Canonical frontal circulation patterns in terms of Green s functions for the Sawyer-Eliassen equation

Transcription

1 Q. J. R. Meteorol. SOC. (2001), 127, pp Canonical frontal circulation patterns in terms of Green s functions for the Sawyer-Eliassen equation By G. J. HAKIM * and D. KEYSER2 University of Washington, USA University at Albany, State University of New York, USA (Received 25 September 2000, revised 29 January 2001) SUMMARY Green s function solutions for the Sawyer-Eliassen equation are derived and compared for quasi-geostrophic and semi-geostrophic two-dimensional frontal dynamics in vertically bounded domains. The semi-geostrophic Green s functions provide a framework for understanding canonical frontal circulation patterns arising from interactions, or coupling, between upper- and lower-tropospheric jet-front systems. Previously documented conceptual models for these circulation patterns are reproduced by the solutions. A noteworthy finding is that vertical coupling produces a deep and narrow region of ascent when the upper- and lower-tropospheric point sources for the Green s functions share the same sign and absolute momentum surface. KEYWORDS: Fronts Green s function Jet streaks Sawyer-Eliassen equation Vertical motion 1. INTRODUCTION In a landmark paper on vertical circulations in frontal zones, Eliassen (1962) formalized and extended the work of Sawyer (1956), who introduced an equation for the ageostrophic stream function in the vertical plane normal to a two-dimensional front. This equation, now called the Sawyer-Eliassen equation (hereafter, the SE equation), provides the foundation for understanding ageostrophic circulations in the vicinity of frontal zones. Here we revisit the Green s function solutions mentioned by Eliassen (1962) to illustrate their utility for understanding and interpreting canonical circulation patterns in the vicinity of frontal zones. Historically, interest in frontal circulations has been motivated primarily by the frequent coincidence of cloud and precipitation features with frontal zones. Given the importance of these weather elements, vertical air motions have enjoyed a prominent role in the study of fronts. This role has been overshadowed in recent times by the widespread acceptance of potential-vorticity (PV) dynamics since, under adiabatic conditions, the vertical motion can be considered an implicit function of the PV. Nevertheless, even within the PV framework, vertical motions require explicit consideration in the presence of diabatic processes, such as latent heating. Just as there are elemental building blocks that can be superposed in the PV framework to recover the instantaneous horizontal flow field (e.g. Bishop and Thorpe 1994), there are building blocks that can be superposed in the vertical-motion framework to recover the instantaneous vertical-motion field (Clough et al. 1996). In both cases the building blocks correspond to Green s function solutions for the respective governing equations for the stream function and the vertical-motion fields, which Clough et al. (1996) consider in the context of the quasi-geostrophic (QG) system. For the more general semi-geostrophict system (hereafter, SG), Eliassen (1 962, section 3) describes qualitatively the properties of the Green s function for the SE equation. Further details on the Green s function are given in Eliassen (195 I), where a similar circulation equation (his * Corresponding author: Department of Atmospheric Sciences, University of Washington, Box , Seattle, WA , USA. hakim@atmos.washington.edu t The term semi-geostrophic is normally reserved for the set of equations resulting from application of the geostrophic-momentum approximation in geostrophic coordinates (Hoskins 1975). For simplicity, the term semigeostrophic is used here for both the geostrophic-coordinate and physical-space forms of the governing Royal Meteorological Society,

2 1796 G. J. HAKIM and D. KEYSER Eq. (29)) is analysed for heat and momentum point sources applied to a steady, stratified, baroclinic vortex in unbounded and semi-infinite atmospheres. Emanuel ( 1980, section 5) also derives Green s function solutions for the SE equation for a stratified, baroclinic, semi-infinite atmosphere and discusses the parametric dependence of the solutions. Two-dimensional circulation equations similar to the SE equation have been derived and applied to tropical cyclones (e.g. Willoughby 1979; Shapiro and Willoughby 1982), squall lines (Lu et al. 1997), the Hadley cell (Hack et al. 1989), and zonally averaged, planetary-scale meridional circulations (e.g. Pedlosky 1987, section 6.14; Holton 1992, section 10.2). In the case of tropical cyclones, Shapiro and Willoughby (1982) and Schubert and Hack (1982) emphasize the importance of inertial stability in modulating the strength of the balanced vertical circulation. Specifically, an intensifying tropical cyclone is associated with increasing inertial stability and a weakening balanced vertical circulation. The rising branch of the weaker circulation produces less adiabatic cooling for a given convective heating, resulting in increased net warming and enhanced development (Schubert and Hack 1982, section 3(a)). The Green s functions for the SE equation treated qualitatively by Eliassen (1962) are well known and widely cited in the synoptic-dynamic meteorology literature. Accordingly, the present analysis is offered to provide a quantitative basis for understanding and interpreting frontal circulations in an idealized context. Further motivation for this analysis derives from a survey of reviews of the SE equation (e.g. Eliassen 1990; Carlson 1991, section 14.4; Bluestein 1993, section 2.5.2; Keyser 1999, section 3), which emphasize the relationship between vertical circulation patterns and spatial distributions of the circulation source function. In contrast, relatively little attention has been devoted to investigating the parametric dependence of the circulation patterns associated with localized distributions of the source function; Green s functions are ideally suited for this purpose. In section 2 the SE equation is solved for QG and SG dynamics; our results extend the solutions of Eliassen (195 1) and Emanuel(l980) to vertically bounded domains. Properties of the QG and SG solutions, including parametric sensitivity, are discussed, and the solutions are scaled in a manner that elicits simplified geometrical descriptions of the circulations. The utility of the Green s functions is then illustrated in section 3, where analytical solutions are derived for four canonical frontal circulation patterns involving upper- and lower-tropospheric jet-front systems. Conclusions are provided in section SOLUTIONS OF THE SAWYER-ELIASSEN EQUATION The SE equation results from invoking the dynamical constraint of thermal-wind balance to relate the Lagrangian time tendency of the vertical shear of the along-front wind to the Lagrangian time tendency of the cross-front buoyancy gradient (e.g. Eliassen 1962). Here we study the Boussinesq SE equation in an (x, z) coordinate frame for a domain extending infinitely in x and bounded by rigid lids at z = 0 and z = H; the former marks the earth s surface and the latter is taken as a crude representation of the tropopause. In the QG system, the SE equation is whereas in the SG system, the SE equation is N2qxx - 2S2rlr,, + F2qz, = -2Q.

