An Isentropic Vertical Coordinate Model: Design and Application to Atmospheric Frontogenesis Studies

Transcription

1 Meteorol. Atmos. Phys. 50, (1992) Meteorology and Atmospheric Physics 9 Springer-VerIag 1992 Printed in Austria Department of Atmospheric Sciences, University of California, Los Angeles, CA, U.S.A An Isentropic Vertical Coordinate Model: Design and Application to Atmospheric Frontogenesis Studies A. Arakawa, C. R. Mechoso, and C. S. Konor With 12 Figures Received December 30, 1991 Revised August 6, 1992 Summary The isentropic vertical coordinate model developed at UCLA is briefly reviewed. The review includes an outline of the approach used to overcome technical difficulties in handling model layers with small mass. The model's performance is demonstrated by simulating the evolution of a middle-latitude baroclinic disturbance. During the evolution of the disturbance, sharp frontal zones are generated in the upper and middle troposphere with realistic tropopause folding. The extent to which different dynamical processes contribute to frontogenesis is analyzed. While the model successfully simulates frontogenesis in the upper and middle troposphere, it has a difficulty in simulating surface fronts. The difficulty arises due to the lack of degrees of freedom in surface temperatures since an isentropic vertical coordinate model requires a large number of vertical layers to obtain a high horizontal resolution at the lower boundary. This suggests the potential of a hybrid vertical coordinate, which approaches 0 at upper levels and cr at lower levels. 1. Introduction The purpose of this paper is to demonstrate the merit of using isentropic surfaces as coordinate surfaces in simulating frontogenesis in the upper and middle troposphere. Most advantages of an isentropic vertical coordinate come from the following two properties of the coordinate: i) Since the entropy (or the potential temperature 0) is a thermodynamic state variable, the horizontal pressure gradient force is an irrotational vector. It then follows that the force does not yield a horizontal circulation along a coordinate surface. ii) The coordinate surfaces are material surfaces under adiabatic processes. Then the advection takes place along the coordinate surface and, therefore, a three-dimensional motion may be formally treated as a two-dimensional motion. As a consequence of i) and ii), (the quasi-static version of) Ertel's potential vorticity has the simplest form (see (2.5))with an isentropic vertical coordinate. Moreover, as a consequence of ii), the problem of discretizing vertical advection is virtually eliminated. Due to the close similarity between this coordinate system for the atmosphere and the shallow-water equation system for an incompressible homogeneous flow, finite-difference schemes developed for the latter system can be almost directly applied to the former system. There is, for example, no room for the "internal symmetric computational instability" discussed by Hollingsworth et al. (1983), which occurs when a certain type of finite-difference schemes for the shallow-water equations are used in a three-dimensional model of the atmosphere based on the p- or o--coordinate.

2 32 A. Arakawa et al. Another consequence of ii) is that the thermodynamic equation is used only for diagnosing the material time change of the potential temperature, instead of predicting the local time change of the potential temperature as is done with the p- or a-coordinate. Thus, as far as the thermodynamic equation is concerned, the relation between the material time change of the entropy (or the potential temperature) and heating is all that is needed and, therefore, there is no discretization error originating from the horizontal advection term in that equation. Also from ii), the mass between two adjacent coordinate surfaces can become arbitrarily small. In this way, an isentropic vertical coordinate model can automatically create fine vertical resolution in a region of vertical confluence. This feature, which is in sharp contrast to any other commonly-used vertical coordinate, is definitely advantageous in simulating frontogenesis and associated dynamical processes. In addition, since fronts develop more or less along isentropic surfaces, an isentropic vertical coordinate model permits the use of a relatively coarse horizontal resolution even within frontal zones, compared to models based on other vertical coordinates. Isentropic coordinates have disadvantages as well. Here we are particularly concerned with the following two features: 1) Small mass between two coordinate surfaces in a frontal layer, which is an advantage as described above, may cause computational difficulties. 2) The lack of explicit prediction of the (potential) temperature near the lower boundary may raise a problem in simulating surface frontogenesis. In "this paper we assess the impact of these advantages and disadvantages using a 0-coordinate model developed at UCLA (Hsu and Arakawa, 1990; Arakawa and Hsu, 1990) by evaluating the model performance in simulations of frontogenesis. This topic is selected in view of the difficulties shown by models with the a-coordinate in producing realistically strong tropopause folding. Section 2 of this paper describes the model, including the approach followed to overcome the computational difficulties described in 1). Section 3 presents the results of an application of the model to a simulation of frontogenesis. Section 4 presents a summary and further discussions on the use of isentropic coordinate models, with remarks on our approach in constructing a hybrid a-o coordinate model. 2. Description of the Model The model developed by Hsu and Arakawa (1990) is based on the following governing equations: <) + Vo.(mv) + ; (too) = 0, (2.1) o Ov _ qk x v* - O ~_v _ VoM, (2.2) & ~0 and ~0 - cv. (2.3) Here m is the mass per unit increment of 0 given by m - (2.4) go30' g is the gravitat!onal acceleration, v is the horizontal velocity., 0 is the material time derivative of 0 given by 0 = Q/%(p/po) ~, Q is the diabatic heating per unit mass, Po is a standard pressure, q is the potential vorticity given by f +k'vo q ~, (2.5) m f is the Coriolis parameter, k is the vertical unit vector, v* is the horizontal mass flux given by v* = my, (2.6) and M is the Montgomery potential given by M = cl, T+ 9z. (2.7) At the top of the atmosphere, it is assumed that mo = 0. (2.8) As in Bleck (1978), the coordinate surfaces intersecting the lower boundary are extended along the boundary by introducing "massless layers" (or, in practice, layers with small m). This approach formally eliminates the difficulties in treating the intersections. Other difficulties, however, appear near the edges of the massless layers. At these edges the horizontal gradient of m along a coordinate surface generally has discontinuities, giving a computational difficulty in discrete

