Generation of mountain wave-induced mean flows and turbulence near the tropopause

Transcription

1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 139: , July 2013 B Generation of mountain wave-induced mean flows and turbulence near the tropopause John McHugh a * and Robert Sharman b a University of New Hampshire, Durham, NH, USA b National Center for Atmospheric Research, Boulder, CO, USA *Correspondence to: J. McHugh, University of New Hampshire, Kingsbury Hall, 33 Academic Way, Durham, NH 03824, USA. john.mchugh@unh.edu Gravity waves interacting with the tropopause are investigated using linear and nonlinear numerical simulations. The tropopause is modelled as the interface between two layers of constant Brunt Väisälä frequency. The simulations are twodimensional with uniform horizontal flow, the background rotation is ignored, and the gravity waves are generated by flow over an idealized isolated obstacle shape at the surface. The nonlinear simulation results show a horizontal wave-induced mean flow at the tropopause similar to previous results treating horizontally periodic internal waves impinging on a density-gradient interface. The mean flow created by the impinging gravity waves is increased over the background wind below the tropopause and decreased above the tropopause. This effect is not present in the linear simulations. The nonlinear effect is felt more strongly for cases with higher mountain heights and larger values of the stability in the upper layer. The final steady mountain-wave flow appears to permanently retain this mean-flow change. The deceleration region above the tropopause results in a patch of slow-moving fluid near the interface which induces local regions of reduced Richardson number and may help explain some observational results of higher turbulence intensities near the tropopause over mountainous regions. Key Words: tropopause: turbulence; gravity waves Received 6 June 2011; Revised 1 August 2012; Accepted 8 August 2012; Published online in Wiley Online Library 16 November 2012 Citation: McHugh J, Sharman R Generation of mountain wave-induced mean flows and turbulence near the tropopause. Q. J. R. Meteorol. Soc. 139: DOI: /qj Introduction Abundant empirical evidence (e.g. Partl, 1962; Hopkins, 1977; Chandler, 1987; Lester, 1993; Worthington, 1998; Pavelin et al., 2001, 2002; Whiteway et al., 2003; Wolff and Sharman, 2008) shows that higher levels of turbulence exist near the tropopause region. Many of these observations (Worthington, 1998; Pavelin et al., 2001, 2002; Whiteway et al., 2003) not only register more turbulence at the tropopause region, but also link the turbulence to atmospheric waves. Worthington (1998) measured turbulence and other quantities in the troposphere and lower stratosphere using ground-based radar. He found higher levels of turbulence in the tropopause region and showed that breaking mountain waves were likely responsible. This is consistent with the MU (Middle and Upper atmosphere) radar observations over Japan by Gavrilov and Fukao (2004). Pavelin et al. (2001) made similar measurements using both ground-based radar and radiosondes. They observed strong wave activity, but argue that there were no mountain waves on this occasion. They conclude that the observed waves were inertia gravity waves caused by geostrophic adjustment processes. They measured a layer of strong turbulence very near the tropopause altitude, much stronger than all other altitudes considered, which they argue was caused by breaking of the inertia gravity waves near the tropopause. c 2012 Royal Meteorological Society

2 Mountain Wave-Induced Effects Near the Tropopause 1633 Pavelin et al. (2002) measured turbulence from an aircraft, along with ground-based radar and radiosondes. The measurements show a strong horizontal jet flow beneath the tropopause, and again strong turbulence near the tropopause. They also measured strong gravitywave activity. The strong turbulence occurs near the tropopause, which is also at the upper edge of the jet. The authors attribute the turbulence to a Kelvin Helmholtz phenomenon associated with the jet shears, rather than breaking mountain waves. Whiteway et al. (2003) also performed airborne measurements along with ground-based radar. The radar measurements indicated strong mountain-wave activity with wave overturning only at the tropopause altitude. The aircraft measurements were made at this wave-breaking altitude and provide strong evidence that mountain waves may break at the tropopause and are a probable source of turbulence there. McHugh et al. (2008a, 2008b) performed experiments over Hawaii using balloon-borne thermosondes. The results again show that the tropopause region had strong levels of turbulence, although other altitudes also experienced strong turbulence. These results also show very strong updraughts only at the tropopause region, indicated by large increases in the ascent rate of the balloons at that altitude and attributed to mountain-wave activity. Altitudes with strong turbulence also showed that the background velocity was near zero, and there were very sudden shifts in wind direction with increasing altitude, both features pointing to the existence of a critical layer at these altitudes. All of these observations clearly indicate higher levels of turbulence at or near the tropopause which