## Abstract

An equivalent-barotropic (EB) description of the tropospheric temperature field is derived from the geostrophic empirical mode (GEM) in the form of a scalar function Γ(*p, ϕ*), where *p* is pressure and *ϕ* is 300–850-mb thickness. Baroclinic parameter *ϕ* plays the role of latitude at each longitudinal section. Compared with traditional Eulerian-mean methods, GEM defines a mean field in baroclinic streamfunction space with a time scale much longer than synoptic variability. It prompts an EB concept that is only based on a baroclinic field.

Monthly GEM fields are diagnosed from NCEP–NCAR reanalysis data and account for more than 90% of the tropospheric thermal variance. The circumglobal composite of GEM fields exhibits seasonal, zonal, and hemispheric asymmetries, with larger rms errors occurring in winter and in the Northern Hemisphere (NH). Zonally asymmetric features and planetary deviation from EB are seen in the NH winter GEM. Reconstruction of synoptic sections and correlation analysis reveal that the tropospheric temperature field is EB at the leading order and has a 1-day phase lag behind barotropic variations in extratropical regions.

## 1. Introduction

A proper specification of mean field is key to our understanding of inhomogeneous turbulence and nonlinear eddy-mean flow interaction. In a review of equilibrium statistical mechanics, Holloway (1986) points out the nontrivial problem of distinguishing mean and fluctuating fields: because geophysical flows are often characterized by continuous and red spectra, the definition of mean field becomes arbitrary and the dynamical relation between mean and eddy fields is elusive.

In atmospheric studies, the most common choice of mean field is zonal mean. A time-averaged zonal-mean climate in physical space comprises both barotropic and baroclinic properties. Its dynamical relevance depends upon the degree of spatiotemporal symmetry of the system: if there is large-amplitude stationary wave or singular climate trend, such Eulerian mean could deviate significantly from actual synoptic fields. Reader may refer to Peixoto and Oort (1992) for a traditional description of the atmospheric mean state. To diagnose longitudinally dependent phenomena, a more robust definition of mean field is needed.

Recent analyses of hydrographic data in the Antarctic Circumpolar Current (ACC) have revealed a relatively steady state of scalar distribution in baroclinic streamfunction space (Sun and Watts 2001, hereafter SW01; Watts et al. 2001). A gravest mode accounts for more than 97% of the hydrographic variance in the ACC and is represented by empirical function Γ(*ϕ, p*), where *p* is pressure and *ϕ* is a baroclinic streamfunction parameter. The function, named geostrophic empirical mode (GEM), is a stationary baroclinic solution and constitutes a baroclinic coherent structure (Sun 2001). It distinguishes the ACC from a three-dimensional turbulence scenario, rendering a low-dimensional perspective to oceanic problems such as water mass formation, poleward heat flux, and low-frequency variability (Sun and Watts 2002a, b,c).

Similar coherent structure exists in atmospheric temperature data. Banacos and Bluestein (2004) mentioned that to a first-order approximation, straight hodographs are observed away from the tropopause and planetary boundary layer (PBL), which implies isotherm orientation does not change with height and geostrophic vertical shear is unidirectional in the mean. Such parallel configuration leads to GEM formulation, as discussed in the appendices.

The observation thus points to a definition of mean field in baroclinic streamfunction space through GEM function Γ(*ϕ, p*). With vertically parallel isopycnals, GEM mean field reflects a barotropization of 3D scalar distribution. Whether or not such a low-dimensional structure acts as a physical constraint in geophysical fluids needs theoretical analysis. The present paper, however, is focused on the statistical aspect of GEM and aims to provide a general diagnosis of tropospheric mean temperature.

The paper is organized as follows. Section 2 establishes a proper equivalent-barotropic (EB) concept for baroclinic fluids and discusses the connection between GEM and existing conceptual models of the atmosphere. The statistical basis for GEM construction is given in section 3. In section 4 we diagnose tropospheric temperature GEM and examine its spatiotemporal characteristics.

## 2. Baroclinic mean field

In meteorology and oceanography, equivalent-barotropic refers to a flow regime where isotherm and height contours are parallel on isobaric surfaces and winds do not change direction with height. For mathematical description, the wind speed at any level is taken as some fraction of that at a mean level (Charney et al. 1950; Killworth 1992):

where *A*(*p*) is a fixed vertical structural function.

