## Abstract

Foehn winds often depend on topographical features of a scale that is not sufficiently resolved in numerical models. Consequently, a successful foehn forecast has crucially depended on the experience of bench forecasters. This study provides a method for an objective, probabilistic forecast of foehn occurrence and strength, based on an operational global model (ECMWF). Because model topography differs from real topography, forecasted wind is not a reliable indicator of a foehn. Instead, using the larger-scale fingerprint of foehn from cross-barrier pressure differences and the descent of isentropes is more successful. These foehn predictors were tested over a period of 3 yr for the subgrid-scale Wipp Valley in the central Alps, which is instrumented sufficiently for objectively diagnosing the occurrence and strength of a foehn. The joint probability from pressure differences and isentropic descent is better at diagnosing a foehn from model analyses than from the distributions of the individual parameters. The larger the pressure difference and the isentropic descent, the higher the foehn probability. As wind speed and pressure gradient are directly connected by the Bernoulli equation, the cross-barrier pressure difference in the model proved to be a suitable predictor for the strength of the foehn. Despite being a small-scale weather phenomenon, the skill of a objective foehn forecast out to 3 days degrades little compared to the analysis. Afterward, the predictability decreases progressively.

## 1. Introduction

This study provides a method for an objective, probabilistic forecast of foehn occurrence and strength based on operational global numerical weather prediction (NWP) models. It is a general way for diagnosing and predicting foehn winds in topographical settings insufficiently resolved in numerical models. It can be therefore used in global models, in general circulation models (GCM), and in regional GCMs—wherever the spatial resolution of the topography or the initial state is too coarse.

Extensive investigations spanning nearly one-and-a-half centuries (e.g., Hann 1866; Ficker 1931; Schweitzer 1953; Seibert 1990; Zängl 2003) and culminating in the Mesoscale Alpine Program (MAP) in 1999 (e.g., Bougeault et al. 2001; Mayr et al. 2004; Mayr et al. 2007) have clarified the mechanisms leading to a foehn. For this article, we have adopted the WMO (1992) definition of a foehn as “wind (which is) warmed and dried by descent, in general on the lee side of a mountain.” As the foehn air plunges down in the lee of an obstacle (Fig. 1), it is warmed by adiabatic compression. Additionally, turbulent mixing occurs both at the surface and at the stable interface separating the foehn air from the flow farther aloft. Consequently, an increase in potential temperature along the surface of the lee slope is observed. The most vigorous mixing takes place where the shooting foehn air adjusts to the air mass farther downstream in the atmospheric equivalent of a hydraulic jump. Where the foehn air reaches its level of neutral buoyancy (which is not shown in Fig. 1), that is, where it becomes lighter than the downstream air mass, it separates from the slope. The separation point often coincides with the top of nocturnally formed cold pools. Especially during the onset and ending stages and downstream of hydraulic jumps, respectively, foehn wind speeds are relatively low. Details about the dynamics of foehn and its varieties are given in Armi and Mayr (2007), Mayr et al. (2007), and Richner et al. (2006). Foehn winds occur downstream of most major and minor mountain ranges in the world and sometimes bear their own regional names like “chinook” east of the North American Rockies, “bora” (or also “bura”) west of the Dinaric Alps, and “puelche” and “zonda” downstream of the Andes.

Forecast issues arise from the change of air mass as well as high wind speeds associated with foehn winds. There are applications for a forecast of foehn occurrence and of foehn strength. The higher potential temperature of foehn air can cause higher levels of air pollutants by reducing the available mixing volume as shown in Fig. 2. In locations where the foehn air does not completely reach the ground, it forms an effective lid that traps the pollutants below (Fig. 2, middle panel). This mechanism is more frequent during cold seasons. Another area affected by warmer temperatures from foehn winds are ski resorts. To minimize costs and guarantee optimal skiing conditions, temperatures need to be known in advance for artificial snow making, which is particularly widespread in Europe, and for the preparation of the slopes. Forecasts of foehn strength are needed for safety-related issues. In aviation, paragliders and hang gliders are already affected by relatively weak foehn winds, whereas larger commercial airliners are put at risk only by the strongest of foehn events or in particular topographical situations, for example, in the proximity of the airports of Juneau, Alaska (Colman and Dierking 1992; Bond et al. 2006), and Hong Kong (Shun et al. 2003). Foehn winds might not only aggravate air pollution but also drastically lower pollutant concentrations near the surface (Fig. 2, right panel). When a foehn erodes a cold pool, it will turbulently mix pollutants that were trapped in it into a much larger volume and also transport them away thus ending high-pollution periods.