3 FRONTAL CIRCULATIONS 1797 In (1) and (2), + is the stream function for the transverse circulation (ua, w), with the horizontal ageostrophic wind given by ua = $, and the vertical motion given by w = -+,; the subscripts x and z indicate partial differentiation. We denote the crossfront (x) wind by u and the along-front (y) wind by v; subscripts g and a denote geostrophic and ageostrophic winds, respectively. The absolute momentum is given by rn = f x + vg, where f is the (constant) Coriolis parameter. The potential temperature is given by O = Oo(z) + (Om/g)b, where &(z) is the reference-state potential-temperature profile, Om is a constant, g is the acceleration due to gravity, and b is the perturbation Boussinesq buoyancy. Accordingly, the reference-state and local static stability are Ni = (g/om)oo, and N2 = N i + b,, respectively. The baroclinicity and inertial stability are measured by S2 = fvgz = b, and F2 = f(f + ugx), respectively. We distinguish between reference-state (external) parameters, Ni, f and H, and local (internal) parameters, N2, F2 and S2. The parameters N2, F2 and S2 can vary for a fixed reference state; the SG solutions are influenced by such changes, whereas the QG solutions are not. In the (x, z) plane, the source function takes the form Q = -fjxz(ug, vg) + 1/2(3t, - fa,), where J is the Jacobian operator, 3t is a heat source, and D is an along-front momentum source. In order to simplify the subsequent analysis and to facilitate physical interpretation of the resulting solution of the SE equation, constant coefficients are assumed in (1) and (2). This assumption implies constant slopes in the (x, z) plane for the isentropic and absolute momentum surfaces of -(S2/N2) and -(F2/S2), respectively. Before proceeding with the solution, it proves useful to express (1) and (2) in the standard form of a Poisson equation with unit coefficients. This objective is achieved by non-dimensionalizing all variables in (1) and (2), and by transforming to geostrophic coordinates in (2). The vertical length-scale is given by the depth of the troposphere H, the horizontal length-scale is taken to be the Rossby radius, and the magnitude of the source function Q is given by 6. In the QG case, the horizontal length-scale is given by LQG = q$ H/ f 2, the stream function is scaled by 6 H2/ f *, and the reference-state PV is given by qqg = f 2Ni. For this definition of qqg, LQG reduces to the conventional expression for the Rossby radius in the QG system, NoH/f. Denoting dimensionless dependent (independent) variables with (without) superscripted asterisks, (1) becomes, +: i-, $: = -2Q*. (3) In the SG case, the horizontal length-scale is given by LSG = qigh/f2, the stream function is scaled by 6H2/F2, and the PV is given by qsg = F2N2 - S4.t To achieve standard form, a transformation to geostrophic coordinates, (X, 2) = (x + ug/f, z), eliminates the mixed-derivative term in (2): +$x + +kz = -2Q*. For both (3) and (4), the rigid-lid condition demands that the vertical velocity should vanish at z = 2 = (0, l), which is satisfied by $*(x, z = (0, 1)) = $*(X, 2 = (0, 1)) = 0. A classic technique in solving (3) and (4) exploits Green s identities and the homogeneous boundary conditions to re-express the differential equation in integral t Positive qqc and qsg are required, respectively, for (1) and (2) to be elliptic. These conditions correspond to gravitational stability in the QG case and to symmetric stability in the SG case. Moreover, qsg > 0 is equivalent to Ri > f2/f2, where Ri = f2n2/s4 is the Richardson number (e.g. Hoskins 1974). (4)

