## 1. Introduction

The mountain torque of the earth–atmosphere system plays a major role in the exchange of angular momentum between the atmosphere and the solid earth. An accurate calculation of mountain torque is essential for analyzing the balance of global atmospheric angular momentum in observations (Huang et al. 1999; Egger et al. 2003) and in general circulation model (GCM) simulations (Boer 1990; Lejenäs et al. 1997). Such an analysis has become a common practice for climate monitoring (Madden and Speth 1995; Weickmann et al. 1997) and for diagnosing model biases (Huang et al. 1999; Brown 2004). This note aims to explore the sensitive dependence of the computed mountain torque on the detail of numerical schemes, a subject that is rarely discussed in the literature. Not intending to exhaust all possible schemes, we will illustrate by examples what are reasonable schemes to use and what should be avoided.

The global mountain torque *τ _{M}* is a function of the surface pressure

*p*and surface topography

*h.*The discussion here is restricted to the axial component of the torque and, further, to the case when both

*p*and

*h*are available on global, regular, or Gaussian grids. These cases cover most meteorological applications using global reanalysis datasets or GCM outputs. Ideally, for the calculation of

*τ*for a GCM simulation one could choose the numerical scheme that is most consistent with the architecture of the particular model. For an effective diagnosis and comparison of the mountain torques for a large set of GCM outputs [e.g., in the Atmospheric Model Intercomparison Project (Gates 1992) and its follow-ups], this approach might be impractical given the complexity of the numerical schemes in different models. A simpler strategy would be to apply a uniform and reasonably accurate numerical scheme to all of the model data. In this spirit, this note will provide a general discussion about the sensitivity of

_{M}*τ*to the numerical scheme without a specific knowledge of the origin of the

_{M}*p*and

*h*fields used in the calculation. Numerical examples will be constructed using the surface pressure and topography from the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) reanalysis (Kalnay et al. 1996).

## 2. Results

### a. Convergence of finite-difference schemes

*p,*and surface topography,

*h,*from the NCEP–NCAR reanalysis are used to provide numerical examples in our discussion. They are available on a 192 × 94 Gaussian grid or, equivalently, in triangular 62 (T62) spherical harmonic coefficients. The mountain torque,

*τ*can be evaluated by(

_{M},*λ*,

*ϕ*) are (longitude, latitude),

*a*is the radius of the earth, and

*X*= −

*p*∂

*h*/∂

*λ*or

*h*∂

*p*/∂

*λ*. We will first consider the former. The integration in (1) is straightforward (e.g., using Gaussian quadrature when the data is on a Gaussian grid). We will focus on the evaluation of (2), which involves the differentiation in longitude. For gridded data on

*N*equally spaced points in longitude, (2) will be approximated byThis is equivalent to adopting the trapezoidal rule for the integration in (2), incorporating the periodic boundary condition,

*X*

_{N}_{+1}=

*X*

_{1}. Although this choice is relatively simple, it preserves the value of the integration under a shift of the domain of integration. [The

*T*evaluated by (3) remains the same whether the integral in (2) is defined from 0 to 2

_{M}*π*or from

*L*to

*L*+ 2

*π*, where

*L*=

*n*Δ

*λ*.] The critical step of the above calculation is to evaluate

*X*= −

_{i}*p*(∂

_{i}*h*/∂

*λ*)

*. Six finite-difference schemes, in the general formare considered with their coefficients*

_{i}*b*, and the coefficients are summarized in Table 1. The result is compared to a benchmark calculation using the spectral method, which provides the most accurate evaluation of the longitudinal differentiation given the periodic boundary condition for

_{j}*p*and

*h.*

Figure 1a shows the calculated daily values of *τ _{M}* using the second-, fourth-, and sixth-order centered finite-difference schemes (C2, C4, and C6 in Table 1) and the spectral method for the evaluation of

*X*= −

_{i}*p*(∂

_{i}*h*/∂

*λ*)

*, for the year 1990 using daily-averaged*

_{i}*p*and

*h*from the NCEP–NCAR reanalysis. (Our conclusions are not affected if 4 times daily data are used instead.) Figure 1b shows the errors of the finite-difference schemes as the departure from benchmark. All three finite-difference schemes captured the detailed daily fluctuation of

*τ*. A notable error (using the benchmark as the perfect answer) is a negative bias that becomes larger for the lower-ordered schemes. For the second-order scheme the bias is on the order of several Hadleys (1 Hadley = 10

_{M}^{18}kg m

^{2}s

^{−2}). However, a large part of the error is systematic and is removable by removing the long-term mean. Thus, the error is greatly reduced if one is only concerned with the anomaly of the mountain torque, useful for many applications (e.g., those related to the diagnosis of the intraseasonal variability of the torque). The computed values of

*τ*converge rapidly when the second-order scheme is upgraded to the fourth- and sixth-order ones. In general, the fourth-order scheme provides a good approximation to the spectral method.