A foehn is a small-scale (meso *γ*) phenomenon, which strongly depends on topographical features (e.g., Lilly and Klemp 1979; Ryan and Burch 1992; Clark et al. 1994; Gohm and Mayr 2004; Weissmann et al. 2004). When such small-scale topography is not sufficiently resolved in operational NWP models, a direct objective prediction of a foehn is impossible. This is currently arguably true for most foehn regions for predictions with global models, and for some foehn areas in the case of limited area models. Despite a nominal resolution as measured by grid distance of 3–12 km in current operational mesoscale models, the model topography is coarser: 2Δ*x* and 4Δ*x* topographical features have to be smoothed to avoid triggering disturbances from the topography that cannot be filtered by numerical diffusion (Pielke 2002). As a result, typical foehn regions (like the Wipp Valley, Austria) are only represented by small indentations. A successful simulation of a foehn in the Wipp Valley requires grid meshes on the order of 1 km and below (Gohm et al. 2004; Zängl et al. 2004).

The classical approach of objective forecasting through model output statistics (MOS) (Glahn and Lowry 1972) would sift through dozens or hundreds of potential predictors that minimize the variance to the observed value. Because the foehn is so strongly topographically forced and its underlying mechanism is well understood, another approach is to use the mesoscale fingerprint that the foehn has and which NWP models can resolve.

Bench forecasters are apt at spotting these fingerprints as well. With sufficient experience, their skill in forecasting a foehn is high—at least from anecdotal and personal experience. We are not aware of any published quantitative studies. Similarly, as far as we were able to find out, (almost) no reviewed literature concerning the *objective* prediction of foehn winds exists. Widmer (1966) developed a forecast scheme to predict (south) foehn winds in Altdorf, Rheuss-Valley (Switzerland). This scheme was simplified by Courvoisier and Gutermann (1971) for use during operational forecasting at the Swiss Met Service. Originally, the method was based on observations in order to predict foehn winds in a deterministic way for the following two days. Since 1971, it has been based on model forecasts of the gradients of the geopotential heights at 500 and 850 hPa across the Alps.

In North America, forecasting studies of foehn winds in the Rocky Mountains and the Appalachians target the prediction of damaging wind gusts in connection with downslope windstorms instead of the onset and demise of such a storm. For the eastern side of the Montana Rockies, Oard (1993) developed a regression equation to anticipate downslope windstorms at least 3–12 h in advance. This scheme is mainly based on the cross-barrier difference of the geopotential height at 700 hPa and its temporal changes. The verification of his scheme yielded good results in predicting wind gusts of more than 13 m s^{−1}. Nance and Colman (2000) tested a nonlinear, two-dimensional model to forecast downslope windstorms for seven regions in the United States. In terms of threshold prediction (high wind speeds), their tool revealed promising results. Though the false warning rate was high for some regions, a large number of observed peak gusts were met by their numerical simulations. However, they did not evaluate the model’s performance for nonwindstorm events. Schultz and Doswell (2000) analyzed lee cyclogenesis and the associated strong winds in the Rockies. They suggested watching for minima in pressure time series at upstream stations to anticipate lee cyclogenesis some hours in advance. For the Cascade Mountains in Washington, Colle and Mass (1998) investigated the sensitivity of downslope windstorms to the strength of crest-level flow, cross-barrier pressure gradient, the height of critical level (which is a region of wind reversal), and crest-level stability. They were able to show that crest-level flow and the pressure gradient have the main impact on downslope windstorms. Their focus, however, was on the simulation of observed wind storms, not on a forecast scheme.

An increasing number of numerical modeling studies of foehn reveal promising results. Even small-scale foehn structures like pressure and wind fields, gravity waves, and hydraulic jumps are successfully simulated (e.g., Zängl et al. 2004; Gohm et al. 2004). However, the resolution used in these simulations is about tenfold finer than that used in the limited area models for routine forecasts.

Although a foehn depends on small-scale topographical features, conditions on a scale larger than that of foehn regions have to be right for a foehn to occur. Choosing appropriate variables describing these conditions may thus be used to forecast a foehn. This approach is pursued in this paper. Because of the strong topographical dependence, we tested and verified the method for a particular geographical setting: the Tyrolean Wipp Valley in the central Alps.

In general, the physically based foehn classification method, which is described hereafter, allows a clear separation between foehn and no-foehn events. For transition periods only (onset or end of a foehn event), the classification may be difficult or even impossible to make because of oscillations between the foehn winds and other wind regimes. Apart from this—compared with the total foehn time—short period on the order of one hour, foehn is a binary event, for which probabilistic forecasts provide more value than traditional categorical forecasts (e.g., Katz and Murphy 1997). Therefore, the objective forecasting scheme for foehn occurrence and strength is formulated in a probabilistic way. An objective foehn forecasting system can also be used to *diagnose* foehn winds in GCMs and regional GCMs.