seems to be related to gravity-wave or inertial gravity-wave activity. Also, in many cases these waves appear to be topographically induced since they occur over bottom topography that includes significant mountains, and the turbulence occurs directly over the mountains (rather than downstream, for example). This suggests that mountain waves can either directly cause the turbulence by breaking, or that mountain waves have some other role, along with background shear, that ultimately results in tropopausal turbulence. Gravity waves impinging on the tropopause (defined here as the interface between two layers of constant Brunt Väisälä frequency or stability N)frombelowwillbe partially reflected, as was first shown by Scorer (1949). Scorer considered mountain waves over a symmetric obstacle using the Witch of Agnesi obstacle shape. Further results in a two-layer flow with and without shear are reviewed e.g. by Baines (1995). Numerical simulations of free internal waves impinging on a density-gradient interface were recently considered by McHugh (2008). The waves were periodic in the horizontal and the amplitude was modulated in the vertical, creating a vertically propagating wave packet. The results show that as the wave packet transits the interface from below, a horizontal wave-induced mean flow is generated at the interface. These results suggest that this wave-induced mean flow causes a region of lowered Richardson number, and may even be strong enough to form a critical layer, causing further incident waves to amplify and break just beneath the interface. In either case, high levels of turbulence would result near the tropopause. Uniform internal waves in the same two-layer flow were considered by McHugh (2009). The wave amplitude was assumed to be small but finite, and constant throughout the domain (no envelope). The results showed that higher harmonics (beyond the first few) accumulate at the interface, and are evanescent elsewhere, providing another possible explanation for nonlinear wave behaviour that is focused only at the tropopause. However, the wave-induced mean flow for these waves does not have a localized component near the interface. Horizontally confined internal wave packets have been considered previously by Bretherton (1969), Sutherland (2001), and Tabaei and Akylas (2007) for the case of uniform stratification (no tropopause). The waves were free travelling waves that were also confined in the vertical. Bretherton (1969) and Tabaei and Akylas (2007) discuss amplitude equations and the more general theory for slowly varying waves. Sutherland (2001) provides numerical simulations without the restriction to slowly varying waves. All indicate the presence of a wave-induced mean flow that is primarily driven by the details of the packet shape. Sutherland s results also show that for large-amplitude waves, an instability develops within the packet, suggested to be the result of a resonance between the incident waves and the wave-induced mean flow. Both Bretherton (1969) and Tabaei and Akylas (2007) indicate the presence of a mean-flow wake behind the wave packet. Mountain waves over isolated obstacles are limited horizontally, different than the horizontally periodic waves considered in McHugh (2008). However, the basic underlying gravity wave dynamics is the same for mountain waves, implying that mountain waves will also induce meanflow changes near the tropopause, resulting in reduced Richardson numbers, and this phenomenon is considered here. The flow over an isolated obstacle includes a spectrum of wave numbers rather than a single wave number, as is well-known, and this phenomenon is therefore much more complicated than the monochromatic forcing case of McHugh (2008). A spectrum of internal waves impinging on the tropopause was discussed previously by VanZandt and Fritts (1989). They point to several observations that show enhanced mean flows and dissipation near the tropopause, as here, and suggest that the cause is an increase in shear-driven instability as a result of the sudden decrease in the vertical wavelength above the tropopause. In contrast, the simulations of horizontally periodic waves by McHugh (2008) do not show a shear instability above the tropopause, but instead show a localized mean flow that develops at the tropopause. The present results also show an enhanced mean flow at the tropopause. Hence the argument of VanZandt and Fritts (1989) does not appear to fully explain this phenomenon. The mean flow induced by mountain waves for an isolated obstacle is considered here using numerical simulations of the inviscid Boussinesq equations for a stratified atmosphere. Mountain waves are different than the packets considered by Bretherton (1969), Sutherland (2001), and Tabaei and Akylas (2007) for several reasons. The traditional problem of steady mountain waves ignores the starting conditions and therefore has no wave packet. However, if the waves are initiated at the bottom in a flow that initially has no waves, as here, then a packet shape is present during the initiation stage. However, as the flow is allowed to approach a steady mountain-wave solution, there is no termination to the packet shape, making the concept of a wake behind this packet ill-defined. Furthermore, flow over mountains creates a spectrum of incident waves, as mentioned above.