3 An Isentropic Vertical Coordinate Model 33 models. This difficulty, however, is not unique to the "massless layer" approach of handling the lower boundary. Layers with small mass can form spontaneously within the model atmosphere as fronts are generated. In fact, as we pointed out earlier, having this possibility is one of the most important advantages of an isentropic vertical coordinate model. In order to maintain this advantage, we need a rather sophisticated discretization scheme for the continuity equation. In the rest of this section, we briefly review the vertical and horizontal discretizations used in the model, including the approach used to overcome the difficulties associated with "massless layers". The distribution of model variables in the vertical is shown in Fig. 1. Prognostic variables are v for each model layer identified by an integer index and p at each interface of the layers identified by. a half-integer index. The vertical mass flux mo is calculated diagnostically at the interface between layers using the thermodynamic equation for given heating. Recall that with the 0-coordinate, it is the distribution of p in the 0-space, rather than 0 in the p-space, that determines the thermodynamical structure of the model atmosphere. With the vertical grid shown in Fig. 1, p is predicted where the vertical mass flux is calculated. In this sense, the grid corresponds to that of Charney-Phillips for the p-coordinate, in which 0 is predicted where the vertical mass flux is calculated (Charney and Phillips, 1953). In contrast, the Lorenz grid, which is widely used for the primitive equation models, predicts 0 at levels that carry the horizontal velocity (Lorenz, 1960). As shown by Arakawa and Moorthi (1988), a drawback of this grid is an extra degree of freedom in the potential temperature and quasi-geostrophic potential vorticity, which causes spurious baroclinic instability for short waves. The. grid we are using for the 0-coordinate model (see Fig. 1) is free from such spurious instability. As described in Hsu and Arakawa (1990), the vertical discretization used in the model satisfies the four integral constraints introduced by Arakawa and Lamb (1977). These constraints are (I) the vertically integrated pressure gradient force does not produce a circulation along a closed contour of topography; II) the energy conversion terms have the same form with opposite sign in the thermodynamic and kinetic energy equations so that the total energy is conserved under adiabatic and frictionless processes; III) the total potential enthalpy is conserved under adiabatic processes; IV) a function of the potential temperature (other than the potential temperature itself) integrated over the entire mass is conserved under adiabatic processes. With an isentropic coordinate, satisfying constraints (III) and (IV) is more or less automatic. By satisfying constraints (II), (III) and (IV), we can rigorously define the available potential energy even in the discrete system (see Hsu and Arakawa, 1990). In the horizontal, the model is based on the "C-grid" (Arakawa and Lamb, 1977). Since the model has massless layers, the scheme for the continuity equation must be positive definite to avoid generation of negative mass as a result of discretization errors. It is equally important for the scheme to control dispersion errors that can be generated near the edges of the massless layers. The scheme used in the model is based on the advection scheme designed by Takacs (1985), which has third-order accuracy when the advecting current is uniform and time is continuous. Hsu and Arakawa (1990) modified this scheme in such a way that the mass flux out of a grid point automatically approaches zero as the mass at the grid point approaches zero. This is done while maintaining the third-order accuracy. The resulting scheme is a positive definite scheme with relatively small dispersion error. The scheme is 3/2 l-1/2,~... U V... g+1/2,- ~ P L-3/2 L - I ~ L-1/2L L+1/2 P Fig. 1. Vertical discretization used in the 0-coordinate model