The paper is organized as follows. Section 2 explains the general method for the objective detection of foehn winds from *observations* and the derivation of suitable *model* diagnostics and predictors. In section 3 the objective forecasting of a foehn and its strength is demonstrated for the prediction of a south foehn in the Wipp Valley. The study concludes with a summary and discussion in section 4.

## 2. Foehn diagnostics

Although foehn winds are often accompanied by striking changes of temperature and humidity (e.g., Oard 1993), distinguishing them from other winds, especially from strong nocturnal down-valley winds but also from thunderstorm outflows, may be difficult.

### a. Direct foehn diagnostics: Small scale

The common method for detecting foehn winds from observational data is to *subjectively* analyze temporal changes of wind, temperature, and humidity by the so-called three-point method (Conrad 1936; Osmond 1941; Obenland 1956). The onset of a foehn is diagnosed, when the turning of the wind to the appropriate foehn wind direction is accompanied by an increase of the wind speed and the gustiness, as well as a rise of the temperature, and the decrease of the relative humidity. The end of a foehn event is characterized by the opposite signals. However, these signals may be faint or ambiguous, for example, if the advection of warm air replaces cooler air masses.

The objective classification method used in this paper was developed within MAP (Vergeiner et al. 2002). It exploits the fact that foehn air descends downstream of a mountain range (cf. Fig. 1), turbulently mixing in air from aloft (e.g., Mayr et al. 2004; Smith 1985; Farmer and Armi 1999; Farmer and Armi 2001; Armi and Farmer 2003) so that the potential temperature at a weather station downstream is at least as high as at a weather station at the crest. Apart from potential temperature, the appropriate foehn wind direction range and a minimum wind speed (typically 1–2 m s^{−1}) are required at both crest and valley/plain stations. The method is explained in more detail in the appendix.

### b. Proxy foehn diagnostics: Mesoscale

The topography in models from GCMs to regional NWP models differs too much from the actual topography to be able to apply the foehn classification for observations to the diagnosis of foehn in the models. Therefore, its characteristics on a scale large enough to be resolved by the models must be exploited. A necessary precondition is that some trace of the mountain range in question be in the model—however smoothened or changed compared with the actual topography. Two of these characteristics are a pressure difference across the mountain range and the descent of isentropes downstream of the crest.

Flow over the mountain range will launch gravity waves and the resulting standing wave will displace air upward upstream of the crest, causing a positive pressure anomaly there, and downward downstream of the crest, causing a negative pressure anomaly. For large mountains and ensuing nonlinear flow, the magnitude of the negative pressure anomaly is much larger than the upstream one (e.g., for reviews see Smith 1979, 1989; Gill 1982) Air masses of different temperature on both sides of the mountain range may, additionally or on their own, hydrostatically create a cross-barrier pressure difference. The mesoscale pressure difference Δ*p* controls the acceleration of the downslope flow. Together with the isentropic descent it constitutes the coarse “fingerprint” of a foehn.

While the basic pattern of the fingerprint is the same at all foehn locations all over the globe, the details are shaped by the underlying topography. We therefore want to apply the general principles to a specific region, namely the Wipp Valley in the central Alps, in order to obtain an objective forecast of a south foehn.

## 3. Objective diagnostics and forecasting of a south foehn: Central Alpine Wipp Valley example

The Wipp Valley in the central European Alps frequently experiences a south foehn. It has been a focal point for foehn research for more than 150 yr (e.g., Hann 1866; Ficker 1931; Schweitzer 1953; Vergeiner et al. 1982; Gohm and Mayr 2004). Since the field experiment of MAP in 1999 (Mayr et al. 2004), the area has been well instrumented for the objective diagnosis of actual foehn periods. With its subgrid-scale topography (width on the order of kilometers, length of approximately 30 km), the Wipp Valley is an ideal region to test the model foehn diagnostics.

### a. Forecast region and data

The Brenner Pass, as the lowest pass through the central Alpine crest (Fig. 3a), connects the northern Wipp Valley and the southern Eisack Valley. With an axis almost perpendicular to the west–east-oriented main Alpine crest, the valleys are favorable for foehn flow both from the north and the south. In this paper, we deal only with a *south* foehn. While the Eisack Valley is open to the south, where it finally runs into the plain of the Po, the Wipp Valley runs into the Inn Valley, where the mountain range north of Innsbruck, with its axis aligned in a west–east direction, builds up a large barrier for foehn flow. The vertical contraction of the Brenner Pass is staggered in the vertical: the narrow lower gap with a width of only a few hundred meters is at approximately 1400 m above mean sea level (MSL); it is topped by an upper gap at approximately 2100 m MSL, which is roughly 15 km wide. The main Alpine crest continues on both sides with an average elevation of 3100 m MSL. The cross-sectional differences between the lower and upper gap allow about 10 times more mass to flow through the upper one during foehn (Mayr et al. 2004).

