## 1. Introduction

**Q**-vector partitioning is a useful tool in the study of the physics of frontogenesis and vertical motion, since that partitioning helps isolate processes and structures of meteorological interest that are difficult to identify by only examining a “total” **Q**-vector display. A standard procedure is to decompose **Q** in a natural coordinate system that follows the isotherms. This system, first used in the seminal paper of Hoskins et al. (1978), allows the separate evaluation of the geostrophically forced intensification and rate of turning of the horizontal thermal gradient. The intensification is measured by the cross-isotherm component of **Q**, while the rate of turning is measured by the along-isotherm component of **Q**.

Keyser et al. (1992) found that the decomposition of **Q** into along- and cross-isotherm components leads to an interesting scale separation of the vertical motion pattern associated with a baroclinic disturbance. While the distribution of the **Q** component normal to the isotherms has the scale of a frontal zone and exhibits a banded structure, the **Q** component parallel to the isotherms shows a synoptic-scale distribution and a cellular structure. This **Q** partition is widely used in the synoptic literature (see, e.g., Kurz 1992; Barnes and Colman 1993, 1994). Davies-Jones (1991) obtained a **Q**-vector partition that has the notable property of being “intrinsic”—that is independent of any reference system. One of the components of that partition, **Q**_{S}, previously found by Hoskins and Pedder (1980), is parallel to the isotherms and is associated with the vorticity advection by the thermal wind. The other component, **Q**_{Λ}, is related to thermal deformation. Schär and Wernli (1993) use a partition that is essentially the same of that of Davies-Jones.

In this paper, a new partitioning is proposed that consists of splitting **Q** into along- and cross-isohypse components. Specifically, the **Q** vector is decomposed in the natural coordinate system that follows the geostrophic wind. The rationale for this choice lies in the possibility of relating the **Q** vector at any given point to the geometrical and kinematical properties of the geostrophic flow in the neighborhood of that point. This relationship, unobtainable from the previously mentioned modes of partitioning, allows a quantitative evaluation of conventional idealized models of vertical motion forcing, even for real synoptic situations. These idealized models have so far only been used for qualitative discussions.

One of those idealized models describes the so-called curvature effect according to which a downstream increase (decrease) in the cyclonic curvature of the isohypses induces subsidence (ascent). This model gives an account of the zones of alternate ascent and descent accompanying the waves in the westerlies. (Bjerknes and Holmboe 1944). Another conventional idealized model represents the effect of confluence and diffluence of the geostrophic wind on the vertical motion and constitutes the physical basis for the “four quadrant” distribution of vertical velocity around a jet streak (Riehl et al. 1952).

While the two idealized models just mentioned are sufficient to explain the geostrophically forced vertical velocity in a strictly equivalent barotropic atmosphere, other forcing mechanisms must be added when there is a significant intersection of isotherms and isohypses on an isobaric surface. One of those mechanisms is the alongflow stretching/contraction of isotherm spacing, a process that always operates in the presence of geostrophic confluence/diffluence. A second mechanism to be added when the atmosphere is not equivalent barotropic is the thermal advection by the horizontal geostrophic shear [or simply “shear advection,” one of the designations proposed in Keyser and Pecnick (1985)]. The latter mechanism was introduced in the literature by Eliassen (1962) and has been widely utilized in the field of upper front dynamics, but not in the analysis of synoptic-scale phenomena. It explains the subsidence (ascent) associated with cold (warm) advection along a jet axis due to differential lateral cooling (warming) and was proposed by Shapiro (1981) as a crucial factor in the inception of tropopause folds that penetrate deep in the troposphere. This has been supported by the study of Keyser and Pecnick (1985).

It should be noticed that the decomposition of **Q** into along- and cross-isohypse components is not Galilean invariant. In fact, the application of a constant geostrophic wind alters the geometry of the geopotential field (a low, e.g., can become a trough). Therefore, both the intensity and the direction of each **Q** component in a natural coordinate system based on the isohypses is also modified by the application of a constant geostrophic wind. However, the lack of Galilean invariance does not affect the diagnostic value of this decomposition.

In section 2, the **Q**-vector partitioning in the natural coordinate system based on the geostrophic wind is explained and it is shown that each of the resulting **Q** components is associated with one of the idealized models described above. The new **Q**-vector partitioning is applied to the gridded analysis of a real synoptic situation in section 3. Finally, section 4 presents a summary of the main results and suggests possible extensions of the present study.

