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  • View in gallery

    Frequency–wavenumber domains occupied by various types of acoustic and gravity waves in an isothermal atmosphere.

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    Dispersion relation for the anelastic approximation.

  • View in gallery

    The Boussinesq and dynamically rigid approximations.

  • View in gallery

    The hydrostatic approximation. Note that the buoyancy oscillation regime does not exist in the H approximation; here it is shown only as it appears in the fully compressible case.

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    The hydrostatic–Boussinesq and hydrostatic–dynamically rigid approximations.

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    The unified and Boussinesq–dynamically rigid approximations.

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    The pseudo-incompressible approximation.

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    Frequency as a function of vertical wavenumber at for nine approximations: A, B, H, DR, H–B, H–DR, B–DR, PI, and U. The fully compressible case is dashed.

  • View in gallery

    Oceanic gravity wave dispersion relations for nonhydrostatic approximations. Shown are the first six vertical modes of the A, B, DR, B–DR, and PI approximations. The fully compressible case is dashed but this is obscured by the approximate cases that cannot be distinguished from each other or from the exact case.

  • View in gallery

    As in Fig. 9, but for hydrostatic approximations. Shown are the first six vertical modes of the H, H–B, and H–DR approximations. The fully compressible case is dashed. In this case, the error of the approximate cases, which cannot be distinguished from each other, is quite distinct.

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Evaluation of Various Approximations in Atmosphere and Ocean Modeling Based on an Exact Treatment of Gravity Wave Dispersion

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Abstract

Various approximations of the governing equations of compressible fluid dynamics are commonly used in both atmospheric and ocean modeling. Their main purpose is to eliminate the acoustic waves that are potentially responsible for inefficiency in the numerical solution, leaving behind gravity waves. The author carries out a detailed study of gravity wave dispersion for seven such approximations, individually and in combination, to exactly evaluate some of the often subtle errors. The atmospheric and oceanic cases are qualitatively and quantitatively different because, although they solve the same equations, their boundary conditions are entirely different and they operate in distinctly different parameter regimes. The atmospheric case is much more sensitive to approximation. The recent “unified” approximation of Arakawa and Konor is one of the most accurate. Remarkably, a simpler approximation, the combined Boussinesq–dynamically rigid approximation turns out to be exactly equivalent to the unified approximation with respect to gravity waves. The oceanic case is insensitive to the effects of any of the approximations, except for the hydrostatic approximation. The hydrostatic approximation is inaccurate at large wavenumbers in both the atmospheric and oceanic cases because it eliminates the entire buoyancy oscillation flow regime and is therefore to be restricted to low aspect ratio flows. For oceanic applications, certain approximations, such as the unified, dynamically rigid, and dynamically stiff approximations, are particularly interesting because they are accurate and approximately conserve mass, which is important for the treatment of sea level rise.

Corresponding author address: John K. Dukowicz, Group T-3, MS B216, Los Alamos National Laboratory, Los Alamos, NM 87545. E-mail: duk@lanl.gov

Abstract

Various approximations of the governing equations of compressible fluid dynamics are commonly used in both atmospheric and ocean modeling. Their main purpose is to eliminate the acoustic waves that are potentially responsible for inefficiency in the numerical solution, leaving behind gravity waves. The author carries out a detailed study of gravity wave dispersion for seven such approximations, individually and in combination, to exactly evaluate some of the often subtle errors. The atmospheric and oceanic cases are qualitatively and quantitatively different because, although they solve the same equations, their boundary conditions are entirely different and they operate in distinctly different parameter regimes. The atmospheric case is much more sensitive to approximation. The recent “unified” approximation of Arakawa and Konor is one of the most accurate. Remarkably, a simpler approximation, the combined Boussinesq–dynamically rigid approximation turns out to be exactly equivalent to the unified approximation with respect to gravity waves. The oceanic case is insensitive to the effects of any of the approximations, except for the hydrostatic approximation. The hydrostatic approximation is inaccurate at large wavenumbers in both the atmospheric and oceanic cases because it eliminates the entire buoyancy oscillation flow regime and is therefore to be restricted to low aspect ratio flows. For oceanic applications, certain approximations, such as the unified, dynamically rigid, and dynamically stiff approximations, are particularly interesting because they are accurate and approximately conserve mass, which is important for the treatment of sea level rise.

Corresponding author address: John K. Dukowicz, Group T-3, MS B216, Los Alamos National Laboratory, Los Alamos, NM 87545. E-mail: duk@lanl.gov

1. Introduction

The governing equations of compressible fluid dynamics contain acoustic waves. In low Mach number flows, typical of atmospheric or oceanic conditions, acoustic waves play no essential role and yet they severely constrain the time step in numerical modeling. Thus, there has long been an interest in approximating the governing equations to eliminate or “filter out” acoustic waves. We shall consider such approximations but not alternative methods for avoiding the problem, such as by an implicit discretization of the equations.

Generally speaking, acoustic waves involve an interaction between the density ρ, dynamic pressure p, and dilatation . There are three such interactions:

  1. the conversion in the continuity equation;

  2. the conversion in the momentum equation; and

  3. the conversion in the equation of state.

Interfering with any one of these interactions has the potential to eliminate acoustic waves. There already exist several well-known approximations that follow this route to filter acoustic waves while hopefully not affecting slower dynamics. The Boussinesq approximation and a variety of anelastic approximations operate by modifying the continuity equation (type A). The hydrostatic approximation modifies the vertical component of the momentum equation (type B) to eliminate vertically propagating acoustic waves (but not Lamb waves). In the atmospheric literature, there are two approximations that operate by the type C mechanism: the “pseudo incompressible” model of Durran (1989) and the “unified” model of Arakawa and Konor (2009). Here we propose two additional approximations of this type, to be called the “dynamically rigid” and “dynamically stiff” approximations because they either eliminate or reduce the dynamic component of compressibility in the equation of state. Similar modifications of the equation of state are used in oceanic modeling but always in conjunction with the hydrostatic approximation, which effectively masks their own ability to filter acoustic waves.