A foehn was classified with 10-min averages from automatic weather stations. Because of the vertically staggered crest, two crest stations were used (cf. the appendix): Sattelberg (SAB) at the upper gap and Brenner (BRE) at the lower gap. The downstream valley station is Ellboegen (ELB). Cross-barrier pressure differences were computed between the upstream station Freienfeld (FRF) and ELB (cf. Fig. 3a).

The model applied in this study was the operational, global forecast model T511 of the European Centre for Medium-Range Weather Forecasts (ECMWF). With a horizontal mesh size of approximately 40 km, the Alps consist of one large banana-shaped mountain range without any valleys. There is a saddle or incision in the mountain range between Innsbruck (situated at the northern slope of the model orography) and Bozen (situated at the southern slope). With a height of approximately 1880 m, this saddle is flanked by two orographical maxima of approximately 2400 m in the west and 2100 m MSL in the east, which is 600–1000 m below the actual orography (Fig. 3b). As it roughly corresponds to the actual upper gap of the Brenner cross section, the first grid point south (GP − 1) of the gap grid point (GP + 0) was taken for upstream conditions, and the first grid point north (GP + 1) for downstream conditions (right-hand side in Fig. 3b). Examining values at grid points up to a distance of roughly 200 km (GP ± 5) from the crest confirmed the finding from theoretical, numerical, and observational studies that the fingerprint of the foehn is strongest close to the crest, where the strongest descent of isentropes, and thus the strongest pressure decrease, occur.

To determine the descent of isentropes downstream from the crest, data along a terrain-following model level approximately 300 m above the model topography were examined. The model surface descends approximately 200 m from GP + 0 to GP + 1. Provided that the atmosphere is stably stratified, there is a difference of ΔΘ between GP + 0 and GP + 1 along that level (Fig. 4a). While positive values (ΔΘ > 0) indicate no or only weak descent, neutral and negative values (ΔΘ ≤ 0) indicate the descent of air along that model level or from even higher (Fig. 4b).

The grid point closest to Innsbruck (GP + 1) is roughly 25 km east of and, at 1630 m MSL, nearly 1000 m higher than the actual city. The grid point GP − 1 is about 25 km east of Bozen at 1660 m, which is nearly 1400 m above the actual height. Although the actual orography is poorly represented by the model, the drag from the subgrid-scale orography is well parameterized by using standard deviation, the anisotropy, the slope, and the main orientation of the subgrid-scale orography (White 2003). This reproduces the influence of the orography on cross-barrier flow and achieves realistic magnitudes of pressure differences across the crest.

ECMWF T511 analyses and forecast data from the 0000 and 1200 UTC run with lead times of +24, +48, +72, +96, and +120 h were used. Both the horizontal and vertical resolutions of the model remained unchanged over the whole period. Data from April 2001–March 2004 were used.

### b. Direct diagnosis of foehn winds

The objective classification scheme (cf. section 2a and the appendix) was used with Sattelberg and Brenner as crest stations and Ellboegen as a downstream valley station. The threshold values for wind and potential temperature differences needed by this method are given in Table 1. A detailed description for the determination of these thresholds is given in Föst (2006). The required minimum wind speed of 2 m s^{−1} is above the level of weak nocturnal downslope flows and clearly above the starting speed of the anemometer used.

No classification was possible for 5.2% of the investigated period because of missing data. There was a foehn during 19.7% of the remaining time. A foehn event lasted, on average, 13 h. The diurnal variation of a foehn was weak because the local topography is not favorable for the buildup of nocturnal cold pools.

Compared with the good temporal resolution of the observations, the available resolution of the model analyses is much coarser: 3 h. The analyses were divided into “foehn” and “no foehn” classes, depending on whether the reference classification indicated foehn during a window of ±1 h around the model validation time. All data within the foehn periods will be called foehn members, while those outside these periods will be considered no-foehn members. Because of the coarser temporal resolution, the statistics for the 3-hourly analysis dataset differ slightly from the 10-min observations. They contain 22.9% foehn and 71.8% no-foehn members. No classification was possible for 5.3% of the analysis data because of missing observations.

