1. Introduction

The distribution of precipitation in mountainous terrain remains an important problem in regional climate and ecology, water resources and flood prediction, paleoclimate and glaciology. One interesting “prototype” area for the study of orographic precipitation is the state of Oregon. The suitability of the Oregon mountain ranges for scientific study arises from several factors. First, the east–west climate and vegetation gradients across the coastal ranges and the Cascades are unusually uniform and strong (Bastasch 1998; Taylor and Hannan 1999). Second, as we will verify, the weather systems that cause precipitation share common characteristics and are easy to represent. Third, Precipitation-Elevation Regressions on Independent Slopes Model (PRISM), one of the most widely used rainfall interpolation schemes was developed and tested in this region (Daly et al. 1994, 2002. Finally, a tradition of scientific observation and interpretation in the Oregon region, extending from Hobbs et al. (1973) and Fraser et al. (1973) to the recent field experiment Improvement of Microphysical Parameterization through Observational Verification Experiment-II (IMPROVE II; Garvert et al. 2003; Stoelinga et al. 2003), provides a basic understanding of airflow and precipitation physics in the region.

The central scientific questions concerning orographic precipitation regard airflow dynamics, and cloud physics. Each of these issues involves scale dependence; the impact of distance on determining the dominant physical processes. Concerning airflow dynamics, an important question is the depth and intensity of the orographically forced ascent. If the moist layer is deep (e.g., 4 km), ascent caused by small-scale hills will not reach the upper portions of the layer and some water vapor will cross the mountains “untouched” by forced ascent. If the phase lines of the forced ascent are tilted upstream, condensation will start ahead of the terrain. Concerning cloud physics, the central question is how quickly and completely cloud water can be converted to rain and snow and how quickly can these hydrometeors fall to earth. If conversion is slow, small-scale hills will form “wave clouds” or “cap clouds” in which cloud water that condenses on the upwind side mostly evaporates on the lee side. If the fallout is slow, hydrometeors will be carried into areas of leeside descent, where they evaporate. In contrast, if the conversion and fallout are fast, and hills are broad, precipitation removes condensed water before lee-side descent begins.

In section 2, we review the physical processes and events that lead to Oregon’s climate transition. In section 3, we recall three simple models of orographic precipitation. Precipitation patterns predicted by the new linear model are discussed in section 4. Observed and predicted spatial patterns of precipitation are compared in section 5. Satellite-derived proxies for precipitation are introduced in section 6. River discharge is analyzed in section 7. Stream water isotope data are used to estimate the drying ratio in section 8, leading to an estimate of the cloud physics delay time in section 9.

2. Precipitation and water vapor flux

As a foundation for the current study, we summarize the forcing and response of the Oregon hydrologic system. Good descriptions of the system are provided by Bastasch (1998) and Taylor and Hannan (1999). In the winter season, frontal cyclones move against the coast from the west. In their trailing fronts, strong moist winds transport water vapor eastward and northward as “atmospheric rivers” (Zhu and Newell 1998). These frontal regions, already precipitating due to frontal circulations, produce heavy rainfall when the coastal range and the Cascades force the air to rise more quickly. The frequency and intensity of these events varies from year to year (Bond and Vecchi 2003).

The amount of lifting required to cross the Oregon terrain is illustrated in Fig. 1. North of 44°N, the ascent over the coastal range exceeds about 700 m before starting the descent into the Willamette Valley. The subsequent ascent over the Cascades takes the air to about 2000 m, except near volcanic peaks such as Mount Adams (3571 m) or Mount Hood (3472 m), or through the Columbia Gorge near 45.6°N. Farther south, the air rises more abruptly near the coast to 1300 m and continues an irregular rise to 2000 m over the Cascades. In central Oregon at 120°W, east of the Cascades, there is a considerable north–south elevation gradient. Westerly flow in northern Oregon climbs over the coastal and Cascade ranges and descends strongly to altitudes below 1000 m. South of 44°N, westerly flow crossing the Cascades descends only slightly to a plateau at 1700 m.

We can verify the dominance of southwesterly flow in producing precipitation in several ways. In Fig. 2, we show, with a “wind rose,” the statistics of wind speed and direction at a point near the Oregon coast (44°N, 124°W) and an altitude of 775 hPa. The data are taken from the 6-hourly 1992–2002 European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-40 reanalysis. Only periods of time with stratiform precipitation in Oregon are included in these statistics. A mean wind direction (with rain) of about 240 true (T) is evident, with some variation that includes winds from directions 200T to 290T. Wind speeds increase strongly with altitude and vary widely from case to case. Speeds exceeding 15 m s⁻¹ are common. The strongest precipitation events are related to the strongest winds. Note the complete absence of easterly wind, verifying the common wisdom that orographic lifting in westerlies dominates the region’s precipitation.

A second verification of westerly flow dominance is shown in Fig. 3. Here we plot together the 6-hourly eastward water vapor flux density across the Oregon coast (in kg m⁻¹ s⁻¹) and the daily discharge in the Alsea River (in m³ s⁻¹). The Alsea watershed lies on the western slopes of the coastal range. The two traces correlate reasonably well (CC=0.64). There is no appreciable time lag or lead between the peaks of the two curves, but the discharge persists longer. The extreme discharge associated with the IMPROVE event on 13 and 14 December 2001 is evident.

3. Models relating water vapor flux to orographic precipitation

The simplest quantitative relationship between water vapor flux and orographic precipitation is the upslope model (UM)

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According to (1), the precipitation at each point is the vector product of the surface-level horizontal water vapor flux density F = ρq_υ0U and the terrain gradient ∇h(x, y). The derivation of (1) requires the assumptions of saturated moist neutral air column, uniform wind with height, airflow streamlines parallel to the terrain at all levels, instantaneous condensation, conversion, and fallout (Smith 1979).