While acoustic waves are in general entirely negligible, the effects of the approximations may not be. It is therefore important to evaluate the nature and magnitude of their errors. Perhaps the simplest way to accomplish this is to study the effects of approximation on the linear waves that are supported by the equations, such as gravity waves. We choose to focus on gravity waves because, for one, they play an important role in geostrophic adjustment for the atmosphere and the ocean and because it is possible to analyze them either analytically or at least to very high accuracy without further approximations to the boundary conditions or to the equations themselves. Perturbation methods to obtain linear wave solutions of the equations of compressible fluid dynamics have a long history, particularly for the atmosphere but also for the ocean (e.g., Eckart 1960, and references therein). However, there are few studies that use linear waves to evaluate approximations. One example is the study by Davies et al. (2003) for the atmospheric case. However, it does not include more recent approximations, such as those by Durran (2008) and Arakawa and Konor (2009). Arakawa and Konor (2009) have a limited study of this type that does include these approximations. There are no studies that look at combinations of approximations. Needless to say, there is a complete lack of such studies involving the ocean. Moreover, such studies have always been approximate. The study by Davies et al. (2003), for example, employs an approximate upper boundary condition, a rigid lid at a fixed high altitude. This creates discrete modes, contrary to what one finds in the free atmosphere. The oceanic problem is greatly complicated by the need to accommodate a Lagrangian upper boundary, which has necessitated approximations that result in an incorrect barotropic mode. One objective of this paper, therefore, is to correct such shortcomings and accurately evaluate a wide set of approximations in both atmospheric and oceanic settings. On the other hand, we attempt to simplify the analysis as much as possible. Thus, for example, we neglect rotation and the effects of global curvature. This of course eliminates important phenomena such as Rossby waves; this is not a serious limitation since gravity waves suffice for our purpose.

2. Acoustics-free approximations

Different acoustics-free approximations may be incorporated into the set of compressible adiabatic equations for fluid dynamics by the use of seven binary switches () that select various approximations. The seven switches denote the anelastic (A), Boussinesq (B), hydrostatic (H), dynamically rigid (DR), dynamically stiff (DS), pseudo-incompressible (PI), and unified (U) approximations, respectively. They take on the values of 0 or 1, such that the value of 0 switches on the corresponding approximation by eliminating or modifying one or more terms in the equations. Thus, results in the fully compressible Euler equations that contain acoustic waves, while setting one or more of the switches to 0 eliminates acoustic waves by means of one or more of the mechanisms discussed earlier. Note that we shall henceforth refer to the approximations by their abbreviations in most cases.

Let us first consider the seven individual approximations. To start, there are three “classical” approximations: namely, the A, B, and H approximations. There may be different variants of these approximations; here we shall consider the simplest possible variant that captures the essence of each approximation. Usually, the distinction between variants disappears in the linearized equations.

  • (i) The anelastic approximation is exemplified by neglect of the time derivative in the continuity equation, as in the models of Ogura and Phillips (1962) and Bannon (1996), for example. The approximation is justified when the time scale of interest is much greater than the time scale associated with acoustic wave propagation. Note that this is really a class of approximations as there are several variants that are primarily used in atmospheric modeling.

  • (ii) The Boussinesq approximation assumes a constant density in the continuity equation, which converts it into an equation for the conservation of volume. Although it is a special case of the A approximation, it is important in its own right both for historical reasons and because of its ubiquitous application in ocean modeling. The classical B approximation (Boussinesq 1903; Spiegel and Veronis 1960), originally formulated to model thermal convection, also assumes that density in the momentum equation is a constant reference value everywhere except in the gravitational force term. This additional approximation is not needed to eliminate acoustic waves and is omitted here. The B approximation is almost invariably combined with the H approximation in z-coordinate ocean modeling although nonhydrostatic Boussinesq ocean models do exist (e.g., Marshall et al. 1997; Matsumura and Hasumi 2008).

  • (iii) The hydrostatic approximation neglects the inertial terms in the vertical momentum equation, usually as a result of a low aspect ratio assumption. This is perhaps the most prominent of the various approximations considered here because it is the basis for the ubiquitous primitive equation models used in both atmospheric and ocean modeling. As mentioned, the H approximation is generally combined with the B approximation in ocean modeling except in the notable case of isopycnic models (e.g., Bleck and Smith 1990), which are hydrostatic but not Boussinesq.

Next, we have the four remaining approximations that operate by the type C mechanism. These generally involve replacing the pressure in the equation of state (EOS) by an approximate pressure, thus modifying the dynamic component of compressibility. There are two options. In the first option a pressure is obtained from hydrostasy,
e1
where is some assumed, time-independent density. Since does not depend on time there will be no pressure fluctuations in the equation of state and therefore no pressure-induced density fluctuations: the EOS is said to be dynamically rigid. For present purposes a natural choice for is the background density distribution. For ocean applications a suitable density distribution is easily obtained by using the special properties of the equation of state of seawater (Dukowicz 2001). For the second option an approximate pressure is obtained self-consistently by means of hydrostatic balance using available potential temperature θ and salinity distributions, as follows:
e2

In effect, this is precisely the pressure one obtains in a primitive equation ocean or atmospheric model. It includes the relatively slow variations of the potential temperature and salinity while omitting the rapid fluctuations due to acoustics. Since dynamic compressibility is not entirely eliminated (density does respond to potential temperature and salinity-induced pressure fluctuations) the fluid is said to be dynamically stiff rather than rigid; see additional discussion in connection with Table 1. Thus, the four remaining approximations are described as follows:

  • (iv) The pseudo-incompressible approximation , introduced by Durran (1989, 2008), and the unified approximation , introduced by Arakawa and Konor (2009), represent the present state of the art in atmospheric acoustics-free modeling. Their distinguishing characteristic is the use of two different densities, ρ and ρ*. Both approximations use the standard density in the momentum equation, as in the fully compressible case, while the PI approximation uses the rigid approximate density and the U approximation uses the stiffened approximate density in the continuity equation. Note that use of two densities is acceptable in atmospheric modeling where evaluation of the ideal gas EOS is inexpensive but possibly not so in ocean modeling where the EOS is more complicated.

