extended period of time. In the preceding reversible moist-adiabatic vortex NLS Forthcoming analysis of wave-mean flow interaction will model, the perturbation to diabatic forcing can be written as follows: set both F and F to zero in Equation (4). The latter NLS B approximation goes beyond that made in the reduced linear S0 v p_ 0 v0p_ v0p_ 0: (8) h ¼ b þ b þ model for symmetric perturbations [(3a)–(3e) with n 0] by ¼ The rightmost term in Equation (8) involving the product of assuming that F is much smaller than a second-order cor- B two perturbation fields presumably has minimal effect on rection to the dynamics. the weak disturbances of interest (see SM07 for caveats). Furthermore, the middle term would be negligible in a 2.2. Parameterisation of the influence of moisture cloudy vortex whose basic state had no secondary circulation. Keeping only the first term in Equation (8),assuming The definition of our chosen thermodynamic variable o p_ 0 qbgw0, and letting v~b qbgvb= zhqb would yield [h p qR P ] implies that À À q = d o ð Þ Sh0 v~b zhqbw0 _ _ ¼ o (9a) cvdqv cpd qt hq Shn v~b zhqbwn: Sh s_ d ; (6) ! ¼ ¼ þ e qv À 1 qt cpd þ þ There is no firm reason to believe that a parameterisation in which e Rd =Rv, Rv is the gas constant of water of the diabatic forcing anomaly based on Equation (9a) vapour, sd cpd ln T Rd ln pd is the specific entropy of would be quantitatively accurate for realistic tropical cyclo- ¼ À dry air, T is absolute temperature, pd p= 1 qv=e is nes that have pronounced secondary circulations with pre- ¼ ½ þ the partial pressure of dry air, qv (qt) is the mixing cipitating clouds of both liquid and solid hydrometeors. On ratio of water vapour (total water content), and the the other hand, for the class of parameterisations propor- overdot represents a material derivative minus any ten- tional to w0, Equation (9a) provides a reasonable starting dency directly connected to small-scale turbulence. To point for a process of systematic adjustment toward a INSTABILITIES OF TROPICAL CYCLONES 5

decent fit with experimental data. Sensitivity of results will exemplified by ordinary acoustic oscillations. It so hap- be assessed by using the more flexible formula pens that such rapidly oscillating modes have either sub- e o dominant or negative growth rates in our applications of Shn vv~b zhqbwn (9b) ¼ the linear model. Purists might reasonably argue that e and letting v vary between 0 and 1.1. For the majority the fast modes should be filtered out of the dynamical of calculations in this paper, vb will be evaluated assum- system for consistency. However, filtering out the acous- ing ice/liquid condensate above/below the freezing level in tic modes alone is somewhat complicated and seems to the troposphere. The reader is referred to Appendix A have negligible effect on the main tropical cyclone for details on how vb is extracted from a numerically instabilities that are investigated in this paper simulated tropical cyclone, and for further commentary (Appendix B). on the relation S0 w0. h / A more general linearised parameterisation of the per- 2.3. Parameterisation of small-scale turbulence turbation to Sh may have the form The influence of small-scale turbulence on the velocity F F S0 r; u; z; t d~rdu~ d~z~rG ~r; u~ ; ~z; r; u; z L^ F0 ~r; u~ ; ~z; t ; hðÞ¼ j ðÞj ðÞperturbation is parameterized with a linear eddy viscosity F;j ððð X ÂÃscheme that incorporates a modification of the oceanic (10) surface drag. The velocity tendencies associated with tur- ^ F in which F denotes a prognostic fluid variable, Lj is the bulence can be expressed as follows: jth member of a generic set of linear operators (including o o m o2 m F 2 m u0 Kh u0 2Kh u0 differential operators) acting on F0; G is an integration D rK j u0 r or h or r2 ou2 r2 kernel paired with that operator, and the volume integral ¼ þ À (12a) 3Km ov Km o2v os is taken over the entire domain of the system. Equation h 0 h 0 rz ; 2 o o o o ^ w À r u þ r r u þ z (9a) can be obtained from (10) by letting L w0 w0, 1 ½ ¼ o o m o2 o o w o F 1 m v0 2Kh v0 1 m u0 G1 v~b ~r; ~z zhqb ~r; ~z d r ~r d u u~ d z ~z =~r, and Gj D rK rK ¼ ð Þ ð Þ ð À Þ ð À Þ ð À Þ ¼ v0 ¼ r or h or þ r2 ou2 þ r2 or h ou 0 for F w or j 1. As usual, the symbol d has been 6¼ 6¼ 2Km ou o v 1 o os used to represent the Dirac distribution. Note that h 0 Km 0 Kmv uz ; r2 ou h or r r or h 0 oz Equation (10) is somewhat restrictive; neither the inte- þ þ À þ ÀÁ grals nor kernels involve time. On the other hand, (12b) m 2 Equation (10) includes parameterisations that relate the 1 o ow0 K o w0 1 o ou0 D rKm v rKm w0 r or v or r2 ou2 r or v oz perturbation of diabatic forcing at a point r; u; z in the ¼ þ þ (12c) ð Þ Km o2v o ow free troposphere to the perturbation of vertical velocity at v 0 2 Km 0 ; r ozou oz h oz a point rc r; u; z ; u r; u; z ; zc at the top of the bound- þ þ ½ ð Þ cð Þ ary layer (z zc). A simple example that maintains the in which the momentum eddy diffusivities (Km and Km) ¼ h v dynamical independence of the azimuthal Fourier trans- are assumed to be functions of only r and z. Azimuthal forms of the perturbation fields (in linear theory) might and temporal dependencies of the diffusivities are have an integration kernel of the form neglected for simplicity. The rz and uz components of the w stress tensor appearing in Equations (12a) and (12b) are G1 Cr; z d rc r; z ~r d u c r; z u u~ d zc ~z =~r ¼ ðÞ ðÞÀ ðÞþ À ðÞÀ given by ÂÃÂÃ(11) w ^ F ou0 ow0 paired with the operator L1 w0 w0, while Gj 0 for all Km z > 0 ½ ¼ ¼ v oz þ or other combinations of F and j. Note that we have let s 8 rz > o 2 uc u c u. Exploration of the preceding type of param- ¼ > Cd u ubu0 ubvbv0 ¼ þ <> Cd u u0 j j þ z 0; eterisation will be deferred to future study. ½j j b þ o u u ¼ b b > j j j j One potential deficiency of the foregoing parameter- > (13a) :> isations [Equation (9b); Equation (11)] is their neglect of o o m v0 1 w0 any direct response of moist convection to small Kv z > 0 oz þ r ou s 8 enhancements or reductions of surface enthalpy fluxes uz o 2 ¼ > Cd u ubvbu0 vbv0 coinciding with surface wind speed perturbations. Such <> Cd u v0 j j þ z 0; ½j j b þ o u u ¼ a response could be incorporated into Equation (10), j j b j jb > (13b) but the importance of such inclusion to mature tropical :> cyclone instabilities is presently unclear. Note also that in which u pu2 v2. Unless stated otherwise, the drag the parameterisation used for this study [Equation (9b)] j j þ coefficient is given by is not designed for high frequency perturbations ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 6 D. A. SCHECTER

Cd0 u < U0 in which ‘max’ returns the greater of its two arguments at j j Cd1 Cd0 each point in the r-z plane. The variables Km r; z and Cd 8 Cd0 À u U0 U0 u < U1 (13c) h;smð Þ ¼ þ U1 U0 ðÞj jÀ j j Km r z in Equation (16) are obtained directly from the > À v;sm ; < Cd1 u U1; ð Þ j j simulation (sm) that generates the tropical cyclone under > 1 1 in which:> U0 5msÀ , U1 25 m sÀ and Cd1 Cd0. consideration. In particular, they correspond to the hori- ¼ ¼ Note that the velocity fields in all formulas pertaining to zontal and vertical momentum eddy diffusivities averaged the surface stress [Equations (13a)–(13b) at z 0; over u (if the simulation is 3D) and over the time period ¼ e Equation (13c)] are evaluated at the lowest active grid that is used to define the basic state. The multiplier k is level above the ocean in our numerical version of the lin- allowed to deviate from unity for sensitivity tests. The m m constants Km and Km are lower limits of the diffu- ear model. Specifications of Kh ; Kv , Cd0 and Cd1 are h;min v;min forthcoming. sivities to be specified in due course. The previously Several remarks are in order regarding the preceding unspecified parameters associated with the drag coeffi- e e representation of turbulent transport in the velocity equa- cient are given by Cd0 0:001 k and Cd1 0:0024 k for ¼ ¼ tions. To begin with, Equations (12a)–(12c) above the the primary instability analysis in Section 5 of this paper. surface are equivalent to a parameterisation of the form The preceding formulas permit consistency with the simulation that generates the basic state when ek 1 (see 3 o ov o ¼ m j0 vi0 Section 4). For further consistency, the thermal eddy dif- Di0 Kij ; (14) ox ox ox h m ¼ j 1 j "# i þ j! fusivities are given by K K , so that the Prandtl X¼ h=v ¼ h=v number is unity throughout the domain of the lin- in which D is the tendency associated with turbulence in i0 ear model. the prognostic equation for the ith component of the velocity perturbation (vi0) in a Cartesian coordinate system x1; x2; x3 in which x3 z. Specifically, it is assumed 2.4. Additional simplifications ð Þ that Km Km for i; j 1; 2 ; Km Km, and Km Km ij ¼ h 2f g 33 ¼ h 3j ¼ j3 ¼ Perturbations to radiative transfer and dissipative heating Km for j 3. Bear in mind that such a parameterisation v 6¼ are neglected in forthcoming sections of this paper. The does not follow from direct linearisation of a typical non- potential impact of radiation on the development of linear model. Direct linearisation would produce add- instabilities has been examined to some extent by adding itional terms accounting for perturbative variations of the Newtonian relaxation of the form eddy diffusivities. Note also that the usual density factors r S hqn=sr (17) have been neglected. Despite such imperfections, hn ¼À Equations (12a)–(12c) are believed to provide a reason- to the perturbation of diabatic forcing in several sensi- able framework for estimating how inhomogeneous tivity tests. The dominant instabilities considered anisotropic turbulent viscosity should influence the per- herein normally have shorter time scales than a typical turbation dynamics. 12-h value of the radiative adjustment time sr. Moving on to the thermodynamic equation, the effect r Accordingly, Shn is normally found to have negligible of small-scale turbulence on hq0 is parameterized by consequence. 2 1 o oh0 Kh o h0 o oh0 The perturbation to Sq in the mass continuity equation D rKh q h q Kh q ; (15) h0 o h o 2 o 2 o v o is difficult to properly model without explicit moisture ¼ r r r þ r u þ z z equations. The present study simply lets h in which Kh=v depends only on r and z. For simplicity, Sqn 0 (18) the perturbation to the surface flux of hq is set to zero ! (see Section 2.5). The loose application of a simple diffu- under the provisional assumption that it is of minor con- sion scheme to the density potential temperature perturb- sequence to the main instabilities of a tropical cyclone. ~ ation is deemed adequate for the present study. It is Equation (18) reduces Sqn to the artificial damping term provisionally assumed that any subtle imprecision in for- activated near the upper and outer boundaries of the mally representing Dh0 by Equation (15) does not affect domain of the dynamical system. an instability analysis more than moderate variation of the e parameter defined below. k 2.5. Boundary conditions Several remaining formulas are required to complete the turbulence parameterisation in the linear system. To The linear model employs a standard set of boundary begin with, the momentum eddy diffusivities are given by conditions for a fluid in a rigid cylindrical enclosure. At m m m r 0, K max ekK ; K ; (16) h=v ¼ h=v;sm h=v;min ¼ ÀÁ INSTABILITIES OF TROPICAL CYCLONES 7

