a. Mountain torque

The mountain torque results from pressure drag due to horizontal variations of pressure and surface height, and is comprised of form and wave drag. Tables 1 and 2 in Davies and Phillips (1985) provide a summary of previous investigations into the pressure drag mechanisms. No attempt is made here to evaluate the components of the pressure drag that produce the mountain torque. Rather, the components are considered to act as one force. The torque is then computed from the surface pressure field and the orography. The total zonal mountain torque on the atmosphere over a limited region is defined in this study as

View Expanded

where a is the mean radius of the earth, Δp is the zonal pressure difference across the mountains (defined as “east minus west”), and ϕ and z are latitude and height above sea level respectively. With this definition, negative T_M leads to a decrease of AAM and will occur when the pressure on the western side of the mountains is higher than on the eastern side. This sign convention is opposite of that used by White (1949).

Computation ofT_M requires that the orography and surface pressure be consistent. To achieve this compatibility, forecasts from the 16-layer version of the National Centers for Environmental Prediction (NCEP, formerly known as the National Meteorological Center) Eta model are used (Mesinger et al. 1988; Black 1988). Equation (1) is evaluated over a limited domain for three lee cyclones. The region of study, outlined in Fig. 1, roughly encompasses the Rocky Mountains of the United States.

The Eta model uses the semistaggered Arakawa-E formulation to define the horizontal grid. The alternating height and velocity points have a latitudinal spacing of (14/26)° and a longitudinal spacing of (15/26)°, where latitude and longitude have been transformed such that the equator passes through the center of the model’s domain. The average diagonal distance between grid points carrying the same variable is about 87 km in the regional domain, a 53 × 53 subset of the entire Eta grid.

The model’s vertical coordinate (Eta), a variation of standard sigma coordinates, is defined by

View Expanded

where p_ref(z) is the U.S. Standard Atmosphere pressure at z, and p_top is 100 hPa. Eta is quasi horizontal in the free atmosphere and is used to determine the vertical boundaries of the terrain. In essence, the three-dimensional grid boxes of the Eta model define the basic “building blocks” used to construct the orography. Model “blocks” or “steps” are allowed in the lowest 7 of the model’s 16 layers where layer 1 is uppermost and layer 16 is nearest sea level. The smallest element of the mountains is bounded by the velocity points surrounding a cluster of four neighboring height points that have been assigned the same elevation. The use of four height points to define a mountain block prevents the creation of “windless” valleys: isolated height points surrounded on all sides by wall velocity points.

Figure 1 indicates the model layer containing the highest block at individual height points within the domain. Table 1 lists the thickness of the mountain blocks in each layer and the “surface height” if a given layer contains the highest block. Figure 1 and Table 1, used together, indicate the model’s surface heights. For example, the highest surface heights are located in south central Colorado at 3416 m along the interface of layers 9 and 10.

Thus, the earth–atmosphere interface in the Eta model is along the horizontal and vertical faces of the highest blocks. The Δp in (1) for a given layer is computed as the pressure difference at midlayer between the east and west faces of the mountains, where adϕdz in (1) defines the area of a block’s lateral face. The torque was computed in each of the 53 rows of the limited domain and, when summed, provides an estimate of T_M over this portion of the Rocky Mountains.

Wei and Schaack (1984) suggest that computation of T_M over a limited region extend from ocean to ocean in order that the upslope and downslope cross-sectional areas be equal. Ignoring this recommendation introduces ambiguity into the calculation. Since the geographical focus of this study is the Rocky Mountains, the domain was chosen to extend from just west of California’s Pacific coast to (roughly) the Mississippi River. However, as seen in Fig. 1, the western and eastern borders of the domain are not at equal elevations. About two-thirds of the eastern border is “underground” in layer 16. To maintain equal cross-sectional areas, the vertical integration of (1) only extends from layers 15 through 10. This approach is similar to that used by Davies and Phillips (1985), who carried out their computation of mountain pressure drag on a section of the Alps for elevations above 400 m. Excluding layer 16 from the integration has the effect of reducing the magnitude of the T_M estimates. Careful examination of Fig. 1 will reveal that two of the eastern border points are also underground in layer 15. Extrapolated pressure values were used at these two points in order to include layer 15 in the vertical integration.

b. Friction torque

Friction torque results from the tangential stress air parcels experience as they move along near the earth’s surface. In this study, the zonal friction stress (τ) for an air parcel is approximated by (e.g., Bluestein 1992)

