The evolution of vortices in vertical shear. 11: Large-scale asymmetries

Q..?. R. Meteorol. Soc. (2000), 126, pp. 3137-3159 The evolution of vortices in vertical shear. 11: Large-scale asymmetries By SARAH C. JONES* Universitat Munchen, Germany (Received 28 April 1999; revised 19 April 2000) SUMMARY The role of large-scale asymmetries in the evolution of a tropical-cyclone-like vortex in vertical shear on an f-plane is investigated. Idealized numerical calculations using a primitive-equation model are used to illustrate the development of the asymmetries and their role in the evolution of the vortex tilt. The asymmetries develop in the outer region of the vortex, well outside the radius of maximum wind. For the vortex profile used in most of the calculations, the region where the asymmetries develop has anomalously negative potential vorticity (PV) compared with the undisturbed environment, and the flow fields associated with the asymmetries consist of largescale anticyclonic gyres. The orientation of the gyres changes with height and they are located on opposite sides of the vortex at lower and upper levels. The influence of the asymmetries on the vortex evolution depends on both the structure and location of the asymmetries and on the orientation of the tilted vortex within the asymmetries. It is hypothesized that the asymmetries arise due to the distortion of the initially symmetric vortex by the horizontally sheared flow associated with the vertical projection of the tilted PV anomaly. Calculations using a barotropic model are presented in support of this hypothesis. The sensitivity of the large-scale asymmetries to the initial vortex structure is investigated. For a vortex profile in which the tangential wind decreases more slowly with radius, the asymmetries form in a region of positive PV anomaly and the associated flow field contains large-scale cyclonic gyres. The implications of the large-scale asymmetries for tropical-cyclone motion and intensity change are discussed. KEYWORDS: Large-scale asymmetries Potential vorticity Tropical cyclones Vertical shear I. INTRODUCTION Environmental vertical shear can influence the motion, the structure, and the Intensity of a tropical cyclone. Recent idealized studies have isolated several mechanisms by which the motion of a tropical cyclone may deviate from the motion implied by the environmental steering flow. Tropical-cyclone motion may be influenced by the interaction between the tropical-cyclone vortex and environmental potential-vorticity (PV) gradients associated with vertical shear (Shapiro 1992; Flatau et al. 1994), by the differential advection of the upper-level anticyclone and the lower-level part of the tropical cyclone (Shapiro 1992; Wu and Emanuel 1993; Flatau et al. 1994; Dengler and Reeder 1997), and by the interaction between the vertical shear and the part of the tropical-cyclone vortex with cyclonic tangential flow (Jones 1995, hereafter Part I; Smith et at. 2000). One influence of vertical shear on the structure of a tropical cyclone is to tilt the tropical-cyclone vortex away from the vertical. Part I investigated the structure of a tilted tropical-cyclone-like vortex and showed that wave-number-one asymmetries in the vertical velocity and potential temperature develop. The orientation of these asymmetries is directly related to the vertical tilt of the cyclonic vortex. DeMaria ( 1996) showed that the potential-temperature asymmetries would lead to a warming over the surface centre and increased stability to convection near the vortex core which might result in less intense eye-wall convection. He proposed that the increased stability in the inner core associated with vortex tilting could explain the decrease in tropical-cyclone intensity observed in environments with vertical shear. The asymmetric vertical circulation described in Part I may affect tropical-cyclone intensity as well. Asymmetries in the eye-wall convection are observed in modelling studies of tropical cyclones in vertical shear where convection is parametrized (Wang and Holland 1996; Bender 1997; Frank and Ritchie 1999). It is possible that this asymmetrically distributed convection might influence the intenqity of a tropical cyclone. * Corresponding address Meteorologisches Tnstitut, Universitat Munchen, Thereuensti 37, 80333 Munchen, Germany 3137

3138 S. C. JONES Since the structural changes which take place in a tropical cyclone in environmental shear are related to the vortex tilt, it is important to understand the mechanisms which determine the tilt. A useful tool to help understand these mechanisms is PV thinking (Hoskins et al. 1985). In this framework the evolution of a tropical cyclone in environmental vertical shear can be described in terms of the interaction between different PV anomalies. Thus, it is necessary to know the PV distribution of both the tropical cyclone and its environment. There are not many detailed observational analyses of the PV structure of a tropical cyclone. One exception is the study of hurricane Gloria (1985) by Shapiro and Franklin (1995). From this study, and from modelling studies and theoretical considerations, we can gain a broad idea of the PV structure of a tropical cyclone. A tropical cyclone is characterized by strong cyclonic flow throughout the troposphere with weak anticyclonic flow at larger radii just below the tropopause. The PV structure of the cyclonic part of the tropical-cyclone vortex consists of a strong deep positive PV anomaly (relative to the environmental PV) at small radii. The positive PV anomaly is typically surrounded by a much weaker negative PV anomaly in the lower and middle troposphere that extends out to large radii. The upper-level anticyclone is characterized by a broad but shallow negative PV anomaly. Model calculations of tropical cyclones using an idealized representation of convective processes exhibit such structure (e.g. Moller and Smith 1994). Observations of tropical cyclones show a broad range of structures and intensities suggesting that the detailed PV structure is likely to vary from storm to storm. The PV structure of an environment containing vertical shear will depend on the vertical profile of the environmental wind. If the shear is constant with height the environment will be characterized by horizontally uniform PV. If the flow is in thermal wind balance there will be a horizontal potential-temperature gradient, which can be considered to be equivalent to a horizontal PV gradient at the surface and the tropopause (Bretherton 1966a,b). If the shear varies with height, for example in association with an upper-level westerly jet, and the environmental flow is in thermal wind balance, the environment will contain horizontal PV gradients. The interaction between vertical shear and the cyclonic part of the tropical-cyclone vortex was described in Part I for initially barotropic vortices. The initial effect of the environmental shear flow is to tilt the PV anomaly of the cyclonic vortex away from the vertical. The balanced flow associated with the tilted PV anomaly penetrates upwards and downwards and leads to the upper and lower portions of the anomaly rotating cyclonically about the mid-level vortex centre. The rotation rate depends on the parameters which determine the Rossby penetration depth (Khandekar and Rao 1974; Jones 1995; Smith et al. 2000). For large penetration depths the upper and lower portions of the cyclonic vortex can execute many rotations. In this case the vortex track contains small-scale oscillations about the direction of the deep-layer-mean flow. For small penetration depths the vortex tilt increases steadily with time, and little or no rotation occurs. Two mechanisms are discussed in Part I by which the behaviour described above reduces the destructive action of the vertical shear on the cyclonic vortex. Firstly, the flow associated with the vertical projection of the tilted PV anomaly is itself vertically sheared, and can oppose the environmental shear flow for certain directions of vertical tilt. Secondly, if the rotation of the upper- and lower-level centres about their mid point continues such that the PV anomaly tilts upshear, the subsequent advection by the environmental flow will tend to reduce the vertical tilt rather than increase it. Both of these mechanisms are more effective for larger Rossby penetration depths since this quantity determines the strength of the aforementioned rotation. For an intense

