Evolution of stratosphere troposphere singular vectors

Transcription

1 Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 137: , January 2011 B Evolution of stratosphere troposphere singular vectors Hylke de Vries a,b *, Jan-Otto Hooghoudt a andw.t.m.verkley a a Royal Netherlands Meteorological Institute (KNMI), De Bilt, Netherlands b Department of Meteorology, University of Reading, UK *Correspondence to: Dr Hylke de Vries, KNMI, PO Box 201, 3730 AE De Bilt, Netherlands. hylke.de.vries@knmi.nl Stratospheric conditions affect the tropospheric circulation on a variety of timescales. This article addresses two issues in stratosphere troposphere interaction that are relevant for time-scales up to a week: the basic growth mechanisms involved and the role of the stratospheric shear. Idealized simulations are conducted of initially stratospheric perturbations that maximize tropospheric total disturbance energy with a lead time of 5 days, assuming linear quasi-geostrophic dynamics and simplified zonal flow. The evolution is analysed in terms of the basic interactions between three Rossby-wave components. On the f -plane (constant Coriolis parameter), the threecomponent analysis is very accurate (the error stays within 1% of vertically integrated total energy) because all available growth mechanisms are captured. Energy growth occurs initially through the Orr mechanism, subsequently through resonance and finally through (normal-mode) shear instability. On the β-plane (Coriolis parameter varying linearly with latitude), errors increase because westward retrogression of the untilting potential vorticity structure is neglected by the three-component model. Further study reveals that the tropospheric energy growth strongly depends on the value of the stratospheric shear. Copyright c 2011 Royal Meteorological Society Key Words: baroclinic development, stratosphere troposphere coupling Received 11 December 2009; Revised 1 November 2010; Accepted 11 January 2011; Published online in Wiley Online Library Citation: De Vries H, Hooghoudt J-O, Verkley WTM Evolution of stratosphere troposphere singular vectors. Q. J. R. Meteorol. Soc. 137: DOI: /qj Introduction Recently there has been considerable interest in understanding the impact of the stratosphere on the tropospheric circulation (e.g. Hartley et al., 1998; Baldwin and Dunkerton, 1999, 2001; Polvani and Kushner, 2002; Charlton et al., 2003; Hooghoudt and Barkmeijer, 2007; Wittman et al., 2007; Hinssen et al., 2010). If stratospheric precursors turn out to be robust predictors for tropospheric circulation changes at a given lead time, weather prediction could potentially benefit from this predictability. The present article addresses two issues that appear to be relevant if time-scales are on the order of a few days up to a week, yet are not fully resolved: the growth mechanisms behind stratosphere troposphere interaction and the sensitivity of the interaction to the stratospheric wind conditions. Note that these time-scales are significantly shorter than those usually discussed in papers on stratosphere troposphere interaction (e.g. Baldwin and Dunkerton, 2001). For short-time evolution, baroclinic instability is likely to play a role. Existing literature indeed confirms that stratospheric zonal wind conditions influence the structure, growth rate and propagation rates of the normal modes (Müller, 1991; De Vries and Opsteegh, 2007a; Wittman et al., 2007; Smy and Scott, 2009). However, it can be expected that the initial-value problem differs significantly from the evolution of a pure growing normal mode (e.g. Orr, 1907; Case, 1960; Pedlosky, 1964; Farrell, 1982; De Vries et al., 2009). Hooghoudt and Barkmeijer (2007) (HB07 below) showed that non-modal growth occurs if one computes stratosphere troposphere singular vectors (STSVs), initially stratospheric disturbances, that optimize Copyright c 2011 Royal Meteorological Society

2 Stratosphere Troposphere Singular Vectors 319 lower-tropospheric perturbation energy at a given lead time ( optimization time ). Although the STSV is synthetic by construction and will not occur in isolation, it provides an upper bound on the system s possibilities for generating stratospherically induced lower-tropospheric disturbance energy. The first aim of this paper is to discuss the growth mechanisms that are relevant to the development of such STSVs or optimal perturbations. The second aim of this article relates to another aspect in the theory of stratosphere troposphere coupling: the role of the vertical shear of the zonal wind in the (lower) stratosphere. The stratospheric zonal wind shows a strong annual cycle at midlatitudes (e.g. Andrews et al., 1987). In the Northern Hemisphere (NH) summer, the zonal jet has a positive maximum near the tropopause of about 15 m s 1, and decreases continuously aloft. In winter, the zonal jet also attains a maximum near the tropopause, but this maximum is much stronger than in summer (about 30 m s 1 ). The zonal wind strength decreases with height until km, but from there increases again with height, attaining a maximum of about m s 1 in the higher stratosphere. Stratosphere troposphere interaction appears to be much more pronounced in the boreal winter (e.g. Baldwin and Dunkerton, 2001; Siegmund, 2005). Charney and Drazin (1961) (CD61 below) have studied the conditions under which tropospheric disturbances may have an impact in the stratosphere. They have derived conditions under which normal-mode solutions are wavelike in the vertical (i.e. φ(z) exp(imz) withre(m) 0). Their results show that only sufficiently long waves can propagate vertically in this sense, and that it may simply be impossible for tropospheric wave disturbances originating from baroclinic instability to reach the higher stratosphere if the stratospheric shear is sufficiently negative or strongly positive (i.e. no waves with Re(m) > 0 are found). While CD61 provide valuable insight for cases where the perturbations are forced for long times (they argue long-wave stationary forcing is the most