τρu²∗(3)

where ρ is the air density and u∗ is the zonal friction velocity. In the Eta model, u∗ is parameterized by (Janjic 1990; Z. Janjic 1996, personal communication)

View Expanded

where K_MS is the vertical turbulent exchange coefficient for momentum in the surface layer, u_Lm is the zonal wind component in the lowest layer above the terrain, z_Lm is the height above ground of the lowest model level, z_c is a critical height level of 2 m, and z₀ is the roughness height. The friction torque is then computed as

View Expanded

where A is 1 (−1) for east (west) surface winds. An east wind will experience an eastward torque, which represents a transfer of eastward momentum to the atmosphere, that is, an increase of AAM.

c. AAM budget

The absolute angular momentum of the atmosphere is the sum of the relative component due to zonal winds (M_R) and the planetary component due to mean rotation (M_P), which are defined as (after Salstein and Rosen 1994)

View Expanded

where g is the acceleration due to gravity, u is the zonal component of the wind, Ω is the earth’s mean angular velocity, and the limits of integration reflect the use of a limited domain. The Eta model’s forecasts of zonal wind and surface pressure are integrated across the regional domain every 3 h.

For the global atmosphere, the tendency of AAM is forced by the mountain and friction torques. For a limited atmospheric volume V, the flux of angular momentum into V across its lateral boundaries and the torques due to pressure and viscous stresses from the atmosphere surrounding V must also be considered. The flux of AAM into V across the lateral boundaries of V is given by

View Expanded

where n is normal to the lateral boundaries of V directed into V and U is the wind velocity. With the assumption that the shear stresses within the atmosphere are small, the torque due to viscous stresses from the surrounding atmosphere can be neglected, leavingthe torque due to the pressure stress of the surrounding atmosphere on V. This torque is approximated by the net zonal pressure force across V multiplied by the lever arm and is given by

View Expanded

where z_top is the height at the top of the domain and Δp is the pressure difference (east minus west) across the volume.

The budget for AAM is now expressed as

View Expanded

where RESID is the residual. The temporal resolution of the simulations is 3 h and so the lhs of (10) is a simple 3-h time difference while T_M^t, T_F^t, T_P^t, and FLUX^t are time means for the 3-h intervals.

d. Description of cases

Once cases were selected, 48-h simulations were performed initialized 24 h prior to the arrival of the observed cyclone in the lee of the Rockies. Case 1 (M88) is the 48-h simulation from 0000 UTC 10 March 1988 to 0000 UTC 12 March 1988, case 2 (N89) is the 45-h simulation from 1200 UTC 26 November 1989 to 0900 UTC 28 November 1989, and case 3 (M91) is the 48-h simulation from 1200 UTC 11 March 1991 to 1200 UTC 13 March 1991. Technical problems resulted in the loss of the hour 48 fields at NCEP for the N89 case. The absence of this time period does not impact the usefulness of the results since the N89 cyclone had already moved east of the domain before this time.

The track and trace of each simulated cyclone’s minimum mean sea level pressure (MSLP) are presented in Fig. 2. The pressure traces for the observed cyclones are also presented for comparison, although the accuracy of the forecasts is not a focus of this study as are the track and trace for a sigma-version simulation of the M91 case to be discussed later.

Figure 3 presents the MSLP field at hours 00, 12, 24, and 36 of each Eta simulation. All three cyclones begin as rather weak features that rapidly intensify by hour 24. The storms are all characterized, to varying degrees, by asymmetric pressure distributions with a trough extending equatorward from each and by high pressure along the Pacific coast. The cyclones also feature troughs to their north at some point, though this feature is not as prominent as the southern trough.

a. Angular momentum budgets

The budget of AAM from (10) for the M88 case, the most intense of the cyclones, is presented in Table 2. All values are in Hadleys, where 1 Hadley = 10¹⁸ N m. The tendencies of M_R and M_P are included in the table. The terms in the rhs of (10) for all three cases are presented graphically in Fig. 4.

Generally, the largest terms in all three cases are T_P^t and FLUX^t. In Fig. 4, their sum is plotted in lieu of the individual terms since it represents the net effect of those processes related to the use of a limited volume in the integrations. These terms would each vanish if the domain were global. However, strong resemblance between the evolution of ∂(AAM)/∂t (not shown) andT_P^t + FLUX^t for each case gives evidence of their importance on regional scales.

