The Roles of the Horizontal Component of the Earth's Angular Velocity in Nonhydrostatic Linear Models

Akira Kasahara National Center for Atmospheric Research,* Boulder, Colorado

Search for other papers by Akira Kasahara in
Current site
Google Scholar
PubMed
Close
Full access

Abstract

Roles of the horizontal component of the earth's rotation, which is neglected traditionally in atmospheric and oceanographic models, are studied through the normal mode analysis of a compressible and stratified model on a tangent plane in the domain that is periodic in the zonal and meridional directions but bounded at the top and bottom. As expected, there exist two distinct kinds of acoustic and buoyancy oscillations that are modified by the earth's rotation. When the cos(latitude) Coriolis terms are included, there exists another kind of wave oscillation whose frequencies are very close to the inertial frequency, 2Ω sin(latitude), where Ω is the earth's angular velocity.

The objective of this article is to clarify the circumstance in which a distinct kind of wave oscillation emerges whose frequencies are very close to the inertial frequency. Because this particular kind of normal mode appears only due to the presence of boundary conditions in the vertical, it may be appropriate to call these waves boundary-induced inertial (BII) modes as demonstrated through the normal mode analyses of a homogeneous and incompressible model and a Boussinesq model with thermal stratification. Thus, it can be understood that the BII modes can coexist with the acoustic and inertio-gravity modes when the effect of compressibility is added to the effects of buoyancy and complete Coriolis force in the compressible, stratified, and rotating model.

Corresponding author address: Dr. Akira Kasahara, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000. Email: kasahara@ucar.edu

Abstract

Roles of the horizontal component of the earth's rotation, which is neglected traditionally in atmospheric and oceanographic models, are studied through the normal mode analysis of a compressible and stratified model on a tangent plane in the domain that is periodic in the zonal and meridional directions but bounded at the top and bottom. As expected, there exist two distinct kinds of acoustic and buoyancy oscillations that are modified by the earth's rotation. When the cos(latitude) Coriolis terms are included, there exists another kind of wave oscillation whose frequencies are very close to the inertial frequency, 2Ω sin(latitude), where Ω is the earth's angular velocity.

The objective of this article is to clarify the circumstance in which a distinct kind of wave oscillation emerges whose frequencies are very close to the inertial frequency. Because this particular kind of normal mode appears only due to the presence of boundary conditions in the vertical, it may be appropriate to call these waves boundary-induced inertial (BII) modes as demonstrated through the normal mode analyses of a homogeneous and incompressible model and a Boussinesq model with thermal stratification. Thus, it can be understood that the BII modes can coexist with the acoustic and inertio-gravity modes when the effect of compressibility is added to the effects of buoyancy and complete Coriolis force in the compressible, stratified, and rotating model.

Corresponding author address: Dr. Akira Kasahara, National Center for Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000. Email: kasahara@ucar.edu

1. Introduction

Current atmospheric models for weather prediction and climate simulation are mostly based on the hydrostatic primitive equations (e.g., Phillips 1973) that are derived from the Eulerian system of equations with some “traditional” assumptions. One of the major assumptions is referred to as “shallowness approximation” based on the notion that the vertical extent of the atmosphere of our interest is rather small compared with the earth's radius. Another assumption is that the vertical acceleration is negligible in the vertical equation of motion. This is a reasonable approximation as far as large-scale motions are concerned and has a benefit of eliminating the vertical propagation of acoustic waves, so that a small vertical grid increment does not overly restrict the choice of time step in explicit discretized calculations.

With the increase of computing power and data storage capacity, we will be able to refine the model resolutions to incorporate smaller-scale motions, as well as to extend the top of model atmosphere to improve the calculation of flows and the quality of observational data assimilations. An extensive discussion is presented by White and Bromley (1995) as a critique to hydrostatic primitive equation (HPE) formulations. A similar critical review is presented also by Marshall et al. (1997) concerning the use of hydrostatic and shallowness approximations for ocean modeling. One of the dynamical consequences of the shallowness approximation is the omission of Coriolis terms involving 2Ω cos(latitude), where Ω is the angular rotation rate of the earth, which appear in the vertical and zonal equations of motion (Phillips 1966). White and Bromley (1995) argue through a scale analysis that the effects of cos(latitude) or simply cosine Coriolis terms may attain magnitudes of as much as 10% of major terms for both planetary-scale and diabatically driven tropical motions. These terms are now included in the numerical prediction models developed at the Met Office (Davies 2000). However, no physical mechanism is offered through which the cosine Coriolis terms may play important roles.

The dynamical effects of cosine Coriolis terms on fluid motions have been studied mostly in connection with boundary layer flows. For example, Wippermann (1969) and Leibovich and Lele (1985) investigated these effects in the problem of Ekman layer instability. Since vertical and horizontal perturbation speeds are comparable in the Ekman layer, the cosine Coriolis terms are on the same order of magnitude of the sine Coriolis terms and both are relevant to the stability problem, although the cosine Coriolis terms play little role in the unperturbed basic Ekman layer velocity field. Wang et al. (1996) considered the effect of cosine Coriolis terms on entraining equatorial ocean boundary layers. They concluded that the importance of cosine Coriolis terms may depend on the thermal stratification of the boundary layer. If stratification is neutral or unstable without a thermocline, the effects of cosine Coriolis terms can be significant, as suggested by Hassid and Galperin (1994).

