1. INTRODUCTION

 A three-dimensional (3D) time-dependent anelastic line convection model with a three-class ice phase parameterization (Schlesinger, 1998; henceforth S98) is used to simulate the structure and bulk microphysics of a well-developed narrow cold-frontal rainband (NCFR). NCFR's are among the most active weather producers in midlatitude cool-season cyclones (Houze et al. 1976). Imbedded along the synoptic-scale parent front, they are dominated by mesoscale to microscale structure, sometimes showing "precipitation cores" alternating with "gaps" of lighter rain (Hobbs 1978, Locatelli et al. 1995). NCFR's can produce very heavy surface rain rates and strong wind gusts comparable to those found in some severe thunderstorms (Carbone 1982, Rutledge 1989), despite being typically much narrower and shallower (Hobbs 1978) and drawing upon marginal buoyant energy in environments with low- and mid-level lapse rates barely exceeding moist-adiabatic (Roux et al. 1993). Previous studies have shown that ice-phase microphysics is crucial to precipitation processes in NCFR's, including graupel growth by riming (Hobbs and Persson 1982, Rutledge and Hobbs 1984). The 3D modeling study reported herein investigates the impact of graupel on the structure and bulk microphysical characteristics of NCFR convection under conditions closely based on the environment of a notably vigorous winter NCFR in central California (Carbone 1982, Rutledge 1989, Parsons 1992).

2. EXPERIMENTAL CONDITIONS

2.1 Model  Properties

 The model domain is Cartesian, 148.2 km in the line-normal coordinate x; 40.8 km in the line-parallel coordinate y which is assumed periodically repeating as in Rotunno et al. (1988); and 13.6 km deep in the vertical coordinate z, with 247¥68¥34 grid cells of dimensions 0.6¥0.6¥0.4 km³. Finite-difference transport schemes and subgrid flux parameterizations are as in Schlesinger (1988). A fourth-order horizontal damping operator -K_D(d⁴/dx⁴ + d⁴/dy⁴) is applied lightly to the 3D wind  to suppress aliasing as in S98, here setting the constant coefficient K_D to 2.5¥10⁷ m⁴s^-1. Boundaries are open and aperiodic in x and rigid flat lids in z as in Schlesinger (1990) with Neumann conditions on the elliptic diagnostic pressure equation (Schlesinger 1975), specifying a unique solution by setting the domain-averaged value of the transformed pressure perturbation P to zero. Coriolis terms are included in the model code, but are turned off in the experiments reported here. Above 10 km, gravity wave reflection off the rigid top at 13.6 km is suppressed via Rayleigh damping of V and q toward the base state as in Durran and Klemp (1983), setting their parameter a to 5¥10^-3 s^-1.

 Aside from some minor changes, the model uses the three-class ice microphysics parameterization described in S98. Adapted from Ferrier (1990, personal communication), Lin et al. (1983) and Rutledge and Hobbs (1983, 1984), the parameterization generally allows for rain, snow and graupel. The 40 microphysical transfer rates in the model are evaluated after the heat and moisture fields are advected and diffused, using an algorithm of Ferrier (1994) to suppress oscillations in graupel and snow content due to alternate melting and freezing at successive time steps.

 Cloud particles are assumed to be non-precipitating, with zero terminal fallspeed and monodisperse size distributions. Cloud phase is diagnosed vs temperature T as all liquid if T > 0°C and all ice if T < -40°C, assuming that the ice fraction at intermediate temperatures varies parabolically with T so as to reach 75% at T = -20°C, based closely on mixed-phase cloud observations cited by Peppler (1940). For cloud-vapor transfers, the same technique weights the saturation vapor pressure (SVP) between its values over liquid water and ice, and the latent heat coefficient between vaporization and sublimation. Frozen precipitation can grow by deposition when T < -40°C, but any supersaturation is condensed out when cloud condensation/evaporation is performed at the end of the time step as a saturation adjustment. All precipitation types are treated as constant-density spheres, with inverse exponential size spectra and power laws for terminal fallspeed vs diameter D. The power law coefficients and particle densities are set as in Ogura and Takahashi (1971) for rain, Rutledge and Hobbs (1984) for graupel and snow. The intercept n₀ is assumed 8¥10⁶ m^-4 for rain, and 4¥10⁶ m^-4 for both graupel and snow. As in S98, autoconversion of cloud water to rain adapts the parameterization of Kessler (1969).