kt un ivndn1; x a X e ; (23a) ¼À n k k vn 1 dn1 orvndn1 0; ¼ k ðÞÀ þ ¼ (19) X wn 1 dn0 orwndn0 0; ðÞÀ þ ¼ in which hqn 1 dn0 orhqndn0 0; ðÞÀ þ ¼ L Xk ; xn t 0 in which dnm equals 1 for n m and is otherwise 0. At the a h ðÞ¼ i ; (23b) ¼ k L o o Xk ; Xk outer boundary radius rB, un 0, r vn=r 0, and rFn h i ¼ ð Þ¼ ¼ Nt 0 for Fn wn; hqn . At the surface and upper boundary i i 2f g y; x yÃ x ; and (23c) (z 0 and z ), w 0 and o F 0 for F u ; v ; h . B n z n n n n qn h i i 1 ¼ ¼ ¼ 2f g ¼ Consistent boundary conditions for qn are implicit in the † LX L M X kÃX : (23d) linear model. Note that all constraints imposed on the n k k i i th perturbation fields at r rB and z zB are incidental The symbol x (yÃ )inEquation (23c) denotes the i ¼ ¼ † when sponge damping is activated. Note further that the element of x (yÃ). The symbol Mn in Equation (23d) rep- free-slip conditions ozun ozvn 0 at z 0 are replaced resents the conjugate transpose of the matrix Mn.The ð ¼ ¼ Þ ¼ with the surface drag parameterisation when Cd > 0. As a eigenmode associated with the greatest positive value of final remark, the velocity fields of the basic state are kR will eventually dominate the right-hand side of assumed to satisfy ub vb 0atr 0, ub 0atr rB, Equation (23a). Should there exist no eigenmodes with ¼ ¼ ¼ ¼ ¼ and w 0atz 0 and z . positive kR, transient or sustained algebraic growth of b ¼ ¼ B the perturbation may still occur (Smith and Rosenbluth, 1990; Nolan and Farrell, 1999; Antkowiak and 3. Instability modes Brancher, 2004). Examination of such nonexponential 3.1. General theory growth in the linear model at hand is deferred to future study. Let x denote the state vector of the linearised system, n So as not to be lost in abstraction, it is worth remark- with each element representing the value of one of the ing that the physical perturbation corresponding to a prognostic perturbation fields (un, vn, wn, hqn,orqn)ata inu kt complex eigenmode is usually given by 2R akXke þ . specific point in the r-z plane. In practice, each field Fn is ½ In other words, if akXk unk; vnk; wnk; hnk; qnk , the phys- represented on a grid with NrF points in r and NzF points ð Þ ical perturbation has the form u0 2 unk r; z cos nu in z. It follows that xn has a total of Nt F NrF NzF ele- kRt ¼ j ð Þj f þ kI t arg unk r; z e and likewise for all other fields. ments, in which the sum is over all 5 prognostic variables. þ ½ ð Þg P The coefficient 2 is replaced by 1 if n 0 and both k and The preceding discretization transforms the continuous ¼ akXk are real. linear model [Equations (3a)–(3e)] into a system of the Suppose that the system is initially perturbed with a form single asymmetric (n 0) eigenmode. Consideration of 6¼ dxn Equation (4) suggests that the discretized symmetric com- Mnxn; (20) dt ¼ ponent of the disturbance will be governed by in which Mn is an Nt Nt non-Hermitian matrix of com- dx 0 2kRt Â M0x0 b e ; (24a) plex coefficients. dt À ¼ k The eigenmodes of the discretized linear system are sol- 2 in which b ak is the time-independent part of a forc- utions to Equation (20) of the form k /j j ing vector obtained by evaluating the right-hand side of kt kt xn akXke ; (21) Equation (5a) with the eigenmode solution xm akXke ¼ ¼ for m n and xm 0 otherwise. In addition to neglecting in which Xk is the time-independent complex eigenvector ¼ ¼ LSF and F , the foregoing simplification of Equation associated with the complex eigenfrequency k kR ikI , N B þ (4) assumes that all asymmetric modes initialised to zero and ak is an arbitrary complex amplitude. Substituting remain subdominant over the time period of interest. Equation (21) into Equation (20) and switching the left Equation (24a) is readily solved by the method of and right sides yields Laplace transforms and the calculus of residues after M X kX : (22) n k k expanding both sides in the eigenvectors Xm of M0. The ¼ f g result for x0 0att 0 is given below: Under ordinary circumstances, there are Nt independent ¼ ¼ a a solutions to Equation (22) composing a complete eigen- m mt m 2kRt x0 Xme Xme ; (24b) basis of the wavenumber-n linear system. It follows that ¼ m 2kR þ 2kR m m À m À the solution to a generic initial value problem can be X X in which written 8 D. A. SCHECTER

L Xm ; bk tropical cyclone simulations. Furthermore, the time am h L i : (24c) X ; Xm required to compute a complete eigenbasis on a modern h m i simulation grid is excessive. Grids of lower resolution The second term on the right-hand side of Equation should be avoided, because they are prone to introduce (24b) will eventually dominate if m < 2k for all eigenfre- R R spurious eigenmodes with dominant growth rates. quencies m of the linear symmetric system. The second f g Moreover, grids of higher resolution are desirable to check term is merely the particular solution to Equation (24a) for convergence of the numerics. 2kRt given by x0 Xpe , in which ¼ The present study employs a less ambitious approach 2kR M0 Xp b : (24d) that begins by extracting the dominant eigenmode from a ðÞÀ ¼ k The particular solution is considered herein to be the solution of the initial value problem. The discretized lin- intrinsic response of the mean vortex to an asymmetric ear model [Equation (20)] is set up on a dense mesh [see instability mode. It is reasonable to consider the intrinsic Section 5.1] and integrated forward in time with a 4th- order Runge-Kutta algorithm. The initialisation involves response to be an essential part of the mode itself. assigning small random values to the real and imaginary

parts of hqn at each grid point; all other fields contained

3.2. Computation of the main instability modes in xn are initialised to zero. It is provisionally assumed Each fluid variable in the linearised model is discretized that the preceding disturbance excites the main instability on a rectangular grid in the r-z plane with nonuniform modes of a tropical cyclone and eventually evolves into a spacing in both coordinates, as in earlier studies such as state dominated by the most unstable member of the NM02. Finer resolution generally exists near the surface group. The real and imaginary parts of the eigenfre- and within the core of the tropical cyclone. The discre- quency k of the most unstable eigenmode are readily obtained from the late time series of a selected element of tized representations of vn, hqn and qn share the same xn. The right-hand eigenvector Xk is very well approxi- grid. The representation of un (wn) is radially (vertically) mated by the late spatial structure of xn. The validity of staggered with respect to vn. The basic state variables and eddy diffusivities are defined on all of the staggered grids. the mode is generally cross-checked against the output of Boundary values are not explicitly stored but are incorpo- a standard sparse-matrix eigensolver (eigs) packed into rated into computations where necessary. Scientific Python (SciPy). Validation is efficiently com- The following simple formulas are normally used for pleted by searching exclusively for the eigenmode of Mn finite differencing and linear interpolation of a generic with k closest to that obtained from the initial value field F : problem. The SciPy eigensolver is also used to find the n L corresponding left-hand eigenvector Xk . Because the oFn F F nþÀ nÀ ; (25a) restricted searches are fast, they are usually repeated on a o‘ 0 ¼ d‘À d‘þ ‘ þ grid with twice the original resolution (in both r and z)to d‘ F d‘ F F 0 À nþ þ nÀ (25b) slightly improve the accuracy of presented results. n ‘ þ ; a ðÞ¼ d‘À d‘þ Suppose that the eigenfrequencies kð Þ are ordered þ 1 2 3 f g 0 in which ‘ represents either r or z, ‘ denotes ‘ at the evalu- such that kRð Þ > kRð Þ > kRð Þ ... In principle, if all eigenmo- 6 ation point, d‘ is the distance from ‘0 to the nearest stag- des with a < b are known, a minor variant of the fore- gered grid point in the positive ( ) or negative ( ) going procedure can be repeated to obtain eigenmode b. 6 þ 6 06 6 À The variant involves filtering out all eigenmodes with direction, and Fn is the value of Fn at ‘ ‘ d‘ . For 0 6 a < b from the initial condition of the state vector that is example, if ‘ represents r (z) and ‘ is on the v-grid, then Fn 6 and ‘ are on the u-grid (w-grid). Formulas for second- integrated forward in time; that is, letting order derivatives and bilinear interpolations are generally L X a ; Y kðÞ x t 0 Y h i X a ; (26) obtained through repeated applications of Equations (25a) n L kðÞ ðÞ¼ ¼ À X a ; X a a < b k kðÞ and (25b). Implementation of more accurate discretization X h ðÞ i techniques will be explored at a future time. in which Y is an arbitrary vector. One may reasonably

The computation of the complete eigenbasis of a finely- assume that the time asymptotic solution of xn will be structured tropical cyclone is usually too expensive to dominated by the eigenmode labelled b. All eigenmodes achieve with confidence of correct results. Although Mn is of interest can thus be found iteratively. Note that the sparse and has a storage requirement proportional to Nt, unfiltered initialisation vector Y need not be random after the eigenbasis Xk has a storage requirement proportional the first iteration; the approach taken here is to let Y 2 f g to Nt . The consequent demand on memory becomes diffi- equal the end-state of xn from the preceding time integra- cult to handle for grids comparable to those used in modern tion used to find the eigenmode labelled b 1. À INSTABILITIES OF TROPICAL CYCLONES 9

Fig. 1. Selected fields associated with the basic state. (a) The azimuthal velocity vb (colour), density potential temperature hqb (solid black contours; K) and Exner function Pb (solid white contours). (b) A measure of gradient imbalance [Dgb defined by Eq. (27)]. (c) The potential vorticity distribution. (d) The relative vertical vorticity distribution. (e) The radial vorticity (contours) and azimuthal vorticity (colour) distributions. The solid yellow and dashed white curves in (b)–(e) correspond to the principal AM isoline, which is commonly shown for spatial reference in the contour plots of this paper. Note that vbm is located where the radius of the isoline is minimised.