3 1634 J. McHugh and R. Sharman And finally, the waves are not slowly varying, making it difficult to separate the waves from the mean flow. A two-layer background state is chosen with constant buoyancy frequency, N, in each layer. The background velocity, U, is constant throughout. Previous simulations by Durran (1995) also treated the mean flow induced by mountain waves, including a case with constant N and U, although this particular case did not have a model of the tropopause. Durran (1995) also treated several cases that included a model of the tropopause. However, background shear and a jet stream were also included to mimic previous observations. Durran s results for the case with a tropopause show that the waves cause the jet stream to increase in strength while the background flow outside the jet decreases in strength; however, there was no mention of localized mean-flow changes near the tropopause, and with the other features of the background state, it is difficult to identify any feature as solely caused by the presence of the tropopause. The results given here will show that mountain waves in the simplest context will generate a region of wave-generated mean flow near the tropopause, which leads to regions of local reductions in the Richardson number, favouring the production of turbulence. The article is organized as follows: the next section provides the basic equations and a discussion of the numerical model used. Following that, section 3 presents some results of the simulations. Finally, a brief summary and conclusions are provided in section Governing equations and numerical methods The governing equations are the nonlinear Boussinesq equations in two dimensions: u t + x (u2 ) + (uw) = P z x, w t + x (uw) + z (w2 ) = P + gσ, (1) z σ t + x (uσ ) + N2 (wσ ) + z g w = 0, u x + w z = 0, where (u, w) are the horizontal and vertical velocity components, P = p /ρ 0 is the pressure deviation from a base state in hydrostatic balance, σ is the normalized potential temperature deviation, θ /θ 0,andg is the gravity constant. Additional terms are included for a sponge layer at the top of the domain using Rayleigh friction. These equations are solved numerically with finite differences using the same methods as described in Sharman and Wurtele (1983). The spatial derivatives are second order on an Arakawa-C grid. Time differencing is treated with the leap-frog method, except for Rayleigh friction terms which are treated implicitly. In all cases the mountain shape is specified by the commonly used Witch of Agnesi profile: h(x) = H 1 + x2 A 2, (2) where H is the maximum obstacle height and A is the obstacle half-width. In order to isolate nonlinear effects in the vicinity of the tropopause from nonlinear effects introduced by flow over high obstacles, the lower boundary condition used prescribes the vertical velocity at z = 0 rather than on the topography through: w 0 = ( U h ), (3) x z=0 where U is the constant mean-flow velocity. Including the nonlinear or non-separable lower boundary condition would make the results more complex by adding vertical harmonics, as discussed by Smith (1977), and perhaps result in steeper waves, but would not change the fundamental conclusions concerning wave-induced mean flows at the tropopause. In all cases to be presented, the simulation grid spacing is x =100 m and z = 16 m. The time step was adjusted according to the grid spacing, and is typically t =1.5s. The background speed U was fixed at a constant value for all simulations. The tropopause is defined by an abrupt change in the constant value of N from its lower layer value N = N 1 to a larger value in the upper layer of N = N 2. The obstacle and therefore the imposed vertical velocity at z = 0 was gradually increased by letting the obstacle height H increase linearly over a fixed number of time steps until its final value, after which the obstacle height remained constant. Several values for the ramp-up time were tested, and this feature did not appear to strongly affect the final results. For all results given here, this ramp-up time was fixed at 1500 s (0.42 h). Typically the computational domain contained 2049 grid points in the horizontal, 1876 grid points in the vertical, and the upper third of this contained the Rayleigh damping layer with viscosity increasing linearly from its base to the model top. An important parameter for mountain waves is the nondimensional obstacle height or inverse Froude number, NH/U (e.g. Baines, 1995). Many previous results have shown thatlarger valuesof NH/U result in steeper streamlines and isentropes, and ultimately wave overturning. The critical value of NH/U for overturning depends on the configuration and is often difficult to determine. Miles and Huppert (1969) assumed steady mountain waves in an infinite Boussinesq fluid with constant N and U, and the Witch of Agnesi obstacle profile (Eq. (2)), and estimated the critical value for overturning using the hydrostatic approximation but with the exact nonlinear lower boundary condition to be NH/U = Values of NH/U greater than 0.85 will allow overturning streamlines at a sequence of altitudes above the obstacle while lower values will produce no overturning at any altitude. Lilly and Klemp (1979) showed that the solution to the same problem with the linear lower boundary condition gives a critical value of NH/U = 1.0. Another important parameter is H/A or NA/U. For narrow obstacles with H/A > O(1), the critical value of NH/U > 1(Milesand Huppert, 1969; Laprise and Peltier, 1989). In all cases, the critical values pertain to steady mountain waves. The present results focus on the initial stages of wave development, so that the exact critical value is unclear. For most cases considered here, NH/U = 1.0 and H/A = 1.0. The present simulations show that the dynamic effects at the tropopause develop at these values of NH/U and H/A without overturning elsewhere in the domain. However, larger values of NH/U > 1 did produce overturning in the lower layer before the waves became fully developed, and this has been avoided by keeping NH/U atunity.