VORTEX EVOLUTION IN VERTICAL SHEAR. I1 3139 atmospheric vortex the Rossby penetration depth can be defined as where fioc = f + 2UT/r, f is the Coriolis parameter, UT the tangential wind, r the radius, is the vertical component of relative vorticity, L the horizontal scale of the PV anomaly and N the Brunt-Vaisala frequency (Hoskins et al. 1985; Shapiro and Montgomery 1993). Thus, larger, more intense tropical cyclones at higher latitudes should be less susceptible to vertical shear than their smaller, weaker counterparts at low latitudes. DeMaria (1996) correlated vertical shear with intensity change for all named tropical cyclones in the Atlantic from 1989 to 1994. He showed that during this period, high-latitude storms were less susceptible to vertical shear than low-latitude storms, hurricanes were less susceptible than tropical storms and larger storms less susceptible than smaller storms. This suggests that the mechanisms described above are relevant for real tropical cyclones. The one group in DeMaria s study which did not support the role of the Rossby penetration depth in intensity change was that containing tropical depressions, which appeared to be less sensitive to vertical shear than hurricanes. In Part I the influence on the vortex tilt of the vertically penetrating flow associated with the tilted PV anomaly was discussed in terms of the flow across the vortex centre. The discussion concentrated on the rotation of the upper- and lower-level vortices about the mid-level vortex without considering the possibility of the vortices changing shape. Thus, it was assumed implicitly that the flow across the vortex at a given level due to the vertical penetration of the tilted vortex is relatively uniform in the horizontal. However, the tangential wind of the initial vortex has strong radial shear and we would expect the vertically penetrating flow associated with the tilted PV anomaly to be strongly sheared in the horizontal. The mutual rotation of the upper- and lower-level anomalies could be accompanied, therefore, by a change of shape of the anomalies which might give rise to large-scale PV asymmetries. In this paper we describe the development of large-scale asymmetries in the PV and discuss their influence on the vortex tilt. These asymmetries were not discovered during the study described in Part I. Although we calculated PV asymmetries, these were dominated by the strong, small-scale asymmetries associated with the distortion of the vortex inner core (see Part I, section 4). The presence of these strong asymmetries masked the much weaker large-scale asymmetries. As in Part I we present calculations with initially barotropic vortices. The final paper in this series (Jones 2000) extends the study to include baroclinic vortices. We do not include diabatic processes in this study. The presence of diabatic processes will undoubtedly change the evolution of tropical-cyclone-like vortices in vertical shear. In particular, we might expect that the vertical tilt will be reduced in the presence of innercore convection. However, we expect that what we learn from these adiabatic studies will enable us to better interpret and understand tropical-cyclone behaviour in more realistic models and in nature. The paper is organized as follows. The details of the numerical calculations are given in section 2. The development of large-scale PV anomalies is described in section 3 for two calculations with differing evolutions of the vertical tilt. Section 4 discusses the mechanism by which the asymmetries develop and section 5 considers the influence of the initial vortex structure on the large-scale asymmetries. Section 6 gives a summary and discussion. 2. EXPERIMENTAL DESIGN The model used for the numerical calculations is the three-dimensional primitiveequation model on an f-plane described in Jones and Thorpe (1992) in the configuration

3 140 S. C. JONES TABLE 1. DE'I'AILS OF THE PRIMITIVE-EQUATION MODEL CALCULATIONS Vertical Shear urnax rmax rcui rwid Calculation (S-') Profile equation and parameters (ms-') (kin) (kin) (km) S(Stroi1gerShenr) 6 x 10 Eq.(I) uo:71.521 u=0.3398 h=s.377x 40 100 1000 100 W(WeakerShear) 4x Eq.(l) ug=71.521 a=0.3398 h=5.377 x 40 100 1000 100 NPI (New Profile I) 6 x 10- Eq. (2) u* = 87.1372 (I = 0.3585 p = 0.4722 40 100 1000 I00 NP2 (New Profile 2) 6 x lop4 Eq. (2) u* = 60.0331 (I = 0.9350 ~4 =0.0336 30 100 5000 500 The profile equations are given in Jones (1995) (Part I). IJ,,, is the maximum tangential wind, rmax the radius of maximum wind, rcut is the radius at which the tangential wind is zero and!',id determines the distance over which the langential wind decreases to zero (see Part 1, p. 824). used in Part I. The environmental flow is purely zonal and is given by: - u = uo i- u,z (2) where Uo and U, are constants and z is the pseudo height (Hoskins and Bretherton 1972). In contrast to Part I we use a westerly shear for the present calculations although the conclusions here and in Part I are valid regardless of the direction of the vertical shear. The environmental flow is in thermal wind balance with a stably stratified temperature field. The initial environmental Brunt-Vaisala frequency, N, given by N2 = (g/oo)(do/dz), is constant, where 8 is the potential temperature, 00 is a reference potential temperature of 300 K, and g is the acceleration due to gravity. The densityweighted PV of the environment is constant. An axisymmetric barotropic vortex is superimposed on the environmental flow. The vortex profiles used, and details of the model initialization are given in Part I and Table 1. The tangential wind is cyclonic out to large radii. The relative vorticity has large cyclonic values in the inner-core of the vortex and much weaker anticyclonic values at larger radii (greater than about 200 km for the calculations described in sections 3 and 4). If we define a PV anomaly relative to the PV of the undisturbed environment there is a strong positive PV anomaly in the inner core and a weak negative PV anomaly at larger radii. The calculations described in sections 3 and 4 use a model domain of 3840 km in the zonal direction, x, by 2880 km in the meridional direction, y, by 12 km in the vertical direction, z. The horizontal resolution is 10 km and there are 14 levels in the vertical with a horizontal resolution of 857.14 m. The time step is 30 s. The model has sixthorder horizontal and second-order vertical diffusion for the total wind and temperature fields. An additional second-order diffusion is added to the divergent part of the flow, as discussed in Part I. The constant diffusion coefficients are 7 x 1019 m6 s-l for the sixth-order horizontal diffusion and 1 m2 s-l for the second-order vertical diffusion of the total-wind and temperature fields. The constant coefficients for the diffusion of the divergent part of the flow are 5 x lo4 m2 s-' for the horizontal second-order diffusion and 50 m2 s-l for the vertical diffusion. 3. THE ROLE OF LARGE-SCALE ASYMMETRIES IN THE MOTION AND THE VERTICAL TILT This section compares two calculations which differ only in the magnitude of the vertical shear. The initial vortex profile is that referred to as the standard profile in Part I. This profile has a maximum tangential wind of 40 m s-l, a radius of maximum wind of 100 km and a radius of gale-force winds (15 m s-') of 300 km. Both calculations are at 20"N with N2 = 1.5 x s-*. The calculation with stronger shear (S) has U, = 6 x lop4 s-l. In the calculation with weaker shear (W) U, = 4 x lop4 s-l. Both