4. The basic state of a mature tropical cyclone lengths are given by CM1-formulas tailored for tropical The primary basic state considered herein corresponds to cyclones in an axisymmetric framework or on grids that a mature tropical cyclone simulated with Cloud Model 1 are deemed insufficiently dense for a standard large-eddy- (CM1-r19.4) in an energy-conserving axisymmetric simulation scheme. The horizontal mixing length mode of operation [Bryan and Fritsch, 2002; Bryan and increases from 100 m to 1 km as the underlying surface Rotunno, 2009 (BR09)]. The model is configured with a pressure decreases from 1015 to 900 hPa. The vertical variant of the two-moment Morrison microphysics par- mixing length increases asymptotically to 100 m with ameterisation (Morrison et al., 2005, 2009), having grau- increasing z. The resulting eddy diffusivities will be dis- pel as the large icy-hydrometeor category and a constant cussed in due course. Heating associated with frictional cloud-droplet concentration of 100 cm–3. Radiative trans- dissipation is activated. Surface fluxes are parameterized fer is not explicitly calculated, but potential temperature with bulk-aerodynamic formulas. The drag coefficient (h) is relaxed toward its ambient value on a 12-h time conforms to Equation (13c) with Cd0 0:001 and –1 ¼ scale with a rate not to exceed 2 K d in magnitude. The Cd1 0:0024, based roughly on the findings of Fairall ¼ influence of subgrid-turbulence above the surface is repre- et al. (2003) and Donelan et al. (2004). The enthalpy sented by an anisotropic Smagorinsky-type scheme resem- exchange coefficient is given by Ce 0:0012 based on ¼ bling that described in BR09. The nominal mixing Drennan et al. (2007). 10 D. A. SCHECTER

(a) Dunion (2011) moist tropical sounding. The sea-surface 30 temperature Ts is 27 C, and the Coriolis parameter f is 5 1 5 10À sÀ . After approximately 7 days of intensifica- 25 Â tion, the maximum azimuthal velocity of the tropical cyc-

20 lone remains steady over an extended period of time. The basic state variables ub; vb; wb; hqb; q ; Pb; v and princi- ð b bÞ 15 pal eddy diffusivities Km ; Km appearing in the linear ð h;sm v;smÞ model are obtained by averaging 25 consecutive hourly 10 snapshots starting 2 days into the aforementioned period

5 of steady intensity. Figure 1a shows that the basic state of the tropical cyclone exhibits typical warm-core baroclinic structure throughout most (but not all) of the troposphere. The

absolute maximum azimuthal wind speed (vbm) of 84.9 m (b) 30 1 sÀ is located 36 km from the centre of the vortex and 25 nearly 1 km above the surface. The maximum azimuthal 1 wind speed of 61.2 m sÀ at the lowest grid level 20 (z 25 m) is indicative of a category-4 hurricane. It is ¼ 15 worth remarking that the primary circulation does not 10 robustly satisfy gradient balance (Fig. 1b). The fractional error defined by 5 2 o vb=r fvb cpd hqb rPb Dgb þ À ; (27) cpdhqborPb

Fig. 2. The secondary circulation of the basic state. (a) is most pronounced (66%) in the vicinity of vbm. Magnitude (colour) and streamlines of the velocity field (ub, wb) Figure 1c shows that throughout the lower and middle in the r-z plane. The streamlines are shaded white or black if the troposphere, the potential vorticity distribution is gener- magnitude of the local velocity vector is respectively less than or 1 ally peaked off centre within the area bounded by the greater than 5 m sÀ . The dashed black curve is the principal AM principal angular momentum (AM rv fr2=2) isoline. isoline. (b) Magnified view of the secondary circulation in (a) in þ the lower troposphere. Here, the potential vorticity is defined by PV f h=q, a Ár in which f f^z v is the absolute vorticity vector. a þrÂ The principal AM isoline is defined so as to pass through The computational domain extends radially to rB ¼ vbm. By analogy to the behaviour of dry vortices in gradi- 1061:25 km and vertically to zB 29:5 km. Free-slip ¼ ent and hydrostatic balance, the radially nonmonotonic boundary conditions are imposed at the upper and outer variation of PV suggests that the tropical cyclone is sus- boundaries, but they are largely incidental. Rayleigh ceptible to vortex Rossby wave instability mechanisms damping is activated above z 25 km and within 100 km ¼ (see Montgomery and Shapiro, 1995). The pocket of nega- of rB. Such damping not only minimises the reflection of upward and outward propagating waves, but also pre- tive PV extending up to 6 km above sea level slightly out- vents the circulation of the tropical cyclone from expand- ward of the principal AM isoline suggests that (neglecting ing to the outer wall. The spatial resolution is fairly viscous dissipation) the vortex may also be susceptible to typical for modern tropical cyclone simulations. The inertial instability mechanisms in the lower-to-middle radial grid spacing is 250 m for r 87:5 km, and grad- tropospheric region of its core (see Eliassen, 1951). ually stretches to 10 km as r approaches the boundary Figure 1d demonstrates that the distribution of relative vertical vorticity (f ) in the lower and middle tropo- radius rB. The vertical grid spacing increases from 50 to zb 500 m between the sea-surface and z 5.5 km, whereupon sphere basically resembles that of PV. The most notable ¼ it remains uniform up to zB. deviation is seen where the PV distribution is thermally The mature tropical cyclone is generated with a stand- enhanced at the top of the boundary layer in the eye of ard spinup procedure on the oceanic f-plane. The model the storm. Note that the primary circulation also pos- is initialised with a surface-concentrated mesoscale cyc- sesses appreciable radial vorticity associated with its verti- lone in gradient-wind and hydrostatic balance as in cal shear (Fig. 1e). Evidently, the radial vorticity achieves

NS14. The ambient atmosphere is initialised with the magnitudes greater than fzb near the surface. INSTABILITIES OF TROPICAL CYCLONES 11

or slightly supersaturated. The dashed black-and-white

100 contours correspond to sp for liquid-only condensate Ã (Bryan, 2008). It is seen that the angular momentum and

liquid-only sp contours passing through the location of 80 Ã vbm are congruent as they ascend along the eyewall up to the freezing level. At higher altitudes, the angular momen-

60 tum contour appears to stay closer to the dotted-blue sp Ã contour calculated under the assumption of ice-only condensate (Hakim, 2011). The preceding observations sug- 40 gest that the eyewall updraft region is in a state of approximate slantwise convective neutrality with respect <20 to appropriately defined pseudoadiabatic thermodynamics. Such a state of affairs is consistent with the classical

Fig. 3. The moist-thermodynamic structure of the basic state, steady state theory expounded by Emanuel (1986). illustrated primarily by the distributions of relative humidity The cloud structure of the basic state is important to

(shading) and sp for liquid-only condensate (dashed black-and- the linear model insofar as it determines the proportional- Ã 1 –1 white contours; J kgÀ K ). The dotted blue curve is a selected ity between Sh0 and w0 in the local parameterisation of contour of sp computed under the assumption of ice-only diabatic processes given by Equation (9b). When the Ã condensate. The red curve is the principal AM isoline. The cyan aforementioned parameterisation is activated, there are curve traces the freezing/melting level across the vortex. two terms proportional to wn on the right-hand side of Figure 2 illustrates the secondary circulation of the Equation (3d) that may be unified as follows: oh h 2 basic state. The maximum wind speed ( u2 w2 31 m qb w S qb N~ w ; (28a) b þ b ¼ oz n hn g n 1 À þ !À sÀ ) of the surface inflow is relativelyq strongffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi but not much greater than typical observations pertaining to in which major hurricanes (Zhang et al., 2011). The inflow inten- o ~ 2 e g hqb N 1 vv~b : (28b) sity is greatest slightly outward of the corner flow region, ðÞÀ hqb oz where the streamlines rapidly turn upward into the eye- 2 A typical value of e v~ between zero and one reduces N~ wall cloud. Note that the azimuthal vorticity associated v b and thereby diminishes the negative/positive Eulerian with the secondary circulation in the vicinity of the cor- change in h associated with a perturbative updraft/down- ner flow (Fig. 1e) is comparable in magnitude to the peak q draft. While not precisely the conventionally defined vertical and radial vorticities associated with the primary 2 moist static stability, N~ has a similar significance. Figure circulation. Note also that the streamline associated with ~ 2 the deep updraft and outflow passing through the loca- 4 compares the distribution of N in the approximate dry limit (ev 0) to the moist variant with ev 1. It is seen tion of vbm is virtually congruent with the principal AM ¼ ¼ isoline. Such a condition is to be expected for a nearly that incorporating the cloud coverage of the basic state ~ 2 equilibrated axisymmetric vortex. As usual, the secondary reduces N up to an order of magnitude in the eyewall circulation in the eye is dominated by weak subsidence. updraft. Significant reduction is also found over much of The streamlines are somewhat less coherent at larger radii the depicted area within and underneath the upper out- ~ 2 between the surface inflow and upper tropospheric outflow of the tropical cyclone. By contrast, N exhibits flow. Concerns that such incoherence may indicate a sig- minimal change in the virtually cloud-free region of the nificant departure from equilibrium are alleviated by eye situated above the boundary layer. noting that the regional wind speeds are minute com- As explained earlier, the eddy diffusivities used by the pared to peak values. linear model are linked to those regulating the basic state. Figure 3 illustrates the moist-thermodynamic structure Figure 5 shows the eddy diffusivities that are defined by e e m of the basic state. Contours of saturated pseudoadiabatic Equation (16) with k 1. Choosing k 1 lets Kh=v m ¼ ¼ ¼ entropy (sp ) are shown superimposed on the relative Kh=v;sm throughout much of the inner core. The maximum Ã humidity distribution. Relative humidity is calculated values are somewhat large but have orders of magnitude with respect to liquid water if the absolute temperature consistent with those inferred from observations (Zhang satisfies T > T0 273:15 K, but is otherwise calculated and Montgomery, 2012; Rogers et al., 2013). To reduce the with respect to ice. It should come as no surprise that the potential for spurious or uninteresting small-scale instabil- eyewall and outflow regions are predominantly saturated ities where the eddy diffusivities in the CM1 simulation are 12 D. A. SCHECTER

=0 =1 (a) 16 (b)

14 5 5

12 2.0 4 2.0 4 10 3 3 8 2 2 6

4 1 1 2.0 2.0 2 0 0

3076128.4 3 0 76 12 8 .4 0 25 50 75 r(km)100 125 ~ 2 Fig. 4. Distributions of N for (a) ev 0 and (b) ev 1. The dashed white curves in (a) and (b) correspond to the principal ¼ ¼ AM isoline.

m 2 1 exceptionally small, the lower limits Kh;min 200 m sÀ rigorous compliance with the assumptions of our theoret- m 2 1 ¼ and K 20 m sÀ are imposed on the distributions. ical framework would not change forthcoming depictions v;min ¼ of the spatial structure of x0 or the dependent kinetic 5. Linear instability analysis of a mature energy perturbation dKE defined later. tropical cyclone Finally, although the physics parameterisations are varied, the domain size and peripheral sponge-layer of the linear The present section of this article examines the instability model used to find the instability modes do not change from of the tropical cyclone described in Section 4. The pri- one calculation to the next. As in the CM1 simulation used mary objective is to elucidate the dependence of the dom- to generate the basic state, the invariant domain of the linear inant instability mode on the parameterisation of the model extends radially to rB 1061:25 km and vertically ¼ perturbation of diabatic forcing. Sensitivity to the param- to zB 29:5 km. The sponge damping coefficient is given ¼ eterisation of small-scale turbulence is also addressed. by c 2 tanh r rc =drc tanh z zc =dzc =2sc,in ¼f þ ½ð À Þ þ ½ð À Þ g The analysis concludes with an assessment of the rele- which rc 961:25 km, drc zc 25 km, dzc 0:75 km ¼ ¼ ¼ ¼ vance of 2D instability theory. and sc 300 s. Further computational details are provided ¼ A few preliminary remarks are warranted. Henceforth, in due course. the meaning of F0 is subtly changed from the exact difference F Fb to the first-order perturbation of the generic À 5.1. Sensitivity to the parameterisation of field F obtained from the linear model [Equations diabatic forcing (3a)–(3e); Equation (20)]. The new meaning of F0 applies to both figures and text. Moreover, the amplitudes of dis- The dominant instability of the tropical cyclone under played instability modes are invariably chosen to render consideration is sensitive to the degree of diabatic forcing the maximum value of v0 (2 vn ) equal to one-tenth of v . allowed in the linear model. The sensitivity is illustrated j j bm The preceding convention amounts to letting below by adjusting ev in Equation (9b) for Shn while kRts ak 0:1vbm= 2Vke , in which Vk is the azimuthal vel- keeping turbulent transport consistently parameterized j j¼ j j ocity element of Xk with the greatest magnitude, and t is with ek 1. A value of the diabatic forcing parameter s ¼ the time of the snapshot. In some cases, the second-order (ev) in the neighbourhood of unity has some basic cred- change to the mean vortex (x0) that will have attended ibility (Section 2.2) but may not coincide with the best the creation of such a state from a weaker disturbance by representation of reality. A smaller value between 0 and 1 way of Equation (24a) is found to have winds moderately seems plausible if, say, the eyewall were to become non- stronger than v0 in certain areas of the flow. Such a result uniformly saturated around an azimuthal circuit. Values indicates that the arbitrarily chosen mode amplitude is of ev very close to 0 or appreciably greater than 1 seem slightly beyond the threshold for the quantitative accur- difficult to justify on physical grounds, but are of theoret- acy of Equation (24a). Choosing a smaller amplitude for ical interest. INSTABILITIES OF TROPICAL CYCLONES 13

solid contours: K (m2 s 1)

