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  • View in gallery

    The EAAM budget. The solid (dashed) curve corresponds to the right-hand side (left-hand side) of (4), and the long-dashed curve is the globally integrated friction torque. All quantities are divided by 1020 kg m2 s−2

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    The power spectrum of the 90°E component of the EAAM vector in the climate model

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    The anomalous surface pressure field regressed against the Greenwich meridian component of the EAAM tendency at (a) lag − 4 days, (b) lag − 2 days, (c) lag 0 days, (d) lag + 2 days, and (e) lag + 4 days. Lag 0 corresponds to the time of the maximum value for the Greenwich meridian component of the EAAM tendency. The contour interval is 30 N m−2, and dense (light) shading indicates positive (negative) t values that exceed the 95% confidence level. Solid contours are positive, dashed contours negative, and the zero contour is omitted

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    The anomalous 250-mb geopotential height field regressed against the Greenwich meridian component of the EAAM tendency at (a) lag − 4 days, (b) lag − 2 days, (c) lag 0 days, (d) lag + 2 days, and (e) lag + 4 days. Lag 0 corresponds to the time of the maximum value for the Greenwich meridian component of the EAAM tendency. The contour interval is 4 m, dense (light) shading indicates positive (negative) t values that exceed the 95% confidence level. Solid contours are positive, dashed contours negative, and the zero contour is omitted

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    The EAAM amplitude regressed against itself divided by 108 kg m2 s−1

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    The anomalous 250-mb streamfunction regressed against the EAAM amplitude at (a) lag − 4 days, (b) lag − 2 days, (c) lag 0 days, (d) lag + 2 days, and (e) lag + 4 days. The contour interval is 2.5 × 10−7 m2 s−1, and dense (light) shading indicates positive (negative) t values that exceed the 95% confidence level. Solid contours are positive, dashed contours negative, and the zero contour is omitted

  • View in gallery

    The anomalous surface pressure regressed against the EAAM amplitude at (a) lag − 4 days, (b) lag − 2 days, (c) lag 0 days, (d) lag + 2 days, and (e) lag + 4 days. The contour interval is 30 N m−2, and dense (light) shading indicates positive (negative) t values that exceed the 95% confidence level. Solid contours are positive, dashed contours negative, and the zero contour is omitted

  • View in gallery

    The anomalous precipitation (shading), anomalous streamfunction (contours) and anomalous wind (vectors) regressed against the EAAM amplitude at (a) lag − 4 days, (b) lag − 2 days, and (c) lag 0 days. The contour interval is 1.0 × 10−7 m2 s−1, and shaded values exceed a magnitude of 0.07 cm day−1. Solid contours are positive, dashed contours negative, and the zero contour is omitted. Dark (light) shading denotes positive (negative) precipitation anomalies. All fields are truncated at zonal wavenumber 7

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The Dynamics Associated with Equatorial Atmospheric Angular Momentum in an Aquaplanet GCM

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  • 1 EMS Environment Institute, The Pennsylvania State University, University Park, Pennsylvania
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Abstract

The dynamical processes that drive intraseasonal equatorial atmospheric angular momentum (EAAM) fluctuations in a 4000-day aquaplanet GCM run are examined. The all-ocean lower boundary has a sea surface temperature field that is both independent of longitude and symmetric across the equator. Because of the absence of topography, the model includes an equatorial bulge and friction torque, but not a mountain torque. The methodology adopted is to regress variables such as surface pressure, streamfunction, precipitation, and the two torques against individual components and the amplitude of the EAAM vector.

The results indicate that the phase of the EAAM vector is associated with the westward propagation of a zonal wavenumber-1 midlatitude Rossby wave. This wave has characteristics that closely match those of a normal mode of the GCM and also those of the first antisymmetric rotational mode of the shallow water model on the sphere. Fluctuations in the amplitude of the EAAM vector are found to be related to the presence of a zonal wavenumber-1 mixed Rossby–gravity wave in the Tropics. The structure of the precipitation anomalies suggests that the latent heat release associated with the mixed Rossby–gravity wave excites poleward Rossby wave propagation, which alters the EAAM amplitude. The above dynamical processes are also found to determine the phase and amplitude of the equatorial bulge torque. It is this torque that dominates the driving of the EAAM. Lastly, the properties of the friction torque are discussed.

Corresponding author address: Dr. Steven B. Feldstein, EMS Environmental Institute, The Pennsylvania State University, 2217 Earth-Engineering Science Building, University Park, PA 16802. Email: sbf@essc.psu.edu

Abstract

The dynamical processes that drive intraseasonal equatorial atmospheric angular momentum (EAAM) fluctuations in a 4000-day aquaplanet GCM run are examined. The all-ocean lower boundary has a sea surface temperature field that is both independent of longitude and symmetric across the equator. Because of the absence of topography, the model includes an equatorial bulge and friction torque, but not a mountain torque. The methodology adopted is to regress variables such as surface pressure, streamfunction, precipitation, and the two torques against individual components and the amplitude of the EAAM vector.

The results indicate that the phase of the EAAM vector is associated with the westward propagation of a zonal wavenumber-1 midlatitude Rossby wave. This wave has characteristics that closely match those of a normal mode of the GCM and also those of the first antisymmetric rotational mode of the shallow water model on the sphere. Fluctuations in the amplitude of the EAAM vector are found to be related to the presence of a zonal wavenumber-1 mixed Rossby–gravity wave in the Tropics. The structure of the precipitation anomalies suggests that the latent heat release associated with the mixed Rossby–gravity wave excites poleward Rossby wave propagation, which alters the EAAM amplitude. The above dynamical processes are also found to determine the phase and amplitude of the equatorial bulge torque. It is this torque that dominates the driving of the EAAM. Lastly, the properties of the friction torque are discussed.