While T_P^t + FLUX^t is positive in all cases except early in M91, T_M^t is negative at all times and its trace in Fig. 4 is generally smoother than those of the other fields. The mountain torque is a “real” forcing mechanism for AAM. Unlike T_P and FLUX, T_M is not a result of the use of a limited domain. The average magnitude of T_M^t exceeds that of ∂(AAM)/∂t in all cases and its sign indicates the transfer of AAM to the earth. The strongest value of T_M^t is found with the M88 cyclone (−47 Hadleys) and the weakest with the N89 case (−4 Hadleys). Examination of Figs. 2 and 4 shows that the magnitude of T_M depends upon more than just the minimum MSLP of the associated lee cyclone. In each case, the occurrence of the maximum negative T_M precedes the occurrence of the cyclone’s lowest central pressure.

The T_F estimates suggest that its magnitude is smaller than that of T_M in cases of lee cyclones. The T_F is negative throughout the N89 and M91 cases, and during most of M88. However, positive T_F is computed during the latter half of the M88 case as easterly winds dominate the surface flow in the region in response to an intense pressure gradient northwest of the cyclone (see Fig. 3).

Studies of the time-mean mountain and friction torques from the 1950s through the early 1970s, as summarized by Wahr and Oort (1984), suggest that at most latitudes T_F may be as much as three to four times larger than T_M and of the same sign. However, more recent studies, summarized by Rosen (1993), indicate that the relative importance of the two torques may be a function of timescale with T_M being dominant on the timescale of days. Since this study uses Rocky Mountain lee cyclones, the very nature of the investigation (i.e., the terrain and pressure fields used in the calculations) would tend to maximize T_M relative to T_F as compared to what might be found in a time-mean budget or may be diagnosed over more uniform terrain.

The budget residuals are negative except for the first time interval in M88, are mostly positive in the M91 case, and transition from positive to negative in N89. The magnitudes of all the residuals are quite comparable except for those early in the M91 case, which are relatively large. In evaluating the integrity of a budget that uses model data, one must consider those processes that occur within the model integration but cannot be accounted for by the budget relation. The version of the Eta model that produced the data for this study applied a nonlinear fourth-order diffusion scheme to the temperature, specific humidity, and wind components in each Eta layer after each adjustment time step of 240 s (Black 1988). Additionally, the divergent component of the wind was suppressed after eachadjustment time step during the first 9 h of integration. Both of these adjustment processes will contribute to a budget residual since each impacts the evolution of the momentum field but neither is accounted for by a term in the AAM budget. The residual will also contain the mountain torque in layer 16, the viscous torque from the surrounding atmosphere, and all errors from the finite differencing and averaging techniques that use time and space resolutions different from those used in the model integration itself.

The average T_M (in Hadleys) for each of the cases, based on the 3-hourly values used to compute the time means displayed in Table 2 and Fig. 4, is −40.1 (M88), −21.5 (N89), and −23.8 (M91). Direct comparison to other studies is difficult because of the dependence upon the area of integration. Oort (1989) computed mean T_M in 5° × 5° boxes over the globe during the month of April for the years 1963–73. The peak value was −1.0 Hadleys in a box located over the Rocky Mountains between 35° and 40°N. Wolf and Smith (1987) computed an average T_M of 20.6 Hadleys on the Rockies based on 6-hourly calculations for the period from 1 December 1982 to 28 February 1983. Their value is positive since this period was characterized by persistent high pressure over the plains east of the Rockies combined with a series of strong Pacific cyclones that lashed California. Wei and Schaack (1984) computed monthly averaged T_M for the months of January, April, July, and October 1979 and obtained values of −136.7, −143.8, −141.1, and −141.3 Hadleys, respectively, over the western half of North America. Newton (1971) also examined these midseason months but used aerological atlases as his data source. He obtained values of 8, 2, 3, and 8 Hadleys, respectively, over all of North America and Greenland. Newton notes in his study that while the net torque was positive, the Rockies alone exert a momentum drain in midlatitudes.

b. Distribution of the mountain torque

The results of the AAM budgets suggest that the diagnostics provide reasonable estimates with which to examine the spatial distribution of T_M. Meridional time sections of zonal T_M per degree latitude are presented in Fig. 5. These values exclude the zonal component of the model-relative meridional T_M. The decision to exclude was based upon the lack of a clear technique for assigning this component to a specific row or rows. The volume integral of this component is small relative to the model-relative zonal T_M and thus its absence does not diminish the usefulness of Fig. 5. The torque is plotted as a function of row and hour for each simulation. The rows are not coincident with earth-relative latitude but rather are model relative. For reference, Fig. 5d depicts several rows in the domain to assist in interpreting the ordinate in Figs. 5a–c. The small open circles indicate the row in which the minimum MSLP is found at each time.

