The Diagnosis of Vertical Motion within Dry Intrusions - AMS Journals

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Oct 1, 2003 - A diagnosis of ascent within the dry intrusion is obtained from ... air (the dry intrusion) spirals in toward the center of the ..... Therefore if suit-.
OCTOBER 2003

JAMES AND CLARK

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The Diagnosis of Vertical Motion within Dry Intrusions RICHARD P. JAMES

AND

JOHN H. E. CLARK

Department of Meteorology, The Pennsylvania State University, University Park, Pennsylvania (Manuscript received 21 May 2002, in final form 4 March 2003) ABSTRACT Dry intrusions play an important role in modulating precipitation patterns both in the midlatitudes and in the Tropics. The lifting of unsaturated air aloft often leads to destabilization and the enhancement of precipitation rates, and may occasionally contribute to the occurrence of severe weather. A method for qualitatively diagnosing vertical motion in a region of elevated dry advection is presented. The procedure measures the rate of propagation of relative humidity isopleths relative to the flow and deduces the sign of the vertical velocity. Changes in static stability are inferred, leading to the possibility of improved short-term forecasting of precipitation associated with dry intrusions. The procedure is illustrated with a case study involving heavy snowfall associated with a dry intrusion in the mid-Atlantic region. A diagnosis of ascent within the dry intrusion is obtained from satellite imagery and confirmed using numerical model output.

1. Introduction A phenomenon commonly observed as part of the life cycle of the midlatitude cyclone is the formation of a ‘‘dry intrusion.’’ The advent of satellite imagery in the early 1960s led to the identification of a frequently occurring ‘‘vortex cloud pattern’’ wherein relatively clear air (the dry intrusion) spirals in toward the center of the extratropical cyclone (Leese 1962). The dry intrusion is a major component of the structure of most extratropical cyclones and has been the subject of considerable scrutiny (Browning 1997). The presence of a midlatitude dry intrusion has been shown to favor the occurrence of deep convection under some circumstances. Carr and Millard (1985) demonstrated that the dry (i.e., unsaturated) air advecting into the rear side of the ‘‘comma-cloud’’ system undergoes strong ascent, leading to adiabatic cooling aloft. Destabilization frequently results, because the dry intrusion, characterized by low wet-bulb potential temperature u w , often overrides a low-level region of high u w , leading to potential instability. The leading edge of the dry intrusion constitutes the ‘‘upper cold front’’ of Browning and Monk (1982), although ‘‘the cooling aloft which accompanies the passage of the upper cold front is—as in the case of the surface cold front—more notable in terms of the decrease in u w than u, which does not always fall signifCorresponding author address: Mr. Richard P. James, Dept. of Meteorology, The Pennsylvania State University, 503 Walker Building, University Park, PA 16802. E-mail: [email protected]

q 2003 American Meteorological Society

icantly’’ (Browning and Monk 1982). A similar phenomenon is described by the ‘‘cold front aloft’’ model of Hobbs et al. (1990), who locate the elevated front at the warm edge of ‘‘a larger-than-background temperature gradient in the mid-troposphere above the precipitation region.’’ The cold front aloft is also characterized by advection of drier air and the rapid lowering of cloudtop heights behind the front (Hobbs et al. 1990). The distinction between the Browning and Monk (1982) model and that of Hobbs et al. (1990) lies in the nature of the boundary that intersects the ground. In the former case, the elevated low-u w air overrides a surface cold front and moves above a warm sector; in the latter model, a lee trough acting as a surface warm front is overridden by the cold, dry air aloft. In both cases the upper front introduces potential instability, which may be released owing to large-scale uplift, and deep and/or severe convection can occasionally occur (Carr and Millard 1985; Browning and Golding 1995; Browning and Roberts 1999). Dry intrusions also play an important role in tropical meteorology. The Tropical Ocean Global Atmosphere (TOGA) Coupled Ocean–Atmosphere Response Experiment (COARE) found that lateral advection of dry air into the tropical western Pacific occurs frequently and is of primary importance in the atmospheric water vapor budget (Parsons et al. 2000). Several studies have demonstrated the suppression of deep convection by the dry intrusions (e.g., Lucas and Zipser 2000; Brown and Zhang 1997). This may be attributable either to increased convective inhibition (Parsons et al. 2000) or to the detrimental effects of increased entrainment of

