Boundary layer mechanisms in extratropical cyclones

Transcription

1 QUARTERY JOURNA OF THE ROYA METEOROOGICA SOCIETY Q. J. R. Meteorol. Soc. 33: (7) Published online in Wiley InterScience ( DOI:./qj.3 Boundary layer mechanisms in extratropical cyclones Robert J. Beare* Met Office, Exeter, UK ABSTRACT: This paper revisits the mechanism for the interaction of the boundary layer with extratropical cyclones. Two diagnostic approaches are compared: Ekman pumping and potential vorticity. Ekman pumping derives from the boundary layer stress which induces convergence and ascent. boundary layer potential vorticity contains in a single quantity both the vorticity and stratification. These quantities are compared for an idealized extratropical cyclone life cycle simulated with the Met Office Unified Model. A significant component of the boundary layer stress and thus Ekman pumping at occlusion is forced by the cold conveyor-belt jet in the unstable boundary layer. In contrast, much of the boundary layer depth-averaged potential vorticity is contained within the stable warm-sector region. Inversion of the warm-sector PV indicates a small local deepening of about.5 hpa. Moreover, switching off the boundary layer mixing in the unstable cold sector has much more impact than in the stable warm sector. The sensitivity of the cyclone and its boundary layer to basic-state jet strength is then investigated. The maximum friction velocity scales closely with the initial maximum jet strength. This demonstrates the important role of the largescale flow in organizing the boundary layer structure. Changes in the minimum pressure produced by altering the boundary layer parametrization correspond closely to changes in the surface stress averaged over the cyclone. Different operational changes to the boundary layer scheme produce small and compensating changes to the cyclone minimum pressure over three days. Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Published by John Wiley & Sons, td KEY WORDS Ekman pumping; friction velocity; potential vorticity Received 4 May 6; Revised November 6; Accepted 4 November 6. Introduction Within the lower sections of an extratropical cyclone, the boundary layer turbulence generates significant vertical fluxes of momentum, heat and moisture. Accurately parametrizing these fluxes is important for both climate and numerical weather prediction (NWP). Nevertheless, the mechanism by which the boundary layer interacts with the cyclone is still an area of ongoing research. A popular explanation is Ekman pumping (Holton, 99). Here, the stress generated by the boundary layer acts to generate convergence into the centre of the cyclone which then gives an upward vertical velocity at the boundary layer top (Ekman pumping) via continuity. The Ekman pumping then acts to spin down the cyclone by vortex squashing. Although elegant in its simplicity, the Ekman pumping mechanism is not without its critics. Adamson et al. (6) performed a potential vorticity (PV) diagnosis of the boundary layer for a baroclinic life cycle in an idealized general circulation model. They showed significant PV generation in the warm conveyor belt and warm front regions. Although there was an Ekman pumping contribution to the PV budget, there was more significant * Correspondence to: Robert J. Beare, Met Office, FitzRoy Road, Exeter, EX 3PB, UK. bob.beare@metoffice.gov.uk baroclinic generation. This suggests that the PV mechanism dominates over Ekman pumping. However, there remains the issue of how much boundary layer stress and Ekman pumping is in regions away from those of significant PV. To address this issue, the first aim of the paper is to compare distributions of Ekman pumping with PV for a modelled cyclone life cycle. Furthermore, inversion techniques will be used to determine the effect of boundary layer PV on the cyclone. Such an inversion was not performed by Adamson et al. (6). This study uses an idealized limited-area version of the Met Office Unified Model (UM), configured to simulate cyclone life cycles. It allows better control over the largescale environment, and a more precise determination of the physical mechanism than a full NWP model. To check the relevance of the idealized configuration to the forecast situation, a test case cyclone simulation with the full NWP global model is also performed. The simulations are used to diagnose and thus distinguish the Ekman pumping and PV mechanisms. Typically, the warm sector of a marine extratropical cyclone is stably stratified but the cold sector is neutral or convective. Recently, Persson et al. (5) analysed near-surface observations from the Fronts and Atlantic Storm-Track Experiment (FASTEX) in terms of warm and cold sectors of extratropical cyclones. However, little analysis of the distribution of boundary layer parameters Crown Copyright 7. Reproduced with the permission of the Controller of HMSO.