Fig. 5. Horizontal (colour) and vertical (solid contours) momentum eddy diffusivities in the middle-to-lower tropospheric core of the simulated tropical cyclone. The dashed curve is the Fig. 6. Complex eigenfrequency k of the n 2 MUM of the principal AM isoline. ¼ simulated tropical cyclone of Section 4 versus the diabatic forcing

parameter ev. The real (blue) and imaginary (red) parts of the The present method for computing the primary eigenfrequency are normalised to their respective values instability modes of the vortex follows the general pro- 5 1 3 1 (kR 7:89 10À sÀ and kI 1:30 10À sÀ ) obtained for Ã ¼ Â Ã ¼À Â cedure outlined in Section 3.2. The most unstable eigen- ev 1. The absence of a discernible instability precludes the e ¼ mode (MUM) for a given v and azimuthal wavenumber plotting of data for ev 0:9. Note that the positive ratio kI =kI ¼ Ã n is provisionally equated to that which dominates a per- represents a nondimensional magnitude of the oscillation turbation within 1 day of initialising the linear model frequency; the actual value of kI is negative. All results depicted here and in Figs. 7–12 are for systems in which turbulent [Equations (3a)–(3e)] with random noise in hq0 . The abso- transport is parameterized with e 1. lute MUM (AMUM) is defined to be that which pos- k ¼ sesses the largest growth rate for all n in the closed interval between 0 and 8. The time integration is con- substantially increases both kR and kI . One might rea- j j e ducted on a grid (set of staggered grids) with double the sonably speculate that high sensitivity to variation of v resolution of the CM1 simulation that generated the basic in the neighbourhood of unity is related to approximate state. The aforementioned grid is denoted G2 and holds slantwise convective neutrality with respect to pseudoadiabatic thermodynamics in the eyewall (Fig. 3). Nt 733; 184 values of the prognostic perturbation fields. ¼ All MUMs are confirmed to be solutions of the eigenpro- Figure 7 shows the basic inner-core structure of the n 2 MUM for values of the diabatic forcing parameter blem on a second grid (G4) with quadruple the resolution ¼ below (ev 0:5) and above (ev 1) the apparent stability of the CM1 grid (G1). All eigenfrequencies and eigen- ¼ ¼ functions shown in Section 5 of this paper are taken from point. The left column shows selected views of the asym- the G4 solutions. For those interested, Appendix C dis- metric velocity perturbation. The middle column illus- cusses convergence of numerical results with increas- trates the thermal structure of each mode in terms of hq0 ing resolution. and P0. The right column shows the distributions of dia- Extensive computations reveal that the AMUM corre- batic forcing. The velocity perturbations of the two sponds to n 2 for ev 0; 0:5; 1 . Despite its common modes are qualitatively similar near the surface but ¼ 2f g dominance, the n 2 MUM varies considerably with the clearly differ aloft. Whereas the pressure perturbations ¼ allowed degree of diabatic forcing measured by ev. Figure seem only subtly distinct, disparities in hq0 are pro- 6 shows the variation of the complex eigenfrequency k. nounced. Marked distinctions in the perturbations of the

The growth rate (kR) gradually decays with increasing ev secondary circulation and potential temperature in the until apparently vanishing at 0.9. By contrast, the oscilla- middle and upper troposphere coincide with substantial tion frequency (kI) changes little. The preceding behav- differences in Sh0 . Not only does Sh0 have a greater ampli- iour is similar to that reported by SM07 for the n 3 tude in the MUM corresponding to ev 1, but the two ¼ ¼ MUM of a cloudy vortex resembling a category-3 hurri- spatial patterns diverge considerably above 4 km in the cane with no mean secondary circulation. On the other eyewall updraft region of the vortex. hand, increasing ev from 0.9 to 1 introduces a new mode Figure 8 elaborates on the inner-core structure of each of instability that oscillates slower and grows faster than MUM. The left column shows the intensity of the vertical e any of its predecessors. Further amplification of v to 1.1 vorticity perturbation fz0 , as measured by its maximum 14 D. A. SCHECTER

(a) (b) contours: /10 4 (c)

6 1 0.4

4 0.5 0.2 2

0 0 0

2 0.5 0.2 4

0.4 6 1

(d) (e) contours: /10 4 (f) 1 2 6

4 0.5 1

0 0 0

0.5 1 1 4

6 1 0.01 2

Fig. 7. (a)–(c) Basic inner-core structure of the n 2 MUM for ev 0:5. (a) Vertical slices of the velocity perturbations in the ¼ ¼ azimuthal direction (colour) and in the r-z plane (vectors). (b) Vertical slices of the perturbations of density potential temperature (colour) and the Exner function (contours). (c) Vertical slice of the perturbation to diabatic forcing (colour) and contours of its maximum value along an azimuthal circuit. (d)–(f) As in (a)–(c) but for ev 1. The yellow curve in each plot is the principal AM ¼ 0 isoline. The slices in (a, b, d, e) are at an azimuth where v0 is maximised. The colour slices in (c) and (f) are at an azimuth where Sh is maximised.

(a) max=6.3 10-3 s-1 (b) max=39.1 10-3 s-1 (c) max=1.7 10-3 s-1

2.5 10 2.5

2.0 8 2.0

1.5 6 1.5

1.0 4 1.0

0.5 2 0.5

0.0 0 0.0

(d) max=3.1 10-3 s-1 (e) max=38.0 10-3 s-1 (f) max=3.4 10-3 s-1

2.5 10 2.5

2.0 8 2.0

1.5 6 1.5

1.0 4 1.0

0.5 2 0.5

0.0 0 0.0

Fig. 8. (a)–(c) Maximum values over u (maxu) of (a) the vertical vorticity perturbation, (b) the magnitude of the horizontal vorticity perturbation, and (c) the divergence of the horizontal velocity perturbation of the n 2 MUM for ev 0:5. (d)–(f) As in (a)–(c) but for ¼ ¼ ev 1. The yellow curve in each plot is the principal AM isoline. ¼ INSTABILITIES OF TROPICAL CYCLONES 15

solid contours: max ( )(10 3 s 1) horizontal divergence, defined by r or ru ouv =r. (a) z 0 ½ ð 0Þþ 0 1 In both MUMs, r0 is broadly smaller than the vertical vorticity perturbation near the surface, but is far from negligible. In the middle tropospheric region of the eye-

0.5 wall cloud, the amplitudes of r0 and fz0 are comparable to each other. The MUM corresponding to ev 1 is distin- ¼ guished by having a middle tropospheric peak of r0 that

0 slightly exceeds the inner core maximum of fz0 . Figure 9 illustrates for each MUM how the azimuthal * phase velocity minus the local angular velocity of the primary circulation (cu kI =n Xb) varies over the core -0.5 À À of the tropical cyclone. The dashed green curves repre-

senting the zero contours of cu correspond to where the mode corotates with the mean flow. Negative/positive val- 1 ues of cu indicate locally retrograde/prograde wave propagation in the azimuthal direction. The superimposed 3 1 (b) solid contours: max ( z)(10 s ) vertical vorticity distribution (solid black contours) of 1 the MUM corresponding to ev 1 is concentrated in the ¼ region of retrograde propagation. On the other hand, the 0.8 MUM with weaker diabatic forcing has a middle-to- 0.5 upper tropospheric swath of intense fz0 that extends well into the region of prograde propagation. In both cases, the magnitude of the intrinsic frequency of the mode

0 (ncu) is less than the nominal inertial frequency ( gbnb) where the vorticity anomalies are peaked. While notable, pffiffiffiffiffiffiffiffiffiffi * such local slowness does not necessarily indicate that -0.5 traditional asymmetric balance theory (Shapiro and Montgomery, 1993) would provide an accurate description of the wave dynamics. Bear in mind that the issue is 1 complicated by the moist secondary circulation and the vertical shear in vb. Moreover, even small deviations from balanced dynamics are potentially important to the Fig. 9. (a) Intrinsic frequency ncu of the n 2 MUM ¼ instability mechanism. normalised to the nominal inertial frequency g n for the case b b Moving outward to where r exceeds 100 km, the in which ev 0:5. The dashed green contours show where the ¼ pffiffiffiffiffiffiffiffiffiffi intrinsic frequency is zero. The white area marked with an MUMs acquire intrinsic frequencies that broadly satisfy 2 ~ 2 asterisk coincides with a region where g n 0 and the gbnb ncu N (not shown). The preceding condi- b b < ð Þ normalisation frequency is imaginary. The solid black contours of tion suggests that the intrinsic frequency lies comfortably 0 the amplitude (maximum over u)offz are shown for reference. within the regime of inertia-gravity waves. Consistent e (b) As in (a) but for v 1. with such waves, one finds that f rn beyond the ¼ j znjj j core of the vortex, barring sporadic pockets of violation. The right panels in Fig. 10 convey the basic structure of value over u. In each MUM, the intensity peaks of fz0 the outer waves as represented by w0 in the two MUMs roughly coincide with a subset of regions where the radial under consideration. Although both modes are normal- gradient of basic state potential vorticity is locally ised to have the same inner core maximum value of v0, enhanced (see Fig. 1c). The amplitudes of the peaks differ the outer waves have appreciably stronger vertical veloc- considerably between the two modes, especially in the ities for the case in which ev 0:5. Whether such a dis- ¼ middle-to-upper troposphere. The middle column depicts tinction is relevant to the mechanism of modal growth is the maximum magnitude of the horizontal vorticity per- a question left for future analysis. In theory, seemingly turbation fh0 along an azimuthal circuit. In both MUMs, weak inertia-gravity wave radiation may contribute sig- f0 broadly exceeds the vertical vorticity perturbation. As nificantly to the prevailing low-n instability of an intense j hj before, differences between the two MUMs are mainly tropical cyclone (Menelaou et al., 2016; Schecter and seen in the amplitudes of various peaks of the plotted Menelaou, 2017). However, the author is unaware of any field. The right column shows the circuit-maximum of the existing method for assessing the importance of inertia- 16 D. A. SCHECTER