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dry air (Brown and Zhang 1997; Wei et al. 1998). The entrainment of dry air into tropical cyclones has been shown to lead to the erosion of convective latent heat release and the decrease of tropical cyclone intensity (Rodgers et al. 1998, 2000). On the other hand, there is evidence that the ingestion of dry air by landfalling hurricanes, in particular, can lead to severe convection and tornado outbreaks (Hill et al. 1966; McCaul 1987; Vescio et al. 1996). Curtis (2001) found that seven of nine tornado outbreaks occurring during hurricane landfall were associated with dry intrusions at mid- and upper levels. It is thus intriguing that the arrival of dry air aloft can lead, under differing circumstances, to diverse effects upon convective activity and precipitation patterns. In the majority of cases the advection of dry air is inimical to the continuation of precipitation; but there are exceptions to this rule. This paper presents a method for qualitatively diagnosing vertical velocity within a region of dry advection aloft. Information about the sign of the vertical velocity may enable a forecaster to ascertain whether the potential instability introduced by the dry air aloft is likely to be released and, hence, whether enhancement of precipitation is a possibility. 2. Modification of relative humidity The low relative humidity of the midlatitude dry intrusion arises from deep subsidence earlier in the life cycle of the system of interest (Browning 1997). Analyses of trajectories (e.g., Kuo et al. 1992) reveal that air parcels in the dry intrusion originate at high tropospheric, or even stratospheric, levels and then descend in the region upstream of a developing cyclone. By the time the classic dry intrusion is identified, subsidence has often ceased and ascent may be occurring (Carr and Millard 1985; Kuo et al. 1992; Leese 1962). Similarly in the Tropics, horizontal advection of previously subsided air accounts for the appearance of the dry intrusions observed over the western Pacific (Yoneyama and Parsons 1999). In general, the relative humidity of air can partially reflect the integrated effects of vertical motion at earlier times. To commence a more quantitative discussion, we may write an equation for the Lagrangian rate of change of relative humidity: d(RH) 5 Q, dt

(1)

where RH is the relative humidity and Q is a source term. An equation such as this may be written for any intensive variable, with the source Q containing all terms leading to nonconservation of the variable. In the case of relative humidity, the source term is Q ø RH

1

2

d ln p R T d lne s 12 d y , dt c p dT

(2)

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where p is the pressure, R d is the specific gas constant for dry air, T y is the virtual temperature, c p is the specific heat capacity of dry air at constant pressure, e s is the saturation vapor pressure, and T is the temperature. The derivation of this equation is presented in the appendix. The primary assumptions made are that mixing ratio is conserved following the parcel, and vertical motion is dry adiabatic (thus the equation is only valid for subsaturated conditions). In addition, radiational effects are ignored because of the short timescales under consideration here. Finally, the hydrostatic approximation is made. The term in the brackets in (2) is always negative under observed atmospheric conditions. Consequently, ascent (descent) always leads to parcel moistening (drying), in the absence of other effects such as evaporation of precipitation or mixing. The local rate of change of relative humidity is ](RH) 5 Q 2 v · =(RH), ]t

(3)