likely candidate for getting stratospheric effects from tropospheric disturbances), they do not investigate the transient evolution (non-normal mode problem) nor the reverse problem (stratosphere to troposphere). In order to address the issues of the non-modality (and growth mechanisms) and the sensitivity to stratospheric wind conditions, idealised simulations are studied, assuming linear quasi-geostrophic dynamics and a time-independent purely zonal basic flow. Initial conditions are taken to be the STSVs already mentioned. Rossby-wave (RW) theory is used to reveal the growth mechanisms underlying the growth. RWs are simply waves in potential vorticity (PV). It has been demonstrated (e.g. Bretherton, 1966; Hoskins et al., 1985; Davies and Bishop, 1994; Heifetz et al., 2004) that linear normal-mode baroclinic instability of arbitrary zonal jets can be viewed as a a pair of interacting RWs. In order to quantitatively describe the evolution of singular vectors (SVs) on short times, the continuous spectrum has to be included. We use a recently developed RW framework (Methven and de Vries, 2008; De Vries et al., 2009). The basis for this framework is formed by the aforementioned pair of interacting RWs and a passive PV component that is purely advected with the mean zonal flow. Sections 2 and 3 introduce the model and the RW framework. Evolution of the leading STSV and the interpretation in terms of RW components are described in Figure 1. Profiles of basic-state (a) zonal wind ū, (b) buoyancy frequency N 2 and (c) implied meridional PV gradient, all as a function of height. The dashed line in (c) shows the mean PV gradient if non-zero β and an exponentially decreasing density (scale height 7.5 km) are taken into account. section 4. Section 5 expands on section 4, by first looking in detail at the trajectories of the RW componentsin a growthrate versus propagation-rate plane, and subsequently by focussing on the growth mechanisms. These are investigated by switching off particular interaction channels. Section 6 explores sensitivity of the STSV to the stratospheric shear. A summary and discussion are given in section Quasi-geostrophic potential vorticity dynamics We consider dry adiabatic quasi-geostrophic (QG) evolution of small perturbations of a zonally symmetric basic state described by a buoyancy frequency N 2 (z) and a zonal wind ū(z), independent of the meridional direction (Figure 1). This basic state represents an idealized troposphere stratosphere system. The stratospheric shear is modified in section 6. The perturbation evolution is described by the QG PV equation: where q t +ū q x + v q = 0, (1) y q = 2 h ψ + f 2 0 ρ ( ) ρ z N 2 zψ is the perturbation PV ( h 2 = 2 x + 2 y ), v = ψ/ x is the geostrophic meridional wind, ψ is the perturbation stream function and ρ is a reference density profile. Basic-state quantities carry a bar (except for N 2 ). The basic-state meridional PV gradient, denoted by q y,isgivenby q y = β f 2 0 ρ ( ) ρ ū z N 2, (2) z where β is the planetary vorticity gradient at 45 N (NH conditions are assumed). Equations are made nondimensional using H = (f 0 /N) 2 /β and L = (N/f 0 )H for vertical and horizontal length-scales respectively. A rigid lid is prescribed at z b = 0 and is taken into account by

3 320 H. de Vries et al. setting the vertical velocity w to zero in the thermodynamic equation. Perturbations are assumed to vanish at infinite height. Equation (1) is solved using a Green s function approach (Appendix A). A vertical domain height of 60 km is chosen with 300 interior levels. Varying the domain height between 40 and 70 km did not significantly influence the results below 30 km, which is our main area of interest. 3. Rossby wave framework If the flow is adiabatic and frictionless, PV is materially conserved. Therefore, perturbations that grow from infinitesimal amplitude contain PV that is obtained solely through meridional advection of the basic-state PV contours: q d = η q y, (3) where η is the meridional air parcel displacement that satisfies ( / t + ū / x)η = v. General initial conditions will also contain PV (denoted by q p )thatdidnotarisefrom meridional displacements, but (for instance) from diabatic processes no longer active. Therefore one can write q = q d + q p, (4) and similarly for the meridional wind v = v d + v p. Substituting (4) into (1) and using (3), one gets q d t +ū q d x + v q d = F, (5) y where F = v p q y. It can be shown thatq p satisfies the equation ( t +ū x )q p = 0, implying that q p is advected passively with the flow, i.e. [ q p (x, z, t) = Re q p (z,0)e ik{x ū(z)t}]. Although q p is advected passively, it is dynamically active, in the sense that the winds v p associated with q p will generate q d in regions where q y 0. This comes to expression in the forcing term F in (5) Three-component model Attention is restricted to the evolution of horizontally wavelike perturbations with a particular initial vertical structure. Such baroclinic initial-value problems can often be described in terms of RW components, which are simply waves in PV. Normal-mode baroclinic instability, for example, can be understood as a mutual, reinforcing interaction between two RW components termed counter-propagating Rossby waves (CRWs; Hoskins et al., 1985; Heifetz et al., 2004). Together with the continuous spectrum (Pedlosky, 1964), the CRWs form a complete orthogonal basis with respect to the pseudomomentum inner product (Held, 1985). If initial conditions are more general, more than two components are required to describe the transient evolution. One could choose the continous spectrum modes. However, the number of these modes increases with increasing resolution, and their structure is resolution-dependent. The challenge is therefore to find the best approximate description of the entire evolution with the least number of components. De Vries et al. (2009) (V09 below) developed a method in which all continuous spectrum modes are grouped in one single RW component. They showed that many initial-value problems can be usefully