Concerning the dynamical effects of cosine Coriolis terms on atmospheric oscillations, Eckart (1960) investigated the normal modes of a stratified and compressible atmosphere in Cartesian coordinates on a tangent plane with a complete representation of the Coriolis force. He discussed the solutions of Lamb waves as the external mode, but left the solutions of internal modes unexplored. Eckart (1960, 134–135) concluded,

These calculations are by no means complete. It would be possible to discuss the properties of the simple waves—their phase and group velocities, the impedances, etc.,—but the algebraic complexities would be great. These two sample calculations indicate one thing, however: there are effects that depend on [2Ω cos(latitude)], and these can be very marked for frequencies in the neighbourhood of [frequency = 2Ω sin(latitude)].

It is well known that there are two kinds of oscillations in the stratified and compressible model on a tangent plane, including only the vertical component of the earth's rotation vector, which is treated as constant (e.g., Eckart 1960; Monin and Obukhov 1959; Gill 1982). One corresponds to high-frequency acoustic waves and the other to low-frequency inertio-gravity waves. However, when we carried out the internal mode solutions of the Eckart model with cosine Coriolis terms under an isothermal condition, we found that there is another kind of wave oscillation whose frequencies are very close to the inertial frequency [2Ω sin(latitude)], in addition to the acoustic and inertio-gravity modes. [There are Lamb waves too, in this case, which are somewhat unusual, and Eckart (1960) discussed their properties in detail.] We are not aware of any discussion on this kind of near-inertial frequency waves in the literature, except that Egger (1999) pointed out the necessity of cos Coriolis terms to give rise to the oscillations whose frequencies are very close to the inertial frequency. [See appendix.] We will come back to this point in section 5.

The objective of this article is to clarify the circumstance in which near-inertial frequency waves appear as a distinct kind. We begin our discussion in section 2 on the role of cosine Coriolis terms in a homogeneous and incompressible model on a tangent plane. Through the normal mode analysis of this model, it is shown that there is a possibility of having a distinct kind of wave oscillations due to the presence of boundary conditions in the vertical in addition to inertial waves. In section 3, we analyze the normal modes of a Boussinesq model with thermal stratification. We notice that the distinct kind of modes found in section 2 also appear in the model and their frequencies are very close to the inertial frequency. Therefore, it may be appropriate to call this distinct kind of mode the boundary-induced inertial (BII) modes, because the role of the boundary conditions is essential. In section 4, the stratified and compressible model with a complete representation of Coriolis effects is considered. Since compressibility is now added to the Coriolis and buoyancy effects, it is natural to expect the emergence of acoustic modes in addition to those modes discussed in section 3. The properties of the modal frequencies of the compressible and stratified model are discussed in section 5. Conclusions and further discussions are presented in section 6.

2. Incompressible and homogeneous model

We consider the small-amplitude oscillations of an incompressible and homogeneous (the density ρ = constant) model, including the vertical and horizontal components of the Coriolis vector 2Ω in the Cartesian coordinates (x, y, z, t) on a tangent plane, with x, y, and z directed eastward, northward, and upward, and t being time. The equations of motion for the velocity components (u, υ, w) corresponding to (x, y, z) with the pressure p and the mass continuity equation are written as
i1520-0469-60-8-1085-e21
where
fVϕ,fHϕ,
with ϕ being the latitude of the coordinate center. Here, fV and fH are assumed to be constant.
We consider the solutions of (2.1)–(2.4) in a three-dimensional domain which is periodic in x and y and bounded by the bottom at z = 0 and the top at z = zT, where we assume that
wzzT
Since Eqs. (2.1)–(2.4) are of constant coefficients, we seek wave solutions in the form
i1520-0469-60-8-1085-e27
where U, V, W, P are functions of z only, and m and n are the wavenumbers in x and y. Also, σ denotes the frequency and can be shown to be real. Here, i = −1.
Thus, we obtain from (2.1)–(2.4) the following:
i1520-0469-60-8-1085-e28
From (2.8) and (2.9), we can express U and V in terms of W and P as
i1520-0469-60-8-1085-e212
where we assume that σ ≠ 0 and (f2Vσ2) ≠ 0 to obtain nontrivial transient solutions.
We now eliminate U from (2.10) using (2.12) and get
i1520-0469-60-8-1085-e214
Likewise, we eliminate U and V from (2.11) and get
i1520-0469-60-8-1085-e215
We then derive the vertical structure equation of W by eliminating P between (2.14) and (2.15) and obtain
i1520-0469-60-8-1085-e216
Our task now is to solve (2.16) under the boundary conditions
WzzzT
which are derived from (2.6).
To eliminate the first-order term of W in (2.16), we use the following transformation:
Wzηzi2z
where
i1520-0469-60-8-1085-e219
Substitution of (2.18) into (2.16) leads to the following equation for η:
i1520-0469-60-8-1085-e220
The solutions of (2.20) that satisfy the boundary conditions (2.17) are given by
i1520-0469-60-8-1085-e221
where Ak denotes the coefficient, k the vertical wavenumber, and ki the vertical wave index.
To determine the coefficient Ak, we express P(z) as
Pzξzi2z
By substituting (2.18) and (2.23) into (2.15), and remembering that (f2Vσ2) ≠ 0, we get
i1520-0469-60-8-1085-e224
where
i1520-0469-60-8-1085-e225
Now, substitution of (2.21) into (2.24) yields
i1520-0469-60-8-1085-e226
The value of ξ at z = 0, namely ξ(0), can be obtained from the distribution of ρ−1p at z = 0. Therefore, the coefficient Ak can be determined from (2.26) in terms of ξ(0) as
i1520-0469-60-8-1085-e227
Thus, (2.21) can be expressed, using (2.27) as,
i1520-0469-60-8-1085-e228
Notice that the value of η at the limit of k → 0, which occurs when zT → ∞ in (2.22), is finite and nonzero unless σ = 0. Therefore, k = 0 can be considered the lowest vertical internal mode.
The frequency of normal modes can be determined by substituting (2.21) into (2.20). The result becomes, again remembering (f2Vσ2) ≠ 0,
i1520-0469-60-8-1085-e229
The solutions of (2.29) are given by
i1520-0469-60-8-1085-e230
There are two real solutions σ2 of different magnitudes. Each solution of σ2 has a pair of positive and negative values of σ with the same magnitude.
In the case of fV ≠ 0 and fH = 0, we find from (2.30) that
σ2+f2V
corresponding to the plus sign in front of the radical of (2.30). For the minus sign, we have
i1520-0469-60-8-1085-e232
Obviously, we must reject the σ2+ solutions (2.31), which contradict with the assumption that (f2Vσ2) should not vanish. However, the system (2.8)–(2.11) can allow the solutions of σ2 = f2V if m = n = 0 with P = W = 0 and U = ±V. This kind of solution is referred to as the inertial oscillations (e.g., Durran 1993). Therefore, in the case of fV ≠ 0 and fH = 0, the only valid wave oscillations when m2 + n2 ≠ 0 are represented by ±σ of (2.32) and are referred to usually as the inertial waves (e.g., Tolstoy 1973).