where v is the three-dimensional wind vector. Advection and vertical motion are thus the two effects controlling the evolution of the relative humidity field, under the assumptions stated above. If vertical motion is nonexistent, RH is altered only through horizontal advection. The RH is, however, very sensitive to vertical motion. For instance, ascent of not much more than 50 hPa in the middle troposphere is sufficient to increase the relative humidity from 50% to 100%, without considering the effect of vertical advection. The passage of a dry intrusion is obviously characterized by dry advection by the horizontal wind. However, the leading edge of the dry intrusion is often also a region of ascent because of its position within the cyclone (Browning and Roberts 1999), and therefore the two terms on the right-hand side of (3) are in opposition. Typically the horizontal advection overwhelms the moistening effect of ascent, so that the relative humidity decreases locally over time. This would be observed as an advancing dry intrusion in satellite imagery or on an upper-air analysis. Occasionally the ascent may be strong enough that isopleths of relative humidity do not advance in space, but may remain stationary or even retreat. The key point here is that the movement of isopleths of relative humidity relative to the wind may be used to qualitatively diagnose vertical motion. In other words, if satellite images (or other tools) indicate that a certain isopleth of relative humidity is advancing at the same speed as the flow, then little or no vertical motion is taking place. If the dry intrusion is not moving as rapidly as the wind, then ascent is taking place, the strength of which is suggested by how rapidly the relative humidity isopleths are in fact moving relative to the horizontal wind. If, as may occasionally happen, the dry intrusion is advancing more rapidly than the local wind, then subsidence may be inferred.

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The preceding argument may be expressed mathematically by assuming that a relative humidity isopleth advances at a velocity c, so that the local rate of change of relative humidity may be expressed as ](RH) 5 2c · =(RH). ]t

(4)

Equation (3) may then be rewritten as Q 5 (v 2 c) · =(RH).

(5)

According to (5), a wind component exceeding the forward propagation speed of the isopleth indicates that Q is positive, leading to a diagnosis of ascent. If good estimates of v, c, and the horizontal gradient of RH are available, then a rough calculation of the magnitude of the vertical velocity may be made.

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at the dry-adiabatic lapse rate for a longer time before saturating. Similarly, subsidence of dry air provides a stabilizing mechanism. As the elevated cold/dry front advances, relatively dry air is advected over relatively moist air, causing potential instability. Hence we suggest that careful observation of the rate of advance (retreat) of a dry intrusion relative to the horizontal wind vector supplies useful information about the (de)stabilizing processes taking place near the leading edge. As a nowcasting tool, or even as a forecasting method using numerical model output, this diagnosis may allow a better understanding of an unfolding meteorological event. In the next section we provide an illustration of the utility of the method as a nowcasting tool in a heavy snow event. 4. Case study

3. Practical considerations It may not be uncommon for situations to arise where the midlevel dry intrusion undercuts an upper-level layer of cloud and, therefore, becomes obscured in satellite images [for an example, see the case studied in Browning and Roberts (1999)]. In these instances it becomes impossible to track the movement of the dry intrusion using satellite imagery. However, radar imagery may be sufficient to infer the rate of propagation of the dry intrusion’s leading edge, by tracking the abrupt cutoff in precipitation that is usually observed behind the elevated front. As long as the velocity of the relative humidity isopleths relative to the flow may be estimated by one method or another, the qualitative diagnosis described here may be performed. The wind speed that is chosen for the diagnosis may be estimated from upper-air analyses, wind profiler data, or numerical model output, and should be consistent with the estimated altitude of the dry intrusion. At least two factors combine to make the choice of the level rather subjective. First, the relatively dry air behind the upper front often extends through a deep layer in which the flow speed and direction may vary significantly. Second, ‘‘water vapor’’ satellite images display the integrated water vapor content through a deep layer, and therefore a certain shading threshold corresponds only approximately to a relative humidity isopleth at a single altitude. It is suggested that a wind vector be chosen near the estimated base of the dry intrusion, because the diagnosis of vertical motion is likely to be most important just above the base of the dry air, where the potential instability is greatest. In addition, a false diagnosis of ascent in the dry intrusion is less likely if a wind speed is chosen at a lower level, because wind speed usually increases with height in the midlatitudes. The diagnosis of vertical motion leads naturally to an indication of changes in static stability, given a knowledge of the moisture profile in the column. Lifting of dry air that overlies more moist air generally leads to destabilization, because the dry air aloft is able to cool