4 I 34 A. Arakawa et al. split into the x and y directions to satisfy a nonlinear stability condition. To maintain the advantage of an isentropic vertical coordinate in view of potential vorticity conservation, the model uses a unique finitedifference scheme for the momentum equation (Arakawa and Hsu, 1990). This scheme is based on the vector invariant form of the momentum equation given by (2.2) and required to be equivalent to a well-defined advection scheme for the potential vorticity equation. The scheme is a member of the family of the discrete shallow water equations derived by Arakawa and Lamb (1981) that conserve potential enstrophy and energy when the mass flux is nondivergent. A modification is included to allow dissipation of potential enstrophy while conserving energy. This modification is done because, for large-scale atmospheric motions, it is primarily the potential enstrophy that cascades to smaller scales (e.g., Sadourny, 1984; Sadourny and Basdevant, 1985). The rate of potential enstrophy dissipation is controlled by a time scale, which can be adjusted for the specific dynamical process targeted. When the mass of a layer becomes small, the potential vorticity q given by (2.5) can become infinitely large, unless f + k" I7o x v is also small; but in the continuous system the product qv* is finite (see (2.5) and (2.6)). In discrete momentum equations, however, products of the potential vorticity and the component of mass flux at different grid points may appear. In this way, the infinite potential vorticity in a massless layer can produce a spurious effect on the time change of the velocity at neighboring grid points with finite mass. Recall, however, that v* is formulated in such a way that it approaches zero as the mass at the upstream grid point approaches zero. It is then possible to minimize the spurious effect by choosing a special member of the family of schemes. With that scheme, the kinetic energy change given by v*.6v/6t is guaranteed to be finite even when the values of q at neighboring grid points are not. The reader is referred to Arakawa and Hsu (1990) for more details. Simple physical processes such as surface friction and Newtonian-type cooling are included in the model. The model also includes a scheme for vertical momentum diffusion, but the scheme is designed to become effective only when a model layer becomes essentially massless. 3. A Simulation of Frontogenesis in the Upper and Middle Troposphere The impact of the model design discussed in section 2 to overcome difficulties associated with the use of isentropic coordinates can be best assessed by simulating frontogenesis and analyzing the associated dynamical processes. In this section, we present the results of a simulation that is representative of the model's performance in view of upper and middle level frontogenesis. The model domain used for this simulation is a 5000 km long and 9000 km wide periodic channel on a/~-plane centered at 45 ~ N. The bottom of the domain is at 0 = 220 K, while the top is formally at 0 = oc. The horizontal resolution is 150 km x 150 km, and there are 31 layers in the vertical. The initial conditions for the simulation consist of a single zonal jet centered at 45 ~ N (see Fig. 2) and a small-amplitude perturbation with zonal wavenumber 1. A sequence of maps of Ertel's potential vorticity (PV) and horizontal velocity for the 24th layer of the model is shown in Fig. 3. This layer is bounded by the 276 K and 284 K isentropic surfaces. In the northern part of the domain, the air in this layer lo GO, -~, ~ -, 3_.~ :220 K~ \ \,, M~\ \ / z'15 ~ K- i 372 K- sm-~ , K- I I I I I I I I I I L Y ( lso KM) Fig. 2. Initial condition for the simulation. Solid lines are isentropic surfaces (model coordinate surfaces) and dashed lines are isotachs. Units are K and ms- 1

5 An lsentropic Vertical Coordinate Model 35 II >. J! I2 o~.o x i " / I Z t{ll 8 I I I I 'I I I I i I I I I I I I I I I w --I, t IX \ \ " )- n o d D O C,t O I I i i L I " "lip "i 'i I I I I I I I I [ 1 I I/i I i t t I II n~ w b- J w rr -A ' ' 'to//, i //"v. -,,- - I~i~I 9,-0 O [] d ~.:-,... ::;/; II i... ""' '..\"G" ' llli! :"... \\! t,. 9 0 E 9 II I i I I I I I I 1 I' I I I I I' I sl ~ I I I I I I I I.o ' ~ \ \ \ \!! I ' ' II!i [] [] o [] ~ ~ G G ~ % ~ ~ ~ - I I I I I I I I I I I I' I d - - x~ -- - fi s F I I I "I" I' 'I' I I I I I I I I I b~ It. ~ ~7 r.r-~,o 9 I