VORTEX EVOLUTION IN VERTICAL SHEAR. I1 3141 E * 100. -100. s -200. O. 0. I I 200. 400. 600. 800. X(km) h E v * 200. l ' ' ' I ' ' ' I ~ ' ' I ' ~ ' I ' ' - 0)) 7 100. - - o.:o iw] \% -100.- - - -200. - - I I I I I I I I I I I I I I I I I I I - Figure I. Vortex tracks. Symbols show the location of the potential-vorticity centre at selected model levels every 12 hours for 72 hours. The initial vortex location is at (x, y) = (0, 0). (a) Stronger shear (S (see text): U, = 6 x lop4 s-i). (b) Weaker shear (W (see text): rl, = 4 x s-i). calculations use Uo = 0. Figure 1 shows the vortex tracks in the two calculations. The vortex centre is defined as the location of the PV maximum, hereafter the PV centre. During the first 12 hours both tracks are similar, with the vortex initially tilting downshear (not shown) and the upper- and lower-level centres beginning to rotate cyclonically about the mid-level centre. At 12 h the vortex tilt for S (Fig. l(a)) is slightly larger than for W (Fig. I (b)) as would be expected. After 12 h the vortex tracks in the two calculations differ markedly. In the S calculation the rotation rate decreases with time and the vortex achieves an almost constant direction of vertical tilt. The magnitude of the tilt continues to increase with time. Note that after 24 h the vortex in the S calculation has an upshear tilt. Thus, the increase in the magnitude of the tilt with time cannot occur due to advection by the environmental flow. In the W case (Fig. l(b)) the rotation rate remains nearly constant with time. The magnitude of the tilt increases until about 42 h then decreases slightly. Note that the vortex tilt continues to increase between 12 and 48 h although the vortex has an upshear tilt. In both calculations the overall vortex motion is in the direction of the mid-level flow. The distance moved by the mid-level centre in 72 h is slightly larger than would be expected if the vortex simply moved with

3142 S. C. JONES the mid-level environmental flow. In the S calculation there is a significant southward motion of the mid-level centre, which is not seen in W. The differences between the two vortex tracks described above cannot be explained by considering only the mechanisms discussed in Part I, namely the mutual rotation of upper- and lower-level PV anomalies about their mean position along with advection by the environmental flow. If the upper- and lower-level PV anomalies rotate about a point between their centres without changing shape, the component of flow across the centre of each anomaly, due to the vertical projection of the other anomaly, will be perpendicular to the line joining the centres, and thus cannot give an increase in the magnitude of the vertical tilt. In addition, the increasing upshear tilt seen in Fig. l(a) cannot result from advection by the environmental flow. Thus, we must look for other factors which contribute to the vortex motion. A further contributor to the motion might be the flow associated with large-scale PV asymmetries. Potential-vorticity asymmetries which influence the vortex motion may be several orders of magnitude smaller than the PV of the vortex itself and are, therefore, not easily observed in the total PV field. However, they may be as imporlant as the PV of the vortex as they are coherent over a larger area and can influence the vortex motion significantly. One method of isolating the asymmetries is to subtract the PV associated with the axisymmetric part of the vortex from the total PV. This method was used in the study described in Part I. However, the PV in the inner core of the vortex is distorted as a result of advection by the divergent circulation such that there are strong small-scale asymmetries in the inner core of the vortex. The flow associated with these asymmetries leads to a small-scale oscillation of the PV centre about the average track. The scale on which the flow occurs is too small to advect the vortex as a whole (see Part I, section 4). The strong, small-scale asymmetries mask any weaker large-scale asymmetries which may be present. Our use of the above method to identify PV asymmetries prevented us from discovering the large-scale asymmetries in Part I. Another method of identifying the weaker large-scale PV asymmetries is to contour the PV field for values less than a specified threshold value. This method eliminates the high values of PV associated with the symmetric vortex and the strong asymmetries in the inner core. Figure 2 shows the PV at the lowest and highest model levels for the S case. Potential-vorticity values less than 0.22 PV units (1 P.V.U. = 1 x lop6 K kg-' m2 s-') are contoured. Note that the constant density-weighted environmental PV far from the vortex for this case is 0.23 P.V.U. and the maximum density-weighted PV of the initial vortex is 5.8 P.V.U. The PV centre at the lowest model level is marked by a tropicalcyclone symbol and the centre at the highest model level by a star. At 12 h at the lowest model level (Fig. 2(a)) the outer PV contours are symmetric, but an asymmetry has developed at a radius of about 300 km. The asymmetry develops in the outer region of anomalously negative PV and is located to the west and south-west of the vortex centre. A similar asymmetry forms at the highest model level (Fig. 2(b)) but is located on the eastern side of the vortex centre. At 24 h (Figs. 2(c) and (d)) the asymmetry has increased in size and the location of minimum PV has moved out to a larger radius. The PV is now very asymmetric with a strong wave-number-one component. The asymmetries are still located on the opposite side of the vortex centre at lower and upper levels. Closer to the vortex centre it appears that axisymmetrization is occumng (Melander et al. 1987) as a result of advection of the asymmetric PV anomaly by the cyclonic vortex circulation. This is a process which tends to reduce asymmetries in the inner core of tropicalcyclone-like vortices (Shapiro and Ooyama 1990; Smith et al. 1990). The large-scale asymmetries continue to develop over the next 48 hours. By 72 h (Figs. 2(e) and (0) the scale of the asymmetries has increased further and the vortex core has become more

VORTEX EVOLUTION IN VERTICAL SHEAR. I1 3143 TIME(hrs)= 12. Z(m)=426.67 TIyE(h)=12. Z(km)= 11.67 l ~ l l ~ l l 800. 3 v 0. * -800. - W(hra)=24. I I I I ' I (c) Z(m)=426.67 - X(kd 600. 1 Y 0. * -800. X ( W I M(hra)=72. Z(km)=11.57 800. 3 v 0. * -000. - Figure 2. Potential vorticity for S (see text) contoured for values less than 0.22 potential vorticity (PV) units (P.v.u.). Contour interval is 0.02 P.V.U. (a) Lowest model level at 12 h, (b) highest model level at 12 h, (c) lowest model level at 24 h, (d) highest model level at 24 h, (e) lowest model level at 72 h, (f) highest model level at 72 h. The PV centre at the lowest model level is marked by the cyclone symbol and that at the highest model level by the star. X(W symmetric at the surface. At upper levels the asymmetry has moved further away from the vortex inner-core than at the surface and has a somewhat different shape. The change with height of the orientation of the asymmetries between the surface and upper levels occurs gradually. For example, at 72 h the asymmetry retains approximately the same shape and orientation with height between 0 and 4 km but moves inwards to smaller radii. Above about 4 km the orientation of the asymmetry rotates in an anticlockwise direction. At 8 km the asymmetry has approximately the same orientation

