## 1. Introduction

### a. Potential vorticity and its inversion

Over the last two decades, it has become evident that a good way to encapsulate the dynamics of the atmospheric flow is through consideration of the potential vorticity (PV). This is because PV has two particularly useful properties: conservation and invertibility (Hoskins et al. 1985). The latter property has proven especially important to many studies that have aimed to improve or otherwise alter the initial conditions of a numerical weather prediction (NWP) model. This is because the invertibility principle allows for the recovery of a number of other dynamical variables given a domain-wide PV distribution, a balance constraint, and appropriate boundary conditions on the domain. An omega equation may be used to recover vertical motion as well. A well-known PV inversion technique is that of Davis and Emanuel (1991, the DE method), which invokes the Charney nonlinear balance (Charney 1955) along with an approximate form of the definition of Ertel PV, using the Exner function as the vertical coordinate. Numerous studies have used the DE method; a notable recent example is Martin and Otkin (2004).

Quasigeostrophic PV (QGPV) inversion is also quite popular, both in real-world (e.g., Hakim et al. 1996) and idealized (e.g., Kim and Morgan 2002) situations. QGPV inversion is advantageous because the equations are linear, but the results are less accurate than full Ertel PV inversion. Accordingly, studies aiming to improve model forecasts through direct intervention have tended to shy away from it.

Other inversion procedures are apparently less frequently used. Raymond (1992) derived a nonlinear balance equation in height coordinates and used two different scaling arguments to formulate approximate definitions of PV. The resulting elliptic equations were solved using a multigrid method (Fulton et al. 1986). Of particular note is that a low-pass filter was occasionally applied to keep the solution from oscillating. These methods apparently have not yet been applied to real-data cases, however.

Arbogast and Joly (1998) formulated PV inversion as a minimization problem, first in a two-dimensional context, and then (Mallet et al. 1999) in a real-data case involving an inversion from output of a numerical model in sigma coordinates, albeit using QGPV. Rather than solving elliptic equations, this method uses a cost function composed of two terms, one measuring the inaccuracy of the PV equation, the other the imbalance of the flow. A quasi-Newton method minimizes the cost function and requires the adjoints of the PV and nonlinear balance operators to do so. A particularly nice advantage to this approach is that the PV need not be strictly positive to guarantee a solution. On the other hand, the method may not find the cost function’s global minimum.

Although they do not perform inversion, Schneider et al. (2003) develop PV equations analogous to Maxwell’s equations in electrostatics. In this formulation, PV and its flux determine the rest of the dynamical variables, rather than PV and a balance constraint. Bishop and Thorpe (1994) present a similar approach in the context of QGPV, later extending it to Ertel PV (Thorpe and Bishop 1995). The study of Vallis et al. (1997) is intriguing since they were able to invert the PV of a small-scale flow arising during analysis of convectively generated balanced motion in a large-eddy simulation, a situation for which DE is unlikely to converge. This study used a linearized form of PV and gradient-wind balance in height coordinates.

### b. Applications of PV inversion toward NWP

With the exception of Mallet et al. (1999), to the author’s knowledge all studies to date involving PV modification of initial conditions have employed either DE’s technique or the QGPV inversion of Hakim et al. (1996). Huo et al. (1998) used the piecewise inversion technique of DE to spread the effect of buoy and ship observations over the Gulf of Mexico throughout the lower troposphere as a revised initial condition for a simulation of the March 1993 superstorm. The model’s forecast was significantly improved.

Demirtas and Thorpe (1999) compared PV on an isentropic surface intersecting the tropopause with water vapor imagery from Meteosat. They then altered the PV on a gridpoint-by-gridpoint basis until gradients in the PV matched those seen in the satellite imagery. Upon inverting the modified PV using the same general procedure as DE, they fed the resulting balanced fields into their model’s assimilation system as “bogus obs.” They noted marked improvement in the track and intensity of a cyclone affecting the United Kingdom. In a follow-up study, however, Swarbrick (2001) found that use of this technique on a regular basis produced little improvement. It should be noted, however, that in Swarbrick’s study the regular assimilation scheme was turned off; in effect, altering the PV *was* the assimilation scheme. Thus, the comparison was not fair.