6 36 A. Arakawa et al. has high PV values characteristic of the lower stratosphere. The zone of large PV gradients in the southern part of the domain represents the belt between the intersections of the upper and lower boundaries of this model layer with the earth's surface. If the spacing of the coordinate surfaces becomes infinitesimally small, this zone becomes the line separating the "massless" region and the finite mass region, at which PV values are discontinuous. As the disturbance develops, a tongue of high PV air of the lower stratosphere is advected southward and eastward. There is also a tongue of air extending north from the intersection of the layer with the surface. Lower PV air is also advected by the cyclonic flow. At day 7, an isolated pool of air with high PV appears. At day 10, this pool seems to reattach itself to the east flank of the high PV tongue. These developments correspond well with the formation of a cutoff cyclone analyzed by Hoskins et al. (1985) using observed data. We show the three-dimensional structure of the flow for days 4, 7 and 10 in Figs. 4-6, respectively. These figures combine contour plots of geopotential height and potential temperature at two levels in the lower and middle troposphere (900 and 500 mb), and ageostrophic circulation and potential vorticity in north-south vertical cross-sections. The cross-sections are taken upstream and downstream of the 500 mb trough. The plots at 900 mb show the intensification of the disturbance and the formation of cold and warm fronts. The plots at 500 mb show the disturbance intensifying throughout the period and a sharp frontal zone extending from upstream to downstream of the trough towards the end of the period. The largest magnitude of the horizontal temperature gradient in this frontal zone is about 13K/100km, which is comparable to observational estimates of upper level fronts by Bosart (1970) and Shapiro (1980). The vertical cross-sections reveal three patterns of ageostrophic vertical circulation: 1) upstream of the trough there is a thermally direct circulation with the sinking branch roughly under the jet and weaker than the rising branch, 2) at the trough there is a thermally indirect circulation with the sinking branch roughly under the jet and stronger than the rising branch, and 3) downstream of the trough there are thermally direct and indirect Day~4 ~:~:;i!!... _ ~ _50omb ~. ~ 00~ U /, ',, ',," T T...dL :: D Fig. 4, Contour plots of geopotential height (thick lines) and potential temperature (thin lines) interpolated to two levels in the lower and middle troposphere (500 and 900 mb), and ageostrophic velocities (arrows) and PV (thick lines) in north-south vertical cross-sections for day 4. The cross-sections are taken upstream (U), bottom (B), and downstream (D) of the 500 mb trough. Contour intervals for geopotential height are 50 m at 900 mb and 150 m at 500 rob. Contour interval for potential temperature and PV are 2 K and 2 PVU, respectively

7 p An Isentropic Vertical Coordinate Model 37 5Dt3mb D a ~ / ~ O g ~ u - - -,'.", 11 '' -, ' ' / /.600 ~ob U t,. e r.. 9 " ' 7 7 t ', 9 z ',. " i t t r [3 Fig. 5. As in Fig. 5, except for day 7 D 588mb Day ~ ~ /~ g0~ ~ z r ~b D U Fig. 6. As in Fig. 5, except for day 10 circulations to the north and south of the jet, respectively, with an intense rising branch under the jet. Viewed locally, the vertical ageostrophic circulation in the upper and middle troposphere is thermally indirect in the region of sinking motion upstream of the trough. This is the region of upper level frontogenesis where a tongue of high PV air from the stratosphere is clearly penetrating into the middle troposphere. The intrusion of high PV air reaches down to 700 mb at day 10. Another tongue of high PV air extending from near the surface into the lower troposphere is apparent at day 10. Elongated tongues of high potential vorticity

8 38 A. Arakawa et al. Day 4 I=31 Day 8 I=13 ts2( ~..~ J'~ ~00-- \ \ \.. I ', i ~ ~ \ \ \\\\\i k ~II~ \il \\ \ X\ ' i X '~ -- x \ ' ~\ \ "., \ \ \ \ \ \ \ \ ~ 80D \X ~\ \\ \, \\ X\ \X ~\ 1\ ~ ~ -- :~a< z?,~ \\ \ XX \\ \k \\ \\ \ \\ ~00 4 -\,, Xj.~.,,,, ' \, I t too T I I I I I I... I l I I I ~ Day 9 I=26 Day 10 I=6 X% 3~2< ~SS~ ~ag \ x\ k\ \\ \.~ \~\\\\\~\" --\ ' x\xx\ \\\\ \\\ \ \ ~'\ \~ ~ i[ 32~': ~.sc X X k X \\X\x\\\\\\ \ \ \ ~. X\\ k \ "-, ~kk \ \\\ k \ N \X X X \ ~-- I \\\ ~'~ \\X. ~ ~\ X\Xx\ XX \'~ ~\ X ~-,. \ \ \ I \\k - 3Oa~ \ x \.~ \ "-\,~\ \ \ SO "15 ~IO i LI5 LIO Y [ 150KMI Y (150KM~ Fig. 7. Location of an air parcel (marked with the solid dot) on north-south cross-sections upstream of the trough. Solid lines are PV (PVU), and dashed lines are 0 surfaces (K) air from the stratosphere invading the troposphere characterize tropopause folding events (Danielsen, 1968; Shapiro, 1980; Danielsen et al., 1987). To gain insight into the origin of air parcels in the tropopause folds, we perform a trajectory analysis for the simulation. The locations of an air parcel on a north-south vertical plane from days 4 through 10 are shown in Fig. 7. At day 4, the parcel is at about 350mb far upstream of the trough; at day 8, the parcel is at about 475mb in the frontal zone moving slightly faster than the disturbance; and at day 10, the parcel has