  • (v) The two new approximations are the dynamically rigid and the dynamically stiff approximations. They make use of a single density in both the momentum and continuity equations. The DR approximation uses , while the DS approximation uses . The DR approximation is more accurate but entails the slight additional cost of solving a nonlinear hydrostatic equation (Dewar et al. 1998). It is of interest to note that both the DS and DR approximation conserve energy. Note that both these density approximations are already used in hydrostatic z-coordinate ocean models to simplify calculations while retaining reasonable accuracy.

Table 1.

Effective dynamic compressibility.

Table 1.

Incorporating all approximations, the complete system of compressible adiabatic equations for fluid dynamics under gravity (Gill 1982) consists of 10 equations for the 10 variables, . The analysis is simplified by excluding rotation. Supplemented by (1) and (2), the system is given by
e3
e4
e5
e6
e7
and
e8
In the above, and are the three-dimensional velocity and position vectors in a Cartesian coordinate system, and are the two-dimensional horizontal velocity and position vectors, and and are the three-dimensional and horizontal gradient operators, respectively. Using conventional notation, the material derivative is . The equation of state takes the general form , where salinity appears in the oceanic case only. For the atmosphere the dry ideal gas equation of state will be used, while for the ocean there are several available equations of state that are expressed in terms of potential temperature.
Note that some approximations may be combined to good effect while observing certain restrictions. For example, the continuity equation (3) shows that combining the A approximation with the B approximation has no effect whatsoever. Similarly, (2) and (5) become equivalent when the H and DS approximations are combined; that is, the H approximation implies the DS approximation but not vice versa. Furthermore, the DR and DS approximations, and the PI and U approximations, are related and therefore should not be invoked simultaneously. The various restrictions can be summarized as follows:
e9
In the following we examine the individual approximations, with the exact case used for reference. In addition, we examine the allowable combinations H–B, H–DR, and B–DR.

3. The atmospheric case

We start with the atmospheric case because of its relative simplicity. The bottom boundary is assumed to be a rigid horizontal surface, requiring a no-penetration boundary condition. At Earth's surface, therefore, we have
e10
The atmosphere is a semi-infinite medium, implying that . This is not a boundary condition but rather an acceptability condition. Note that this implies a continuous spectrum of waves in the vertical direction, which is considerably simpler than the eigenproblem that occurs in the oceanic case. In addition, the atmospheric problem can be handled in Eulerian coordinates, which is also a substantial simplification.

a. The background state

Conventionally, the background state for the study of atmospheric waves has been a resting isothermal atmosphere in hydrostatic equilibrium (e.g., Eckart 1960; Davies et al. 2003; Arakawa and Konor 2009). The ideal gas equation of state is
e11
where R is the ideal gas constant, , is the ratio of specific heats, and is a constant reference pressure. The resulting density, pressure, and potential temperature in the background state become
e12
where is the atmospheric surface pressure and H is the pressure or density scale height, . Taking , we obtain
e13
The buoyancy or Brunt–Väisälä frequency is given by
e14
where is the sound speed, so that both are constants. Assuming ≈ 273 K and g ≈ 9.8 m s−2, we obtain
e15
As noted earlier, it is convenient to take ; therefore, from (1) we have
e16
For conciseness, we will henceforth suppress the vertical dependence of background state quantities: that is, , , and .

Remark

The use of an isothermal background for dispersion analysis is not well justified for a number of reasons. For instance, Klein et al. (2010) show that the A and PI approximations are justified only for weak stratifications: that is, for a typical flow Mach number . However, an isothermal atmosphere implies a moderately strong stratification since from (13) we have . Further, a perturbation analysis in an isothermal atmosphere fails at a sufficiently high altitude, as we shall show below. Nevertheless, an isothermal background is not ruled out because solutions based on it are perfectly suitable for comparing different approximations to each other and to the exact case since all are evaluated under identical conditions.

b. Linearizing about the background state

Expanding in small perturbations about the background state,
e17
we obtain a set of linearized equations for the perturbations, corresponding to (2)(8), as follows:
e18
e19
e20
e21
e22
Linearizing (7) and (8) and using restrictions (9), we have
e23
and
e24
The above are seven equations for the seven natural variables,
eq4
This choice of variables leads to a simple system in which all coefficients are constant, including . Equation (18) for the approximate pressure perturbation is obtained from (2) by linearizing and substituting from (23). Also, since (18) is uncoupled from the other equations except in the DS and U approximations, we have simplified it by setting and .

The second and third terms in the perturbation equations of state, (23) and (24), pertain to the notion of dynamic compressibility, as specified by the coefficient of either or . Dynamic compressibility differs from that in the fully compressible case only in the DR, DS, PI, and U approximations. These differences are summarized in Table 1. The DR case has zero dynamic compressibility since both terms vanish because , and since because of (9). The DS case has a relatively large dynamic compressibility since is a function of through its dependence on and since we expect from a scrutiny of (18), (21), (22), and (24) in the exact case that . The PI approximation is dynamically rigid and the U approximation is dynamically stiff with respect to the continuity EOS, and both are fully compressible with respect to the momentum EOS. Note, however, that static compressibility, in contrast to dynamic compressibility, is always fully accounted for as evidenced by the variation of density in the background state.