Corresponding author address: Dr. Steven B. Feldstein, EMS Environmental Institute, The Pennsylvania State University, 2217 Earth-Engineering Science Building, University Park, PA 16802. Email: sbf@essc.psu.edu

1. Introduction

Most studies that investigate temporal fluctuations of the atmospheric angular momentum vector have concentrated on its axial component. The reason for limiting atmospheric angular momentum research to its axial component is, in part, because the axial component of the atmospheric angular momentum vector is closely related to the zonal mean zonal flow, a fundamental variable of the atmospheric general circulation. The two equatorial components of the atmospheric angular momentum vector have, on the other hand, generated much less interest among atmospheric scientists. In part, this may have arisen from the property that the equatorial components of the atmospheric angular momentum vector are related to a planetary-scale zonal wavenumber-1 Rossby wave [see Eqs. (2) and (3) below]. This particular Rossby wave, although of interest to atmospheric scientists, has not commanded the same attention as the zonal mean zonal flow.

Several studies have shown that the two equatorial components of the atmospheric angular momentum vector are closely related to polar motion, which is defined as the displacement of the position of the earth's axis of rotation relative to the earth's crust. For example, on both interannual and intraseasonal timescales, the correlation between polar motion and equatorial atmosphere angular momentum (EAAM) is typically near a value of 0.6 (e.g., Eubanks et al. 1988; Chao 1993; Nastula 1995). The reason that this correlation is not higher, as is the case for the relationship between the axial components of atmospheric and solid-earth angular momentum, is due to an important role played by the ocean (e.g., Ponte et al. 1998; Ponte and Stammer 1999; Celaya et al. 1999; Nastula and Ponte 1999).

The three-dimensional atmospheric angular momentum vector EM, integrated over the entire atmosphere, can first be written as (e.g., see Bell 1994)
i1520-0469-60-15-1822-e1
where ρ is density, r the position vector, u the velocity vector relative to the earth's underlying surface, and a volume element. Inspection of (1) indicates that EM has both a mass contribution, which arises from the atmosphere's rotation with the solid earth's angular velocity Ω and a wind contribution due to atmospheric motions relative to the earth's surface. In studies of both EAAM and polar motion, it is standard to specify the two equatorial components of the EAAM vector as being aligned with the Greenwich meridian and at 90°E. With this representation, the EAAM vector is conveniently written in a coordinate system fixed to the rotating earth, rather than being expressed in an inertial or nonrotating reference frame. As shown in Bell (1994), the two equatorial components of the EM vector in (1), expressed in a coordinate system rotating with the earth, can be written as
i1520-0469-60-15-1822-e2
where the quantities EM1 and EM2 are the Greenwich meridian and 90°E components of EM, respectively. In (2) and (3), ps is the surface pressure; u and υ the zonal and meridional winds in the earth's rotating coordinate system, respectively; λ, ϕ, and p the longitude, latitude, and pressure, respectively; R the earth's radius; and g the gravitational acceleration. The first (second) term on the right-hand side of (2) and (3) is the mass (wind) contribution to EM1 and EM2. In the atmosphere (Eubanks et al. 1988; Bell 1994), fluctuations in the mass contribution are about one order of magnitude larger than those of the wind contribution. As discussed by Bell (1994), many of the dynamical characteristics of EAAM are determined by the ratio of its mass and wind contributions. Given this dominance of the mass terms in (2) and (3), it is immediately apparent that EAAM is primarily associated with wave disturbances in the zonal wavenumber-1 surface pressure field.
The equation for the conservation of EAAM can be written as (for the remainder of this paper, EM will designate only the two equatorial components of the EAAM vector)
i1520-0469-60-15-1822-e4
[For more detail in the derivation of this equation, see Eq. (17) of Bell (1994) and Eq. (2.18) of Egger and Hoinka (2000).] The time derivatives in (4) are for inertial, and rotating coordinate systems, denoted by the subscripts “i” and “r,” respectively. The quantities FT, MT, and BT are the friction, mountain, and equatorial bulge torques, respectively. The equatorial bulge torque can be understood as representing a mountain torque that arises from the spheroidal shape of the earth. Because BT = Ω × EMm [Bell 1994; this can be seen by comparing (2) and (3) with (7) and (8) in section 4], we can rewrite (4) as
i1520-0469-60-15-1822-e5
where EMm and EMw are the mass and wind field contributions to EM, respectively, that is, the first and second terms on the right-hand side of (2) and (3).

The approach to be adopted in this study is to examine EAAM temporal variability in a Geophysical Fluid Dynamics Laboratory (GFDL) general circulation model (GCM) that has an all-ocean lower boundary whose sea surface temperature is specified to be independent of longitude and symmetric across the equator. Such a model is often referred to as an aquaplanet GCM. The advantage of using this idealized model, rather than using atmospheric data, lies with its simplicity. This is because it is likely much easier to understand the dynamical processes that drive EAAM fluctuations in the idealized model, and because the idealized model results may help us develop a coherent dynamical framework for future studies of EAAM in the atmosphere. Thus, our use of the aquaplanet GCM should be viewed as a first step toward the goal of enhancing our understanding of EAAM of the atmosphere. Also, it is important to keep in mind that the zonal uniformity of the lower boundary in the aquaplanet GCM implies that there are limitations to using this idealized model for EAAM research. For example, several studies have identified particular longitudes on the earth's surface where fluctuations in atmospheric mass are more strongly associated with EAAM/polar motion (Eubanks et al. 1988; Salstein and Rosen 1989; Kuehne et al. 1993; Nastula 1995; Nastula and Salstein 1999). Furthermore, Egger and Hoinka (2002) showed that the standard deviation of the globally integrated mountain torque is about one-fifth that of the globally integrated equatorial bulge torque. In another study, de Viron et al. (1999) showed that the globally integrated mountain and equatorial bulge torques are of similar magnitude. These results imply that a more complete understanding of EAAM in the atmosphere requires taking into account the nonuniformity of the earth's surface, a feature not present in the idealized model. These studies also showed that the equatorial bulge torque is one to two orders of magnitude greater than that of the friction torque.