9 An Isentropic Vertical Coordinate Model ~Day~ 4 Confluence 500 mlo Day 7 ~ 500 mb Day 10 Confluence 500 mb R9 R3 "~0 3-/ ~, i0 -- ~Tilting 500rob[ Day7 ~ ~ ~Dayl0 Tilting ~--500mb[! I I I I I l I I I I i i I I I i I I I I I ' I l i I I i i I I 5s58 Day452 R9 ~ ~o-- ~-~ ~ _ " '~ ~ 28 ~ 25 ~. ~ 22 _ i--i i i i i [ I i i i =1 tl ls "1 l = ls t-I X (tso I(M) X (150 KM] X (150 KM) Fig. 8. Contour plots of confluence and tilting as defined in the text (thick lines), and 0 (thin lines) interpolated to 500 mb from days 4, 7 and 10. Units for confluence (tilting) and 0 are K (105m)- i day- 1, and 3 K, respectively descended to about 900mb. During its journey, the parcel loses PV slightly due to finite-difference errors and diabatic effects through the Newtonian-type cooling used in this version of the model. It is clear that the model is able to produce very intense tropopause folding. Since this feature is more realistic than that obtained using a model with the o--coordinate (Hines and Mechoso, 1991), we proceed to analyze the dynamics of flow to gain insight into the differences between frontogenetical mechanisms simulated by the two models. To quantify the contribution of processes involved in upper and middle troposphere frontogenesis, we compute the terms of the frontogenetical function (Miller, 1948). With the 0-coordinate, as in Buzzi et al. (1977), the frontogenetical function

10 40 A. Arakawa et al. T= ~ DRY T= R DRY 2, kd ne L0 03 td cr 0-1 R Y C150KH) a Y (150KH) b Fig. 9. (a) Ageostrophic circulation transversal to the flow, and (b) co (thin lines) for the cross-section marked in Fig. 8 (day 4). Also shown are isotachs of the flow perpendicular to the cross-section (thick lines) and isentropic surfaces (dashed lines). Units for co and for horizontal wind and mb day- 1 and ms- 1, respectively can be expressed as Di~7ljOi=OOilTopl-1 { 1~(6~P~ 2 (ap]21 Lc~x c~y] 2L\&/ \~/ / [ou,.} papaw ap a<o ] (3.1) The first three terms on the right hand side of (3.1) involve horizontal processes, which we will refer to as "confluence", and the last term involves vertical processes, which we will refer to as "tilting". These terms can be evaluated from the model output for each model layer. Figure 8 shows the results interpolated to 500 mb for days 4, 7 and 10. We can see that the dominant frontogenetical process varies in space: 1) far upstream of the trough confluence is t'rontogenetical while tilting is frontolytical, 2) downstream of this region but upstream of the trough both confluence and tilting are ffontogenetical, and 3) near and at the trough tilting is frontogenetical and confluence is frontolytical. The ageostrophic circulation in the vertical cross-section marked in Fig. 8 is shown in Fig. 9. The circulation in this cross-section is mostly confined to the northern half of the domain, with the maximum subsidence directly under the jet. An insight into the driving mechanisms of the vertical motion can be obtained through a generalized m-equation. With the p-coordinate, the equation may be written in the form, I Vy/ - K(p) "t, + ay/ II (3.2) where L is a linear (and usually elliptic) differential operator and ~(P)~ K(p)= p~?p Poo " (3.3)