It is now useful to introduce nondimensional variables, to be denoted by a tilde. Thus, we write , where we have normalized using H as the length scale and buoyancy frequency as the inverse time scale. In general, the problem is characterized by two independent nondimensional parameters, which we choose to be
e25
The dimensionless parameter represents the ratio of gravitational to acoustic effects, and represents the ratio of buoyancy to gravitational effects. However, in an isothermal atmosphere these parameters are not independent (i.e., , ), and therefore there is really only one nondimensional parameter: namely, γ.
Since coefficients in the system of equations are constant, solutions will be in the form of horizontally propagating waves, as follows:
e26
where are horizontal wavenumbers, is a vertical coefficient, and is the frequency ( are assumed real, while may be complex). The nondimensional variables are
e27
Substituting (26) into (18)(24), after considerable algebra we obtain a cubic polynomial for in terms of and , where . There are three different cases to consider. First, for the exact case and all approximations other than the DS and U approximations there is a factor in the polynomial that corresponds to the dummy variable . After stripping this off, we have a quadratic equation,
e28
whose coefficients are given by
e29
Coefficients take the value 0 or 1 as determined by the approximation switches. The explicit form in terms of approximation switches is complicated so, instead, their values are summarized in Table 2 for the different cases considered presently.
Table 2.

Coefficients defined in (29) for all cases except for the DS approximation.

Table 2.

Remark

The fully compressible case is characterized by . All approximate, acoustics-free cases are characterized by .

The two roots of (28) are
e30
These may be real or complex conjugate. Real roots represent horizontally propagating waves that either grow or decay in the vertical direction depending on their sign. Solutions that grow in the vertical direction are nonphysical and must be discarded. Solutions given by complex conjugate roots represent three-dimensional gravity or acoustic waves that decay in the vertical direction since the real part is always negative.
The hydrostatic equation (18) plays an essential role in the DS and U approximations and therefore the complete cubic equation in must be considered. However, in the U approximation the cubic equation can be factored,
e31
Thus, there is a physically meaningful real root, , that corresponds to an as yet unidentified horizontally propagating evanescent wave. The quadratic factor is otherwise the same as (28). This approximation will be discussed in more detail at a later stage.
In the DS approximation, the cubic dispersion polynomial is not factorizable: that is,
e32
The three roots, , are either all real or there is one real root and a pair of complex conjugate roots. However, the product of the three roots is always positive: that is, . This means that at least one of the real roots is positive and so this case always contains an unphysical, vertically growing solution. As a result, the DS approximation is not viable and will not be discussed further in the atmospheric case.

c. The fully compressible case

There are several different wave types in the fully compressible case. The ones of primary interest are the three-dimensionally propagating acoustic and gravity waves. There are also horizontally propagating evanescent waves of both types and regions in parameter space that imply nonphysical, vertically growing waves. Finally, there is a set of special, horizontally propagating acoustic waves, called Lamb waves. The domains these waves occupy in frequency–horizontal wavenumber space are shown in Fig. 1.

Fig. 1.
Fig. 1.

Frequency–wavenumber domains occupied by various types of acoustic and gravity waves in an isothermal atmosphere.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

1) Acoustic and gravity waves

Three-dimensional acoustic and gravity waves occur when , in which case the roots in (30) are complex conjugate, as follows:
e33
where is a vertical wavenumber, given by
e34
and is the scale height for vertical attenuation. Equation (34) may be viewed as a general dispersion relation that accommodates both acoustic and gravity waves. The fact that is real and positive delimits the region in parameter space occupied by these waves, shown shaded in Fig. 1 as regions A and D. Equation (34) may alternatively be written as a quartic polynomial in as follows:
e35
where
e36
The solution of (35) has two branches: an acoustic wave branch and a gravity wave branch given by the dispersion relations,
e37
and
e38
respectively. Note that, in contrast to the oceanic case, these are continuous functions of the vertical wavenumber , meaning that the atmosphere supports a continuous spectrum of acoustic and gravity waves.
Remark

Keeping in mind that , the fact that frequency is real implies that . Thus, the boundary between the acoustic and gravity wave branches is . Substituting this in (37) and (38), the boundary simplifies to . This is plotted in Fig. 1 as the separation line between the gravitationally dominated regions A, B, and C and the acoustically dominated regions D, E, and F.

From (33), we note that the attenuation scale height is the same for both acoustic and gravity waves: namely,
e39
The positive and negative signs in (33) correspond to upward and downward propagation directions, respectively. Taking boundary condition (10) into account, the solution for the vertical momentum density takes the form
e40
and therefore all remaining natural variables retain the vertical factor
eq6
Thus, all perturbations decay with altitude, as expected. However, other physically meaningful quantities, such as potential temperature and velocity components, grow and become unbounded as altitude increases,
eq7
This is disconcerting. On the other hand, for kinetic energy density we have
eq8
which remains constant with altitude, meaning that these solutions are marginally acceptable. However, background state pressure and density distributions attenuate faster than the perturbations since . Thus, pressure and density perturbations do not stay small relative to background as altitude increases, contrary to the assumption. These results are a direct consequence of the isothermal atmosphere. Nevertheless, as noted earlier, the analysis remains valuable for evaluating the approximations in comparison to the exact case.