2 5 54 R. J. BEARE with stability has been done in a modelling context. Here, the diagnosis of both the PV and Ekman pumping will be analysed in terms of stability. Also, the influence of the boundary layer fluxes in the warm and cold sectors will be determined by separately switching them off in stable and convective stability regimes. The author is not aware of such an experiment being performed before. The previous diagnostics address the issue of how the boundary layer interacts with the cyclone. However, the boundary layer does not exist in isolation, and often neglected is the degree to which the large scale influences and controls the boundary layer structure. Martin et al. () examined the role of the upper-level trough decay in cyclolysis events, and argued that the boundary layer convergence was being organized by this large-scale feature. Although it is not possible to completely separate the boundary layer from the large scale, some progress can be made by running a model with a range of largescale forcings. To this end, here a number of simulations with a range of upper-level jet strengths are used to study how the closely the boundary layer friction varies with it. Another way to determine the boundary layer mechanism is to measure the sensitivity of the cyclone to various aspects of the boundary layer parametrization scheme. It also helps quantify the model error associated with the boundary layer scheme. A further aim of the paper is to investigate the sensitivity of the cyclone and its boundary layer to the boundary layer mixing scheme. Idealized dry models have proved valuable in furthering understanding of the cyclonic boundary layer. For example, Keyser and Anthes (98) used a twodimensional (D) baroclinic wave model to examine the role of boundary layer diffusion models. They demonstratedanincreaseinlow-levelshearandekmandamping in the presence of boundary layer diffusion. angland et al. (995) applied adjoint techniques to an idealized cyclone life cycle and found sensitivity to sea-surface temperature anomalies and surface exchange coefficients in the warm sector region. Doyle (995) used a mesoscale model configured with an idealized jet to compare coupling to a wave model with simpler prescriptions of surface roughness. Becker et al. (996) used a D model to compare the boundary layer structure in the presence of either a low-level or upper-level jet, demonstrating enhanced boundary layer Ekman pumping for the low-level jet. Thompson and Williams (997) used a primitive-equation model to examine the role of the boundary layer in a 3D baroclinic life cycle cyclogenesis. The presence of boundary layer diffusion resulted in more realistic frontal structure.. Method.. Idealized model configuration The UM was configured on a limited-area domain with fixed lateral boundary conditions over a sea surface. The model was made dry by setting the surface latent heat flux and initial moisture fields to zero. The radiation scheme was also switched off. The horizontal resolution was.4 (approximately 44.5 km), with a time step of minutes. The domain was 4 38 grid points (in the x,y and z directions respectively), approximately 8 km in the x direction and 9 km in the y direction. The significant horizontal extent of the domain was designed to allow the development of baroclinic waves in the interior which were not inhibited by the lateral boundary conditions. The grid varied quadratically in the vertical such that the grid spacing was about m near the bottom of the boundary layer and 8 m in the upper troposphere. For simplicity, the Coriolis parameter was kept constant. The semi-agrangian dynamical core in the UM is described by Davies et al. (5), and was used without any additional numerical diffusion besides the boundary layer scheme. The initial state was a zonal mean jet (Figure ), similar to that adopted by Beare et al. (3), but extended to include a stratosphere (see Equation (A.)). Prescribing the initial jet in this way permitted its strength to be varied easily in the sensitivity experiments. Superimposed on the jet was an initial perturbation consisting of a vortex with width 3 km, close to the maximum Eady growth-rate wavelength of 34 km, and maximum amplitude (A) ofms at the tropopause, the amplitude decaying exponentially in the vertical away from Zonal mean potential temperature (K) Zonal mean wind (m/s) Height (m) Height (m) N atitude S S N N atitude S Figure. Vertical cross-sections of initial zonal mean potential temperature and zonal wind. Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

3 CYCONE BOUNDARY AYERS 55 the tropopause (Equations (A.) and (A.3)). The upper vortex was applied 39 km from the western boundary of the domain. The large amplitude of the upper-level vortex was designed to trigger rapid cyclogenesis. The initial potential temperature was calculated from the prescribed wind field using the thermal wind balance, Equation (A.4). The sea-surface temperature was uniform at 9 K and the initial potential temperature above the surface was adjusted (by adding a constant to the potential temperature throughout the domain) so that the boundary layer was initially neutral in the centre of the domain, stable to the south, and unstable to the north. The model was run for five days, and a single cyclone life cycle was identified for the first 6 hours of the simulation. This life cycle will be the focus of the study. The control run (CONTRO) used the same version of the boundary layer scheme (ock et al. ) as in the operational NWP system... Diagnostics Here the diagnostic framework is described. The friction velocity, u, quantifies in a single scalar quantity both the x and y components of the surface stress, τ x and τ y : (τx u = + τ y ) ρ, () where ρ is the surface density. The cyclone-averaged surface boundary layer stress (CASS) is defined as: CASS = ρ u dx dy, () A cyc p 996 where the integral is performed over the area of the cyclone (with pressure p less than 996 hpa) and A cyc is the area of the cyclone. The boundary layer depthaveraged PV, [P ], is given