The magnitude of T_M is dependent upon the strength of both the windward anticyclone and the leeward cyclone, and upon their relative positions across the terrain. In the M88 case, the high–low pressure couplet is oriented west–east across the terrain at hour 00 (Fig. 3) but has begun to shift to a northwest–southeast orientation byhour 12. This factor alone acts to enhance the torque across the northern portion of the domain. However, as seen in Fig. 5a, T_M grows more quickly to the south of the MSLP minimum than to its north, a result of the leeward pressure trough south of the cyclone and the slight southward movement of the storm into the lee of the highest terrain. Figure 6 is a profile, or silhouette, view of the Eta model terrain looking from the western boundary of the volume toward the east. During the latter half of the M88 case, the maximum negative T_M is located in rows 30 and 31 that coincide with the northern edge of the highest surface heights in Fig. 6. Note that while the maximum negative value of T_M in Fig. 5a occurs at hour 36, 3 h after the cyclone reaches its lowest central pressure (see Fig. 2), the maximum negative T_M^t for the entire domain occurs between hours 21 and 24 (Fig. 4).

The N89 case (Fig. 5b) is characterized by a more symmetric distribution of T_M, a result of the west–east orientation of the pressure couplet throughout the simulation. However, the maximum negative T_M in Fig. 5b is found to the south of the location of the MSLP minimum at most times. The eastward track of the N89 MSLP center in Fig. 2 is north of those rows containing the highest mountains but the presence of the broad trough south of the cyclone (see Fig. 3) enhances the torque across the steepest orography over what would be found with a cyclone whose pressure distribution was more symmetric. The maximum negative T_M for the entire domain occurs between hours 18 and 21 in Fig. 4, nearly concurrent with the maximum row value at hour 18 in Fig. 5b. The lowest MSLP, however, occurs later at hour 39 after the cyclone’s center has moved east of the domain. The rapid eastward movement of the low is reflected by the weakening torque in Fig. 5b after hour 24 as high pressure builds into the lee region in the wake of the cyclone.

Weak positive T_M is diagnosed in the M91 case over the northern portion of the domain (Fig. 5c). This results from a Pacific cyclone over the northwestern quadrant of the domain and an anticyclone in the northeastern quadrant. The Pacific low gradually fills, and negative T_M is diagnosed everywhere after hour 42. The maximum negative row value of T_M (Fig. 5c), the maximum negative time average volume integral of T_M (Fig. 4), and the occurrence of the lowest MSLP (Fig. 2) are all nearly concurrent in this case. At each time, the maximum negative T_M is found in row 30, 31, or 32, that is, across the northern portion of the highest orography and in very close proximity to the row where the MSLP minimum is located.

Equation (2) indicates that η is simply a generalization of σ and setting η_S = 1 allows the Eta model to be run in σ coordinates with all predictive equations and surface heightsessentially the same as for the η version. A σ version of the MSLP fields was obtained for the M91 case. The MSLP track and trace for this case are included in Fig. 2. Differences in the path and depth of the cyclone become apparent between the simulations, suggesting that the evolving pressure field and thus the distribution of T_M about a lee cyclone are sensitive to the form of the orography. Bates (1990) previously demonstrated this dependence by comparing mountain–no mountain simulations for a cyclone passing over the Rockies.

The magnitude of T_M is dependent upon the interaction of three factors: a) the pressure difference across the orography, b) the height of the orography, and c) the lever arm to the earth’s axis of rotation. The first two factors are of primary importance to T_M, while the third is secondary in a regional domain where the variation in lever arm is somewhat limited. In each of the cases examined, the cyclone is characterized by an asymmetric pressure distribution with a trough extending to the south of the low center. This feature has been noted in other investigations of Rocky Mountain lee cyclones. For example, Fawcett and Saylor’s (1965) composite of 21 Colorado cyclones clearly depicts a strong pressure gradient to the north of these features with a pressure trough extending equatorward. On the planetary scale, the trough minimizes the surface pressure in the lee of some of the highest orography in the Rocky Mountain chain and serves to maximize T_M relative to that which would occur with a symmetric lee cyclone whenever the pressure is relatively high on the windward versus the leeward mountain slopes. Note in Fig. 5 that the meridional distribution of T_M is fairly symmetric about the maximum value. This is true even though the height of the orography in Fig. 6 decreases more rapidly to the south than to the north of the location of maximum torque, which is generally found across the northern edge of the highest orography. The effect on T_M of the decreasing terrain heights south of the cyclone is largely balanced by an enhanced pressure difference across the orography in association with the lee trough and the greater lever arm on which the net force acts. In the context of the planetary balance of AAM, this distribution increases the pressure difference across the earth’s orography where the lever arm is relatively great, thus maximizing the mountain pressure torque associated with a given lee cyclone.