3 144 S. C. JONES 600. 1-0. * -600. -~l,,,,, I, X(km) 4 TME(hm)-12. Z(km)=l1.67 I ( I l I I I - (bi 1 - X ( W!lTKE(hrs)=72. Z(m)=428.67 I ~ I I ~ I I ~ 600. 3 Y 0. * -600. XfW Figure 3. As Fig. 2 for W (see text). as at 12 km, but the minimum PV occurs at a smaller radius. Between 8 and 12 km the location of minimum PV moves gradually outwards. The development of the large-scale PV asymmetries in W is illustrated in Fig. 3. The asymmetries develop in the same manner as in S. At 12 h (Figs. 3(a) and (b)) and 24 h (Figs. 3(c) and (d)) the asymmetries are markedly similar to those seen in Fig. 2. By 72 h the shape and orientation of the asymmetries differs between the W (Figs. 3(e) and (f)) and S (Figs. 2(e) and (f)) calculations. Near the surface the asymmetry in the W calculation is rotated cyclonically when compared with the S case. At upper levels the same change in orientation is seen and the asymmetry remains closer to the vortex centre than in the S calculation.

VORTEX EVOLUTION IN VERTICAL SHEAR. 11 3145 TIME(hrs)=24... Z(m)-428.57.. l ~ l l ~ l l /- - a TIYE(hra)=24. Z(bn)=l1.67 l l [ l ' - X(k) TIy6(hrs)=72. Z(m)=428.57 ' - - ' - : * o o " 000..... 000. -O - 1 v 0. * x(km) Figure 4. Relative vorticity for S (see text). The contours are chosen to be equivalent to the contours of potential vorticity (PV) in Fig. 2 assuming N2 = 1.5 x lop4 sp2, where N is the Brunt-Vaisala frequency. Dashed contours indicate negative relative vorticity. The PV centre at the lowest model level is marked by the cyclone symbol and that at the highest model level by the star. (a) Lowest model level at 24 h. Maximum value -1.0468 x 10 s-', contour interval 4.8833 x s-'. (b) Highest model level at 24 h. Maximum value -1.7866 x Inp5 s-', contour interval 4.6679 x lo-' s-'. (c) Lowest model level at 72 h. Maximum value - 1.0468 x 10@ s-', contour interval 4.8833 x lot5 s-i. (d) Highest model level at 72 h. Maximum value -1.7866 x contour interval 4.6679 x lop5 s-i. SKI, We would expect the flow field associated with the negative PV anomalies described above to be anticyclonic. In order to obtain quantitative information about the flow field associated with the PV asymmetries seen in Figs. 2 and 3 it would be necessary to perform a piece-wise PV inversion using a balance assumption, such as that based on the nonlinear balance equations (Charney 1955) or on the asymmetric balance theory (Shapiro and Montgomery 1993). Piece-wise PV inversion for tropical cyclones has been performed using these assumptions (Wu and Emanuel 1995a,b; Shapiro 1996; Wu and Kurihara 1996; Moller and Jones 1998; Shapiro and Franklin 1999). Using such techniques for the calculations shown here would be a major undertaking and is beyond the scope of this paper. We can obtain some indication of the flow field associated with the PV asymmetries, however, by considering the relative-vorticity field. Figure 4 shows the relative-vorticity field for the stronger shear case at 24 and 72 h. The maximum value contoured and the contour interval are equivalent to the values of PV contoured in Fig. 2, assuming N2 = 1.5 x s-~. A comparison of these fields with the PV plots (Figs. 2(c)-(f)) shows that the asymmetries of PV and relative vorticity have very similar structures. This suggests that the PV asymmetries are dominated by the contribution from the relative vorticity and the contribution from potential-temperature variations is small, as is typically the case for large-scale motion in the tropics. We can

3146 S. C. JONES get some indication of the flow field associated with the large-scale PV asymmetries by calculating the stream function, @, and the horizontal wind field, u = (u, v), from the relative vorticity, {, using { = v2@ and u = (-@)), @x). The flow field calculated from the relative-vorticity asymmetries for the two calculations is shown in Figs. 5 and 6. Only values of relative vorticity less than the maximum values contoured in Fig. 4 are used so that the relative vorticity of the vortex innercore is not included in the calculation. Figure 5 shows the anticyclonic gyres associated with the relative-vorticity asymmetries at 24, 48 and 72 h for the S case. The centre of the surface gyre is located roughly south-west of the vortex centre. At upper levels the centre of the anticyclonic gyre is almost due east of the upper-level vortex centre. The separation between the centre of the gyre and the vortex centre increases with time both at the surface and at upper levels. The flow across the vortex centre due to the anticyclonic gyres is of the order of a few metres per second. The effect of the asymmetries on the vortex tilt can be seen by considering the flow across the vortex centre at the surface and at upper levels due to the asymmetries. At 24 h the flow at the surface advects the surface vortex south-eastwards (Fig. 5(a)) while the flow at upper levels advects the upper-level vortex north-westwards (Fig. S(b)). As the vortex tilt is approximately south-east-north-west at this time, most of the flow due to the asymmetries is in the plane of the tilt and acts to increase the magnitude of the tilt without changing the direction of tilt substantially. During the following 48 hours (Figs. S(c)-(f)) neither the direction of vortex tilt nor the orientation of the anticyclonic gyres changes significantly. The strength of the flow across the vortex centre decreases over this time period as the centre of the anticyclonic gyre moves away from the vortex inner core. We hypothesize that the constant upshear tilt occurs because of advection by a combination of the environmental flow, the vertically penetrating flow associated with the tilted PV anomaly, and the flow associated with the large-scale anticyclonic gyres. Each of these flows can be resolved into two components, one in the plane of the vortex tilt, which tends to increase the magnitude of the tilt, and one perpendicular to the plane of the tilt, which acts to change the direction of tilt. The flow across the vortex centre associated with the anticyclonic gyres is predominantly in the plane of the tilt. The direction of vertical tilt does not change significantly with time. Thus, the component of the environmental flow perpendicular to the plane of the vortex tilt must approximately balance the flow due to the vertical penetration of the tilted vortex. Figure 6 shows the anticyclonic gyres for the W case. At 24 h (Figs. 6(a) and (b)) the gyres are similar in shape and strength to those that occur in the stronger shear calculation (Figs. 5(a) and (b)). The orientation of the tilted vortex with respect to the gyres is slightly different, but the flow due to the anticyclonic gyres is still predominantly in the plane of the vortex tilt. The parameters which enter the penetration depth given in Eq. (1) are the same for the S and W calculations. Thus, we can assume that the strength of the flow associated with the vertical penetration of the tilted vortex is similar at 24 h in both cases. In the S case, this flow is balanced approximately by the environmental shear flow and the direction of tilt remains constant with time. In the W case, the environmental flow is weaker so that the cyclonic rotation of the upper- and lower-level vortex centres about the mid-level centre continues. The component of the flow due to the anticyclonic gyres tends to increase the magnitude of the vertical tilt which could explain why the vertical tilt does not decrease between 12 and 48 h despite the vortex having an upshear tilt. At 48 h (Figs. 6(c) and (d)) the anticyclonic gyres still resemble those observed in the S case (Figs. S(c) and (d)). However, the direction of the vortex tilt is north-south for the W case so that the flow due to the anticyclonic gyres is almost perpendicular to the direction of vertical tilt. This flow component acts in the opposite