## 2. Q-vector partitioning

**Q**-vector dependence on the geostrophic wind and thermal fields [Hoskins et al. (1978), Eq. (6)] is adapted here to standard isobaric coordinates as

**Q**

**v****∇**_{g}

**∇***α,*

*α*is the specific volume and

**v**

_{g}=

**k**×

**∇***ϕ*/

*f*

_{0}is the geostrophic wind vector, with

**k**being the vertical unit vector,

*ϕ*the isobaric geopotential, and

*f*

_{0}a reference Coriolis parameter. In (2.1) and throughout the rest of this paper, all horizontal and time derivatives without subscript are on an isobaric surface.

**Q**on the fields of

*ϕ*and

*α*also involves the kinematic constraint of nondivergence on

**v**

_{g}, namely,

**∇****v**

_{g}

This constraint, a direct result of the assumption that the Coriolis parameter is constant, is a reasonable approximation for a study restricted to vertical motion only [see Hoskins et al. (1978), at the end of their section 4]. It should be noted that the last constraint is not a good approximation at the time of inferring ageostrophic winds, as pointed out by Blackburn (1985) and also by Lim et al. (1991).

*ω,*the vertical velocity in pressure coordinates, is (Sanders and Hoskins 1990) where

*p*denotes pressure,

*σ*≡ −

*α*

_{0}Θ

^{−1}∂Θ/∂

*p*is a stability parameter, and

*α*

_{0}and Θ are, respectively, a reference state specific volume and potential temperature, each depending on pressure only. The rhs of (2.3), twice the convergence of the

**Q**vector, is the geostrophic forcing of

*ω*that under fairly general conditions is associated with ascent (subsidence) when positive (negative). For brevity, the geostrophic forcing of omega will be called

*F*; that is,

*F*

**∇****Q**

**Q**can be readily computed by expanding (2.1) in arbitrarily oriented, rectangular Cartesian coordinates

*x*and

*y.*The result of the expansion is [see Hoskins and Pedder (1980), their Eq. (5)] where

**i**and

**j**are the unit vectors in the

*x*and

*y*directions and

*u*

_{g}and

*υ*

_{g}are the

*x*and

*y*components of the geostrophic wind.

*x*axis parallel to the isotherms and the

*y*axis pointing to the thermal gradient. With this convention, Eq. (2.5) becomes The first term of (2.7) represents the rate of turning of the horizontal thermal gradient by the geostrophic wind;it is parallel to the isotherms and points toward increasing values of thermal advection. The second term of (2.7) describes the strengthening/weakening of the horizontal thermal gradient by the geostrophic wind; it is normal to the isotherms and points to the warm air for strengthening (frontogenesis case) and to the cold air for weakening (frontolysis case).

*s*and its associated unit vector,

**t**. The axis normal to the geopotential contour lines is denoted by

*n,*and its unit vector by

**n**that points toward a direction such that the triplet (

**t, n, k**) is right handed. Note that

**k**is the vertical unit vector. With this notation, the Cartesian expansion of (2.5) for

**Q**is transformed into where

*s*

_{g}, the speed of the geostrophic wind, replaces

*u*

_{g}of (2.5). The partial derivatives of

*υ*

_{g}in (2.5) are replaced in (2.8) by curvature terms, namely, (see Petterssen 1956, section 2.8). Here

*K*

_{s}is the curvature of the geopotential contour lines and

*K*

_{n}is the“normal” curvature of the geopotential contour lines—that is, the curvature of the lines orthogonal to the geopotential contours.

It is noted that *K*_{s} > 0 for counterclockwise motion and *K*_{s} < 0 for clockwise motion. The curvature *K*_{s} is“cyclonic” or “anticyclonic” if its sign is the same or opposite, respectively, to that of the Coriolis parameter *f*_{0}. The normal curvature *K*_{n} is positive for diffluence and negative for confluence.

It should be emphasized that (2.8) is identical to (2.5) but viewed from the perspective of a natural coordinate system. In this framework, the components of **Q** have a particular meaning that will be elucidated right below.