_{M}The above calculations are repeated by using the first-, second-, and third-order noncentered finite-difference schemes (N1–N3 in Table 1). The results are not shown in Fig. 1 because these schemes produced extremely large errors, on the order of hundreds of Hadleys, compared with the benchmark. The convergence of the results from N1 to N3 is very slow and we did not proceed further to higher-order schemes. The large errors associated with the noncentered schemes can be understood by a simple example of a triangular-shaped mountain centered at *i* = *I*, with *h _{I}* = 1,

*h*

_{I}_{−1}=

*h*

_{I}_{+1}= 1/2, and

*h*= 0, otherwise, and surface pressure of the same pattern,

_{i}*p*= −

_{i}*Ch*, such that analytically

_{i}*τ*= 0. In this case, all even-ordered centered schemes produce an antisymmetric (w.r.t.

_{M}*i*=

*I*) pattern of (∂

*h*/∂

*λ*)

*such that*

_{i}*τ*remains zero. A noncentered scheme effectively shifts the mountain in longitude; the pressure loading on the top of the mountain,

_{M}*i*=

*I*, that otherwise should exert no mountain torque, is now acting on the eastern or western slope, producing a large artificial mountain torque. This error is inherent to all noncentered schemes. It is severe enough that a third-order noncentered scheme performed far worse than a second-order centered scheme.

### b. Discussion

The above calculations had used *X _{i}* = −

*p*(∂

_{i}*h*/∂

*λ*)

*in (3). To see the effect of using the alternative,*

_{i}*X*=

_{i}*h*(∂

_{i}*p*/∂

*λ*)

*, it is noted that the*

_{i}*T*in (3) can be written as a quadratic form,

_{M}*T*,

_{M}*= −*

_{1}**p**

^{T}𝗤

**h**, where

**h**= {

*h*} and

_{i}**p**= {

*p*} are the vectors of the topography and surface pressure and 𝗤 is a matrix whose elements depend on the numerical scheme. For example, for the C2 scheme, we have tridiagonal (

_{i}*Q*,

_{i}

_{i}_{−1},

*Q*,

_{i}*,*

_{i}*Q*,

_{i}

_{i}_{+1}) = (−1/2, 0, 1/2) for all

*i*,

*Q*

_{1,}

*= −1/2, and*

_{N}*Q*

_{N}_{,1}= 1/2. When using

*X*=

_{i}*h*(∂

_{i}*p*/∂

*λ*)

*, one obtains an alternative form of*

_{i}*T*as

_{M}*T*

_{M}_{,2}=

**h**

^{T}𝗤

**p**. The difference,

*T*,

_{M}*−*

_{2}*T*,

_{M}*=*

_{1}**h**

^{T}(𝗤 + 𝗤

^{T})

**p**, is identically zero as long as 𝗤 is antisymmetric (𝗤 = −𝗤

^{T}), as is satisfied by all even-ordered centered finite-difference schemes and the spectral method. Thus, our results in Fig. 1 remains intact with the alternative formulation. For the noncentered schemes, 𝗤 is not antisymmetric such that

*T*

_{M}_{,1}and

*T*

_{M}_{,2}can differ by any value depending on the contents of

**h**and

**p**. This is another undesirable property of the noncentered schemes.

Among the even-ordered centered schemes, Fig. 1 shows that there is a generally negative offset in the computed mountain torque that increases in magnitude as one adopts a cruder (e.g., going from fourth to second order) scheme. The error in *τ _{M}* contributed by a grid point, (

*λ*,

_{i}*ϕ*), can be written as

_{j}*E*(

*i*,

*j*) = −

*p*(∂

_{i,j}δ*h*/∂

*λ*)

*Δ*

_{ij}*A*, where Δ

_{ij}*A*is the area of the grid box and

_{ij}*δ*(∂

*h*/∂

*λ*) = (∂

*h*/∂

*λ*)

_{SECOND ORDER}− (∂

*h*/∂

*λ*)

_{BENCHMARK}. Figure 2 shows a map of

*E*(

*i*,

*j*) for an arbitrarily chosen day from Fig. 1. The curve at right is the annual mean of the summation of

*E*(

*i*,

*j*) over

*i*for each latitude and the red shading indicates its range (minimum to maximum) for the year 1990. A close examination of

*δ*(∂

*h*/∂

*λ*) (not shown) indicates that, along the steepest slopes, the finite-difference scheme always underestimates the

*magnitude*of the slopes—the cruder the scheme, the more severe the underestimation—such that

*E*(

*i*,

*j*) is always positive over the western slope and negative over the eastern slope of a mountain, as reflected in the alternating brown and green stripes in Fig. 2.