As an example of (1), imagine a 15 m s⁻¹ airflow forced to rise by Oregon’s terrain. In Oregon, the Cascades rise to 2 km within 300 km of the coast, giving a mean slope of 0.007. Locally, however, some slopes approach one-to-one. We will choose an intermediate slope value of |∇h| = 1/20 = 0.05. If the sea level air temperature is 5°C, giving q_υ0 = 5.5 g kg⁻¹, and ρ = 1.0 kg m⁻³, the rain rate is P = 0.004 13 kg m⁻² s⁻¹ = 15 mm h⁻¹. This value is generally too high. As pointed out by Smith et al. (2003), the use of (1) with unsmoothed high-resolution terrain gives unreasonably large total regional precipitation; even exceeding the influx of water vapor. Several of the key assumptions in (1) may break down in unsmoothed complex terrain. One problem is that (1) “double counts” precipitation from repeated cycles of uplift. It lacks a proper memory of previous lifting events.

Another classical model of orographic precipitation is the parcel model (PM), introduced in textbooks on atmospheric thermodynamics (e.g., Wallace and Hobbs 1977). In this model, one assumes that air parcels crossing the terrain follow a pseudoadiabatic law during saturated ascent, with instantaneous condensation, conversion, and fallout. During descent or unsaturated ascent, no condensation occurs. The precipitation is given by the vertical integral of the condensation rate above each point. Computation requires a prescription of how the forced ascent decays aloft, and a “memory” of upstream lifting and drying events. The total amount of water lost from each air parcel is determined from the maximum lifting. Using the preceding example, consider a saturated parcel at sea level with T = 5°C and q_υ0 = 5.5 g kg⁻¹ of water. Along a pseudoadiabat, the value of q_υ drops to 4.2, 3.0, 1.9, and 1.1 g kg⁻¹ at pressures of 900, 800, 700, and 600 hPa, respectively. If we define the drying ratio as

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its value proceeds from 20% to 40%, 60%, and 80% as lifting continues from 1000 to 900 hPa and on to 800, 700, and 600 hPa, respectively; corresponding to altitude gains of about 1, 2, 3, and 4 km. When the precipitation from parcels is integrated vertically, the parcel model is similar to the upslope model as long as the terrain is simple and low.

The amount of lifting that occurs along the Oregon coast can be estimated from Fig. 1, showing the crest lines as a function of latitude. Using PM with the example given earlier, the drying ratio for the Cascades will approximate 40% for the low troposphere. The drying ratio for parcels aloft is poorly constrained, even for the PM, as the amount of lifting is unknown.

Both classical models (UM and PM) predict that precipitation will fall only on windward slopes, due to their assumption of instantaneous precipitation. Both models have unsatisfactory or unspecified treatments of airflow dynamics. Neither model captures scale-dependent processes.

The model we use in this paper is the linear theory (LT) of orographic precipitation proposed by Smith and Barstad (2004). This model solves for the airflow patterns using linear mountain wave theory and solves for the resulting precipitation field using a linear cloud physics representation. In this representation, ascent creates cloud water that advects downstream while converting to hydrometeors on a time scale (τ_c). Hydrometeors also advect downstream while falling to earth on a time scale (τ_f). Descent evaporates cloud water and, if the air becomes subsaturated, hydrometeors evaporate as well.

The linear model is well suited for orographic situations where air passes over complex terrain, so that only a fraction of condensed water precipitates before evaporating on lee slopes. Disadvantages of the LT model include the simplification of vertical structure by vertical integration, linearization of the fluid and cloud dynamics, and the lack of a full water budget. Far downstream, all perturbation quantities return to zero, implying a return to a saturated state with a background precipitation rate. The linear theory is more complicated than either UM or PM, but it may be more realistic in complex terrain if the drying ratio (a measure of thermodynamic nonlinearity) is less than 50%. Further description of LT is given by Smith (2003), and Barstad and Smith (2005).

The LT model can be succinctly represented by the equation

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in which the double Fourier transform of the terrain function

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is multiplied by a transfer function (3) to obtain the double Fourier transform of the precipitation field [P̂(k,l)]. The spatial pattern of precipitation is recovered from an inverse Fourier transform followed by truncation of negative values:

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In (3), C_w is a condensation coefficient depending on the surface humidity and the lapse rate; H_w is the depth of the ambient moist layer. The symbols τ_c and τ_f represent the time scales for conversion and fallout. In the case of τ_c, it describes both the rate of conversion of cloud water to hydrometeors and the rate of evaporation of hydrometeors when the air is subsaturated. The background precipitation caused by synoptic-scale uplift is P_∞. The intrinsic frequency is σ = Uk + Vl and the vertical wavenumber is the proper root of

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where N_m and f are the moist stability frequency and Coriolis parameter.

The three brackets in the denominator of (3) describe respectively: airflow dynamics, advection during conversion of cloud droplets to hydrometeors, and advection of hydrometeors during gravitational fallout. The first bracket shifts the pattern upstream while the second and third brackets shift the pattern downstream. All three brackets generally decrease the precipitation amount. The max function in (5) sets negative values equal to zero in regions of strong descent, capturing the effect of descent and drying. The transfer function (3) is sensitive to scale. Vertical motions from small horizontal scales (D < 1 km) do not penetrate substantially into the moist layer and have little influence on precipitation. Intermediate scales (D ≈ 5 km) generate lee wave clouds but little precipitation. Longer scales (D > 15 km) are increasingly efficient at producing precipitation. The LT model (3)–(6) reduces to the simple upslope model (1) when C_w = 1, the vertical motion penetrates the moist layer (mH_w ≪ 1) and the cloud conversion and fallout are instantaneous (τ_c = τ_f = 0).