c. Effect of a Boulder wind storm

A subgrid-scale process of note in the lee of the Rockies is the severe downslope wind storm, such as is observed in Boulder, Colorado. Brinkmann’s (1974) climatology of 20 such events noted that at the storms’ peak, mean hourly winds in Boulder were principally westerly at 20 m s⁻¹. The average pressure difference across the mountains west of Boulder was about 8 hPa, which produced a mean torque of −0.424 Hadleys at the latitude of Boulder. With the simple assumption that all other factors would remain constant, this would enhance the regional T_M estimates at each time period by an average of 1% (M88), 3% (N89), and 2% (M91). The west winds would also contribute to negative T_F. To estimate this contribution, T_F was recomputed at the four grid points closest to Boulder in the Eta model by substituting 20 m s⁻¹ for u_LM in (4) if the modelvalue was less. The T_F over the area bounded by these four grid points was enhanced by an average of 308% (M88), 140% (N89), and 274% (M91) at each time. Once again assuming that all other factors did not change, this would enhance the regional T_F estimates at each time period by an average of 0.7% (M88), 0.2% (N89), and 0.4% (M91).

d. Comparison of regional T_M to global AAM

Hemispheric and global AAM series, calculated from the NCEP/National Center for Atmospheric Research Reanalysis Project (Kalnay et al. 1996), were obtained to help place the results of this study in perspective. The tendency of global AAM calculated from 6-hourly data is presented in Fig. 7 for March 1988. Recall that the M88 case covers 10–11 March 1988. The raw tendencies are plotted in the figure as well as a distance-weighted least squares (DWLS) fit to the values. The DWLS line suggests that the tendency of global AAM was negative during most of the first half of March 1988 with the most negative tendencies being concurrent with the M88 cyclone.

The average tendencies of global AAM, Northern Hemispheric AAM, and Northern Hemispheric M_R during each simulation as calculated from the 6-hourly NCEP/NCAR data are presented in Table 3. Two average values are listed in the table for each of the tendencies; the first is based upon a DWLS fit to the tendencies, while the value in parentheses is the average of the actual tendencies. Also presented are the average and peak values of regional T_M based on the 3-hourly values used to compute the time means in Table 2 and Fig. 4. The average and peak T_M are comparable in magnitude and equal in sign to all the tendencies in the M88 case, the most intense of the three cyclones examined. The Northern Hemispheric tendencies of AAM and M_R correspond fairly well to T_M in the weaker M91 case, but all tendencies for the N89 case are opposite in sign to the N89 average and peak T_M values. The apparent lack of any qualitative agreement between the forcing and the global response in the N89 case may be related to the swift passage of the cyclone through the lee of the Rockies.

Swinbank (1985) has suggested that, globally, T_M may dominate AAM variability on short timescales. Madden and Speth (1995) note that on timescales of over 5 days about 70% of the variation in global AAM is explained by the global T_M in a 396-day period, including the M88 case, and that during March 1988 the largest change in T_M in the 15.1° to 55.5°N band is over the Rocky Mountain region. They do, however, note that the overall coherence between T_M and AAM falls off considerably at periods shorter than five days.

Simple linear regression indicates that, for the M88 case, 44% of the variability in regional AAM is explained by the regional T_M. However, linear regression may misrepresent the actual relationship since T_M would lead AAM by one-quarter of a cycle if it were the principal forcing mechanism. While the relationship between regional T_M and AAM displays linear characteristics (Pearson correlation of 0.66), the nonlinear relation between regional T_M and global AAM in the M88 case suggests a lagged response to the forcing. The brevity of the Eta simulations placeslimitations on spectral analysis of the relation between the regional T_M and the global AAM. A more rigorous, quantitative evaluation of this relation is required before the cause of the sharp cusp in AAM tendency in Fig. 7 can be attributed to the Rockies. Wavelet analysis may prove most promising in this regard and is the basis for further work on the M88 case.