Applications of PV inversion extend beyond model initialization or the analysis of flow fields. For example, in a study by Roebber et al. (2002), the sensitivity of the 3 May 1999 tornado outbreak over Oklahoma to a particular potential vorticity anomaly off the California coast was assessed. The PV anomaly had been hypothesized to be important to the development of the outbreak, and this was tested by changing the magnitude of the anomaly, inverting the various resulting PV distributions, and rerunning numerical forecasts using the inverted PV for the initial conditions.

### c. Objective

The objective of the research presented here is to generalize previous PV inversion techniques to the case of the terrain-following coordinate of a particular NWP model. This paper focuses on one of the necessary parts in such an endeavor, namely, the development of an appropriate nonlinear balance constraint for the terrain-following coordinate. It is believed that this will lead to an improvement in the PV inversion techniques currently available, and therefore make the applications of PV inversion mentioned above even more amenable to study. In addition, the study of the balanced (or imbalanced) nature of the flows simulated by the NWP model may be worthwhile in its own right. This work most directly builds on the PV inversion techniques of Davis and Emanuel (1991) and Arbogast and Joly (1998) and the model initialization study of Sundqvist (1975).

The roadmap of this paper is as follows. Section 2 describes the methods used while carrying out this study, including the test cases used and the calculation of streamfunction. Section 3 derives the nonlinear balance equation for the terrain-following vertical coordinate, and section 4 presents the results of applying that equation to model forecasts. Section 5 provides the conclusions.

## 2. Methods used

### a. Structure of the WRF model

The NWP model chosen for use is the Weather Research and Forecasting (WRF) model using the Advanced Research WRF (ARW) dynamical core (Skamarock et al. 2005). The WRF model is formulated in a terrain-following coordinate that is closely related to traditional sigma coordinates. In the horizontal plane, Arakawa C staggering is used, while Charney–Phillips staggering is employed in the vertical plane. Figure 1 shows how the basic variables of the model are distributed among the grid boxes and also indicates the indexing convention used in this paper.

It should be noted that the nonhydrostatic mesoscale model (NMM) dynamical core of the WRF model uses different coordinates and grid staggering schemes. Therefore, the techniques developed herein cannot be applied to the NMM core without additional work.

### b. Test cases

Throughout the course of this study, multiple test cases, ranging in complexity from an idealized run to a (contiguous United States) CONUS-scale high-resolution run, have been utilized, and they are summarized in this section.

#### 1) WRF idealized baroclinic wave

The simplest test case (hereafter called test case 1) is distributed with the WRF model under the name “em_b_wave.” It simulates a baroclinic wave growing on a baroclinically unstable jet on an *f* plane. Periodic boundary conditions are used along the east and west boundaries, with symmetric boundary conditions along the north and south boundaries. A 100-km grid spacing is used, with 41 grid boxes in the zonal direction, 81 grid boxes in the meridional direction, and 64 vertical layers. All map factors are identically one in this test case, so any errors in the implementation of the various operators with respect to the map factors will not be apparent. Furthermore, with no terrain in the model, each terrain-following coordinate closely approximates an isobaric surface.

Figure 2 shows how the baroclinic wave grows exponentially during the simulation. What is a hardly noticeable perturbation up to 33 h into the run (Figs. 2a,b) becomes a strong shortwave by hour 75 (Figs. 2c,d) before becoming an extremely potent closed low only 42 h later (Figs. 2e,f).

Test case 1 was used to provide input data for the wind field partitioning test described in section 2c.

#### 2) North Pacific system

For a more complicated test case (hereinafter called test case 2), a real-world domain was chosen over the North Pacific Ocean, centered on 40°N, 150°W. This area of the world is known for frequent cyclones, yet the absence of topography allows for further testing to be performed in a simplified environment. Again the grid spacing was set to 100 km, and a relatively small domain of 34 × 26 grid boxes was used, with 20 equally spaced vertical levels. This small domain reduced the amount of computation time needed during testing and implementation.