The aim of this paper is twofold: first, to document the dynamical processes that are associated with intraseasonal fluctuations (period <90 days) of EAAM in the aquaplanet GCM, and second, to present hypotheses that may account for these dynamical processes. Although there are several previous papers that do describe the properties of EAAM, for example, Salstein and Rosen (1989) and Egger and Hoinka (2000), as far as this author is aware, there are no previous papers that examine the dynamical processes that drive EAAM fluctuations.

Although the focus of this study is on EAAM, the results will have implications for the study of polar motion. This is the case even though the equatorial angular momentum of the solid earth in the model does not change with time. If the equatorial angular momentum of the solid earth in the model were allowed to vary with time, the impact of these small changes on the model's EAAM would be negligible. This implies that the same processes that drive the model's EAAM fluctuations would also drive polar motion in the model, if it were allowed to take place.

The methodology is described in section 2. This is followed in section 3 by an analysis of the dynamical processes, which determine the phase and amplitude of the EAAM vector. An investigation of the equatorial bulge and friction torques is presented in section 4, followed by the discussion and conclusions in section 5.

2. Methodology

As stated in the introduction, we will use an aquaplanet GCM to examine EAAM variability. The length of the model integration is 4000 days, and the truncation is at rhomboidal 30 resolution. Since the lower boundary of this model is entirely ocean, the model includes the equatorial bulge and friction torques, but it does not include a mountain torque. This longitudinal uniformity of the lower boundary both simplifies the properties of the torques and implies that any statistical differences between the two components of EAAM must be due to sampling. The model retains all the complex physical processes, such as radiation, convection, and the hydrological cycle, of a full GCM. There are several important criteria that a model must satisfy if it is to be useful for examining EAAM. These include 1) that the model conserves EAAM; 2) because the mountain torque is absent, that the EAAM budget be dominated by the equatorial bulge torque (Bell et al. 1991; de Viron et al. 1999; Egger and Hoinka 2000, 2002); and 3) that the mass contribution to the EAAM is one order of magnitude greater than the wind contribution. The latter two criteria are based on observations of EAAM in the atmosphere. For this model, as we will see, the EAAM budget is extremely well balanced, the equatorial bulge torque is much larger than the friction torque1 (not shown), and the mass contribution is indeed one order of magnitude greater than the wind contribution. Thus, this particular GCM, which represents a balance between simplicity and complexity, does appear to be appropriate for investigating the fundamental physical processes that drive EAAM fluctuations.

The primary methodology adopted is to linearly regress relevant atmospheric variables against the individual components of the EAAM vector EM1 and EM2, the amplitude of the EAAM vector EM21 + EM22, and the tendency of the EAAM vector both in nonrotating and rotating coordinate systems. In the regression equations, the amplitude of the anomaly (in the study, an anomaly is defined to be the deviation of a quantity from its time-mean value) is always specified to be one standard deviation. Included in this regression analysis are the equatorial bulge and friction torques, 250-mb geopotential height, 250-mb streamfunction, surface pressure, and precipitation. These regressions extend over lags from 30 days before to 30 days after the maximum value of the quantity being regressed against.

In this paper, all quantities examined will remain temporally unfiltered. We concentrate on variability at intraseasonal timescales because more than 50% of the observed EAAM variance occurs for timescales less than 90 days (Salstein and Rosen 1989).

3. Phase and amplitude of EAAM vector

Before examining the properties of the phase and amplitude of the EAAM vector, we first verify that the EAAM budget is well balanced. For this calculation, we regress the Greenwich meridian component of both the left- and right-hand sides of (4), that is, (dEM1/dt)rΩEM2 and FT1 + BT1, respectively, against (dEM1/dt)r (see Fig. 1). This regression extends over lags from ±30 days. As can be seen, the budget is indeed well balanced at most lags. Very similar results are obtained for the regression against (dEM2/dt)r. Also, Fig. 1 shows that all terms in the EAAM budget fluctuate with a period near 10 days. This 10-day period is discussed in detail in the following section on the phase of the EAAM vector. One noticeable feature in Fig. 1 is that the friction torque (the friction torque will be examined in more detail in section 4) is much smaller than the other two terms in (5), which implies that the tendency of the individual EAAM components in the rotating coordinate system are primarily determined by the wind field contribution to Ω × EM (recall that MT = 0 in this model).

a. Phase of EAAM vector

In this section, we examine the dynamical processes associated with the phase of the EAAM vector, in a coordinate system fixed with the rotating earth. In section 4a, which deals with the equatorial bulge torque, we will also discuss the phase of the EAAM vector as seen in a nonrotating coordinate system. We begin by evaluating the power spectrum of the individual EAAM components, followed by a lagged regression analysis of the surface pressure and upper-tropospheric geopotential height.

We first examine the power spectrum of EM2, the 90°E component of the EAAM vector (due to the zonal symmetry of the lower boundary, the power spectrum of the Greenwich meridian component EM1 is extremely similar; see Fig. 2). The distinctive 10-day spectral peak implies that the EAAM vector includes a component that rotates with a 10-day period, relative to the rotating earth. Furthermore, a lag correlation calculation between the two EAAM components indicates that EM2 leads EM1 by 2 to 3 days with a maximum value of about 0.3. This lag correlation, which exceeds the 99% confidence level, indicates that this 10-day rotation at in a westward direction. These properties of the EAAM vector also imply that EM1 and EM2 must grow and decay within a period of 10 days.

To examine if this 10-day spectral peak occurs in observational data, power spectra of EM1 and EM2 were calculated using 17 yr (1979–95) of the National Centers for Environmental Prediction–National Center for Atmospheric Research (NCEP–NCAR) Reanalysis dataset. Although not as distinctive as in Fig. 2, the analysis did show the presence of strong oscillations at a period near 10 days in both EM1 and EM2. Similar periods were also reported by Brzezinski (1987). In addition, with lag autocorrelation calculations Egger and Hoinka (2002) also found a 10-day period for both components of the observed EAAM vector. These results give further confidence that the essential underlying dynamics describing EAAM in the model may also be taking place in the atmosphere.