To illustrate the application of the LT to Oregon terrain, we first consider a smooth one-dimensional idealization of the coastal and Cascade ranges. Referring to the actual terrain in Figs. 1 and 4a, we represented the two ranges with 750- and 1500-m Gaussian ridges, 200 km apart. Each ridge has a width scale of 30 km. The coordinate system is centered on the crest of the coastal range. East of the Cascades, the terrain forms a slowly descending plateau. As a reference run, we select ρ = 1.2 kg m⁻³, q_υ0 = 0.0044, U = 15 m s⁻¹, N_m = 0.003 s⁻¹, H_w = 3000 m, f = 10⁻⁴ s⁻¹ τ_c = τ_f = 2400 s. The resulting precipitation pattern from (3) and (5) is shown in Fig. 5, prior to adding a background value (P_∞) or truncating the negative values. Maximum values of 1.4 and 3.4 mm h⁻¹ nearly coincide with the highest terrain. Unless the background precipitation exceeded 1 mm h⁻¹ dry regions would be found downstream of the coastal range and over most of the plateau east of the Cascades. The precipitation drops to zero at a point about 75 km downstream of the crest. The dry plateau is the evaporative effect of dynamic subsidence. Several modifications to the reference run are described next.

When the Coriolis and nonhydrostatic terms are neglected in (6), the changes are too small to perceive. The horizontal scales in our “idealized Oregon” are too large for vertical acceleration effects and too small for the Coriolis effect to be important. When the coastal range is removed, the predicted precipitation over the Cascades is nearly unchanged (Fig. 5). This is so because the dynamic effect of the coastal range decays on the scale of the ridge (30 km) and the cloud effect decays on the advection scale Uτ = 15 m s⁻¹ × 2400 s = 36 km while the ridges are 200 km apart.

When the cloud delay times are reduced from 2400 to 1200 s, the change is much more significant (Fig. 5). The two precipitation maxima nearly double in magnitude and they move upstream of the ridge crest slightly. The dry zone begins 30 km downstream of the crest. The choice of the shorter τ = 1200 s amplifies the precipitation so much for this smooth high terrain that it violates the assumption of linear theory. The total precipitation in the 2400- and 1200-s runs are 140 and 240 kg m⁻¹ s⁻¹, respectively, while the estimated influx of vapor is F = ρq_υ0UH_m = (1.2)(0.0044)(15)(3000) = 216 kg m⁻¹ s⁻¹. Thus, in the 1200-s run, the total precipitation exceeds the incoming flux!

In Fig. 5, we can see some constraints on the cloud delay times. To avoid dry peaks τ must exceed about 500 s. To avoid a wet plateau, τ must not exceed about 5000 s. Below, we consider the three-dimensional and multiscale aspects of the Oregon terrain.

4. Spatial patterns of precipitation

We ran the linear model in two modes with real terrain: event mode and climate mode. In event mode, single values for the environmental values are chosen. Such a run might simulate a few hours of a rainfall event with constant environmental parameters. In climate mode, an ensemble of LT computations is run with a variety of environmental parameters. The precipitation fields for these runs are averaged together, weighted by the frequency of occurrence of this type of event. A climate run could simulate a set of storm events with altering wind direction and other factors both within and between the storms. The biggest difference between “event” and “climate” runs is the smoothness of the fields. Climate runs give smoother fields. Both types of runs are carried out with 1-km spatial resolution using the terrain shown in Fig. 4a. In a later section we compare the climate runs with ground truth fields in Figs. 4b–d. In all the runs, we used a moist layer depth of 3000 m, a moist stability frequency of 0.005 s⁻¹, and a grid increment of 1 km. Also, to reduce the number of free parameters, we set the two cloud time scales equal (τ_c = τ_f = τ).

We start with a description of how specific model parameters influence the precipitation field computed from the Oregon terrain (Figs. 6a,b,c). Figure 6a shows the linear theory precipitation field computed from (3)–(6) with a wind direction of 225T and relatively short cloud time delay (See Table 1). We note the dramatic small-scale structure in the LT field. In spite of the effect of cloud delays (τ = 600 s), there is a lot of variability on scales near 20 km. With a wind speed of 12 m s⁻¹, we anticipated that the system could respond to an orographic forcing scale as small as d = Uτ = 7.2 km. A large portion of the region is completely dry (P = 0) in this run, including some ridge crests, reflecting the quick action of precipitation and downslope evaporation in the model.

The LT runs in Figs. 6b and 6c are chosen to illustrate the sensitivities to background precipitation (P_∞) and wind direction (WD). First, the role of background precipitation is seen by comparing Fig. 6a with P_∞ = 0 with either Fig. 6b or Fig. 6c with P_∞ = 1 mm h⁻¹ (see Table 1). With a background precipitation of 1 mm h⁻¹, many of the leeside dry areas fill in. Some zero areas remain, however, downwind of the largest hills. The mean and maximum values increase (Table 1).

The precipitation patterns are quite sensitive to wind direction. Wind from the SSW (Fig. 6c) gives a prevailing east–west alignment in patterns of wet and dry. Wind from the WSW (Fig. 6b) gives a prevailing north–south pattern alignment. This anisotropic property of wet and dry zones is caused by hydrostatic gravity wave dynamics (Smith 1980). Wave components with phase lines perpendicular to the airflow penetrate more deeply aloft (6) and generate more upslope condensation (and more descent on the lee side) than oblique wave components.