described in this way. Following V09, a partitioning into three RW components is chosen. Two of these are the CRWs. The third component is the passive PV, q p, which possibly has intricate vertical structure, but is highly predictable (simple advection with basic-state wind). In the next section we derive evolution equations for the RW components Evolution equations and the PAR-PV approximation By projecting (5) onto the CRW structures using the pseudomomentum inner products (details in Appendix B), equations are derived that describe the evolution of the complex CRW wave amplitude α j = A j exp(iɛ j )[j = 1, 2], where A j is the positive definite amplitude of CRW-j,andɛ j its phase, [ ] α i (t) = ik c ij α j (t) + f i (t), i, j = 1, 2. (6) j The factors c ij describe propagation of and interaction between CRWs, f i (t) describes the excitation of CRW-i by the winds associated with the evolving q p.thecrw equations (6) are exact for a given choice of q p (V09): any left-over PV q n = q q p q crws will not project onto the CRWs (q crws = α 1 q 1 + α 2 q 2 ). By setting q n = 0 at all times, all PV that does not project onto the CRWs is advected with the basic-state wind, defining the so-called passively advected residual PV (PAR-PV) approximation (V09). V09 described a number of examples where the PAR-PV approximation yields an accurate description of the evolution. Known situations in which the PAR-PV approximation will fail are if there are no growing normal modes, or if the initial PV distribution is located far away from the regions where the CRWs have maximal amplitudes. If there are growing normal modes in the system, the PAR-PV approximation will perform at least as well as a conventional modal projection onto growing and decaying normal modes. In many cases it will however give a much improved description, because by construction the error is identically zero at initial time and at very long times. The maximal deviation is reached typically at some intermediate time. 4. Evolution of STSV STSVs have been constructed for a total energy norm and an optimization time of 5 days (details in Appendix C). At initial time, the STSVs are constrained to have zero perturbation total energy below an altitude of 15 km (NB the tropopause is located near 10 km altitude; Figure 1). The STSVs optimize perturbation total energy below 5 km (lower troposphere) at t = 5 days. The STSVs were given a meridional and zonal wavelength of both 6000 km.forthe chosen optimization time and norm, only the leading STSV has a singular value larger than unity and exhibits significant For an imposed meridional meridional wavelength 4500 km, the largest singular values were obtained for zonal wavelengths around 6000 km.

4 Stratosphere Troposphere Singular Vectors 321 Figure 2. Zonal height cross-sections of the evolution of RW components,showing PV (thin solid contours with shading on high values), boundary PV (high PV denoted by black circles) and meridional flow (bold solid contours positive and dashed contours negative values). From top to bottom: Full PV, q; passive PV, q p ;CRW-2PV,q 2 ;CRW-1PV,q 1.Fromlefttoright:timeat1,2,3,4,5days.TheframeofreferencemoveswiththegNMphase speed. Vertical units are 10 km, and horizontal units kx.att = 0, all PV is put into q p. tropospheric growth. For this reason, only results for the leading STSV are discussed. Two experiments are conducted. In the first experiment, the planetary vorticity gradient β is set to zero, and the density ρ is taken constant. These choices imply that q y is only non-zero at the ground and in a finite region near the tropopause. In the second experiment, β assumes a characteristic midlatitude value, and the density decreases exponentially with height, thus leading via (2) to a PV gradient that is non-zero throughout the entire domain (the dashed line in Figure 1). Note that the decreasing density in combination with negative shear results in a q y β in the stratosphere (and q y β in the mid-troposphere). After the dominant STSV has been obtained, (1) is solved, to obtain the full PV, q. TheCRWequations(6) are solved using the entire initial PV structure as q p (0). This yields expressions for the CRW amplitudes and phases. The neglected PV follows as q n = q q p q crws. It has to be kept in mind that the results to be shown below depend on parameters such as the optimization time and the minimal altitude of the initial STSV. Larger optimization times mostly lead to increased finite-time growth rates. On the other hand, if the minimal STSV height would be above 25 km, hardly any tropospheric development occurs within the first week. Sensitivity to the choice of stratospheric shear is discussed in section 6. A more complete sensitivity study is considered beyond the scope of the present article. V09 described several methods to do the initial partitioning. In many cases, assigning the entire initial PV to q p gives reasonable results. In the present case, the choice is motivated also by the fact that the CRWs have only small PV amplitude at the height of the initial STSV Experiment 1: f -plane Figure 2 shows the STSV evolution in terms of the RW components described in the previous section. At initial time t = 0 (not shown), perturbation PV only exists above 15 km altitude and is tilted strongly against the (negative) stratospheric shear. To guarantee that the perturbation winds vanish below 15 km, a large-amplitude PV anomaly exists near 15 km height, which is of opposite sign to that of most of the PV directly aloft. The constraint of optimal projection into the lower troposphere results in the initial PV structure decaying rapidly with height in the stratosphere. As time evolves, the perturbation develops new PV extrema, first near the tropopause (day 2), and later (day 3) also at the surface. After 5 days the STSV is structurally similar to the growing normal mode (gnm) in the troposphere, but in the stratosphere near 15 km (the region where the initial PV was maximal) there is still significant additional circulation. We now describe the evolution of the RW components. Since q y = 0 in the region of the initial PV, it is impossible to generate that PV by advection (i.e. q d (0) 0). Therefore, the entire initial PV q(0) can be attributed unambiguously to the passive PV component q p. As discussed above, q p (second row) evolves in a robustly predictable way: it is advected by the basic-state zonal wind. As the PV structure q p untilts as a result of the shear, perturbation kinetic energy increases rapidly at upper levels and peaks near the optimization time (day 5). Shortly after day 1, the circulation induced by q p near the tropopause region is strong enough to make CRW-2 appear in the figure. Its initial phase agrees with what one would expect from PV thinking (the negative maximum appears first where the northward winds due to