VORTEX EVOLUTION IN VERTICAL SHEAR. 11 3147 800. 400. 0. -400. -800. 800. 400. 0. -400. 800. "'... (e)~: ' ' ' _''. 1. 'I. ' ' ' "'... 400. A 0. E zc v *-400. -800. - Figure 5. Wind vectors calculated from the relative-vorticity asymmetry for S (see text). (a) Lowest model lcvel at 24 h, maximum vector is 7.25 m s-i; (b) highest model level at 24 h, maximum vector is 7.08 m s-'; (c) lowest model level at 48 h, maximum vector is 7.5 m s-l; (d) highest model level at 48 h, maximum vector is 6.5 m s-' ; (e) lowest model level at 72 h, maximum vector is 7.48 m s-l; (0 highest model level at 72 h, maximum vector is 6.06 m s-'. The potential-vorticity centre at the lowest model level is marked by the cyclone symbol and that at the highest model level by the star.

3148 S. C. JONES 400. - 0. E x v *-400. -800. - -800. 0. 800. Figure 6. As Fig. 5 for W (see text). (a) Maximum vector is 6.94 m s-', (b) maximum vector is 6.85 m s-', (c) maximum vector is 6.99 m s-l, (d) maximum vector is 6.79 m s-i, (e) maximum vector is 7.11 m sp1, (f) maximum vector is 6.61 m sp1.

VORTEX EVOLUTION IN VERTICAL SHEAR. I1 3149 direction to the environmental shear flow and tends to oppose the cyclonic rotation of the upper and lower vortex centres about their mean position. At 72 h the location and orientation of the anticyclonic gyres in the W case (Figs. 6(e) and (f)) differs noticeably from the S case (Figs. 5(e) and (0). In particular, the centre of the surface gyre is almost due south of the surface centre and that of the upper-level gyre is located to the north of the upper-level centre in the W case. In the S case these locations are approximately south-west and east of the respective centres. The vortex tilt in Figs. 6(e) and (f) is close to east-west so that the flow associated with the anticyclonic gyres tends to reduce the vortex tilt and opposes the cyclonic rotation due to the vertical penetration effect. Note that the much smaller horizontal separation of the upper- and lower-level PV anomalies at 72 h for the W case means that the vertical penetration effect will be stronger than in the S case, where the horizontal separation is of the order of the horizontal scale of the positive PV anomaly. The above discussion implies that both the magnitude and direction of the vertical tilt is influenced by the presence of the large-scale anticyclonic gyres. The manner in which this occurs depends not only on the location, shape and strength of these gyres, but also on the orientation of the vortex tilt relative to the gyres. Although the anticyclonic gyres appear rather similar in both calculations, the orientation of the tilted vortex within the gyres differs. Thus, the evolutions of the tilted vortices in the S and W calculations are significantly different. 4. A MECHANISM FOR THE DEVELOPMENT OF THE ASYMMETRIES This section discusses the mechanism responsible for the development of largescale PV asymmetries such as those seen in Figs. 2 and 3. As discussed above, the vertically penetrating flow associated with the PV of the tilted vortex core has strong horizontal shear. We hypothesize that the mechanism responsible for the development of the large-scale asymmetries is the differential advection of the PV at a given level by this horizontally sheared flow. Studies of barotropic vortex motion have shown that an initially symmetric vortex in a horizontally sheared flow develops relative-vorticity asymmetries (e.g. Ulrich and Smith 1991). However, the question remains as to whether advection by the flow associated with the tilted PV anomaly can result in the PV asymmetries seen in Figs. 2 and 3. This question can be addressed using a barotropic model as follows. The barotropic model is initialized with the symmetric vortex used in the three-dimensional calculations. A source term is introduced into the non-divergent barotropic vorticity equation to represent the advection of the symmetric vortex in the barotropic model by the flow associated with the vertical penetration of an identical symmetric vortex at a different level and with a different centre. The barotropic vorticity equation is then given by: Here up is the horizontal wind field of the initial symmetric vortex, but with its amplitude multiplied by a factor ap and its centre located a distance (Xd, Yd) from the location of maximum vorticity in the barotropic model. The factor ap represents the reduction in strength of the vortex due to the vertical penetration. The magnitude of ap depends on the Rossby penetration depth. This method has been used to study the evolution of the polar stratospheric vortex (Juckes and McIntyre 1987; Juckes and O Neill 1988). The development of the largescale asymmetries on the tropical-cyclone-like vortex described in this paper resembles