**Q**. The rhs of (2.10) originates from the application of (2.9), and indicates that the alongstream stretching (contraction) is invariably accompanied by confluence (diffluence). The magnitude of

**Q**

_{alst}is the value of the advection of

*α*by the geostrophic alongstream stretching. Its direction is along the geopotential contours, pointing downstream for warm (cold) advection and diffluence (confluence) and upstream for cold (warm) advection and confluence (diffluence). As a rule of thumb,

**Q**

_{alst}always points to the warm (cold) air in the case of diffluence (confluence).

**Q**

_{curv}is a generalization to an arbitrarily oriented thermal gradient of the same vector found in Sanders and Hoskins (1990) and depicted in their Fig. 4. It can be inferred from (2.11) that if

*α*increases (decreases) in the direction of

**∇***ϕ*[i.e., if the geostrophic flow is positively (negatively) sheared with height],

**Q**

_{curv}points downstream (upstream) for cyclonic and upstream (downstream) for anticyclonic curvature.

**Q**. It is depicted in Fig. 1c, where the shear is recognized by the nonuniform spacing of straight parallel isohypses. Here

**Q**

_{shdv}is normal to the isohypses. Its direction is the same (opposite) of

**∇***ϕ*for warm (cold) advection and cyclonic shear, and is the opposite (the same) of

**∇***ϕ*for warm (cold) advection and anticyclonic shear. As a rule of thumb,

**Q**

_{shdv}always points toward increasing (decreasing) warm (cold) advection. Here

**Q**

_{shdv}isolates the physical process by which thermal advection by the horizontal wind shear generates vertical motion.

**Q**. Here

**Q**

_{crst}is always normal to the isohypses. It can be deduced from (2.13) that if

*α*increases (decreases) in the direction of

**∇***ϕ*[i.e., if the geostrophic wind is positively (negatively) sheared with height],

**Q**

_{crst}has the same (opposite) direction of

**∇***ϕ*for confluence (diffluence).

**Q**components occurs. Mutual cancellation is a real possibility since the projection of

**Q**on either direction

**t**or

**n**comes from two

**Q**components, as indicated by (2.8). These components oppose each other and even can cancel each other out, for certain distributions of

*ϕ*and

*α.*These distributions can be found by applying the following two formulas, whose derivation is given in appendix A: which represents the projection of

**Q**parallel to the

*ϕ*-contour line, and the normal component to the same contour lines:

In the last two expressions, *J* is the usual symbol for the Jacobian. In (2.14), *β* is the angle between the geostrophic wind and a reference geographical line, the local latitude circle, for instance. From (2.14) it can be seen that in regions where isotherms and isogons of **v**_{g} are parallel to each other, the alongflow stretching component of **Q** is exactly opposite to the curvature component, as illustrated in Fig. 2a. In addition, it can be inferred that in areas where isotherms and isotachs of **v**_{g} are parallel to each other, the cross-flow stretching component of **Q** is exactly opposite to the shear advection component. This is illustrated in Fig. 2b.

**Q**and represented by

**Q**

_{st}. Here

**Q**

_{st}can be determined directly from the knowledge of

*ϕ*and

*α*fields by applying (2.10), (2.13), and (2.9). The result is

The magnitude of **Q**_{st} is the product of the magnitudes of the stretching deformation and of the gradient of *α.* The direction of **Q**_{st} at a given point is illustrated in Fig. 3 and can be inferred from the fields of *ϕ* and *α* by noticing that the vector on the rhs of (2.16) has the direction of the mirror image of **∇***α* (−**∇***α*) about the isohypse passing for that point, for diffluence (confluence).

*F*of the vertical velocity, as indicated in (2.4), is twice the convergence of

**Q**. Because this is a linear operation on

**Q**,

*F*is given by the superposition of the partial forcings

*F*

_{alst},

*F*

_{curv},

*F*

_{shdv}, and

*F*

_{crst}; the alongflow stretching, curvature, shear advection, and cross-flow stretching components, respectively. The analytic expressions for those partial forcings are where the identities

**·**

**∇****t**=

*K*

_{n}and

**·**

**∇****n**= −

*K*

_{s}have been applied. The quantity

*ζ*

_{g}is the geostrophic relative vorticity.

The practical computation of each component of **Q** and its respective forcing is explained in appendix B.

## 3. Application to a real weather situation

In this section, the preceding method of partitioning the **Q** vector is applied to a real synoptic situation. The **Q** vector is computed from the gridded analysis provided by the European Centre for Medium-Range Weather Forecasts (ECMWF) valid for 0000 UTC 10 January 1979. The analysis is at 2° lat × 2.5° long resolution, which is coarser than the original. The time/date chosen is arbitrary as well as the region of analysis. In this example, the eastern part of North America and western Atlantic waters are shown. This is a region of pronounced baroclinicity, where the **Q** vectors are most meaningful.

