Semigeostrophic Invertibility Experiments with TAMEX Data

The balanced atmospheric response to a squall line as a moving heat source is computed. Specifically, we consider the permanent m o d ifi cat io n s to the large-scale balanced flow (geostrophy) rather than the transient gra vi t y · inertia wave motion. The potential vorticity anomaly, horizontal and vertical wind shears in terms of a dimensionless parameter a and swept-through dis-­ tance of the squall line are presented. The physical meanings of the parameter a are discussed. Observational cases from mid-latitude, subtropics and trop­ ics are g iv en in terms of the squall line speed and a. This cl ass i fi ca t ion is based on the balanced atmospheric response that the squall line induced. Namely, we emphasize the concept of potential vorticity and balanced dynamics to c l as s if y the squall lines. The a-c classification will provide a measure of the squall line force in dynamic models. The computed balanced solutions give a reference base to monitor the geostrophic adjustment processes. Moreover, the a-c observations enable us to interpret the model result related to dif­ ferent observational squall lines. The invertibility computations from squall lines during Taiwan Area Mesoscale EXperiment (TAMEX) are shown. The implication of the results and future research are discussed.·


INTRODUCTION
The semigeostrophic system is a filtered set of equations providing remarkably accurate descriptions of'many phenomena which lie beyond description by the quasi-geostrophic equa tions. Traditionally. the phenomena studied include surface and upper tropospheric fronts. jets and occluding· baroclinic waves. The first exploitation of the semigeostrophic system was the two-dimensional frontogenesis studies of Hoskins (1971) and Hoskins and Brether ton (1972). Imposing a horizontal deformation field to force the frontogenesis as in the study made by Hoskins (1972). the semigeostrophic system is able to generate infinite temperature gradient at the surface in less than 12 hours. This is owing to the feedback of geostrophic rel ative vorticity (9 and the ageostrophic secondary circulation in the semigeostrophic system, which is missing in the quasigeostrophic system. By combining the geostrophic momentum 180 TAO, Vol.4, No.2, June 1993 approximation (Eliass en, 1948) and the geostrophic coordinate transformation (Yudin, 1955). a comprehensive semi-geostrophic theory in three dimensions was worked out by Hoskins (1975) and Hoskins and Draghici (1977). Later frontogenesis studies by Hoskins (1976), Hoskins and West (1979) and Hoskins and Reckley (1981) involved the forcing from a de veloping baroclinic wave with Eady waves and uniform potential vorticity flows. A study of the energy cascade predicted by the semigeostrophic theory of frontogenesis can be found in Andrews and Hoskins (1978).
In these mid-latitude baroclinic wave and frontogenesis studies with semigeostrophic system, the potential vorticity field is of central attention. The potential vorticity previously has been used as a passive tracer, for example, to trace the entrainment of stratospheric air into the troposphere (Danielsen, 1968). Thus, only the conservation property of Rossby-Ertel potential vorticity has been emphasized. The more recent view has added dynamics to it the concepts of balance dynamics, invertibility and transformed horizontal coordinates. It provides the simplest way to diagnose or predict balanced dynamics through the use of the Rossby-Ertel potential vorticity on isentropic or entropy surfaces. The acceptance of the modem view is in large part due to the discussion of Hoskins et al. (1985), who point out that "PV thinking" can lead to increased insight into such phenomena as the formation of cut off cyclones and blocking anticyclones, Rossby wave propagation, and baroclinic/barotropic instability.
The semigeostrophic equations in terms of "potential vorticity modeling" involve two main mathematical principles, the conservation of potential pseudo-density (the inverse of potential vorticity) and the principle of invertibility. The conservation of pseudo-density serves as the fundamental prediction equation of the model. The invertibility serves as the diagnostics to obtain the balanced wind and mass fields from the predicted potential pseudo density. Along the path of "potential vorticity modeling", Schubert et aL (1989) studied the balanced atmospheric response to the squall line in mid-latitude by the semigeostrophic model.
In this paper, we study the atmospheric equilibrium (geostrophy) response to squall lines during Ta iwan Area Mesoscale Experiment (Kuo and Chen, 1990) with a model siniilar to that of Schubert et aL (1989). We present the semigeostrophic model with the entropy ( cp In( 9 / 80 )) vertical coordinate. We discuss the functional dependence of semigeostrophic solutions with an ideal moving heat source in two dimensions. We reinterprete the a param� eter introduced by Schubert et al. (1989) and stress the importance of the a parameter and the squall line speed c to classify the squall line. Observational squall lines are presented in terms of a and the squall line speed c.
In section 2 we review the three-dimensional and two-dimensional invertibility princi ples in semigeostrophic theory with the entropy vertical coordinate. The potential vorticity anomaly and wind shears as a functions of a and swept-through region by the squall line in semigeostrophic theory are given in section 3. Section 3 also contains a class ification of squall lines and the balanced response of atmosphere to the squall lines during Taiwan Area Mesoscale EXperiment (TAME){, Kuo and Chen, 1990). Section 4 gives the concluding remarks.