3150 S. C. JONES the evolution of the polar vortex in their calculations. In both cases the forcing on the right-hand side of Eq. (3) is a dipole. However, the horizontal scale and temporal evolution of the forcing differs, as our forcing represents the vortex tilt and the forcing of the polar vortex models the effects of planetary waves which propagate from the troposphere into the stratosphere. Calculations using the barotropic model as described above are carried out using a domain of 3000 km by 3000 km with a resolution of 10 km. The vortex centre is initially at the centre of the domain. The wind field, up, is updated at hourly intervals. The values of (Xd, Yd) are taken from the baroclinic calculations described in the previous section, whereby (Xd, Yd) is the distance between the location of maximum PV at the highest model level and that at the lowest model level. Thus, the calculation represents the advection of the surface vortex by the flow associated with the downward penetration of the upper-level vortex. We do not expect to reproduce the asymmetries in the baroclinic model exactly in this calculation, since all the levels above the surface must contribute to balanced flow at the surface. Rather, we wish to illustrate that the sort of asymmetries seen in the baroclinic model can be produced by the mechanism in question. A crude estimate of the magnitude of ap can be obtained as follows. We assume that during the first few hours of the calculation the asymmetries are small in magnitude and that the meridional vortex motion is entirely due to the vertically penetrating flow associated with the tilted vortex. (Remember that the environmental flow is purely zonal.) We calculate the meridional speed of motion of the vortex centre at the surface and upper levels from the position of the vortex centre at hourly intervals for the first 6 hours. We then calculate the meridional flow which occurs across the vortex centre if it is placed in the symmetric vortex at the location implied by the relative positions of the upper- and lower-level vortex centres in the model calculation at the given time. The ratio of the speed of motion to the flow across the vortex centre gives us an estimate for up. The average value of ap obtained as described above is 0.06. We do not expect this value to be a very accurate one, due to the errors involved in its estimation, especially as the distance moved by the vortex centres in one hour is only of the order of the model grid length. However, it gives us an idea of the range of values of ap which should be investigated. Calculations are performed with values of ap ranging from 0.02 to 0.6. For values of alp of 0.4 or greater the symmetric vortex becomes strongly distorted. Figure 7 shows the relative vorticity from a calculation with ap = 0.2. The vortex centres from the S case are used to define (Xd, Yd). The contours used are the same as in Fig. 4(a). After 12 hours of the barotropic calculation an anticyclonic asymmetry is beginning to form to the south-west and west of the vortex centre (Fig. 7(a)). This is the location where the PV asymmetry is seen in Fig. 2(a). At 24 h the asymmetry is much stronger (Fig. 7(b)). A comparison of Fig. 7(b) with Fig. 4(a) shows a striking similarity between the relative vorticity in the baroclinic calculation and that from the barotropic model. This supports the hypothesis that the large-scale asymmetries arise due to the distortion of the initially symmetric PV anomaly by the horizontally sheared flow associated with the vertical penetration of the tilted vortex. The sensitivity of the asymmetries in the barotropic model to the value of ap is illustrated in Fig. 8. A calculation with ap = 0.1 is shown in Fig. 8(a) for 24 h. The asymmetry in this case is weaker than for ap = 0.2 (Fig. 7(b)). For up = 0.02 no asymmetry can be seen at 24 h (Fig. 8(b)). Thus, the strength of the asymmetries depends strongly on the value of ap and hence on the Rossby penetration depth. The extent to which the asymmetries influence the vortex motion will depend on the relative magnitudes at the vortex centre of up, and the flow associated with the asymmetries.

VORTEX EVOLUTION IN VERTICAL SHEAR. 11 3151 ' t -1500.L'''" " I -1500. -900. -300. " 300. I, 900. ' ' 1500 Figure 7. Relative vorticity from the barotropic model. Contours as in Fig. 4(a). (a) ap = 0.2 at 12 h, (b) ap = 0.2 at 24 h. (See text for further details.) Figure 8. As Fig. 7. (a) ap = 0.1 at 24 h, (b) ap = 0.02 at 24 h. Smith et al. (2000) showed that the evolution of quasi-geostrophic vortices in vertical shear can be well represented by an analogue model which assumes asymmetries, such as those shown here, are negligible. Our calculations suggest that their assumption is justified for cases where the penetration depth is sufficiently small. The quasigeostrophic penetration depth is a constant given by Lf/N. For the primitive-equation calculations the penetration depth given by Eq. (1) varies with radius. Both the modified Coriolis parameter (fioc) and the vortex relative vorticity are much larger in the inner core than in the environment. Thus, for a given L the penetration depth will be larger in the inner core than the quasi-geostrophic equivalent. The definition of L is a factor which further complicates the calculation of the penetration depth as L might also be a function of radius.

3152 S. C. JONES Radius (km) Figure 9. Radial profile of relative vorticity s-i) versus radius (km) for the standard profile (dashed) and the NP1 profile (see text, solid). The insert shows the anticyclonic part of the profile in greater detail. 5. THE INFLUENCE OF VORTEX STRUCTURE ON THE ASYMMETRIES The large-scale asymmetries form in the outer region of the vortex. Since we hypothesize that the asymmetries form through advection of the PV of the initial vortex, the structure of the asymmetries should be sensitive to the details of the initial vortex profile at larger radii. We present here two calculations which investigate the influence of the initial vortex structure by using a different vortex profile: namely, that given by Eq. (2) in Part I. In the first calculation we change the structure of the outer part of the vortex whilst keeping the vortex structure in the inner core as similar as possible to that in the previous calculation, so as not to change the Rossby penetration depth significantly. Thus, we use the same maximum tangential wind, radius of maximum wind and radius of gale-force winds as for the previous calculations. In the second calculation, we use an initial profile in which the tangential wind decreases much more slowly with radius than in the previous calculations. In this case the positive PV anomaly extends to larger radius and the negative PV anomaly is much weaker. The parameters used in the first calculation with the new profile (NPl) are given in Table 1. All other parameters used are identical to those in the S calculation. The relative vorticity of the two different profiles is shown in Fig. 9. At first sight the profiles appear very similar, although the maximum relative vorticity in the new profile is approximately 15% larger than in the standard profile. However, careful inspection of the region of anticyclonic relative vorticity (insert in Fig. 9) shows that the structures of the two vortex profiles in this region differ. The minimum value is approximately 23% lower for the new profile and the magnitude of the relative-vorticity gradient larger for radii greater than about 250 km. Figure 10 shows the vortex track for the NP1 calculation. The vortex evolution is similar to that in the S calculation (Fig. l(a)). However, after 72 hours the surface centre is 110 km further south and the upper-level centre is 350 km further north and 25 km further east in the NP calculation. The upper- and lower-level PV asymmetries at 24 h

VORTEX EVOLUTION IN VERTICAL SHEAR. I1 3153 5 z=o.429 km 0 z=6 km Q z=11.57 km 400. - 200. E 4: v * 0. s -200. Figure 10. As Fig. 1 for the NP1 calculation (see text). TIME(hrs)=24. Z(m)=428.57 TM3(hrs)=24. Z(km)=11.57 I I I I I I I I 800. 1 Y 0. * -800. - -800. I- Figure I I. Potential vorticity for NPI, contoured as in Fig. 2. (a) Lowest model level at 24 h, (b) highest model level at 24 h. are shown in Fig. 11 for the NP1 calculation. The asymmetries form in the region of anomalously negative PV, have a similar structure to those in the S case (Figs. 2(c) and (d)), but are somewhat stronger. This would imply that the flow associated with the large-scale asymmetries should be stronger for the NP1 case. The orientation of the asymmetries is such that the flow across the vortex centre will be predominantly northerly at the surface and southerly at lower levels. Thus, a stronger flow associated with the asymmetries could account for the larger meridional vortex tilt seen in the NP1 calculation. There are two factors which make it difficult to demonstrate quantitatively that the different vortex structure in the anticyclonic region is responsible for the different vortex evolution in the S and NPl calculations. Firstly, the difference between the meridional displacement of the vortex centre after 24 hours in the two calculations is