11 An Isentropic Vertical Coordinate Model z49 ~48 J43 ~ z Day 4 ( I ) I, ~ v -3 IZ 3~ ~n 3z 28 Day 4 ( I I ) l ~S zl6 z l O - ~ -3 '-13 --Z3 3 ~ > s I0 7 z4 l / I I I I I I I I I I z ' z-I >_ LI 1 I I I I I [ I I I I ls q X (150 KM) X (150 KM) Day 4 (I) Geostrophic Day 4 ( ] I ) Geostrophic ~9 zt8 '43 ' 'q ~ 2~s-~ 3,4 ~ 3t ~ I0 I '4 l I I I I I I I I I I z4 7 I z ~ ] I I I I I I I I I N 7 I0 13 t X (150 KM) X [150 KM) Fig. 10. Terms (I) and (II) in (3.2) calculated at 500rob for day 4 (upper panels). The lower panels show the corresponding values computed using the geostrophic component of the wind. Units are 10-is Pascal-is-3. Shaded area corresponds to horizontal potential temperature gradient more than 2.7 K (105m) - 1 day-1

12 42 A. Arakawa et al 00 ~E ~OD R =,~ o~y (I) ~ 30 iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii!!!_!!l! aaz~ afar,....czz r=. oav (II) O-- n~ a:s~ ~ ~oo~ 03 (.o L~ rr god i xxx xx x~x \ \ xx \x\ Xxx \x ~K ~2~ D-- ~ looo-- 90D-- ooo..... Z'_ _-_-:... :, zsok It00 I I I I I I I q ltoo I I I I I I I q Y (150KM) Y [15DKM) T= "fl DRY (I) Geostrophic iiiiiiiiiiiiiiiiiiiiiiiiiii iiiiiiii!i!! I00-- as2~: ~etk ~72K oo as6k 200--_... 3,~olc q00-- -I1!I T= q o~y (II) Geostrophic... II!I iiiiiiiiiiiiiiiiii!ii!!!!!iiiiiiii!ii_..., as6k mo~ 600-2~Z~ "~x x x I ze~k ~TsK ~GBK IgDl ~i z~ok ii00 I I I I I I I q II00 i 1 I I I L i 7 I IS Y (150KM) Y (150KM) Fig. 11. Terms (I) and (II) in (3.2) at day 4 for the cross-section marked in Fig. 10 (upper panels). The lower panels show the corresponding values computed using the geostrophic component of the wind. Units are Pascal- is-3 Here we have included the advection by ageostrophic wind in the terms on the right hand side of (3.2). The values of these terms at day 4 and the corresponding values but with the geostrophic component only are shown for 500 mb in Fig. 10. To produce these values we used the simulated fields interpolated to pressure levels at 30 mb intervals. Figure 10 shows that along the frontal region the term (I) has larger magnitudes than the term (II). It also shows that the contribution of the geostrophic component to the term (I) is relatively

13 An Isentropic Vertical Coordinate Model 43 small. This is not the case at later times when the ageostrophic contribution is weaker. Note that these features do not apply near the trough, indicating a different local dynamics. The same fields for a vertical cross-section marked in Fig. 10 are shown in Fig. 11. Here we see that the largest magnitudes of both (I) and (II) appear near the jet. In both Figs. 10 and 11, the temperature advection at day 4 is almost entirely by the geostrophic component of the wind. Again with the p-coordinate, the vorticity equation may be written as D~ --= -- S + T, (3.4) Dt where S= -(f + f) + =(f + _(~o)& &oou) (3.6) T= \ ~3 O y -@p " We refer to the terms S and T as "twisting" and "stretching", respectively. Analysis of these terms (not shown) reveals that the vorticity gradient across the jet intensifies due to twisting at middle levels, and primarily due to stretching and partially due to twisting at upper levels. The following picture of the flow upstream of the trough emerges from these results. In the upper troposphere, sinking motion under the jet stretches the cyclonic vortex tubes on the cold side of the jet and the anticyclonic vortex tubes on the warm side of the jet in the upper troposphere. Consistently, the magnitude of cyclonic and anticyclonic vorticities increases. In the middle troposphere, the tail of the jet extends downward, resulting in cyclonic and anticyclonic vorticity increases on the cold and warm side, respectively. The anticyclonic vorticity is mainly advected by the cross-jet ageostrophic wind, and its maximum is located right above the sinking maximum. According to (3.2), cross-jet anticyclonic vorticity advection increasing in height is associated with sinking motion under the jet. These results suggest that the following positive feedback mechanism between frontal and synoptic scales, of the type proposed by Mudrick (1974), may be at work in this simulation: 1) large-scale deformations drive a thermally direct ageostrophic vertical circulation transverse to the flow, 2) anticyclonic vorticity is advected across the jet primarily by the ageostrophic flow, 3) sinking motion is enhanced under the jet, 4) cross-jet vorticity gradients increase due to enhanced twisting and stretching associated with sinking under the jet, and 5) the anticylonic vorticity advection increases with enhanced vorticity gradient, therefore, sinking motion under the jet increases further. The work of Hines and Mechoso (1991), on the other hand, does not find clear evidence that this mechanism is present in simulations with the a-coordinate, which produce substantially weaker ageostrophic contributions to frontogenesis and, consistently, weaker upper-level frontogenesis starting from similar initial conditions even using a higher horizontal resolution. 4. Summary and Conclusions Following a brief discussion of the advantages and disadvantages of isentropic vertical coordinates, a 0-coordinate model developed at UCLA is reviewed. The model's performance is demonstrated by simulating the evolution of a middlelatitude baroclinic disturbance and associated frontogenesis. Results from an analysis of the extent to which different dynamical processes contribute to frontogenesis are presented. With an isentropic vertical coordinate, the coordinate surfaces are material surfaces under adiabatic processes and the mass between two coordinate surfaces may become arbitrarily small. Consequently, the model can automatically create fine vertical resolution in regions of vertical confluence. Clearly this feature is advantageous for simulating frontogenesis and associated dynamical processes. In addition, since fronts develop more or less along isentropic surfaces, an isentropic vertical coordinate model permits the use of a relatively coarse horizontal resolution even within frontal zones. To maintain the advantage of isentropic vertical coordinates described above, the model must be able to handle layers with small mass, which we referred to as "massless" layer. Such layers, however, can easily produce computational difficulties as reported in Zhu et al. (1992). To overcome these difficulties without sacrificing the advantages of isentropic vertical coordinates, the model developed at UCLA uses a rather sophisticated discretization scheme for the continuity

