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Rossby wave dispersion on the C-grid

Authors

  • J. Thuburn

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    1. Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter, EX4 4QE, UK
    • Mathematics Research Institute, School of Engineering, Computer Science and Mathematics, University of Exeter, Exeter, EX4 4QE, UK.
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Abstract

Some spatial averaging of the Coriolis terms is unavoidable on the staggered C-grid, and the resulting artifical slowing of near-grid-scale inertial waves in numerical models is well-known. A poor treatment of near-grid-scale Rossby waves might also be expected. It is shown here that numerical Rossby wave dispersion on the C-grid is sensitive to the details of the discretization of the Coriolis terms, and that quite good Rossby wave dispersion can be obtained even for near-grid-scale meridional wavelengths. Copyright © 2007 Royal Meteorological Society

1. Introduction

Previous studies have shown that the dispersion properties of various kinds of waves in numerical models of stratified, rotating fluids, such as the atmosphere and ocean, can be improved by storing the prognostic variables on a staggered grid (e.g., Arakawa and Lamb, 1977; Williams, 1981; Arakawa and Moorthi, 1988; Fox-Rabinovitz, 1991, 1996; Randall, 1994; Ringler and Randall, 2002; Ničković et al., 2002; Thuburn and Woollings, 2005; Thuburn, 2006). Various grid staggerings are possible. In the horizontal, the C-grid arrangement of variables (Fig. 1) is known to capture well the propagation of inertio-gravity waves, and hence the process of geostrophic adjustment, provided the Rossby radius is well resolved (Arakawa and Lamb, 1977). The disadvantage of the C-grid is that the u and v wind components are not colocated, so some averaging is unavoidable in the Coriolis terms. For this reason the C-grid poorly represents near-grid-scale inertio-gravity waves when the grid spacing is greater than the Rossby radius, for which the Coriolis terms are dominant.

Figure 1.

Schematic showing the placement of the prognostic variables u and v (the x and y components of the velocity respectively) and Φ (the geopotential) on the C-grid

The Coriolis terms, in particular, the northward gradient β of the Coriolis parameter f, are also crucial for the dynamics of Rossby waves. It might therefore be expected that the unavoidable averaging of Coriolis terms on the C-grid will inevitably lead to a poor representation of the propagation of near-grid-scale Rossby waves. Indeed, previous theoretical studies (Wajsowicz, 1986; Dukowicz, 1995; Gavrilov and Tošić, 1998, 1999) seem to support this, predicting that Rossby wave frequencies will rapidly approach zero as the wavelength approaches twice the grid spacing. However, some numerical calculations of normal mode frequencies for C-grid discretizations of the shallow water equations (Thuburn and Staniforth, 2004) found quite good representation of Rossby wave propagation, with none of the predicted retardation for short-meridional wavelengths. There is a strong motivation for trying to understand this apparent improvement of the numerical results over the exisiting theory, as this could allow the systematic construction of numerical models with improved Rossby wave propagation properties; that is the topic of this article.

There were several differences between the previous theoretical studies cited and the Thuburn and Staniforth (2004) numerical study. The theoretical studies all used quasigeostrophic theory in β-plane geometry, whereas the numerical study did not make the quasigeostrophic approximation and used spherical geometry. However, as discussed in detail below, the crucial difference is in the details of the way the Coriolis terms are discretized: The Thuburn and Staniforth (2004) work was based on energy-conserving discretizations of the Coriolis terms, whereas the theoretical studies were not. In section 2 below an analytical approximation to the dispersion relation for the linearized β-plane shallow water equations is derived, from which approximate frequencies for the inertio-gravity and Rossby wave frequencies are obtained. The derivation is then repeated in section 3 for various C-grid discretizations, showing that the numerical Rossby wave propagation is indeed sensitive to the details of the way the Coriolis terms are discretized. Some of the ramifications are discussed in section 4.