We next examine the phase propagation characteristics of the EAAM vector, relative to the rotating earth. This is performed by regressing the anomalous surface pressure field against (dEM1/dt)r. [As expected, the regression against (dEM2/dt)r was found to give extremely similar spatial patterns, except for the 90° phase difference between the EM1 and EM2 surface pressure fields. Also, although the phase could be evaluated with regression against EM1, we chose (dEM1/dt)r because its emphasis on large phase changes in time yielded slightly clearer results than regression against EM1.] The results illustrated in Fig. 3 indicate that the rotation of the EAAM vector coincides with a wave that also propagates westward with a 10-day period. Calculations at other model levels show that this 10-day westward-propagating wave exhibits an equivalent barotropic vertical structure. For example, in Fig. 4, a very similar spatial pattern can be seen in the anomalous 250-mb geopotential height field. Also, as can be seen from Figs. 3 and 4, the wave field associated with anomalous EAAM is a zonal wavenumber-1 disturbance in both hemispheres that has a 180° change in phase across the equator. The fact that this wave takes on these features arises from the constraints implied by the form of the first term on the right-hand side of (2) and (3); that is, such a wave structure maximizes the EAAM. Furthermore, the distinct 10-day period of this wave provides further support for the idea that this wave does not arise from random projections onto the spatial patterns in Figs. 3 and 4, but that the wave corresponds to a coherent mode of the model.

The factors determining the westward phase propagation of the wave in Figs. 3 and 4 are examined in the context of normal modes of the shallow water model equations on the sphere. This approach is motivated by Eubanks et al. (1988), Bell (1994), and Egger and Hoinka (1999), who suggest that fluctuations in EMm may be interpreted as arising from atmospheric normal modes. For our calculations, we obtain orthonormal Hough mode solutions to Laplace's tidal equations [see Kasahara (1976) for details of the methodology used in this study]. These equations correspond to those when the shallow water model equations are linearized. This solution can be written symbolically as
i1520-0469-60-15-1822-e6
where W = (U1n, −iV1n, Z1n); the Θ1n are the Hough vector modes; the C1n are the expansion coefficients; i = (−1)1/2; and U1n, V1n, and Z1n denote the complex zonal and meridional velocities, and height, respectively (see Kasahara 1980). The subscript indicates the meridional mode index, which corresponds to (n − 1) nodes between the North and South Pole, and the superscript denotes zonal wavenumber 1. Since the wave field in Figs. 3 and 4 is antisymmetric across the equator, the index n in (6) is restricted to even numbers [to see the spatial structure of these modes, see Fig. 5 in Kasahara (1976)]. For these calculations, the value chosen for the equivalent depth is 5.8 km. This particular value is selected because the n = 2 Hough mode, which has a meridional structure very similar to that shown in Figs. 3 and 4, is found to have a period of westward propagation of 9.8 days, very close that seen in these two figures. For the next two antisymmetric Hough modes, that is n = 4 and n = 6, these waves are found to propagate westward with a period of 19.0 and 31.9 days, respectively.

The relationship between the Hough mode solutions of Laplace's tidal equation and the EAAM is examined by performing linear correlations. For these calculations, the individual EAAM component time series are correlated with time series obtained by projecting the daily surface pressure data onto individual Hough mode spatial patterns. This projection gives a time-varying amplitude for the Hough modes in (6). The correlation values are 0.97 and 0.52 for the n = 2 and n = 4 Hough modes, respectively, with all higher-order modes yielding much smaller values, typically being less than 0.1. Power spectra calculations of these Hough mode amplitude time series show that the n = 2 mode has the same very strong, 10-day spectral peak as for the individual EAAM components shown in Fig. 2. This implies that the anomalous surface pressure field in Fig. 3 shows properties that are extremely similar to the n = 2 mode of the shallow water model. In contrast, because it is found that the amplitude time series for the n = 4 Hough mode lacks any strong spectral peaks, a connection cannot be made between the anomalous surface pressure field associated with EAAM and the n = 4 normal mode of the shallow water model.

Similar projection calculations and linear correlations are performed with the 250-mb height field data. The correlation between the individual EAAM component time series with the amplitude time series of the n = 2 and n = 4 Hough modes is 0.70 and 0.55, respectively, with the correlation values for all the higher-order Hough modes, again, being much smaller. Furthermore, the linear correlation between the EAAM component time series and a multiple regression time series consisting of both the n = 2 and n = 4 amplitude time series yields a value of 0.98. This indicates that, as for the surface pressure field, in the upper troposphere, both the n = 2 and n = 4 Hough mode spatial patterns dominate the EAAM variability. However, in contrast to the surface pressure field, the n = 2 Hough mode plays a less prominent role in the upper troposphere. The power spectra analysis of the 250-mb height n = 2 and n = 4 amplitude time series is very similar to that for the surface pressure field, in the sense that only the n = 2 amplitude time series has a strong spectral peak, again at a period near 10 days.

These linear correlation results for both the surface pressure and 250-mb height fields suggest that the n = 2 Hough mode is a normal mode of this GCM, implying that the 10-day westward phase speed of the wave associated with EAAM corresponds to this normal mode. Thus, this wave propagation can be understood as simply arising from the propagation of a free Rossby wave. Furthermore, these results suggest that the fluctuation in the amplitude of the EAAM vector, to be discussed in section 3b is also associated with this normal mode of the model.