14 44 A. Arakawa et al. equation. In the scheme, the mass flux out of a grid point automatically approaches zero as the mass at that grid point approaches zero. In addition, the scheme is relatively free from computational dispersion of errors, which may be generated near the edges of the "massless" layers. The model also uses a unique finite-difference scheme for the momentum equation. The scheme belongs to a family of schemes with which the potential enstrophy is either conserved or dissipated, when the mass flux is nondivergent, while the total energy is conserved. A special member of the family is chosen for use in the model to minimize the spurious effect of the "massless" layers. Because the model is designed to handle "massless" layers within the model atmosphere, artificial "massless" layers are introduced to simplify the lower boundary condition by extending the intersecting coordinate surfaces along the boundary. The model's performance is demonstrated by simulating the evolution of a middle-latitude baroclinic disturbance. During this evolution, sharp frontal zones are generated in the upper and middle troposphere and a realistic tropopause folding develops. As the folding progresses, air parcels penetrate from the lower stratosphere into the lower troposphere. The model is also able to capture many other observed features of frontogenesis. We have presented evidence that a positive feedback between frontal and synoptic scales is at work in the simulation. This positive feedback acts to intensify the horizontal temperature gradient and to sharpen the jet as a result of the enhanced subsidence under the jet. Previous work with a-coordinate models (Hines and Mechoso, 1991) provided no clear evidence of this mechanism since simulated fronts and associated ageostrophic circulations were too weak. While the scheme for handling layers with small mass is successfully tested in this simulation, the difficulty in defining temperatures at the lower boundary still remains. This is due to the fact that the number of degrees of freedom in the distribution of surface temperatures is determined by the number of coordinate surfaces intersecting the surface. In an isentropic coordinate model, therefore, the effective horizontal resolution of surface temperatures can be increased only through ~-0 Fig. 12. Schematic of the heights of model coordinate surfaces for the a-0 hybrid coordinate increasing the "vertical" resolution. The version of our model used for the simulation described above does not seem to have a sufficiently high "vertical" resolution to realistically simulate surface fronts. This situation is also responsible for difficulties in obtaining sufficiently smooth distributions of the Montgomery potential along coordinate surfaces above. This lack of degrees of freedom also gives a difficulty in computing the potential vorticity (PV) near the lower boundary. Where a layer meets with the lower boundary, its mass becomes small, giving artificially high PV values, even before the layer becomes a massless layer. In view of these difficulties with an isentropic coordinate model and its incapability of handling unstably stratified layers, it is highly desirable to construct a hybrid a-0 coordinate model. The model should maintain the advantages of isentropic vertical coordinates away from the lower boundary, its capability of having layers with small mass in particular, and a smooth transition between the two coordinate systems as shown in Fig. 12. Constructing a model that satisfies these two requirements is a difficult task and, consequently, compromises in one way or another had to be made in the existing hybrid coordinate models (e.g., Uccellini et al., 1979; Zhu et al., 1992). It seems possible, however, to develop such a model and we are currently making progress toward that objective. Acknowledgements This research was supported by NSF under Grants ATM , and and by NASA under Grant NAG The simulations were performed at the computer facilities of NCAR and UCLA-OAC.