Following a brief synoptic overview, the description of the total geostrophic forcing of the vertical motion at 700 mb will be given. The choice of 700 mb is based on the observation that this pressure surface reflects both the upper patterns and the structure of the low-level synoptic systems. No effort has been made to invert the *ω* equation (2.3), since the discussion involving vertical motion will be centered only on the geostrophic forcing of *ω,* not on *ω* itself. The degree of realism of the geostrophic forcing will be subjectively assessed by contrasting the forcing with the model output of vertical motion and by applying the usual assumption that the sign of *ω* opposes that of its forcing; that is, that a positive (negative) forcing induces ascent (subsidence). This comparison between the analyzed vertical motion and its geostrophic forcing will be followed by a detailed description of the **Q**-vector components and their respective forcing contributions for this synoptic situation.

### a. Synoptic overview

The meteorological situation in the lower half of the troposphere over eastern North America and adjacent Atlantic waters at 0000 UTC 10 January 1979 is shown in Fig. 4. The fields of sea level pressure and 500 mb–1000-mb thickness (Fig. 4a) show an incipient frontal low, with central pressure less than 1004 mb east of Newfoundland and a weaker, quasi-barotropic depression southeast of Hudson Bay, with pressure values greater than 1012 mb. These lows are separated by a ridge of high pressure that crosses the St. Lawrence River. The frontal low is embedded in a southwesterly current at 500 mb, associated with a longwave trough over the eastern Great Lakes and a longwave ridge south of Greenland.

The 700-mb geopotential and thermal fields (Fig. 4c) confirm that the longwave pattern is well represented at this pressure level, and that signatures of surface systems are also present. These include the shortwave trough, which is related to the frontal low and the thermal ridge over eastern Canada, which corresponds to the warm sector of the frontal low, and the high pressure ridge that crosses St. Lawrence River. The thermal ridge separates cold front–related cold advection over eastern Canada and warm front–related warm advection over the Davis Strait. The isotachs of the geostrophic wind (Fig. 4c) depict two jet streaks. The jet streak centered over southeastern Newfoundland exceeds 35 m s^{−1}, is located at the base of the shortwave trough, and is embedded in the cold advection region described above. In this jet streak, a remarkable asymmetry is noted between the well-defined entrance and a rather loose exit that exhibits weak diffluence. The thermal distribution is in turn asymmetric about the jet axis since the bulk of the horizontal *α*’s variation is concentrated on the cyclonic-shear side of the Newfoundland jet. The other jet streak, with a maximum wind speed exceeding 25 m s^{−1}, is located slightly downstream of the axis of a thermal trough. Warm air advection occurs at its entrance and cold air advection occurs at its exit.

### b. Comparison of geostrophic forcing and analyzed vertical motion

The forcing *F* is computed from (2.4). The zonal and meridional derivatives of *α* and those of the geostrophic wind components *u*_{g} and *υ*_{g} are approximated by centered finite differences. The data used to compute *F,* in units of 10^{−18} m kg^{−1} s^{−1}, are the gridded analysis of isobaric height and temperature and the Coriolis parameter at 45°N. Figure 5a shows the geostrophic forcing of *ω* at 700 mb. Ascent is forced over the southwestern Labrador Sea, with a maximum of 20 units. Forced subsidence is distributed over a larger area that extends between New England and Newfoundland with a minimum of −18 units. Offshore from Nova Scotia, a region of positive forcing is found with values less than 10 units. Finally, a pair of forcing centers of opposite sign extends between Lake Huron and Quebec.

The model-computed vertical velocity is depicted in Fig. 5b. The centers of vertical motion forcing described above can be recognized in this figure. Despite obvious differences in extent and position between the the analyzed upward and downward motion and the respective zones of positive and negative geostrophic forcing, it is apparent that quasigeostrophic processes constitute a significant component of the dynamics of the primitive equation model from which the analyzed *ω* has been obtained.