4 1798 G. J. HAKIM and D. KEYSER form (e.g. Courant and Hilbert 1953, pp ): +*(r) = /: Q*(r )G(r, r) da. Here, r = (x, z), and G is the Green s function, with primes denoting a variable point of integration, for the two-dimensional Poisson equation: 1 G(r, r) = -- log Ir - rl + h(r, r). (6) n Note that h(r, r) is harmonic in the solution domain S2 and ensures that the boundary conditions are satisfied. The logarithm in (6) is often referred to as the two-dimensional free-space Green s function, since in the absence of boundaries h(r, r) = 0. Since G(r, r) is the solution of (5) for a 8-function distribution of Q*(r ), G(r, r) is taken as the fundamental building block for constructing more complicated solutions. Solutions that follow assume a point source with amplitude Q: located at (xf, zf). Boundary conditions are satisfied by choosing the appropriate h(rf, r). Here we use the method of images, which employs virtual point sources exterior to S2. For one boundary, a single image of opposite sign to that of the interior source satisfies homogeneous Dirichlet boundary conditions. However, the current problem is complicated by the presence of a second boundary, necessitating an infinite sequence of images. By combining the free-space logarithm in G with the infinite sequence of logarithms in h, we find: An alternative solution for the stream function +* in terms of an eigenfunction expansion is given in the appendix, which provides a consistency check of the numerical evaluation of (7). Considering only the free-space (n = 0) contribution to (7) shows that the solution takes the form of concentric circular contours of constant stream function centred on (xf, zf), with the stream-function value of a contour of unit radius equal to zero. The radius of any non-zero stream-function contour greater (less) than unity is inversely (directly) proportional to the source amplitude, such that the magnitude of the radial gradient of + increases with increasing source amplitude. This simple geometrical interpretation applies to appropriately scaled QG solutions and to scaled SG solutions in geostrophic coordinates. (a) QG solution In terms of dimensional variables, the QG solution becomes

5 FRONTAL CIRCULATIONS 1799 and (1 1) where nh +zf, n even, = [ (n + 1)H - zf, n odd. In dimensional physical space (i.e. (x, z)), the free-space contribution takes the form of highly eccentric ellipses, with the ratio of minor to major axes, the circulation aspect ratio, scaled by H/LQG = f 2/qkt = f/no. For future reference in the interpretation of the SG solutions (section 2(b)), it is observed that H/Lw also may be expressed as q$/ni. Vertical motion (10) is inversely proportional to the square root of the reference-state PV, whereas the ageostrophic wind (11) is inversely proportional to the reference-state vorticity squared. Equations (10) and (11) also indicate that w (Ua) decays more slowly in x (z) than Ua (w). When boundaries are present, terms for n # 0 in (9)-( 11) become important, particularly when the point source is located near a boundary. As an example, for a given source amplitude, ua becomes increasingly large as the point source approaches a boundary. Other than Qo, all quantities determining the amplitude and structure of the circulation are external parameters pertaining to the reference state. For a fixed latitude (f) and domain depth (H), only the reference-state static stability (Ni) can affect the circulation through the reference-state PV (recall that qqc = f 2Ni). Specifically, large No acts to decrease the circulation aspect ratio and to reduce w. These stringent restrictions are relaxed in the SG solution, where internal parameters modulate the circulation. (b) SG solution In terms of dimensional variables, the SG solution becomes The vertical motion and ageostrophic wind are obtained from which give

6 1800 G. J. HAKIM and D. KEYSER and where The interpretation of + and w for the SG solution in geostrophic coordinates is the same as for the QG solution, with the exceptions that the QG parameter f is replaced by F, and the PV and the Rossby radius take their SG forms. The interpretation of Ua in the SG case is modified relative to the QG case by an additional term contributing to structure in X (cf. (16) with (1 1)). To complete the SG solutions, (1 3), (1 5), (1 6) and (1 7a,b) must be transformed back to physical space ((X, Z) + (x, z)). This transformation is achieved analytically by observing that (Z - Zf) = (z - zf) and that the following exact relation arises from a Taylor series expansion of vg as a consequence of uniform F2 and S2 : F2 (X- Xf) = -(X S2 - Xf) + -(z f2 f2 - Zf). (18) In dimensional physical space, the SG solution becomes and where nh + zf, n even, = [ (n + l)h -zf, n odd. Although the amplitude coefficients for +, w, and ua are unchanged from the SG solution in geostrophic coordinates, the transformation to physical space introduces

7 FRONTAL CIRCULATIONS 1801 significant structural modifications to the solution. In the slope of w = 0 is 6z I free-space case (n = 0), the which coincides with the slope of absolute momentum surfaces, and the slope of Ua = 0 is which coincides with the slope of isentropic surfaces. The slopes of these respective contours may be contrasted with the QG solution, where the w = 0 contour is vertical and the Ua = 0 contour is horizontal. Further consideration of the free-space contribution to the SG solution in physical space reveals that a contour of constant stream function is described by the standard formula for a rotated ellipse: F4x2 + 2S2F2xz + F N z = C. For convenience, we have taken (xf, zf) = (0,O) and C denotes a constant. The mixed term, 2S2 F2xz, can be eliminated by rotating coordinates through angle a, where (25) Considering typical midlatitude values of internal parameters, we note that F2 < S2 << N2. These inequalities motivate a small-angle approximation for ct in (26), such that tan a * -(S2/N2). Accordingly, the major axis of the rotated ellipse slopes ap- proximately along an isentropic surface, consistent with the analysis of Emanuel(l980, section 5). For small a, the circulation aspect ratio is qsg 1 /2/N2 F/N, which compares to qht/ni = f/no for the QG solution. Given the inherent difficulty of interpreting and visualizing the free-space contribution to the SG solution in dimensional physical space, where the slopes of isentropic surfaces are small, we proceed to scale x by d: = NH/F and z by H. This particular choice of length-scales results in unit coefficients for the first and third terms on the lefthand side of (2); it is of interest that J corresponds to the Cartesian limit (i.e. r + 00) of the natural length-scale ( NH/{(f + <)(f + 2~/r)}'/~) that arises in the analysis of a balanced symmetric baroclinic vortex (e.g. Shapiro and Willoughby 1982, section 2; Hoskins et al. 1985, section 3). For this scaling and for positive S2 (i.e. warmer air for increasing x), the major axis of the ellipse is rotated by -n/4 with respect to the horizontal. Furthermore, the non-dimensionalized slopes of absolute momentum surfaces and isentropic surfaces are reciprocals of one another (-(FN/S2) and -(S2/FN), respectively); therefore, the major axis of the ellipse bisects the angle between m and 8. Finally, the circulation aspect ratio is given without approximation by q:t/(fn + S2) = ((1 - S2/FN)/(1 + S2/FN)}'/2. As in the unscaled SG case, the circulation aspect ratio is directly proportional to the square root of the PV.