If foehn winds were simplistically determined from down-valley wind speeds only (Fig. 5a), the distinction between nocturnal, thermally induced down-valley winds and weak foehn winds up to 4–5 m s^{−1} would be difficult or even impossible. The model analyses on the first grid point from the crest (GP + 1) make the discernment even more difficult because the downslope winds during no-foehn events are even more frequent and especially because there is a small but significant number (5%) of foehn events with *upslope* winds predicted by the model. Contrary to the real topography with two west–east-oriented crests, the model has only one long northern slope (cf. Fig. 3). Especially in cases where shallow cold air pools at the upstream side are leading to foehn flow through the actual incisions, thermally driven upslope winds are found along the northern slope of the model topography.

### c. Proxy diagnosis of foehn winds

Because neither a direct classification via the down-valley wind speed nor a method similar to the reference classification schemes works well with global model data, proxy foehn diagnostics (cf. section 2b) will be used.

#### 1) Observations

In the majority of our cases, foehn air reaching the downstream station (Ellboegen) originates from a level clearly above the lower crest (Brenner). This can be seen from the probability density function (PDF) of ΔΘ between the mountain reference station and the downstream station (Fig. 5b, gray lines) with the peak of the distribution between −0.5 and −3 K. During no-foehn periods, the differences in the potential temperature are mainly positive because of stable stratification. The small part of the negative differences is caused by strong surface heating leading to superadiabatic lapse rates in the immediate vicinity of the surface. The usability of a particular variable as a foehn diagnostic depends on its ability to sharply separate the distribution of foehn from the one of no foehn. A smaller region of overlap means fewer ambiguous cases. The overlap region for the potential temperature difference from crest to valley stations encompasses 15%. The pressure difference between upstream and downstream stations has an overlap area of 18%. The pressure difference distribution peaks at slightly more than 2 hPa (over a distance of approximately 34 km). Using the Bernoulli equation, which is valid only along a stream surface and without (turbulent) friction, for a rough consistency check, this pressure difference would be equivalent to an acceleration of the wind speed by 20 m s^{−1} as an upper limit (because of the violation of the assumptions), which is about three times higher than the observed values. Very few foehn events have pressure differences exceeding 8 hPa.

#### 2) Model analyses

Despite the large difference in topography between nature and the ECMWF model, the PDFs for potential temperature and pressure difference (Figs. 5b and 5c), respectively, qualitatively resemble the PDFs of the observations. The distributions are narrower for both foehn and no-foehn situations and for both variables compared with the observations. The extreme tails for foehn are especially shorter, an indication of the much smoother model topography. The overlap areas and thus the ability to clearly distinguish foehn from no foehn is comparable to observations with 16% for ΔΘ and 19% for Δ*p*. The distribution of ΔΘ extends from +5 to −5 K, with no-foehn members mainly in the positive, and the foehn members mainly in the negative range. The peak of the distribution for a foehn between −2 and −1 K (36%) shows that the air often does not descend exactly along the chosen terrain-following model level, but originates from somewhat higher at GP + 0. The distribution of Δ*p* spans from −3 to +5.5 hPa. All foehn members show the higher pressure upstream (Δ*p* > 0 hPa). The no-foehn members of Δ*p* are centered around 0 hPa.

### d. Foehn probability from a single proxy parameter

Computing the frequency distribution of the mesoscale pressure difference across the (model) crest and the descent of the isentropes downstream confirmed their expected suitability as proxy parameters for diagnosing the occurrence of foehn winds from model *analyses,* which closely resemble distributions of observations. For the reasons described at the end of section 3b, the distributions of the downstream wind speed, *υ*, have a large overlap of the flat foehn and the no-foehn curves (cf. Fig. 5a). Therefore, using the direct diagnostics of downstream wind speed proved to be less successful.

To achieve the goal of forecasting the probability of foehn occurrence, the proxy parameters were divided into discrete intervals. The probability was computed for each interval separately by calculating the ratio of foehn members to all (foehn and no foehn) members in the respective interval. The width of the interval *W* was chosen using an empirical formula for determining ideal widths for histogram bins (Scott 1979):

where *σ* is the standard deviation of the diagnostic parameter and *n* is the total number of values (Table 2).

#### 1) Observations

The ideal result would be to reach probabilities of 0% and 100%, respectively, at the ends of the curves. Actually, the probability of foehn is nearly 0% for observed differences of the potential temperature exceeding +1.5 K or pressure differences lower than −1 hPa (Fig. 6, gray lines). While the maximum attained probability with the proxy diagnostic Δ*p* > 4.5 hPa is almost 100%, it stagnates at 90% for ΔΘ ≤ −3 K. The discrepancy of 10% is caused by superadiabatic lapse rates in cases of strong surface heating.