The WRF model was initialized at 1200 UTC 5 April 2006, and a 24-h forecast was produced. The initial time (Figs. 3a,c) was characterized by a large-scale trough at 500 hPa along the northern portion of the domain. Within this larger trough, smaller shortwaves and absolute vorticity maxima were evident. Near the surface (Fig. 3c), a cyclone was located along the northern boundary directly below the 500-hPa height minimum. This fact, along with the lack of temperature contrast along the trough extending southeastward from the 1000-hPa height minimum, indicated that the cyclone was well occluded. The triple point was located just north of 40°N, 140°W (Fig. 3c), and its location just east of a 500-hPa vorticity maximum (Fig. 3a) suggested that a secondary development could occur in that area. Trailing southwest from the triple point was a cold front and broad baroclinic zone, marked by a height trough especially in the southwest.

Twelve hours into the forecast (Figs. 3b,d), the westernmost vorticity maximum at 500-hPa started to become dominant (Fig. 3b), while at 1000 hPa (Fig. 3d), the primary cyclone filled as the secondary development almost became closed off at the peak of the warm sector. A long cold front continued to extend toward the southwest, and although the temperature gradient remained strong in the immediate vicinity of the front, the broad baroclinic zone to the north began to diffuse. All subsequent references to test case 2 will focus on this forecast time.

#### 3) Continental United States

A final test domain was created that covers the continental United States with 24-km grid spacing. This domain consists of 210 grid boxes in the zonal direction, 136 grid boxes in the meridional direction, and 49 vertical layers. The pronounced topography and relatively high resolution of this domain serves as a more stringent test of the nonlinear balance operator described below. One forecast (test case 3) has been run on this domain, initialized at 1200 UTC 24 March 2006.

Results from this test case will focus on forecast hour 36 (Fig. 4). At that time, rapid cyclogenesis was forecast off the East Coast, while a potent shortwave was progressing across the Intermountain West. A strong intermountain cold front (Shafer and Steenburgh 2008) can be discerned along the Utah–Nevada border. These features were concomitant with extensive reports of convective wind damage across southern Idaho around this time (NOAA 2009). The presence of two notable weather features, one over the ocean, the other over mountains, allows for ample investigation into the effects of the terrain on the balance equation. Finally, note the oscillatory nature of the height contours at 500 hPa (Fig. 4a) over the west, indicating that mountain waves are likely present in the model forecast.

### c. Streamfunction calculation

As we shall see, computing the extent to which the flow is balanced requires the streamfunction of the mass-weighted flow as one of two dependent variables (geopotential being the other). Values of streamfunction must be specified along the lateral boundaries of the domain, and interior values of streamfunction serve as a first guess. The technique used to compute the streamfunction is derived from the algorithm of Bijlsma et al. (1986). Their algorithm uses a horizontal staggering similar to the WRF, except an additional half grid box is added to the north and east sides of the domain such that the northern (eastern) edge of the domain contains values for the *x* (*y*) component of the wind.

If we were using a global domain, the Helmholtz theorem tells us that any vector field can be decomposed into two unambiguous components, one that is nondivergent (defined by the streamfunction), and the other that is irrotational (defined by the velocity potential). However, on a limited-area domain, such as the domains used here, there is a third component, commonly called the harmonic component (e.g., Loughe et al. 1995), that is both nondivergent and irrotational. There are an infinite number of ways to assign the harmonic component, in whole or in part, to the nondivergent and irrotational wind components. The method described here assigns all of the harmonic component to the nondivergent component, since the inverted PV should capture as much of the total flow as possible.

*ψ*is the streamfunction and

*χ*is the velocity potential. From Eq. (1), the Laplacian of the streamfunction (velocity potential) can be equated to the vorticity (divergence) of the mass-weighted wind, and the following boundary conditions may be derived:where

**s**and

**n**are unit vectors in the directions tangential and normal to the boundary, respectively, and

*s*and

*n*are distances along those directions. The second-order accurate discrete forms for the relationships between

*ψ*,

*χ*, the vorticity

*ζ*, and the divergence

*m*represents map factor values at the vorticity points, and Δ

^{ψ}*x*is the nominal grid spacing. Using these equations to solve for

*ψ*and

*χ*on the interior of the domain requires knowledge of

*ψ*and

*χ*along the boundaries, that is, when

*i*= 0,

*i*= nx,

*j*= 0, or

*j*= nx. (Refer to Fig. 1 for a depiction of the grid discretization.) However, the values of