8 1802 G. J. HAKIM and D. KEYSER (c) Limiting cases for the free-space SG solutions Limiting cases of the SG solution are now considered for variations in S2, F2, and N2. First, we note that the QG limit is recovered for F2 + f 2, N2 + Ni, and S2 + 0, which gives qsg + qqg, and (LsG, 8) + LQG. The remaining limits are discussed from a PV perspective. For given F2 and N2, qsg is maximized for S This is the barotropic limit, with orthogonal absolute momentum and isentropic surfaces. In the 63-scaled framework, lines of constant stream function form circles. Minimum PV, qsg + 0, occurs when S2 + F2N2, which places an upper bound on S2 given F2 and N2, or a lower bound on F2N2 given S2. In the 8-scaled framework, the limit of vanishing qsg is associated with a rotation of the absolute momentum and isentropic surfaces towards the major axis of the ellipse, accompanied by the circulation aspect ratio decreasing to zero. Maximum PV, qsg +. 00, occurs for both F or N2 + 00, and there are different solutions in these two limits. For both limits, the absolute momentum and isentropic surfaces are orthogonal in the 8-scaled framework. For F2 + 00, there is no motion: (+, w, ua, OC) In contrast, N results in w +. 0; however, In this limit , and in dimensional physical space the circulation consists of purely horizontal ageostrophic motion along horizontal isentropic surfaces. (d) Properties of the fill solutions To examine the structure of the full QG and SG solutions, we must account for the boundary contributions (i.e. terms with n # 0). Below we briefly explore the impact of variations in the internal parameters and point-source location for a given reference state (i.e. choice of Ni, f and H). For ease of interpretation, the source term Qo is taken to be independent of the other parameters; it is assumed to be due to gradients in cross-front geostrophic wind, diabatic heating, or friction. In all cases, the magnitude of Qo is specified to be 3 x s-~. The reference-state parameters used for all solutions are representative midlatitude synoptic-scale values: Ni = s-~, f = s-', and H = 10 km. These values yield qqg = 1 x s - ~ and LQG = 1000 km; the corresponding amplitudes for +, w, and Ua for the QG solution, given by the leading coefficients of (9)-(1 l), respectively, are 9.55 x lo3 m's-', cm s-', and m s-l. Internal parameter values, along with the corresponding values for QSG, 8, the scaled circulation aspect ratio, and the amplitudes for +, w, and Ua for the SG control and test cases, are listed in Table 1. In all cases, the infinite sum in the solutions is truncated at n = 1000, for which the maximum residual + at the boundaries is less than 10 m2s-l. For all figures that follow, the plotting domain is 1000 km wide and 10 km deep; the point source is positioned at zf = 5 km, except where noted otherwise. The I) field for the QG solution exhibits a characteristic elliptical structure, with the major axis in the cross-front direction (Fig. l(a)). A thermally direct circulation is apparent in the w and ua fields, which exhibit a simple dipole structure, without tilt, in

9 FRONTAL CIRCULATIONS 1803 TABLE 1. PARAMETER VALUES FOR SG CONTROL AND TEST CASES Case F~ s2 N~ ~ S G L 9 J/o wo uao Control 1 3 I 0.91 lo Large F Large S lo Large N Dimensional values are given for the internal parameters, F2 S2 (10-7s-2), and N2 (10-4s-2), along with corresponding values for the potential vorticity, qsg (10-'2s-4), horizontal length-scale, L = NH/F (km), scaled circulation aspect ratio, 9 = {(I - S2/FN)/(1 + S2/FN)]'/2, and the amplitudes of the stream function, J/o (lo3m2s-'), vertical velocity, wo (cm s-i), and ageostrophic wind, U,O (m s-l). The amplitudes of the stream function, vertical velocity, and ageostrophic wind are given by the leading coefficients of (19)-(21), respectively. In these calculations, H = 10 km and Qo = 3 x lo-'' s-~. Figure 1. (ahc) QG and (dhf) SG control cases for a point source at zf = 5 km. The stream function is given in (a) and (d) by thick lines every 2 x lo3 m2sp1, vertical motion is given in (b) and (e) by thick lines (negative values dashed) every 2 cm s-', and ageostrophic wind is given in (c) and (f) by thick lines (negative values dashed) every 2 m s-'. Potential temperature is given by thin solid lines every 4 K, and absolute momentum is given by thin dashed lines every 30 m s-' throughout. The plotting domain is lo00 km wide and 10 km deep. the vertical plane (Figs. l(b) and (c)). The QG solution for w (Fig. l(b)) complements the QG solutions of Clough et al. (1996, see e.g. their Fig. 7), who solve for Green's functions of the omega equation in the absence of an upper boundary. In contrast to the QG solution, the SG control case exhibits a prominent tilt (Figs. l(d)-(f)). As inferred from the analysis in section 2(b), the w = 0 contour slopes along an absolute momentum surface (Fig. l(e)), while the ua = 0 contour slopes approximately along an isentropic

