The comparative effects of frictional convergence and vertical wind shear on the interior asymmetries of a hurricane

Transcription

1 1 The comparative effects of frictional convergence and vertical wind shear on the interior asymmetries of a hurricane Hongyan Zhu and Roger K. Smith Meteorological Institute, University of Munich, Munich, Germany April 29, 2004 Corresponding author/present address: Dr. Hongyan Zhu, Department of Meteorology, University of Reading, P.O. Box 243, Earley Gate, Reading RG6 6BB, England. hongyan.zhu@reading.ac.uk

2 2 Abstract A minimal three-dimensional hurricane model formulated on an f-plane is used to investigate the asymmetries that develop when a model storm is subjected to ambient vertical wind shear. The asymmetries that form in the moist version of the model have a different structure to those that form in a dry version. In the moist case there are two competing factors that influence the inner-core asymmetries: vertical wind shear and frictional convergence in the boundary layer. We show that the relative importance of these factors is different in the different stages of vortex evolution and different also in the core region compared with the outer region of the vortex. In the developing stage, the patterns of vertical velocity and temperature deviation above the boundary layer are primarily determined by the shear. When saturation occurs in the core region, the vortex rapidly intensifies and the upper and lower portions of the vortex become strongly coupled so that there is little tilt of the core region. In the mature stage, ascent associated with frictional convergence in the core tends to dominate the vertical motion field induced by the shear, but there are individual times when the patterns of ascent at the top of the boundary layer and at the upper level differ significantly. These times coincide mostly with fluctuations in the vortex track, which, in turn, must be influenced by asymmetries in the horizontal flow in the vortex. Even though the inner core of the vortex becomes upright with the onset of deep moist convection, the outer regions continue to have a significant tilt. Outside the core region the asymmetries in the pattern of vertical motion above the boundary layer are associated primarily with the tilt of the outer vortex.

3 3 1. Introduction The effect of vertical shear on the dynamics of even a dry vortex is rather complicated and the shear is known to produce significant asymmetries in the flow. One of the earliest studies is that by Jones (1995), who carried out a series of numerical simulations of barotropic tropical cyclone-scale vortices in a dry model without surface boundary layer. She found that the shear causes a vortex to tilt, initially in the downshear direction. As the vortex evolves, the direction of tilt alters and the vortex develops an azimuthal wavenumber one asymmetry in the temperature and vertical motion pattern, the orientation of which is related to the direction of tilt. Jones interpreted the behaviour in terms of potential vorticity (PV) thinking (Hoskins et al. 1985), since a vortex can be regarded as a PV anomaly at each level relative to the environmental flow far from the vortex centre at that level. For a dry vortex, the subsequent evolution of the PV at any level can be attributed to advection by the balanced flow associated with the entire PV distribution. When a vortex is exposed to shear the PV anomalies are displaced horizontally, initially in the direction of the shear. Further, the balanced flow associated with the PV anomaly at a given level penetrates upwards and downwards. Subsequently the downward penetration of flow associated with the upper anomaly and the upward penetration of that associated with the lower anomaly leads to a rotation of the tilt axis as the vortex is advected downshear. The balance constraint can account also for the patterns of induced vertical motion and temperature perturbations that occur in the numerical simulations. Pointing to some results of Raymond (1992), Jones argued that in a balanced configuration, the isentropes must be raised in the downtilt direction and lowered in the uptilt direction. Then, as the strong rotational flow moves along the sloping isentropes, ascent occurs on the downtilt right side and subsidence on the downtilt left side. Since the motion is adiabatic, air cools in the downtilt direction and warms in the uptilt direction. The strength of the interaction between the upper and lower PV anomalies of the tilted vortex depends on the the Rossby penetration depth, which for a rapidly-rotating, nearly axisymmetric vortex is given by (f loc (f + ζ)) 1/2 L/N, where L the horizontal length scale of the anomaly and N the Brunt-Väisälä frequency, f loc = f + 2V T /r, f is Coriolis parameter, r is the radius, V T is the tangential wind speed and ζ is the vertical component of relative vorticity (Hoskins et al. 1985; Shapiro and Montgomery 1993). Jones (1995) examined the sensitivity of the vortex evolution to the Rossby penetration depth and showed that for higher values of Rossby penetration depth, the vortex is much less susceptible to the destructive influence of the vertical shear than for smaller depths. Bender (1997) simulated the asymmetric structure of tropical cyclones using the Geophysical Fluid Dynamics Laboratory high resolution triply nested movable-mesh hurricane model. His study focussed mainly on the quasi-steady component of the flow asymmetries in the eyewall region. An analysis of terms in the equation for the vertical component of relative vorticity showed that the rate-of-generation of symmetric cyclonic (anticyclonic) vorticity at low levels by differential advection associated with the vertical shear is approximately opposed by the compression (stretching) of cyclonic vorticity associated with the divergence (convergence) of the asymmetric wind. The analysis showed also that other terms in the symmetric vorticity equation are small in comparison with