In the general case of fV ≠ 0 and fH ≠ 0, we see from (2.30) that two kinds of solutions, represented by the higher value of σ2+ and the lower value of σ2, exist unless the meridional mode n = 0. If n = 0, only the traditional type of (2.32) with n = 0 is obtained. This can be seen easily from (2.29) as the fH appears only in association with n.

It may be instructive to derive approximations to the two kinds of the solutions of (2.29) when f2Vf2H and n2, m2, and k2 are on the same order of magnitudes. Since σ2+ is close to f2V and σ2 to k2f2V/(m2 + n2 + k2), we can find approximate expressions of σ2+ and σ2 by a perturbation method as follows:
i1520-0469-60-8-1085-e233
By comparing (2.33) with (2.31), we find that ±σ+ are now valid wave solutions because of the presence of fH terms.
The actual values of σ2+ and σ2 of the general case can be calculated from (2.30). The maximum of σ2 becomes f2V + f2H = (2Ω)2 for m = k = 0 with n ≠ 0, and the minimum of σ2 approaches zero for large values of m and n. Thus, two kinds of σ2 represented by ±σ+ and ±σ together produce a spectrum of discrete frequencies, spanning from 2Ω to −2Ω, namely
2σ2+σ2
for various combinations of zonal, meridional, and vertical wavenumbers.
We should emphasize that the emergence of the two kinds of σ2 in the general case arises from the imposition of the boundary condition (2.17). If we disregard these boundary conditions, then the plane wave solutions of Eq. (2.16) for W can be expressed proportionally to exp(ikz) and substitution of this form into (2.16) yields only that
i1520-0469-60-8-1085-e236
This dispersion equation is clearly different from that given by (2.29). It is worthwhile to note that the frequency σ, as given by (2.36), depends on the sign of k/n if fH ≠ 0. Therefore, upward and downward propagating plane wave solutions of the form exp [i(mx + ny ± kz)] cannot be combined to satisfy the boundary condition W = 0 at z = 0 because they have different frequencies. Thus, the solutions of (2.16) must have the form exp[i(mx + ny) + i2 ± k)] to satisfy the boundary condition W = 0 at z = 0, because then the frequencies of incident and reflected waves at the boundary z = 0 are identical as given by the solutions of the dispersion equation (2.29). In the case of fH = 0, (2.36) is reduced to (2.32) and the imposition of the boundary conditions does not create the need of different solutions from the plane wave solutions.
Before leaving this section, we should point out that, if σ = 0, the system of (2.8)–(2.11) is satisfied by the following steady-state “geostrophic” solutions:
i1520-0469-60-8-1085-e237
On the other hand, if σ2 = f2V and m2 + n2 ≠ 0, then we can derive a first-order homogeneous equation for W from (2.8)–(2.11). But, due to the boundary condition that W = 0 at z = 0, we find that W = 0. Then, we can show that P = 0, V = 0, and U = 0, successively. If σ2 = f2V and m = n = 0, we find that U = ±V from (2.9), W = 0 from (2.8). Then, P can be determined from (2.10) as the solution of dP/dz = fHU with a suitable boundary condition on z (Kamenkovich and Kulakov 1977).