#### 2) Model analyses

Using the analysis dataset to compute the probability of foehn winds, the curves increase for both proxy parameters from roughly 0% to almost 100% for 1.5 K ≤ ΔΘ ≤ −3 K and +0.5 hPa ≤ Δ*p* ≤ 4.5 hPa, respectively (Fig. 6, black lines). Compared with the observations, the probability curve is slightly steeper for ΔΘ. However, the maximum attained probability does not stagnate for ΔΘ ≤ −3 K, but reaches almost 100%. This means that in the model these lapse rates are also produced by a foehn and not only by surface heating. Because of the narrower distribution of the pressure difference analyses, the increase in the probability of a foehn with increasing Δ*p* is much steeper compared with the observations. For both datasets, the curves approach 100% with Δ*p* ≥ 4.5 hPa.

#### 3) Forecast and its uncertainty

In the same way as described above, probability curves were also calculated for the foehn proxy parameters of the forecast data. Because they are used for forecasting foehn winds, they will be called predictors henceforth. The probability curves of the forecast dataset show a pronounced increase in the foehn probability within a narrow range of both predictors, very similar to the use of the analysis dataset (Fig. 7). However, with increasing lead time, forecast uncertainty also increases. First, this is expressed by the up and down variations at both ends of the curves. With a decreasing number of members in the intervals at the tails of the distributions, a few incorrect predictions cause these large variations. Second, the increasing forecast uncertainty is expressed by a flattening of the curves: For both predictors, ΔΘ and Δ*p*, the curves of the forecast validity times of +24 and +48 h do not differ much from those of the analysis dataset. With forecast lead times ≥+96 h for ΔΘ and ≥+72 h for Δ*p*, the probability of a foehn decreases, especially in the range where it jumps from approximately 50%–90% in the analysis data. For example, at ΔΘ = −2 K and Δ*p* = 3 hPa, the probability of a foehn is some 20%–30% lower for the forecasts valid at +120 h than for the analysis data. In contrast, we find an increase of the probability of roughly 10%–20% points at the low ends of the curves, where the probability should be about 0%. The decrease in the probability at the top and increase at the low ends of the curves with increasing lead time are more pronounced for the predictor ΔΘ than for Δ*p*.

### e. Foehn probability from combined proxy parameters

Because the proxy parameter Δ*p* represents the driving force for a foehn and ΔΘ represents the possibility of foehn air to descend toward the surface, combining both parameters is a logical step. It also eliminates the ambiguity for the forecast probability one obtains depending on which proxy parameter is used.

Again, the probability is calculated by the ratio of the foehn members to the total number of members (foehn plus no foehn); this time within an interval of ΔΘ *and* an interval of Δ*p*. To still have a significant amount of data in the joint interval, the width of the intervals had to be set to 1 hPa for Δ*p* and 1 K for ΔΘ. As outliers can grossly distort the results, the probability was only calculated for interval combinations with more than three members, which means in our case on average more than one occurrence in a year.

#### 1) Observations

The investigation of the observational data revealed a generally strong dependence of the foehn probability on both predictors Δ*p* and ΔΘ (Fig. 8). Similar to the single predictors, the probability surges from a large region of less than 10% to a region of more than 80% for ΔΘ < 0 K and relatively large pressure differences (Δ*p* > 3.5 hPa). For a fixed value of one parameter, the foehn probability can run through several classes of the joint probability when the second predictor changes, for example, for a pressure difference of 2 hPa, the foehn probability increases from less than 10% for ΔΘ > 2.5 K to more than 80% for ΔΘ ≤ −1.5 K. In contrast, when Δ*p* is used as a single diagnostic, the foehn probability is roughly 55% at 2 hPa. This clearly shows the advantage of using a joint probability over the probability of a single diagnostic. For 1.5 hPa < Δ*p* < 3.5 hPa and ΔΘ < −5 K, the probability decreases. Similar to the single diagnostics where a stagnation of the maximum attained foehn probability was found, this is caused by few cases of strong superadiabatical lapse rates due to solar heating without a foehn.

#### 2) Model analyses

The distribution of the joint probability computed with the analysis data (Fig. 9a) qualitatively resembles that computed with the observations (Fig. 8). Though the ranges of both diagnostics are narrower, the probability sharply increases from roughly 10% (for Δ*p* < 1 hPa and ΔΘ > 1 K) to more than 80% (for Δ*p* > 3 hPa and ΔΘ < −2 K). Outliers at the boundary region (e.g., at 2 hPa/−3 K, or at 4.8 hPa/−0.2 K) are caused by foehn periods lying immediately outside of the 2-h window around the model verification time. Because the model does not produce lapse rates by solar heating similar to those actually observed, there is no decrease in the joint probability at the boundary as described above.