2) Atmospheric edge waves

Another class of waves arises whenever (i.e., is imaginary), which results in a pair of real roots, given by
e41
where are corresponding scale heights such that is positive while may be of either sign. When is negative the solution is unphysical since the wave amplitude grows without bound. Such solutions correspond to regions C and F in Fig. 1. On the other hand, positive produces horizontally propagating evanescent waves; these waves occupy regions B and E in Fig. 1. Following Garrett (1969), we shall refer to these as atmospheric edge waves. The vertical momentum density is able to satisfy boundary condition (10) and solutions take the form
e42
The remaining variables retain the same vertical dependence, analogous to (40). Just as for the case of three-dimensional waves, density and pressure perturbations decay while potential temperature and velocity components grow with increasing elevation. On the other hand, kinetic energy density grows with height, making these waves unphysical, at least in an isothermal atmosphere. Perhaps this is the reason that these waves are generally ignored in atmospheric studies. Additionally, these waves may well be restricted to flat horizontal boundaries since they propagate horizontally. However, this issue is beyond the scope of the paper and will not be discussed further.

3) Lamb waves

In the fully compressible case there exists a set of anomalous waves, called Lamb waves, which do not appear in the dispersion polynomial (28) or dispersion relations (37) and (38). These consist of horizontally propagating, hydrostatically balanced, evanescent acoustic waves (Bretherton 1969; Gill 1982). Lamb waves are obtained by postulating zero vertical velocity, , in (18)(24). The Lamb wave solution is then easily found, as follows:
e43
The corresponding dispersion relation , or , indicating pure acoustic waves, is shown by the dashed curve in Fig. 1.
The density and pressure in (43) attenuate with the scale height , and, as found with other waves, horizontal velocity components grow exponentially with increasing altitude. In contrast to the other cases, however, the horizontal kinetic energy density attenuates in the vertical direction: that is,
eq9
Thus, Lamb waves are better behaved. However, we again find that density and pressure perturbations attenuate more slowly than the background since , so the perturbation analysis fails above some height.

Lamb waves are typically associated with an isothermal atmosphere. However, Bretherton (1969) has shown that they may exist under more general conditions provided variations in sound speed and horizontal velocity are not too great. Bretherton (1969) also remarks that a rigid, horizontal boundary is essential. Thus, similar to the case of atmospheric edge waves, Lamb waves may well be restricted to flat horizontal boundaries.

d. Acoustics-free approximations

We have noted that the approximations have the effect of setting , which retains only the gravitationally dominated branch of solutions: that is, regions A, B, and C in Fig. 1. Here we will focus on the physically more interesting gravity and Lamb wave solutions and omit regions B and C. As noted earlier, all approximations other than the DS approximation contain the quadratic factor (28) associated with gravity waves. The U approximation, however, also contains a linear factor associated with the real root . The interpretation of this extra solution is not immediately obvious. We shall return to this question in section 3d(2).

1) Acoustics-free gravity waves

The dispersion relation in this case retains only the gravity wave branch (38), so there is no need to retain the subscript g for frequency, and it takes the simple form
e44
where is given by (36). The attenuation scale height is again given as , where comes from (29) and is always positive. As before, after satisfying boundary condition (10) we have
eq10
Thus, density and pressure attenuate while the potential temperature and velocity components grow with height: that is,
eq11
since are negative in all approximations. On the other hand, the sign of the scale height for kinetic energy density depends on the approximation,
eq12
Dimensional values of for the different approximations are shown in Table 3. Positive scale heights indicate attenuation with height and negative values indicate growth; an infinite scale height implies constant amplitude.
Table 3.

Various scale heights for the growth or attenuation of atmospheric gravity waves. Negative values indicate growth with altitude.

Table 3.
(i) Remark

The fact that kinetic energy density grows with height for the A, DR, and H–DR approximations is concerning. Again, this does not rule out these three approximations for the reasons mentioned in section 3c(1). However, this is yet another measure for judging the accuracy of approximations. By this measure, the H, U, PI, and B–DR approximations are particularly accurate since they match the exact case.

The gravity wave dispersion relation (44) is plotted in Figs. 27 for five representative values of the vertical wavenumber, , for all approximations that we consider, in comparison with the fully compressible case. To emphasize that the dispersion relation represents a continuous spectrum in vertical wavenumber, Fig. 8 plots frequency as a function of vertical wavenumber at a constant value of horizontal wavenumber: namely, . Another quantity of interest is the phase speed, . Thus, in Table 4 we show nondimensional values of the asymptotic phase speed, , which is constant in the longwave limit, .

Fig. 2.
Fig. 2.

Dispersion relation for the anelastic approximation.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

Fig. 3.
Fig. 3.

The Boussinesq and dynamically rigid approximations.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

Fig. 4.
Fig. 4.

The hydrostatic approximation. Note that the buoyancy oscillation regime does not exist in the H approximation; here it is shown only as it appears in the fully compressible case.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

Fig. 5.
Fig. 5.

The hydrostatic–Boussinesq and hydrostatic–dynamically rigid approximations.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

Fig. 6.
Fig. 6.

The unified and Boussinesq–dynamically rigid approximations.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

Fig. 7.
Fig. 7.

The pseudo-incompressible approximation.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

Fig. 8.
Fig. 8.

Frequency as a function of vertical wavenumber at for nine approximations: A, B, H, DR, H–B, H–DR, B–DR, PI, and U. The fully compressible case is dashed.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

Table 4.

Asymptotic phase speed for long atmospheric gravity waves.

Table 4.

(ii) Remarks

  1. The gravity wave dispersion relation (44) is identical for the B and the DR approximations and also for the H–B and the H–DR approximations. Thus, we only show a single dispersion plot for each pair in Figs. 3 and 5. This is remarkable since these approximations are not equivalent as seen in Table 2 and confirmed by their different vertical scale heights shown in Table 3.

  2. Gravity waves in the U and the B–DR approximations are identical, as may be seen in Table 2. Thus, Fig. 6 shows only a single dispersion plot for this pair of approximations. Notice also that all other properties of these two approximations are identical, as seen in Tables 3 and 4. This is unexpected since these approximations are not at all similar. For instance, the B–DR approximation conserves volume while the U approximation conserves an approximate mass.