The slow westward propagation of the Rossby wave in Fig. 3 indicates that the EAAM vector, as observed in an inertial coordinate system, rotates eastward with an angular velocity slightly less than that of the rotating earth. The 10-day Rossby wave westward phase speed implies that the angular velocity of the EAAM vector in an inertial coordinate system must be approximately 0.9Ω. Also, this constant changing in the direction of the EAAM vector requires that a torque always be present. As we will see in section 4, these directional changes in the EAAM vector are accomplished by the equatorial bulge torque.

b. Amplitude of EAAM vector

We next examine the dynamical processes associated with fluctuations in the amplitude of the EAAM vector. Since amplitude is a scalar quantity, it must be the same in both the rotating and nonrotating coordinate systems. As discussed in section 2, the technique adopted involves the regression of various quantities against the amplitude EM21 + EM22 of the EAAM vector. Because the phase of the wave anomaly field is arbitrary, due to the zonal uniformity of the model's lower boundary, a longitudinal phase shifting to a common longitude must be done for each day prior to performing the regression analysis. Otherwise, this arbitrariness in phase would result in a regression pattern that is entirely noise. This longitudinal phase shifting is accomplished by first determining the longitude λmax, which maximizes the projection of the daily surface pressure field ps(λ, ϕ, t) onto the pattern sinϕ cos2ϕ cos(λλmax). This particular spatial pattern represents a combination of the integrands of the first term on the right-hand side of (2) and (3). Then, the surface pressure field is longitudinally phase shifted to ensure that λmax coincides with the Greenwich meridian. All other variables are specified to undergo the same amount of phase shifting as does the surface pressure field. The regression of these longitudinally phase-shifted variables yields stationary anomaly patterns. That is, longitudinally phase-shifted anomalies can be understood as being viewed in a coordinate system that moves with the anomalous surface pressure wave field.

Before investigating the dynamical processes associated with fluctuations in the EAAM amplitude, we first calculate the anomalous EAAM amplitude regressed against itself (see Fig. 5). It is important to keep in mind that this regression shows the anomalous EAAM amplitude, that is, EM21 + EM22EM21 + EM22, where the overbar denotes a nonzero, time-mean value. As can be seen in Fig. 5, the anomalous EAAM amplitude undergoes a life cycle of growth and decay in approximately 10 days, similar to the period of the individual EAAM components EM1 and EM2.

We next examine the anomalous 250-mb streamfunction field regressed against the EAAM amplitude (see Fig. 6). This particular variable, rather than geopotential height, is evaluated in order to better examine dynamical processes that may take place in the Tropics. The evolution of the streamfunction anomalies in Fig. 6 suggests that EAAM amplitude growth is associated with anomalous poleward Rossby wave propagation from the Tropics to high latitudes. This direction of propagation can be seen by examining the lag − 2 day streamfunction anomalies (see Fig. 6b), which shows that the anomalies tend to take on a northeastward (southeastward) orientation in the Northern (Southern) Hemisphere. This can also be seen from the temporal evolution of the amplitude of the low- and high-latitude streamfunction anomalies. For example, between lag − 2 and lag 0 days (see Figs. 6b and 6c), the tropical anomalous streamfunction amplitude decreases, while over the same time interval the high-latitude anomalous streamfunction amplitude increases. A similar comparison between Figs. 6c and 6d suggests that the decrease in the EAAM amplitude is associated with equatorward Rossby wave propagation.

The corresponding anomalous surface pressure field is illustrated in Fig. 7. As expected, a middle- and high-latitude spatial pattern is seen to grow, reach its maximum at lag 0, and then decay. The absence of a strong signal in the tropical anomalous surface pressure field can be understood by noting that the Rossby penetration depth fL/N, where f is the Coriolis parameter, L the horizontal length scale, and N the Brunt–Väisälä frequency, is small in low latitudes (Hoskins et al. 1985). The impact of the Rossby penetration depth on the surface pressure field and the EAAM is discussed in more detail below.

The anomalous precipitation, streamfunction, and velocity fields are shown in Fig. 8. One of the more striking features about the anomalous precipitation field is that, at lag − 2 days, when the growth in the EAAM amplitude is largest (Fig. 8b), there is a large zonal wavenumber-1 contribution that straddles the equator. This anomaly pattern is present, but weaker, at lag − 4 and lag 0 days (Figs. 8a and 8c). As can be seen in Figs. 8a and 8b, the anomalous streamfunction and wind velocity fields near the equator are also dominated by zonal wavenumber 1. These anomaly characteristics are broadly consistent with those of a zonal wavenumber-1 mixed Rossby–gravity wave confined to the Tropics (Wheeler and Kiladis 1999; Wheeler et al. 2000). However, there are some differences between the anomalies in Figs. 8a and 8b and those expected for a mixed Rossby–gravity wave as derived from shallow water theory (Matsuno 1966). For example, the centers of the streamfunction anomalies in Figs. 8a and 8b are not located on the equator, as in shallow water theory; rather, they are found at between and 5° and 7° off the equator. Also, if we assume that precipitation and divergence anomalies should coincide, then there are further differences from shallow water theory since the anomalous precipitation field is not antisymmetric across the equator. On the other hand, when we examine the location of the precipitation anomalies relative to the streamfunction anomalies, an important similarity to shallow water theory can be found. According to shallow water theory, there should be four centers of divergence and convergence, with all four centers attaining their maximum values at the nodes of the anomalous streamfunction pattern (Matsuno 1966; also see Fig. 14 in Wheeler et al. 2000). The location of the four precipitation anomalies in Fig. 8b denoted by “+” and “−” signs do fit this general pattern.

The above difference in spatial structure between the theoretical mixed Rossby–gravity wave, which is symmetric across the equator, and the anomalies shown in Fig. 8, likely arises both because midlatitude influences extend into the Tropics, for example, the strong equatorward wind anomalies north of the equator near 180° longitude, and because the 4000 days in the model run provides for too small a sample size. Presumably the results in Fig. 8 would show improved symmetry properties if the model integration were longer.