2.2 Initial Soundings and Experimental Layout (Fig. 1)

Figure 1. Common initial sounding for the two NCFR simulations. (a) Profiles of temperature (°C, solid curve) and dew point (°C, dotted curve), with dry adiabat for q = 300 K (dashed curve) for reference. (b) Respective line-normal and line-parallel wind earth-relative components u' and v' (m s^-1).

 Two 168-min simulations are run. The first retains the full microphysics and the second turns off all processes involving graupel, under conditions that are otherwise identical including the initial profiles of temperature, humidity and winds. These profiles, for temperature T and dew point T_d (Fig. 1a) along with the respective line-normal and line-parallel earth-relative wind components u' and v' (Fig. 1b), are closely adapted from a rawinsounding taken at Sheridan in the central California valley on 5 February 1978 at 1800 UTC ahead of a notably vigorous NCFR with very heavy convective rain, peak updrafts of 15-20 m s^-1 and a small tornado (Carbone 1982, Rutledge 1989). Conditional instability in Fig. 1a is marginal, with CAPE ~ 107 J kg^-1 based on undiluted parcel ascent from 0.2 km. Below 10 km, the line-normal wind is sheared strongly through ~1-3 km and ~6-7 km and less markedly at other levels, while the line-parallel wind shows mainly weak shear.

 At the surface, base-state pressure is 1000 mb, and the earth-relative winds are set equal to those at the lowest interior grid level (z = 0.2 km) for consistency with the free-slip lower boundary imposed in this study. To keep the main convection reasonably near mid-domain where it is triggered (see section 2.3) and avoid excessive along-line migration of convective elements in the model reference frame, the grid translates at a steady velocity c = c_xi + c_yj whose respective line-normal and line-parallel components are c_x = 19.4 m s^-1 and  c_y = 18.0 m s^-1.

2.3 Initial Forcing

 With w=0 everywhere, an initial cold dry pool is prescribed for 0 < z < 4.8 km from mid-domain rearward (x decreasing), akin to the heat and moisture sink used by Redelsperger and Lafore (1988) to spin up a simulated tropical squall line. Its leading edge is a sine wave 40.8 km long and 12 km wide rear-to-front (RTF) spanning 69.9 < x  < 81.9 km. The potential temperature deficit in this pool tends asymptotically front-to-rear (FTR) to 5.0 K for 0 < z  < 3.2 km, decaying linearly with z to zero for 3.2 < z  < 4.8 km. Throughout the pool, level for level, the relative humidity is set to 0.8 times the ambient value.

3. RESULTS

3.1 Time Variation of Vertical Velocity Extrema (Fig. 2)

Figure 2. Time variation of spatial maximum (WMAX) and minimum (WMIN) of vertical velocity w (m s^-1) in the two NCFR simulations.

 Figure 2 shows the domain-wide spatial vertical velocity maximum WMAX and minimum WMIN vs time in the two NCFR simulations. In conjunction with tabulations (not shown) of the altitudes for WMAX and WMIN, Fig. 2 further shows that:

•  In response to the strong deep initial cold pool described in section 2.3, both runs quickly spin up a strong quasi-steady system. WMAX reaches 10 m s^-1 before 8 min en route to higher values that span the remainder of the simulation time, and WMIN reaches its absolute minimum of -6.4 m s^-1 at ~10 min. This early strong downdraft peak reflects compensatory downflow adjacent to the updraft, as both WMAX and WMIN are then at 2 km.
• The absolute peak of WMAX is very similar in both runs, 15.3 m s^-1 with graupel vs 15.7 m s^-1 without, but much earlier in the former case than in the latter, at ~49 vs ~142 min. Prior to 60 min, the difference between WMAX in the two cases is minor, with little persistence of sign, but after 60 min WMAX is somewhat enhanced without graupel at all times, as is WMIN almost uninterruptedly. During 108-168 min, the time interval of main interest here, WMAX averages stronger without graupel than with it by ~23% (14.30 vs 11.58 m s^-1), and WMIN by ~14%  (-3.98 vs -3.50 m s^-1).
• During 108-168 min in the full-physics NCFR simulation, WMAX occurs at a very low altitude (~1.2-2 km), and WMIN somewhat higher (~2-3.5 km). Omitting graupel raises these altitudes appreciably, to ~2-3.5 km for WMAX and ~3-5 km for WMIN.