Figure 6d is the climate run with τ = 1200 s. For this field, we averaged together fields from runs with WD = 200, 225, and 260T with weights 0.25, 0.5, and 0.25, respectively (see Fig. 2). We also increased the wind speed to 18 m s⁻¹, corresponding to Fig. 2. The result of all these changes is to produce a much smoother field. Only a few totally dry pixels remain. In spite of the stronger winds (WS) and higher surface temperature (T_s), the mean and maximum values of precipitation in Fig. 6d are reduced substantially compared with Figs. 6b and 6c (Table 1). The decrease in the mean precipitation by about half occurs because the increased τ allows condensed water to reach the lee slopes, increasing the downslope evaporation. We will compare this climate run with other datasets in the next section.

5. Comparison with PRISM

In this section, we compare the patterns of precipitation predicted by the linear model climate run (Fig. 6d) with annual averaged rain gauge data (Fig 4b). The interpolated rain gauge data are part of a gridded 1961–90 mean monthly and annual precipitation dataset for the conterminous United States (USDA-NRCS 1998). The dataset was developed with the well-known PRISM system designed by Daly et al. (1994, 2002. PRISM is a sophisticated data interpolation technique, based on observed point values of climatological variables, such as precipitation and temperature. It employs a moving-window regression of climate versus elevation that is calculated for each grid cell in a digital elevation model (DEM). Stations surrounding the grid cell provide data points for the regression. The heart of the model is the spatial climate knowledge base that calculates station weights upon entering the regression function. These weights are based on each station’s perceived climatological similarity to the grid cell being estimated. In addition to a distance weighting function, the knowledge base currently accounts for spatial variations in climate due to: 1) elevation, provided by a DEM; 2) terrain orientation, from topographic facet (areas of common aspect) grids; 3) orographic effectiveness (bulk and steepness) of terrain, from a terrain profile grid; 4) moisture regime, from output of a straight-line trajectory model (used mainly in data-sparse applications); 5) coastal proximity, from output of a coastal trajectory model; and 6) atmospheric layer, from an inversion height or moisture depth grid. The disadvantages of PRISM include its ad hoc nature and the number of arbitrary constants that must be specified. Bootstrap tests of PRISM show that it has better skill than kriging and cokriging methods. The version of PRISM used to produce the precipitation data used here made no use of information about wind direction and speed, the equations of airflow dynamics, or the properties of clouds.

While both models (LT and PRISM) use a DEM to construct spatial patterns of precipitation, they differ in their other inputs. PRISM uses observed climatologically averaged precipitation at station locations, as well as several other input grids (discussed earlier). The LT uses parameters describing the environment for a particular event (wind speed and direction, surface temperature and lapse rate, background nonorographic rain rate) and information about environmental variability. In some respects, the PRISM field can be treated as data to test physical models. In other respects, however, PRISM is also just a model that may have errors and inconsistencies.

When comparing PRISM and LT fields, we note that they have different units. PRISM annual precipitation (Fig. 4b) is in mm yr⁻¹ while LT is expressed in mm h⁻¹ (Fig. 6d). Over the domain shown, the two fields have mean values of 988 mm yr⁻¹ and 0.94 mm h⁻¹, respectively. The two means would agree with each other if the events represented by LT existed for a period T = 988/0.94 = 1051 h yr⁻¹ = 44 days each year.

As shown in Table 2, the correlation between LT-climate and PRISM is respectable: CC = 0.78. One pattern difference between LT and PRISM (Figs. 4b and 6d) is that LT-climate has a higher relative precipitation on the western slopes of the Cascades compared to the Coastal Range (Fig. 6).

As a caution against overinterpretation of the correlations in Table 2, we have included a field composed of each pixel’s distance from the Pacific coast (C-distance; not shown in figures or other tables). This field represents perhaps the simplest of all “models” of precipitation; namely that the precipitation decreases away from the water vapor source region of the Pacific Ocean. Some of the highest correlation magnitudes arise between this field and the various precipitation fields. The high negative correlation between a featureless C-distance field and highly detailed PRISM field, for example, suggests that the correlation measure is not responding as our eye and brain do to similar spatial details in the precipitation pattern.

Surprisingly, the terrain elevation (DEM) correlates poorly with precipitation; whether PRISM (CC = −0.32), or LT-climate (CC = −0.13). While terrain generates most of the precipitation, it does not do so simply by forcing more precipitation on the highest mountains. Both correlations are significantly negative, as can be confirmed by visual inspection of the large-scale patterns in Figs. 4a,b and 6d. We note that on the largest scale, the broad high plateaus of south central Oregon are very dry.

On the small scale, by contrast, the precipitation and terrain height correlate positively and much more closely. To illustrate this, we zoom into a small region of the coastal range between 43.65° and 44.68°N. The terrain (Fig. 7a) has lots of finescale structure. The PRISM field (Fig. 7b) is coarsely resolved, but agrees qualitatively with the smoother LT-climate field (Fig. 7c). Both precipitation patterns exhibit similarity with the terrain. In the case of linear theory (Fig. 7c), the elevation–precipitation correspondence is caused by an approximate cancellation of upstream shift caused by wave dynamics and downstream shift caused by cloud time delays. In the case of PRISM, the correspondence is partly due to the raw data and partly due to the interpolation scheme that assumes a local linear relationship between elevation and precipitation.

6. Landscape proxies for precipitation

Rain gauge data, even with PRISM interpolation, cannot provide accurate precipitation fields on small scales. No existing rain gauge network is sufficiently dense to capture the detailed precipitation patterns forced by multiscale terrain. Furthermore, most rain gauge networks are spatially biased, with most stations located in convenient valley locations. Remote sensing might be of some assistance. Long-range weather radars have frequently been used to map detailed patterns, but over mountainous terrain the radar beam is often blocked and it fails to see low-level precipitation enhancement or evaporation (Browning et al. 1974).