The above results suggest that fluctuations in the EAAM amplitude are triggered by the anomalous precipitation field associated with a zonal wavenumber-1 mixed Rossby–gravity wave. We can understand the dynamical processes associated with this relationship between EAAM amplitude and anomalous tropical precipitation by first noting that the anomalous precipitation must be accompanied by anomalous latent heat release in the upper tropical troposphere. For large-scale motions in the Tropics, this anomalous latent heat release is balanced by anomalous divergence (e.g., Holton 1992). As found in many studies, for example, Hoskins and Karoly (1981), the anomalous divergence can excite Rossby waves that propagate poleward in the upper troposphere toward middle and high latitudes. Furthermore, as shown by Hoskins et al. (1985), through the use of potential vorticity dynamics, upper-tropospheric wave anomalies induce a secondary circulation that extends downward to a distance of the Rossby penetration depth D, where D = fL/N, as defined above. This secondary circulation acts to maintain the flow in thermal wind balance. Therefore, as long as D is comparable to the depth of the troposphere, which is the case in middle and high latitudes, upper-tropospheric wave anomalies can excite surface pressure anomalies, hence changes in the EAAM. In summary, the results of the above calculations suggest that fluctuations in EAAM amplitude are first triggered by anomalous tropical precipitation within a zonal wavenumber-1 mixed Rossby–gravity wave, followed by poleward wave propagation, and then by the inducement of surface pressure anomalies.

4. Torques

We next examine the properties of the equatorial bulge and friction torques. The two components of the equatorial bulge torque can be written as (see Bell 1994)
i1520-0469-60-15-1822-e7
where BT1 and BT2 are the Greenwich meridian and 90°E components of the equatorial bulge torque. As mentioned in section 1, and as can be seen by comparing (7) and (8) with (2) and (3), respectively, it is straightforward to show that BT = Ω × EMm. Thus, for individual components, we can write
1Ω2,m2Ω1,m
where EM1,m and EM2,m are the mass contributions to the Greenwich meridian and 90°E components of EM.
The expression used for the friction torque, which is based on Barnes et al. (1983), is
i1520-0469-60-15-1822-e10
where FT1 and FT2 are the Greenwich meridian and 90°E components of the friction torque. Calculation of the equatorial bulge torque is straightforward, since it only depends upon the surface pressure field. For the friction torque, since the surface stresses were not included by GFDL in the model output, it is necessary that they be estimated. For this purpose, we use the same formulation for the frictional stress as in this and other GFDL GCMs (Gordon and Stern 1982). That is, for the frictional stresses in the zonal and meridional directions Fλ and Fϕ, respectively, we use the bulk parameterization scheme Fλ = −ρCD|U|u and Fϕ = −ρCD|U|υ, where |U| is the magnitude of the wind vector. However, unlike in the GCMs, where the value for the drag coefficient varies with time, a constant value of CD = 1.0 × 10−3 is specified. A constant value is chosen for CD since the time variation of this quantity was not saved in the model output. Given that the EAAM budget in Fig. 1 is well balanced, it is likely that the use of a constant value for CD is a reasonable estimate.
We first examine the magnitude of the torques and the other terms in (4), the equation for the conservation of EAAM. This is accomplished by regressing each of the terms in (4) against (dEM/dt)i = (dEM/dt)r + Ω × EM. It is found that Ω × EM is one order of magnitude greater than (dEM/dt)r. The fact that Ω × EM is one order of magnitude greater than (dEM/dt)r can be understood by recalling from section 3 that the period for both components of the EAAM vector in a coordinate system rotating with the earth is approximately 10 days, that is, an angular velocity of Ω/10. The regression of the torques indicates that BT is two orders of magnitude greater than FT (recall that MT is zero in the model). The same difference between the magnitude of the equatorial bulge and friction torques was found in the observational study of de Viron et al. (1999). These results imply that (4) can be approximated as
i1520-0469-60-15-1822-e12
Thus, the temporal variation of the EAAM vector is almost entirely due to the equatorial bulge torque. We will use each of these simplified properties in (12) to further investigate the properties of the torques.

a. Equatorial bulge torque

In the last paragraph of section 3a, we discussed the property that the EAAM vector rotates eastward in a nonrotating coordinate system with an angular velocity 0.9Ω, slightly less than that of the earth. We can both quantify this eastward rotation of the EAAM vector and attain a better understanding of the equatorial bulge torque that drives this rotation by considering the individual components of (12), which can be written in a nonrotating coordinate system as
i1520-0469-60-15-1822-e13
where the subscript “i” denotes the vector component in a nonrotating coordinate system. Solution of (13) indicates that, in the nonrotating coordinate system, the EAAM vector rotates eastward with an angular velocity of Ω, that is, the diurnal frequency of the rotating earth. The fact that the actual angular velocity of the EAAM vector deviates slightly from Ω, must be related to the neglecting of the (dEM/dt)r and Ω × EMw terms in (13). It is these terms that are associated with the westward Rossby wave propagation in Figs. 3 and 4.

Since the integrand for the equatorial bulge torque in (7) and (8) is identical to that for the EM mass contribution in (2) and (3), except for the quadrature phase relationship, and because EM is dominated by its mass contribution, (13) implies that the same anomalous surface pressure field that describes the EAAM vector also accounts for the equatorial bulge torque that drives the eastward rotation of the EAAM vector. This quadrature relationship between the EAAM vector and equatorial bulge torque in (13) is straightforward to understand physically when one interprets the equatorial bulge torque as a mountain torque acting on the earth's bulge. For example, consider the anomalous surface pressure field in Fig. 3d, a lag when the EM1 is near its maximum value. As can be seen in Fig. 3d, the zonal wavenumber-1 meridional pressure gradient, whose maxima at the equatorial bulge occurs at both the Greenwich meridian and the dateline, drive torques that result in an EAAM increase 90° toward the east. In this manner, the same anomalous zonal wavenumber-1 surface pressure field can account for both the EAAM and the torque.