The dispersion diagrams in Figs. 27 show that gravity waves span two limiting regimes. The boundary between these two regimes depends on the value of the horizontal wavenumber in the denominator of (44). At low values of horizontal wavenumber, frequency is proportional to the vertical wavenumber, while at high values frequency becomes constant. The separation between the two regimes (i.e., the inflection point in a particular dispersion curve) occurs approximately when the denominator in (44) is equal to or, equivalently, when , as indicated by the horizontal dashed line in Figs. 27. The first regime, called a pure gravity wave regime, is typified by a constant phase speed at long wavelengths. The second regime, called the buoyancy oscillation regime, is typified by frequency equal to the buoyancy frequency, and therefore zero phase speed, at short wavelengths (thus, gravity waves cease to propagate and only stationary oscillations remain). These two regimes exist in all approximations except in the H approximation, which supports only the gravity wave regime.

Approximation errors are generally confined to the region of small horizontal and vertical wavenumbers (except for approximations involving the H approximation where the error is also large at high horizontal wavenumbers). This is the opposite of discretization errors where the error is typically largest at high wavenumbers. The asymptotic phase speed in this region is too high in most approximations, as seen in Table 4, except in the H, B–DR, and U approximations, where it is exact. Overall, the B–DR and the U approximations are the most accurate, coinciding with the exact solution at both low and high values of horizontal wavenumber for all values of the vertical wavenumber. The B and DR approximations and their hydrostatic versions are the least accurate. Thus, it is remarkable that in combination the B and DR approximations are so highly accurate, despite being among the least accurate individually; somehow their errors must compensate. The A and PI approximations are of intermediate accuracy. However, Achatz et al. (2010) show that the PI approximation has a broader range of applicability than the A approximation. These results also apply to the attenuation scale heights in Table 3, except for the PI case where the attenuation scale height is exact.

The H approximation is quite different in nature from all other approximations; it is very accurate at low horizontal wavenumbers but very inaccurate at high wavenumbers, as is seen in Fig. 4. Accuracy is retained provided , which is equivalent to requiring a small flow aspect ratio. The poor accuracy at large wavenumbers is passed on when the H approximation is combined with other approximations, making the H–B and H–DR approximations the least accurate and even less accurate than the H approximation itself.

2) Lamb waves in acoustics-free approximations

One may ask whether Lamb waves survive in any of the approximations. Making the same assumption as in the exact case [i.e., substituting into (18)(24)], we find that the H approximation supports Lamb waves in the form given by (43), while approximations other than the U approximation do not. Recall that the dispersion polynomial for the U approximation (31) yields three solutions: two gravity wave solutions from the quadratic factor and an unidentified solution, , from the linear factor. The existence of three solutions also indicates a potential problem: it is not clear how three solutions can simultaneously satisfy the single boundary condition (10). Let us again assume that in the U approximation version of (18)(24). Assuming horizontally propagating waves , we obtain
e45
These equations agree with (43) provided . This will indeed be true since each of the pressures satisfies the same equation and the same boundary condition. This confirms that Lamb waves also exist in the U approximation. The third root, , can now be associated with the Lamb wave solution. There is also no issue with the bottom boundary condition since the solution satisfies this boundary condition all by itself, freeing the other two solutions to satisfy the boundary condition as before. Thus, the U approximation mimics the exact case even more closely due to the presence of Lamb waves.

4. The ocean case

The development of the oceanic case parallels the atmospheric case in many respects. There are two major differences, however. First, the ocean upper boundary is a Lagrangian free surface on which the pressure is specified: that is, , where is the surface elevation and is a constant atmospheric pressure at the ocean surface. To properly capture such a boundary it is necessary to incorporate it into a vertically Lagrangian coordinate system, which is a significant complication. Failure to do this has serious consequences for the barotropic mode. Second, due to the presence of an upper boundary the ocean case involves the solution of an eigenproblem, in contrast to the atmospheric case. The bottom boundary condition specifies no penetration, the same in both cases, as follows:
e46
where now is the depth of the ocean in the resting state.

a. Transformation to Lagrangian vertical coordinates

Let us temporarily revert to dimensional variables. Define a coordinate system that follows Lagrangian interfaces in the vertical direction, analogous to oceanic isopycnic coordinates (Bleck and Smith 1990), as follows: , , etc., where and are the horizontal coordinates and time in the Lagrangian coordinate system, and is a label for Lagrangian interfaces that corresponds to their resting vertical position. The transformation to this coordinate system (De Szoeke and Bennett 1993) is given by
e47
where . Lagrangian motion of -constant surfaces is ensured by the requirement that
e48
Transforming the original equations (1)(6), we obtain
e49
e50
e51
e52
and
e53
where
eq15
is the transformed Lagrangian derivative. These equations are supplemented by (7) and (8), which require no transformation.

b. Background state

We again consider fluctuations about a hydrostatically balanced resting state in a horizontally infinite, flat bottom domain. The density distribution is assumed to be exponential with depth,
e54
where is a constant scale depth and is the density at the ocean surface. The corresponding pressure is hydrostatic and is given by
e55
We also assume that sound speed is a constant, a good approximation for the ocean, in which case a plausible equation of state for seawater is
e56
where is some function of potential temperature and salinity. For conciseness, we adopt the notation , as in the atmospheric case. The background state Brunt–Väisälä or buoyancy frequency is now given by
e57
which is also a constant.

The use of constant buoyancy frequency and sound speed is very useful for simplifying calculations, as in the isothermal atmosphere, and has a long history (e.g., Eckart 1960). A constant buoyancy frequency is not representative of actual oceanic conditions but it is not unphysical in the sense that plausible resting distributions satisfying (56) may be constructed.