On 8–9 March 1999, an extratropical cyclone affected a broad region of the northern and eastern United States. Snowfall occurred over a wide area and was locally heavy, especially over portions of the mid-Atlantic region. Surface and upper-air analyses for 1200 UTC 9 March appear in Fig. 1, and Fig. 2 shows a surface analysis at 1800 UTC. The fronts are drawn according to the National Centers for Environmental Prediction (NCEP) analysis, although we note that surface temperatures did not fall off rapidly behind the ‘‘cold front’’ and the boundary might be analyzed more properly as a trough (Locatelli et al. 2002). The dry intrusion associated with the system is clearly seen in the 7.0-mm infrared imagery shown in Fig. 3. Infrared radiation with wavelengths in the range 5–7 mm is strongly absorbed by water vapor but not by other absorbing gases in the atmosphere (Salby 1996). Therefore the emission to space at these wavelengths gives a good indication of the distribution of water vapor in the mid- to upper troposphere. According to Soden and Fu (1995), emission at wavelengths around 6.7 mm is sensitive primarily to the average relative humidity in a deep layer centered at about 400 hPa, for a typical midlatitude temperature and moisture profile. Thus bright areas in Fig. 3 represent regions of high relative humidity at mid- to upper levels, while darker gray areas indicate low levels of moisture (note that the black shading indicates intermediate moisture levels). It can be seen that dry air at mid- to upper levels formed a cyclonic spiral pattern ahead of the closed low (Fig. 1d), which moved eastward and northeastward over time. The axis of driest air at 1200 UTC extended from Mississippi to the spine of the Appalachians, and then northwestward from extreme western Virginia to central Indiana. The narrow northwestward extension of the dry intrusion was not sampled by the radiosonde network, but the presence of dry air farther south was confirmed by the dewpoint depression reports at 700 and 500 hPa from Jackson, Mississippi, and Birmingham, Alabana (Figs. 1b,c). The leading edge of the dry air aloft co-

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FIG. 1. (a) Surface analysis valid at 1200 UTC 9 Mar, showing sea level pressure (hPa), station reports (conventional plotting), and surface features (conventional symbols); (b)–(d) upper-air analyses valid at 1200 UTC 9 Mar, showing geopotential heights (dam) and station reports (conventional plotting): (b) 500 hPa; (c) 700 hPa, heavy solid line marks approximate position of dry intrusion; and (d) 850 hPa.

incided approximately with the arrival of weak cold advection at 700 and 500 hPa. The position of this boundary aloft was well ahead of the surface trough (Fig. 1a), and hence this system fits within the cold front aloft conceptual model of Hobbs et al. (1990). Radar reflectivity images from the morning of 9 March (Fig. 4) indicate that a curved band of precipitation existed ahead of the elevated frontal boundary. This arrangement is consistent with the expected structure of the cold front aloft type cyclone (Locatelli et al. 2002). As the elevated front advances, lifting is gen-

FIG. 2. As in Fig. 1a but at 1800 UTC 9 Mar.

erated at and ahead of the leading edge, producing a band of precipitation that is tied to the frontal boundary (Locatelli et al. 1995). The signature of the 9 March band in the infrared imagery (Fig. 3) was a curved region of high, cold cloud tops that advanced east-northeastward just ahead of the leading edge of the dry intrusion. At the surface, light to moderate snow was reported over a wide area, with some reports of heavy snow close to the upper frontal boundary (e.g., Dayton, Ohio, at 1200 UTC). Of particular interest for the purposes of this study was the sequence of events that occurred in the northern mid-Atlantic region during and after the passage of the upper cold front. Hourly 6.7-mm satellite images are shown in Fig. 5; this wavelength differs from that shown in Fig. 3 because of data availability restrictions. Both channels provide an indication of mid- to upper-level relative humidity. Surface observations at the closest hour are superimposed in Fig. 5. At 1315 UTC (Fig. 5a) the cold cloud band was located across West Virginia and central Virginia, with much drier air positioned a short distance to the southwest. The boundary between the dry air and the cold cloud band has a shape consistent with that seen in Fig. 3c and marks the approximate position of the cold front aloft. Heavy snow was reported over southeastern Ohio, close to the strong moisture gradient aloft.