Potential Pseudo-density Equation
We begin with the /-plane system of equations with the geostrophic momentum ap proximation (Eliassen, 1948;Hoskins, 1975) and proceed with an analysis similar to that of Schubert et al. (1989). but rather than using potential temperature 8 as the vertical coordin-8.te.
(2.9 a) (2.9b) The equivalence of c2.sa) and c2.sb > ronows easily rrom the fact that a{/ ax + a11 / ay + fJ( / fJs = 0. The significance of (2.8b) is discussed by Haynes and Mcintyre (1987), who draw attention to the fact that for the primitive equation in the isentropic coordinate a flux form such as (2.8b) leads directly to the theorem that even with diabatic heating and frictional forces "there can be no net transport of Ros.shy-Ertel potential vorticity across any isentropic surfaces" and that "potential vorticity can neither be created nor destroyed within a layer bounded by two isentropic surfaces." From (2.8b) we conclude that the primitive equation result of Haynes and Mcintyre also holds when we make the geostrophic momentum approximation with entropy as vertical coordinate.
We can eliminate the isenttopic divergence between (2.5) and (2.8a) to obtain (2.10) is the potential pseudo-density.
To gain a deeper understanding of the physical meaning of the potential pseudo-density, consider a small cylindrical element of fluid spinning cyclonically relative to the earth (i.e., ( > /); this fluid is }?ounded on the bottom and top by two entropy surfaces. If the flow is adiabatic, we conclude from (2.10) that a * is conserved, so that the pseudo-density a must decrease as the absolute vorticity ( decreases, i.e. the pressure difference between the bottom and top isentropic surfaces must decrease as ( is reduced to f. One can imagine a rearrangement process in which mass moves outward in a divergent fashion without crossing both the bottom and top entropy surfaces, leading to a simultaneous reduction of a and (. Of course, if the fluid element were originally spinning anticyclonically relative to the earth (i.e., ( < /), the rearr angement would require mass to move inward while a and ( simultaneously increased. In either case we can interpret the potential pseudo-density a * as the pseudo-density the fluid element would acquire if ( were changed.to f 11Il4er a frictionless and adiabatic rearrangement process. Since ( can be expressed in terms of u 9 and v 9 and hence M through geostrophic balance (2.6), and since u can be expressed in tenns of T and hence M through hydrostatic balance (2.4), there exists a second-order partial differential equation relating M and a * .
This equation, along with its assoc iated boundary conditions, is usually referr ed to as the invertibility principle. Thus, we have (2.10) as a predictive equation for u * and an assoc iated invertibility principle from which we diagnose M from a known a*. As we shall see from the simplicity of (2.35), a * seems to be a more convenient variable than its inverse, the commonly used potential vorticity. In fact, Schubert et aL (1991) show that u * is a much better variable than the potential vorticity to dynamically define the tropopause in the tropics.
The D / Dt as is expressed in physical space by (2;7), (2.10) involves advection by the total wind, in which case the predictive equation for a * and the invertibility principle do not form a closed system. This is the point at which geostrophic coordinates ( X, Y, S, T) = ( x + v 9 / f, yu 9 / f, s, t) entered. Derivatives in ( x, y, s, t) space are related to derivatives in (X, Y, S, T) space by a ax a aY a a at = at ax + 8t aY + aT' . a ax a aY a ax = fJx ax t fJx aY ' a ax a BY a ay = ay ax + ay aY' a ax 8 BY a a as = as 8X t as aY + as· Inverting (2.13) and (2.14) to obtain are what might be called "vonex coordinates." To express the geostrophic and hydrostatic equations in (X, Y, S, T) space, it is con� venient to introduce the Bernoulli function Applying (2.15), (2.16) and (2.17) to the Bernoulli function, it can be shown that geostrophic and hydrostatic relations in (X, Y,.S, T) take the form (2.20) which are identical to the form taken in ( x, y, s, t).
The transformation relations (2.12 )-(2.15) imply that the total derivative given by (2. 7) can also be written as (2.21) where X = DX/ Dt and Y = DY/ Dt. In the frictionless aunosphere, it can be easily shown that (2.22) (2.23) A major advantage of the transformation from ( x, y, s, t) space to ( X, Y, S, T) space is the change from advection by ( u., v) to advection by ( X, Y), which reduces to advection by ( u 9, v 9) in the frictionless case.
With the help of (2.18), the potential pseudo-density equation (2.10) can be simplified as Du*