#### 3) Forecast and its uncertainty

Processing the forecast data in the same way as the analysis data, the first and most important result is that the pattern of the pronounced increase of the foehn probability with the decrease of ΔΘ and the increase of Δ*p* still remains (e.g., Figs. 9b–d). Even the forecasts of +120 h lead time exceed a probability of 90% for a pressure difference of more than 3.5 hPa and descending isentropes (ΔΘ < 0 K).

However, the region of pronounced increase becomes more and more blurred and bumpier with increasing lead time. The ranges of ΔΘ and Δ*p*, in which a foehn is possible at all, and especially with a high probability, become smaller for the joint probability of the forecasts. Particularly for lead times ≥+72 h, disturbances in the joint probability are found; that is, there is an increase in foehn probability despite low pressure differences and positive values of ΔΘ. For example at Δ*p* ≈ 0 hPa and ΔΘ ≈ 3.5 K the 72-h forecast (0000 UTC run) has a foehn probability of 20%–30% compared with 0%–10% in the analysis. Additionally, foehn probability decreases in spite of high pressure differences and negative values of ΔΘ. For example at Δ*p* ≈ 4.5 hPa and ΔΘ ≈ −1.5 K the 120-h forecast has a foehn probability of only 30%–40% compared with 70%–80% in the analysis. These disturbances are caused by forecast uncertainty.

Using Δ*p* and ΔΘ at given values as single predictors provides two foehn probabilities, which are not necessarily identical. Each of the two probability values is an average value, containing the appropriate occurring spectrum of the second predictor. Thus, the combination of Δ*p* and ΔΘ provides a clear improvement over using single predictors.

### f. Probabilistic forecast of foehn strength

While forecasting the occurrence of a foehn is important for issues like air quality and artificial snow making in ski areas, foehn winds also carry the potential for damage resulting from high wind speeds and associated turbulence. We provide a method for predicting maximum gusts as a way of further estimating the risk of damage. In contrast to the prediction of foehn occurrence, the prediction of foehn gusts can rely on one predictor alone. Wind speeds (via kinetic energy) and pressure difference are related, as expressed in the Bernoulli equation. Consequently, the model predictor Δ*p* was connected with wind gusts observed during foehn at the downstream station Ellboegen. The derivation of the forecast relationship for foehn strength is similar to the one for foehn occurrence described above.

#### 1) Model analyses

For each foehn member of the model analysis, the maximum foehn gust *observed* within a time window of ±1 h around the model’s validity time was determined. The probability that actual gusts will exceed a threshold level was computed as a function of pressure difference Δ*p* in the model analysis. Threshold levels of the gusts were set according to the Beaufort scale. No pressure differences larger than approximately 6 hPa occurred between the two model grid points over the investigated period of 3 yr. As Δ*p* increases from 0 to 6 hPa, the probability of foehn gusts exceeding 5 on the Beaufort scale (a fresh breeze at approximately 10 m s^{−1}) rises from less than 10% to more than 90% (Fig. 10a). Strong gales (9 on the Beaufort scale, at least 21 m s^{−1}) occur with a probability of 80% when the pressure difference is 5 hPa. For an 80% probability of storm strength (10 on the Beaufort scale), the pressure difference must be almost 6 hPa. The strongest *observed* foehn gust in these 3 yr was 29.2 m s^{−1}.

#### 2) Forecast and its uncertainty

The slope of the percentile curves for the gust probabilities as a function of model-predicted cross-barrier pressure difference remains nearly the same for the first 3 days, similar to the probabilistic forecast of foehn occurrence. For example, the probability of a gust reaching 7 on the Beaufort scale remains a little above 90% when the model predicts a pressure difference of 4 hPa. The +5-day forecast resolves the possible gusts significantly worse, the percentile curves are flatter. Interestingly, the range of pressure differences, which are predicted often enough to contour the field, is smaller for the forecasts than for the analysis.

## 4. Discussion

This work has presented a method to probabilistically diagnose or predict a binary event, foehn or no-foehn winds, as well as the expected gusts for locations where the relevant topography is subgrid scale in the prediction and general/regional circulation models. Our scheme is based on statistical postprocessing of model output. Unlike MOS, the diagnostic variables were not chosen from dozens or hundreds of possibilities. Instead, two features of foehn winds, which are its larger-scale fingerprint, were exploited: the pressure decrease downstream of the crest and the descent of the isentropes. Despite the small spatial scale of the foehn phenomenon, a reliable forecast several days in advance is possible. The skill of identifying a foehn from the larger-scale fingerprint obtained from the joint probability of cross-barrier pressure difference and isentropic descent in the ECMWF-T511 decreases only slightly between the model analysis and the 3-day forecasts. A similar level of skill is achieved for a probabilistic forecast of foehn strength represented by expected maximum gusts, which requires not two but only one predictor: the cross-barrier pressure difference.