An alternative remote sensing approach is to use the spatial patterns of vegetation, derived from satellites, as a proxy for precipitation. A standard method is to use reflected red and near-infrared light to form a normalized difference vegetation index (NDVI; see Mather 2001). There are two known limitations to this approach. First, in regions where rainfall is plentiful, dense forests with large leaf area index may develop. These forests “saturate” the satellite-derived NDVI signal, and thus we lose sensitivity to rainfall amount. Further, in these regions, trees are not stressed by water shortage so NDVI retains little value as a rainfall proxy.

In arid regions, the vegetation density is much more sensitive to rainfall and the NDVI method is more useful. For example, the gradient in NDVI accompanying the transition from dense forest to open steppe is dramatic. Unfortunately, vegetation density may also be influenced by other climatic and soil factors. Altitude has some effect, expressed through temperature. The lower temperatures at higher altitudes reduce evaporation and stunt tree growth, effects that only partly cancel.

The primary advantage of the remote sensing approach is the potential for high spatial resolution. With 1-km MODIS imagery, the extremely sharp leeside climate gradient can be fully resolved.

A second satellite-derived proxy for precipitation is the infrared brightness temperature derived from the radiance in the infrared “window” near 10 μm. On a clear sunny summer day, the evaporation from plant leaves has a significant effect on the brightness temperature. Areas with wet microclimates appear cooler than dry areas. Forests are also cooler because of their larger roughness, avoiding a heated surface layer. Extraneous factors influencing brightness temperature include the lapse rate on mountain slopes and cool lake surfaces. Like NDVI, it provides a good view of the sharp leeside precipitation gradient.

We used a warm season MODIS satellite image of Oregon to derive these two proxies (Figs. 4c,d). Channels 1 (620–670 nm) and 2 (841–876 nm) on MODIS are used to compute the NDVI. Received radiance in MODIS channel 11 is converted to brightness temperature (BT) using the Planck function. The pixel size in the image is 1 km. The image of Oregon was taken at 1840 UTC (1040 a.m. local time) 27 June 2003 (Julian day 178). The correlation coefficients for all our fields are given in Table 2.

The most obvious feature of the satellite field is the sharp discontinuity between 121° and 122°W; adding credence to the PRISM interpretation of this feature. The minimum in the Willamette Valley and the decay of the Cascade influence south of 43°N are also evident. The LT captures these patterns too (Fig. 6d).

The apparent pattern agreement between NDVI, BT, and our various precipitation estimates is reasonably good, but some differences are apparent. The most obvious discrepancies arise from special land surface types. In southwestern Oregon, for example, a large area of clear-cut forest has low NDVI and warm BT in spite of large precipitation. On the peaks of the Cascades, permanent snow fields give low NDVI. Along the Columbia River in central Oregon and Washington, large irrigated fields give high NDVI and cool BT in spite of low precipitation. Lakes on the eastern slope of the Cascades are cool, with low NDVI. These special areas degrade the utility of our proxy-precipitation method somewhat, but it still remains qualitatively valid over most of the domain.

Another striking point of agreement between PRISM, satellite proxies, and the LT model is the extremely dry V-shaped zone east of the Cascades (Figs. 4b–d and 6d). This zone starts near 45°N, 121°W and extends northeastward and southeastward. The ability of the LT (3) to predict this pattern from the given terrain (Fig. 4a) is notable.

The correlation between PRISM and the satellite proxies are quite high: 0.70 and −0.77. The correlations between LT-climate and the satellite proxies are not as good: 0.59 and 0.61. The correlation between PRISM and LT-climate (0.78) was discussed in section 4. An important feature in all three fields is the transition to a dry climate east of the Cascades. In the PRISM and satellite fields (Figs. 4b–d), this transition lies between 121° and 122°W depending on latitude. In the τ = 1200 s LT climate run (Fig. 6d), the transition lies 25–50 km farther west, depending on which precipitation-rate contour one associates with the transition. To fit this observed feature, the cloud delay time should be increased in LT to a value between 1800 and 2400s (recall Fig. 5). Generally, we conclude that satellite data adds to our confidence in PRISM and thus encourages a fine-tuning of the cloud delay time in the linear model.

7. The Alsea watershed

As the linear model predicts precipitation patterns on a rather small scale, we can discuss the distribution and amount of precipitation in relation to a small watershed such as the Alsea River, flowing into the Pacific Ocean at Waldport. The Alsea stream gauge [United States Geological Survey (USGS) 14306500] is located upstream of Waldport at 44°23′10″N, 123°49′50″W. The gauge is shown by the circle in Fig. 7. We delineated the Alsea catchment using the 30-m-resolution National Elevation Dataset (NED) shown in Fig. 7d. Our estimate of the catchment area (856 km²) agreed with the USGS value within 0.2%.

According to the time series data in Fig. 3, the average discharge rate over the 78-day period is D = 122 m³ s⁻¹ while the average (positive) vapor flux is F = 72 kg m⁻¹ s⁻¹. The use of average flux values over several weeks reduces the effect of water storage in the basin on our rainfall estimates. We neglect any evaporation from the basin. Using these flux values, a convenient flux ratio can be defined

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With units of meters, the flux ratio (R) can be viewed as an equivalent width of the vapor flux stream. A vertical gate along the coast, 1694 m in width, will encompass an eastward water vapor flux equal to the river discharge flowing westward past the Alsea stream gauge. This flux ratio value is the target for the theory.