### c. The* Q*-vector analysis

Typical values of the magnitude of **Q** are of the order of 10^{−12} m^{2} kg^{−1} s^{−1} and the associated values of the forcing −2** ∇**·

**Q**are of the order of 10

^{−18}m kg

^{−1}s

^{−1}. The geostrophic forcing

*F*(Fig. 5a) is a result of the

**Q**-vector distribution (Fig. 6). It can be seen that areas of divergence (convergence) of

**Q**in Fig. 6 corresponds to areas of forced subsidence (ascent) in Fig. 5.

Figure 7a shows the pattern of the 700-mb alongstream component **Q**_{alst} (presented on a background of 700-mb isohypses, *α*-contour lines, and a shaded representation of the geostrophic speed field to aid interpretation). Significant values of **Q**_{alst}, whose magnitude is the advection of *α* by the geostrophic alongstream stretching, can be seen over the Gulf of Saint Lawrence. In this region, the geostrophic wind speed *s*_{g} increases downstream while crossing *α* isopleths. The effect of **Q**_{alst} is clearly frontolytic and for that reason points toward colder air. The partial forcing −2** ∇**·

**Q**

_{alst}(Fig. 8a) is very weak in relation to the total forcing and promotes ascent over the Gulf of St. Lawrence and subsidence over Newfoundland.

Figure 7b presents the distribution of the 700-mb curvature component **Q**_{curv}. The background fields are *ϕ* and *α* at 700 mb. Since ∂*ϕ*/∂*n* has the same sign as ∂*α*/∂*n,* **Q**_{curv} is parallel to the geostrophic wind where the curvature of the isohypse is cyclonic, and is directed opposite to **v**_{g} where the curvature is anticyclonic. As a result, convergence of **Q**_{curv}—that is, forcing of ascending (descending) motion occurs between a trough and its downstream (upstream) ridge—this result conforms to the classical theory (Bjerknes and Holmboe 1944) and is confirmed by inspection of Fig. 8b.

The field of the 700-mb shear advection component **Q**_{shdv} is shown in Fig. 7c, and its contribution to *ω*’s forcing is given in Fig. 8c. In Fig. 7c, two regimes of **Q**_{shdv} are present on the cyclonic-shear side of the Newfoundland jet streak. These regimes are separated by the axis of the thermal ridge. Southwest of this ridge, there is cold advection diminishing northwestward, which is the direction of **Q**_{shdv}. To the northeast of the thermal ridge, warm advection increasing southeastward prevails, giving the direction of **Q**_{shdv} in that region. In Fig. 8c, a four-quadrant distribution of the partial forcing −2** ∇**·

**Q**

_{shdv}is evident on the cyclonic-shear side of the jet, where the thermal gradient is significant. The four quadrants are defined by the intersection of the warm ridge axis and the line of maximum shear.

Figure 7d shows the field of the 700-mb crosstream component **Q**_{crst}. The observation that *α* increases in the direction of **∇***ϕ* explains (see section 2) that **Q**_{crst} has the direction of **∇***ϕ* in confluent regions (as observed southwestward from the Newfoundland jet streak) and has the direction of −**∇***ϕ* in diffluent regions (as seen northeastward from the same jet and eastward from the Great Lakes jet). The partial forcing −2** ∇**·

**Q**

_{crst}, portrayed in Fig. 8d around the Newfoundland jet streak, has a pattern similar to the classical four-quadrant model of Riehl et al. (1952). Ascent (subsidence) is forced on the anticyclonic- (cyclonic-) shear side of the entrance and on the cyclonic- (anticyclonic-) shear side of the exit. In addition, the forced subsidence over the Gulf of St. Lawrence takes place where both jets show a slight lateral coupling. In this sense, the area of forced descent in Fig. 8d can be considered a “subsidence version” of the mechanism suggested by Uccellini and Kocin, according to which, ascending motion is significantly enhanced by the collocation of the anticyclonic-shear side entrance of a jet streak and the cyclonic-shear side exit of a neighboring jet. This mechanism is associated with the generation of snowstorms in the eastern (Uccellini and Kocin 1987) and northern (Hakim and Uccellini 1992) United States. Atlas et al. (1994) also found that this mechanism involving lateral coupling of upper jets is relevant to the intensification of low-level, orographically generated jets. However, despite its importance, the mechanism has not been separately assessed for real synoptic situations.

On the anticyclonic-shear side of the entrance in Fig. 8d, the expected positive forcing zone can be seen; however, it is centered on the jet axis rather than on the anticyclonic-shear side. This constitutes a departure from the four-quadrant model. This departure implies a shift of the center of the direct circulation associated with the jet entrance toward the cyclonic-shear side of the jet. This shift should be attributed to the pronounced asymmetry in **∇***α,* a factor that has so far not been given much attention in the literature.