10 1804 G. J. HAKIM and D. KEYSER Figure 2. Semi-geostrophic solutions for: (a) large inertial stability, (b) large baroclinicity, (c) large static stability, and (d) near-boundary point source (zf = 1 km, see text). The stream function is given by thick lines every 2 x lo3 m2s-', potential temperature is given by thin solid lines every 4 K, and absolute momentum is given by thin dashed lines every 30 m s-'. Parameter values are given in Table 1; the values for the control case apply in (d). surface (Fig. l(f))*. For completeness we note that, aside from dimensional differences, the w and ua fields give the free-space distributions of stream function for point sources of heat and momentum, respectively (cf. our Figs. l(e) and (f) in the vicinity of the point source with Figs. 8 and 11 in Eliassen (1951), respectively). Test cases are now examined by varying internal parameters to illustrate their effect on the SG control solution; limiting cases have been discussed in the previous subsection. When the inertial stability parameter F2 is doubled, there is a factor of 2.1 increase in the PV and the amplitudes of $, w, and ua are reduced significantly (Figs. 2(a), 3(a) and 4(a); Table 1). The horizontal length-scale decreases to 707 km from 1000 km * Although the analysis in section 2(b) establishing the correspondence between the w = 0 contour and an rn surface is restricted to the free-space contribution to the SG solution, inspection of (20) shows that this correspondence carries over to the full solution because of the independence of the numerator on n. In contrast, inspection of (21) shows that the free-space correspondence between the ua = 0 contour and a 0 surface does not cany over to the full solution. In the full solution, the ua = $z = 0 contour must deviate from the 0 surface passing through the point source, since the boundaries force the line along which = 0 to deflect below (above) this 0 surface on the cold (warm) side of the point source.

11 FRONTAL CIRCULATIONS 1805 Figure 3. As in Fig. 2, except vertical motion is given by thick lines (negative values dashed) every 2 crn s-'. for the control case. Doubling the baroclinicity parameter S2 does not change the horizontal length-scale, but produces a circulation where the slope of the w = 0 contour is shallower than in the control case. While the amplitudes of + and Ua are the same as in the control (refer to (19) and (21)), the amplitude of w increases by 19% (Figs. 2(b), 3(b) and 4(b); Table 1). When the static stability parameter N2 is doubled, the amplitudes of 1+4 and ua are unchanged from the control. However, the amplitude of w decreases by 31% and the horizontal length-scale increases to 1414 km (Figs. 2(c), 3(c) and 4(c); Table 1). When the point source is located at zf = 1 km, a pronounced asymmetry in the vertical direction is introduced by the homogeneous boundary conditions (Figs. 2(d), 3(d) and 4(d)). Moreover, the circulation is strongly damped in x, although the horizontal length-scale is the same as in the control case. Despite the fact that the amplitude coefficient for w is the same as in the control case, there is a considerable increase in vortex stretching w, at the lower boundary. This increase in vortex stretching is consistent with synoptic experience, which suggests that, for all other factors being equal, localized forcing near the earth's surface is more conducive to surface development (i.e. cyclogenesis) than is localized forcing in the middle troposphere.

12 1806 G. J. HAKJM and D. KEYSER Figure 4. As in Fig. 2, except ageostrophic wind is given by thick lines (negative values dashed) every 2 m s-l. 3. APPLICATION TO CANONICAL FRONTAL CIRCULATION PATTERNS Clough et al. (1996) emphasize that significant weather elements associated with vertical air motions frequently are highly localized in space and often arise from interactions involving upper- and lower-tropospheric weather systems. Because of its inherently localized nature, the Green s function provides a natural structure that, through superposition, allows for construction of ageostrophic circulation patterns attributable to specific weather systems. This technique is applied here to four canonical ageostrophic circulation patterns in observed jet-front systems: vertically uncoupled and coupled upper- and lower-level circulations in jet-streak exit regions situated above surface frontal zones; vertically coupled upper- and lower-level circulations in jet-streak entrance regions situated above surface frontal zones; and horizontally coupled circulations arising from laterally separated jet-streak entrance and exit regions. Prior to applying the Green s function technique to these realistic circulation patterns, we remind the reader that, strictly speaking, the solutions pertain to two-dimensional flows having uniform PV for SG dynamics. Despite these restrictive assumptions, the subsequent application illustrates that the solutions offer considerable conceptual utility in situations more general than those for which they are formally valid.