3.2 Time Variation of Near-Surface Cold Pool Strength (Fig. 3)

Figure 3. Time variation of spatial minimum (THMIN) of potential temperature deviation q' (K) in the two NCFR simulations, plotted only when this minimum is located at z = 0.2 km as an indicator of near-surface cold pool strength.

 To indicate how the strength of the cold pool evolves near the ground, Fig. 3 displays time series of the domain-wide spatial minimum THMIN for the perturbation potential temperature q' in both NCFR simulations when this minimum is located at z = 0.2 km, the lowest interior grid level at which q' is computed. THMIN is indeed located there except during ~12-21 min, when the greatest cooling occurs in an upper gravity wave near 10 km.

 After 22 min, the evolution of THMIN in the two NCFR simulations shows marked contrasts. The most obvious contrasts are the weaker evolution and much longer wait for sustained enhancement without graupel than with. More specifically,

•  From 22 to 35 min, the cold pool strengthens in both cases, but about one-third more slowly in the no-graupel vs full-physics simulation.
•  Aside from a transient absolute peak of 6.7K near 47 min, the maximum cooling in the full-physics run is strongest during 40-96 min when it continuously exceeds 6.2K, and then remains nearly steady between 6.0 and 6.3K through the end of the experiment.. But without graupel, the cold pool mostly weakens during 35-75 min and then meanders narrowly about its initial strength of 5.0K until ~115 min, and the cooling does not continuously exceed 5.8K until 149 min, vs 30 min with full physics.

3.3 Mean Line-Normal NCFR Structure (Fig. 4)

 Figure 4 depicts the mature simulated NCFR's at 108 min via comparative line-normal vertical (x-z) cross-sections of selected fields averaged in y (line-parallel) over one period. Only the centermost 70.2 km of the 148.2-km domain length are shown.

 Relative to the NCFR propagation, Fig. 4 shows wind vectors u_yavi + v_yavj, where ( )_yav denotes the y-averaging operator, along with the corresponding streamfunction y defined at any given location and time by integrating r₀u_yav vertically up from the surface to the location in question. Superimposed are the cloud boundary (scalloped curve) and selected isopleths of the y-averaged contents for rain (solid curves), graupel (dotted curves) and snow (sawtoothed curves) as specified in the figure caption.

 NCFR motion was estimated from a least-squares quadratic curve fit for its position vs time, defining this position as the x-coordinate of the grid column containing the updraft maximum WMAX plotted vs time in Fig. 2. According to this definition, both simulated NCFR's propagate rapidly akin to density currents, at close to the observed speed of 21.7 m s^-1 cited by Carbone (1982): 21.4 and 20.9 m s^-1 as of 108 min in the respective full-physics and no-graupel runs, and accelerating slightly over time, a little more in the former case (4.2¥10^-4 m s^-2) than in the latter (3.5¥10^-4 m s^-2).

 Both simulated NCFR's show several common features in Fig. 4:

•   Despite the strength of the updraft (Fig. 2), the cloud top reaches only ~6 km, as did the observed radar echo tops (Rutledge 1989).
•  The convective portion of the rain region, marked by a bulge in its top, is only a few kilometers wide, considering that the y-averaging process ìsmearsî features laterally because, as revealed by horizontal cross-sections (not shown), the (periodically repeating) NCFR segment still retains ~1/5 of the 12-km FTR excursion imposed on the sinusoidal initial cold pool edge.
•  The rain volume reaches only ~4 km in the convective zone -- as did the 40-dBz radar echo in the observed NCFR, whose maximum reflectivities were 50-55 dBz (Carbone 1982) -- and ~2.5 km in the leading stratiform rain region.
•  As in the observed storm (Carbone 1982), a layer of marked FTR storm-relative flow feeds the updraft between the surface and ~2.5 km, ascends through the forward flank of the convective rain region, and continues rearward into the trailing stratiform region above the frontal zone at heights of ~2.5-5 km, while the storm-relative flow below 2 km has a much weaker line-normal component  behind the frontal zone than ahead of it.
•  Maximum y-averaged rain water contents in the convective region of the NCFR are nearly identical with full physics (Fig. 4a) and without graupel (Fig. 4b), 1.27 and 1.26 g m^-3 respectively.
•  Dominance of "cold rain" processes in the simulated NCFR's, as will be documented more extensively in section 3.5, is implied by the abundant frozen precipitation in the upper half of the convective rain region, with y-averaged concentrations up to ~3 g m^-3 near 3 km.