To predict R, two climate runs of the linear model were used; with cloud time delays of τ_c = τ_f = 600 and 1200 s. In both cases the background precipitation rate is specified as P_∞ = 1 mm h⁻¹ and the mean surface temperature is 275 K. The linear precipitation model is run with the 1-km Global 30 Arc-Second Elevation Dataset (Gtopo30) terrain. According to the simple physics incorporated in the linear model, the precipitation patterns would not be influenced by the higher resolution (NED) terrain shown in Fig. 7d. The 1200-s pattern is shown in Fig. 7b. The field is quite smooth on this scale, and shows a correlation with terrain height. The Willamette Valley in the eastern part of the figure is quite dry.

In the τ = 600 s case, the minimum, maximum, and mean rainfall rates in the catchment are 0.65, 3.0, and 1.5 mm h⁻¹. In the τ = 1200 s case, these values are 1.4, 2.3, and 1.7 mm h⁻¹. Note that the longer cloud time delay gives a smoother field and a larger mean value. The increase in the mean value with τ indicates that much of the Alsea flow is due to spillover. While the Alsea catchment lies in the coastal range, it is somewhat sheltered by high ground upstream so it relies in part on spillover.

In the climate run, the vapor flux is

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when the typical wind component perpendicular to the coast is used. Using the catchment area, and the mean precipitation rate of 1.7 mm h⁻¹ (for τ = 1200 s), the flux ratio (7) is R = 2580 m; too large by half.

At least three tuning factors are available to adjust the theory to fit the data: cloud delay time, background precipitation, and wind speed. As mentioned earlier, the average precipitation in the Alsea catchment is rather insensitive to cloud delay time, as it receives a mixture of upslope and spillover precipitation. A large change in tau would be required to correct a small offset. Alteration in the background precipitation has more effect. A reduction of P_∞ from 1 to 0.5 mm h⁻¹ would give reasonable agreement. A change in wind speed has no first-order effect as both the precipitation and the vapor flux increase; while their ratio (7) changes little. There is a second-order effect, however. A drop in the wind speed parameter would improve the prediction, as it would decrease spillover.

Accounting for evaporation from the basin would also reduce our error. Independent of the factors mentioned earlier, the loss of one-third the precipitation to evaporation would bring our LT prediction in line with the discharge measurement. In fact, an evaporative loss of only 15% is probably more reasonable (i.e., an evapotranspiration of 50 mm month⁻¹). These uncertainties prevent us from quantitatively testing the linear model to better than about ±30%. While the agreement between observed and predicted Alsea discharge is acceptable, the insensitivity to τ prevents us from using this kind of data as a constraint on τ. This situation reminds us of the need for accurate environmental parameters when testing mesoscale models.

8. Isotopic estimates of the atmospheric drying ratio

The determination of the bulk drying ratio,

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is helpful for evaluating models and setting adjustable coefficients. For steady flow, the definition (9) is equivalent to (2). In this section we use deuterium and oxygen-18 isotopic data to estimate DR.

The decrease in isotope ratio in precipitation with distance from the sea was recognized by Friedman et al. (1964) and Dansgaard (1964) among others. More recent work has verified that distance from the sea is equally important as altitude, suggesting that progressive airmass transformation is occurring (Cortes and Farvolden 1989; Zwally et al. 1998; Giovinetto et al. 1997). Other studies have used isotopes to reveal dominant cloud physics processes (Warburton and DeFelice 1986; Smith 1992).

According to Rayleigh fractionation theory (Friedman et al. 1964; Dansgaard 1964), the ratio of heavy to light isotope in condensate is given by

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where the fractionation coefficient α(T) = Esat_L/Esat_H > 1 is the ratio of vapor pressures for the light and heavy species, and R_V is the isotope ratio in the vapor. As progressive condensation proceeds, the isotope ratio of the remaining vapor becomes “lighter” according to

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where R_V0 is the initial isotope ratio in the vapor and Θ is the fraction of the vapor remaining. In (11), we have assumed that α is constant. If we apply (10) to both the initial and the end point condensation, we obtain a relationship between isotopic concentration in precipitation and the fraction of vapor remaining in the atmosphere,

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From the definition of the drying ratio (2) or (9)

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To apply this formula to orographic precipitation, we require the isotope ratio in precipitation in the extreme upwind and downwind locations. Ideally, samples for this analysis could be obtained from collectors during the rainfall event. An easier method, yielding more representative samples, is the collection of stream water. Stream water will retain the isotope ratio of rainfall if evaporation of soil water is minimal or if evaporation is nonfractionating. It is generally believed that in forested terrain, little fractionation occurs as soil water is taken up by the roots of trees. According to Kendall and Coplen (2001), “. . . if the isotopic composition of baseflow is thought to be a good representation of mean annual precipitation (Fritz 1981), then δ(¹⁸O) and δ(²H) of rivers sampled during low flow can integrate the composition of rain over the drainage areas and be useful for assessing regional patterns in precipitation related to climate.” Stream water has an advantage over precipitation as it represents a mixture of water from several events, especially if one samples “base flow” rather than peak flow.

Two river water datasets were analyzed for this project. First, a set of river samples were collected along an east–west transect during the recent period 25 June–3 July 2003. The samples were analyzed on a mass spectrometer by Iso-Analytical Ltd, in Cheshire, England, Both hydrogen and oxygen ratios were determined (Table 3).

Second, data from 11 samples reported by Friedman et al. (1964) for Oregon were reviewed (Table 4). These samples were collected in a random spatial pattern during the period September–October 1956. Only deuterium ratios are available from the earlier dataset.