4 4 these two terms. Bender invoked this approximate balance to infer the pattern of lowlevel asymmetric divergence, but did not explain why such a balance is to be expected. For example, in the f-plane case with a 5 m s 1 easterly vertical shear, he showed that there is ascent at the front of the vortex and subsidence at the rear associated with the foregoing pattern of asymmetric divergence, whereas in westerly shear, the pattern of asymmetric vertical motion is reversed. In the former case, the ascent is located where there is enhanced low-level frictional convergence, whereas in the latter case the region of enhanced frictional convergence is located where there is asymmetric divergence. Bender considered only asymmetric motion in a vertical plane in the direction of vortex translation, but the maximum precipitation in his calculation is located in the front left quadrant of the storm (his Fig. 11, right bottom). Thus the precipitation asymmetry cannot be understood in terms of the foregoing theory. Bender speculated that the precipitation asymmetry might be a result of the tilt of the vortex axis in the direction of the shear vector. He argued that this tilt would increase the horizontal vorticity in the direction of the shear vector, enhancing the vertical motion to the left and decreasing it to the right. However, this association of vertical motion to tilt is opposite to that obtained and explained by Jones (1995, 2001). Frank and Ritchie (1999, henceforth FR99) carried out a series of numerical simulations of idealized tropical cyclones using the Pennsylvania State University National Center for Atmospheric Research fifth-generation Mesoscale Model (MM5), which is a nonhydrostatic, three-dimensional model that includes both parameterized and explicitly resolved moist processes. For simplicity their simulations were performed on an f-plane. They compared dry and moist simulations with similar initial conditions to examine the behaviour of a sheared vortex when latent heating was allowed to occur, but considered only a weak value of shear (3 m s 1 ). Their dry calculations produce a deep region of ascent in the sector of downshear right of the centre, as in the calculations of Jones (1995). However, in the moist calculations, once saturation occurs in the inner core and rapid intensification starts, the maximum upward motion and rainfall both move to the downshear left side. FR99 attributed this change to the latent-heat release associated with resolved moist ascent of slightly unstable air, which eliminates the downshear cold anomaly that forms as a result of dry dynamics. Frank and Ritchie (2001) further investigated the effects of vertical wind shear on the intensity and structure of simulated hurricanes using MM5 with a finer horizontal resolution and a fully-explicit representation of moist processes. They showed that when a large-scale vertical wind shear is applied to a mature tropical cyclones, the storm quickly develops a wavenumber one asymmetry with upward motion and rainfall concentrated on the left side of the shear vector looking downshear, in agreement with their 1999 study. They stated that... the mechanism responsible for this asymmetry appears to be the differential vorticity advection with height caused by the shear, following the interpretation given by Bender (1997). Jones (2000a) extended her previous work by investigating the role of large-scale PV-asymmetries in the evolution of a dry tropical-cyclone-like vortex in vertical shear on an f-plane. She found that the asymmetries result from the distortion of the initially symmetric vortex by the horizontally sheared flow associated with the vertical projection of the tilted PV anomaly. In a companion paper she investigated the evolution of

5 5 baroclinic vortices in environmental vertical shear (Jones 2000b). Smith et al. (2000) showed that some features of the interaction between the upper and lower PV anomalies of a sheared vortex may be understood in terms of a simple two-layer analogue model in which the vortex in each layer is advected without change in shape by the sum of a representation of the flow associated with the vortex in the other layer and by the flow in the layer, itself. For weak shear and/or strong vortices and strong coupling, the vortices rotate around each other as their mean centre translates with a fraction of the mean zonal flow. For strong shear and/or weak coupling, the vortices undergo a partial rotation while they are in proximity, but become progressively separated by the shear. The early calculations by Jones (1995) fall into the latter regime, but an example of a calculation that falls into the former regime is described by Jones (2004). In addition to vertical shear, asymmetries in low-level convergence associated with boundary layer friction may influence also the location of the inner-core asymmetries of vertical motion in a hurricane. Shapiro (1983) used a slab boundary layer model of constant depth to investigate the effects of the translation of a hurricane on the distribution of boundary layer winds and on the organization of convection. His calculations show that for a slowly moving hurricane, convergence is large in a broad arc ahead of the storm and that for a fast moving storm, the maximum convergence lies ahead of and to the right of the storm. He hypothesized that the maximum convection in the hurricane would coincide with the region where the boundary layer convergence is a maximum. An observational study of the relationship between motion, vertical wind shear and convective asymmetries in Atlantic hurricanes was reported by Corbosiero and Molinari (2002, 2003) based on lightning data. They found a strong signature in the azimuthal distribution of lightning with respect to vertical wind shear. The relative importance of asymmetric friction and vertical wind shear on the azimuthal asymmetry of convection was determined by examining circumstances in which the two effects would place maximum lightning in different quadrants. They concluded that without exception, the influence of vertical wind shear dominated the distribution. They stated that Although asymmetric friction creates vertical motion asymmetries at the top of the boundary layer, these apparently do not produce deep convection if vertical wind shear induced circulations oppose them. Two possible reasons were given: first, in a strong sheared environment the mean current in the boundary layer is less than the motion; secondly, the vertical wind shear-induced circulations oppose the boundary layer friction-induced vertical motion asymmetries at the top of the boundary layer, which is shown in an integration on a β-plane with no external shear in Bender (1997) and in an f-plane simulation by FR99. Recent theoretical studies of the resiliency of tropical-cyclone-scale vortices to vertical shear have emphasized the importance of dry dynamical processes (Jones, 2004, Reasor et al. 2004). Jones showed a calculation in which a dry vortex remains nearly upright during 4 days under the influence of shear. Thus it is possible for even a dry vortex to resist the destructive effects of vertical shear if the vortex is sufficiently intense and the shear is not too large. She showed also that the outer portion of the vortex tilts more strongly than the inner core and that the pattern of vertical velocity is related to the