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FIG. 3. Geostationary satellite (GOES-8) 7.0-mm infrared images, 9 Mar: (a) 0946, (b) 1146, (c) 1346, and (d) 1647 UTC.

Over the next 5 h (Figs. 5a–f), the cold cloud band progressed rapidly toward the east-northeast, reaching the Virginia coast by about 1630 UTC. However, heavy snow continued to be observed in northern Virginia and extreme eastern West Virginia well after the passage of the cold cloud band. In particular, stations in the Washington, D.C., area reported heavy snowfall at 1700 and 1800 UTC, after the elevated frontal boundary had passed at about 1630 UTC. A careful examination of the infrared images in Fig. 5 reveals that the driest air (black shading) was substantially moistened as it advanced within the midlevel southwesterly flow. The nose of the dry intrusion gives the appearance in the imagery of being pinched off. (Note that the rapid departure of the cold cloud band permitted an unobscured view of the dry intrusion in the satellite imagery.) The result of this moistening process was that over a period of 5 h the relative humidity isopleth approximately represented by the edge of the darkest shading in Fig. 5 failed to advance northward at all, despite a strong southerly component to the midlevel flow (Fig. 1b). Hence, based on the argument in section 2, we qualitatively diagnose that ascent was occurring at mid- to upper levels within the dry intrusion, behind the elevated cold front. In order to further investigate the processes occurring

over the mid-Atlantic region on 9 March, numerical simulations were performed using the fifth-generation Pennsylvania State University–National Center for Atmospheric Research (PSU–NCAR) Mesoscale Model (MM5; Dudhia 1993). Initialization fields and lateral boundary conditions were obtained from the operational NCEP Eta Model runs from 8–9 March. Observational nudging, based upon the standard radiosonde network reports, was used to increase the fidelity of the simulations. A two-way nested simulation was performed, with 15-km grid spacing used for the inner nest over the central Appalachian region. The model performed well in terms of the synoptic evolution over the central and eastern United States; it also reproduced the dry intrusion with an appearance similar to that in the satellite imagery (Fig. 6b), although the position of the dry intrusion’s leading edge lagged approximately 4 h behind the observed position. The lifting associated with the elevated front was reproduced in the model and led to significant precipitation in approximately the same locations as the observed cold cloud band (Fig. 7). The relative humidity field in the model displayed an evolution similar to that inferred from the satellite imagery in Fig. 5. Between 1800 and 2200 UTC in the model simulation, the northwestward extension of the dry intrusion was entirely eliminated (Figs. 6b–d). This

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FIG. 4. Radar reflectivity composite images with shading levels beginning at 5 dBZ and with 5-dBZ increments, 9 Mar: (a) 1000, (b) 1100, (c) 1200, and (d) 1300 UTC.

moistening process was accomplished by an ascent of magnitude 0.05–0.1 m s 21 (Fig. 8). The lifting and moistening of the midlevel dry intrusion is hypothesized to have led directly to destabilization, because of the presence of potential instability. Vertical profiles of temperature produced by the model simulation are consistent with this hypothesis, as illustrated in Fig. 9, which shows a model sounding through the center of the dry intrusion at a time when it was almost completely eliminated by lifting. The sounding contains a layer from 700 to 580 hPa that is approximately neutral to moist displacements. Further lifting would cause the lapse rate within the layer to slightly exceed the moist-adiabatic lapse rate, leading to convective overturning. Rapid ice dendrite growth would occur, because temperatures were close to 2158C near the top of the unstable layer (Fukuta and Takahashi 1999). As supporting evidence that convective processes were indeed responsible for the postfrontal precipitation, a high-resolution visible satellite image from 1702 UTC is presented in Fig. 10. The textured pattern of cloud tops over northern Virginia suggests that convective overturning was taking place, with cloud tops reaching approximately 400 hPa based upon the infrared imagery. Finally, we further illustrate, using the MM5 output, the method involving the movement of relative humidity