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(2.37) Expressing x and yin terms of u.9 and v9, and then using the equation of state p = p R T, geostrophic and hydrostatic balances of ( 2.20), we can write (2.37) as If the upper boundary is an entropy surface with entropy Stop and the temperature Tt o p is specified there (e.g. T is constant for an isothermal top), the upper boundary condition for where Stop = cpln(Btop/Bo). Likewise the lower boundary is the isentropic surface with potential temperature() = Bo and the geopotential is specified there (e.g. </Jb o t = 0 for a fiat surface), then M = cpT + ¢bot at S = 0. Written in terms of M *, this lower boundary condition becomes (2.38c) Together with appropriate lateral boundary conditions, equations (2 .35), (2.20) and (2.38) form a closed system for three dimensional invertibility. The computational scheme is as fol lows:knowing u *, solve (2.38) for M *; use (2.20) to compute ( u 9, v 9 ) ; use these geostrophic wind winds in (2.35) to predict a new a*. (2.39) Because the geostrophic flow does not advect u* in the two-dimensional case, we can now solve (2.39) for u* independent of the solution of (2.40).

Analytical Solution of Potential Pseudo-density
In our calculation the squall lines will be viewed as a moving heat source. Thus. a moving heat source with a Gaussian shape in X and a mid-b'opospheric maximum in two dimension and propagating at constant spee d c is considered. The source is Let us assume that X -+ ±oo, u* approaches the constant uo. Since u* sinZ is constant along each characteristic, we can write the solution of (2.39) as Although ( In summ ary, (3.6) along with the auxiliary relations (3.4) and (3.5) give the analytic solution of (2.39) with the specified heating (3.1). With the known a and c, we then have u*(X, S, T).
Similar derivation was presented in Schuben et al. (1989). · 3.2 Importance of a Parameter and Squall Line Speed c Schubert et al. (1989) interpreted a as the ratio of the convective overturning time to the squall line passage time of an air pareel entering the squall line. In general, a is small for heavily raining, wide, slowly moving squall lines while it is large for weakly raining, narr ow, fast moving squall lines. To interpret a differently, we write a = (Stopc)/(rrS0X0) which is the ratio of time rate of area (mass) in (X, S) space span by the moving squall line to the time rate of entropy change by the squall line heat source. Since the potential vorticity is the mass inside X and S, a may also be interpreted as the inverse of the potential vorticity production or reduction rate for the region that the squall line swept through. Thus, the above interpretations link potential vorticity anomaly to a. To gether with the invertibility equation (2.40). we have the squall line heat source as a physical analog of the static electric field. The squall line heat source· charges the air parcel that flows into it with potential vorticity .. The balanced flow induced by the potential vorticity anomaly corresponds .to the electrical fi eld intensity, the potential vorticity corresponds to charge density and the Bernoulli function M* corresponds to the electrical potential. The only difference lies in the linear elliptical (poisson) equation involved in a static electrical field calculation while the invertibility calculation in (2.40) is nonlinear. · To substantiate the idea ·that a is the inverse of the potential vorticity production or reduction rate for the region swept-through by the squall line, we have examined the extreme PV anomaly produced by the heat source (3.1) in the semigeostrophic solution. OUr calcula tion indicates that the extreme PV anomaly value produced by a moving heat source (squall line) is indeed a function of a only. It is not a function of squall line speed c or time. The maximum q/ f (normalized potential vorticity) as a function of a in the lower atmosphere and the minimum q / f as a function of a in the upper atmosphere are presented in Figure   1. In addition to a fixed a, we have found that the balanced response of the atmosphere  3.S depends only on the swept-through distance of the heat source. The permanent modifications to the large-scale balanced flow by .the moving heat source depend on the PV anomaly and the size of the anomaly region. For example. the response of c = 5m/ s at hour 10 is the same as the response of c = 1 Om/ s at hour S for a fixed a. Figures 2 and 3 give the maximum horizontal wind shear across the PV anomaly in the upper and lower atmosphere. Figure 4 presents the maximum vertical wind shear (holding X fixed) in the unit of ms-1 hP a-1 This classification is based on the balanced atmospheric response induced by the squall line.
Thus. the concept of potential vorticity and balanced dynamics are used. Table 1 gives the a values during the TAMEX and squall lines observed during the January-March period of 1988. Because all of the squall lines in TAMEX are over the Taiwan Strait, the heating rate Q0 (thus So) in these cases are evaluated based on the total precipitation and asswning that the vertical heating profile takes the form of a half sine wave as in (3.1). This half sine wave profile is an idealization for a typical cumulus apparent heat source. The total precipitation is calculated from the radar data (Chen and Chou, 1993). The X0 and care tabulated from the OMS satellite pict:ufe as well as from radar data. Observational squall lines from COPT81 (Roux, 1988). GATE (Houze, 1977), TAMEX (Chen and Chou. 1993) Fig. 4. The maximum vertical wind shear (holding X fixed) in the unit of ms-1 hPa-1 as a function of the swept-through distance of heat source (ordinate) and a (abscissa).
Oklahoma (Schubert et aL 1989 and are given in terms of squall line speed c and a in Figure 5. Also included are two cases of the squall lines observed near northern Taiwan on March 4, 1988. The much larger a cases in Table 1 are omitted for plotting clarity. The case studied by Schubert et al. (1989) is the typical Oklahoma squall line of which a = 2.3. The TAMEX cases are different as compared to mid-latitude Oklahoma cases in that a and c in TAMEX are smaller. This is due to the high humidity environment in the TAMEX cases as compared to the much drier mid-latitude Oklahoma cases. The very hwnid background probably leads to a stronger precipitation heating rate (thus a smaller a) and a lack of cold air behind the squall line (thus the slower moving speed). The March 4 squall lines have much larger a and smaller c than the TAMEX cases. These squall lines had less rain than the TAMEX cases. The COPT81 and GATE cases share similar a value as the TAMEX cases and possess a faster moving speed. According to Figures 2, 3 and 4, the TAMEX squall lines tend to produce larger balanced horizontal wind shears in the upper atmosphere than the COPT81 and the GATE squall lines. On the other hand, the COPT81 and the GATE squall lines have twice as much balanced vertical wind shears.  (Roux, 1988), GATE (Houze, 1977), TAMEX (Chen and Chou, 1993) and mid-latitude Oklahoma (Schu bert et aL 1989 and Hertenstein and Schubert, 1991) as a function of squall line speed c( ms-1) and parameter a.