6 6 vertical tilt of the outer portion of the vortex, rather than the shear directly. She showed further that the average vertical shear across the centre of the vortex depends on both the vortex tilt and the presence of large-scale PV asymmetries in the outer region of the vortex. It follows that the initial environmental shear might not be a reliable measure of the vertical shear felt by the vortex at later times. Reasor et al. proposed an alternative explanation for the resiliency of tropical cyclone vortices to vertical shear based on the damping of vortex Rossby waves excited by the shear. They suggest that the diabaticallydriven secondary circulation of an intensifying vortex increases both Rossby number and the radial gradient of azimuthal-mean potential vorticity, which, in turn, increase the efficiency of the mechanism and hence the vertical rigidity of the vortex. In a recent study of vortex evolution in a quiescent environment on an f-plane using a minimal three-layer, three-dimensional tropical cyclone model, Zhu and Smith (2003, henceforth referred to as ZS) showed that even starting with an initially-symmetric vortex, significant flow asymmetries develop when latent heat release occurs. This is particularly true when the model is formulated with a Lorenz-grid in the vertical, a grid that is commonly used in atmospheric models, including all of the models described above. The spurious development of asymmetries is not just a feature of the threelayer model, but occurs also, for example, in multi-level tropical-cyclone simulations using the MM5 model (S. Nguyen, personal communication). For this reason some of the asymmetries in previous numerical studies of tropical cyclones might be affected by the computational mode when moist processes are involved, even though the models have a higher vertical resolution than the ZS-model. ZS showed that the spurious asymmetries were very much weaker if a Charney-Phillips (CP-) grid is used for vertical differencing. Since the relative importance of vertical wind shear and boundary-layer friction on the asymmetries of vertical motion in a hurricane is a topic of much current interest, we are motivated to revisit the problem using the foregoing model developed by ZS. The present version of the model includes explicit and parameterized representations of moist processes and uses the CP-grid to minimize any numerical influence on the formation of asymmetries. The aim is to further explore the mechanisms leading to hurricane asymmetries in the presence of vertical shear. 2. The numerical model The minimal hurricane model is that described in ZS. It is three-dimensional and based on the hydrostatic primitive equations formulated in σ-coordinates (x, y, σ). It uses the Charney-Phillips grid (CP-grid) for the vertical differencing (see Fig.1). The model equations and the advantages of CP-grid are discussed in ZS. The model is divided vertically into four layers of unequal depth in σ: the lowest layer has depth 0.1 and the three layers above have depths 0.3. The present calculations are carried out on an f-plane. Newtonian cooling is used to represent the effect of radiative cooling. The turbulent flux of momentum to the sea surface and the fluxes of sensible heat and water vapour from the surface are represented by bulk aerodynamic formulae. The surface drag coefficient, C D, is calculated from the formula used by Shapiro (1992): C D = ( R F u b ) 10 3, (1)