3. Boussinesq model

In order to include the effect of buoyancy in the previous example, we consider the following Boussinesq model often adopted in oceanography in connection with the study of inertial motions (e.g., Munk and Phillips 1968; Pollard 1970), but we include both the vertical and horizontal components of the Coriolis vector:
i1520-0469-60-8-1085-e31
where
i1520-0469-60-8-1085-e36
and g denotes gravity constant. The boundary conditions are the same as (2.6).
We seek the solutions of (3.1)–(3.5) in the form
i1520-0469-60-8-1085-e38
By substituting (3.8) into (3.1)–(3.5), we obtain
i1520-0469-60-8-1085-e39
The boundary conditions are the same as (2.17).
From this point, we assume that N in (3.13) is constant. After elimination of the variables U, V, S, and P, we can derive the vertical structure equation for W in a similar manner as we derived (2.16). The result is
i1520-0469-60-8-1085-e314
It is clear that (3.14) reduces to (2.16) in the case of N = 0, corresponding to the reduction of the Boussinesq model (3.9)–(3.13) to the homogeneous model (2.8)–(2.11). Thus, by substituting (2.18) into (3.14), we obtain
i1520-0469-60-8-1085-e315
The solutions of (3.15) that satisfy the boundary conditions (2.17) are given by the same form as (2.21). Therefore, by substituting the eigensolutions in the form of (2.21) into (3.15) and remembering that (f2Vσ2) ≠ 0, we get
i1520-0469-60-8-1085-e316
Since (3.16) is a quadratic equation for σ2 similar to (2.29), we can easily obtain the explicit form of solutions like (2.30). However, because usually N2 ≫ Ω2, we can obtain fairly accurate solutions by approximation.
For the high-frequency approximate solutions of (3.16), we get
i1520-0469-60-8-1085-e317
It is clear that these frequencies correspond to the inertio-gravity modes, modified by the fH terms, as explained below.
In the case of fV ≠ 0 and fH = 0, we can factor out (3.16) by (σ2f2V), which should not vanish. Therefore, we have the only solutions in the form
i1520-0469-60-8-1085-e318
corresponding to the inertio-gravity modes (e.g., Monin and Obukhov 1959; Gill 1982).
For the low-frequency solutions of (3.16), because normally N2 ≫ Ω2, we expect by inspection of (3.16) that σ2 is close to f2V. Therefore, by substituting σ = fV + Δσ into (3.16) and retaining only the first-order terms of Δσ, we obtain the following approximate solutions:
σIfVf2Hn2E−11
where
E1m2n2N2f2Vn2f2H

Because (3.19) is an approximate solution, σI does not depend on the vertical wavenumber k at this level of approximation. Nevertheless, as can be verified numerically, the formula (3.19) gives accurate results in general. Moreover, the fact that σI is present only when n ≠ 0 is the exact result. If n = 0, (σ2f2V) can be factored out from (3.16). Since we have assumed that (σ2f2V) should not vanish, no wave solution corresponding to σI exists if n = 0. The magnitude of σI is very close to |fV|, because the correction term f2Hn2E−11 in (3.19) is normally on the order of 10−4.

We should again reiterate that the imposition of the boundary conditions is responsible to produce the two kinds of solutions in this model. If we disregard the boundary conditions, then the plane wave solutions of Eq. (3.14) can be expressed proportionally to exp(ikz), and substitution of this form into (3.14) gives only
i1520-0469-60-8-1085-e321

These frequencies correspond to those of inertio-gravity waves including the effect of fH terms. However, when the vertical boundary conditions are imposed, the eigensolutions of (3.14) must take the form proportional to exp[i2 + k)z] to satisfy the boundary conditions and the dispersion equation becomes (3.16), which is clearly different from (3.21). Note that (3.21) does depend on the sign of k/n if fH ≠ 0, while (3.16) does not. Moreover, the high-frequency solutions σ2g of (3.16) have the form similar to (3.21), while the low-frequency solutions σ2I emerge as a variation to the forbidden wave solutions of σ2 = f2V. Since the wave oscillations corresponding to σ2I are uniquely associated with the boundary conditions, it may be appropriate to refer to this kind of normal mode as the BII mode.

4. Compressible and stratified model

Finally, we consider small-amplitude oscillations of a stratified and compressible model in the Cartesian coordinates (x, y, z, t) on a tangent plane, including the vertical and horizontal components of the Coriolis vector. This problem was initiated by Eckart (1960) who discussed the solutions of Lamb waves, but the discussion on the internal modes was not completed. Here, we obtain the solutions of internal modes of this problem using the same approach as in the previous two sections, but the inclusion of the compressibility of fluid adds some complexity in treatment.

The basic system consists of linearized equations for the momentum, the mass continuity, and the law of thermodynamics. The basic states are assumed to be at rest with temperature To(z), pressure po(z), and density ρo(z) that are in hydrostatic equilibrium dpo/dz = −ρog, where To is defined through the equation of state, po = ρoRTo, and R denotes the gas constant.

The basic equations for the perturbation variables of the velocity components (u, υ, w), the pressure p, and the density ρ are expressed by
i1520-0469-60-8-1085-e41
where a new variable s is defined by
i1520-0469-60-8-1085-e46
The use of the variable s instead of ρ by (4.6) helps the derivation of perturbation energy equation, and s is related to perturbation of the logarithm of potential temperature (Gill 1982).
The basic-state parameters are
i1520-0469-60-8-1085-e47
where Cs denotes the speed of sound, with γ defining the ratio of the two specific heat values at constant pressure and at constant volume γ = Cp/Cυ, and N is the Brunt–Väisälä frequency.
We seek the solutions of (4.1)–(4.5) in the form
i1520-0469-60-8-1085-e48
and obtain
i1520-0469-60-8-1085-e49
where the parameter Γ is defined by
i1520-0469-60-8-1085-e414
By eliminating U, V, and S, we obtain
i1520-0469-60-8-1085-e415
where
i1520-0469-60-8-1085-e417
Equations (4.15) and (4.16) are solved under the boundary conditions that
WzzzT
From this point we will consider the case of isothermal basic state. Thus, the following basic parameters become constant:
i1520-0469-60-8-1085-e422
where
i1520-0469-60-8-1085-e423
Now, (4.15) and (4.16) become equations of constant coefficients. After elimination of P between them, we obtain
i1520-0469-60-8-1085-e424
By introducing the transformation
WηzeiΓ2z
into (4.24), we obtain for η(z) in the form
i1520-0469-60-8-1085-e426
As seen from the two previous sections, the solutions of (4.26) that satisfy the boundary conditions (4.21) are proportional to sin(kz). Thus, substitution of (2.21) into (4.26) gives the following equation for frequency σ, using the definition of Γ1 by (4.17), λ by (4.19), and μ by (4.20),
i1520-0469-60-8-1085-e427