5 322 H. de Vries et al. Figure 3. As Figure 2, but for the case with non-zero β and exponentially decreasing density. The bottom row shows PV that is neglected by the PAR-PV approximation (q n ). The frame of reference moves with the gnm phase speed. At t = 0, all PV is put into q p. q p are maximum). Similarly, the lower CRW, CRW-1, is excited π out of phase with CRW-2, again initially primarily by q p, despite the large vertical separation. However, beyond two days CRW-2 already has a large effect on the growth of CRW-1. The maximal instantaneous error, measured as the total energy contained in the perturbation difference field q n, is less than 1% of the full model total energy at all times. For this reason q n is not shown in Figure 2 and will not be discussed Experiment 2: β-plane In the second experiment, non-zero β and exponentially decreasing density are included. For the same zonal and meridional wavelength, the eastward propagation speed of the gnm is reduced due to the effects of β, butthegnm growth rate is slightly larger. The meridional PV gradient above 15 km is small but non-zero (Figure 1), thereby making it less obvious how to partition the initial condition into q p and q d. For comparison with experiment 1, the choice was made to attribute the entire initial PV q(0) to q p. The evolution is shown in Figure 3 and is very similar to Figure 2, although the CRWs have a more complex vertical structure. The maximal error made by using the threecomponent model is on the order of 15%. It can be seen that in the neglected PV component q n,showninthebottom row in Figure 3, a substantial cause of error is that the PAR-PV approximation neglects (westward) retrogression of the untilting initial stratospheric PV (Yamagata, 1976; Boyd, 1983). In the stratospheric region where the initial PV is located (far away from boundary and tropopause), the neglected PV q n is approximately in quadrature with the passive PV q p. Note that also some PV in the tropopause region is not explained by the CRWs. 5. Growth mechanisms 5.1. Phase-space trajectories To examine the perturbation evolution more quantitatively, we return to the CRW equations (6). By decomposing (6) into their real and imaginary parts, one obtains expressions for two key properties of the CRWs: their instantaneous propagation rate ɛ i and growth rate Ȧ i /A i : [ A j ɛ i = k c ij cos(ɛ i ɛ j ) + F ] i cos(ɛ i η i ), A j i A i [ Ȧ i A j = k c ij sin(ɛ i ɛ j ) + F ] i sin(ɛ i η i ), A i A i A i j where F j (t) andη j (t) are the amplitude and phase of the forcing f j (t)forcrw-j.sinceboth ɛ i and Ȧ i /A i are functions of time only, the evolution of CRW-i can be represented as a trajectory [X i (t), Y i (t)] = [ ɛ i, Ȧ i /A i ] in the two-dimensional

6 Stratosphere Troposphere Singular Vectors 323 of CRW-2 remains small compared to the self-propagation of CRW-2. The effects of q p become negligible at long times (QP ends at the origin). Also the growth of CRW-1 (bottom panel) initially is due to q p, which also contributes to the phase speed of CRW- 1. However, after about 2 days, the influence of CRW-2 (Q2) has become so strong that it dominates the growth of CRW-1, while simultaneously slowing down the eastward propagation of CRW-1. As for CRW-2 in the upper panel, the contribution from q p gradually decreases. Nevertheless it takes approximately 5 days for CRW-1 to amplify and propagate the gnm growth rate and phase speed Reduced models Figure 4. CRW trajectories (denoted by Z i (t)) in the propagation rate (x-coordinate of each point) growth rate (y-coordinate of each point) plane for (a) CRW-2 and (b) CRW-1, in units of f /N. Alsoshownare the different contributions to these trajectories: QP contribution from q p, Q1 contribution from CRW-1, Q2 contribution from CRW-2. The time between the plotted points is 24 h and only the first 5 days are shown. propagation rate growth rate plane. At a particular time T, the x-coordinate X i (T) ofapoint[x i (T), Y i (T)] on the plane gives the propagation rate of CRW-i, and theycoordinate Y i (T) its instantaneous growth rate. Since both growth and propagation rate asymptote to constant values, the trajectories end at a fixed point. The actual propagation and growth rate of each CRW (i.e. the trajectory in propagation rate growth rate space) is the net result of the CRW self-propagation and of the interactions with the other RW components (other CRW and passive component). The latter will generally affect both the CRW propagation and growth rate. The contributions of all these interactions (terms in the R.H.S. of the above equations) are compared quantitatively, by displaying them as separate trajectories (such that the sum results in the true CRW trajectory). By zooming in on the evolution of the individual CRWs, the trajectory analysis further deepens the understanding of the perturbation evolution. Trajectories have been calculated for the STSV evolution of the first experiment (zero β, constant ρ). The trajectories obtained for the experiment with non-zero β are qualitatively similar and are not shown. The results are in Figure 4; CRW trajectories are labelled as Z i (t). Starting with the contributions to CRW-2 (top panel), the first thing to notice is that CRW-2 (Q2) does not contribute to its own growth but only to its propagation. The same holds for CRW-1 in the bottom panel. The reason is that by construction a CRW has a vertically untilted PV structure and a meridional wind which is in quadrature with the PV. The contribution from q p to the growth of CRW-2 (labelled QP) dominates for a long time, almost up to the optimization time of 5 days. The contribution from CRW- 1 (Q1) to the growth of CRW-2 remains smaller than that of q p up to day 4; its contribution to the phase-propagation In the previous sections it was shown that the CRW equations (6) reproduced most aspects of the STSV evolution remarkably well, both qualitatively and quantitatively. Therefore, in this particular case, one may well approximate the full model STSV evolution (integrating (1)) by the reduced model STSV evolution under the PAR- PV approximation (integrating (6)). In this section, we examine the role of the different growth mechanisms on the perturbation energy growth. This is assessed by switching off some of the interaction channels that define these mechanisms. Three main mechanisms leading to energy changes are (V09): I PV-unshielding (Orr mechanism). Passive advection of the initially against-shear tilted q p structure by the zonal wind results in an increase of energy (not of enstrophy). II Excitation of CRWs by q p (Resonance). The winds associated with the passive component q p excite CRW- 2 and, to less extent, also CRW-1. In (6) this process is represented by the f i terms. III CRW-interaction (Shear instability). The CRWs interact (the c ij terms in (6)) and may reach a mutually reinforcing configuration (gnm). The advantage of having (6) available as the exact CRW equations, is that it is easy to switch off a particular interaction channel (growth mechanism). For example, if f i is set to zero, CRW-i cannot be excited via resonance (II). Similarly, by setting c ij (i j) tozero,thecrwswill not be able to interact, therefore effectively prohibiting the formation of the gnm through shear instability (III). Figure 5 shows the results of various experiments that have been conducted. Here vertically integrated total energy of the truncated model is compared to the total energy of the full model. The crosses show what happens if only q p is retained, and no CRWs or any other structures are taken into account. Clearly, since no resonance or shear instability can occur, the energy increases only via the Orr (1907) mechanism (I), and total energy at optimization time (5 days) is much less than that of the full model simulation. Nevertheless, it is seen that retaining q p only gives a very good approximate value of vertically integrated total energy up to day 2.5, with the largest error occurring in the troposphere (not shown). The remaining lines display what happens if some terms related to the CRW excitation and interactions are included. The open circles show the evolution if the CRWs are allowed to be excited via resonance (II), by taking f i 0, but subsequently are not allowed to interact with each other, i.e. c 12 = c 21 = 0. In this case, shear instability (III)

7 324 H. de Vries et al. Figure 6. Effect of changing the shear of the zonal wind in the stratosphere on the leading singular values (SV1) and the sum of non-leading singular values (SV2+SV3+...). Figure 1 corresponds to the case where s = 1. Figure 5. Evolution of the vertically integrated total energy as obtained with various truncated models (indicated in the legend), and measured relative to the full model total energy: (a) model with β = 0 and constant density, (b) model with non-zero β and an exponentially decreasing density. The optimization time occurs at t = 5 days (dashed lines). The dash-dotted line in (b) shows the energy associated with an evolving gnm, obtained by modally projecting the initial condition onto the gnm. is effectively excluded, which has a dramatic effect on the energy growth beyond 3 days. Perhaps the most interesting case is shown as the solid grey line. It displays what happens if CRW-2 is not allowed to grow via resonance (f 2 = 0), but the CRWs are allowed to interact. In this case the STSV will havetoprojectfirstontocrw-1beforeitcanexcitecrw-2. This indirect way is fatal to the perturbation growth. The total energy growth is reduced dramatically, and is almost as little as having no shear instability available at all up to 5 days. Somewhat suprisingly, if CRW-1 is not allowed to grow via resonance (dashed grey), the system initially amplifies too rapidly. This is partly because the CRWs are excited by q p initially in opposite phase. Clearly, in this RW perspective all three mechanisms play an important role. Similar results are obtained if β and density are included (Figure 5(b)). As stated earlier, the PAR-PV approximation performs less well, with maximum total energy errors of about 15%, which are reached slightly beyond the optimization time and decrease thereafter. However, in terms of the energy evolution, essentially the same features ofthecasewithoutβ and density are reproduced, showing a dramatic decrease of total energy at optimization time if either the upper CRW is not allowed to grow via resonance (grey line), or if the CRWs are not allowed to interact. 