*ψ*and

*χ*along the boundaries. For example,

*ψ*is updated along the left boundary usingwhere

*υ*is the component of

*y*direction, and

*m*is the map factor at the

^{υ}*υ*points. Note that

*ζ*and

Putting it all together, then, we begin by setting *ψ* and *χ* to zero everywhere and then solve Eq. (2) to determine values for *ψ* around the boundaries, keeping *ψ*_{0,1} = 0 always. This is followed by solving Eq. (4) for interior values of *ψ* using successive overrelaxation for one iteration only. Boundary values of *χ* are maintained at zero so that all of the harmonic component is assigned to the nondivergent flow. Equation (5) is then solved (again for only one iteration) for interior values of *χ*. This ends one step of the procedure, which may be repeated [by solving Eq. (2) again] as many times as necessary. The solution is deemed converged when the metric ^{−3} m s^{−1}, where the prime represents the wind reconstituted from the fields of *ψ* and *χ* according to Eq. (1). The process of updating the boundary conditions as *ψ* and *χ* are being solved for leads to good convergence properties (Bijlsma et al. 1986).

As a test, the process above was carried out using the initial field of

The closed low and downstream ridge are captured very well by the streamfunction in Fig. 6a. The velocity potential (Fig. 6b) indicates divergent flow downstream of the trough and convergent flow farther upstream. Since this example uses data from approximately the 500-hPa level, this makes sense synoptically. Figures 6c,d clearly show that the nondivergent wind captures the bulk of the flow, with a small residual given by the irrotational wind. Again, this is not unexpected in typical large-scale midlatitude flow.

Calculations were performed for many different fields of ^{−11} s^{−1} for vorticity and 10^{−19} s^{−1} for divergence. As an additional check, the divergence of the nondivergent wind and the vorticity of the irrotational wind were both computed. The maximum absolute value of these quantities, which should be identically 0, rarely exceeded 10^{−20} s^{−1}.

### d. PV inversion

When knowledge of a potential vorticity distribution is used to initialize the WRF model, it must be inverted to provide new values for the WRF state variables (e.g., geopotential, wind, and temperature). One might think that the popular DE method would be well suited for this purpose. This is problematic, however, for a variety of reasons. First, the DE method is carried out in Exner function (i.e., pressure) coordinates, whereas the WRF model uses a terrain-following vertical coordinate. Although interpolation to pressure coordinates is possible, it is not optimal since some loss in accuracy is inevitable, and the forecast domain may include mountainous terrain such as the Rocky Mountains, making the lower boundary perhaps difficult to handle. It should be noted, however, that some authors have used the DE method over highly variable terrain [e.g., Chang et al. (2000) over China] with no apparent problems. The author’s experiences with using the DE method in areas of steep terrain suggest, on the other hand, that fields such as the geopotential heights derived from the inversion can oscillate wildly in an exaggerated diurnal cycle, as can be seen in Fig. 3.44 of Decker (2003). An additional problem with the DE method is that it frequently does not converge when the grid spacing is less than about 100 km. Since modern NWP is performed at grid spacings on the order of 10 km, this would result in additional interpolation, this time between small- and large-scale grids.

**X**containing values of streamfunction and geopotential at every grid point in the model domain for which they are defined (recall that, according to Fig. 1, streamfunction and geopotential are staggered relative to each other), the application of the PV operator on

**X**results in a new vector representing the value of the potential vorticity in each WRF grid box. A second operator, the nonlinear imbalance operator, is also defined that operates on

**X**. Application of this operator on

**X**results in a new vector representing the degree to which each grid box is in nonlinear balance. A cost function is then defined to bewhere

*P̃*is the PV operator,

**P**is the given PV distribution (i.e., the PV to be inverted), ε is a constant to give each term in the cost function equal weighting, and

*B̃*is the nonlinear imbalance operator, which measures the degree to which the flow is not in balance. The inner product is taken to be a simple dot product. The gradient of

*J*with respect to the state vector

**X**is thenwhere the asterisks denote the adjoints of the linearized operators. This gradient is then used to minimize

*J*using the method of steepest descent, or, more generally, any quasi-Newton method. The portion of this procedure that is the focus of this paper is the construction of the nonlinear balance operator and its adjoint.