5. Frequency equations for the compressible and stratified model

a. General case with fV ≠ 0 and fH ≠ 0

In order to derive the algebraic equation of σ from (4.27), we must assume that f2Vσ2. By multiplying (4.27) by C2s(f2Vσ2)2, we obtain
i1520-0469-60-8-1085-e51
The above is a sixth-order equation of σ, and we must resort to solving it numerically. Before doing so, we consider some simpler cases to identify the nature of oscillations associated with the six roots of (5.1).

b. Traditional case of fV ≠ 0 and fH = 0

If fH = 0, Eq. (5.1) can be factored out by (f2Vσ2), which is assumed to be nonzero. Therefore, the following quartic equation of σ is obtained:
i1520-0469-60-8-1085-e52
Equation (5.2) has the following two pairs of solutions:
i1520-0469-60-8-1085-e53
where σa corresponds to the solutions with the plus sign in front of the radical and σg with the minus sign.

The characteristics of (5.3) have been discussed extensively by Eckart (1960), Monin and Obukhov (1959), and Gill (1982). The high-frequency oscillations corresponding to ±σa are known as acoustic modes. The low-frequency oscillations corresponding to ±σg are referred to as inertio-gravity modes. Of course, the acoustic modes too are modified by the presence of fV in (5.2).

For a later discussion, it is useful to give the solutions of (5.2) in the case of m = n = 0, which is reduced to
i1520-0469-60-8-1085-e54
The solutions of (5.4) are
i1520-0469-60-8-1085-e55
However, the low-frequency solutions σ2g are not acceptable, because these violate the basic assumption of σ2f2V. Thus, only vertically propagating acoustic waves are present, if m = n = 0, but k ≠ 0.

c. Special case of fV = 0 and fH ≠ 0

At the equator, fV vanishes and fH becomes 2Ω. In this case, the frequency equation (5.1) is reduced to the following quartic equation by again assuming that σ does not vanish:
i1520-0469-60-8-1085-e56
It is obvious that there are two pairs of large and small values of σ having positive and negative signs with slightly different magnitudes. These high- and low-frequency pairs correspond to the acoustic and gravity waves, respectively, modified by the earth's rotation.
For the zonal motions m = 0, the roots of (5.6) can be expressed by
i1520-0469-60-8-1085-e57
Furthermore, in the case of n = 0 in addition to m = 0, (5.7) yields
i1520-0469-60-8-1085-e58
Thus, only vertically propagating acoustic waves are present if m = n = 0, but k ≠ 0.

d. Special case of fV ≠ 0 and fH ≠ 0 with m = n = 0 but k ≠ 0

Egger (1999) examined the characteristics of oscillations in a stratified and compressible model identical to ours, including the fV and fH terms, except that he only treated the case of both zonal and meridional wavenumbers being zero; that is, m = n = 0 but k ≠ 0. In this special case, Eq. (5.1) is factored out by (f2Vσ2) which should not vanish, and the remaining fourth-order equation becomes
i1520-0469-60-8-1085-e59
This agrees with the quartic equation derived from Eq. (18) of Egger (1999). To compare the form of (5.9) with his Eq. (18), we introduce
i1520-0469-60-8-1085-e510
for κ = 2/7. The quantity σa0 is referred to as the acoustic cutoff frequency (e.g., Tolstoy 1973; Gossard and Hooke 1975). Hence, (5.9) becomes
i1520-0469-60-8-1085-e511

As noted by Egger (1999), the above equation gives two pairs of frequencies with plus and minus sign. One pair of high frequencies can be approximated by ±(4Ω2 + σ2a0 + C2sk2)1/2. They represent the vertically propagating acoustic waves modified by the earth's rotation. The other pair of low frequencies are close to ±|fV| and represent a kind of inertial wave. As Egger (1999) stated, the role of cos Coriolis terms, as well as compressibility and gravity are needed to recover inertial waves in the case of m = n = 0 but k ≠ 0. This point can be seen by comparing (5.9) with (5.4). The presence of f2H in (5.9) is responsible for giving a slight shift of σ2 away from f2V. Thus, in contrast to the situation of (5.5), the vertically propagating inertial oscillations can emerge even in the case of m = n = 0 but k ≠ 0.

e. Boundary-induced inertial modes

Earlier in section 3, we noted that there are two kinds of normal modes in the Boussinesq model. One kind represents the inertio-gravity modes and an approximate form of the solutions is given by (3.17). The other represents the BII modes whose frequencies are very close to fV and an approximate form of frequencies is given by (3.19) with (3.20). Therefore, it is likely that there exist two roots of (5.1) whose values are close to ±|fV|. We can derive an approximate form of these roots under normal conditions of N2 ≫ Ω2. The result is
σIfVf2Hn2E−12
where
i1520-0469-60-8-1085-e513

We can see from (5.12) that one σI corresponds to fV and another σI corresponds to −fV. Their magnitudes are slightly smaller than |fV|, because the terms f2Hn2E−12, which is assumed to be much smaller than unity, is normally positive. Moreover, the magnitude of positive σI is slightly larger than the magnitude of the corresponding negative σI, except for the case of m = 0 in which case both magnitudes are equal.

It is clear that the forms (3.19) and (5.12) are similar, and E1 and E2 are related. In fact, in the Boussinesq model (Cs → ∞ and Γ = 0), E2 reduces to E1. Therefore, the waves represented by the frequency σI of (5.12) are the BII modes in the sense defined in section 3, modified now by the presence of compressibility and Γ.