The relationship described by (9), together with the property that the EAAM vector is dominated by its mass contribution, implies that regression of surface pressure against either component of the equatorial bulge torque, to examine its phase, or against the amplitude of this torque, must yield results that are almost identical to those presented in section 3 for the phase and amplitude of the EAAM vector. As required by (9), the primary differences in the anomalous surface pressure fields when regressed against either BT or EM are found to be in phase, that is, the patterns are in quadrature, not in the spatial structure of the patterns themselves (not shown). These results imply that the same dynamical processes that account for the phase and amplitude of the EAAM vector are also associated with the phase and amplitude of the equatorial bulge torque vector. That is, the phase of the equatorial bulge torque is associated with westward Rossby wave propagation, and fluctuations in the amplitude of the equatorial bulge torque are initially triggered by precipitation anomalies associated with the occurrence of a zonal wavenumber-1 mixed Rossby–gravity wave in the deep Tropics.

b. Friction torque

Although the friction torque is two orders of magnitude smaller than equatorial bulge torque, and thus plays a negligible role in the model's EAAM evolution, we nevertheless briefly discuss results from calculations of the surface stress and friction torque regressed against (dEM/dt)i. The primary motivation for examining the friction torque is to lay the groundwork for future study of EAAM in the atmosphere, where because of the tendency for there to be large cancellation between the equatorial bulge and mountain torques (de Viron et al. 1999), it is possible that for some time periods the friction torque may play a nonnegligible role in the atmosphere.

Calculations of FT1 and FT2 regressed against (dEM/dt)i find that the friction torque vector also rotates westward relative to the rotating earth with an approximate period of 10 days. We also examine the contribution to the friction torque individually by both the zonal and meridional frictional stresses, that is, Fλ and Fϕ in (10) and (11). It is found that the contributions by Fλ and Fϕ strongly oppose each other for both the Greenwich meridian and the 90°E components of the friction torque vector (not shown). For example, the globally integrated friction torque with either Fλ and Fϕ set to zero is typically about 3 times as large as the globally integrated friction torque with both surface stresses included. Thus, for the atmosphere, where the surface drag coefficient is spatially nonuniform, it is possible that there will be lesser internal cancelation between the contributions from Fλ and Fϕ, resulting in a relatively larger friction torque than in the model.

5. Discussion and conclusions

In this study, we examine the atmospheric dynamical processes that are associated with fluctuations in EAAM in an idealized aquaplanet GCM. This model has an all-ocean lower boundary with sea surface temperatures specified to be both zonally symmetric and symmetric across the equator. Thus, although equatorial bulge and friction torques are present, the simplicity of the lower boundary precludes the presence of mountain torques. The EAAM was found to be closely associated with a zonal wavenumber-1 disturbance that extends through the entire troposphere and is antisymmetric relative to the equator. Solutions were found to Laplace's tidal equations in order to determine whether the wave field associated with the EAAM is related to the normal modes of the shallow water model on the sphere. This was accomplished by projecting both the surface pressure and 250-mb geopotential height fields onto the Hough mode solutions of the shallow water model. The results revealed that the wave field associated with the EAAM is dominated by the first antisymmetric rotational normal mode relative to the equator in the shallow water model, and that this same mode is a normal mode of the aquaplanet GCM.

There are two primary results. First, the phase of the EAAM vector arises from the westward propagation of the normal mode described in the above paragraph. Thus, the phase of the EAAM vector can be understood in terms of Rossby wave propagation. Within the earth's coordinate system, this is manifested as a 10-day westward rotation, whereas in an inertial coordinate system, this indicates that the EAAM vector rotates eastward with an angular velocity of 0.9Ω, slightly less than that of the earth. Second, in addition to the direct relationship between the amplitude of the EAAM vector and that of the above-normal mode, it was found that fluctuations in the amplitude of the EAAM vector are associated with equatorial precipitation and wind anomalies that have characteristics typical of a zonal wavenumber-1 equatorial mixed Rossby–gravity wave.

It was shown that the equatorial bulge torque dominates the driving of the EAAM fluctuations. Furthermore, the dominance of the EAAM fluctuations by their mass contribution implied that the same anomalous surface pressure field that describes the EAAM vector also accounts for the equatorial bulge torque, which drives the eastward rotation of the EAAM vector in an inertial coordinate system. This indicates that changes in the phase and amplitude of the equatorial bulge torque are due to the same physical processes that drive the phase and amplitude fluctuations of the EAAM vector.

Analyses of other variables, together with the above findings, suggest that fluctuations in the EAAM amplitude are initially triggered by the latent heating associated with the anomalous precipitation within the mixed Rossby–gravity waves. This latent heating appears to generate poleward-propagating Rossby waves. When these waves reach middle and high latitudes, they are easily capable of inducing waves in the surface pressure field. At the same time, these waves, which extend throughout the troposphere, will project onto the above-normal mode of the model. This results in a change in the amplitude of both the equatorial bulge torque and the EAAM vector. This link between EAAM and mixed Rossby–gravity wave is perhaps not surprising, given that the surface pressure field associated with EAAM and the precipitation fields associated with the mixed Rossby–gravity waves are both associated with zonal wavenumber-1 disturbances that are antisymmetric across the equator.

The results presented in this study suggest a rather simple picture for the processes that drive EAAM fluctuations in the model. The next step in our research of EAAM is to examine the degree to which the processes that drive the model's EAAM fluctuations also drive fluctuations in both the EAAM observed in the atmosphere and in polar motion of the solid earth. Because the study of de Viron et al. (1999) found that the observed equatorial bulge and mountain torques are of similar magnitude and are strongly anticorrelated, it is plausible that the same physical picture presented in this study will apply to the atmosphere, with the mountain torque acting primarily to dampen the influence of the equatorial bulge torque.