(a) (b) 16 3 -1.5 -1 -0.5 00.51 1.5 w (m/s) 14 2 12

10 1 eyewall 8 0

6 1

4 outer vortex 2 2 3 0 100 150 200 250 300 350 r(km)400 450 (c) (d) 16

-1.5 -1 -0.5 0 0.5 1 1.5 3 w (m/s) 14 2 12

10 1 eyewall 8 0

6 1

4 outer vortex 2 2 3 0 100 150 200 250 300 350 r(km)400 450

Fig. 10. (a, b) Slices of the vertical velocity perturbation of the n 2 MUM in (a) the inner core and (b) the outer region of the vortex ¼ for the case in which ev 0:5. The dotted black contours in (a) correspond to w0 0. The solid (dashed) white contours in (b) ¼ 1 ¼ correspond to w0 7( 7) cm sÀ . Note that the units of the colorbar labels differ between (a) and (b). (c, d) As in (a, b) but for ¼ À ev 1. In all plots, the dashed black contour corresponds to the principal AM isoline. The azimuth of the top (bottom) row of the ¼ figure is equivalent to that of Fig. 7a (7d).

gravity wave emission to the growth of a multifaceted notably includes a band of eddies along the eyewall instability mode of a convective vortex with the geomet- updraft. The bands associated with the two MUMs are rical complexity of a realistic hurricane. distinguishable in part by having opposite rotational ten- Figures 11a,d show changes to the mean flow that dencies at various locations. attend the growth of each instability mode from an Figures 11b,e show the perturbation of kinetic energy asymptotically small disturbance. The symmetric compo- density dKE associated with the growth of each MUM. 2kRts nent of the perturbation is given by x0 Xpe , with ¼ To second-order in the asymmetric mode amplitude, X given by Equation (24d). The growth of either p q q instability mode modestly reduces the u-averaged azi- dKE u2 v2 w2 b u2 v2 w2 2 ðÞþ þ À 2 b þ b þ b muthal wind speed at the initial location of maximal 2 2 2 q ubu0 vbv0 wbw0 q un vn wn intensity while accelerating the cyclonic rotation of the ¼ bðÞþ þ þ b j j þj j þj j q0 2 2 2 inner eye, at least in the lower troposphere. The middle u v w 2R ubun vbvn wbwn qÃ ; þ 2 b þ b þ b þ ðÞþ þ n tropospheric patterns of symmetric azimuthal acceleration ÂÃ(29) and deceleration are clearly dissimilar inward of the principal AM isoline. Moreover, the MUM affected by in which the overline denotes an azimuthal average. It weaker diabatic forcing (ev 0:5) induces greater positive has been verified that the bottom line in the second ¼ and negative azimuthal accelerations of the mean flow in equality involving the density perturbation is negligible the upper-outer part of the eyewall updraft. The perturb- (not shown). Moreover, it is seen that the distribution of 2 2 2 ation of the symmetric secondary circulation u0; w0) that dKE — here divided by KEb q u v w =2 —is ð bð b þ b þ bÞ emerges during the growth of either instability mode similar to that of v0 regardless of whether ev is 0.5 or 1. INSTABILITIES OF TROPICAL CYCLONES 17

Fig. 11. (a) The symmetric velocity perturbation that attends the growth of the n 2 MUM for the case in which ev 0:5. Colours ¼ ¼ depict v whereas vectors depict (u , w ). (b) The perturbation of kinetic energy density associated with the n 2 MUM and the 0 0 0 ¼ attendant symmetric modification of the vortex for the case in which ev 0:5. The perturbation is expressed as a positive or negative ¼ percentage of the local kinetic energy density of the basic state. The dotted black contours correspond to dKE 0. (c) The distribution ¼ of KEn associated with the n 2 MUM for the case in which ev 0:5. The white and black contours correspond to 3 ¼ ¼ KE 0:6; 3:2; 6:3; 13; 19 JmÀ . (d)–(f) As in (a)–(c) but for ev 1. The yellow or red curve in each plot is the principal AM isoline. n ¼½ ¼ The thick black or blue line drawn from the location of vbm to the surface [in all plots but (b) and (e)] shows where vb is maximised with respect to variation of r in the boundary layer. The thin black curves in (a) and (d) trace the edges of the unperturbed eyewall updraft, where wb is 2.5% of its maximum positive value.

The contribution to dKE from the asymmetric fields is in which well approximated by the following positive definite meas- oX ov 2 2 2 b R b R ure of local wave intensity: KEn q un vn wn . PC 2qbr unvnÃ 2qb wnvnÃ ; (30b) bðj j þj j þj j Þ À or À oz Figures 11c,f show the spatial distributions of KEn for the oub 2 ÂÃub 2 ÂÃowb 2 two MUMs under present consideration. Both MUMs SC 2qb o un 2qb vn 2qb o wn À r j j À r j j À z j j (30c) have their greatest values of KEn near the surface, inward ou ow 2q b b R u w ; of the radius of maximum wind, in the vicinity of where the b oz or n nÃ À þ ½ vertical vorticity of the basic state (fzb) has a pronounced ~ o 2qbg R Pb R maximum. The middle-to-upper tropospheric peaks of KEn BNC wnhqÃn 2cpd qb unhqÃn ; (30d) hqba À or are found in distinct locations. Above the surface perturb- o ÂÃo oÂÃo e 1 rubKEn wbKEn KEn 1 rqbub qbwb ation, the distribution of KEn corresponding to v 0:5 AFX ðÞðÞ ðÞðÞ; ¼ r or oz q r or oz has relatively strong peaks outward of the central part of À À þ b þ the eyewall. The instability mode that results from allowing (30e) greater diabatic forcing (ev 1) has its principal middle o rq h R u P Ã o q h R w P Ã 2cpd b qb n n b qb n n ¼ PFX 2cpd tropospheric maximum of KEn well within the eye- Àr noorhiÀ noozhi wall updraft. o o 1 rqbhqbun inqbhqbvn qbhqbwn 2c R P Ã ; Differences between the MUMs are also evident in pd r or r oz n þ ÀÁþ þ ÀÁ various terms that formally contribute to the growth rate (30f) of KEn. Equations (3a)–(3c) imply that TRB 2q R unD~ Ã vnD~ Ã wnD~ Ã : (30g) b un þ vn þ wn hi oKEn The term labelled PC combines tendencies proportional PC SC BNC AFX PFX TRB; (30a) ot ¼ þ þ þ þ þ to the radial and vertical shear of the primary circulation 18 D. A. SCHECTER

Fig. 12. Domain integrals of the individual contributions to o KE [Equations (30b)–(30g)] and their sum for the n 2 MUM with t n ¼ (left to right) ev 0 to 1.1. The value of each integral is normalised to that of o KE . The contributions from AFX and PFX are ¼ t n combined into APFX. The PC contribution is decomposed into the radial shear component proportional to orXb (r, dark red) and the vertical shear component proportional to ozvb (v, light red). The TRB contribution is decomposed into the primary part attributable to turbulent dissipation (dark cyan) and the much smaller part attributable to sponge damping (light cyan cap). of the basic state. SC combines tendencies proportional strongest peaks of the asymmetric kinetic energy density to ub and the spatial derivatives of the velocity fields of from the surface to the middle troposphere (not shown). the secondary circulation. BNC is linked to the vertical The attendant structural change coincides with notable and radial buoyancy accelerations. AFX primarily repre- modifications to the global KEn-budget. For example, the sents the convergence of the advective flux of KEn. The vertical shear component of the PC integral becomes included correction is attributable to the small but non- dominant. Moreover, the integral of BNC becomes nearly zero divergence of the momentum density of the basic equal to that of PC. state. PFX primarily represents the convergence of the flux vector associated with forcing by the perturbation of 5.2. Sensitivity to the parameterisation of the pressure-gradient. The included correction is attribut- turbulent transport able to the small but nonzero divergence of the approximate momentum perturbation weighted by hqb. TRB is The MUM associated with arbitrary n generally varies associated with turbulent momentum transport and (to a with the parameterisation of small-scale turbulence. lesser extent) sponge-damping near the upper and outer Sensitivity to the intensity of turbulent transport is illus- edges of the computational domain. It is worth pointing trated herein by reducing the value of ek defined in out that substantial cancellations of the tendency terms Section 2.3. The minimum value of ek to be considered often result in a local value of otKEn 2kRKEn that is will be 0.0625, which is slightly below the limit of 0.07 ¼ m m much smaller than its individual parts. (0.08) that guarantees Kh (Kv ) will uniformly equal the m m Figure 12 illustrates how the value of the diabatic forc- value specified for Kh;min (Kv;min)inSection 4. ing parameter (ev) affects the volume-integrals of the KEn Figure 13 shows how reducing ek affects the complex tendency terms pertaining to the n 2 MUM of the trop- eigenfrequencies of the MUM and the second most ¼ ical cyclone. The volume integrals are over the entire unstable eigenmode (SMUM) of linear systems with n 2 ¼ domain of the linear model. The results are similar for all and ev 0:5; 1 . Results are shown for ek 1, 0.25 and 2f g ¼ ev 0:75. The integral of PC provides the greatest posi- 0.0625. As before, the MUM is provisionally equated to the tive contribution to the sum. The component of PC asso- prevailing instability mode that emerges during a time inte- ciated with the radial shear of the basic state is gration of the linear model initialised with a random distri- dominant. The integral of SC is smaller than that of PC, bution of hq0 on G2. The SMUM is provisionally equated to but often greater than the integral of all terms combined. the prevailing instability mode of a continued integration The integrals of both BNC and TRB are negative and that filters out the MUM [see Equation (26)]. Both modes substantial. On the other hand, the integrals of AFX, are verified to solve the eigenproblem on G4. The displayed PFX and their displayed sum are negligible. The budget data are obtained from the G4 eigensolutions. corresponding to ev 1 has several distinctive features. Consider first the group of linear systems that allow a ¼ The difference between the PC and SC integrals is appre- medium degree of diabatic forcing (ev 0:5). Section 5.1 ¼ ciably reduced. Moreover, the BNC integral is positive. thoroughly described the dominant MUM when ek 1. ¼ Of lesser significance, the combined integral of AFX and The corresponding SMUM has a lower oscillation fre-

PFX is discernibly negative owing mostly to the correct- quency and is structurally distinct in having KEn concen- ive component of PFX. Increasing ev to 1.1 moves the trated in the middle troposphere (not shown). Reducing INSTABILITIES OF TROPICAL CYCLONES 19

same growth rate. Reducing ek to 0.0625 switches the ordering of the preceding instability modes without changing their top-tier status. The new mode is distinguished by having a greater oscillation frequency and a

dissimilar distribution of KEn above the boundary layer (Fig. 14c). Moreover, the global KEn budget of the new mode is distinguished by having a greater vertical shear component of PC, and a negative contribution from BNC (Fig. 14d). It is worth remarking that decreasing the eddy diffusivity often magnifies the importance of higher wavenumber

MUMs. For example, reducing ek to 0.25 in a system with ev 1 allows an n 3 MUM (Figs. 14e,f) to chal- ¼ ¼ lenge its n 2 counterpart for dominance among instabil- ¼ ity modes with substantial KEn near the surface. While the former oscillates approximately 1.6 times faster than the latter, both MUMs have growth rates of 1:1 4 1 Â 10À sÀ .