15 An Isentropic Vertical Coordinate Mode[ 45 References Arakawa, A., Lamb, V. R., 1977: Computational design of the basic dynamical processes of the UCLA general circulation model. General Circulation Models of the Atmosphere (Methods in Computational Physics, 17), edited by J. Chang, Academic Press, Arakawa, A., Lamb, V. R., 1981: A potential enstrophy and energy conserving scheme for the shallow water equations. Mon. Wea. Rev., 109, Arakawa, A., Moorthi, S., 1988: Baroclinic instability in vertically discrete systems. J. Atmos. Sci., 45, Arakawa, A., Hsu, Y.-J.G., 1990: Energy conserving and potential-enstrophy dissipating schemes for the shallow water equations. Mon. Wea. Rev., 118, 196(~1969. Bleck, R., 1978: On the use of hybrid vertical coordinates in numerical weather prediction models. Mon. Wea. Rev., 106, Bosart, L., 1970: Mid-tropospheric frontogenesis. Quart. J. Roy. Meteor. Soc., 96, Buzzi, A., Nanni, T., Tagliazucca, M., 1977: Mid-tropospheric frontal zones: Numerical experiments with an isentropic coordinate primitive equation model. Arch. Meteor. Geophys. Biokl., A26, Charney, J. G, Phillips, N. A., 1953: Numerical integration of the quasi-geostrophic equations for barotropic and simple baroclinic flows. J. Meteor., 10, Danielsen, E. F., 1968: Stratospheric-tropospheric exchange based on radioactivity, ozone and potential vorticity. J. Atmos. Sci., 25, Danielsen, E.F., Hipskind, R.S., Gaines, S.E., Sachse, G. W., Gregory, C. L., Hill, G. F., 1987: Three-dimensional analysis of potential vorticity associated with tropopause folds and observed variations of ozone and carbon monoxide. J. Geophys. Res., 92, Hines, K. M., Mechoso, C. R., 1991: Frontogenesis processes in the middle and upper troposphere. Mort. Wea. Rev., 119, Hollingsworth, A., Kalberg, P., Rennet, V., Burridge, D. M., 1983: An internal symmetric computational instability. Quart. J. Roy. Meteor. Soc., 109, 4t Hoskins, B. J., Mclntyre, M. E., Robertson, A. W., 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, Hsu, Y.-J. G., Arakawa, A., 1990: Numerical modeling of the atmosphere with an isentropic vertical coordinate. Mon. Wea. Rev., 118, Lorenz, E. N., 1960: Energy and numerical weather prediction. Tellus, 12, Miller, J.E., 1948: On the concept of frontogenesis. J. Meteor., 5, Mudrick, S. E., 1974: A numerical study of frontogenesis. J. Atmos. Sci., 31, Phillips, N.A., 1957: A coordinate system having some special advantages for numerical forcasting. J. Meteor., 14, Sadourny, R., 1984: Entropy coordinate, quasi-geostrophic turbulence and the design of lateral diffusion in general circulation models. Numerical Methods for Weather Prediction, 1, ECMWF Seminar 1983, Sadourny, R., Basdevant, C., 1985: Parameterization of subgrid scale barotropic and baroclinic eddies in quasigeostrophic models: Anticipated potential vorticity method. d. Atmos. Sci., 42, Shapiro, M. A., 1980: Turbulent mixing within tropopause folds as a mechanism for the exchange of chemical constituents between the stratosphere and troposphere. J. Atmos. Sci., 37, Takacs, L. L., 1985: A two-step scheme for the advection equation with minimized dissipation and dispersion errors. Mon. Wea. Rev., 113, Uccellini, L. W., Johnson, D. R., Schlesinger, R. E., 1979: An isentropic and sigma coordinate hybrid numerical model: Model development and some initial tests. J. Atmos. Sci., 36, Zhu, Z.-X., Thuburn, J., Hoskins, B. J., Haynes, P. H., 1992: A vertical finite-difference scheme based on a hybrid o-o-p coordinate. Mon. Wea. Rev., 120, Authors' address: A. Arakawa, C. R. Mechoso, and C. S. Konor, Department of Atmospheric Sciences, University of California, Los Angeles, CA 90024, U.S.A.