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Figures 6-11 give results of non-dimensionalized q, the geostrophic flow and the dis turbed pressure at T = 16hr with the c and a of the TAMEX cases given in Table 1. The 8 is used in the plotting rather than the entropy S. It is interesting to note that the induced geostrophic balanced flow covers a region greater than the region swept through by the squall line. Behind the squall line there is a decrease in u (stabilization) in the lower troposphere and an increase in u (destabilization) in the upper troposphere. This is a reflection of the mutual adjustment between the wind and mass fields. This adjustment happens in such a way that q = ( / u is large in the lower part and small in the upper part. We have larger ( and smaller u for large q and the opposite for small q. Whether this upper destabilization will provide a favorable background for the upper stratiform precipitation is beyond the scope of our semigeostrophic model. Although the solutions have similar structure as compared to the mid-latitude case studied by Schubert et aL (1989), the TAMEX cases (with the exception of IOP2 case) apparently possess a much stronger wind shear. It is possible that these large horizontal wind shears cause a barotropic type of.instability.

1054.
1437.    There are several ways to extend the research presented here. A natural extension is the semi-geostrophic f3 plane themy as proposed by MagnusdoUir and Schuben (1990). In addition, the abnospheric response is very sensitive to the vertical heating profile of the squall line. The heating profile has yet to be determined from observations. Also, latent heat from the strati.form region has not been considered at all in this study. We need to incorporate this heating in our definition of a.
Raymond and Jiang (1991) discuss the possible relationship of potential vorticity anomaly and the long-lived storm. To look into this effect, we need not only good numerical advection schemes, but also fast three-dimensional invertibility solvers. To consider the effect of inter section of entropy surface with a lower boundary , the "massless" layer approach as proposed by Hoskins et al. (1985) needs also to be considered. These pose serious challenges for "IPV modeling". In addition to the numerical difficulty in developing the fast elliptical solvers for the invertibility principle, the advection of potential vorticity requires efficient and accurate schemes. Namely, one must cope with sharp gradient of potential vorticity and guarantee the positive definiteness of potential vorticity. In our paper. the advective pan of computation is bypassed by the analytical method. In a realistic physical situation that potential vorticity anomaly may be moved by large scale flow and trigger a new convection. A fast invertibility three-dimensional solver has been developed by Fulton and Taft (1991). We are currently testing the positive definiteness advection schemes proposed by Hsu and Arakawa (1990) and Smolarkiewicz (1983).
Finally, we note that the latent heat release assoc iated with the squall line produced a reverse gradieat of potential vonicity on a isenttopic surface. This barotropic/baroclinic instability deserves more attention. For most cases in TAMEX with a less than 1, squall line beating is distributed in a small region. The relevance of the instabilities to the actual weather needs to be investigated. The stability analysis of the balanced wind shear is the starting point for future studies.