7 7 where R F = 0.8 is used to reduce the boundary layer wind, u b, to the 10-m level. The surface exchange coefficients for moisture and heat are assumed to be equal to each other and to C D. Explicit condensation occurs when the air becomes supersaturated at a grid point. At such points the excess water is assumed to precipitate out while the latent heat released increases the air temperature. The scheme, which is described in detail by Zhu et al. (2001; henceforth referred to as ZSU), involves an iterative procedure. The parameterization of deep cumulus convection is based on a mass flux approach suggested by Arakawa (1969) and detailed in ZSU. Implementation of the scheme in the CP-grid model is described in ZS. 3. The Calculations The initial vortex is axisymmetric and baroclinic. The tangential wind profile is that used by Smith et al. (1990), but with different parameters: the maximum tangential wind speed is 20 m s 1 at level-4 at a radius of 180 km and its magnitude reduces to 14.5 m s 1 at level-1. The initial mass and thermal fields are obtained by solving the inverse balance equation (see ZSU). The far-field temperature, geopotential height and humidity structure are based on the mean West Indies sounding (Jordan 1957). The horizontal grid spacing of the model is 20 km and the integration time step is 12 s. The ocean surface temperature is 28 o C. The experiments are performed on an f-plane at 20 o N. Although the use of a variable f would be more realistic, it would introduce an additional mechanism for generating asymmetries: the restriction to an f-plane provides a cleaner thought experiment for comparing the effects of vertical shear and boundary layer friction on the inner core asymmetries of a hurricane. Five experiments are carried out as detailed in Table 1. The first experiment excludes moist processes and boundary layer friction. A unidirectional linear easterly shear of 5 m s 1 is added above the boundary layer at the initial instant (t = 0). The second experiment is also dry simulation with the same shear as Expt. 1, but includes boundary layer friction. Experiment 3 is like Expt. 2, but includes a cumulus parameterization scheme as well as an explicit scheme for latent heat release when saturation occurs at a grid point. Experiments 4 and 5 differ from Expt. 3 only by the inclusion of a uniform westerly flow of 2 m s 1 and 6 m s 1, respectively. The five experiments are designed to compare the role of the boundary layer convergence and vertical shear on the inner core asymmetries of a hurricane in the dry and moist simulations. As the asymmetries of vortex in the dry simulation have been investigated by many previous workers, we focus here mainly on the vortex asymmetries in the three moist simulations. 4. Experiment 1 This experiment is designed to ascertain the extent to which the model is capable of reproducing the results of the dry simulations of previous authors. Panel (a) of Fig. 2 shows the 6 hourly positions of the vortex centre (defined for the present as the location of maximum relative vorticity) in the top layer (layer 1) and in the boundary layer (layer 4). Note that the vortex centre moves steadily northwestwards in the boundary layer and

8 8 steadily southwestwards in the top layer so that the relative vortex core becomes tilted in the northeast-southwest direction and not in the downshear direction. These tracks are consistent with the results of previous studies (Jones 1995, 2000b, FR99, Smith et al. 2000). As explained by Jones (1995), the PV distribution is tilted by the shear. Then the flow associated with the upper PV anomaly penetrates downwards and leads to a surface cyclonic circulation to the west of the lower PV anomaly. The resulting flow advects the low-level vortex northwards. Likewise, the flow associated with the lower PV anomaly penetrates upwards and advects the upper-level vortex southwards. The temperature deviation from its environmental value in the middle troposphere (level ) shown in Fig. 3 at 24 h and the corresponding vertical velocity distribution at the same level is shown in Fig. 4. The corresponding patterns in the upper troposphere (level ) and at the top of the boundary layer (level ) are similar to these and are not shown. These patterns are consistent with those shown by Jones (1995). In this experiment the vortex weakens very slightly due to both the tilting of the vortex away from the vertical and presumably also to numerical diffusion. The maximum tangential wind speed in the boundary layer decreases by 1.4 m s 1 and the minimum surface pressure increases by 1.7 mb after 48 h of integration. 5. Experiment 2 Experiment 2 is like Expt. 1, but the effects of surface friction are included in the lowest layer. Then the wind speed together with the centrifugal and Coriolis forces are reduced, leaving a net inward pressure gradient force in this layer (see e.g. Smith, 1968; 2003). This unbalanced force field leads to convergence in the boundary layer and to divergence above it, so that the vortex decays (Smith, 2000). Indeed, the maximum tangential wind speed in the boundary layer decreases by 13.2 m s 1 and the minimum surface pressure increases by 8.6 mb after 48 h of integration whereupon the decay rate is much greater than that in Expt. 1 as expected. The tracks of the vortex centre in the top layer and in the boundary layer (panel (b) of Fig. 2) are similar to those in Expt. 1, with a northwestward motion in the boundary layer and southwestward motion in the top layer. The presence of a boundary layer has little influence on the vortex track, an aspect not discussed by FR99. Figure 5 shows the pattern of temperature deviation at the middle level (level ) in Expt. 2, which should be compared with that in Expt. 1 shown in Fig. 5. These patterns have a similar structure with cooling on the downtilt (southwestern) side and warming on the uptilt (northeastern) side. Figure 6 shows the vertical velocity distribution at the middle level and at the top of the boundary layer (level ) after 24 h. As in Expt. 1, the region of ascent at the middle level is located downtilt right, and the region of descent is located downtilt left (compare Figs. 4 and 6a). The pattern of vertical velocity in the upper troposphere is the same as that in Expt. 1 also (figure not shown), but that at the top of the boundary layer is significantly different as shown in Fig. 6b. There, the ascent is mainly near the