by: h [P ] = ζ θ dz, (3) h ρ where P is the PV, square brackets imply averaging over the boundary layer depth, h is the boundary layer depth (the depth at which the boundary layer stresses fall to zero), ζ is the 3D absolute vorticity, is the 3D gradient operator, z is height, θ is potential temperature, and ρ is the density. Twomeasureswereusedtodeterminehowthefriction velocity and [P ] were distributed with stability. Stable (unstable) boundary layers were defined as those where the surface sensible heat flux, H s, was less (greater) than zero. The stable boundary layer region also defined the warm sector for CONTRO. The bulk stability of the boundary layer was given by the measure h/, where is the Obukhov length defined by: = θ u 3 κgh s. (4) κ is the von Kármán constant (set to.4), g is gravity, and θ the surface potential temperature. The measure h/ is positive for stable and negative for unstable boundary layers. It combines the depth of the boundary layer with the surface fluxes..3. Ekman pumping and PV mechanisms In order to set the context for later in the paper, the Ekman pumping and boundary layer PV mechanisms are outlined here. Assuming that the turbulence is in equilibrium with the mean profile gives the following momentum balance: f k u ag = z ( ) τ, (5) ρ where f, k, u ag, τ denote the Coriolis parameter, vertical unit vector, horizontal ageostrophic wind vector and boundary layer stress vector, respectively. In order to compare with the PV analysis of Cooper et al. (99), a uniform density, ρ, is assumed, as well as linear profiles of heat flux, H, and stress, τ: H = H s ( z h ) ; τ = τ s ( z h ) for z<h, (6) where τ s is the horizontal surface stress vector. Substituting the stress profile from Equation (6) into Equation (5) gives: f k u ag = τ s ρ h. (7) The equation of continuity is given by: u ag = w z. (8) Taking k of Equation (7) and substituting for u ag from Equation (8) gives: w z = ρ f h k τ s. (9) Integrating Equation (9) over the boundary layer depth gives: w h = k τ s, () ρ f where w h is the vertical velocity at the top of the boundary layer, the Ekman pumping or suction, which influences the large-scale flow by vertical vortex squashing or stretching. The Ekman pumping is a function of the surface stress but not its vertical profile in the boundary layer. This justifies a focus on the distribution of the surface stress using friction velocity later in the paper. Following Cooper et al. (99), the boundary layer depth-averaged PV tendency is given by: [ DP Dt ] = w hf θ ρ h ξ hh s ρ h τ s k θ h ρ h U (k H s) ρ h, () Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

4 56 R. J. BEARE where DP/Dt is the agrangian time derivative of PV, θ is the change of potential temperature across the boundary layer, ξ h and θ h are the vertical component of absolute vorticity and potential temperature respectively at the boundary layer top. U is the vector change in the horizontal wind across the boundary layer. Note that the Ekman pumping or suction (w h in Equation ()) is also included in the first term of Equation (). However, since the first term also depends on θ, its distribution is not necessarily the same as that of the Ekman pumping. For example, a neutral boundary layer ( θ = ) might have significant Ekman pumping but zero contribution to the PV generation. In Equation (), the last two baroclinic terms are significant in regions of large horizontal gradients of heat flux and potential temperature, typically the frontal zones of a cyclone. In contrast, the first two barotropic terms are not confined to just the frontal zones. Adamson et al. (6) demonstrated that, in addition to the PV flux from the top of the boundary layer, Equation () can be used to interpret the generation of [P ], and further reference to it will be made later. Explicit calculation of the individual terms will not be made however as this was done by Adamson et al. (6)..4. PV inversion In the course of the work, it became necessary to perform an inversion of the warm sector boundary layer PV. This enabled a direct evaluation of its influence on the surrounding flow. Such a calculation requires the specification of a balance condition and suitable boundary conditions (Davis and Emanuel, 99). Typical balance conditions are the quasi-geostrophic balance or the more accurate nonlinear balance. Within the boundary layer, however, these balances do not apply. In fact, the balance between the stress divergence, the Coriolis force and the pressure gradient expressed in Equation (5) is more appropriate. Above the boundary layer, a balance condition is approximately valid. In order to invert the warm sector PV, a depth-averaged PV anomaly, P a, needs to be defined. This is done by expanding Equation (3) about a reference potential temperature, θ r, and density, ρ r, which are constant with height, and also restricting the anomaly to the area of the cyclone, defined by surface pressures less than 996 hpa: P a = f h (θ θ r ) dz = f ρ r h z ρ r h θ for p<996 hpa. () This approximates the PV anomaly as a stratification anomaly, in agreement with Adamson et al. (6). However, instead of making a heuristic assessment of the impact of the anomaly in terms of Eady growth rate (as done by Adamson et al. 6), here the direct impact of the anomaly is made by PV inversion. Above the boundary layer (depth assumed constant at km to simplify this calculation), the quasi-geostrophic balance is used: ψ x + ψ y + f Ntrop ψ = for z h, (3) z where ψ is the geostrophic stream function and N trop is the tropospheric Brunt Väisälä frequency. The lower boundary condition for Equation (3) is the vertical gradient of ψ (Davis and Emanuel, 99), which is related to boundary layer depth-averaged PV anomaly: ψ g θ f (z = h) = = ghρ rp a. (4) z θ r f θ r The stream function is assumed to asymptote to zero at large distances from the anomaly. The geostrophic wind components (u h g, vh g ) at the top of the boundary layer are given by: u h g = ψ y (z = h); vh g = ψ (z = h). (5) x Assuming a barotropic geostrophic wind (a reasonable