In Tables 3 and 4 and in Fig. 8, the isotope data are presented as delta values

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where R_SMOW is the isotope ratio in standard mean ocean water. Both datasets show similar trends; a strong reduction in heavy isotope concentration from west to east. The recent data show a smoother trend than the 1956 data, perhaps because of the earlier season, the shorter time span, or the more regular sampling pattern. We use the westernmost and easternmost samples from the recent collection for the following calculation (Table 3). On the coast of Oregon, precipitation is characterized by deuterium concentrations of δD₀ = −51.98 corresponding to R_P0/R_SMOW = 0.948 [using (14)], while the precipitation east of the Cascades has δD₁ = −106.34 corresponding to R_P1/R_SMOW = 0.894. Note that R_P/R_P0 = (R_P/R_SMOW)(R_P0/R_SMOW). Substituting these values into (13), using α_D = 1.106 (for T = 0°C) gives

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Some added confidence in our results can be gained from the oxygen isotope data from the nine recent samples. According to Dansgaard (1964) and Friedman (1953) oxygen-18 and deuterium fractionate in a similar way to a first approximation. In Table 3, we see that the oxygen and deuterium data correlate extremely well. On the coast of Oregon, precipitation is characterized by oxygen values of δ(¹⁸O)₀ = −7.66 corresponding to R_P0/R_SMOW = 0.992 38 [using (14)], while the precipitation east of the Cascades has δ(¹⁸O)₁ = −14.27 corresponding to R_P1/R_SMOW = 0.9857. Substituting these values into (13), using α₁₈ = 1.0111 (for T = 0°C) gives

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A further check on the representativeness of our data can be done by comparison with the continental-scale river isotope maps of Kendall and Coplen (2001). Their maps were constructed from datasets that included neither the Friedman et al. (1964) nor the present data. They show a decrease in isotope ratio across Oregon of similar magnitude to ours. Their gradients are slightly weaker than ours, however, perhaps due to less dense sampling and enhanced smoothing. Their values of deuterium excess run from about −10 on the coast to −2 in eastern Oregon, a gradient we do not see (Table 3).

There are several reasons for caution concerning these estimates of drying ratio (15), (16). For example, Eqs. (10) and (12) assume that water is being condensed from a well-mixed reservoir. In the orographic precipitation system, there may be moist layers aloft that are not influenced by the forced uplift or mixed with those layers that are lifted. These “passive” layers would play no role in the drying or fractionation process. The drying ratio we compute would be relevant only to the “active” layer that feels the thermodynamic processes associated with cross-mountain airmass transformation.

Our results depend on the choice of cloud temperature: T = 0°C. A colder cloud temperature (e.g., −10° or −20°C) would give larger fractionation coefficients causing smaller drying ratios for a given isotope shift. For example, if the active cloud processes occur on the average at −10°C, the fractionation factor for hydrogen/deuterium becomes α_D = 1.123 and

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compared to DR = 0.42 from (15). The additional assumption of equilibrium fractionation has been made in (10). The error in this assumption caused by kinetic effects is discussed by Merlivat and Jouzel (1979) and Cappa et al. (2003). If kinetic effects decreased the effective fractionation coefficient during condensation by the faster diffusion of the light species, the given isotope data would imply a greater drying ratio.

Large values of drying ratio such as DR = 0.42 or 0.45 cast some doubt on the utility of the linear theory. The linear theory implicitly assumes that only a small fraction of the total water will be rained out. Values of DR = 0.1 or 0.2 would be more consistent with the model assumption of near saturation. Nonetheless, we proceed below to determine a cloud physics delay time using the linear theory and the isotope-based value of DR.

9. Estimating the cloud physics delay time

In this section, we examine the relationship between cloud delay time and drying ratio. We hope to tune the value of τ in the LT model to give a reasonable value of drying ratio. Generally, large values of τ will force orographic clouds to retain their condensed water long enough for air parcels to reach the region of leeside descent. The condensed water will then evaporate instead of precipitate. Short values of τ allow clouds to quickly precipitate their condensed water before parcels reach the lee slope. To handle the wide variety of atmospheric parameters that would occur over a rainy winter season, we do a larger ensemble calculation with 24 simulations for each specified value of τ = τ_c = τ_f ranging from 100 to 2000 s. These 24 members of the ensemble comprise four wind speeds (10, 15, 20, 25 m s⁻¹) and six wind directions (210, 230, 250, 270, 290, 310T). Ensemble members are weighted according to the wind rose shown in Fig. 2. In the derivation of DR, the LT model precipitation is integrated from the sea to 120.2°W over a latitude range from 44.02° to 45.3°N (nearly 150 km). The computational speed of the LT model allows the ensemble runs to be quickly completed. The calculation is done with P_∞ = 0. The drying ratio would be increased if positive values of P_∞ were used.

The strong influence of cloud delay time is clearly seen (Fig. 9). With no time delay (not shown) the drying ratio (DR) predicted by the linear model slightly exceeds unity. This unphysical result was expected in complex terrain such as western Oregon as the model double counts repeated uplift events (Smith et al. 2003; Smith and Barstad 2004). A short delay time of τ = 100 s gives very efficient precipitation and large DR; in the range of 0.7–0.9. A value of τ = 2000 s gives DR values mostly from 0.16 to 0.3. The delay value of about 600 s, gives DR values near to 0.4, agreeing with the values from (15), (16), DR = 0.42 and 0.45.

There are several levels of uncertainty in our inference of the cloud delay time. First, for a fixed τ, the value of DR varies with meteorological conditions. For example, faster winds give a reduced DR. The variations within the ensemble are represented by the plus and minus one standard deviation curves in Fig. 9. The second difficulty is that the linear theory breaks down for cases that give a large value of drying ratio. The assumption of small perturbation is violated.