•  Turning off graupel results in vastly more snow production, with a maximum y-averaged snow  content  of 3.58 g m^-3 in Fig. 4b vs 0.64 g m^-3 in Fig. 4a, within a deeper and far wider region of significant snow  content.

Figure 4. Line-normal vertical (x-z) sections of y-averaged vectors for the vertical and line-normal wind components relative to NCFR motion (UW-REL, m s^-1) with contours of the associated streamfunction y (PSI, 10⁶ g m^-1 s^-1) superimposed, at 108 min for: (a) the full-physics run CF-TST17, (b) the no-graupel variation CF-TST19. Ticks are Dx = 0.6 km apart horizontally and Dz = 0.4 km apart vertically with overall dimensions of 70.2 km ¥ 13.6 km, windowed to 39 < x < 109.2 km within the full 148.2-km line-normal length of the domain. CI represents contour intervals for y, with zero and positive contours solid and negative contours dashed. SCALE is the maximum wind speed (m s^-1), scaled to two horizontal tick separations. For y-averaged hydrometeor fields in both experiments, also superimposed, the thick scalloped curve is the cloud boundary and the thick solid curves from outermost to innermost are the rain water content isopleths for 0, 0.5 and 1 g m^-3; the thick sawtoothed curves from outermost to innermost  are the snow content isopleths for 0.2, 0.4 and 0.6 g m^-3 in the full-physics run, and for 1, 2 and 3 g m^-3 in the no-graupel variation. For the full-physics run only, the respective outer and inner thick dotted curves are the 1 and 2 g m^-3 isopleths for y-averaged graupel content. The ratio of vertical to horizontal scaling is 2.25, with wind vector orientation adjusted to suit.
 

•  The full-physics NCFR has a secondary low-level patch of y-averaged rain water contents > 1 g m^-3 a few kilometers behind the primary one within whose forward flank overlaps the updraft (Fig. 4a), whereas the no-graupel NCFR lacks this secondary feature (Fig. 4b).
•  The top of the trailing stratiform rain volume, behind the convective bulge, slopes gradually downward FTR in the full-physics run (Fig. 4a) but is flat when graupel is absent (Fig. 4b).

3.4 Some Further Remarks Concerning the Full-Physics Model NCFR

 Due to space limitations, no horizontal cross-sections of the simulated NCFR's are shown. Much of their basic structure emerges in Fig. 4, and the main updraft was predominantly two-dimensional (2D) in the large, especially at low levels. Nevertheless, while not achieving the observed full-fledged alternation of precipitation "cores" and "gaps" (Carbone 1982), the model hydrometeor fields still were sufficiently 3D that the along-line averaging in Fig. 4 damped them by ~30-50%, and three-dimensionality was more pronounced behind the front. Thus, for the full-physics run at 108 min (Fig. 4a), the respective maxima of the y-averaged cloud water, rain, graupel and snow fields were 1.27, 1.27, 2.48 and 0.64 g m^-3, whereas their corresponding local maxima in the 3D model grid were 1.94, 2.39, 3.65 and 0.93 g m^-3; the respective updraft and downdraft extrema at this time were 6.32 and -0.88 m s^-1 y-averaged, vs 11.25 and -3.73 m s^-1 locally.

 Also, while not evident from Fig. 4a, sporadic filaments of strong cyclonic vertical vorticity z about 1 km wide occurred near the ground along the frontal windshift line, vaguely hinting at (though not amounting to) a brief weak tornado such as was observed. At 0.2 km, the level maximum value of z was 1.23¥10^-2 s^-1 at 108 min, similar to vertical vorticity magnitudes in supercell thunderstorm mesocyclones, though this parameter was considerably smaller at 48 min (0.75¥10^-2 s^-1) and 168 min (0.70¥10^-2 s^-1).