9 9 vortex centre, with a maximum in the northwest quadrant. This pattern is broadly in agreement with the results of Shapiro (1983) for a moving vortex. These calculations indicate that the boundary layer effect in the dry simulation is relatively shallow and influences only the patterns of vertical motion asymmetries at the top of that layer. Thus the frictionally-induced divergence must be confined to the lower troposphere. Moreover, the temperature deviation and vertical velocity patterns at levels above the boundary layer are still determined by the balanced theory described by Jones (1995). These results are similar to those obtained by FR99 in their dry vortex simulation. 6. Experiments 3 The main calculation in this paper is Expt. 3, which has the same initial condition and environmental shear as in Expts. 1 and 2, but with both surface friction and moist processes included. The aim of this experiment is to compare the structure of the innercore asymmetries in the dry and moist simulations and to explain the differences. By inner-core we refer to the region extending out to about 150 km from the vortex centre. (a) Stages of evolution Figure 7 shows a time series of the maximum boundary-layer wind speed during the 96 h integration period in Expt. 3. The heating of the vortex core by parameterized deep convection leads to a secondary circulation with low-level convergence and upper-level divergence. Thus, in contrast to Expts. 1 and 2, in which there is no latent heat release, the primary (tangential) circulation strengthens at low levels and weakens aloft. The intensification is slow at first characterizing a gestation period lasting about 24 h, but becomes rapid later when the inner core saturates on the grid scale. After about 6 h the tangential wind speed reaches a maximum of 54 m s 1. Thereafter the vortex settles down to a quasi-steady state in which the intensity fluctuates about some mean value that is a little less than the earlier maximum. We refer to the period beyond 42 h as the mature stage. (b) Vortex track and tilt Figure 8 shows successive positions of the vortex centre in the boundary layer (layer- 4) and in the upper troposphere (layer-2) in Expt. 3. These centres begin to separate during the gestation period with the boundary-layer centre moving slightly south of west and that in the upper troposphere moving north of west. This movement is different from that in the dry simulations of Expt. 1 and 2 and different, apparently from that obtained by FR99, although they provide only a few details of this aspect, saying only that the vortex drifted slowly to the northwest. During the period of rapid intensification, the two centres converge and subsequently track together, moving generally westwards in the mature stage. The differences in the vortex tracks during the gestation period compared with the dry simulations must arise from the implementation of a cumulus parameterization

10 10 scheme in Expt. 3, and the differences between our results and FR99 s are presumably due to differences in the schemes used to represent deep convection. The effects of the scheme used here may be understood as follows. First, the strengthening of the lower tangential circulation and the weakening of that aloft reduce the mutual advection of the upper and lower PV anomalies by the upward (downward) projection of the tangential circulation at lower (upper) levels as happens in the dry case. However, in the presence of vertical shear the parameterized heating is not symmetric and leads to asymmetries in the vortex circulation that have an important influence on the track. Panel (a) of Fig. 9 shows the middle level (level ) temperature deviation at 9 h, a time within the gestation period. We have seen that initially, in the dry case, vertical shear leads to cooling on the downshear side and as the vortex tilts, the maximum cooling shifts downtilt, i.e. to the southwest. In Expt. 3, however, the initial downshear cooling together with the increased surface heat flux ahead of the storm increases the convective instability in that region. The result is to enhance subgrid-scale convection as evidenced by the pattern of cloud-base mass flux in the cumulus parameterization scheme as shown in Fig. 10. The convective heating provides an additional source of positive (negative) PV at low-levels (high-levels) ahead of the storm. Figure 11 compares the asymmetric vorticity distributions in the upper and lower troposphere in Expt. 1 (left panels) and Expt. 3 (right panels) after 9 h of integration. In Expt. 3, the convective heating induces a positive relative vorticity at the front of the storm in the lower troposphere and a negative in the upper troposphere. The absolute value of relative vorticity in Expt. 3 is about 3 times than that in Expt. 1. We examined to what extent the motion of the vortex centres in the upper layer and in the boundary layer in Expt. 3 are consistent with the asymmetric wind vectors, calculated by inverting the relative vorticity distribution at each level. The method, successfully employed by Jones (2000a) for an initially-barotropic vortex, would assume that the flow is barotropic at each level. The agreement in our case was not very good suggesting that the barotropic assumption is not valid in our case. A more detailed analysis would require the use of partial-pv inversion (Davis, 1992), which is beyond the scope of the paper. The tracks in Fig. 12 show that there is very little vortex tilt during the mature stage, at least in the core region (but see subsection (d) below), indicating that there is much stronger coupling between the upper and lower portions of the vortex in this region. (c) Inner-core asymmetries The distribution of vertical velocity at the top of the boundary layer and at the middle level at 9 h in Expt. 3 is shown in Fig. 12. The vertical velocity pattern at the top of the boundary layer is quite asymmetric (Fig. 12a), with maximum ascent occurring southwest of the centre, approximately in the direction of vortex motion at this time. In FR99 s calculation, the maximum ascent at early times is in the northwestern sector, but again approximately in the direction of vortex motion. At the middle level, the maximum ascent occurs in the northwestern sector (Fig. 12b) and there is no well-defined region of subsidence on the opposite side to the ascent as occurs in the dry calculations. At 18 h, the broadscale pattern of temperature deviation at the middle level (Fig. 9b) is similar to that at 9 h, but a warm core has begun to develop at radii less than about 100 km