5.3. Relationship to 2D instability theory It is common practice to explain the instability of the primary circulation of a tropical cyclone in the context of a two-dimensional nondivergent barotropic model (see Appendix D). The foregoing analysis casts doubt on the general adequacy of such an approach. That is to say, the preceding results suggest that the three-dimensionality of the tropical cyclone under present consideration has a major impact on the prevailing mode of instability. The Fig. 13. Variation of the complex eigenfrequency k of the n 2 evidence includes MUMs with substantial horizontal vor- e ¼ MUM and SMUM with the small-scale turbulence parameter k for ticity and divergence. The evidence also includes major systems with (top) ev 0:5 and (bottom) ev 1. The real (blue) ¼ ¼ contributions from SC and/or the vertical shear compo- and imaginary (red) parts of each eigenfrequency are normalised 5 1 nent of PC to the volume integrated time-derivative of to their respective values (kR 7:89 10À sÀ and kI Ã Ã 3 1 ¼ Â ¼ asymmetric kinetic energy (KEn). 1:30 10À sÀ ) obtained for the MUM when ev ek 1. À Â ¼ ¼ Further insight is gained by directly comparing 2D and 3D instability theory. The 2D analysis requires reduction ek introduces a faster instability that overtakes both of of the basic state to a circular shear-flow characterised by the aforementioned eigenmodes. The greater growth rate a 1 D vertical vorticity profile f r . Because the asym- bð Þ (kR) of the new MUM coincides with a greater oscillation metric kinetic energy density of the instability usually has frequency ( kI ). The new MUM is also structurally dis- greatest amplitude in the lower troposphere, f will be j j b tinct in having KEn largely confined to a shallow layer extracted from the q -weighted vertical average of f r; z b zbð Þ near the surface (Fig. 14a). Moreover, the global KEn (Fig. 1d) between the sea-surface and z 2 km. The kine- ¼ budget is distinguished from that of the original MUM matic viscosity K2d will be varied between 0 and by having a greater vertical shear component of PC, and 4000 m2s–1. The upper limit is roughly 1.4 times the peak m a minimal contribution from SC (Fig. 14b). value of K in the 3D model when ek 1(Fig. 5). h ¼ Consider next the set of linear systems that allow rela- The nonmonotonic radial variation of fb facilitates a tively strong diabatic forcing (ev 1). As before, the variety of algebraic and exponential instabilities. An alge- ¼ reader may consult Section 5.1 for a thorough description braic instability is expected to dominate the n 1 compo- ¼ of the dominant MUM when ek 1. The corresponding nent of an arbitrary disturbance (Smith and Rosenbluth, ¼ SMUM is similar to that of the equally diffusive system 1990). The exponentially growing eigenmodes associated with ev 0:5. Reducing ek to 0.25 modestly accelerates with greater azimuthal wavenumbers are readily obtained ¼ the instability associated with the original MUM and from a complete numerical solution to the eigenproblem leads to the appearance of a new SMUM with nearly the on a stretched radial grid comparable to that of G2. For 20 D. A. SCHECTER

(a) (b)

(e) (f)

Fig. 14. (a) Spatial distribution of otKEn 2kRKEn for the n 2 MUM with ev 0:5andek 0:0625. The white and black ¼ 3 3 ¼ ¼ ¼ contours correspond to o KE 0:1; 0:5; 1:0; 2:0; 4:0 10À WmÀ . (b) Domain integrals of the individual contributions to o KE for t n ¼½ t n the n 2 MUM with ev 0:5 and e 0:0625. (c, d) As in (a, b) but for ev 1ande 0:0625. (e, f) As in (a, b) but for the n 3 ¼ ¼ k ¼ ¼ k ¼ ¼ MUM with ev 1ande 0:25. The red curves in (a, c, e) correspond to the principal AM isoline; the dashed green curves show ¼ k ¼ where cu 0; the blue lines show where v is maximised with respect to variation of r in the boundary layer. The plots in (b, d, f) are ¼ b completely analogous to those in Fig. 12. INSTABILITIES OF TROPICAL CYCLONES 21

(a) The two modes are distinguished by their virtually invari- 0m2/s ant oscillation frequencies that differ roughly by a factor 2x 103 4x 103 of 2. Decreasing the viscosity from its maximal value is seen to unleash the instability of the high-frequency mode, K2d such that it transitions from SMUM to MUM status as 2 –1 K2d drops below 2500 m s . Despite the reordering of growth rates, neither the low-frequency mode (Fig. 16a) nor the high-frequency mode (Fig. 16b) radically changes

structure with variation of K2d over the interval under consideration. Except for moderate radial smoothing of

(b) 3.0 low-freq mode the vorticity wavefunction, the unshown modifications high-freq mode 2.5 linked to greater viscosity are difficult to discern with a n =2 casual glance. 2.0 To some extent, the low-frequency mode of the 2D sys- 1.5 tem that prevails under conditions of high viscosity 1.0 resembles the lower tropospheric section of a typical 3D 0.5 MUM that dominates under moderate diabatic forcing e 0.0 when turbulent transport is parameterized with k 1. (c) ¼ 1.5 Figure 16c (16e) depicts the lower tropospheric structure

2.0 of the 3D MUM corresponding to ev 0:5 (1). As in the ¼ 2.5 low-frequency mode of the 2D system, the strongest per- low-freq mode 3.0 turbation eddies are centred on the outer edge of the high-freq mode 3.5 n =2 main vorticity annulus. In similar agreement, the promin- 4.0 ent inner and outer waves of fz0 are close to being diamet- 4.5 rically out of phase at azimuths where the amplitudes are

5.0 peaked. On the other hand, seemingly subtle differences

3 2 cannot be ignored. To begin with, the radii at which the 2d (10 m /s) 2D and 3D modes corotate with the circular shear flow Fig. 15. (a) Azimuthal wavenumber (n) dependence of the (shown by the dashed green circles) do not coincide. In growth rate of the 2D MUM for several values of K ,as 2d principle, even a slight displacement of a corotation indicated in the legend. Computed MUMs with growth rates of 6 1 radius can substantially affect the impact of locally order 10À sÀ or less (at high n and appreciable K2d) are excluded from the plot, because they are considered virtually enhanced (potential) vorticity stirring on the growth of neutral and have questionable accuracy. (b) K2d dependencies of an instability mode. The nature of any delicate imbalance the growth rates of the two most unstable 2D eigenmodes of various growth and decay mechanisms may also be associated with an n 2 perturbation. (c) As in (b) but for the sensitive to small variations in the relative amplitudes and ¼ oscillation frequencies. The extended blue ticks on the vertical phases of the primary inner and outer vorticity waves. axes of (b) and (c) mark the growth rate and oscillation Moreover, the horizontal velocity perturbations associ- frequency of the n 2 MUM of the 3D system with ev 0:5 and ¼ ¼ ated with the depicted 3D instability modes have nonne- e 1; the red ticks are the same but for ev e 1. k ¼ ¼ k ¼ gligible divergence. Figures 16d,f illustrate the divergent (irrotational) components of the modal flow fields K2d 0, the AMUM corresponds to n 3, but all ¼ ¼ obtained from a standard Helmholtz decomposition as MUMs with 2 n 5 have growth rates within 8% of explained in Appendix E. The maximum divergent wind 2 the maximum (Fig. 15a). Increasing K2d toward 4000 m speed for ev 0:5 (1) is an appreciable 16% (27%) of the 1 ¼ sÀ diminishes the growth rate of each MUM with maximum nondivergent wind speed. greater effect at larger n; ultimately, the exponential insta- It is notable that (for n 2) the low-frequency instabil- ¼ bilities are confined to azimuthal wavenumbers 2 and 3. ity modes of both the 2D system and the 3D systems The preservation of n 3 dominance (or shared domin- studied in Section 5.2 are superceded by higher frequency ¼ ance) with increasing viscosity appears to be at odds with modes as viscosity tends toward zero. The low-viscosity e e the 3D model. For k 1 and v 0; 0:5; 1 , the wave- 3D MUM corresponding to ev 0:5 and ek 0:0625 ¼ 2f g ¼ ¼ number-3 instability modes of the 3D model were found (Fig. 14a) is fairly similar to its 2D counterpart (Fig. to be subdominant. 16b). To begin with, the 3D MUM is confined to a shal- Figures 15b,c show how the complex eigenfrequencies low layer near the surface. Moreover, unshown analysis of the two most unstable n 2 eigenmodes vary with K2d. of the horizontal flow in the lower troposphere ¼ 22 D. A. SCHECTER

(a) max | |: 4.8 m/s 2D low-freq mode (b) max | |: 4.8 m/s 2D high-freq mode

40 0.75 40 1.0

0.50 20 20 0.5 0.25

0 0.00 0 0.0

0.25 20 20 0.5 0.50

40 0.75 40 1.0

20 0 x(km)20 40 20 0 x(km)20 40

0.50 0.50 20 20 0.25 0.25

0 0.00 0 0.00

0.25 0.25 20 20 0.50 0.50

0.75 0.75 40 40 1.00 1.00

20 0 x(km)20 40 20 0 x(km)20 40

(e) max | |: 4.8 m/s MUM: =1, k=1 (f) max | |: 1.2 m/s MUM: =1, k=1

40 0.75 40 0.75

0.50 0.50 20 20 0.25 0.25

0 0.00 0 0.00

0.25 0.25 20 20 0.50 0.50

40 0.75 40 0.75

20 0 x(km)20 40 20 0 x(km)20 40

Fig. 16. (a) Vertical vorticity (red and blue), streamlines (black) and corotation circles (dashed green) of the low-frequency mode of 3 2 1 the 2D system with K2 10 m sÀ . The streamline thickness is directly proportional to the local magnitude of the horizontal velocity d ¼ perturbation u0. (b) As in (a) but for the high-frequency mode. (c) As in (a) but for vertically averaged fields associated with the MUM of the 3D system with ev 0:5 and e 1; the averaging is over a 2 km layer adjacent to the sea-surface. (d) As in (c) but with the ¼ k ¼ streamlines corresponding to the irrotational component of u0. (e, f) As in (c, d) but for ev e 1; note that segments of the ¼ k ¼ corotation circle can be found at the corners of both plots. In all subfigures, the axis labels x and y denote horizontal Cartesian coordinates measured from the center of the vortex. INSTABILITIES OF TROPICAL CYCLONES 23

(a) nondivergent wind speed. On the other hand, the 2D model does not provide an entirely accurate picture of 2D theory the low viscosity perturbation dynamics. The oscillation 3D theory frequency of the 3D MUM is 0.8 times that of the 2D CM1 sim MUM, and the growth rate is 0.4 (0.6) times that pre- 2 –1 dicted by the 2D model with K2d 0 (1000 m s ). ¼ Greater facilitation of diabatic forcing (ev 1) at low vis- ¼ cosity (ek 0:0625) leads to much greater disparity ¼ between the 3D and 2D MUMs. The 3D MUM (Fig. 14c) exhibits complex vertical structure deep into the free troposphere, and the perturbation fields near the surface have more features in common with the low-frequency SMUM of the 2D model (Fig. 16a). Consistently, the oscillation frequency of the 3D MUM is 0.5 times that of the 2D MUM.