The vector field **Q**_{st} in Fig. 7e represents the stretching component of **Q**. As seen previously, **Q**_{st} is the vector sum of **Q**_{alst}, the alongstream stretching component, and **Q**_{crst}, the cross-flow stretching component. These components are not independent of each other. Therefore, **Q**_{st} is the vector field logically appropriate to represent the effect of the geostrophic stretching on the thermal field. It should be remembered that **Q**_{st} can be obtained without previous knowledge of **Q**_{alst} and **Q**_{crst}. In fact, (see section 2) **Q**_{st} has the direction of the mirror image of **∇***α* or −**∇***α* about the isohypses for diffluent or confluent geostrophic flow, respectively.

In spite of being dependent on one another, the explicit display of **Q**_{alst} and **Q**_{crst}, is useful, especially in cases where they cancel or reinforce the components **Q**_{curv} and **Q**_{shdv}. In the weather situation presented here, an important cancellation takes place between **Q**_{crst} and **Q**_{shdv} north and northeast of the jet core at Newfoundland, where the isotherms are practically parallel to the geostrophic isotachs (cf. Figs. 7c and 7d).

## 4. Summary

In this paper, quasigeostrophic vertical motion forcing mechanisms that have been identified in previous studies are for the first time evaluated separately for a real synoptic situation. This was made possible by transforming the **Q**-vector expansion from the rectangular Cartesian form (2.5) into the geostrophic wind–natural coordinate form (2.8). Each **Q**-vector component in the new coordinate system is named according to the wind distribution that the component describes: (a) **Q**_{alst} is the alongstream stretching component, (b) **Q**_{shdv} is the shear advection component, (c) **Q**_{curv} is the curvature component, and (d) **Q**_{crst} is the cross-stream stretching (or confluence/diffluence) component. The **Q** components (a) and (c) are parallel, whereas (b) and (d) are normal to the geopotential contour lines. The components (a) and (b) act on the alongflow thermal gradient, while components (c) and (d) affect the cross-flow thermal gradient. The two stretching components (a) and (d) are not independent of each other, as they are linked by the constraint of nondivergence on the geostrophic wind. The partial forcing that comes from any given **Q** component is given the same name as the component and is obtained by applying the linear operation −2** ∇**· to the same component.

This study shows that the shear advection is important enough to be considered in any analysis of the large-scale vertical motion forcing. The relevance of the advection by the geostrophic wind shear in shaping the vertical motion has the important consequence that the confluence/diffluence mechanism is not always sufficient for explaining the vertical motion around a jet streak, as anticipated by Keyser et al. (1989). In fact, **Q**_{shdv} can either reinforce or counteract the confluence/diffluence component **Q**_{crst}. The cancellation between both components is total where isotherms are parallel to the isotachs of the geostrophic wind, as demonstrated in section 2 and illustrated in section 3, in the discussion of a real meteorological situation.

The shear advection process is ubiquitous. This process is always present in the generation of vertical motion accompanying the migrating cyclones and anticyclones that develop along the main baroclinic zones in middle latitudes. In fact, one important reason for the ascent (subsidence) that accompanies warm (cold) advection is the horizontal wind shear, because a uniform wind acting on a uniform temperature gradient cannot induce vertical motion, as shown in Hoskins et al. (1978).

The thermal advection by alongflow stretching is insignificant in comparison with the curvature effect in inducing vertical motion at 700 mb, in the weather situation presented in section 3. Here **Q**_{alst} is more significant at 900 mb, the lowest level above the boundary layer (not shown).

The relative merits of using either isotherms or isohypses as reference lines to select the natural coordinate system for **Q**-vector partitioning depends on the objective of the analysis. If the objective is to evaluate the geostrophically induced changes of both magnitude and direction of the horizontal temperature gradient, separately, the appropriate choice is isotherms (or equivalently, isentropes or isopycnics). However, if the objective of the analysis is to relate vertical motion to flow geometry, the system adopted in this paper, using isohypses as reference lines is more advantageous. For instance the curvature effect, which has been isolated in the present work, cannot be evaluated separately when the reference lines are isotherms, unless the isotherms are parallel to the isohypses.