We have verified through the numerical solutions of (5.1) that the formula (5.12) with (5.13) gives accurate approximations to the numerical values that correspond to the remaining two roots of (5.1) besides the two pairs of acoustic frequencies σa and inertio-gravity frequencies σg. If n = 0, (σ2f2V) can be factored out from (5.1), and the roots of the remaining quartic equation are identified as the two pairs of acoustic and inertio-gravity modes with n = 0. Since we have assumed that (σ2f2V) should not vanish, no solution corresponding to σI exists if n = 0.

f. Lamb waves

For isothermal atmospheres there are additional special solutions of Eqs. (4.15) and (4.16) with the boundary conditions (4.21). They are
WPe−(Γ1iΓ2)z
Equation (4.16) is satisfied by μ = C2s. Hence, using the definition (4.20) we find
σ2C2sm2n2f2V
In order to ensure that the eigensolutions, known as Lamb waves, decrease exponentially with height, we must choose Γ1 > 0, namely
σfHmC2sm2n2
This condition has been pointed out by Eckart (1960). Note that the waves corresponding to negative σ satisfy (5.16) automatically, but the waves with positive σ must satisfy the following condition:
C2sm2n2f2V1/2fHmC2sm2n2
which is obtained by combining (5.15) and (5.16).
Eckart (1960) examines this unique character of Lamb waves. Note that the amplitude of Lamb waves decreases exponentially, but the direction of propagation is not horizontal. The angle α with the horizontal is given as
i1520-0469-60-8-1085-e518
by using (4.18) and (5.15). For the terrestrial atmosphere, the condition (5.17) is well satisfied and the angle α is negligibly small except for very large-scale motions, never exceeding the matter of few degrees.

6. Conclusions and further discussion

Motivated by the desire to quantitatively assess the role of the horizontal component of the earth's rotation, which is neglected in primitive equation models, normal mode analysis has been performed for a compressible and stratified isothermal atmosphere that includes the cos (latitude) Coriolis terms, referred to as fH terms. It was found that there are three kinds of internal modes. Each kind consists of positive and negative frequencies whose magnitudes are slightly different unless the product mΓ vanishes. The three kinds are acoustic, inertio-gravity, and boundary-induced inertial (BII) modes.

A question has been raised why there are six wave frequencies for a system that consists of five time-dependent equations for five dependent variables. Phillips (1990) analyzed the same system of equations by expressing the solutions in the form of plane waves proportional to exp[i(mx + ny + kzσt)] and obtained the dispersion equation for σ that is a fifth-order polynomial. One of the roots of the frequency equation is σ = 0, which corresponds to steady-state solutions. Four remaining roots represent two pairs of acoustic and gravity waves, both of which are modified by the presence of fV and fH terms.

This article is written to clarify the reasons why there are three kinds of normal modes in a compressible, stratified, and rotating fluid bounded by rigid horizontal surfaces, but only two kinds of plane wave if the fluid is unbounded.

A unique role of the boundary conditions in influencing the normal modes of rotating fluid motions in contrast to plane wave solutions may be explained succinctly using an incompressible and homogeneous model presented in section 2. First, the vertical structure equation (2.16) of W was derived. Because (2.16) contains the first-order derivative term, we introduced transformation (2.18) to eliminate the first-order term and obtained Eq. (2.20), which is solved under the boundary condition (2.17). The origin of a fourth-order term in the frequency equation (2.29) can be traced to the square term Γ22 in Eq. (2.20). The vertical structure functions W are given in the form of exp[i2 ± k)z].

If the same problem is solved without the boundary conditions to obtain plane wave solutions, the frequency equation becomes (2.36), and the vertical structure function W is given by exp(ikz). Note that the frequency σ, as given by (2.36), depends on the sign of k/n if fH ≠ 0. Therefore, upward and downward propagating plane waves cannot be combined to satisfy the boundary condition W = 0 at z = 0 because they have different frequencies. In contrast, the incident and reflected waves of the form exp[i2 ± k)z] can be combined to satisfy the vertical boundary conditions, because the frequency σ, as given by (2.30), does not depend on the sign of k/n.

The same methodology is used to analyze the normal modes of a Boussinesq model in section 3. With the addition of thermal stratification effect in the incompressible and homogeneous model, the dispersion equation became again quartic due to the presence of the boundary conditions in the case of fH ≠ 0. One unique aspect of the Boussinesq model is that, under a normal condition of N2 ≫ Ω2, two sets of the quadratic roots of (3.16) tend to separate well. They are expressed approximately by (3.17) corresponding to the inertio-gravity modes and by (3.19) as the BII modes. Once the circumstance in which the cause of BII modes is understood in the Boussinesq model, it is logical to expect in the compressible, stratified, and rotating model the acoustic modes to emerge in addition to the inertio-gravity and BII modes.

It may be pertinent to comment here on the influence of boundary conditions on the normal modes of rotating fluid. A well-known example is the case of Kelvin mode in a shallow, rotating fluid in a partially bounded channel, oriented parallel to the x axis. Pedlosky (1987, section 3.9) presented a step-by-step procedure to derive the eigensolutions and associated eigenfrequencies and discussed the emergence of the Kelvin mode in addition to inertio-gravity modes, known as the Poincaré waves. Pedlosky also examined the physical significance of the third apparent solution of the dispersion equation as an oscillation whose frequency is the Coriolis parameter fV and showed that such a solution is spurious. This example points out the desirability of examining the influence of the lateral conditions, as well as the vertical conditions on the normal modes of fluid models with a complete representation of the Coriolis force. However, such studies may lead to very complicated analytical problems.