11 11 the vortex centre on account of the explicit release of latent heat. The left panels of Fig. 12 show that this warm core is surrounded by an annular region of enhanced ascent at both the middle level and at the top of the boundary layer, with pronounced maxima to the northwest of the vortex centre at each level. The results suggest that at this stage of development, the frictional convergence induced by the boundary layer determines the pattern of ascent aloft in the core region, dominating the effect produced by shear in this region. To investigate the inner-core asymmetries during the mature stage, we show the mean vertical velocity averaged from 42 h to 72 h in the core region in Fig. 13. The mean is based on averages of the respective fields every 3 h. During this period, the storm is moving slightly south of west. Again the mean vertical velocity at the middle level (Fig. 13a) has a similar pattern to that at the top of the boundary layer (Fig. 13b). At both levels, the maximum ascent is located to the west of the core region. Consistent with the asymmetry in the vertical velocity field, the explicit precipitation rate has a maximum in the western quadrant of core (Fig. 14). Thus, in the mature stage also, the inner-core asymmetries of vertical velocity at all levels and the precipitation distribution are determined primarily by the frictionally-induced convergence in the boundary layer as hypothesized by Shapiro (1983) and as found also by FR99. One may ask how representative the patterns of average vertical motion are of those at individual times? Examination of the plots each 6 h for this experiment shows that the most significant departure from the mean picture occurs at 48 h, when the largest ascent at the upper level occurs to the southsoutheast of the vortex centre, coinciding with one of two local maxima in the boundary layer, the other one being in the direction of motion. At other times the maximum ascent at both levels lies in the sector between southwest and northwest, except at the upper level at 72 h where the maximum lies to the northnorthwest. The differences in the inner-core asymmetries in the different developing stages in Expt. 3 can be explained in terms of the Rossby penetration depth (Jones, 1995). During the gestation period, the vortex is relatively weak and becomes slightly tilted, but the tilt is different from that in Expt. 2 because of the additional generation of PV by the presence of (parameterized) convective heating. During the period of rapid intensification, the Rossby penetration depth and hence the vertical coupling increase in the innercore region on account of the increase in the relative vorticity and the decrease in the static stability arising from saturation in that region. Therefore in the mature stage the hurricane becomes vertically upright in the core region and the boundary layer friction dominates the effects of shear in determining the asymmetric distribution of vertical motion above the boundary layer. (d) Broadscale vortex tilt Up to this point our analysis has focussed on the inner core of the vortex, the location of which is characterized reasonably well by that of the maximum PV or minimum perturbation geopotential. Recently, however, Jones (2003) noted that the locations of Note that our simple explicit moisture scheme removes all liquid water as soon as it is generated. In FR99 s more sophisticated treatment, rain water is advected horizontally as it falls so that the rainfall maximum is displaced cyclonically from the vertical velocity maximum.