4 Mountain Wave-Induced Effects Near the Tropopause 1635 Table 1. Parameter values for the U =10ms 1 case. Parameters Value Mountainheight (H) 1000 m Mountainwidth (A) 1000 m Mean flow (U) 10m/s N in troposphere (N 1 ) 0.011/s N in stratosphere (N 2 ) /s Tropopause height m NH/U 1 NA/U 1 Horizontal grid spacing ( x) 100 m Vertical grid spacing ( z) 16m 3. Results An idealized case that is representative of observational values at midlatitudes is U =10ms 1 with N 1 =0.01s 1 in the lower layer, N 2 = s 1 in the upper layer, and this case is considered first; a complete list of simulation parameters is provided in Table 1. Results for this case are given in Figures 1 and 2, each showing contours of horizontal velocity and local Richardson number for a model time of s (5 hours) from initiation. Figure 1 is a linear simulation using a linearized version of the governing Eqs (1) while Figure 2 includes nonlinear effects in the equations of motion. Both simulations use the linear lower boundary forcing (Eq. (3)). The local Richardson number (Ri)isdefinedby N 2 Ri = (du/dz) 2, (4) where the stability in the numerator and the shear in the denominator contain the effects of both mean flow and wave-induced perturbations. Note that Ri has been smoothed once in Figures 1 and 2 using a five-point smoothing function. Also in Figures 1 and 2, the obstacle is located at the origin (x = 0), the tropopause is located at the vertical centre of the figure, and the computational domain is significantly larger than the region shown. The contours in Figures 1 and 2 indicate that the mountain wave pattern is nearly fully developed, except in the very far field. In this case NA/U = 1.0, smaller than the value required for approximate hydrostaticity (e.g. Smith, 1979; Laprise and Peltier, 1989), and some lee-wave activity is present, as can be seen in Figures 1 and 2. Larger values of NA/U were considered and generally show a weaker nonlinear mean flow at tropopause (discussed below). The contours of horizontal velocity in Figures 1 and 2 demonstrate a complex behaviour at the tropopause downstream of the obstacle, including a change in tilt of the phase lines, but this appears to be a minor feature. The contours of Ri for the nonlinear simulation in Figure 2 provide stronger evidence that this region has significant complexity, and that there are important dynamic processes in play near the tropopause. The contours of Ri for the linear case in Figure 1 do not show this complexity, indicating that nonlinear effects are responsible for the increased complexity near the tropopause. Figure 3 shows profiles of total horizontal velocity at a fixed horizontal position 6543 m downstream of the obstacle centreline for a sequence of times for the nonlinear simulation shown in Figure 2. Each subsequent profile in Figure 3 is shifted Figure 1. Contours of total horizontal velocity u and Ri for a linear simulation. The mountain is located at the origin (x = 0), while the tropopause is located at the vertical centre. Time is s, U=10 m s 1 and H=1000 m. Thin solid lines in are the u=10 m s 1 contours, while thick solid lines have u <10 m s 1, and dashed lines have u >10 m s 1, incremented by 2 m s 1.ContoursinhaveRi values of 1, 10 and 100. by 10 m s 1 for display. The horizontal position of the profiles in Figure 3 is chosen as the position where the value of horizontal velocity near the tropopause reaches a minimum. The waves for the first (leftmost) profile are not quite fully developed, as can be seen in Figure 3 by the small wave amplitudes in the upper layer. The second (centre) and third (rightmost) profiles correspond to later times when the waves are more developed. The important feature in Figure 3 is that the horizontal velocity is a minimum at the mean position of the tropopause and presents a cusp-like feature in the vertical profile there. This is true for the second and third profiles in Figure 3, and all later profiles (not shown). The simulation was terminated at a time of s for practical reasons, and the wave pattern at this stage had not quite reached the steady mountain-wave solution throughout the entire computational domain. It is not clear from these results if this peak in horizontal velocity is muted for very long times. A corresponding steady-state solution to Long s equation for the same basic two-layer flow was considered by Durran (1992), but he does not provide profiles of horizontal velocity. It is not clear whether the numerical solution generated here should match the steady solution of Durran (1992), as the solution generated here may have a mean-flow component that is related to the initiation of the mountain waves, not included in Durran s steady