6. Stratospheric shear The stratospheric shear s is an important parameter. Figure 6 shows the effect of changing s (other conditions unaltered) on the obtained singular values. The singular value changes significantly when s is increased from 1 to 0. The β-plane case (dashed line) reaches its maximum outside the domain investigated. Also shown is the sum of all non-leading singular values. They account for only a small fraction of the growth. Only singular values larger than unity will result in perturbations with significant tropospheric development at optimization time. Over the range of s values studied, the normal-mode growth rate exhibits changes of the order of 5 10% (dashed contours in Figure 8, further discussed in section 7) and therefore cannot account for the strong decrease of the singular value. However a RW interpretation for the change can be given. In order to excite the upper CRW, the circulation associated with q p (i.e. the untilting initial condition) not only has to reach the levels where the upper CRW resides, it also has to have roughly the same phase speed. If the phase-speed difference is too large, destructive interference will quickly terminate the growth through resonance. For increasing values of the stratospheric shear, it becomes more and more difficult to match the phase speeds. Therefore the excitation of the CRWs by q p (II, resonance) becomes (strongly) suppressed as s increases and this explains to a large extent the grey reduction of the singular value. For the current set of parameters, the maximum singular value is obtained when s 1. More generally, the maximum will be reached for that value of s where the zonal wind speed at the lowest admitted initial height (in our case 15 km) is approximately equal to the normal-mode phase speed, thereby maximizing the growth due to resonance (II). While the above-mentioned phase-speed argument explains the occurrence of the maximum and part of the decrease of the singular value, there is another reason why the singular value attains a minimum. This has to do with the Orr mechanism (the energy changes induced by the untilting of the initial condition with the shear) and with the special properties of the initial condition (no perturbation energy below a certain height). Figure 6 shows that the f -plane STSV produces no growth at all if s = 0. In that case the entire perturbation is advected (barotropically) with a constant speed (no Orr mechanism). Since the initial condition does not have any circulation below 15 km, there will be no further development. On the β-plane, there is a similar situation (now for positive stratospheric shear) where the singular value attains a minimum. In this case there is an

8 Stratosphere Troposphere Singular Vectors 325 approximate balance between the shear trying to untilt the PV, and q y trying to restore this by westward retrogression. Figure 7 shows the evolution of the STSV obtained for experiment 2 (non-zero β and exponential density) but for a basic state with s = 0.5. The evolution confirms that hardly any development occurs in the troposphere during the first 5 days of the development. Please note that such conditions would typically be modelled incorrectly with the three-component model. The results obtained here are different from CD61, who found that stratospheric effects of tropospheric perturbations (i.e. the reverse problem) are unlikely to be relevant if the stratospheric shear is negative. In CD61 necessary conditions are derived for the existence of (mostly stationary and neutral) normal modes with wave-like vertical structure, with the argument that structures that are evanescent in the vertical will not be able to penetrate energy deeply into the stratosphere. Wave-like vertical structure for such normal modes is not possible in the stratosphere if the shear is negative, nor for the wavelengths typically associated with baroclinic instability (CD61). The STSVs discussed in this article have a shorter horizontal wavelength than the external mode of CD61, but amplify rapidly inthetropospherebecausetheyinvolveexcitationofthe gnm. The non-modal excitation of the gnm appears to be much more effective for small negative shears than for positive stratospheric shear, for reasons explained above. However, the gnm does have a non-propagating vertical structure in the sense of CD61 and it will be difficult to penetrate energy deeply into the stratosphere. In contrast, even an initially small gnm projection coefficient (resulting from the modal projection of the purely stratospheric initial condition on the gnm) will quickly lead to a gnm with large amplitudes in the entire troposphere. Thus, one could argue that there is significant impact from the stratosphere onto the troposphere, but less impact from troposphere to stratosphere for these wavelengths. In this sense, stratosphere troposphere interaction is asymmetric and the troposphere may be more sensitive to largeamplitude stratospheric perturbations than the other way around. 7. Summary and discussion Recent studies have shown that stratospheric conditions influence the tropospheric circulation on a variety of temporal and spatial scales. Stratosphere troposphere singular vectors (STSVs) are perturbations of the basic state, that start in the stratosphere and maximize lower tropospheric disturbance energy at a given lead time (HB07). STSVs can be used to determine an upper bound on the stratosphere s possibilities to generate lower-tropospheric energy. Note that the relevance of SVs, their dynamics and the provided upper bound on the growth, depends on the realism of the assumed basic state and linear dynamics. In this study they are computed assuming quasigeostrophic dynamics and a simple time-independent basic state resembling troposphere and stratosphere (Figure 1). The two objectives are to improve understanding of the mechanisms reponsible for stratospherically induced tropospheric development and to explore the sensitivity of the STSV growth to the vertical shear of the zonal wind in the stratosphere. It is shown that the evolution of the dominant STSVs can be compactly described by the propagation and interaction of three RW components (V09). The first of these is advected with the basic-state wind, thereby giving rise to energy changes through the Orr mechanism (Orr, 1907). In cases where rapid tropospheric growth does occur, the winds associated with the untilting PV structure, subsequently excite predominantly two other RW components via a resonance-like mechanism (De Vries and Opsteegh, 2007b). These two RW components are termed counterpropagating Rossby waves (CRWs) after Heifetz et al. (2004) and their interactions describe the long-term linear evolution. The three-component model gives nearly exact results on the f - plane (maximum instantaneous total energy error