2. Analytical dispersion relation

To obtain an analytical dispersion relation we will work in planar geometry, and require a set of constant-coefficient governing equations so that solutions proportional to exp{i(kx + ly − ωt)} exist, where x and y are the horizontal coordinates with k and l the corresponding wavenumbers, where t is time, and ω is frequency. One approach is to make the f-plane approximation, i.e. assume constant f, to examine the inertio-gravity waves, and make the quasigeostrophic β-plane approximation (e.g. Wajsowicz, 1986; Dukowicz, 1995; Gavrilov and Tošić, 1998, 1999) to examine the Rossby waves. An alternative is to make a rather weaker, WKB-like approximation, which allows both sets of waves to be examined under the same approximation. We will follow the second approach, because it is instructive and does not appear to be widely known. Both approaches lead eventually to the same answers.

Start with the shallow water equations, linearized about a state of rest and mean geopotential Φ0:

equation image(1)
equation image(2)
equation image(3)

where u and v are the velocity components in the x and y directions respectively, Φ is the geopotential perturbation, f is the Coriolis parameter, which for now is regarded as a function of y, and subscripts x, y and t indicate partial derivatives. Taking ∂/∂x of (2) minus ∂/∂y of (1) gives an equation for the relative vorticity ξ,

equation image(4)

while taking ∂/∂x of (1) plus ∂/∂y of (2) gives an equation for the divergence δ,

equation image(5)

where β = fy.

Having brought out the β terms, we would now like to set f and β to constants to facilitate analytical solution. However, the resulting system would no longer be energy-conserving and would support unstable solutions. The problem can be avoided (White, 2002) if, at the same time, we replace βv in (4) by βvrot and βu in (5) by βudiv, where subscripts ‘rot’ and ‘div’ indicate the rotational and divergent components of the velocity respectively. These replacements can be justified if the horizontal scale of the velocity field is much smaller than the scale over which f varies, for then |βvdiv|≪|fδ| and |βurot|≪|fξ|. Thus, this approach has something of the flavor of a WKB approximation.

The resulting constant coefficient equations then have solutions proportional to exp{i(kx + ly − ωt)}. Substituting such solutions in (4) and (5) then rearranging leads to modified equations for u and v that include β terms:

equation image(6)
equation image(7)

where K2 = k2 + l2.

Equations (6) and (7) may be combined to obtain separate expressions for u and v in terms of Φ. Substituting these in the mass continuity equation then leads to the dispersion relation:

equation image(8)

This is a cubic equation for ω; for any given values of the other parameters, there are three roots for ω, corresponding to eastward and westward propagating inertio-gravity waves and a westward propagating Rossby wave. The inertio-gravity wave solutions can be isolated by setting β = 0 and ignoring the ω = 0 root, leading to

equation image(9)

which is exactly the result obtained by making the f-plane approximation in (1)–(3). The Rossby wave solution can be isolated by neglecting (ω+ kβ/K2)2 compared to f2 in (8), leading to

equation image(10)

which is exactly the result obtained by making the quasigeostrophic β-plane approximation in (1)–(3). Note that the β term in (10), which is crucial for determining the Rossby wave frequency, comes from the last term in (8), which in turn comes from a y-derivative of fv.

3. Dispersion relations for C-grid discretizations

We now repeat the derivation of section 2 for various C-grid discretizations of (1)–(3). We assume a uniform grid interval d in both the x and y directions. For solutions proportional to exp{i(kx + ly − ωt)}, a finite difference x-derivative, e.g.

equation image(11)

where j is the x-index, is equal to Sx times the true derivative ikΦj where Sx = sin(kd/2)/(kd/2). Similarly, a finite difference y-derivative picks up a factor Sy = sin(ld/2)/(ld/2), while any term averaged in the x-direction or y-direction picks up a factor Cx = cos(kd/2) or Cy = cos(ld/2) respectively. Of particular interest is the behavior of near-grid-scale waves. As kd→π, Sx→2/π, whereas Cx→0, with similar results for small scales in the y-direction. Thus, the appearance of Cx or Cy in dominant terms of the numerical dispersion relation will indicate poor representation of near-grid-scale waves.