5 1636 J. McHugh and R. Sharman Figure 2. Same as Figure 1 except for a nonlinear simulation. contours of horizontal velocity u while Figure 4 shows contours of Ri, as in Figure 2. The patch is centred near a position approximately 5000 m downstream of the obstacle centreline. The horizontal velocity achieves its minimum value for the entire field within this patch, which for this case is 4.15 m s 1 (compared to the free stream velocity of 10 m s 1 ). The shape of the patch evolves as the mountain-wave pattern approaches steady state, and appears to converge to the shape shown in Figure 4 and remain a permanent feature of the steady solution. Again, it is not known from the simulations if extremely long time histories would result in the disappearance of these patches, but this seems unlikely. Note in Figure 4 that some small-scale motion appears at the core of this patch for later times. This patch of decelerated flow is surrounded by regions of enhanced shear. Wave-induced changes to the stability also occur. Together these two effects will modify the local Ri resulting in patches of relatively lower Ri as shown in Figure 4. The patches of lowest Ri appear above and below the decelerated fluid. These regions are areas that are more likely to have complex behaviour resulting from a Kelvin Helmholtz type of motion, and more turbulence would be expected here on average. Note that Ri achieves its minimum value, apart from the value at the ground, within the patch in Figure 4 that is above the region of decelerated fluid. In this case, the lowest value of Ri in the tropopause region is Ri = 1.75, which occurs approximately 300 m above the mean position of the tropopause. The mean velocity for horizontally periodic waves is a welldefined quantity, merely being the horizontal integral over a single wavelength divided by the wavelength. However, for mountain waves there is no clear definition of mean, and all practical definitions have flaws. The primary reason that the mean flow is difficult to define is that the length scales of the mean flow and the waves are similar for mountain waves. The mean velocity adopted here is the horizontal integral of horizontal velocity u over the entire width of the computational domain, after subtracting the background velocity U, divided by the mountain width (twice the half-width: 2A): u = x 2A M (u i U), (5) i=1 Figure 3. Vertical profiles of total u at a position 6543 m downstream of the centreline of the mountain. The upstream velocity is U=10 m s 1.The times are 3000 s (leftmost), s (centre), and s (rightmost profile). Each profile is shifted by 10 m s 1 for display. solution. For example, McHugh s (2009) two-layer solution for uniform waves did not have a mean flow at the interface, while in contrast, the unsteady two-layer numerical solution of McHugh (2008) showed such a mean flow that persisted after the waves were fully developed and even after they exited the tropopause region. The region of decelerated flow just above the tropopause is concentrated in patches. Figure 4 shows these patches for a sequence of four time steps. Figure 4 shows where M is the total number of horizontal grid points in the computational domain. This definition is not a true average, and it may seem that a true average over the computational width would be a better definition; however, the mean determined with a true average is artificially small in magnitude. This is because the mountain waves at any chosen altitude are zero upstream and approach zero again downstream, and a true average over the entire computational width means padding the averaging integrand with zeros and then dividing by a large length, hence producing an artificially small value. A true average over a shorter distance could also be used, but integration lengths that are too short have oscillating values that are difficult to interpret. Other choices for the integration length, such as the horizontal positions where the vertical velocity is zero, seem useful; however, these positions depend on altitude. It is important to be able to compare the mean horizontal flow at different altitudes without concern that differences may be due merely to the

6 Mountain Wave-Induced Effects Near the Tropopause 1637 Figure 4. Contours of total u and Ri near the tropopause for several time steps with U=10 m s 1. The thick solid line is the position of the tropopause. Solid contours in have u <5ms 1, while dashed contours have u >5ms 1, incremented by 1 m s 1. Contours in have Ri = 2, 3.5, 5 and 10. definition of the mean, and our simple definition achieves this. Figure 5 shows profiles of the < u > for the U = 10 m s 1 case specified in Table 1. Figure 5 compares the nonlinear two-layer results (the solid line) to the same two-layer case with nonlinear effects excluded, and also a case with N 2 set to the same value as the lower layer. Figure 5 shows that the tropopause does indeed experience a region above the undisturbed position of the tropopause with decelerated mean flow. This deceleration increases in strength as the waves develop and then appears to reach a steady state and become a permanent feature. This flow compares well with previous simulations of horizontally periodic waves (McHugh, 2008), which showed a very similar waveinduced mean flow at the tropopause altitude. The linear case in Figure 5 does not show this mean-flow feature, indicating that this is a nonlinear effect. The single-layer case in Figure 5 does not exhibit an enhanced mean flow either, confirming that this effect is a result of the sudden change in N at the tropopause. The nonlinear case also has a downstream component below the undisturbed position of the tropopause that approximately matches the upstream component (discussed below), a feature that also does not appear in the linear or the single-layer case. Figure 5 shows the nonlinear mean-flow profiles for several different values of stability in the upper layer; all other parameters are the same as in Table 1. Figure 5 shows that the upwind component of the mean flow at the tropopause altitude increases sharply as the stability in the upper layer increases. The mean flow at larger distances away from the tropopause region is not greatly affected. Thus the wave-induced mean-flow effect is confined to the mean tropopause region. The mean-flow profiles in Figure 5 for the nonlinear cases show small-scale features in the upstream flow above the undisturbed tropopause altitude. This small-scale feature