Extensive numerical calculations were conducted to solve the sixth-order polynomial frequency equation (5.1) for various values of wavenumbers, m, n, and k under atmospheric conditions (Kasahara 2001, unpublished manuscript). As expected, the effects of all Coriolis terms have little influence on the frequency of acoustic modes. One interesting result is that the fH terms have significantly more effect than the fV terms on the frequency of inertio-gravity modes, except for very large scale motions. {The degree of influence of the earth's rotation on the frequency of the inertio-gravity waves is measured by the ratio of [σ(fV or fH) − σ(0)]/σ(0), where σ(fV) and σ(fH) denote, respectively, the value of σg at the pole (fV = 2Ω and fH = 0) and the value of σg at the equator (fV = 0 and fH = 2Ω) with σ(0) indicating the gravity wave frequency without the effect of earth's rotation.}

Because of normal conditions of N2 ≫ Ω2, the impacts of earth's rotation on gravity waves are relatively minor. Nevertheless, if accurate calculations of the divergent wind components are desired, then it is unreasonable to neglect the role of fH terms. Likewise, it is not justifiable to neglect the role of fH terms in a stratified model if our interest is in the wave motions whose frequencies are close to the Coriolis frequency. It is not the purpose of this article to analyze the properties of the BII modes, nor to discuss the possibility of this kind of wave motion to exist in the atmosphere and oceans.

Acknowledgments

The author is grateful for encouragement and advice from Professor George Platzman throughout the course of this study. Thanks are also due to Joseph Tribbia, William Large, and Theodore Shepherd for their interest and stimulating discussions on this study. Comments received from two anonymous reviewers were very helpful and contributed to improve the text. The author also appreciates Geoffrey Vallis for his sensible advice and editorial assistance. The manuscript was typed by Barbara Ballard at NCAR.

REFERENCES

  • Davies, T., 2000: Some aspects of high resolution NWP at the Met Office. Proc. ECMWF Workshop on Developments in Numerical Methods for Very High Resolution Global Models, Reading, United Kingdom, European Centre for Medium-Range Weather Forecasts, 35–46. [Available from ECMWF, Shinfield Park, Reading RG2 9AX, United Kingdom.].

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., 1993: Is the Coriolis force really responsible for the inertial oscillation? Bull. Amer. Meteor. Soc., 74 , 21792184.

  • Eckart, C., 1960: Hydrodynamics of Oceans and Atmospheres. Pergamon Press, 290 pp.

  • Egger, J., 1999: Inertial oscillations revisited. J. Atmos. Sci., 56 , 29512954.

  • Gill, A., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Gossard, E. E., and W. H. Hooke, 1975: Waves in the Atmosphere. Elsevier, 456 pp.

  • Hassid, S., and B. Galperin, 1994: Modeling rotation flows with neutral and unstable stratification. J. Geophys. Res., 99 , 1253312548.

    • Search Google Scholar
    • Export Citation
  • Kamenkovich, V. M., and A. V. Kulakov, 1977: Influence of rotation on waves in a stratified ocean. Oceanology, 17 , 260266.

  • Leibovich, S., and S. K. Lele, 1985: The influence of the horizontal component of earth's angular velocity on the instability of the Ekman layer. J. Fluid Mech., 150 , 4187.

    • Search Google Scholar
    • Export Citation
  • Marshall, J., C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res., 102 (C3) 57335752.

    • Search Google Scholar
    • Export Citation
  • Monin, A. S., and A. M. Obukhov, 1959: A note on general classification of motions in a baroclinic atmosphere. Tellus, 11 , 159162.

  • Munk, W., and N. Phillips, 1968: Coherence and band structure of inertial motion in the sea. Rev. Geophys., 6 , 447472.

  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer, 710 pp.

  • Phillips, N. A., 1966: The equations of motion for a shallow rotating atmosphere and the “traditional approximation.”. J. Atmos. Sci., 23 , 626628.

    • Search Google Scholar
    • Export Citation
  • Phillips, N. A., 1973: Principles of large scale numerical weather prediction. Dynamic Meteorology, P. Morel, Ed., Reidel, 1–96.

  • Phillips, N. A., 1990: Dispersion processes in large-scale weather prediction. WMO 700, World Meteorological Organization, 126 pp.

  • Pollard, R. T., 1970: On the generation by winds of inertial waves in the ocean. Deep-Sea Res., 17 , 795812.

  • Thuburn, J., N. Wood, and A. Staniforth, 2002a: Normal modes of deep atmospheres. I: Spherical geometry. Quart. J. Roy. Meteor. Soc., 128 , 17711792.

    • Search Google Scholar
    • Export Citation
  • Thuburn, J., N. Wood, and A. Staniforth, 2002b: Normal modes of deep atmospheres. II: f- F-plane geometry. Quart. J. Roy. Meteor. Soc., 128 , 17931806.

    • Search Google Scholar
    • Export Citation
  • Tolstoy, I., 1973: Wave Propagation. McGraw-Hill, 466 pp.

  • Wang, D., W. G. Large, and J. C. McWilliams, 1996: Large-eddy simulation of the equatorial ocean boundary layer: Diurnal cycling, eddy viscosity, and horizontal rotation. J. Geophys. Res., 101 , 36493662.

    • Search Google Scholar
    • Export Citation
  • White, A. A., and R. A. Bromley, 1995: Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Quart. J. Roy. Meteor. Soc., 121 , 399418.