assumption within the warm sector), the boundary layer winds are calculated using the Ekman spiral solution for the balance in Equation (5) (Sorbjan, 989): u + iv = (u h g + ivh g [ ) exp { ( + i) πz }]. (6) h.5. Global NWP model simulation The idealized model differed from the Met Office operational forecast model in two main ways: the exclusion of moist physics, such as large-scale precipitation and moist convection, and idealized initial conditions. The degree to which these differences were important was tested by comparing the idealized simulation with the Met Office operational global forecast model (January 4 version) initialized with the UTC analysis for 4 January 4. The operational model was run for 8 hours and produced a deep extratropical cyclone in the Western Atlantic..6. Sensitivity to the large scale The basic state jet maximum, U jet, was varied in coarse increments of m s to determine how the large-scale environment influences the boundary layer structure. The Eady model growth rate (Eady, 949) provides a simple model of the baroclinic growth rate, σ eady : σ eady =.3 f U jet N trop H. (7) A simple modification, σeady, to include the magnitude of the upper-level trough amplitude (A = m s )gives σ eady =.3f (U jet + A) N trop H ; T = σ eady, (8) Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

5 CYCONE BOUNDARY AYERS 57 where T is a typical time-scale for baroclinic development. These scalings will be used to non-dimensionalize the data in order to explore the link between the jet and subsequent boundary layer development..7. Sensitivity to boundary layer fluxes As in several previous studies, the boundary layer and surface fluxes were turned off completely (NO B) to identify their overall role in the cyclone. In addition, the boundary layer fluxes were switched off for just stable (NO SB) and unstable (NO CB) boundary layers to explore the separate contributions from stable and unstable regions respectively. The sea-surface roughness (typically 3 4 m) in CONTRO, determined by the Charnock (955) formulation, was replaced by a typical land surface roughness of. m (AND). The previous modifications aid our understanding of the role of boundary layer fluxes in cyclone dynamics. However, they would not typically be considered as changes to an operational NWP model. Such operational changes are often much finer adjustments to the boundary layer scheme formulation. Thus, sensitivity to the following such modifications were tried: SHARP. The stability functions in the stable boundary layer are often configured to give enhanced mixing at large Richardson number. Beljaars and Viterbo (998) argued that these stability functions are required to give sufficient Ekman pumping in extratropical cyclones. However, a sharper fall-off with Richardson number (called SHARP) is more justified from detailed largeeddy simulations (Beare et al., 6) and comparison with observations over sea (Brown et al., 5). NG STRESS. Brown and Grant (997) demonstrated with large-eddy simulations that a local diffusion scheme tends to under-represent the mixing of momentum in convective conditions. This justified a nongradient stress parametrization (NG STRESS) which produces more well-mixed momentum profiles and greater boundary layer stress in convective conditions. NO OC CB. For convective boundary layers, the boundary layer scheme in CONTRO performs a maximum operation on an exchange coefficient which depends on the local gradient Richardson number and a non-local coefficient which scales with the boundary layer depth (ock et al., ). This is a pragmatic solution to ensure sufficient mixing in neutral boundary layers. In order to see how much effect this maximum operation has on a cyclone, the local scheme is turned off in convective boundary layers (NO OC CB). DOUB CHAR. The Charnock (955) formulation for wave-induced roughness is employed in CONTRO with a Charnock parameter of.. However, there is still a range of uncertainty in this parameter (Moon et al., 4), making it reasonable to double the Charnock parameter (DOUB CHAR). CU HOR; QU and QU. In addition to the boundary layer diffusion, a significant source of effective diffusion is from the interpolation scheme adopted to find departure points in the semi-agrangian advection scheme. The control model uses cubic agrange interpolation in the vertical and horizontal, however other schemes are more accurate, but more computationally expensive. The configurations tested here are quintic interpolation in the vertical and cubic agrange in the horizontal (CU HOR; QU), and quintic interpolation in both the horizontal and vertical (QU). 3. Results 3.. Evolution of control run Figure shows the evolution of CONTRO. In Figure, the PV on 3 K shows the initial upperlevel trough in thermal wind balance with the tropopause level vortex (Equations (A.) and (A.3)) and there is little disturbance in the m potential temperature at initial time. By 48 hours (Figure ), the coupling between the tropopause and surface causes development of a significant baroclinic wave. The m potential temperature pattern indicates a seclusion in the region of the low centre. Figure (c) shows a very deep low (956 hpa minimum pressure) by 48 hours, with weaker highs and lows developing upstream and downstream. The low is very asymmetric with much stronger pressure gradients on the southwest flank than to the east, giving a cold-conveyorbelt jet. Both the seclusion and the cold-conveyor-belt jet are in accord with the conceptual model of Shapiro and Keyser (99). Since the Shapiro and Keyser (99) model is useful in understanding many real extratropical cyclone cases (for recent examples, see Ahn et al., 5, Clark et al., 5), this idealized low has particular relevance. It will be the focus of much of the paper. 3.. boundary layer diagnosis of control run Figure 3 shows the time evolution of the friction velocity in CONTRO. Although there is a background friction velocity greater than. m s throughout the region of the low pressure system, there are also regions of enhancement indicated by the shading. At 4 hours, the friction velocity is largest in the cold and warm front regions. Although the friction velocity peaks outside the stable warm sector, there are still significant values on the warm side of the cold and warm fronts. As the cyclone develops (36 hours), the friction velocity pattern intensifies in the cold- and warm-front regions and elongates in the east west direction, in sympathy with the occluding low. By 48 hours, the two regions have merged into a wrap-around type of pattern peaking to the south of the the warm seclusion. The peak friction velocity (values greater than.5 m s ) is almost entirely in the convective boundary layer (to the north of the thin dashed line). Figure 4 shows the Ekman pumping for CONTRO at 48 hours diagnosed using Equation (). There is a narrow region of ascent wrapping around the low with Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