The automated postprocessing of model output data in the described manner is simple to implement. It should therefore be possible to apply this forecasting method to regions of foehn and downslope winds all over the world. A requirement for the numerical model is that the obstacle be represented at least in a rudimentary way. Shape and maximum height may differ. The most stringent constraint is on the observational side: the need for suitably located automatic weather stations from which foehn can be diagnosed. However, with the increasing availability of mesonets and relatively low cost of automatic weather stations, that constraint can be met. This will allow a bench forecaster to get information on foehn probability and the expected gusts in the case of a foehn instead of, or in addition to, a time-consuming study of relevant parameters on weather maps, like pressure distribution, stability, wind, and solar heating. The probabilistic information for foehn occurrence and strength can also be provided to interested customers, for example, authorities responsible for keeping air quality at a healthy level, operators of ski resorts, road authorities, and aviation administrators.

## Acknowledgments

Funding for the weather stations was provided by the Austrian Science Fund (FWF 13489). We thank one of the anonymous reviewers for suggesting the addition of the prediction of foehn strength to the prediction of its occurrence.

## REFERENCES

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### APPENDIX

#### Objective Classification of Foehn

An objective method of diagnosing foehn winds that takes advantage of the current understanding of foehn mechanisms will be described here briefly. More details are given in the thesis of Vergeiner (2004). A prerequisite for this method are automatic weather stations at the crest of the obstacle and at the downwind location(s) where one wants to detect the occurrence of foehn winds. The detection principle is simple and uses wind and a nearly conserved variable. Wind at the crest has to blow in a direction that makes the lower station a downwind station. The directional range has to be chosen depending on the small-scale topographical details of the crest station but can usually be close to 180°. Because air accelerates down the slope, wind speed constraints can be kept weak. They need to be clearly above the starting speed of the anemometer, though. The directional range of the wind at the downwind station needs to be chosen by again taking into account the local topography, especially channeling. Because horizontal wind speeds at the beginning of the foehn, after hydraulic jumps, and close to the location of flow separation might be weak, only a speed threshold clearly above the starting speed of the anemometer is required.

During foehn winds, air crossing the crest and reaching the downwind location will approximately keep its potential temperature. Foehn air therefore will have to fulfill

where the subscript *c* indicates the crest, subscript *d* indicates the downwind location, and subscripts *o* indicates an offset. The offset will generally and ideally be zero. In real-world situations, however, the temperature sensors at the crest and at the downwind station might not be perfectly intercalibrated. Then, the offset has to be determined statistically from a series of foehn events. Additionally, diabatic processes might change the potential temperature of the air on its way from the crest. It will actually have increased in most foehn regions due to turbulent mixing at the surface, in hydraulic jumps, and at the interface separating the foehn layer from the flow aloft (cf. Fig. 1). Turbulence assists in fulfilling the potential temperature criterion of Eq. (A.1). Detrimental diabatic processes are phase changes of liquid or frozen water to vapor, and radiative cooling. Because of the short travel time for the air from the crest to the downwind station of typically 1 h or less, radiative cooling is negligible. When clouds are present upstream, the energy needed for evaporation will diminish the warming of the descending air. The most extreme case is that of foehn air descending and remaining saturated whence, for example, the pseudoequivalent potential temperature would have to be used instead of the dry-adiabatic potential temperature in Eq. (A.1). The moisture content of the upstream clouds is typically low enough for the effects of evaporation to be only measurable immediately downstream of the crest and a foehn wall cloud, respectively.

The most common no-foehn situation with downslope flow is during the night when radiative cooling of the surface induces a thermally driven flow down the slope. The wind criterion may be fulfilled but not the potential temperature criterion. In such a situation, the air on the downslope side is stably stratified and air descending from the crest will quickly reach its level of neutral buoyancy and separate from the slope. The potential temperature difference between the crest and downwind stations will be negative, contrary to a foehn.

The algorithm can deal with more complex terrain, like that of the Alpine Wipp Valley, where the foehn flow passes through both a very narrow lower gap and a higher upper gap, as long as the different airstreams at the crest are measured by an automatic weather station.

## Footnotes

*Corresponding author address:* Susanne Drechsel, Institute of Meteorology and Geophysics, University of Innsbruck, Innrain 52, A-6020 Innsbruck, Austria. Email: susanne.drechsel@student.uibk.ac.at