 As documented in section 3.3, the modeled NCFR in the full-physics simulation reproduced several observed features reasonably well including the overall morphology, rapid propagation, limited vertical extent and strong updraft. Also, the maximum snow content in the model domain was between 0.8 and 1.1 g m^-3 from ~14 min onward, while the maximum graupel content was >4 g m^-3 during 13-97 min, including >5 g m^-3 from 18 to 68 min, still averaging 3.98 g m^-3 during 108-168 min. Thus, maximum snow and graupel contents were comparable to those cited by Rutledge (1989) in a 3D kinematic-microphysical simulation of the 5 February 1978 NCFR case using a time-independent flow field derived from dual Doppler radar winds. However, the extreme observed peak local rainfall rates of ~250 mm h^-1 (Rutledge 1989) were not closely approached. The rain rate in the full-physics experiment, after peaking early (90 mm h^-1 at 48 min), was 48 mm h^-1 at 108 min. Barth and Parsons (1995) also cite a similar figure (43 mm h^-1) in their 2D simulation of the same NCFR case using a fairly similar microphysical parameterization and slightly finer grid in comparison to ours.

3.5 Domain-Integrated Bulk Microphysical Features (Table 1)

 Table 1 lists selected bulk microphysical properties of the two NCFR simulations integrated over the volume of the model domain, time-averaged over the final 60 min of each experiment, i.e., 108-168 min, during which both vertical velocity extrema are quasi-steady in each run (Fig. 2). As in an analysis by Lin and Wang (1997) for a 3D subtropical convective storm simulation, these properties are: (a) the mass of each hydrometeor class, (b) the percentage contributions of each class to the total hydrometeor mass, and (c) for each precipitating hydrometeor class, the percentage contributions of its leading sources and sinks to its total production and depletion respectively.

 Table 1 reveals several qualitative microphysical similarities between the two NCFR simulations:

•  Cloud contributes far less mass than precipitation does.
•  Cloud ice is over an order of magnitude less abundant than cloud water, as might be anticipated from the low cloud tops.
•  The precipitation mass is much more ice than liquid, and rain is the second most abundant hydrometeor of the five that the model includes.
•  Accretion of cloud water is a significant secondary source of rain.
•  Evaporation and accretion by snow are mutually comparable major rain sinks.
•  Conversion from cloud ice strongly dominates snow production, while riming and deposition rank as appreciable secondary snow sources.
•  Sublimation is a secondary snow sink.

 However, the no-graupel NCFR simulation also differs significantly from the full-physics case in several respects:

•  Total hydrometeor mass is over twice as large, though the rain mass is somewhat less.
•  The most abundant hydrometeor is snow rather than graupel, and much more lopsidedly so, contributing an order of magnitude more mass; this is largely because, in the full-physics run, accretion of snow is by far the largest graupel source (and the main reason for the limited snow mass).
•  The top source of rain, which is graupel melt in the full-physics NCFR, is instead snowmelt when graupel is absent.
•  Conversion of cloud ice to snow is less strongly dominant as a snow source, while the secondary roles of riming and deposition are substantially more important than in the full-physics experiment.
•  Whereas snow depletion in the full-physics case is overwhelmingly due to accretion by graupel while sublimation is over 40 times slower, snow in the absence of graupel is instead lost mainly through melting and it sublimates only about three times slower than it melts.

 The simulations reported herein were sponsored by National Science Foundation (NSF) Grant ATM-9633424. Computations were performed on a Cray J90 computer at the National Center for Atmospheric Research (NCAR; Boulder, Colorado) which is supported by NSF. Thanks to Stanley Trier of NCAR for providing a tabulated version of the pre-NCFR sounding (subsequently preprocessed by the author to arrive at the initial profiles plotted in Fig. 1), and especially to Prof. Pao Wang of the authorís home department (Department of Atmospheric and Oceanic Sciences, University of Wisconsin -- Madison) for encouraging the modeling study highlighted in this paper.

5. REFERENCES

Barth, M. C., and D. B. Parsons, 1995: Microphysical processes associated with intense frontal rainbands and the impact of evaporation and melting on frontal dynamics. Preprints Conf. on Cloud Physics, Dallas, TX, January 15-20, 1995, Amer. Meteor. Soc., 226-230.

Carbone, R. E., 1982: A severe frontal rainband. Part I: Stormwide hydrodynamic structure. J. Atmos. Sci., 39, 258-279.

Durran, D. R., and J. B. Klemp, 1983: A compressible model for the simulation of moist mountain waves. Mon. Wea. Rev., 111, 2341-2361.