## 3. Derivation of the balance equation using the WRF vertical coordinate

### a. Expansion of divergence equation

*p*is the dry hydrostatic pressure, and

_{d}*p*and

_{s}*p*refer to values of the dry hydrostatic pressure at the surface and model top, respectively. The dry hydrostatic pressure is a measure of the weight of the dry air in an atmospheric column above a given point. In particular,

_{t}*μ*≡

*p*−

_{s}*p*is directly proportional to the weight of the dry air in an atmospheric column above a surface grid box.

_{t}*u*and

*υ*are the zonal and meridional winds, respectively, on the computational grid,

*η̇*≡ (

*dη*/

*dt*),

*α*is the specific volume of dry air,

*p*is the total pressure, Φ is the geopotential, and

*x*and

*y*are the computational coordinates. The map-scale factors

*m*are given by the quotient of Δ

*x*divided by the true earth distance, where Δ

*x*is the constant horizontal grid spacing. Finally,

*χ*≡ (1 +

*q*+

_{υ}*q*+

_{c}*q*+

_{r}*q*+ · · ·)

_{i}^{−1}, where

*q*

_{υ,c,r,i}are the mixing ratios for water vapor, cloud, rain, ice, and so on.

*ϵ*

_{i,j,k}is the permutation symbol, the indices not associated with

*ϵ*

_{i,j,k}can take values of 1 or 2,

*u*

_{1}≡

*u*,

*u*

_{2}≡

*υ*,

*x*

_{1}≡

*x*, and

*x*

_{2}≡

*y*. We can simplify this equation via normalization and the elimination of

*α*and

*p*. Also used is the continuity equation, which can be written asAs a first step, we divide both equations by the constant

*p*

_{0}−

*p*, and multiply Eq. (12) by

_{t}*u*. The resulting equations areandwhere

_{i}*M*≡ (

*p*−

_{s}*p*)/(

_{t}*p*

_{0}−

*p*) =

_{t}*μ*/(

*p*

_{0}−

*p*) and

_{t}*p*

_{0}is a reference pressure (say, 1000 hPa).

### b. Scale analysis

*Ḣ*for

*η̇*. The starting point is the scaling for

*ω*≡ (

*Dp*/

*Dt*), the pressure velocity, that Sundqvist (1975) givesIt can be shown that the dry pressure velocity is related to

*η̇*in the following way:By definition, 0 ≤

*η*≤ 1. In addition,

*ω*/(

_{d}*p*−

_{d}*p*) ∼

_{t}*ω*/(

*p*−

_{d}*p*) ∼

_{t}*ω*/

*p*∼

_{d}*ω*/

*p*∼ 10

^{−6}s

^{−1}using Eq. (18). As shown in Fig. 7a,

*M*≈ 1 and varies on the order ε = 10

^{−2}when topography is not an issue. Thus, (∂

*M*/∂

*t*) ∼ (1/

*M*)(∂

*M*/∂

*t*) ∼ (ε/

*τ*) = ε(

*C*/

*S*) =

*f*εRo, where the time scale

*τ*is given by

*τ*=

*S*/

*C*,

*S*is the horizontal scale,

*C*is the velocity scale, and the Rossby number, Ro, is defined as Ro =

*C*/(

*fS*). Using values of 10

^{−4}s

^{−1}for the Coriolis parameter and 10

^{−1}for the Rossby number, (∂

*M*/∂

*t*) ∼ (1/

*M*)(∂

*M*/∂

*t*) ∼ 10

^{−7}s

^{−1}. The term (1/

*M*)

**V**·

**∇**

*M*requires special consideration of the horizontal scale, which we interpret in this case to be the scale over which fluctuations of

*M*occur. Figure 7 shows that

*S*∼ 1000 km over flat terrain, but

*S*∼ 100 km over topography. Additionally, when topography is present, ε

_{m}*∼ 10*

_{m}^{−1}. Together, these considerations imply that (1/

*M*)