The results of this study are for an earthlike parameter regime for which the mass contribution to the EAAM vector is an order of magnitude greater than the wind contribution. To examine the importance of this mass-to-wind ratio in a more general context, we rewrite the equation for EAAM conservation with the friction and mountain torques neglected; that is,
i1520-0469-60-15-1822-e14
As shown in Bell (1994),
i1520-0469-60-15-1822-e15
where σ is the frequency of the wave field observed from the rotating earth. In the aquaplanet model, the 10-day period of the Rossby wave field associated with the EAAM corresponds to σ = 0.1Ω. Thus, the dominance of the third term on the left-hand side of (14), as we found in this study, is a manifestation of the large mass to wind EAAM ratio in the model. These mass-to-wind EAAM ratio properties have interesting implications when we consider EAAM for non-earthlike parameter settings. For example, compared to the earth, the value of σ will be of much larger magnitude for planets with a much stronger westerly jet. Such a stronger jet can arise on a planet with a much larger meridional temperature gradient, or a much larger planetary radius (S. Lee 2002, personal communication). For such planets, the third term on the left-hand side of (14) would no longer dominate, and the rather simple relationship between the EAAM vector and the equatorial bulge torque found in this study would no longer hold. Thus, to the extent that the findings of this study do apply to the atmosphere, these results suggest that the earth's atmosphere is in a conceptually simpler part of the parameter space, at least from a geophysical fluid dynamical viewpoint.

Acknowledgments

This research was supported by the National Science Foundation through Grant ATM 0224870. I would like to thank Drs. Sukyoung Lee, Christian Franzke, and David Salstein, and an anonymous reviewer for their beneficial comments. I would also like to thank the NOAA/Geophysical Fluid Dynamics Laboratory for providing me with the data for this study.

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Fig. 1.
Fig. 1.

The EAAM budget. The solid (dashed) curve corresponds to the right-hand side (left-hand side) of (4), and the long-dashed curve is the globally integrated friction torque. All quantities are divided by 1020 kg m2 s−2

Citation: Journal of the Atmospheric Sciences 60, 15; 10.1175/1520-0469(2003)060<1822:TDAWEA>2.0.CO;2

Fig. 2.
Fig. 2.

The power spectrum of the 90°E component of the EAAM vector in the climate model

Citation: Journal of the Atmospheric Sciences 60, 15; 10.1175/1520-0469(2003)060<1822:TDAWEA>2.0.CO;2

Fig. 3.
Fig. 3.

The anomalous surface pressure field regressed against the Greenwich meridian component of the EAAM tendency at (a) lag − 4 days, (b) lag − 2 days, (c) lag 0 days, (d) lag + 2 days, and (e) lag + 4 days. Lag 0 corresponds to the time of the maximum value for the Greenwich meridian component of the EAAM tendency. The contour interval is 30 N m−2, and dense (light) shading indicates positive (negative) t values that exceed the 95% confidence level. Solid contours are positive, dashed contours negative, and the zero contour is omitted

Citation: Journal of the Atmospheric Sciences 60, 15; 10.1175/1520-0469(2003)060<1822:TDAWEA>2.0.CO;2

Fig. 4.
Fig. 4.

The anomalous 250-mb geopotential height field regressed against the Greenwich meridian component of the EAAM tendency at (a) lag − 4 days, (b) lag − 2 days, (c) lag 0 days, (d) lag + 2 days, and (e) lag + 4 days. Lag 0 corresponds to the time of the maximum value for the Greenwich meridian component of the EAAM tendency. The contour interval is 4 m, dense (light) shading indicates positive (negative) t values that exceed the 95% confidence level. Solid contours are positive, dashed contours negative, and the zero contour is omitted

Citation: Journal of the Atmospheric Sciences 60, 15; 10.1175/1520-0469(2003)060<1822:TDAWEA>2.0.CO;2

Fig. 5.
Fig. 5.

The EAAM amplitude regressed against itself divided by 108 kg m2 s−1

Citation: Journal of the Atmospheric Sciences 60, 15; 10.1175/1520-0469(2003)060<1822:TDAWEA>2.0.CO;2

Fig. 6.
Fig. 6.

The anomalous 250-mb streamfunction regressed against the EAAM amplitude at (a) lag − 4 days, (b) lag − 2 days, (c) lag 0 days, (d) lag + 2 days, and (e) lag + 4 days. The contour interval is 2.5 × 10−7 m2 s−1, and dense (light) shading indicates positive (negative) t values that exceed the 95% confidence level. Solid contours are positive, dashed contours negative, and the zero contour is omitted

Citation: Journal of the Atmospheric Sciences 60, 15; 10.1175/1520-0469(2003)060<1822:TDAWEA>2.0.CO;2

Fig. 7.
Fig. 7.

The anomalous surface pressure regressed against the EAAM amplitude at (a) lag − 4 days, (b) lag − 2 days, (c) lag 0 days, (d) lag + 2 days, and (e) lag + 4 days. The contour interval is 30 N m−2, and dense (light) shading indicates positive (negative) t values that exceed the 95% confidence level. Solid contours are positive, dashed contours negative, and the zero contour is omitted

Citation: Journal of the Atmospheric Sciences 60, 15; 10.1175/1520-0469(2003)060<1822:TDAWEA>2.0.CO;2

Fig. 8.
Fig. 8.

The anomalous precipitation (shading), anomalous streamfunction (contours) and anomalous wind (vectors) regressed against the EAAM amplitude at (a) lag − 4 days, (b) lag − 2 days, and (c) lag 0 days. The contour interval is 1.0 × 10−7 m2 s−1, and shaded values exceed a magnitude of 0.07 cm day−1. Solid contours are positive, dashed contours negative, and the zero contour is omitted. Dark (light) shading denotes positive (negative) precipitation anomalies. All fields are truncated at zonal wavenumber 7

Citation: Journal of the Atmospheric Sciences 60, 15; 10.1175/1520-0469(2003)060<1822:TDAWEA>2.0.CO;2

1

Because the GCM neglects both ellipticity (the model is spherical) and the centripetel acceleration, the equatorial bulge torque is implicitly present in the model equations (Bell 1994). See also Egger and Hoinka (1999).

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