remains less than 1%) because all growth mechanisms are captured. On the β-plane and with non-constant density, however, the maximal errors increase (order 15%), due in part to westward retrogression of the untilting initial PV structure (Yamagata, 1976; Boyd, 1983), which is neglected in the three-component approach (V09). Stratospheric shear s plays an important role in constraining the STSV growth. For the spatial scale that has been studied (zonal and meridional wavelength of 6000 km), the STSV singular value is maximized at a negative value of s and decreases rapidly as s is increased. From a RW perspective, this decrease occurs because the initial disturbance fails to excite the CRWs (through resonance) due to a mismatch in phase-speed. A discussion is now given of how the results obtained in this paper relate to other studies, along with some limitations and possible generalizations. One difference between the STSV and the conventionally computed tropospheric SV (TSV) is its rather modest amplification. A modal explanation for this difference is that the STSV only weakly projects onto the gnm, whereas the gnm projection coefficient of the TSV is much larger. An explanation in terms of RWs is that the untilting initial PV excites mainly the upper CRW in the case of the STSV (which subsequently has to excite the lower CRW, before shear instability and normal-mode growth can take over). In case of the TSVs, both CRWs are rapidly excited, after which they further interact to form the gnm. This threestage development scenario has been previously suggested for the evolution of SVs in the Eady model (e.g. Morgan, 2001; De Vries and Opsteegh, 2007b), where the CRWs take the simple form of boundary edge waves. Here it is shown that this scenario also applies in the present case, where the CRWs have a much more complex structure and the basic-state meridional PV gradient is non-zero in large parts of the interior. Another point, already briefly discussed in the main text, relates to the choice of the minimal altitude of the initial STSV. If this minimal altitude is not too high up in the stratosphere, the f -plane and β-plane cases give similar results (as shown in this article). However, the differences become much more pronounced if the minimal altitude lies above 25 km (depending on horizontal wavelength). If STSVs are computed for such conditions (and a longer optimization time of say 15 days), the β-plane model does produce strong tropospheric development, whereas there is virtually no growth on the f -plane. The difference could be explained from the structural change of the underlying continuous spectrum (e.g. De Vries, 2009). An explanation in terms of RWs is that excitation of CRWs (and the

9 326 H. de Vries et al. (a) 2.5 T=0 (b) 2.5 T= π π/2 0 π/2 π 0 π π/2 0 π/2 π Figure 7. As Figure 3 (non-zero β, non-constant density), but for the case with s = 0.5, at time (a) 0 days and (b) 5 days, showing the full PV (shaded contours) and meridional wind (bold and dotted contours). The frame of reference moves with the gnm phase speed. accompanying strong tropospheric growth) is only possible if the initial PV can propagate downwards (i.e. if PV can be generated below the initial PV perturbation by advecting the mean PV contours with the winds associated with the initial PV) to reach the tropopause CRW. This downward propagation is impossible if, as in the case of the f -plane, q y = 0 in the stratosphere. The downward propagation is also not captured by the three-component model and the PAR-PV approximation will fail for these cases. Certain aspects of stratosphere troposphere interaction can not be described properly if the model is over-idealized. Observed stratosphere troposphere interaction occurs preferably in the boreal winter. In January, the zonal jet at northern midlatitudes does not show a continuous decrease with altitude above the tropopause, as it does in summer (up to 70 km), but increases with height above 25 km. Section 6 shows in some sense the opposite, i.e. a strong decrease of the singular value for increasing stratospheric shear. One possibility is that the difference is caused by the seasonal variation of the tropospheric shear, t,whichis nearly twice as strong in winter than in summer. Figure 8 shows the dependence of singular value and gnm growth rate as a function of both tropospheric and stratospheric shear. The normal-mode growth rate is approximately a linear function of t, but hardly varies with s. In contrast, the singular value strongly depends on both t and s. Approximate summer and winter tropospheric and stratospheric shear values were obtained using the lowest 30 km of the profiles shown in Charney and Drazin (1961). For the choice of parameters (basic-state wind and buoyancy profiles, horizontal wavelength and initial minimal height and optimization time of the initial STSV), the approximate winter and summer zonal-wind profiles give similar singular values. An explanation for the observed seasonal dependence should therefore be sought in other directions, for instance a more realistic zonal-wind profile that differs from the simple two-layer configuration used in this paper, a longer spatial and time-scale and an increased minimal height of the initial STSV. A comparison of STSVs computed for two such more realistic situations (i.e. taking the observed shear values for troposphere and stratosphere up to 70 km, an optimization time of 15 days and an initial height above 30 km) confirms (not shown) that the tropospheric growth then is indeed (much) larger for the typical winter case than it is for the summer case. Further study is required to examine this aspect in more detail. Note that the results in sections 4 and 5 Figure 8. Singular value (shaded solid contours) and gnm growth rate (dashed contours) as a function of tropospheric and stratospheric shear (zonal and meridional wavelength 6000 km). The approximate summer (S) and winter (W) conditions based on Charney and Drazin (1961) are indicated. This figure is available in colour