**V**·

**∇**

*M*∼

*C*ε

*/*

_{m}*S*∼ 10

_{m}^{−5}s

^{−1}over topography, but is two orders of magnitude smaller over level terrain. Given these scalings, Eq. (19) implies that

*Ḣ*ranges from 10

^{−6}s

^{−1}to 10

^{−5}s

^{−1}, depending on terrain. One can use the continuity equation [Eq. (12)] to show that

When dealing with horizontal motions, it is important to realize that topography may also impose a horizontal scale on the flow that is on the order of the scale of the topography itself. Since our model will ultimately have a grid spacing on the order of 10 km, this imposed horizontal scale may also be on the order of 10 km, or about one hundredth the synoptic horizontal scale. Thus, we use Ro ∼10 in the scaling that follows.

As a check to see how accurate the above scalings are, each term was analyzed for each test case at a variety of levels. For each term, two metrics were computed, the maximum absolute value of the term at any grid point (to be referred to as the “maximum value”), and the average absolute value of the term over all of the grid points (to be referred to as the “average value”). The results are summarized in Table 1. Some terms vary from case to case (usually increasing in magnitude as the resolution and amount of topography is increased), but other terms do not. In particular, terms A4, A5, *C*, and *D* show little change (excluding the fact that *D* is identically zero in test case 1), while term B2 shows the most change. Despite these variations, many of the terms have typical values that are within an order of magnitude of what was expected by the scale analysis. Exceptions include terms B1, B2, *C*, *D*, and *E*, which were overestimated.

A discussion of the A and B terms that are of higher order proves useful in interpreting the results from the simplified balance equation shown later, so these terms are now presented. All terms are plotted for test case 3 at hour 36 for *η* = 0.8375, which is far enough above the ground for the effects of friction to generally be negligible yet close enough for the coordinate surface to have substantial horizontal variations in height.

Terms A1–3 are displayed in Fig. 8. The first term (Fig. 8a), the time derivative of mass-weighted divergence, is fairly noisy, with the highest magnitudes in the vicinity of the coastal storm and throughout the west. The term does not appear to be tied too directly to topography, as absolute values are just as high over the Atlantic Ocean as they are over mountainous terrain. The maximum value of this term is 5.92 × 10^{−8} s^{−2} at this level. It should be noted that, because output from these test cases was only available every hour, the true magnitude of this term (using the model’s time step instead) is likely higher. This issue also affects other terms involving time derivatives, namely A4 and A5, but those terms are expected to be small in any case. The second term (Fig. 8b), related to vertical advection of divergence, is generally positive. This is due to the fact that upward (downward) vertical motion at this level will usually be situated above a region of convergence (divergence) at lower levels by the continuity equation. This terms shows a greater affect from topography, but again the term is about the same magnitude as the vigorous ascent associated with the coastal storm. The maximum value of this term is 4.15 × 10^{−7} s^{−2}. The third term (Fig. 8c), involving vertical shear, is generally large where the other A terms are large, with a maximum value of 1.84 × 10^{−7} s^{−2}. Terms A1 and A3 have good agreement with the scale analysis, while term A2 is somewhat larger (but still smaller than the other two on average), likely because of the intense vertical motions resolved by the model at this scale.

The B terms, which generally represent curvature in the flow (an analog to the centripetal force in the gradient wind balance), are shown in Fig. 9. Term B1 (Fig. 9a) exhibits patterns that are on a larger scale than the other terms, especially along the East Coast. Although we are somewhat removed from the 500-hPa level, the positive regions of this term bear some resemblance to the vorticity distribution shown in Fig. 4. This makes sense given the connection between this term and curvature, and the connection between curvature and vorticity. The maximum value of Term B1 at this level is 1.51 × 10^{−7} s^{−2}. This is smaller than the magnitude estimated above, suggesting the smaller scales forced by the topography are not affecting this term as much as anticipated. Term B2 (Fig. 9b) is clearly affected by the topography, with a maximum value of 1.84 × 10^{−7} s^{−2}. Unlike the other terms, the coastal storm has no appreciable reflection in this term. Despite the topographical influence, B2 is again smaller than anticipated across most of the domain. Term B3 (Fig. 9c) is the noisiest of the three, with numerous dipole structures in close proximity, generally elongated perpendicular to the flow. This term is of the magnitude we expected on average, but its maximum value of 1.08 × 10^{−7} s^{−2} is quite large. Term B3 is strongly associated with divergence, which implies that gravity waves (including mountain waves) are likely contributing to this term, whether they be generated by cyclogenesis over level terrain or flow over mountains. Also of note are the relatively high values along the boundaries, suggesting the WRF boundary conditions are generating spurious regions of divergence. Since we will end up neglecting this term, the fine-grained structure attributable to it will be absent from the balanced flow.