12 12 these quantities at different levels is not a suitable indicator of the vertical tilt of the overall vortex and she showed an example in which the inner core of the vortex remains upright whilst the outer regions are more strongly tilted. To highlight these differences she proposed an alternative definition of the vortex centre, the centre of PV, analogous to the centre of mass of a solid body, which characterizes the centre of the broadscale circulation. The centre of PV is defined as: (x CP V, y CP V ) = ( xq (x, y)dxdy, yq (x, y)dxdy) q (x, y)dxdy) (2) where q is the perturbation PV and the double integrals are evaluated over the area within a circle of radius R centred on the location of maximum PV at the given level. To investigate the tilt of the broadscale tilt of the vortex in Expt. 3, we plot the track of upper layer vortex centre and boundary layer vortex centre by using Eq. 2, but with q replaced by the relative vorticity at the relevant level. Two different values of R are chosen: 100 km to characterize the inner-core region and and 500 km, to characterize the broadscale vortex. These tracks are shown in Fig. 15. As expected tracks for R = 100 km, shown in Fig. 15a are similar to those in Fig. 8 with the upper level vortex moving westnorthwestwards and the boundary layer vortex moving westsouthwestwards in the gestation period and the two centres merging together and moving westwards in the mature stage. In contrast, with a R = 500 km the vertical tilt is larger in the mature stage than during the gestation period with the vortex centre in the boundary layer moving predominantly northwest and upper vortex centre moving southwest (Fig. 15b). Therefore, the broadscale vortex shows a significant tilt as in the dry calculations, even though the central core region is nearly upright. The vertical tilt outside the vortex core region is illustrated by the relative vorticity fields at 96 h in the upper layer (level-2) and boundary layer (level-4), shown in panels (a) and (b) of Fig. 16. At this time the vortex in the outer core region is tilted in the north-south direction, while the regions of large vorticity in the core region at the two levels are essentially collocated. In other words, the core region has become displaced relative to the centre of the outer circulation at upper levels. The dashed lines in panel (c) of Fig. 16 show the corresponding distribution of vertical p-velocity, ω, at the middle level. The maximum vertical velocity outside the core is located in the southwestern sector, which is to the right of the tilt direction of the broadscale vortex. In contrast, the maximum ascent in the inner-core region (the shaded area in Fig. 16c) occurs slightly to the south of west from the vortex centre, which is close to the direction motion at this time (see upper panel of Fig. 15). 7. Experiments 4 and 5 Experiments 4 and 5 are identical to Expt. 3 except for the inclusion of a uniform westerly flow. These experiments are designed to further investigate the effects of the motion on the low-level asymmetries of vertical motion and to determine whether these asymmetries extend to higher levels as in Expt. 3. In Expt. 4, the strength of the easterly flow (2 m s 1 ) is chosen so that the displacement of the vortex centre in the boundary layer is relatively small during the integration period. In Expt. 5 the strength (6 m s 1 ) is chosen to produce an eastward displacement of this centre, comparable with the westward displacement in Expt. 3. The tracks of the upper- and lower-level vortex centres in these

13 13 two experiments are shown in Fig. 17 and the average distributions of vertical p-velocity at the upper level and at the top of the boundary layer for the mature stage (42-72 h) of the two experiments are shown in Fig. 18. In Expt. 4 the mean asymmetry at the top of the boundary layer (Fig. 18c) is much more symmetric than in the other Expts. 3 and 5, consistent with the vortex motion at that level being weak, and although the asymmetry in the upper layer is less symmetric, the region of maximum ascent is co-located with that at the top of the boundary layer (Fig. 18a) and lies in the direction of mean motion. In Expt. 5 the average vertical motion at the top of the boundary layer has a maximum to the east of the vortex centre, a little to the left of the direction of mean motion, while the mean ascent at the upper level has a maximum in the southeast, a little to the right of the mean motion (Figs. 18b,d). These results broadly support the findings for Expt. 3 that the inner-core asymmetries of vertical velocity at all levels are determined primarily by the frictionally-induced convergence in the boundary layer. Nevertheless it should be noted that there some variability in the fields at individual times during the mature stage and by considering fields at a particular time one might come to a different conclusion. As space considerations prevent us from showing more fields, we restrict ourselves to a brief summary of diagrams similar to Fig. 18 showing the fields for selected times. In Expt. 4, for example, at 42 h, maximum ascent at both levels lies south of the vortex centre consistent with the previous 6 h motion towards the south, but at 48 h, following an abrupt change in the direction of motion, the asymmetries are not consistent with the motion. Nevertheless, the patterns of ascent at the two levels are similar with two local maxima. During the subsequent 24 h period, where the vortex motion is weak, there is a strong tendency for the ascent at the top of the boundary layer to become axisymmetric, but with a weak superimposed wavenumber-4 asymmetry. This asymmetry does not rotate and is probably a result of using a square grid on which to represent an approximately symmetric flow. At the upper level, the asymmetry in the vertical motion is less symmetric and has a predominantly wavenumber-2 asymmetry. In Expt. 5, at 42 h, the maximum ascent at both levels lies southsoutheast of the vortex centre, broadly consistent with the previous 6 h motion towards the southeast (Fig. 17b), but at 48 h, following a change in the direction of motion to the eastsoutheast, the maximum ascent lies to the right of the track while that in the boundary layer lies to the east consistent with the motion. A similar pattern persists at 54 h, despite a track change towards the northeast. During the period of weak motion, between 54 and 60 h, the ascent out of the boundary layer tends to become symmetric while asymmetries remain at the upper level with local maxima to the west and north of the vortex centre and a minimum to the east. During the subsequent 12 h, the vortex has a mean motion to the southeast and the boundary and upper layer asymmetries of vertical motion are similar and are consistent with the motion, being approximately in the direction of motion. 8. Relationship to previous studies The results of dry calculations are consistent with those of Jones (1995, 2003) and FR99. In particular, the patterns of temperature deviation and vertical velocity above the boundary layer show ascent and cooling to the right of the direction of tilt and subsidence and warming to the left. Also, in agreement with FR99, the inclusion of a frictional boundary layer has little influence on the asymmetries of temperature deviation