6 58 R. J. BEARE 5 Pot. temp at m; PV on 3K h 5 Pot. temp at m; PV on 3K 48 h y (degrees) y (degrees) x (degrees) x (degrees) 5 MSP (hpa) 48 hh y (degrees) (c) x (degrees) Figure. Horizontal cross-section of m potential temperature (solid contours, interval 3 K) and the PVU contour on the 3 K potential temperature surface (dashed) for CONTRO at hours and 48 hours. PVU = 6 Kkg m s. (c) mean surface-level pressure (contour interval 8 hpa) at 48 hours, with values > hpa dotted. The cyclone which is the focus of the paper is marked. a maximum of.9 m s to the south of the centre. The ascent region coincides with the friction velocity gradients and is situated mainly in the unstable boundary layer. There is some Ekman ascent in the northern part of the stable warm sector. There are also broad surrounding regions of descent (Ekman suction). In contrast, Figure 4 shows [P ] to be mainly positive and confined to the stable warm sector. Applying Equation (), the stable warm sector has negative surface fluxes (H s < ) and potential temperature increasing with height ( θ > ), giving positive barotropic terms in this regions. The peak of [P ] is to the south of the warm sector, in the region of the fractured cold front and warm conveyor belt. Adamson et al. (6) also show significant PV generation in this region, originating from the baroclinic terms (terms 3 and 4 of Equation ()). The location of the friction velocity is quite different from [P ] with only a slight overlap to the north of the warm sector. The overlap region is likely to be associated Ekman generation of PV (term in Equation ()). There is also negligible negative [P ]. Figure 5 aims to clarify how the maximum in boundary-layer friction velocity is forced. Figure 5 shows the friction velocity with the low-level (.9 km) potential temperature and upper-level (8.4 km) wind speed. Figure 5 then shows the vertical profiles of the wind speed at the point of maximum friction velocity (X), maximum upper-level jet (Y) and over the low centre (). The maximum friction velocity is forced by a distinct low-level jet, the so-called cold-conveyor-belt jet (Figure 5). Figure 5 then indicates that this jet lies in a region of low-level potential temperature increasing towards the low centre, giving southeasterly shear and wind speed decreasing with height by thermal wind balance. Combined with some support from the northern flank of the upper-level jet, the cold-conveyor-belt jet provides optimal conditions for maximum stress. In contrast, the point of the maximum upper jet does not lie over low-level potential temperature increasing to the north. The point over the low does not have the upperlevel support. Figure 6 shows the PV-inverted flow for the warm sector anomaly using Equations () (6). The top of boundary layer stream function (Figure 6) shows a maximum deepening of.5 hpa, near the apex of the warm sector, but separate from the low centre. The inverted stream function is also distinct from the wraparound friction velocity pattern. Figure 6 shows the vertical cross-section of the inverted winds showing a cyclonic circulation with a maximum value of about 7.5 m s on the eastern side of the warm sector. Although this is a significant magnitude of velocity, it is distinct from and much smaller than the low-level jet. Figure 7 shows scatter plots of friction velocity, Ekman pumping and [P ] against stability (h/). These provide more insight into stability dependence of the boundary layer diagnostics. The friction velocity (Figure 7) is strongly peaked at h/.5 onthe unstable side of neutral stability. Its distribution decays more rapidly on the stable (h/ > ) than the unstable (h/ < ) side. The Ekman pumping peaks near the neutral stability vertical line (h/ = ), and is more symmetric. In stark contrast, the [P ] has a negligible Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

7 CYCONE BOUNDARY AYERS 59 4 h 36 h 48 h 9 January 4 due to the relatively long integration time (8 hours), otherwise it had a similar occluded structure. Also the resolution was approximately half that in the zonal direction of the idealized simulation. Figure 8 shows the friction velocity and the line dividing stable and unstable boundary layers. In this low pressure system, the peak friction velocity has a similar wrap-around structure to the idealized simulation. Figure 8 demonstrates that the peak friction velocity is also in the weakly neutral convective boundary regime and with a similar shape of stability distribution. Thus, the relevance of the idealized set-up to a full forecast scenario is confirmed Sensitivity to large-scale forcing Figure 9 shows the evolution of minimum mean sea level pressure (MSP) and maximum friction velocity for the first 7 hours of simulation and for different basic state jet strengths, U jet. Clearly, the larger the initial jet, the faster the growth of the cyclones, consistent with the Eady growth rate (Equation (7)). Also, the maximum friction velocity increases with initial jet strength, suggesting that there may be a scaling which collapses the curves. Therefore, Figure 9(c) shows the friction velocity scaled by U scal (where U scal =.