13 FRONTAL CIRCULATIONS 1807 Figure 5. Schematic illustrations of (a) and (b) vertically uncoupled, and (c) and (d) coupled upper- and lowerlevel jet-front systems. (a) and (c) show plan views of the location of the upper-level jet-streak exit region with respect to the surface frontal zone. Isotachs are given by thick solid lines, with the solid arrow denoting the axis of the upper-level jet streak, surface isentropes are given by thin dashed lines, and the open arrow denotes the axis of the low-level jet. (b) and (d) show cross-sections along AA in (a) and BB in (c), respectively. Isotachs are indicated by thick dashed lines surrounding the upper- and lower-level jets, frontal boundaries by thin solid lines, the tropopause by thin double lines, the moist boundary layer by the stippled region, and the transverse ageostrophic circulation by solid arrows. From Keyser (1999), as adapted from Shapiro (1982). Ageostrophic circulations associated with upper-level jet streaks have long been recognized (e.g. Beebe and Bates 1955; Uccellini and Johnson 1979), with the anticyclonicshear-side entrance and cyclonic-shear-side exit regions favoured for midlevel ascent in the case of straight jet streaks in the absence of appreciable horizontal temperature advection (Shapiro 1982, his Fig. 6). Uccellini and Johnson note that coupling between upper- and lower-level jet streaks can be important for providing ascent needed to trigger convection. From a circulation point of view, the question can then be asked: What effect does vertical and horizontal alignment have on coupling? Based on analyses of the superposition of upper- and lower-level jet-front systems, Shapiro (1982) proposes schematic configurations of jet-streak exit regions that are unfavourably (Figs. 5(a) and (b)) and favourably (Figs. 5(c) and (d)) aligned with respect to a surface frontal zone for the release of convective instability. These respective configurations are modelled using a superposition of the SG Green s function solution applied to two sources: the upper-level source, located at zf = 5 km, associated with the jet-streak exit region (negative source), and the lower-level source, located at zf = 1 km,

14 1808 G. J. HAKIM and D. KEYSER Figure 6. Semi-geostrophic solution for an upper-level jet-streak exit region overlying a surface frontal zone (refer to Figs. 5(a) and (b)). Values of F2, S2, and N2 coincide with those for the control case (see text). Stream function is given by thick lines (negative values dashed) every 2 x lo3 m2s-'; positive values of vertical motion w are shaded every 2 cm s-i starting at 1 cm s-i, and absolute momentum is given by thin dashed lines every 30 m s-'. Vectors depict the ageostrophic circulation, with w scaled by a factor of 100 commensurate with the inverse aspect ratio of the physical dimensions of the domain, which is square in the plotting coordinates. associated with the surface frontal zone (positive source). When the upper- and lowerlevel sources are vertically aligned, the respective circulations destructively interfere (Fig. 6), such that the circulations are uncoupled and upward motion in the warm air is suppressed as in Fig. 5(b). When the upper-level source is displaced 200 km to the warm-air side of the lower-level source, the respective circulations constructively interfere (Fig. 7), such that the circulations are coupled and an upright column of deeptropospheric ascent results as in Fig. 5(d). The foregoing configuration of a jet-exit region aligned above a surface frontal zone may be modified to apply to a jet-entrance region. A synoptic example is a zonally oriented quasi-stationary front situated beneath a confluent jet-entrance region (e.g. Kocin and Uccellini 1990, Figs. 116 and 118). This configuration is modelled here by assigning positive upper- and lower-level sources to a common absolute momentum surface in order to achieve optimal vertical coupling; the respective sources are located at zf = 5 km and 1 km, and separated in x by 240 km. Recall that the w = 0 contour slopes along an m surface and, in the present case, this slope also yields vertical coupling of ascent and descent patterns associated with the respective sources (Fig. 8). Contours

15 FRONTAL CIRCULATIONS 1809 Figure 7. As in Fig. 6, except for an upper-level jet-streak exit region ahead of a surface frontal zone (refer to Figs. 5(c) and (d)). of constant stream function are nearly upright in the lower troposphere immediately to the warm side of the absolute momentum surface containing the point sources, as compared with the nearly horizontal orientation in the control case in the absence of the low-level source (cf. Figs. 8 and 1 (d)). The solution shown in Fig. 8 provides an explanation for why the anticyclonic-shearside entrance region of a jet streak is a favourable location for the formation of clouds and precipitation, particularly when a low-level frontal zone is present. Furthermore, the enhanced vertical coupling that results when upper- and lower-level point sources share a common absolute momentum surface supports synoptic conventional wisdom that in a strongly baroclinic environment the upward direction is along an absolute momentum surface. The fourth canonical circulation pattern applies to observations showing that the rising branches of jet-streak circulations can also couple horizontally (Uccellini and Kocin 1987; Hakim and Uccellini 1992). Specifically, coupling occurs when the rising branch of a thermally indirect circulation in the exit region of one jet streak (negative source) coincides with the rising branch of a thermally direct circulation in the entranceregion of a second jet streak (positive source), where the second jet streak is positioned polewards of the first (Fig. 9). A solution using a superposition of Green s functions at zf = 5 km and separated horizontally by 320 km is given in Fig. 10. A sloping updraught, the axis of which coincides with an m surface, occurs between the two