First consider the C-grid scheme analyzed by Wajsowicz (1986), Dukowicz (1995), and Gavrilov and Tošić (1998, 1999). The corresponding momentum equations are

equation image(12)
equation image(13)

where the overbar indicates an average with the superscripts indicating the direction of the average. The Coriolis parameter f is evaluated at u points in the u equation and at v points in the v equation.

As suggested in section 2, the Rossby wave dispersion properties depend crucially on what happens under discretization to the (fv)y term that comes from taking ∂/∂y of (1). In this case, this term must be split up as

equation image(14)

Upon repeating the derivation of section 2 for the discrete case, the dispersion relation is found to be

equation image(15)

where = Sxk and equation image. Picking the Rossby wave branch gives

equation image(16)

in agreement with the results published previously. The heavy averaging of v in the β term in (14) leads to the equation image factor in the numerator of (16), which implies drastic slowing of Rossby wave frequencies for short zonal and particularly short-meridional wavelengths.

This theoretical prediction is confirmed by numerical calculation of normal mode frequencies, following the method of Thuburn and Staniforth (2004) but for a β-plane channel rather than a sphere (Fig. 2). Here a solution proportional to exp(ikx) is assumed and the equations are discretized only in the y-direction. For short-meridional wavelengths the Rossby waves are indeed drastically slowed.

Figure 2.

Numerical normal mode frequencies (s−1, crosses) for a β-plane channel discretized in the y-direction using (12), (13). Increasing latitudinal mode number indicates decreasing latitudinal scale. The channel width is aπ/3 where a = 6371220 m is the Earth's radius; β = 1.619 × 10−11 m−1s−1 and f at the channel centre is 1.031 × 10−4 s−1 corresponding to a tangent plane at (45°N); Φ0 = 105 m2s−2; and the zonal wavenumber k = 2/a. For comparison, the diamonds show the analytical mode frequencies given by (9) and (10). Differences between analytical and numerical frequencies for small mode number are because the analytical frequencies ignore the effects of the channel walls; differences for large mode number are due to discretization errors. Only westward propagating modes are shown

This scheme may be compared with an alternative in which f is defined only at u points and the entire term fu is averaged in the v equation:

equation image(17)
equation image(18)

This form of the Coriolis terms is one member of a general family of energy-conserving forms discussed by Arakawa and Lamb (1981). In this case, the crucial (fv)y term must be split up in a different way (compare (14)) as

equation image(19)

and the corresponding discrete dispersion relation is

equation image(20)

with Rossby wave branch

equation image(21)

The β term in (19) is much less heavily averaged than in (14), so that Cy does not appear in the numerator of (21), and there is no slowing of short-meridional wavelength Rossby waves. Again, this prediction is confirmed by numerical calculation of normal mode frequencies (Fig. 3).

Figure 3.

As in Fig. 2 but using (17), (18)

An alternative energy-conserving form of the Coriolis terms is obtained by evaluating f only at v points, so that the entire fv term is averaged in the u equation:

equation image(22)
equation image(23)

Now the (fv)y term must be split up as

equation image(24)

(where β has been taken as constant) and the corresponding discrete dispersion relation is found to be

equation image(25)

with Rossby wave branch

equation image(26)

For large meridional scales, the factor (equation image) in the numerator of (26) is close to (1), but for short-meridional scales it approaches − 1; thus short-meridional-scale Rossby waves will propagate eastward. Again, this prediction is confirmed by numerical calculation of normal mode frequencies (Fig. 4).

Figure 4.

As in Fig. 2 but using (22), (23). Both eastward and westward propagating modes are shown to highlight the spurious eastward propagation of short-meridional-wavelength Rossby waves

Thuburn and Staniforth (2004) claimed good Rossby wave dispersion for both the f-at-u-points and f-at-v-points energy-conserving forms of the Coriolis terms. The origin of this contradiction with the results presented here was eventually traced to a coding error in the numerical normal mode calculations of Thuburn and Staniforth (2004): the term fvy had in fact been evaluated as

equation image(27)

where here j is the y-index. This erroneous discretization in fact gives far better Rossby wave propagation than the intended discretization (which is why it remained unnoticed). It may be analyzed in the same way as the schemes discussed above. In this case, the (fv)y term must be split up as

equation image(28)

leading to dispersion relation

equation image(29)

with Rossby wave branch

equation image(30)

(which is idential to (21)). Thus, this discretization is predicted to give good Rossby wave propagation even for short-meridional scales, as found in the numerical experiments of Thuburn and Staniforth (2004). Numerical results for the β-plane case are shown in Fig. 5.