6. Comparison of linear instability theory to NS14 (b) Reducing the general uncertainty of linear instability theory will require refinement of the physics parameterisations. Such refinement will require a comprehensive comparison of theory to state-of-the-art cloud resolving numerical simulations. While a comprehensive refinement effort is beyond the scope of this paper, a comparison of our linear model to the results of one of our earlier simulations is easy and worth reporting. The simulation considered for illustrative purposes corresponds to the three-dimensional moist experiment of NS14 distinguished from others by the following ratio of

surface-exchange coefficients: Ce=Cd 0:3. The experi- 2D theory ment examined the evolution of a random perturbation 3D theory of an initially axisymmetric category-2 hurricane in CM1. The disturbance followed an initial pattern of develop- CM1 sim ment similar to that found in all simulations of the study,

including those with larger values of Ce=Cd and stronger vortices. Specifically, the perturbation spurred asymmetric wave growth energetically concentrated near the surface, Fig. 17. Comparison between the prevailing instability modes and the wave growth engendered a ring of five well- of a tropical cyclone simulated with CM1 and two theoretical predictions. (a) Growth rate versus azimuthal wavenumber n as defined mesovortices (Figs. 3 and 6 of NS14). The physics determined by (circles) 2D instability theory, (diamonds) 3D parameterisations utilised in the experiment differed from instability theory, and (squares) the pertinent 3D CM1 simulation those described in Section 4 in several notable ways. To of NS14. (b) As in (a) but for the oscillation frequency. The error begin with, the microphysics parameterisation excluded bars are explained in the main text. ice. So as to keep the ratio of surface-exchange coefficients constant over the entire ocean, the drag coefficient demonstrates that the strongest perturbation eddies are was held fixed (along with Ce)atCd 0:005. Perhaps of m=h ¼ centred on the inner edge of the main vorticity annulus, greatest significance, Kh was an order of magnitude and that a corotation radius lies in between the primary smaller in the vicinity of maximum wind speed. The inner and outer vorticity waves. In good agreement with reader may consult NS14 for further details. a key assumption of the 2D model, the maximum magni- For better compatibility with the model configuration tude of the divergent component of the lower tropo- of NS14, the physics parameterisations used presently in spheric velocity perturbation (averaged over a 2-km layer computing the linear instability modes differ somewhat adjacent to the sea-surface) is merely 6% of the maximum from those used previously. Diabatic forcing is given by 24 D. A. SCHECTER

Equation (9b), but v~b is calculated under the assumption parameterisations of diabatic forcing and small-scale tur- of liquid-only condensate. The drag coefficient is simpli- bulence were separately examined by reducing ev to 0 m m fied to Cd 0:005ek. The variables K and K in and ek to 0.0625 on G2~ and G4~ . The error bar on each ¼ h;sm v;sm Equation (16) are obtained as before, but from the simu- diamond covers the full range of values for kR or kI lation used to generate the basic state of the NS14 vortex. obtained from all configurations of the linear model; the m m Typical values of both Kh;sm and Kv;sm are between 100 smallness of the error bars indicates robust results. Note 2 1 and 200 m sÀ where the pertinent instability modes are that the plotted eigenfrequencies are confined to n concentrated. The lower limits of the eddy diffusivities between 2 and 8, which correspond to eigenmodes ener- m 2 1 m 2 1 are given by K 100 m sÀ and K 40 m sÀ . getically concentrated near the surface (not shown); a h;min ¼ v;min ¼ Figure 17 compares wave growth in the CM1 simula- slower growing middle-tropospheric MUM associated tion to that predicted by linear instability theory. The with n 1 is excluded from present consideration. ¼ squares show (a) the growth rates and (b) the oscillation Figure 17 clearly demonstrates that the eigenfrequen- frequencies of the primary Fourier components of the cies of the theoretical and simulated 3D MUMs are in asymmetric radial velocity field (du u u) in the simula- good agreement where the latter are inferred from the À tion. The measurements are made by a straightforward cleanest monochromatic signals (4 n 7). On the other procedure. To begin with, du is vertically averaged over hand, both the growth rates and oscillation frequencies the interval 0 < z < 1:0 km and expanded into a discrete are smaller than those of the MUMs associated with an Fourier series with respect to the azimuthal coordinate u. analogous 2D vortex (circles). The 2D vortex under con- The wavenumber-n Fourier coefficient of the vertically sideration is modelled after the primary circulation of the averaged field is denoted dun r; t . Time series of the amp- NS14 tropical cyclone averaged in z over a layer of thick- ð Þ litude and phase of dun (for all n between 1 and 8) are ness d adjacent to the surface. The plotted 2D data points obtained from three probes placed 10-km apart on a correspond to the means taken from 6 configurations in 3 2 –1 radial line segment that is centred roughly at the radius which d 1; 2; 3 km and K2d 0; 10 m s .As 2f g 2f g of maximum wind. Each time series is taken over the usual, the error bars extend from the minimum to max- interval 0 t 90 min. Data during the initial adjust- imum values of the data set for each n. ment period and near the end of the interval (when In this particular case study, the insensitivity of 3D lin- incipient mesovortices take form) are generally discarded. ear instability theory to the degree of diabatic forcing The growth rate is obtained from an exponential curve fit allowed in the model is consistent with the concentration to the amplitude data, whereas the oscillation frequency of modal wave activity inward of the eyewall cloud is obtained from a linear regression of the phase data (NS14). Insensitivity to the reduction of ek seems reason- (over 1 oscillation period). The plotted growth rates and able given the short e-folding times of the MUMs (15- oscillation frequencies correspond to their respective 20 min) relative to the minimum applicable time scale for 2 2 means among the three probe measurements; each error turbulent diffusion, sk min l =Kh; l =Kv , in which l ð h v Þ h=v bar covers the full range of probe values. The preceding is the horizontal/vertical lengthscale relevant to the mode measurements are sensibly associated with the complex and Kh=v is the horizontal/vertical eddy diffusivity. Taking 2 –1 3 eigenfrequencies of MUMs provided that a single grow- K 200 m s and l 10 m yields sk 83 min. h=v h=v ing wave dominates dun. Such a condition appears to be satisfied quite well for 4 n 7. The greater error bars 7. Conclusion shown for n 3 and n 8 indicate that the probe signals ¼ ¼ are not as clean. Because the time series for n 1 and This paper has proposed a method to account for dia- ¼ n 2 do not closely resemble those of a single growing batic forcing and inhomogeneous eddy diffusivities in pre- ¼ wave, they are excluded from the plots. dicting and analysing the dominant instability modes of The diamonds in Fig. 17 show the growth rates and numerically simulated tropical cyclones. Excluding expli- oscillation frequencies of the 3D MUMs predicted by the cit moisture equations from the linearised model necessi- linear model for the tropical cyclone simulated in NS14. tated a partly intuitive parameterisation of the diabatic

The MUMs were first identified as perturbations domi- forcing Sh0 . The parameterisation considered herein set Sh0 nating solutions of the initial value problem with ev proportional to w with the modulating coefficient ¼ 0 ek 1 on a grid G2~ with double the resolution of that evv~ ozhqb dependent on the local moist thermodynamic ¼ b used in NS14. The diamonds are centred on values of kR conditions of the basic state. A more general parameter- and kI obtained by recomputing the eigenmodes on a grid isation scheme [Equation (10)] was presented for future G4~ with double the resolution of G2~ . Further sensitivity consideration. to grid spacing was examined by repeating the computa- The instability analysis was illustrated for a mature tions on the original NS14 grid. Sensitivity to the tropical cyclone representative of a category 4 hurricane. INSTABILITIES OF TROPICAL CYCLONES 25

The basic state was generated by an axisymmetric numer- instability predicted by the linear model showed little sen- ical simulation with two-moment cloud microphysics and sitivity to switching ev between 0 and 1. typical settings for the parameterisation of subgrid turbulence. Initial consideration was given to linear systems having vertical and horizontal eddy diffusivities compar- Acknowledgements able to those regulating the basic state. With the diabatic The author thanks Dr. Konstantinos Menelaou and an e forcing parameter v set to a value between 0 and 1, per- anonymous reviewer for their comments on this study turbation growth was commonly dominated by a slowly prior to publication. The author also expresses his growing n 2 eigenmode with deep structure but maximal ¼ gratitude to Dr. George Bryan for providing the cloud intensity (KEn) in the lower tropospheric region of the resolving model (CM1) used to generate the tropical inner core. The complex eigenfrequency, spatial structure cyclones whose instabilities were analysed herein. The and energetics of the n 2 MUM were sensitive to vari- ¼ NS14 study re-examined in Section 6 was made possible ation of e . Increasing e from 0 to 0.9 gradually stabi- v v through computing allocations from XSEDE and the lised the mode. Further amplification of e to 1 v National Center for Atmospheric Research as noted in introduced a new MUM distinguished in part by having the original article. a larger growth rate than any of its predecessors, and by having a slightly positive buoyancy-related contribution to the production of integrated KEn. Disclosure statement Reducing the eddy diffusivities with e fixed at either v No potential conflict of interest was reported by 0.5 or 1 generally changed the nature of the n 2 instabil- ¼ the author. ity. For ev 0:5, the original MUM was ultimately ¼ replaced by a faster surface-concentrated instability mode whose growth of KEn involved a much smaller fractional Funding contribution from the term directly linked to the second- Preparation of this paper was partly supported by the ary circulation. For ev 1, the original MUM was ¼ National Science Foundation under grant AGS-1743854. replaced by a faster instability mode whose growth of

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Shapiro, L. J. and Montgomery, M. T. 1993. A three- phases of moisture. To keep the working estimate of Sh0 dimensional balance theory for rapidly rotating vortices. J. relatively simple, the definition of v that assumes liquid- Atmos. Sci. 50, 3322–3335. doi:10.1175/1520- only (ice-only) condensate is used where T>T0 (T 2.0.CO;2 in which T0 273:15 K. The temporal averaging of v Smith, R. A. and Rosenbluth, M. N. 1990. Algebraic instability that produces vb (see Section 4) helps smooth of hollow electron columns and cylindrical vortices. Phys. discontinuities at cloud edges and the freezing level. Of Rev. Lett. 64, 649–652. doi:10.1103/PhysRevLett.64.649 further note, the value of q substituted into the definition Walko, R. and Gall, R. 1984. A two-dimensional linear stability t of v arbitrarily excludes contributions from relatively fast analysis of the multiple vortex phenomenon. J. Atmos. Sci. 41, 3456–3471. doi:10.1175/1520-0469(1984)041 < 3456: falling hydrometeors such as rain, graupel and snow. The ~ 2 ATDLSA > 2.0.CO;2 very subtle change to the distribution of N [Equation Zhang, J. A. and Montgomery, M. T. 2012. Observational (28b)] resulting from such exclusion is minimal compared estimates of the horizontal eddy diffusivity and mixing length to the effect of adjusting ev as in Section 5.1 of the in the low-level region of intense hurricanes. J. Atmos. Sci. main text. 69, 1306–1316. doi:10.1175/JAS-D-11-0180.1 Section 2.2 indirectly suggested that the condition