16 1810 G. J. HAKIM and D. KEYSER Figure 8. As in Fig. 6, except for an upper-level jet-streak entrance region sharing the same absolute momentum contour as a surface frontal zone. Values of F2, S2, and NZ (see text) coincide with those given in Table 1 for the case of large baroclinicity. sources. This updraught results from the superposition of the respective rising branches associated with each source, in a manner similar to observations (cf. Hakim and Uccellini 1992, their Fig. 13). 4. CONCLUSIONS Green s functions for the ageostrophic circulation stream function provide a useful tool for constructing analytical solutions that explain localized vertical circulation patterns occurring in the vicinity of observed jet-front systems. This approach, whereby the ageostrophic stream-function and vertical-motion fields are attributed to individual sources of circulation in a pointwise manner, complements the attribution of the balanced stream-function and horizontal-motion fields to individual sources of PV. In this paper, Green s function solutions for the SE equation are derived for vertically bounded domains, and are superposed to provide simple models of canonical circulation patterns in observed upper- and lower-tropospheric jet-front systems. Both QG and SG solutions are derived; the former provide a reasonable approximation to the latter when the QG PV is a reasonable approximation to the SG PV. Unlike the free-space stream-function contours in the QG case, which consist of highly eccentric ellipses with major and minor axes oriented in the horizontal and vertical directions, in

17 FRONTAL CIRCULATIONS 1811 Figure 9. Schematic illustration of horizontally coupled jet-streak circulations. Jet streaks are given by long arrows labelled 'JET'; ageostrophic circulations are given by short unlabelled arrows. The rising branch of a thermally indirect circulation located within the exit region of the equatorward jet streak is coupled with the rising branch of a thermally direct circulation located within the entrance region of the poleward jet streak. Representative mean sea-level isobars and a surface frontal analysis also are depicted. From Hakim and Uccellini (1992), as adapted from Uccellini and Kocin (1987). the SG case the circulation ellipses are tilted such that the major axis slopes approximately along an isentropic surface. Interpretation of the free-space circulation in the SG case is facilitated by a judicious scaling whereby a universal rotation angle arises for the orientation of the major axis of the circulation ellipses (-n/4 with respect to the horizontal for positive baroclinicity parameter), and the major axis bisects the angle between the scaled absolute momentum and isentropic surfaces. In this scaled framework, the circulation ellipses reduce to circles in the limit of vanishing barwlinicity and become increasingly anisotropic as the PV tends toward zero. Analytical solutions in the SG case are constructed for four canonical examples of vertical and horizontal coupling between upper- and lower-level jet-front systems. The solutions indicate that a significant distinction exists between vertical coupling in the exit and entrance regions of jet streaks. Upward motion is suppressed and circulations are vertically uncoupled in the case of a jet-exit region overlying the cold-air side of a surface frontal zone. In contrast, a tropospheric-deep pattern of upward motion results from vertical coupling between circulations in the case of a jet-exit region overlying the warm-air side of a surface frontal zone. In the case of a jet-entrance region and a surface frontal zone, a tropospheric-deep pattern of upward motion also results from vertical coupling when circulation sources of the same sign share a common absolute momentum surface. Finally, horizontal coupling occurs when the rising branch of a thermally indirect circulation in the exit region of an equatorward jet streak is juxtaposed with the rising branch of a thermally direct circulation in the entrance region of a

18 1812 G. J. HAKIM and D. KEYSER Figure 10. As in Fig. 6, except for horizontally coupled jet-streak circulations. poleward jet streak. In this case, a sloping updraught results between the two sources in response to the superposition of the respective rising branches associated with each source. On the basis of the analysis conducted in this paper, we close with a synoptic rule of thumb with regard to vertical coupling between upper- and lower-level jet-front systems: Vertical coupling of ageostrophic circulations is optimized when the upperand lower-level sources share the same sign and absolute momentum sugace. The practical utility of this rule may be evaluated by constructing vertical cross-sections of absolute momentum surfaces in regions containing localized distributions of upperand lower-level frontogenetical forcing. The cross-front direction is defined by a deeplayer (e.g hpa) thickness gradient, following conventional practice in the assessment of slantwise convective stability in three-dimensional baroclinic flows (e.g. Emanuel 1994, section 12.3). To the extent that this rule is valid, a sloping region of deep-layer ascent (as indicated in Fig. 8) would be expected when upper- and lower-level maxima of frontogenetical forcing are aligned along a common absolute momentum surface. ACKNOWLEDGEMENTS Work on this project began while the lead author was enrolled in the atmospheric science doctoral program at the University at Albany. The inspiration to bring this