7 1638 J. McHugh and R. Sharman Figure 6. Comparison of vertical profiles of < u > near the tropopause for four values of the vertical grid spacing ( z). Time is s, U=10 m s 1 and H=1000 m. Each profile is shifted by 10 m s 1 for display. Figure 5. Profiles of < u > computed from Eqn (5) near the tropopause for the U=10 m s 1 nonlinear case. Also shown in is the mean profile for a linear simulation and for a nonlinear simulation with uniform N throughout (N 2 = N 1 ). In the same mean profile is compared to corresponding cases with several different values of stability in the upper layer (N 2 ). Time is s. is due to the small-scale motion within the core of the patch shown in Figure 4. To insure that this region was properly resolved, vertical grid spacings ( z) werevaried to test convergence of the results, and Figure 6 shows the resulting mean flow for these test cases. As can be seen, decreasing z from 50 to 25 m does show some difference in the pattern above the tropopause, but using still smaller z does not significantly change the vertical structure. Still smaller z does not appear to be required, and all results reported here use z = 16 m. Decreasing the horizontal grid spacing ( x) did not have a significant effect, only slightly changing the horizontal phase of the waves, hence x is maintained at approximately 100 m and this allows a larger computational domain. Some aspects of mountain wave-induced mean flows are analogous to the mean flows generated by travelling waves. For monochromatic travelling waves that are modulated in the vertical but horizontally periodic, the horizontal component of the mean flow has the same sense as the horizontal component of the group velocity of the incident waves. Furthermore, if the stability is uniform, the waveinduced mean flow is primarily driven by the details of the packet shape. Simulations of a wave packet (Sutherland, 2001; McHugh, 2008) clearly show these characteristics of the mean flow. The group velocity for mountain waves has a vertical component directed upwards, and a horizontal component that is directed upwind, as discussed by Smith (1979). By analogy with travelling waves, the mean flow induced by mountain waves is expected to be upwind, acting locally to reduce the strength of the mean flow, and this was indeed the outcome of the simulations by Durran (1995). Therefore, by this same analogy, the evolution of the mountain-wave amplitude during start-up will produce a mean flow that moves upward with the transient, and the vertical thickness of this mean flow is directly related to the vertical thickness of the wave packet. In contrast, the simulations by McHugh (2008) show that a mean flow created at the interface does not move upward with the wave packet, but is attached to the interface. It is still true however that the direction of the interfacial mean flow in McHugh (2008) had the same sense as the group velocity of the incident waves. For the mountain waves treated here, the deceleration in the mean flow above the tropopause in Figure 5 has the same sense as the incident waves and appears to be directly analogous to the interfacial mean flow found in McHugh (2008). However, the acceleration (downstream) in the mean flow that appears in Figure 5 below the tropopause has the opposite direction of the wave group velocity. This downstream flow is not driven directly by the group velocity of the incident waves, but instead appears to exist to counter the mass flux due to the upstream mean flow above the tropopause. In this manner, the combination of upstream and downstream mean flow may act as an initiator of a large-scale circulation centred at the tropopause, resulting in downward motion upstream and upward motion downstream, on average. Other features of the wave-induced mean flow at the tropopause may be surmised by comparison with horizontally periodic waves. As in McHugh (2008), the vertical thickness of the interfacial mean flow is not simply the vertical thickness of the incident wave packet. Instead, the vertical thickness of the interfacial mean flow appears to be related to the vertical wavelength of the incident waves, implying that the trend in the thickness is related to the vertical wave number for linear waves. The linear solution for uniform flow over a sinusoidal lower boundary with uniform stratification in a single Boussinesq layer (see Smith (1979) and Baines (1995) for a general discussion)