online at wileyonlinelibrary.com/journal/qj were obtained for a generic profile with t = 1and s = 1 (Figure 1). Those results are comparable to evolution from typical winter conditions with a longer optimization time. There are further limitations to the present study, both in terms of geometry and in terms of (balanced) dynamics. An important next step would be to include spherical geometry. CRWs have already been obtained for primitive equations and spherical geometry (Methven et al., 2005a,b). The CRWs have a more complex vertical structure and are meridionally confined, but follow the same CRW evolution equations. STSVs could be implemented straightforwardly and there is no formal limitation to implement the PAR-PV approximation. Applying the ideas discussed in this article and, in particular, isolating the relevant RW components for realistic numerical weather prediction models, will obviously be more complicated. If one has a general (i.e. non-zonally symmetric) zonal wind structure, as used for instance by HB07, the STSVs have an even more confined structure, with many (zonal and meridional) wave numbers playing a role. Nevertheless, one can imagine an analysis, such as the one presented in this article, being possible. A passive PV component could be identified, for instance, as the entire initial STSV, which is subsequently evolved with the winds of the unperturbed background state, used in the

10 Stratosphere Troposphere Singular Vectors 327 tangent linear integration. Identifying the other relevant RW components would be more difficult, but could be based on the existence of the finite-time normal modes discussed in Frederiksen (2000) in combination with pseudomomentum orthogonality (Methven et al., 2005a). It is hoped that the present study will prove to be relevant for understanding the development and the growth mechanisms in such more general geometries. Acknowledgements The authors thank the reviewers for their constructive comments. Discussions with John Methven, Brian Hoskins and Tom Frame were appreciated. HdV acknowledges the National Environmental Research Council (NERC, grant NE/D011507/1). Appendix. Mathematical details A. Green s function formalism If attention is restricted to perturbations that can be described by a single zonal and meridional wavenumber k and l,eq.(1)canberewrittenas: q = ika[q], A[q] =ūq+ q y (z) G(z, z )q(z )dz, (A.1) where the integral is over the entire vertical domain and G(z, z ) is the Green s function for the operator that relates PV to streamfunction. Boundary conditions are included in (A.1) using the Bretherton (1966) approach. In the present study, N 2 varies continuously with height and G(z, z ) is obtained numerically. The solution to the vertically discretized analogue of (A.1) is q(t) = Mq(0), where M = exp( ikat) is the linear propagator. B. Pseudomomentum orthogonality and CRW interaction coefficients Held (1985) showed that neutral normal modes are orthogonal in the sense that <η j, q i > = δ ji (i j), (B.1) forgivennormalmodeswithpvq i and displacement η j.the inner product is defined as <X, Y >= ρx Ydydz,thebar denotes zonal averaging and dy is over one wavelength in the meridional direction. Also the growing and decaying normal modes are pseudomomentum orthogonal to the neutral modes, but not to each other. The CRW framework (Heifetz et al., 2004) combines growing and decaying normal modes into two vertically untilted structures called CRWs, that are orthogonal under the above inner product. Equation (6) follows by writing q(z, t) = α 1 (t)q 1 (z) + α 2 (t)q 2 (z) + q p (z, t) and multiplying (5) from the left with η i, and integrating over the domain (i.e. compute the pseudomomentum inner product), using the orthogonality of modes and the following definition of the interaction coefficients c ij = ũ i δ ij γ ij /k, (B.2) where ũ i = <η i, ūq i > <η i, q i >, γ ij = <η i, γ [q j ]> <η i, q i >, (B.3) with γ [q j ] = k q y (z) G(z, z ) q j (z )dz.theforcingterm f i (t) appearing in (6) is computed as f i (t) = <η i, γ [q p ]> <η i, q i >, (B.4) with q p the passive PV at time t (V09). C. Computation of singular vectors Stratosphere troposphere singular vectors (STSVs) are computed as follows. The domain-integrated total energy is 1 2 <ψ, q>= 1 2 <ψ, G 1 ψ> ψ Eψ, where G 1 is the inverse of Green s function. The local definition of total energy is required because the STSV involves projection operators. The optimization functional is ψ (t)(p f EP f )ψ(t) λ 2[ ψ (t 0 )(P i EP i)ψ(t 0 ) 1 ], (C.1) where P i and P f denote the projection operator at initial and final time respectively. One cannot simply set the first variation of (C.1) with respect to ψ (t 0 )tozeroandsolvethe resulting eigenvalue problem, because (P i EP i)issingular. We reduce the rank of the problem by writing ψ(t 0 ) = (ψ s,0,,0) T, (C.2) where ψ s = P i ψ(t 0 ). The STSVs follow by setting the first variation of (C.1) with respect to ψ s (t 0) to zero. This yields [ E 1 M P f EP f M ] ψ s = λ 2 ψ s, r where [ ] r indicates a reduced-rank matrix and M = GMG 1 is the stream-function propagator, ψ(t) = Mψ(t 0 ). Once the eigenvectors ψ s have been obtained, ψ(t 0 ) follows from (C.2), and q(t 0 ) = G 1 ψ(t 0 ). References Andrews DG, Holton JR, Leovy CB Middle atmosphere dynamics. Academic Press. Baldwin MP, Dunkerton TJ Propagation of the Arctic Oscillation from the stratosphere to the troposphere. J. Geophys. Res. 104: Baldwin MP, Dunkerton TJ Stratospheric harbingers of anomalous weather regimes. Science 294: Boyd JP The continuous spectrum of linear Couette flow with the beta effect.j. Atmos. Sci. 40: Bretherton FP Baroclinic instability and the short wavelength cut-off in terms of potential vorticity. Q. J. R. Meteorol. Soc. 92: Case KM Stability of inviscid plane Couette flow. Phys. Fluids 3: Charlton AJ, O Neill A, Stephenson DB, Lahoz WA, Baldwin MP Can knowledge of the state of the stratosphere be used to improve statistical forecasts of the troposphere? Q. J. R. Meteorol. Soc. 129:

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