The value of background precipitation must also be accounted for. As orographic precipitation events in Oregon usually occur within precipitating cyclonic systems, the background or “nonorographic” component of the precipitation is significant. Even in the absence of mountains, moist frontal systems transporting water vapor northward produce copious precipitation and a corresponding isotope fractionation. In mountainous regions, the orographic and nonorographic precipitation and fractionation reinforce each other. The presence of background precipitation will lift the DR curve (Fig. 9) everywhere, so for a given value of DR the inferred τ is larger. Thus, the τ = 600 s estimate must be viewed as a lower bound. Using the isotope-derived value of DR = 0.43, our estimate of τ could be increased from 600 to 1200 s or even greater. Values ranging from 800 to 1500 s appear reasonable in this context, and agree with earlier estimates (Jiang and Smith 2003; Smith 2003).

Finally, we used the sensitivity of DR to τ to characterize the active terrain scales in Oregon. According to the box model of Jiang and Smith (2003) for a single simple ridge with half-width a, the effect of two equal cloud delays on drying ratio can be represented approximately as

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where T is the time required for a parcel to cross the windward slope, that is, T = a/U. Fitting the model data in Fig. 9 to (16), gives C₁ ≈ 1 and T ≈ 1560 s. If the average wind speed is U = 15 m s⁻¹, this gives an effective terrain width of a = UT = 23.4 km. This short distance scale indicates that the relatively small-scale components of the complex Oregon terrain are playing a role in determining the drying ratio. A similar conclusion was reached by Smith et al. (2003) for the Alps.

10. Conclusions

In this paper, we used a linear theory of orographic precipitation and four datasets to investigate the precipitation gradient across the mountain ranges of western Oregon. The linear theory helps to simplify an otherwise complex problem by identifying the cloud delay times as the primary unknown control parameters. All four datasets provide opportunities to test the model and/or constrain the cloud delay times.

Several physical characteristics of the linear theory are evident when the model is run in “event mode” over the complex Oregon terrain. For horizontal scales in the range 10–30 km, the effect of condensate advection slightly dominates upstream wave tilt, so that the maximum precipitation occurs near the highest terrain, and there is some spillover. There is some anisotropy in the model fields. Wet and dry regions orient themselves across the wind vector. The tendency for these crosswind structures arises from the smaller vertical penetration of oblique wave modes. The dry lee slopes arise from strong wave-induced descent and parameterized evaporation of condensed water. On the larger scales, the precipitation falls upstream of the crests.

When the LT is run in “climate mode,” predicted large-scale patterns agree with the PRISM and satellite patterns in many respects. The coastal and Cascade ranges produce similar large precipitation amounts. The observed Willamette Valley minimum is seen. The sharp precipitation gradient east of the Cascades is captured, both in the north, where the terrain drops quickly and in the south, where the terrain forms a high plateau. Spatial patterns in the lee of the Cascades are also reproduced. These datasets generally confirm the linear model as long as the cloud delay times are kept in the broad range of say 500–5000 s. Tau values less than 500 s would give dry ridge tops. Tau values greater than 5000 s would allow too much spillover onto the dry plateaus east of the Cascades. The adjustment of τ to match the location of the sharp Cascade leeside gradient region yields a value between 1800 and 2400 s.

On the smaller scale, we examined the Alsea watershed. The agreement between LT and river discharge is good, but several uncertainties in environmental parameters limit precision. The LT model indicated a weak positive sensitivity of Alsea rainfall to cloud delay time, suggesting that in spite of its coastal range location, it received some of its precipitation from spillover. Because of the weak sensitivity, it does not provide a useful constraint on τ.

The stable isotope ratios from baseflow stream water samples provide a more quantitative way to estimate atmospheric drying ratio across the Oregon Coastal and Cascade Ranges. Using this method with oxygen-18 and deuterium we obtained DR = 43 ± 5%. From this value we can roughly deduce an average value for the cloud delay parameter in the LT model. Values in the range of 600 s or greater fit the data. The unknown rate of background precipitation limits precision.

Interestingly, a drying ratio of 43% is not so different than one would compute from a simple parcel model (PM; section 3). A smooth 2-km pseudoadiabatic lifting of sea level air would condense about 40% of the original vapor. The parcel model assumes smooth lifting with perfect precipitation efficiency while the LT model describes multiple brief lifting events each with poor efficiency. According to linear theory, the sensitivity of DR to τ is quite strong for real Oregon terrain, greater than the sensitivity we found for ideal terrain (section 3). The sensitivity implies a strong role for condensed water advection and leeside evaporation. The nature of the Oregon terrain can be characterized by the sensitivity of drying ratio (DR) to cloud time delay (τ). According to this measure, the active scale in the Oregon terrain is about 23 km.

Future attempts to estimate cloud delay times from macroscopic data are recommended, although some of the same problems encountered here should be expected. Foremost is the question of whether τ values are constant for a whole domain, and over time. Jiang and Smith (2003) suggest that nonlinearities can give a threshold character to the conversion process, leading to variations in effective cloud delay time. Also, distinguishing conversion times (τ_c) from fallout times (τ_f) is difficult without in situ cloud measurements. For high terrain, the linear theory is at the limit of its applicable range.

Several improved methods can be suggested. A dense surface rain gauge network in a region of spillover might provide a sensitive determination of τ. The discharge rates in small watersheds in pure upslope or pure spillover regions might be sensitive to τ. Monitoring the drying ratio with upstream and downstream soundings or global positioning system (GPS) integrated water sensors might provide estimates of DR and τ. In all cases, accurate estimation of the environmental conditions will be needed.