(U jet + A), wherea is the initial upper-level vortex magnitude) against time scaled by the Eady time-scale, T (Equation (8)). The curves collapse well with the non-dimensionalized value of max(u ) increasing to a maximum value of max{max(u )}.7(U jet + A); t.45t, (9) (c) Figure 3. Time evolution of friction velocity (feint contours with interval.5 m s ). Values >.5 ms (> ms ) are shaded light (dark) grey. The bold dashed and dash double dotted lines show the positions of the surface cold and warm fronts respectively. The thin dashed line divides the stable and unstable boundary layers, with stable boundary layers south of the line. The horizontal region for each time is centred on the low. contribution from the unstable boundary layer and is only significant in the stable boundary layer (h/ > ) Comparison with forecast simulation The Met Office global forecast low is shown in Figure 8. It was hpa deeper than the analysis for UTC on and then remaining fairly constant. The close scaling of friction velocity with large-scale jet-trough strength shows the tight coupling between the two Sensitivity to boundary layer fluxes Figure shows the evolution of the minimum MSP over 6 hours for CONTRO and coarse changes to the boundary layer and surface mixing. CONTRO deepens significantly over this time, by as much as 7 hpa between 4 and 48 hours, occluding at a depth of 95.5 hpa by 6 hours. Turning the boundary layer and surface fluxes off completely (NO B) produces a large deepening of 5 hpa over 6 hours relative to CONTRO. Conversely, replacing the sea-surface roughness with typical land values (AND) produces a 6-hour relative filling of 4 hpa. This illustrates the enhanced spin-down experienced by a cyclone when it passes from sea to land. Switching off the boundary layer mixing in just the stable warm sector (NO SB) and unstable convective regions (NO CB) enables further clarification of the dominant mechanism of spin-down. The NO CB case has much more impact than the NO SB case. The NO CB case produces.5 hpa deepening relative to CONTRO by 6 hours whilst NO SB produces Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

8 5 R. J. BEARE 48 h Ekman pumping 48 h B depth averaged PV Figure 4. Ekman pumping (contour interval. m s, with solid contours positive, dotted negative, and zero contour omitted for clarity), plotted over friction velocity >.5 ms (shaded) and the line dividing stable and unstable boundary layers (dashed). is as but shows the boundary layer depth-integrated PV (contour interval PVU = 6 Kkg m s ) h Y X Wind at: X (solid), Y (dashed), (dotted) 35 3 z(m) wind m/s Figure 5. Wind speed at 8.4 km (bold contours, interval m s ), potential temperature at.9 km (feint dotted, interval 3 K) and friction velocity with values >.5 ms (> ms ) shaded light (dark) grey. Vertical profiles of wind speed at the point of maximum 8.4 km wind (Y in ), point of maximum friction velocity (X in ) and over the low pressure centre ( in ). 5.5 hpa deepening. This further confirms that it is the unstable boundary layer that is producing the most significant stress. Since the Ekman pumping is generated by the unstable boundary layer stress, there is strong evidence that it is the dominant mechanism. Figure shows the corresponding cycloneaveraged surface boundary layer stress (CASS, Equation ()). The runs which produce more deepening (NO B, NO CB, NOSB) tend to have less CASS relative to CONTRO, with the slight exception of the end of the NO SB run. Conversely, the AND run produces less deepening than CONTRO and more CASS. This provides more evidence that the surface stress, and thus the Ekman pumping, plays an important role in determining the cyclone depth at occlusion. Figure shows the changes relative to CONTRO in minimum MSP for fine changes to the boundary layer mixing and dynamical diffusion. The changes span 3 hpa over 6 hours and have compensating effects, i.e. some deepen and some fill relative to CONTRO. The differences tend to become apparent beyond 3 hours, during the occluded phase of each cyclone. There are small changes for the NG STRESS (+.3 hpa relative to CONTRO) and SHARP (.5 hpa). The small magnitude of this change is likely to be related to the fact that the distribution of the friction velocity at occlusion peaks close to neutral stability (Figure 7), whilst the NG STRESS and SHARP changes have particular impact in both the free convective and very stable limits, respectively. The NO OC CB and DOUB CHAR Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

9 3 CYCONE BOUNDARY AYERS height (m) J Figure 6. Horizontal section of the stream function at the top of the boundary layer (contour interval hpa, negative values dotted), friction velocity greater than.5 m s (shaded), and the boundary layer depth-averaged PV anomaly.5 PVU contour (bold). x z section of the inverted winds through the centre of the domain in (contour interval m s, with negative values dotted). The low-level jet (J) wind speed contours >3 m s are shown dashed..5 Friction Velocity (ms )..5 Ekman pumping (cms ) 5. 4 h/ 5 4 h/ 5 B Depth avergaged PV (PVU) (c) 5 4 h/ Figure 7. The variation of friction velocity, Ekman pumping and (c) boundary layer depth-averaged PV with boundary layer stability, h/. PVU= 6 Kkg m s. The neutral line (h/ = ) is marked as a bold vertical line. runs, representing operational changes that are not a function of boundary layer stability, have larger impacts, of order.3 hpa relative deepening and.5 hpa filling respectively. The changes to the interpolation scheme used in the semi-agrangian advection also have a small impact. The cubic horizontal and quintic vertical interpolation (CU HOR; QU) produces only a.5 hpa filling at 6 hours, whereas the quintic interpolation in both the vertical and the horizontal has a.4 hpa deepening. Also the changes in CASS (Figure ) correspond tightly to the changes in MSP, with the exception of the SHARP experiment. For example, the DOUB CHAR run gives a positive change in the MSP relative to CONTRO, and a corresponding increase in CASS. The DOUB CHAR Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