We assume that the ocean is typified by the values
e58
As in the atmospheric case, there are two nondimensional parameters, and . The oceanic values are therefore and . These two parameters are considerably smaller than the atmospheric values. From (57) we see that . This implies that km, a value far larger than the ocean depth so that ocean density is nearly uniform.

c. Linearization about the background state

Expanding in small perturbations about the background state,
e59
we obtain the following set of linearized equations from (49) to (52),
e60
e61
e62
e63
e64
where we have used and to convert these equations into a system with constant coefficients. We have also used from (9) to simplify coefficients. The linearized tracer transport equations (53) are
e65
Using (56) and (65), the linearized equations of state, (7) and (8), become
e66
e67
Thus, here the natural choice for dependent variables is . The corresponding set of boundary conditions is given by
e68
Note the presence of additional boundary conditions at the surface, in contrast to the atmospheric case.
Let us nondimensionalize as previously, using a tilde to denote nondimensional variables. The presence of constant coefficients allows us to assume horizontally propagating waves: that is, . After considerable algebra we obtain a factored quartic polynomial for all cases except for the DS and U cases, as follows:
e69
where the coefficients and are the same as those in (29) and Table 2 in the atmospheric case. This is not surprising since the atmosphere and the ocean satisfy the same governing equations. Only the quadratic polynomial factor is of interest; it corresponds to acoustic and gravity wave solutions as in the atmospheric case. The factor is associated with pressure perturbations , which are not of dynamical interest except in the DS and U approximations. These two approximations give rise to a cubic polynomial factor in (69) instead of a quadratic. The cubic polynomial in the DS case is the same as in the atmospheric case [viz., (32)], but this is not the case in the U approximation since the cubic polynomial does not factor like (31). The occurrence of these cubic polynomials greatly complicates the solution. Anticipating later results showing that dispersive relations for all oceanic nonhydrostatic approximations scarcely differ from each other and from the exact case, so that even the simplest approximation is sufficiently accurate, we find that there is little motivation to consider these two more complicated approximations.

The two roots of the quadratic factor in (69) are given by (30). Acoustic and gravity waves occur when , resulting in a pair of complex conjugate roots given by (33) and (34) such that is again the vertical wavenumber and is the scale height for vertical attenuation. As in the atmospheric case, the gravity wave frequency is given by (38), and we again drop the subscript g since we will only be interested in gravity waves and not acoustic waves. As in the atmospheric case, (38) represents the general dispersion relation provided is known. However, instead of being specified, must now be obtained from an eigenproblem.

d. The oceanic eigenproblem

Up to now the development has closely paralleled the atmospheric case. The principal difference arises from the boundary conditions. Using the bottom boundary condition from (68), the solution for the Lagrangian interface displacement takes the form
e70
This is similar to the atmospheric case. We now need a boundary condition at the upper surface in terms of . Eliminating from (62) and (63) and using (68) to eliminate pressures at the free surface, we deduce that that the boundary condition at the free surface is given by
e71
where, as noted earlier, . We have previously remarked that (71) must be approximated if one tries to solve the oceanic problem in Eulerian coordinates. For example, Gill (1982) derives an approximation that omits the first term in parentheses in (71). As a result, the barotropic mode becomes indeterminate, although baroclinic modes stay reasonably accurate. The most common expedient in this situation has been to discard the barotropic mode by imposing a rigid-lid boundary condition at the surface (i.e., ), and substituting a surface gravity wave for the missing barotropic mode (e.g., Eckart 1960).
Substituting (70) into (71) results in an eigenproblem for , which can be expressed as follows:
e72
where
e73
Solutions of (72), or the normal modes for the problem, correspond to zeros of the sine function: namely,
e74
where is an index corresponding to the sequence of normal modes, starting with the barotropic mode and continuing with baroclinic modes . Thus, for each value of the vertical wavenumber is a specific function of . The solution of (74) for as a function of is obtained by continuation starting from . Having , the dispersion relation is obtained from (38). For continuation we need the value of at . This value is directly related to the asymptotic phase speed for long gravity waves that is obtainable from (38), as follows:
e75
Using (75) in (74), we obtain an implicit equation for the starting value of . The asymptotic phase speed (i.e., in the limit ) is an interesting quantity in its own right and is shown in Table 5 for the first seven eigenmodes.
Table 5.

Asymptotic phase speed for long oceanic gravity waves.

Table 5.

Dispersion relations, obtained from (38) and (74), are plotted in Fig. 9 for the nonhydrostatic approximations (except for the DS and U approximations, as noted earlier) and in Fig. 10 for the hydrostatic approximations, in comparison to the exact case. Note the characteristic “spectral gap” between the barotropic and baroclinic modes due to the relatively small stratification in the ocean. Contrary to the many approximate treatments using a surface gravity wave in place of the barotropic mode (Eckart 1960; Gill 1982), the barotropic mode is clearly a gravity wave whose frequency saturates at the buoyancy frequency in the short wavelength limit. Similarly, the barotropic mode phase speed in the longwave limit is not equal to the surface gravity wave value but varies depending on the approximation (see Table 5). Comparing Figs. 9 and 10, the dispersion error is dominated by the error of the hydrostatic approximation. The error due to nonhydrostatic approximations, whether individually or in combination, is so small that it cannot be visually observed in a comparison with the exact case. Just as in the atmospheric case, the H approximation supports only pure gravity waves, with essentially constant phase speed for each mode. As a result, the H approximation error is small at low wavenumbers and large at high wavenumbers. The effect of different approximations on the longwave phase speed is shown in Table 5, and this is also quite minimal for all approximations, including the H approximation.

Fig. 9.
Fig. 9.

Oceanic gravity wave dispersion relations for nonhydrostatic approximations. Shown are the first six vertical modes of the A, B, DR, B–DR, and PI approximations. The fully compressible case is dashed but this is obscured by the approximate cases that cannot be distinguished from each other or from the exact case.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

Fig. 10.
Fig. 10.