8 Mountain Wave-Induced Effects Near the Tropopause 1639 Table 2. Parameter values for the U =5ms 1 case. Parameters Value Mountainheight (H) 500 m Mountainwidth (A) 1000 m Mean flow (U) 5ms 1 N in Troposphere (N 1 ) 0.01s 1 N in Stratosphere (N 2 ) s 1 Tropopause height m NH/U 1 NA/U 2 Horizontal grid spacing ( x) 100 m Vertical grid spacing ( z) 16m gives m = N 2 U 2 k2 (6) for the vertical wave number, m. Hence for a given horizontal wave number k, the vertical wavelength decreases with decreasing U. This implies that the thickness of the meanflow region at the tropopause will also decrease in thickness as U decreases. The strength of wave-induced mean flows generally increases with wave amplitude, indicated here with the value of NH/U. But the mean is also affected by the direction of the group velocity, and is expected to become stronger for cases where the group velocity vector is more aligned with the horizontal, all other aspects being the same. Again the linear monochromatic wave solution for flow over sinusoidal topography provides an expression for horizontal component of the group velocity c gx ) c g x = U (1 k 2 U2. (7) For a given k, the value of c gx has a maximum at U = 3 1/2 N/k, implying that the wave-induced mean flow will decrease in strength for large or small values of U. Overall, waves over a periodic bottom with large values of U are expected to have a weaker and thicker mean-flow region at the tropopause. As U is decreased, this region will become stronger in magnitude and thinner in vertical extent, both of which might enhance the shear values locally. As U is further decreased, the mean flow is expected to decrease in strength, but continue to become thinner in vertical extent. Flow over an isolated obstacle includes a spectrum of wave numbers rather than a single wave number, and the results are much more complicated in general. However, a sequence of simulations for a range of values of U (5 m s 1 < U < 10 m s 1 ) generally confirm the trends (with U) suggested above. The value of NH/U is maintained at unity for these cases by reducing the mountain height to avoid wave breaking. The results for the case with U =5m s 1 uses the parameters listed in Table 2. Figure 7 shows contours of total u and contours of Ri for this case. The value of NA/U for this case is 2.0, still small enough for non-hydrostatic effects to be important, and lee waves can be seen to be present in Figure 7. Profiles of u are shown in Figure 8 at a position 8439 m downstream of the mountain centreline. These three slices show that the horizontal velocity is a minimum at the tropopause as the N 2 Figure 7. Same as Figure 2 except the model time is s, U=5 m s 1 and H=500 m. Thin solid lines in are the u=5 m s 1 contours, while thick solid lines have u <5 ms 1, and dashed lines have u >5 ms 1, incremented by 1 m s 1. Figure 8. Profiles of total u at a position 8439 m downstream of the centreline of the mountain. The upstream velocity U =5ms 1. The time slices are 9000 s, s, and s. Each slice is shifted by 5 m s 1 for display. initial wave packet traverses the tropopause, as before, and remains a minimum after the waves have become close to fully developed at this horizontal position. The altitude of the minimum value is at the mean position of the tropopause, matching the previous case.

9 1640 J. McHugh and R. Sharman Figure 9. Comparison of vertical profiles of <u > near the tropopause for the U=5 and 10 m s 1 cases. The thick solid line is the case with U=5 m s 1 and H=500 m evaluated at a time of s, while the thin solid line isalinearversionofthesamecase.thedashedlineisthenonlinearcase with U=10 m s 1 and H=1000 m evaluated at a time of s. The profile of < u > with U = 5 m s 1 is shown in Figure 9 (solid line). The profile for < u > for the previous U = 10 m s 1 case is shown with a dashed line for comparison. The narrow region of horizontal mean flow is present with U =5ms 1, only now the upstream flow above the tropopause is muted compared to the downstream component below the tropopause. This feature is likely due to the tuning of the basic mountain wave between the ground and the tropopause (e.g. Baines, 1995; Keller et al., 2012). The vertical scale of the wave-induced mean-flow region in Figure 9 is smaller than for the U =10ms 1 case, following the trend predicted for periodic waves in Eq. (6). However, Figure 5 showed that this vertical scale for a sequence of values of N 2 is close to constant, yet the vertical wavelength in the upper layer changes rapidly with N 2. Hence this vertical thickness is not reflected simply by the vertical wavelength, as suggested by Eq. (6). Note that while the vertical scale in Figure 5 is approximately constant, the strength of the flow is not constant with changes in N 2. The deceleration regions for the U =5ms 1 case again occur in patches, as shown in Figure 10. These patches are qualitatively similar to the previous case, and indicate decelerated flow with patches of low Ri above and below. The minimum value of Ri within these patches is approximately 2.08 for this case, only slightly larger than Ri = 1.75 for the U =10ms 1 case. It appears that this minimum Ri depends mostly on the value of NH/U, which was fixed for both these cases at unity. Both the U =10ms 1 and U =5ms 1 cases show smallscale oscillations in the upstream component of the mean flow. Such small-scale components were also present in the results of McHugh (2008) for horizontally periodic waves, suggesting a more complex feature of the wave-induced mean-flow process. However, another explanation for this small-scale motion may be that shear instability has formed, as suggested by VanZandt and Fritts (1989). The cause of this small-scale motion remains unclear. The mean flow generated by wider mountains, A =5and 10 km, is shown in Figure 11, with all other parameters