Ferrier, B. S., 1994: A double-moment multiple-phase four-class bulk ice scheme. Part I: Description. J. Atmos. Sci., 51, 249-280.

Hobbs, P. V., 1978: Organization and structure of clouds and precipitation on the mesoscale and microscale in cyclonic storms. Rev. Geophys. Space Phys., 16, 741-755.

______, and P. O. G. Persson, 1982: The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. Part V: The structure of narrow cold-frontal rainbands. J. Atmos. Sci., 39, 280-295.

Houze, R. A., Jr., P. V. Hobbs, K. R. Biswas, and W. M. Davis, 1976: Mesoscale rainbands in extratropical cyclones. Mon. Wea. Rev., 104, 868-878.

Kessler, E., 1969: On the Distribution and Continuity of Water Substance in Atmospheric Circulations. Meteor. Monogr., No. 32, 84 pp.

Lin, H.-M., and P. K. Wang, 1997: A numerical study of microphysical processes in the 21 June 1991 Northern Taiwan mesoscale precipitation system. Terres. Atmos. Oceanic Sci., 8, 385-404.

Lin, Y.-L., R. D. Farley and H. D. Orville, 1983: Bulk parameterization of the snow field in a cloud model. J. Clim. and Appl. Meteor., 22, 1065-1092.

Locatelli, J. D., J. E. Martin, and P. V. Hobbs, 1995: Development and propagation of precipitation cores on cold fronts. Atmos. Res., 38, 177-206.

Ogura, Y., and T. Takahashi, 1971: Numerical simulation of the life cycle of a thunderstorm cell. Mon. Wea. Rev., 99, 895-911.

Parsons, D. B., 1992: An explanation for intense frontal updrafts and narrow cold-frontal rainbands. J. Atmos. Sci., 49, 1810-1825.

Peppler, W., 1940: Forschg. u. Erfahrung. Reichsamt f. Wetterdienst., B, No. 1.

Redelsperger, J.-L., and J.-P. Lafore, 1988: A three-dimensional simulation of a tropical squall line: Convective organization and thermodynamic vertical transport. J. Atmos. Sci., 45, 1334-1356.

Rotunno, R., J. B. Klemp and M. L. Weisman, 1988: A theory for strong, long-lived squall lines. J. Atmos. Sci., 45, 463-485.

Roux, F., V. Marécal, and D. Hauser, 1993: The 12/13 January 1988 narrow cold-frontal rainband observed during MFDP/FRONTS 87. Part I: Kinematics and thermodynamics. J. Atmos. Sci., 50, 951-974.

Rutledge, S. A., 1989: A severe frontal rainband. Part IV: Precipitation mechanisms, diabatic processes and rainband maintenance. J. Atmos. Sci., 46, 3570-3594.

______, and P. V. Hobbs, 1983: The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. VIII: A model for the "seeder-feeder" process in warm-frontal rainbands. J. Atmos. Sci., 40, 1185-1206.

______, and ______, 1984: The mesoscale and microscale structure and organization of clouds and precipitation in midlatitude cyclones. XII: A diagnostic modeling study of precipitation development in narrow cold-frontal rainbands. J. Atmos. Sci., 41, 2949-2972.

Schlesinger, R. E., 1975: A three-dimensional numerical model of an isolated deep convective cloud: Preliminary results. J. Atmos. Sci., 32, 934-957.

______, 1988: Effects of stratospheric lapse rate on thunderstorm cloud-top structure in a three-dimensional numerical simulation. Part I: Some basic results of comparative experiments. J. Atmos. Sci., 45, 1555-1570.

______, 1990: Feedback of deep moist convection to its near environment as diagnosed from three-dimensional numerical model output: Results from an early experiment. J. Atmos. Sci., 47, 1390-1412.

______, 1998: Effects of midlevel jet intensity on structure and latent heating profiles in Darwin squall lines during summer monsoon break: Semi-idealized 3D simulations. Preprints 19th Conf. on  Severe Local Storms, Minneapolis, MN, September 14-18, 1998, Amer. Meteor. Soc., 653-658.
_______________
*Corresponding author address: Robert E. Schlesinger, Univ. of Wisconsin - Madison, Dept. of Atmospheric and Oceanic Sciences, Madison, WI  53706; e-mail: schlesin@meteor.wisc.edu.