Such conventional EB concept is associated with geostrophic velocity and geopotential. It offers poor description for synoptic processes since a fundamental property of rotating inhomogeneous fluids, namely the barotropic–baroclinic partition of geostrophic motions, is not reflected. The barotropic component (i.e., the reference geopotential for the thermal wind) is defined as the geostrophic velocity or pressure near the lower boundary. Although mathematically it can be chosen at any level or like in some literature defined as a vertically average property, the choice here is based on physical consideration: the surface pressure reflects the weight of the entire fluid column that varies from the internal thermal structure, and the barotropic–baroclinic variables thus defined are dynamically separable and often out of phase in synoptic variability.

Many conceptual models in synoptic meteorology distinguish barotropic and baroclinic components properly. Charney et al. (1950) noticed that synoptically it is more tenable to assume isotherms, rather than winds, to be parallel at all heights. To describe cyclone development, Sutcliffe and Forsdyke (1950) combined surface pressure and 500–1000-mb thickness as a first approximation to the three-dimensional troposphere. By assuming small variation in the thermal wind direction, they found thickness lines tend to be advected with the 1000-mb geostrophic wind. In the same vein, Sanders (1971) developed an analytical model for synoptic baroclinic waves, essentially assuming no vertical phase difference in the temperature field. The assumptions for vector and scalar fields in these models are kinematically equivalent and all conform to a GEM thermal field, as demonstrated in the appendices.

The GEM solution derived by Sun (2001) is equivalent to the premise that there be no thermal-wind contribution to horizontal temperature advection, that is,

It enables us to extend EB concept to scalar distribution. For midlatitude synoptic-scale processes, it is the baroclinic field (temperature and thermal wind **u**_{BC}), rather than the total geostrophic wind, that approximately holds EB character. A dynamically relevant EB concept for baroclinic fluids should be based on GEM scalar distribution.

We will use GEM function Γ(*ϕ, p*) to define an EB mean field in baroclinic streamfunction space and ascribe the spatiotemporal variation of baroclinic parameter *ϕ*(*x, y, t*) to a barotropic problem. By doing so, the barotropic and baroclinic aspects of geostrophic adjustment are decoupled. The tropospheric temperature field can be expanded in streamfunction space around this mean field as

It contrasts with the quasigeostrophic approximation, which is an asymptotic expansion around a resting state based on Rossby number (Pedlosky 1987),

Horizontal variation of stratification is ignored at the leading order of quasigeostrophy. Its basic equations center on geopotential without distinguishing barotropic–baroclinic components. We shall point out that those models based on the traditional EB concept (e.g., Charney et al. 1950; Krupitsky et al. 1996) are essentially quasigeostrophic, because the structure function, such as *A*(*p*) in (1), does not include horizontal variation. Without a proper baroclinic representation at the leading order, the theoretical effort to improve quasigeostrophy by proceeding to higher-order asymptotic expansion is conceptually deficient.

## 3. Geostrophic empirical mode

In contrast with Eulerian mean methods, the GEM function Γ(*ϕ, p*) defines a streamfunction mean field that is purely baroclinic. The exact functional form is diagnosed from observational or numerical data. The National Centers for Environmental Prediction and the National Center for Atmospheric Research (NCEP–NCAR) have produced a global analysis of atmospheric fields by assimilating various observational data. The daily averaged data from a ten-year period are used. In view of the dominant seasonal cycle in the troposphere, the GEM analysis will be conducted on monthly basis. At this stage the interannual variability and potential secular trend in the data are neglected.

### a. Baroclinic parameter

We choose thickness as the baroclinic parameter for the atmospheric GEM. It is defined as the difference of geopotential height between two pressure surfaces. For example, 300–850-mb thickness is

where *R* is the gas constant for dry air and *g* is a global average of gravity at mean sea level. A daily map of *ϕ*_{300–850} is plotted in Fig. 1, with 7.7 km contours highlighting the axes of westerlies. The large-scale atmospheric flows are primarily zonal, and more so in the Southern Hemisphere (SH) because of the less land–sea temperature contrast.

Figure 2 shows a SH temperature section along 0°. The lowest kilometer of the troposphere is a boundary layer strongly influenced by turbulent fluxes. The stratosphere above the tropopause is close to radiative equilibrium. A deep geostrophic jet appeared between 30° and 50°S as indicated by isentropic tilt through the troposphere. Examination of other daily maps shows that *ϕ*_{300–850} = 7.7 km statistically corresponds to the westerly axis, of which the maximum velocity is located at the tropopause. We will focus on extratropical regions (20°–70°) where midlatitude jet streams are located, and use January and July data from 1991–2000 to represent the two extreme seasons for our analysis.