Figure 5.

As in Fig. 2 but using (22) and (23) with the fvy term replaced by (27)

4. Discussion

Numerical Rossby wave propagation on the C-grid for short-meridional wavelengths is found to be sensitive to the details of the discretization of the Coriolis terms. For a uniform grid, the f-at-u-latitudes energy-conserving discretization has good Rossby wave propagation even for short meridional wavelengths, whereas the f-at-v-latitudes energy-conserving discretization has even worse Rossby wave propagation than the scheme examined by Wajsowicz (1986), Dukowicz (1995), and Gavrilov and Tošić (1998, 1999). With (27) replacing the fvy term in (22), good Rossby wave propagation is restored; however, the resulting scheme is not energy-conserving (though it remains second-order accurate); thus the apparent link between energy conservation and good Rossby wave propagation suggested by Thuburn and Staniforth (2004) is not borne out by closer analysis. On current understanding, the good Rossby wave propagation of some schemes rather than others appears to be nothing more profound than an accident of the algebra.

Note that all of the results shown are for the case of grid spacing smaller than Rossby radius. For grid spacing greater than the Rossby radius, this analysis predicts that all of the C-grid schemes examined will badly handle near-grid-scale Rossby waves, because of the equation image factor in the denominator of the numerical dispersion relation. Numerical calculation of mode frequencies (not shown) confirms this prediction.

Note also that all four of the numerical Rossby wave dispersion relations (16), (21), (26), and (30), include a factor Cx; thus all of these schemes will retard short-zonal-wavelength Rossby waves. There seems to be no possibility of avoiding this problem on the C-grid when f is a function only of y. One way to see this is as follows. The crucial βv term arises in the vorticity equation (4). On the C-grid the natural vorticity points are the cell corners at v-latitudes and u-longitudes. Thus, some longitudinal averaging of v to the vorticity points is unavoidable.

On a rotated grid, however, where the y direction is no longer northward, f would be a function of both x and y. The natural extension of the good f-at-u-latitudes scheme would be to evaluate f only at Φ points; the momentum equations (with energy-conserving Coriolis terms) would then become

equation image(31)
equation image(32)

If we write fx = α, fy = β and repeat the analysis of section 3 then we obtain the numerical Rossby wave dispersion relation

equation image(33)

where = Syl. Some fraction of the Coriolis-gradient effect will be captured unless the wave is near-grid-scale in both the x and y directions.

It is interesting to compare with the B-grid, for which there is no averaging of the Coriolis terms and there is essentially only one sensible choice for evaluating f. Then the numerical Rossby wave dispersion relation is as given by Wajsowicz (1986), Dukowicz (1995), and Gavrilov and Tošić (1998, 1999), which in the present notation becomes

equation image(34)

Just as for the best C-grid schemes, the B-grid can give good Rossby wave propagation even for short-meridional wavelengths, though not for short zonal wavelengths. However, the B-grid works well only when the grid spacing is coarser than the Rossby radius, whereas the C-grid works well only when the grid spacing is finer than the Rossby radius.

Although numerical Rossby wave propagation is sensitive to the details of the discretization of the Coriolis terms, numerical inertio-gravity wave propagation is not so sensitive. All of the schemes examined here have essentially the same inertio-gravity wave dispersion properties as found in the Arakawa and Lamb (1977) f-plane analysis.

Finally, note that the analysis and numerical results presented here are restricted to planar geometry. Additional issues arise when spherical geometry is introduced, as discussed by Thuburn and Staniforth (2004).

Acknowledgements

Discussions with Andrew Staniforth and Nigel Wood were helpful during the course of this work. This work was supported in part by the Met Office.

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