Although not every term has been shown, it is worthwhile to note that the terms deemed small based on scale analysis were generally noisier than the presumably larger terms. If small wavelengths had been filtered out of the numerical solutions (and the WRF, being nonhydrostatic, contains small-scale features such as sound waves in addition to inertia–gravity waves), the disparity in magnitude between terms would likely have been accentuated. Furthermore, these noisy terms (with the exception of B2) are as noisy over level terrain as they are over undulating terrain. Therefore, the noise is not an artifact of the topography per se, but rather a reflection of the small-scale circulations that the WRF model is able to resolve. The simplified balance equation presented below, by virtue of neglecting these noisy terms, will filter out this small-scale noise.

### c. Balance equation

*E*(according to the original scale analysis), Eq. (17) reduces toor, using indicial notation:According to Fig. 10, this reduction in complexity of the lhs of the balance equation does not radically alter its value, except on the smallest scales, as has already been alluded to. The left (right) panels refer to test case 1 (2). The top panels (Figs. 10a,b) show the sum of all of the terms A through

*D*, whereas the middle panels (Figs. 10c,d) show the lhs of Eq. (21). The original lhs tends to be a bit noisier, but there are no important extrema that are not reproduced in the proper locations by the simplified version. The bottom panels (Figs. 10e,f), which show the sum of the terms neglected, verifies that this is the case. However, the neglected terms in test case 2 do have a greater magnitude relative to the full lhs in that case.

Figure 11 shows the corresponding comparison for test case 3. The total lhs (Fig. 11a) is very noisy, and although the simplified form (Fig. 11b) captures some of the small-scale features, the neglected terms are on the same order of magnitude. It is interesting to see that the neglected terms can be large over level ground (e.g., off the East Coast) and small over mountains (e.g., over the Cascades).

*ψ*using the definition

*ϵ*

_{k,i,j}

*ϵ*

_{k,l,m}=

*δ*

_{i,l}

*δ*

_{j,m}−

*δ*

_{i,m}

*δ*

_{j,l}and

*δ*

_{i,i}= 2 have been used. Substituting Eqs. (22), (23), and (24) into Eq. (21) and multiplying by −1 giveswhich in vector notation isDespite the coordinate change, the form of this equation is similar to the balance equations found in previous work, such as McWilliams and Gent (1980) and Gent and McWilliams (1983).

We define the nonlinear imbalance as the lhs of Eq. (25) minus the rhs of Eq. (25). A flow in perfect balance thus has a nonlinear imbalance that is identically zero. The greater the imbalance, the further the flow is from obeying the nonlinear balance equation.

## 4. Analysis of nonlinear imbalance

*l*

_{1}=

**∇**· (

*f*

**∇**

*ψ*):and

*r*=

**∇**· [

*M*

**∇**Φ −

*η*

**∇**

*M*(∂Φ/∂

*η*)], ignoring contributions from water substance for the time being. With these definitions and the appropriate discretization (not shown for brevity), the imbalance present in the test cases can be assessed.

For test case 1, the degree of imbalance in the flow is assessed at hour 117. The left half of Fig. 12 shows the results. The top panel (Fig. 12a), showing the combination of terms *l*_{1}, *l*_{2}, and *l*_{3}, should be compared to Fig. 10c. The difference between these two figures is small, and is due to the fact that, in the current case, the streamfunction is being used rather than the original wind data. [Note that Eq. (25) has been multiplied by −1 relative to Eq. (17).] The impact of this is that the results are smoother when the streamfunction is used. The middle panel (Fig. 12c) shows term *r*, and the bottom panel (Fig. 12e) shows the imbalance, which is quite reminiscent of Fig. 10e, as expected.