Zhang, J. A., Rogers, R. F., Nolan, D. S. and Marks, F. D. Jr, v0p_ v p_ 0 would help justify the assumed b b 2011. On the characteristic height scales of the hurricane proportionality between S and w p_ 0=q g, at least in a h0 0 À b boundary layer. Mon. Wea. Rev. 139, 2523–2535. doi:10.1175/ system governed by reversible moist-adiabatic MWR-D-10-05017.1 thermodynamics. Upon considering the lowest order

terms in a multivariable Taylor expansion of v0, one may Appendix A. Notes on Sh0 formulate a stronger version of the preceding inequality for single-wavenumber perturbations as follows: The parameterisation of diabatic forcing given by o o o Equation (9b) requires formulas for the partial pressure v pn p_ b v Tn p_ b v qtn p_ b u s C j j ; j j ; j j 1; derivatives of h and h appearing in the definition of v op v p_ oT v p_ oqt v p_ q q "#b bj nj b bj nj b bj nj mx [Equation (7)]. SM07 derives the following formulas for (A2) the special case of a system with liquid-only condensate: in which ... mx denotes the maximum order of u o Rd=cpd ½ hq TRd pa 1 qt=e 1 qt=e magnitude among the bracketed items. Violation of o þ 1 þ ; p ! ¼Àcpdp p 1 qt À1 qtcpv=cpd!inequality (A2) evaluated with the basic state and sm;qt þ þ (A1a) perturbation fields of our linear model would cast doubt on the adequacy of having formulated S as a linear o s h0 hq function solely of w0. The forthcoming evaluation will op ! ¼ sm;qt take perturbation amplitudes from the n 2 MUM of the Rd=cpd e ¼ TRd pa 1 qv = linear system with ev e 1, as displayed in Section þ Ã ¼ k ¼ À cpdp p 1 qt 5.1. A conservative estimate will be used for the order-of- þ 2 magnitude of qtn that is not explicitly provided by the qv L q Ã v v j j 1 1 R TÃ linear model. cpdqv þ e þ d 21 Ã 3; e 2 e The following analysis is restricted to the interior þ Rd À qv cpv qt qv cl Lvqv 1 qv = Ã À Ã Ã Ã 6 1 ðÞ ðÞþ 2 7 region of the eyewall updraft, defined to be where 6 þ cpd þ cpd þ RvcpdT 7 1 6 7 wb 2:6msÀ . In this saturated region of the tropical 4 (A1b)5 s cyclone, v oph . The partial derivatives of v at ¼ð qÞsm;qt in which c is the isobaric specific heat of water vapour, p; T; q p ;T ;q are accurately obtained from basic pv ð tÞ¼ð b b tbÞ L T is the latent heat of vapourisation, and c is the finite differencing without the need to derive lengthy vð Þ l specific heat of liquid water. For a system with ice-only analytical formulas. The pressure variables appearing in condensate, the same two equations apply with the (A2) are given by following modifications to the saturated air formula: the h q c p qn n pd p ; (A3a) latent heat of sublimation Ls replaces Lv, and the specific n ¼ h þ q c b qb b vd heat of ice ci replaces cl. As in the main text, the variable o o o o o p_ n tpn ub rpn inXbpn wb zpn un rpb wn zpb; qv p;T represents the vapour mixing ratio at saturation ¼ þ þ þ þ þ Ãð Þ (A3b) with respect to either liquid or ice, depending on the p_ u o p w o p : (A3c) allowed condensate. b ¼ b r b þ b z b As in reality, the simulated tropical cyclone providing Linearising the relation T 1 qt Phq= 1 qv p; T =e ¼ð þ Þ ½ þ Ãð Þ the basic state of Section 4 is complicated by having 3 for air that remains precisely saturated— and using 28 D. A. SCHECTER

standard formulas (Emanuel, 1994) for the partial derivatives of qv —yields Ã

Tb Rd qv b pn hqn qtn Tn eÃ ¼1 Lbqv b=RdTb "#cpd þ pb þ hqb þ1 qtb þ Ã þ (A4) pn hqn jh qtn Tn jp jh ; j j ; !j j j pb þ hqb j 1 qtb "#þ mx in which jh Tb= 1 Lbqv b=RdTb , jp jh Rd=cpd ð þ Ã Þ ½ð Þþ qv b=e , qv b qv pb;Tb and Lb is either Lv Tb or Ls if Tb ð Ã Þ Ã ¼ Ãð Þ ð Þ is greater or less than T0. For simplicity, let us suppose that q e q ; (A5) j tnj q tb in which eq is tentatively set to 0.5, so as to equal 10 times the maximum of v =v . j nj bm Two estimates of C have been calculated. C1 (C2)is obtained by evaluating all 5 variables in each bracketed item of (A2) with their maximum (average) absolute values in the interior region of the eyewall. The Fig. B1. Scatter plot of the complex eigenfrequencies of the marginally reassuring results are C1 0:3 and C2 0:2; ¼ ¼ n 2 MUM in linear systems with (pink) and without (black) smaller values are obtained as e 0. ¼ q ! acoustic filtering. Different symbol shapes correspond to different combinations of ev and ek as shown in the legend in the lower Appendix B. Acoustic filtering left corner of the graph. It is common practice to use an anelastic approximation Equation (B1). The density equation is given by of the equations of motion when acoustic waves are Mvq VF MFq yq deemed unimportant to the instabilities of interest n À n n n "#F u;w (NM02; Hodyss and Nolan, 2008). The effect of acoustic 2Xfg MvF yF MvvVF yF filtering is considered herein by implementing the ¼À n n À n n n (B2c) F u;w;hq F u;w following approximation of Equation (3e): 2fgX 2Xfg ~ ~ ~ ~ o o VF MFF yF VF MFvVF yF : inqbvn 1 rqbun qbwn n n n n n n n (B1) þ F u;w þ F u;w : 2fg 2fg r ¼Àr or À oz F~ Xu;w;hq F~Xu;w 2fg 2fg Details of the implementation are provided below. q The preceding formula for yn is obtained by substituting Recall from Section 3 that the unfiltered linearised (B2b) and (B2a) for F u; w into the prognostic equations of motion can be written dx =dt M x ,inwhich 2f g n ¼ n n azimuthal velocity equation of the unfiltered system. xn is a vector representation of all prognostic perturbation Note that the foregoing acoustically filtered model F fields at all grid points. Let yn denote the subvector of xn applies only for n 1, owing to a derivation from representing field Fn. The acoustically filtered system consists equations where vn and qn are multiplicatively coupled to of three prognostic equations of the form n. An alternative formulation valid for n 0 would seem F dy ~ ~ possible by (say) switching the roles of wn and vn as n MFF yF ; (B2a) dt ¼ n n prognostic and diagnostic variables. F~ u;v;w;hq;q 2fgX As suggested earlier, acoustic filtering is expected to FF~ in which F u; w; hq and Mn is the submatrix of Mn have minimal consequence on the primary inner core 2f g F accounting for the tendency of yn directly dependent on instability of a tropical cyclone. Consider the tropical yF~ . The equations for the perturbations of azimuthal cyclone of Section 4. The unfiltered n 2 MUM is n ¼ velocity and density are diagnostic. The azimuthal described in Section 5 for various combinations of ev and velocity equation is given by ek in the linear model. The acoustically filtered MUM is

v F~ F~ here found (on G2) by initialising Equations (B2a)–(B2c) yn Vn yn ; (B2b) u w hq ¼ with yn; yn and yn of the unfiltered MUM and F~ u;w 2Xfg integrating forward in time over several e-folding periods. u w in which Vn and Vn are the coefficient matrices relating Under ordinary circumstances, the perturbation is quickly v u w yn to yn and yn according to the discretization of dominated by the filtered MUM. Figure B1 verifies that INSTABILITIES OF TROPICAL CYCLONES 29

the complex eigenfrequencies of the filtered and unfiltered The system under present consideration exhibits a instability modes hardly differ, regardless of the selected somewhat complicated sensitivity to grid spacing. The parameters regulating the strengths of diabatic forcing n 1 mode is virtually invariant with increasing ¼ and turbulent transport. resolution. The oscillation frequency (k ) of the n 2 I ¼ mode is also robust, but the growth rate on G1 exceeds that on G4 by 29%. The values of k (for n 2) on G2 Appendix C. Sensitivity of MUMs to the R ¼ and G4 are deemed closer to the continuum limit based computational grid on their modest 2% difference. All of the modes on G1 In Section 5, a MUM was provisionally equated to the that are shown for 1 n 4 have maximal KE near the n eigenmode that dominates a perturbation within 1 day of surface. Whereas the properties of the modes with n 1 ¼ initialising the linear model [Equation (20)] with random and n 2 are fairly insensitive to increasing resolution, ¼ noise on a grid (G2) having double the resolution— in the n 3 and 4 modes on G1 are fragile and superceded ¼ both r and z —of the CM1 grid used to generate the by middle tropospheric instabilities on G2 and G4. All of basic state (G1). The MUMs of G1 are readily found by the dominant instabilities at higher wavenumbers have a similar procedure. Figure C1 displays the complex maximal KEn in the middle troposphere. It is notable eigenfrequencies associated with the MUMs of both G1 that increasing the resolution for cases in which n exceeds (blue) and G2 (green) for 0

model to values of K2d where axisymmetric diffusion and asymmetric perturbation growth have commensurate time scales is technically improper, but is deemed reasonable for the purpose of illustrating modal sensitivity to the magnitude of viscosity. Fig. C1. Scatter plot of the complex eigenfrequencies of the The asymmetric (n 1) eigenmodes of an MUMs of the tropical cyclone of Section 4 as determined by the axisymmetric vortex are perturbations of the form 3D linear model with ev ek 1. The number associated with inu kt ¼ ¼ f0 Z r e c:c:, in which c:c: denotes the complex each data point denotes the value of the azimuthal wavenumber ¼ ð Þ þ þ (n) of the MUM. Data is included for computations on G1 conjugate of the preceding term. Substituting the (small blue), G2 (medium green) and G4 (large red). Dark (light) eigenmode solution into the linearised equations of symbols indicate modes with KEn maximised in the lower motion derived from Equations (D1a)–(D1c) yields (middle) troposphere. Note that the G4 data correspond to n dfb 2 eigenfrequencies nearest to those of the G2 MUMs, but the ik nXb Z W iK2d nZ 0; (D2a) associated modes were not verified to dominate time-asymptotic ðÞÀ þ r dr À r ¼ perturbations on G4. in which X r is the angular velocity corresponding to bð Þ 30 D. A. SCHECTER

Appendix E. The Helmholtz decomposition f , 2 W Z; 2 o r 1o n2=r2, and o o o .A b rn ¼ rn rr þ À rÀ rr r r formal solution for the wavefunction of w0 consistent The Fourier transform of the horizontal velocity with regularity at the origin and u0 0atr rB is perturbation associated with a 3D instability mode can ¼ ¼ be decomposed into the following sum of irrotational 1 rB r n r 2n W r dr~r~ < 1 > Z r~ ; (D2b) (superscript-/) and nondivergent (superscript-w) ðÞ¼À2n r > À rB ðÞ ð0 "# components: in which r < (r > ) is the lesser (greater) of r~ and r / w un un un / þ w (E1a) (Schecter et al., 2000). The second outer boundary vn ¼ v v n þ n condition or v0=r 0 combined with u0 0 amounts to 1 ð Þ¼ ¼ in which Z 2rÀ dW=dr at r rB. Substituting (D2b) into both ¼ ¼ / o w (D2a) and the preceding outer boundary condition un r/n un inwn=r / ; w Ào ; (E1b) W v ¼ in/n=r v ¼ rwn eliminates from the eigenproblem. Subsequent n n discretization of the radial coordinate yields a standard and matrix eigenproblem of the form Mx kx, in which x is ¼ / r a vector containing the values of Z on each grid point. 2 n n : (E1c) rn w ¼ f The 2D results of Section 5.3 and 6 correspond to n zn solutions of the preceding eigenproblem; the outer The boundary conditions at the origin are regularity of the boundary condition on v0 is obviated for computations in velocity potential /n and streamfunction wn;the which K2d 0. Selected results were successfully cross- implemented boundary conditions at rB consistent with ¼ o checked against independent solutions of the 3D model un 0are r/n 0andwn 0. The velocity fields ¼ ¼ ¼ / w set up with a thin barotropic vortex sandwiched in depicted in Figs. 16d,f correspond to u0 u0 u0 ,inwhich inu /=w /=w¼ /À=w inu between rigid free-slip walls at z 0 and 0.5 km. u0 u ;v e c:c: and u0 u ;v e c:c: ¼ ¼ð n nÞ þ ¼ð n n Þ þ