isopleths that was introduced in section 2. The motion of the 80% relative humidity isopleth at the leading edge of the dry intrusion (Fig. 6) was measured over three 1-h periods: 1800–1900, 1900–2000, and 2000–2100 UTC. In each period the component of the wind speed perpendicular to the isopleth was compared to the movement of the isopleth perpendicular to itself. The results are shown in Table 1. It was found that in all three cases the 80% relative humidity isopleth at the leading edge of the dry advection was moving forward less rapidly than the flow in which it was embedded. Equation (5) was used to obtain an estimate of the vertical velocity, and the results are included in Table 1. Small positive vertical velocities were diagnosed, which is consistent with the existence of weak ascent in the model output (Fig. 8). Of course, the diagnosis is entirely unnecessary in the case of the model output because the vertical velocity field is available; but the results are presented to strengthen the argument that the same analysis can be performed using water vapor satellite imagery in order to obtain a qualitative vertical velocity field at the leading edge of a dry intrusion. It should be noted that the quantitative estimate for vertical motion is difficult to obtain using satellite imagery, because the magnitude of the horizontal gradient of relative humidity at a single level is not easily retrievable (see section 4).

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FIG. 5. GOES-8 6.7-mm infrared images with surface station reports (conventional plotting) superimposed, 9 Mar: (a) 1315, (b) 1415, (c) 1515, (d) 1615, (e) 1715, and (f ) 1815 UTC.

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FIG. 6. Relative humidity (%) and wind barbs (m s 21 ) from the MM5 at 500 hPa at (a) 1600, (b) 1800, (c) 2000, and (d) 2200 UTC.

5. Conclusions Broad regions of relatively dry air are commonly observed in both the midlatitudes and the Tropics. On a large scale, dry air is obviously correlated with the absence or suppression of precipitation. However, the lifting of dry air that overlies relatively moist air can lead to the release of the potential instability and the formation or enhancement of precipitation. The key difference is the presence of ascent in the destabilizing

case. The diagnosis of vertical motion in the operational forecasting environment is thus of great importance for this reason, among others. We have shown that an analysis of the rate of propagation of a dry intrusion (as defined by relative humidity isopleths) relative to the flow in which it is embedded can provide qualitative diagnostic information about vertical motion. The presence of potential instability in the typical dry intrusion also leads to an inference about changes in static stability as a result of the vertical motion.

FIG. 7. Precipitation (cm) from the MM5 accumulated from 1700 to 1900 UTC 9 Mar.

FIG. 8. Vertical velocity (m s 21 ) from the MM5 averaged in the layer 600–400 hPa at 2100 UTC 9 Mar.

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FIG. 9. Skew T–log p diagram from the MM5, with winds in kt, at 39.58N, 79.58W, at 2100 UTC 9 Mar.

FIG. 10. High-resolution visible image, 1702 UTC 9 Mar, with 1700 UTC surface precipitation reports (conventional plotting) superimposed.

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TABLE 1. A comparison of the rates of propagation of the 80% relative humidity isopleth and the flow speed, from the MM5 output, together with diagnosed vertical velocity from Eq. (5). Both propagation speeds are taken perpendicular to the isopleth. 1800–1900 1900–2000 2000–2100 UTC UTC UTC 80% RH isopleth propagation speed (m s21) Wind speed (m s21) Diagnosed vertical velocity (m s21)