Turning our attention now to test case 2, the same procedure was carried out, and the analogous results are displayed on the right half of Fig. 12. In this real-world case, regions of relatively high imbalance are more widely dispersed throughout the domain, with hints that areas near the boundaries tend to be less in balance (possibly reflecting the imbalanced nature of the WRF boundary conditions rather than any boundary effects from the imbalance computation itself). In both cases considered thus far, the imbalance is on the order of 10^{−9} s^{−2}.

Finally, the imbalance from test case 3 is shown in Fig. 13. As might be expected from the previous analysis, there is a large degree of imbalance present in this WRF forecast on the smaller scales, whether the flow is over topography or not. However, the imbalance is oscillatory in nature on the interior of the domain, which implies that the imbalance largely cancels out on larger scales. This explains how the imbalance can be so small in the previous test cases. Also becoming quite apparent in test case 3 is a strip of imbalance parallel to each boundary, something that was also observed (to a lesser extent) in test case 2. This suggests that the WRF boundary conditions (or the relaxation process that forces the WRF evolution to match the imposed boundary conditions) are introducing imbalanced flow near the boundaries.

The imbalance calculation can be thought of as a nonlinear operator that takes *ψ* and *ϕ* as input and gives IMB as output. The tangent linear and adjoint forms of the IMB operator have been developed, but will be discussed elsewhere.

## 5. Conclusions

This paper has presented the derivation and testing of a nonlinear imbalance operator (along with its tangent linear and adjoint) that has been crafted to operate on the direct output of the WRF model. These measures of imbalance have value as a way to diagnose the dynamics of a model solution just as gradient-wind balance can. However, it could be argued that the insights gained from such a diagnosis perhaps would not be worth the additional complexity these operators entail. The utility of this work, therefore, may not lie in the direct assessment of the imbalance of a given model solution.

Instead, this tool is intended to be used as part of a PV inversion procedure aimed at providing initial conditions to the WRF model. Such initial conditions could be used during case studies to ask complex “what if” questions. In addition to asking, “What if the PV anomaly were doubled?” as Roebber et al. (2002) did, one would be able to ask “What if the PV anomaly had a different shape?” Additionally, it is hoped that through techniques similar to those used by Swarbrick (2001), the PV inversion procedure could be used as a form of data assimilation. The complete PV inversion scheme is under development.

However, it is clear that for model runs with high enough resolution, the flow fields on the smallest scales will be highly imbalanced. This appears to be true whether topography is present or not. Therefore, a PV inversion scheme employing this balance constraint has no chance of recovering these small-scale features.^{2} This may not be much of an issue, though, to the extent that these features are reflections of inertia–gravity waves, including mountain waves, that are not dynamically relevant to most aspects of a weather forecast.

Finally, the streamfunction calculation technique described in this paper can find use whenever the display of streamfunction on the WRF vertical coordinate is desired, as it is not dependent on knowledge of the nonlinear (im)balance or PV distributions.

The work presented here is a portion of the author’s Ph.D. dissertation, and was funded under NSF Grants ATM-0202012 and ATM-0452318. Discussions with Drs. Jonathan Martin, Michael Morgan, and Grace Wahba were much appreciated during the development of this work. Comments from two anonymous reviewers greatly improved the structure of this paper.

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# APPENDIX

## Details of Balance Equation Derivation

*m*= 1) will be considered for simplicity. In actuality, the result is transformed to the WRF grid. With

*m*= 1, Eqs. (16) and (14) simplify toandrespectively, whereNote that in the dry case,

*P*=

_{i}*M*(∂Φ/∂

*x*) −

_{i}*η*(∂Φ/∂

*η*)(∂

*M*/∂

*x*). The balance equation is derived by subtracting the divergence of Eq. (A2) from the divergence of Eq. (A1) and defining

_{i}The average absolute magnitudes of various terms in the balance equation.

^{1}

Although the WRF model is nonhydrostatic, we use the hydrostatic approximation in this derivation. Thus, the balance equation will not capture any nonhydrostatic part of the flow.

^{2}

The statistical method of Hakim (2008) may do better in this regard.