The error in Fig. 2 comes from mountainous regions (notably the vicinity of the Andes; the nonzero values over the ocean were due to Gibbs ripples in the topography associated with the NCEP–NCAR reanalysis model) and adds to a negative value for almost any given latitude. The error is geographically widespread and does not appear to be associated with a particular local pressure system. As such, we will attempt to interpret it only in a statistical sense. Since the net error is the result of the cancellation between the positive errors on the western slopes and negative errors on the eastern slopes, for it to be negative there need to be, on the average, (i) greater pressure loading on the eastern slopes, and/or (ii) steeper slopes on the eastern side of the mountains. Although the effects of the two are not clearly separable, one can readily corroborate (ii) by a quick survey of the slope, (∂*h*/∂*x*)* _{i}* (

*x*is the east–west distance), evaluated using the spectral method, for the topography of the NCEP–NCAR reanalysis for the mountainous belts of 30°–45°N and 20°–30°S. It reveals that the steepest slopes indeed occur more often over the eastern slopes then over the western slopes; the numbers of grid points with |∂

*h*/∂

*x*| > 1/2000 are 467 and 427, and those with |∂

*h*/∂

*x*| > 1/1000 are 291 and 270 for the eastern and western slopes, respectively. [The cause of the east–west asymmetry in the earth’s topography is the subject of geomorphology (e.g., Willett et al. 2001).] A caveat to the interpretation using (ii) alone is that the presence of a steeper slope on the eastern side of a mountain also implies fewer grid points being located in that side of the mountain (i.e., the ups and downs of the topography add to zero for a latitude circle). In the case when there is no pressure difference across the mountain ranges (take the extreme, if unphysical, case of

*p*=

_{i}*p*= constant along a latitude circle, as was drawn to our attention by the reviewers), the error would be zero regardless of the shape of the mountains. Thus, (i) is also needed, and it is likely the combination of the east–west asymmetry in the pressure field and in the steepness of the topography that lead to the net bias.

## 3. Concluding remarks

Our analysis suggests that the fourth-order centered finite-difference scheme is an accurate, and efficient, approximation to the spectral method. We recommend it for the use in a quick diagnosis of the mountain torque and budget of angular momentum in large datasets of reanalysis or GCM simulations. Using that scheme, the two formulas of *X* = −*p*(∂*h*/∂*λ*) and *X* = *h*(∂*h*/∂*λ*) in (2) should yield identical results. Noncentered finite-difference schemes should be avoided. The accuracy of the finite-difference schemes discussed in this work is with respect to the spectral method and given the *p* and *h* from the GCM outputs. The model biases in the latter may yet lead to a further discrepancy between the mountain torque that is consistent with the model and the true mountain torque. This error is beyond the scope of our study.

The authors appreciate insightful comments from two anonymous reviewers. This paper is a modified version of, and supersedes, a previously privately circulated note. HPH appreciates interesting conversations with Colin Stark that inspired some of the interpretations of the negative bias in section 2. This work was supported in part by NSF Grant ATM-05-43256.

## REFERENCES

Boer, G. J., 1990: Earth–atmosphere exchange of angular momentum simulated in a general circulation model and implications for the length of day.

,*J. Geophys. Res.***95****,**5511–5531.Brown, A. R., 2004: Resolution dependence of orographic torque.

,*Quart. J. Roy. Meteor. Soc.***130****,**3029–3046.Egger, J., , K-P. Hoinka, , K. Weickmann, , and H-P. Huang, 2003: Angular momentum budgets based on NCEP and ECMWF reanalysis data: An intercomparison.

,*Mon. Wea. Rev.***131****,**2577–2585.Gates, W. L., 1992: AMIP: The Atmospheric Model Intercomparison Project.

,*Bull. Amer. Meteor. Soc.***73****,**1962–1970.Huang, H-P., , P. D. Sardeshmukh, , and K. M. Weickmann, 1999: The balance of global angular momentum in a long-term atmospheric data set.

,*J. Geophys. Res.***104****,**2031–2040.Kalnay, E., and Coauthors, 1996: The NCEP/NCAR 40-Year Reanalysis Project.

,*Bull. Amer. Meteor. Soc.***77****,**437–471.Lejenäs, H., , R. A. Madden, , and J. J. Hack, 1997: Global atmospheric angular momentum and Earth–atmosphere exchange of angular momentum simulated in a general circulation model.

,*J. Geophys. Res.***102****,**1931–1941.Madden, R. A., , and P. Speth, 1995: Estimates of atmospheric angular momentum, friction, and mountain torques during 1987–88.

,*J. Atmos. Sci.***52****,**3681–3694.Weickmann, K. M., , G. Kiladis, , and P. D. Sardeshmukh, 1997: The dynamics of intraseasonal atmospheric angular momentum oscillations.

,*J. Atmos. Sci.***54****,**1445–1461.Willett, S. D., , R. Slingerland, , and N. Hovius, 2001: Uplift, shortening, and steady state topography in active mountain belts.

,*Amer. J. Sci.***301****,**455–485.

The coefficients *b _{j}* for the finite-difference schemes.