10 5 R. J. BEARE Global model Z 9//4.5 6N 5N 4N W 5W 4W 3W 99 Friction Velocity (m/s) h/ Figure 8. Met Office global forecast run (Western Atlantic region) centred on an occluded low pressure system, valid for UTC on 9 January 5: friction velocity, with values >.5 ms (> ms ) shaded light (dark) grey, the dashed line dividing stable and unstable boundary layers (the stable warm sector is on the right), and mean-sea-level pressure (feint contours with interval 8 hpa). The variation of friction velocity with h/.. min MSP (hpa) ms 4 ms 5 ms 6 ms 7 ms max u * (ms ) t (h) t (h).8 (max u * )/u scal.6.4. (c) t/t Figure 9. Time series using different initial jets for the central low minimum mean-sea-level pressure and maximum friction velocity. (c) shows the normalized friction velocity versus normalized time. U scal =.(U jet + A), with A = m s the initial upper-level vortex magnitude. Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

11 CYCONE BOUNDARY AYERS 53.8 minimum MSP (hpa) CONTRO NO B NO SB NO CB Cyclone averaged B stress (Nm ).6.4. AND time (h). 4 6 time (h) Figure. Time series of the minimum mean-sea-level pressure and the surface stress averaged over the cyclone for the coarse sensitivity experiments..4 change in min mslp (hpa) SHARP NG STRESS CU HOR; QU QUINTIC NO OC CB DOUB CHARNK change in cyclone ave. stress (Nm ) time (h) 4 6 time (h) Figure. Time series of the change relative to CONTRO of minimum mean-sea-level pressure and the surface stress averaged over the cyclone for the fine sensitivity experiments. run also indicates the importance of the wave-induced stress on the cyclone evolution (Doyle, 995). 4. Discussion This paper aimed to clarify the mechanism of the boundary layer s interaction with extratropical cyclones. Two dynamically motivated approaches were compared: Ekman pumping and PV. On the one hand, Ekman pumping derives from the convergence induced by boundary layer stress. On the other hand, PV combines both the stratification and vorticity in a single quantity, and has proven invaluable in explaining large-scale dynamical mechanisms (Hoskins et al., 985); its utility in explaining boundary layer dynamics is less established, however, although Adamson et al. (6) present a case for it. Whilst the Ekman pumping gives a clear insight into the decay of the cyclone via vortex squashing, the PV needs to be inverted to fully determine its effect on the large-scale circulation. An inversion was not performed by Adamson et al. (6), but was in this study. The distribution of friction velocity, Ekman pumping and boundary layer depth-averaged PV were compared for an idealized extratropical life cycle in the Met Office Unified Model. In agreement with Adamson et al. (6), the PV at occlusion was within the warm sector Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

12 54 R. J. BEARE and warm-conveyor-belt regions. However, the Ekman pumping was mainly in the cold sector, overlapping little with the PV, and forced by the cold-conveyor-belt jet in the wrap-around region of cold air at occlusion. The boundary layer surface stress which forced the Ekman pumping was maximum in a near-neutral region of the unstable boundary layer. The PV on the other hand emphasized stratified boundary layers over well-mixed unstable boundary layers. The inversion of the warm sector PV indicated only a small deepening effect distinct from the low-level jet forcing the Ekman pumping. This result disagrees with Adamson et al. (6), who made a heuristic argument for the boundary layer PV decreasing the cyclone growth rate. Moreover, the deepening from the inverted warm sector PV was opposite to the filling from the stable boundary layer fluxes (demonstrated when they were switched off). This contradiction shows how little the PV represents the boundary layer s role in the cyclone dynamics. The relative importance of warm and cold sector boundary layer fluxes was further investigated. The novel method of separately switching off the boundary layer fluxes in the unstable cold sector and the stable warm sector revealed much more contribution from the cold sector. The Ekman pumping mechanism, forced by the cold conveyor belt, was thus the most significant source of the boundary layer decay. Whilst other studies emphasize the warm sector boundary layer in extratropical cyclone dynamics and predictability (angland et al., 995; Adamson et al., 6), this paper has revealed the important role of the cold sector boundary layer. The friction velocities for different cyclone growths were non-dimensionalized with respect to the tropopause level winds at initial time. An intriguing collapse of the data was achieved. Thus, a close link was established between the strength of the initial upper jet trough system and the maximum friction velocity later in the cyclone life cycle. This demonstrates that the evolution of the boundary layer in a cyclone must be interpreted within the context of the large-scale flow. Take the example of the relative performance of a forecast model s m winds in cyclone life cycles at two different resolutions. The change in m winds could be due to the increased resolution of the baroclinic zones as well as changes due to the boundary layer mixing scheme. Further support for the Ekman mechanism was provided by a close link between the boundary layer surface stress averaged over the cyclone (CASS) and the amount of the deepening. As the CASS increased (by changing the boundary layer scheme), the cyclone deepened less and vice versa. Since the Ekman pumping depends closely on the surface stress, this also implies similar changes to it. The way in which changes to the boundary layer scheme corresponded to changes in CASS demonstrates that Ekman pumping is a good framework for interpreting changes to boundary layer schemes in NWP models. Appendix Initialization The initial state is defined on a limited-area domain using Cartesian coordinates x, y, z. The initial wind field is defined as (U bs + U vort, V vort,),whereu bs is the basic state zonal jet, and U vort,v vort are