As in Fig. 9, but for hydrostatic approximations. Shown are the first six vertical modes of the H, H–B, and H–DR approximations. The fully compressible case is dashed. In this case, the error of the approximate cases, which cannot be distinguished from each other, is quite distinct.

Citation: Monthly Weather Review 141, 12; 10.1175/MWR-D-13-00148.1

To find out if the ocean supports Lamb waves, we substitute into (60)(67). This effectively reproduces (43) and (45), which describe Lamb waves in the atmospheric case. Also, both oceanic boundary conditions [viz., (46) and (71)] are satisfied, but this time with no need to solve an eigenproblem. Thus, the ocean supports Lamb waves in the exact case and in the H and U approximations, just as in the atmospheric case.

5. Summary and conclusions

We have analyzed a number of “soundproof” approximations that are either being used in or may be suitable for atmospheric and oceanic modeling. The analysis is novel in that it is based on an exact, nonapproximated treatment of gravity wave dispersion. The following is a summary of the observations and results from this study:

  1. Although they satisfy the same equations, the atmospheric and oceanic cases are quite different both qualitatively and quantitatively. This is because they satisfy different boundary conditions and operate in different parameter regimes. The major qualitative difference is that the atmosphere does not have an upper boundary, while the ocean has a Lagrangian free upper surface boundary. This difference has important consequences: the atmospheric case accommodates a continuous spectrum of waves and the analysis can be performed in Eulerian coordinates, while the oceanic case contains a discrete spectrum of waves and the analysis must be performed in Lagrangian coordinates to accommodate the free surface. Quantitatively, the difference is primarily due to the very low compressibility of seawater as compared to air. This accounts for the large difference in the magnitude of the atmospheric and oceanic parameters .

  2. Atmospheric waves belong to either an acoustic or gravity wave branch in the frequency–horizontal wavenumber domain, as shown in Fig. 1. Each of these branches contains regions for three-dimensional waves (which are physically more interesting), regions for two-dimensional evanescent (or edge) waves, and excluded regions where waves are not permissible. In addition, there is also a set of anomalous, horizontally propagating acoustic waves known as Lamb waves. The various approximations eliminate the acoustic branch, and in most cases also eliminate Lamb waves, except in the H and U approximations. Analogous waves exist in the oceanic case, including Lamb waves.

  3. Atmospheric waves are analyzed assuming an isothermal atmosphere as background, as is standard practice. However, while extremely convenient for our purpose, this assumption leads to physical or mathematical inconsistencies. In all cases velocity components and potential temperature grow without bound with increasing altitude. This is also true for kinetic energy density in the A and DR approximations. Further, even physically well-behaved variables such as density and pressure attenuate at a slower rate than the background state, implying that perturbations will not stay small relative to the background. Using an entirely different methodology, Klein et al. (2010) have shown that the assumption of an isothermal atmosphere places the anelastic and pseudo-incompressible approximations outside their regime of validity for a typical flow Mach number, . Similarly, in the oceanic case we have made the nonrealistic but physically plausible assumption of a constant buoyancy frequency for reasons of mathematical simplicity. Nonetheless, we argue that the comparison of approximations, which is the main purpose of this paper, is still valid despite the flaws associated with our choice of background.

  4. Some approximations have not been analyzed in detail, such as the DS approximation in the atmospheric and oceanic cases and the U approximation in the oceanic case. This is because the DS approximation gives unphysical results in the atmospheric case and because the mathematical analysis of gravity wave dispersion in the oceanic case is too complicated in both the DS and U approximations. However, in view of the insensitivity of the oceanic case to different nonhydrostatic approximations, one might expect that this insensitivity carries over to the DS and U approximations and therefore that both these approximations should be viable in an oceanic model.

  5. The atmospheric case is much more sensitive to approximation, particularly in the long wavelength part of the spectrum, for all approximations except the H approximation. The oceanic case, on the other hand, is not at all sensitive since there is very little to distinguish the different approximations from the exact case. The H approximation stands out in that it preserves accuracy at long wavelengths (low aspect ratio) but becomes extremely inaccurate at short wavelengths.

  6. The dispersion relation for gravity waves is divided into two sections. One section is dominated by horizontally propagating gravity waves, and the other is dominated by stationary buoyancy oscillations. The demarcation between these regimes takes place at approximately . Thus, the H approximation effectively eliminates the buoyancy oscillation regime.

  7. In the atmospheric case, the various approximations may be roughly divided into three classes according to their overall accuracy with respect to both the dispersion relation and attenuation scale height:

    • poor accuracy: the B and DR approximations and all H approximations;

    • intermediate accuracy: the A and PI approximations; and

    • high accuracy: the U and the combined B–DR approximations.

    • Thus, for example, the A approximation has good accuracy with respect to dispersion but poor accuracy with respect to vertical attenuation. The H approximation, on the other hand, has excellent accuracy with respect to vertical attenuation but very poor dispersion accuracy at high wavenumbers. In the oceanic case, all approximations, except for those involving the H approximation, appear to be highly accurate.

  8. In the atmospheric case, the individual B and DR approximations have exactly the same dispersion relation but quite different attenuation scale heights. The U and the combined B–DR approximations, on the other hand, are completely identical with respect to gravity wave propagation. This is quite remarkable since individually the B and DR approximations are different and have relatively poor accuracy, yet they combine to equal the high accuracy of the U approximation. The U approximation is unique in that it combines high accuracy while preserving Lamb waves.

  9. In the oceanic case, the most interesting approximations are the DR, DS, PI, and U approximations because they avoid hydrostatic errors and feature relatively accurate conservation of mass, important for the treatment of sea level rise.

Acknowledgments

The author wishes to thank Dr. C. S. Konor for helpful discussions regarding the unified approximation.

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