Future work will involve a more detailed analysis of the *ω*’s forcing. In this respect, simple inspection of the formulas for the forcing components at the end of section 2 shows 10 different contributions that can be isolated. A preliminary study shows that some of those contributions are revealing. In view of the linearity of the *ω* equation, the effect of each forcing contribution to the total vertical motion can be also isolated. The **Q**-vector partition proposed in this paper can be applied to a new **Q** vector defined by Davies-Jones (1991) in the context of the alternative balance (AB) approximation. This approximation, more relaxed than QG, is based on the assumption that the total time derivative of the vertical variation of the ageostrophic wind is zero. The new **Q** vector contains the “total” (instead of geostrophic) wind and includes in its definition the vertical wind shear. Standard QG theory can be used to calculate a first approximation of the ageostrophic wind necessary to get the total wind.

## Acknowledgments

The authors would like to thank Dr. Daniel Keyser for his extremely valuable suggestions to this work. Discussions with Drs. Melvyn Shapiro, Stephen Bloom, and Hugo Berberys also contributed greatly. Mr. Joseph Ardizzone and Ms. Laura Rumburg helped the authors with the display of graphics. Finally, the authors wish to thank the reviewers for their helpful comments.

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## APPENDIX A

### Along- and Cross-Isohypse Projections of the Q Vector

*K*

_{s}and

*K*

_{n}are, respectively, where

*β*is the angle between the local latitude circle and the geostrophic wind vector.

By plugging (A2) and (A3) into the resulting expression for (A1) once (2.9) is applied, (2.14) is obtained.

**Q**

_{n}, the cross-isohypse projection of

**Q**, is given by the rhs of formula (2.15), which is obtained by summing up (2.12) and (2.13) as follows:

Substitution of *s*_{g}*K*_{n} by using (2.9) yields the rhs of (2.15).

## APPENDIX B

### Computation on a Rectangular Grid of the Q-Vector Components and Their Respective Contributions to Omega Forcing in the Natural Coordinate System that Follows the Geostrophic Wind

**Q**-vector components in the natural coordinate system that follows the geostrophic wind are evaluated in a local rectangular system

*x, y,*where

*x*is tangent to the local latitude circle and

*y*is tangent to the local meridian. The sphericity of earth is not considered in the present approach. Where necessary, the constraint of nondivergence of the geostrophic wind (2.6) is utilized. The strategy of the computation consists in reducing every expression describing either a

**Q**component or an element of forcing to a set of operations involving only

*x*and

*y*derivatives of specific volume

*α*and/or Cartesian components

*u*

_{g}and

*υ*

_{g}of the geostrophic wind. In pursuing that strategy, the following seven equalities are found to be basic: the expansion of the geostrophic wind speed

*s*

_{g}in Cartesian coordinates,

*s*

_{g}

*u*

_{g}

*υ*

_{g}

**t**

**i**

**j**

**n**

**i**

**j**

**i**and

**j**are, respectively, the zonal and meridional unit vectors; the expansion in Cartesian coordinates of both the tangential- and normal-derivative operators, and convenient expressions for the cosine and sine of

*β,*the angle between the geostrophic wind vector and

*x*:

As it could be anticipated, the rhs of the last expression is identical to that of (B8), but with the sign opposite.

The operations (B4) and (B5) applied to *α,* and the formulas (B2), (B3), (B8), (B9), (B13), and (B17), provide the necessary tools to compute the **Q**-vector components given by (2.10)–(2.13) in the text.

*α*in natural coordinates are where successive applications of either operator (B4) or (B5) have been performed on

*α,*and use of the Cartesian expressions for

*K*

_{s}and

*K*

_{n}have been made.

The product *s*_{g}(∂*K*_{n}/∂*s*) appearing in the first term of the rhs of (2.17) is obtained by substracting (B21) from *K*^{2}_{n}*s*_{g}. The product *s*_{g}(∂*K*_{s}/∂*s*) contained in the first term of the rhs of (2.18) is the difference between (B24) and *K*_{n}*K*_{s}*s*_{g}. Finally *s*_{g}(∂*K*_{n}/∂*n*), that belongs to the first term of the rhs of (2.20), is calculated by adding (B23) and *K*^{2}_{n}*s*_{g} and changing the sign of the result.

The set (B18)–(B24) of second derivative formulas provides the necessary elements to compute the forcing contributions (2.17)–(2.20).