the x,y components of the upper-level vortex. a. Initial jet The initial basic state zonal jet, U bs,isgivenby U bs (y, z) = U jet sinh(mz/h ) sinh(m) [ { ( cos l y + )}] y for z H, y y y, U bs (y, z) = U { jet exp ms ( z ) } H [ { ( cos l y + )}] y for z H, y y y, and U bs = otherwise. (A.) U jet, m, l, H and y are the maximum jet wind speed, a vertical wave number, a horizontal wave number, the depth of the troposphere and the meridional width of the jet, respectively. S = Nstrat /N trop = 4, where N strat and N trop are the stratospheric and tropospheric Brunt Väisälä frequencies. In the control run, m =.7, U jet = 5 m s, y = 3 km, H = 9km, and l = π/ y. The jet is plotted in Figure, along with the zonal mean potential temperature in thermal wind balance with it. b. Upper vortex The upper vortex pattern (U vort, V vort )isgivenby { } πntrop (z H) U vort = Aexp f x y ( ) πr r sin x for z H,r x /, { } πnstrat (H z) U vort = Aexp f x y ( ) πr r sin x for z H,r x /, and U vort = otherwise. (A.) Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj

13 CYCONE BOUNDARY AYERS 55 { } ( ) πntrop (z H) x x πr V vort = A exp sin f x r x for z H,r x /, { } ( ) πnstrat (H z) x x πr V vort = A exp sin f x r x for z H,r x /, and V vort = otherwise. (A.3) f =.9 4 s is the Coriolis parameter, A = m s is the amplitude of the vortex, x = 5 km is the location of the vortex centre in the x direction relative to the centre of the domain, x = 3 km is the width of vortex in the x direction, r ={(x x ) + y }.5 is the radial distance from the centre of the vortex. N trop =.6 s,andnstrat /N trop = 4 c. Thermal wind balance Thermal wind balance is integrated horizontally to calculate a consistent potential temperature, θ, distribution from the prescribed wind fields. The thermal wind balance is given by θ x = f θ V vort g z ; θ y = f θ (U jet + U vort ). g z (A.4) Acknowledgement I would like to thank Richard Forbes of the UK Met Office for help setting up the idealized configuration of the Unified Model. I also appreciated several discussions of the material with Andy Brown, Adrian ock, Roy Kershaw and Mike Cullen of the Met Office during the writing of the paper. References Adamson DS, Belcher SE, Hoskins BJ, Plant RS. 6. boundary layer friction in midlatitude cyclones. Q. J. R. Meteorol. Soc. 3: 4. Ahn JV, Sienkiewicz J, McFadden G. 5. Hurricane-force extratropical cyclones observed using QuikSCAT near real time winds. Mariners Weather og 49: 5/ toc.shtml. Beare RJ, Thorpe AJ, White AA. 3. The predictability of extratropical cyclones: Nonlinear sensitivity to localized potentialvorticity perturbations. Q. J. R. Meteorol. Soc. 9: Beare RJ, MacVean MK, Holtslag AAM, Cuxart J, Esau I, Golaz J-C, Jimenez MA, Khairoutdinov M, Kosovic B, ewellen D, und TS, undquist JK, McCabe A, Moene AF, Noh Y, Raasch S, Sullivan P. 6. An intercomparison of large-eddy simulations of the stable boundary layer. Boundary ayer Meteorol. 8: Becker A, Kraus H, Ewenz CM Frontal substructures within the planetary boundary layer. Boundary ayer Meteorol. 78: Beljaars ACM, Viterbo P Role of the boundary layer in a Numerical Weather Prediction model. Pp in Proceedings of the Academy colloquium Clear and cloudy boundary layers. Holtslag AAM, Duynkerke PG. (eds.), Royal Netherlands Academy of Arts and Sciences: Amsterdam, The Netherlands. Brown AR, Grant AM Non-local mixing of momentum in the convective boundary layer. Boundary ayer Meteorol. 84:. Brown AR, Beljaars ACM, Hersbach H, Hollingsworth A, Miller M, Vasiljevic D. 5. Wind turning aross the marine atmospheric boundary layer. Q. J. R. Meteorol. Soc. 3: Charnock H Wind stress on a water surface. Q. J. R. Meteorol. Soc. 8: Clark PA, Browning KA, Wang C. 5. The sting at the end of the tail: Model diagnostics of fine-scale three-dimensional structure of the cloud head. Q. J. R. Meteorol. Soc. 3: Cooper IM, Thorpe AJ, Bishop CH. 99. The role of diffusive effects on potential vorticity in fronts. Q. J. R. Meteorol. Soc. 8: Davies T, Cullen MJP, Malcolm AJ, Mawson MH, Staniforth A, White AA, M. H. M., Wood N. 5. A new dynamical core for the Met Office s global and regional modelling of the atmosphere. Q. J. R. Meteorol. Soc. 3: Davis CA, Emanuel KA. 99. Potential vorticity diagnostics of cyclogenesis. Mon. Weather Rev. 9: Doyle JD Coupled ocean wave/atmosphere mesoscale model simulations of cyclogenesis. Tellus A 47: Eady ET ong waves and cyclone waves. Tellus : Holton JR. 99. An introduction to dynamic meteorology. Academic Press: New York, USA. Hoskins BJ, McIntyre ME, Robertson AW On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteorol. Soc. : Keyser D, Anthes RA. 98. The influence of planetary boundary layer physics on frontal structure in the Hoskins Bretherton horizontal shear model. J. Atmos. Sci. 39: angland RH, Elsberry R, Errico RM Evaluation of physical processes in an idealized extratropical cyclone using adjoint sensitivity. Q. J. R. Meteorol. Soc. : ock AP, Brown AR, Bush MR, Martin GM, Smith RNB.. A new boundary layer mixing scheme. Part I: Scheme description and single-column model tests. Mon. Weather Rev. 8: Martin JE, Grauman RD, Marsili N.. Surface cyclolysis in the North Pacific Ocean. Part I: a synoptic climatology. Mon. Weather Rev. 9: Moon I-J, Hara T, Ginis I, Belcher SE, Tolman H. 4. Effect of surface waves on air-sea momentum exchange. Part I: Effect of mature and growing seas. J. Atmos. Sci. 6: Persson POG, Hare JE, Fairall CW, Otto WD. 5. Air-sea interaction processes in warm and cold sectors of extratropical cyclonic storms observed during FASTEX. Q. J. R. Meteorol. Soc. 3: Shapiro MA, Keyser D. 99. Fronts, jet streams, and the tropopause. Pp in Extratropical Cyclones: The Erik Palmen Memorial Volume. Newton CW, Holopainen EO. (eds.) American Meteorol. Soc: Boston, USA. Sorbjan Z Structure of the atmospheric boundary layer. Prentice Hall: New Jersey, USA. Thompson WT, Williams RT Numerical simulations of maritime frontogenesis. J. Atmos. Sci. 54: Crown Copyright 7. Reproduced with the permission of the Controller of HMSO. Q. J. R. Meteorol. Soc. 33: (7) DOI:./qj