14 14 and vertical velocity above the boundary layer in the dry experiments. These results indicates that the minimal model can capture the principal dynamical effects of vertical shear. In the moist calculations, the low-level ascent is approximately in the direction of the low-level motion vector as in FR99 (compare their Fig. 15 with Fig. 13b) and in accordance with the results of Shapiro (1983), but the pattern of ascent in the middle troposphere in our calculation (Fig. 13a) is different from that at 700 mb shown in Fig. 16a of FR99. In our case the asymmetry is similar to that at the top of the boundary layer, while in FR99 it the maximum ascent at 700 mb occurs to the left of the track. However one has to bear in mind that FR99 s calculation has a higher vertical resolution than ours and that the fields they show are for a particular time compared with our averaged fields. In their observational study, Corbosiero and Molinari (2002) found different patterns of asymmetry in the inner-core region and outer region of hurricanes, with downshear to downshear left maxima of convection in the core and downshear right maxima in the outer bands. Jones (2003) pointed out that the downshear right location of lightning activity in the outer rainband might be attributed to the tropical cyclone vortex having a larger vertical tilt at the radius of the outer rainbands than in the inner core as in her calculations. The different tilt in different regions of the vortex is a feature also of the moist calculation reported here, but the observations would include any effects of β, or more generally, the broadscale PV-gradient. Corbosiero and Molinari (2003, p375) explore the question: Why does the asymmetric friction effect not show in the convection? They downplay the effect of friction suggested by Shapiro s (1983) study on the grounds that the calculations only predict the vertical motion at the top of the boundary layer only and they cite FR99 s numerical integrations in which... shallow upward motion without convection occurs in the front and right-front quadrants, while strong convection occurs downstream left. The present results would indicate a stronger control of boundary layer friction on the deep upward motion supporting Shapiro s hypothesis, indicating that more work is required to reconcile Corbosiero and Molinari s observations with model results. The VRW-mechanism described by Reasor et al. (2004) is an appealing one for understanding the resilience of vortices to vertical shear. The basic idea is that the shear excites VRWs that cause the vortex to precess, the rate-of-precession depending on the angular rotation rate of the waves, which in turn depends on the vortex profile. If this rate is sufficiently rapid for a given shearing strength, the tilted vortex axis periodically changes its orientation relative to the shear so that during parts of the cycle the shear reduces the vertical tilt (consistent with the arguments given by Jones, 1995). In the moist calculations the vortex profile changes rapidly during the period of rapid intensification and continues to evolve even in the mature stage. As far as we are aware, the theory of VRWs on rapidly varying basic states has yet to be developed and for this reason it is beyond the scope of the present study to investigate the importance of the mechanism in our calculations. Nevertheless, the mechanism could provide a plausible explanation for some of the variability in the asymmetries of vertical velocity described in section 7, since episodic pulsations of convection could be expected to excite such wave modes.

15 15 9. Conclusions The results of the present study complement those of earlier ones. In the dry experiments, the inclusion of a frictional boundary layer has little influence on the asymmetries of temperature deviation and vertical velocity above the boundary layer and little effect on the motion of the vortex centre at different levels. The results of the moist calculation agree broadly with those of previous studies, indicating that the minimal model is able to capture the principal effects of vertical shear. In particular, the effects of vertical shear dominate those of boundary layer friction in determining the inner-core asymmetries of vertical motion above the boundary layer, and of precipitation during the developing stage. However, this situation is reversed as saturation occurs in the core region and the vortex rapidly intensifies. Then the asymmetries of vertical motion at the top of the boundary layer and in the upper troposphere have mostly similar patterns. Exceptions occur when there are large fluctuations in the track, which themselves must be associated with rapid changes in the asymmetry of horizontal motion. When the combination of shear and background flow are such that the vortex translation speed is very small (less than about 0.3 m s 1 ), the ascent at the top of the boundary layer and at the upper level is comparatively symmetric. When the storm translation speed is larger, the maximum ascent at both levels occurs mostly on the side of the storm that is in the direction of storm motion as hypothesized by Shapiro (1983), but there are individual times when this is not the case. We have shown how moist processes affect the motion of the vortex centre at different levels. During the gestation period, the vortex centres in the upper-layer and boundarylayer are displaced horizontally on account of the shear, but the separation of two centres is much smaller than that in the dry simulations. As the grid-scale saturation begins near the vortex centre, there is warming in an annular region around the centre on account of the explicit latent heat release and the earlier pattern of cool and warm temperature anomalies associated with the shear are destroyed in the inner-core region. In the mature steady stage, the centres of the upper-layer vortex and that in the boundary layer merge and remain together. Even though the inner core of the vortex remains upright during this stage, the outer regions are more strongly tilted and there the vertical velocity above the boundary layer is still determined by the tilt of the outer region of the vortex as in the dry calculations. Acknowledgements This work was supported by the US Office of Naval Research through Grant No. N We are grateful Drs. Sarah Jones and Lloyd Shapiro for helpful discussions and for their thoughtful comments on an earlier version of this paper.

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