    • Search Google Scholar
    • Export Citation
  • Wippermann, F., 1969: The orientation of vortices due to instability of Ekman boundary layer. Beitr. Phys. Atmos., 42 , 225244.

APPENDIX

Note on Additional Recent References

During a stage of the review process of this article, the author's attention was called to two articles published recently by Thurburn et al. (2002a,b) related to the normal modes of compressible and stratified atmospheric models with inclusion of both the sin and cosine Coriolis terms. Thuburn et al. (2002a) deals with the model in spherical geometry and the other is concerned with the model in the Cartesian coordinates on a tangent plane in the domain that is periodic in the zonal and meridional directions, but bounded at the top and bottom.

Thus, Thuburn et al. (2002b) deals with exactly the same model as described in section 4 of this article. Therefore, it is no surprise that the results of their normal mode analysis agree in many aspects with those described in sections 4 and 5 of this article and vice versa. For example, the dispersion equation (5.1) in section 5 of this work, which is a sixth-order equation of frequency σ, can be derived from Eq. (4.1) of Thuburn et al. (2002b) by matching the variables, parameters and symbols used in the two works. Also, the boundary-induced inertial modes discussed in section 5e are identical in principle to those of “new modes” discussed in section 5 of their article. Since their method of solution is somewhat different from that used in this work, agreement between the two results will provide independent verification on the existence of this unique kind of wave modes in the present model configuration.

*

The National Center for Atmospheric Research is sponsored by the National Science Foundation.

Save
  • Davies, T., 2000: Some aspects of high resolution NWP at the Met Office. Proc. ECMWF Workshop on Developments in Numerical Methods for Very High Resolution Global Models, Reading, United Kingdom, European Centre for Medium-Range Weather Forecasts, 35–46. [Available from ECMWF, Shinfield Park, Reading RG2 9AX, United Kingdom.].

    • Search Google Scholar
    • Export Citation
  • Durran, D. R., 1993: Is the Coriolis force really responsible for the inertial oscillation? Bull. Amer. Meteor. Soc., 74 , 21792184.

  • Eckart, C., 1960: Hydrodynamics of Oceans and Atmospheres. Pergamon Press, 290 pp.

  • Egger, J., 1999: Inertial oscillations revisited. J. Atmos. Sci., 56 , 29512954.

  • Gill, A., 1982: Atmosphere–Ocean Dynamics. Academic Press, 662 pp.

  • Gossard, E. E., and W. H. Hooke, 1975: Waves in the Atmosphere. Elsevier, 456 pp.

  • Hassid, S., and B. Galperin, 1994: Modeling rotation flows with neutral and unstable stratification. J. Geophys. Res., 99 , 1253312548.

    • Search Google Scholar
    • Export Citation
  • Kamenkovich, V. M., and A. V. Kulakov, 1977: Influence of rotation on waves in a stratified ocean. Oceanology, 17 , 260266.

  • Leibovich, S., and S. K. Lele, 1985: The influence of the horizontal component of earth's angular velocity on the instability of the Ekman layer. J. Fluid Mech., 150 , 4187.

    • Search Google Scholar
    • Export Citation
  • Marshall, J., C. Hill, L. Perelman, and A. Adcroft, 1997: Hydrostatic, quasi-hydrostatic, and nonhydrostatic ocean modeling. J. Geophys. Res., 102 (C3) 57335752.

    • Search Google Scholar
    • Export Citation
  • Monin, A. S., and A. M. Obukhov, 1959: A note on general classification of motions in a baroclinic atmosphere. Tellus, 11 , 159162.

  • Munk, W., and N. Phillips, 1968: Coherence and band structure of inertial motion in the sea. Rev. Geophys., 6 , 447472.

  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer, 710 pp.

  • Phillips, N. A., 1966: The equations of motion for a shallow rotating atmosphere and the “traditional approximation.”. J. Atmos. Sci., 23 , 626628.

    • Search Google Scholar
    • Export Citation
  • Phillips, N. A., 1973: Principles of large scale numerical weather prediction. Dynamic Meteorology, P. Morel, Ed., Reidel, 1–96.

  • Phillips, N. A., 1990: Dispersion processes in large-scale weather prediction. WMO 700, World Meteorological Organization, 126 pp.

  • Pollard, R. T., 1970: On the generation by winds of inertial waves in the ocean. Deep-Sea Res., 17 , 795812.

  • Thuburn, J., N. Wood, and A. Staniforth, 2002a: Normal modes of deep atmospheres. I: Spherical geometry. Quart. J. Roy. Meteor. Soc., 128 , 17711792.

    • Search Google Scholar
    • Export Citation
  • Thuburn, J., N. Wood, and A. Staniforth, 2002b: Normal modes of deep atmospheres. II: f- F-plane geometry. Quart. J. Roy. Meteor. Soc., 128 , 17931806.

    • Search Google Scholar
    • Export Citation
  • Tolstoy, I., 1973: Wave Propagation. McGraw-Hill, 466 pp.

  • Wang, D., W. G. Large, and J. C. McWilliams, 1996: Large-eddy simulation of the equatorial ocean boundary layer: Diurnal cycling, eddy viscosity, and horizontal rotation. J. Geophys. Res., 101 , 36493662.

    • Search Google Scholar
    • Export Citation
  • White, A. A., and R. A. Bromley, 1995: Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Quart. J. Roy. Meteor. Soc., 121 , 399418.

    • Search Google Scholar
    • Export Citation
  • Wippermann, F., 1969: The orientation of vortices due to instability of Ekman boundary layer. Beitr. Phys. Atmos., 42 , 225244.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 423 53 4
PDF Downloads 231 39 2