3154 S. C. JONES 25 km at the surface and 60 km at upper levels. This corresponds to a difference in the meridional flow across the vortex centre of only 0.3 m s-l at the surface and 0.7 m s-l at upper levels over 24 hours. Secondly, once the vortex tilt differs in the two calculations the other contributions to the vortex motion will differ also. For example, at 24 h the meridional vortex tilt is approximately 190 km in the S case and 270 km for the NP1 case. For magnitudes of the vortex tilt greater than the radius of maximum wind, a larger vortex tilt should correspond to weaker flow across the vortex centre due to the vertical projection of the tilted PV anomaly. Thus, the vertical penetration effect should be weaker in the NP1 case. This would explain why the mutual rotation of the upper- and lower-level centres is slightly slower for the NP1 calculation, as can be seen by comparing the direction of tilt in Figs. l(a) and 10. The direction of the vortex tilt determines whether the environmental vertical shear acts to change the direction or the magnitude of the tilt. In the S calculation the environmental shear flow will affect both the magnitude and direction of the vortex tilt. In the NPl calculation the environmental flow will have little effect on the magnitude of the tilt. This could lead to further differences between the vortex evolution in the two calculations. The NP1 calculation shows also that it is difficult to isolate any one contribution to the vortex evolution, even in a very idealized calculation. We have compared two calculations with very similar vortex profiles and observed rather similar vortex evolutions. However, had we made more substantial changes to the vortex profile we would have changed not only the structure of the outer anticyclonic region, but also the strength and horizontal scale of the inner PV anomaly. We showed in Part I that such changes have a strong influence on the vortex evolution. Nevertheless, the profile we use may not be representative of all tropical cyclones, so that it is instructive to study the evolution for a vortex profile which differs substantially from that in the previous cases. In calculation NP2 we use a profile in which the tangential wind decreases much more slowly with radius than in NPl. Thus, the PV anomaly is significantly broader than in the previous calculations so that, for the same vortex strength, the Rossby penetration depth is larger, and the vortex less susceptible to vertical shear. In order to still observe the effect of vertical shear on the vortex tilt we use a weaker vortex with maximum tangential wind of 30 m s-'. The parameters used are given in Table 1. The vortex extends out to larger radii which necessitates using a much larger domain than in the previous calculations. Due to the memory restrictions of the computer available to us, the calculation has to be carried out with lower horizontal and vertical resolution. We use a model domain of 13 000 km in x by 1 1 000 km in y by 12 km in z. The horizontal resolution is 20 km and the vertical resolution 3 km. The constant diffusion coefficients for the sixth-order horizontal diffusion are 4.48 x 10l2 m6 s-l, so that the damping time for the horizontal 2-grid wave is not changed. All other diffusion coefficients are as in calculation S. A horizontal resolution of 20 km is sufficient to capture the evolution of the large-scale asymmetries. The sensitivity to the vertical resolution was investigated for the S calculation and was found to be significant. In particular, a two-level calculation did not capture the evolution of the vortex tilt seen in Fig. l(a). In a calculation with the vertical resolution used in NP2 the evolution was qualitatively similar to that seen in Fig. l(a) but there were quantitative differences. The reason for the sensitivity to vertical resolution will be reported on in a forthcoming publication. The relative vorticity of the vortex profile used in NP2 is shown in Fig. 12. A comparison of this profile with the standard profile (dashed line in Fig. 9) shows that the maximum anticyclonic vorticity occurs at larger radii and is more than an order of magnitude weaker for profile NP2. The radius at which the relative vorticity becomes

VORTEX EVOLUTION IN VERTICAL SHEAR. I1 3155 0. 0. 500. 1000. 1500. Radius (km) Figure 12. Radial profile of relative vorticity (lot4 s-') versus radius (km) for the NP2 profile (see text). The insert shows the anticyclonic part of the profile in greater detail. Note that the scales on both axes are different from Fig. 9. anticyclonic is 395 km for NP2 compared with 180 km for the standard profile. The vortex track for calculation NP2 is shown in Fig. 13. (Note that the star and cyclone symbols are at different heights in Figs. 1 and 13 due to the different vertical grid spacing.) The vertical shear and Brunt-Vaisala frequency are the same as in calculation S. During the first 12 hours the evolution is very similar for both cases, with the upperand lower-level centres rotating cyclonically about their mid point. After 12 hours the vortex acquires an upshear tilt in both cases, but the magnitude of the vertical tilt remains much smaller for the NP2 case, and indeed after 36 hours the vertical tilt decreases with time. After 48 hours the upper- and lower-level centres rotate anticyclonically about their mid point. The large-scale PV asymmetries for the NP2 calculation at 24 hours are shown in Fig. 14. The PV of the undisturbed environment has been subtracted from this figure. As in the previous cases, large-scale PV asymmetries form in the outer region of the vortex and on opposite sides of the vortex centre at lower-levels (Fig. 14(a)) and upper-levels (Fig. 14(b)). In contrast to the previous cases, the asymmetries form in the region where the vortex PV is higher than the environmental PV. Thus, we would expect the flow field associated with the PV asymmetries to consist of cyclonic gyres to the south-east of the vortex centre at lower levels and to the north-west of the vortex centre at upper levels. The effect of these cyclonic gyres at 24 hours would be an anticyclonic rotation of the lower- and upper-level centres about their mid point. Thus, the large-scale PV asymmetries oppose the cyclonic rotation due to the vertical penetration of the tilted vortex, leading to a smaller upshear tilt in the NP2 case than in the S case. We suggest that the different evolutions of vortex tilt for the S and NP2 calculations are a result of the formation of large-scale PV asymmetries which are accompanied by anticyclonic flow in the S case and cyclonic flow in the NP2 case.

3156 S. C. JONES Figure 13. As Fig. 1 for the NP2 calculation (see text). 900. - I I I + 900. (a) - 600. - - 600. - ' I I - (b) - - - - e -,-. 300. - 300. E x v * 0. - m *. - 0.- -300. - e - -300.,,' I 6 - - e - - -600.- I I * -600. C s,..0, ---..a, - I I 4, Figure 14. Potential vorticity (PV) anomuly for NP2 (see text). (a) Lowest model level at 24 h contoured for values less than 0.22 PV units (P.v.u.). Contour interval is 0.02 P.V.U. (b) Highest model level at 24 h contoured for values less than 0.55 p.v.u. Contour interval is 0.05 P.V.U. (These values correspond to the same density-weighted PV as in (a)). In this section we have shown that the evolution of a tropical-cyclone-like vortex in vertical shear is sensitive to the structure of the vortex profile away from the innercore. Sensitivity to the structure of the outer winds has also been found in studies of barotropic vortex motion (e.g. Fiorino and Elsberry 1989; Smith et al. 1990). 6. SUMMARY AND DISCUSSION We have identified an additional mechanism which influences the motion and vertical tilt of a tropical-cyclone-like vortex in vertical shear. This mechanism is complementary to those described in Part I. We used idealized calculations with initially barotropic vortices in vertical shear to show that large-scale wave-number-one PV asymmetries