remaining the same as in Table 1. Figure 11 shows that the wave-induced mean flow (as defined by Eq. (5)) becomes weaker as A increases. Figure 11 also shows that the smallscale oscillations above the tropopause disappear for wider mountains, and the nonlinear effect remains approximately symmetric, with deceleration above and acceleration below the tropopause. For all cases considered, the region of decelerated flow is always present. The region appears early in the development of the waves, and becomes a permanent feature of the flow. Simulations with values of U larger than 10 m s 1 showed relatively weaker and thicker regions of mean flow at the tropopause. The weakness is likely due to the fact that the group velocity of the incident waves is becoming more vertical. The increased thickness may be related to the larger vertical wavelengths of the mountain waves for higher values of U. Simulations with values of U less than 5 m s 1 show a very narrow band and require much higher vertical resolution, and these cases were not pursued further. Cases with larger values of NH/U would have a stronger mean flow, but also resulted in wave breaking at levels below the tropopause, and could not be easily treated. 4. Summary and conclusions The simulation results presented here indicate that nonlinear effects associated with topographically generated gravity waves impinging on the tropopause can cause mean-flow accelerations below and decelerations above the tropopause in a narrow region there, which in turn creates regions of enhanced shear and lowered Ri. This effect is not present in the linear simulations. The narrowness of the region makes this feature difficult to resolve at the vertical grid spacings routinely used in operational numerical weather prediction models. The present results are limited to values of NH/U equal to unity to avoid wave overturning at lower levels, and the wave-induced mean flow at the tropopause appears relatively weak for this case. Larger values of NH/U would likely result in stronger wave-induced mean flows, as with the previous work with periodic waves. However, the wave overturning that may occur at lower altitudes before this mean flow becomes strong makes robust simulations very challenging. The traditional mountain-wave problem addresses steady flow over an isolated obstacle. For such steady mountain waves, Eliassen and Palm (1961) linearized the governing equations about a background velocity profile and showed that the momentum flux, <u w >, is constant with altitude, where u and w are the disturbance velocity components and the angle brackets indicate the horizontal mean. For non-steady flow, the horizontal average of the linearized horizontal momentum equation is <u> t + <u w > z = 0, (8) indicating that a mean flow will only develop when the momentum flux is not constant with height. This suggests that for the present results, the mean flow at the tropopause has developed during the start-up phase of the simulation, and once the mountain waves become approximately steady, this mean flow no longer changes. The simulations confirm this general behaviour, and as the waves become fully developed, the mean flow ceases to change in time and the momentum flux becomes constant with altitude in

10 Mountain Wave-Induced Effects Near the Tropopause 1641 Figure 10. Contours of total u and Ri near the tropopause for several time steps with U=5 m s 1. The thick solid line is the position of the tropopause. Solid contours in have u < 2.5 m s 1, while dashed contours have u > 2.5 m s 1, incremented by 1 m s 1.ContoursinhaveRi = 2, 3.5, 5 and 10. agreement with the theory of Eliassen and Palm (1961). Unless the forcing is allowed to change further with time, the wave-induced mean flow at the tropopause will remain a permanent feature. A solution to the steady equations that avoids the start-up and determines the steady mountain wave directly, as in Durran (1992), does not show a mean flow at the tropopause, providing further evidence that the tropopausal mean flow is created during the unsteady startup period. Furthermore, while this start-up is essential to the creation of this tropopausal mean flow, sensitivity studies (not shown) indicate that the details of the start-up process are unimportant. It should be noted that although this study focused on topographically generated gravity waves impinging on the tropopause, the same effects would be produced for any source of gravity waves, e.g. convectively induced gravity waves. However, in reality the constant wind approximation treated in this study is an oversimplification, since it does not include the presence of jet streams that may often be located in the vicinity of the tropopause in middle latitudes. The presence of the jet will induce environmental shears that reduce the background Ri from the essentially infinite background Ri cases considered here, and this may further encourage turbulence generation (e.g. Lane et al., 2004). The sharp change in the stability used here to model the tropopause is also an idealization. However, Birner (2006) using rawinsonde data showed that the average stability profile does indeed exhibit a sharp jump in stability at the tropopause in midlatitudes, resembling the profile used here. As long as the thickness of the tropopause region is

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