19 FRONTAL CIRCULATIONS 1813 project forward to publication followed a discussion between the authors and Professor Arnt Eliassen of his landmark 1962 paper on frontal circulations at a symposium held in honour of his eightieth birthday at Endicott House, Dedham, MA, in October We thank two anonymous reviewers for their comments, which helped clarify portions of the manuscript. Funding for this research has been provided by the National Science Foundation through Grants ATM and ATM , awarded to the University at Albany, and ATM , awarded to the University of Washington. APPENDIX An alternative solution to (3) (or (4)) employs eigenfunctions to satisfy the boundary conditions. Taking the Fourier transform (denoted by ^) of (3) in x gives: where h 2- $zz(kl Z) - k $(k, Z) = -2G(k, z), (A.1) The solution of (A. 1) is $(k, z) = -2 where the Green s function is Jo 1 &k, z )c(z, z ; k) dz, (A.3) The Wronskian is given by W=det( $1 1, 3 2 $11 $2z and $1 and $2 are homogeneous solutions that satisfy the boundary conditions at z = 0 and z = 1, respectively: &(k, z) = C1 sinh(kz), (A.6a) $2(k, z) = Ca(cosh(kz) - coth(k) sinh(kz)), (A.6b) where C1 and C2 are arbitrary non-zero constants. For a point source, Q*(x, z) = Q:6(x - xf)s(z - zf), a Fourier transform+in x h gives Q(k, z) = Q: e-ikxfs(z - zf). Knowledge of &k, z) and 6(z, 2 ; k) determines $(k, z) through (A.3) and subsequently $*(x, z) through (A.2): where The odd part of the exponential in (A.7) integrates to zero leaving * E(z, zf; k) cos(k(x - xf)) dk. $*(x, z) = -- n Numerical quadrature confirms the equivalence between (A.9) and (7). (A.9)

20 1814 G. J. HAKIM and D. KEYSER Beebe, R. G. and Bates, F. C. Bishop, C. H. and Thorpe, A. J. Bluestein, H. B. Carlson, T. N. Clough, S. A., Davitt, C. S. A. and Thorpe A. J. Courant, R. and Hilbert, D. Eliassen, A. Emanuel, K. A. Hack, J. J., Schubert, W. H., Stevens, D. E. and Kuo, H.-C. Hakim, G. J. and Uccellini, L. W. Holton, J. R. Hoskins. B. J. Hoskins, B. J., McIntyre, M. E. and Robertson, A. W. Keyser, D. Kocin, P. J. and Uccellini, L. W. Lu, C., Ciesielski, P. E. and Schubert, W. H. Pedlosky, J. Sawyer, J. S. Schubert, W. H. and Hack, J. J. Shapiro, M. A. Shapiro, L. J. and Willoughby, H. E. Uccellini, L. W. and Johnson, D. R. Uccellini, L. W. and Kocin, P. J. Willoughby, H. E REFERENCES A mechanism for assisting in the release of convective instability. Mon. Weather Rev., 83,l-10 Potential vorticity and the electrostatics analogy: Quasigeostrophic theory. Q. J. R. Meteoml. SOC., 120, Synoptic-dynamic meteorology in midlatitudes: Volume 11. Oxford University Press, New York Mid-latitude weather systems. HarperCollins Academic, London Attribution concepts applied to the omega equation. Q. J. R. Meteoml. SOC., 122, Methodr of mathematical physics: Volume I. John Wiley and Sons, New York Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astmphysica Norvegica, 5,1940 On the vertical circulation in frontal zones. Geophys. Publ., 24, Transverse circulations in frontal zones. Pp in Extratmpical cyclones, The Erik Palmkn memorial volume. Eds. C. W. Newton and E. 0. Holopainen. American Meteorological Society, Boston Forced and free mesoscale motions in the atmosphere. Pp in Collection of lecture notes on dynamics of mesometeomlogical disturbances. Eds. Y. K. Sasaki, N. Monji and S. Bloom. University of Oklahoma, Norman Atmospheric convection. Oxford University Press, New York Response of the Hadley circulation to convective forcing in the ITCZ. J. Atmos. Sci., 46, Diagnosing coupled jet-streak circulations for a northern plains snow band from the operational nested-grid model. Weather and Forecasting, 7,2648 An introduction to dynamic meteomlogy. Academic Press, San Diego The role of potential vorticity in symmetric stability and instability. Q. J. R. Meteoml. Soc.. 100, The geostrophic momentum approximation and the semigeostrophic equations. J. Atmos. Sci., 32, On the use and significance of isentropic potential vorticity maps. Q. J. R. Meteoml. Soc., 111, On the representation and diagnosis of frontal circulations in two and three dimensions. Pp in The life cycles of extmtmpical cyclones. Eds. M. A. Shapiro and S. GrBnils. American Meteorological Society, Boston 'Snowstorms along the northeastern coast of the United States: 1955 to 1985'. Meteoml. Monogs, No. 44, American Meteorological Society, Boston Geostrophic and ageostrophic circulations in midlatitude squall lines. J. Atmos. Sci., 54, Geophysical fluid dynumics. Springer-Verlag, New York The vertical circulation at meteorological fronts and its relation to frontogenesis. Pmc. R. SOC. London, A234, Inertial stability and tropical cyclone development. J. Atmos. Sci., 39, 'Mesoscale weather systems of the central United States'. CIRES/NOAA Technical Report, University of Colorado, Boulder The response of balanced hurricanes to local sources of heat and momentum. J. Atmos. Sci.. 39, The coupling of upper and lower tropospheric jet streaks and implications for the development of severe convective storms. Mon. Weather Rev., 107, The interaction of jet streak circulations during heavy snow events along the east coast of the United States. Weather and Forecasting, 2, Forced secondary circulations in hurricanes. J. Geophys. Res.. 84,

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