13.5 14.8 0.07

11.6 14.1 0.11

12.4 14.6 0.06

The 9 March 1999 cyclone was examined in order to illustrate the utility of the method. A cold front aloft generated a band of ascent and precipitation, as expected within the Hobbs et al. (1996) conceptual model. However, infrared satellite images suggest that strong moistening of the dry air behind the elevated front preceded a prolonged period of heavy snowfall over the northern mid-Atlantic region. The observation that the relative humidity isopleths were advancing much less rapidly than the flow leads to the diagnosis of significant ascent at the leading edge of the dry air. By examining this imagery in real time, it is likely that a forecaster could have been alerted 2–3 h in advance to the possibility of the postfrontal enhancement of precipitation. Visible satellite imagery indicates the likelihood that the precipitation was produced by convective overturning, which is consistent with the idea that potential instability was released by the lifting of the dry intrusion. Numerical simulations of the event reproduce the lifting, moistening, and (to some extent) destabilization of the dry air, and support the notion that a qualitative diagnosis of vertical motion may be obtained by observing the propagation of the dry intrusion’s leading edge. Cyclones in the central and eastern United States are often characterized by cold fronts aloft (Locatelli et al. 2002), which are in turn usually associated with elevated dry advection. Given that the elevated front, especially in its northern portions, is commonly located in a region of midlevel ascent (Browning and Monk 1982), the destabilization of the dry intrusion may not be unusual. The technique described here may be of use in shortterm forecasting of precipitation events related to this destabilization process. The required tools are infrared satellite imagery of an appropriate wavelength to detect midlevel water vapor content, as well as a knowledge of the wind speed and direction at the level of the dry air. The latter may be obtained from radiosonde reports, wind profilers, or numerical model output. In some cases the dry intrusion may undercut a shield of higher clouds and become obscured in the satellite imagery, leading to difficulty in tracking its motion. It is possible that in these situations radar imagery could still be used to diagnose whether the dry air is advancing more or less rapidly than the flow in which it is embedded. Large-scale uplift of potentially unstable air in the tropical atmosphere will also lead to destabilization and

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the enhancement of precipitation. It is likely that this mechanism contributes to tornado outbreaks within landfalling hurricanes (McCaul 1987). Therefore if suitable satellite images and wind analyses are available, the method presented here may be equally useful in tropical regions. Acknowledgments. The authors are grateful to Mike Fritsch for helpful discussions pertaining to the theme of this work. Technical assistance was graciously provided by George Bryan. The comments of three anonymous reviewers were most helpful and are gratefully acknowledged. Satellite imagery is reproduced by permission of the Cooperative Institute for Meteorological Satellite Studies (CIMSS), University of Wisconsin— Madison. This work was supported by the COMET Cooperative Project (UCAR Award 099-15810). APPENDIX Derivation of the Relative Humidity Equation Using the definition of relative humidity, RH 5

e , es

(A1)

where e is the vapor pressure and e s is the saturation vapor pressure, the Lagrangian rate of change of relative humidity is d(RH) 1 de e de 5 2 2 s. dt e s dt e s dt

(A2)

The vapor pressure may be written [following from Emanuel (1994), Eq. (4.1.4)], ry p « e5 , ry 11 «

(A3)

where r y is the water vapor mixing ratio, p is the pressure, and « 5 R d /R y . Here, R d and R y are the specific gas constants for dry air and water vapor, respectively. Assuming that mixing ratio is conserved following a parcel (i.e., subsaturated motions only, and neglecting mixing), de e dp 5 . dt p dt

(A4)

The rate of change of temperature following a parcel in dry-adiabatic ascent or descent is dT g 5 2 w, dt cp

(A5)

where g is the acceleration due to gravity, c p is the specific heat capacity of dry air at constant pressure, and w is the vertical velocity (in m s 21 ). We then make the approximation

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v[

dp 5 2rgw, dt

(A6)

where r is the density. This approximation involves the use of the hydrostatic approximation, and the neglect of ]p/]t and v h · = h p compared to w(]p/]z), where vh is the horizontal wind vector. In a synoptic-scale environment, w(]p/]z) is at least two orders of magnitude larger than ]p/]t, assuming vertical motions of only 1 cm s 21 . If winds are approximately geostrophic, then v h · = h p is also very small. Using these approximations and the equation of state, we obtain dT RT ø d y v, dt cp p

(A7)

where T y is the virtual temperature. Finally, the equation for relative humidity reads d(RH) RH dp RH R d Ty de s dp ø 2 . dt p dt e s c p p dT dt

(A8)

This simplifies to

1

2

d(RH) d ln p R T d lne s ø RH 12 d y . dt dt c p dT

(A9)

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