VORTEX EVOLUTION IN VERTICAL SHEAR. 11 3157 develop in the outer region of the vortex. The orientation of the asymmetries relative to the vortex centre varies with height. The structure of the asymmetries depends crucially on the vortex profile used. In three of the calculations presented (W, S, NPl) the asymmetries form in the region of the vortex where the PV of the initial vortex is anomalously negative. The flow field associated with the asymmetries consists of broad anticyclonic gyres. Calculations S and NPl were identical in all respects except for the initial symmetric vortex used. The differences between the two calculations can be attributed to the different vortex structures in the region with anticyclonic relative vorticity. In calculation NP2 a vortex profile was used in which the tangential wind decreases much more slowly with radius. In this case the asymmetries form in the region of anomalously positive PV and have an associated flow field consisting of broad cyclonic gyres. We hypothesize that the large-scale asymmetries form after the vortex develops a vertical tilt in response to the environmental shear. They arise due to differential advection of the vortex PV by the horizontally sheared flow associated with the vertical projection of the tilted PV anomaly. The hypothesis is supported by barotropic model calculations which show that the asymmetries seen in the baroclinic calculations could form as a result of the above mechanism. Thus, the structure and strength of the asymmetries depends on the Rossby penetration depth as well as on the tangential-wind profile of the tropical-cyclone vortex. The influence of the large-scale PV asymmetries on the vortex evolution depends not only on the location and structure of the asymmetries, but also on the orientation of the vortex tilt relative to the asymmetries. This determines whether the flow associated with the asymmetries acts to change the magnitude or the direction of the vortex tilt. In the calculations with stronger and weaker shear described in section 3 the large-scale asymmetries have a similar structure during the first 48 hours, but the direction of the vortex tilt is different, and the contribution of the asymmetries to the vortex motion differs between the two cases. Further work is needed to investigate the influence of such large-scale asymmetries in more complicated models and in the atmosphere. Another possible mechanism for the development of the asymmetries is barotropic instability. The relative vorticity of the symmetric vortex is maximum at the vortex centre, decreases to a minimum at a radius greater than the radius of maximum wind and then gradually increases (see Figs. 9 and 12). Thus, there is a change of sign with radius of the absolute-vorticity gradient and the necessary condition for barotropic instability is fulfilled. In the W, S and NPl cases the large-scale asymmetries form in the region where the absolute-vorticity gradient changes sign. Weber and Smith (I 993) investigated the stability of the standard vortex and found no unstable barotropic modes. The stability of the other two vortices used here has not been investigated. Gent and McWilliams (1986) investigated the stability of quasi-geostrophic vortices with a similar profile to those used here, but with anticyclonic relative vorticity at the centre surrounded by a ring of cyclonic relative vorticity. They found an unstable internal mode with azimuthal wave-number one. If such an internal instability exists for the vortices used here it may influence the evolution of the asymmetries, particularly during the early stages of the calculation. Gent and McWilliams (I 986) present a nonlinear calculation using a twolevel quasi-geostrophic model in which their vortex is perturbed with the most unstable internal mode with azimuthal wave-number one. In the early stages of their calculation exponential growth is observed at the linear growth rate. During the phase of linear growth the vortex centres in each level move apart. This motion is consistent with the presence of a wave-number-one normal mode. In the nonlinear phase the growth rate decreases and the vortex centres move further apart. The lower- and upper-level centres of the anticyclonic vortices rotate anticyclonically about their mid point. The rotation

3158 S. C. JONES rate decreases with time. The relative vorticity at the end of the nonlinear calculation exhibits structures which resemble the asymmetries presented in this paper. A possible explanation for the finite-amplitude behaviour observed by Gent and McWilliams is that the vortex tilt develops initially as a result of the linear instability. Once the vortex is tilted the vertical penetration mechanism described here and in Part I leads to the mutual anticyclonic rotation of the vortex centres and influences the evolution of the asymmetries. The sensitivity of the large-scale asymmetries to the initial vortex profile has implications for the bogussing of tropical cyclones in numerical forecast models. The sensitivity to the outer wind structure shows that care must be taken in choosing the form of the profile used to create a bogus. The presence of the large-scale asymmetries suggests that bogusses which include only a symmetric vortex will not be adequate in a vertically sheared environment. The evolution of an adiabatic tropical-cyclone-like vortex with purely cyclonic flow in vertical shear depends largely on three processes. The first process is differential advection by the environmental flow. The second process, discussed in detail in Part I, is the vertical projection of the tilted PV anomaly which results in a cyclonic rotation of the upper and lower parts of the vortex around the mid-level centre. The third process is the formation of large-scale PV anomalies discussed here. The relative importance of each process will depend on the strength of the shear, the Rossby penetration depth, and the tangential-wind profile of the tropical cyclone. All three processes affect the development of the vertical tilt and thus influence both the motion and the structure of a tropical cyclone. As the structural changes of a tilted vortex involve changes in the vertical stability and the vertical circulation, these processes may affect tropical-cyclone intensity. Further work is needed to elucidate the role of vortex tilting in tropical-cyclone intensity change. ACKNOWLEDGEMENTS I gratefully acknowledge discussions with Martin Juckes, Dominique Moller, Lloyd Shapiro and Roger Smith and thank them for their comments on an earlier version of this paper. Additional thanks go to Mike Montgomery and Mark DeMaria for their insightful and valuable comments and for, amongst other things, providing the motivation for calculation NP2. Wolfgang Ulrich kindly gave me the barotropic model used in section 4. This work received support from the US Office of Naval Research under grant NOOO14-95- 1-0394. Bender, M. A. 1997 Bretherton, F. P. 1966a 1966b Charney, J. G. 1955 DeMaria, M. 1996 Dengler, K. and Reeder, M. J. 1997 Fiorino, M. and Elsberry, R. L. 1989 REFERENCES The effect of relative flow on the asymmetric structure of the interior of hurricanes. J. Atmos. Sci., 54,703-724 Critical layer instability in baroclinic flows. Q. J. R. Meteorol. SOC., 92,325-334 Baroclinic instability and the short wavelength cut-off in terms of potential vorticity. Q. J. R. Meteorol. Soc., 92,335-345 The use of primitive equations of motion in numerical prediction. Tellus, 7,22-26 The effect of vertical shear on tropical cyclone intensity change. J. Atmos. Sci., 53,2076-2087 The effects of convection and baroclinicity on the motion of tropical-cyclone-like vortices. Q. J. R. Meteorol. Soc., 123, 699-727 Some aspects of vortex structure related to tropical cyclone motion. J. Atmos. Sci., 46,975-990

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