To illustrate the GEM idea, January 500-mb potential temperature data at 0° are plotted against several meridional coordinates in Fig. 3. The rms residuals (relative to a linear fit) vary in the reference frames of latitude (a geographic coordinate), 500-mb geopotential height (a geostrophic streamfunction containing barotropic component), 100–1000-mb thickness and 300–850-mb thickness (both are baroclinic streamfunction). The baroclinic parameter *ϕ*_{300–850}, which excludes the PBL (dissipative regime) and the tropopause, gives the least variance. The comparison is similar for tropospheric temperature at other levels. It suggests most of the temporal variability in physical space can be removed by a projection onto baroclinic coordinates that represent the geostrophic interior. The relatively large variance in geopotential coordinates, on the other hand, hints that barotropic and baroclinic variations are often out of phase. This will be verified by correlation analysis in section 4c.

It is worthwhile to mention that specific humidity, a thermodynamic variable of the atmosphere, shows no functional relationship with baroclinic streamfunction. This stands in contrast with the adiabatic interior of the ocean, where dissolved oxygen is not a dynamical variable either but has GEM structure, albeit relatively poor (SW01).

### b. The a priori error

The second-order structure function for homogeneous turbulence is not applicable in baroclinic fluids, unless it is defined in baroclinic streamfunction space (SW01). Each temperature measurement *T* contains signal *t* and noise *ε*. If the noise is uncorrelated and the signal has a long correlation length, the second-order structure function for the scalar field becomes

Here we examine the limit in which the station distance *ϕ* in baroclinic streamfunction space is small enough that signal difference (*t _{i}* −

*t*) is negligible. It proffers an estimate of the a priori error,

_{j}The calculation chooses all station pairs with difference of 300–850-mb thickness around value *ϕ*. This is different from the approach in homogeneous turbulence where data are screened in physical space. In a strong baroclinic zone, two samplings taken at the same location may yield very different profiles due to varying fronts.

Figures 4 shows the square root of the structure function at 500 mb as a function of *ϕ*. Because the baroclinic zone has large horizontal temperature gradients, the structure function value increases with *ϕ*, resulting from inclusion of more baroclinic variance. For small *ϕ* values, the influence of the background baroclinicity becomes relatively weak and there is a plateau in the structure function beneath *ϕ* = 10 m, which is used as the a priori error estimate.

### c. Sectional GEM

We now determine GEM function Γ(*ϕ, p*) from the SH January data at 0°. Thickness between 300 and 850 mb, which best represents the tropospheric dynamical regime, is chosen as the baroclinic parameter. In the idealized solution the GEM fields produced with different baroclinic parameters are all equivalent (Sun 2001). To capture nonlinear features, cubic smoothing spline rather than linear fit is applied to the temperature data on each isobaric surface (Fig. 5). The smoothing parameter can vary between 1 (spline interpolation) and 0 (least squares linear fit), producing smaller or larger rms residuals respectively. An appropriate parameter value should yield rms residuals close to the a priori error.

The two-dimensional modal field is plotted in Fig. 5. There is no structural feature corresponding to tropospheric fronts. This differs from the ACC GEM field where the Polar Front is identified by a temperature inversion layer (Sun and Watts 2002a). The oceanic GEM displays richer geometry partly because seawater density is determined by both temperature and salinity. The thermohaline fields can have compensating variations without altering the density field. The GEM solution is ultimately determined by density advection (Sun 2001).

The rms error associated with this GEM field is less than 1 K between 400 and 700 mb (matching the a priori error in Fig. 4), and increases toward the stratosphere and the surface (Fig. 6). It is one order of magnitude larger than the oceanic GEM error. To examine how well the meridional baroclinicity is represented, we calculate a percent variance ratio as described in SW01,

where *σ*^{2} is the total variance of the reanalysis temperature data and *σ _{r}* is the residuals of these data relative to the GEM field. Between 300 and 850 mb this particular GEM field explains more than 95% of the local thermal variance (Fig. 6). The worst

*γ*ratio is found at the tropopause where meridional temperature gradient reverses sign (signal

*σ*

^{2}is small). For comparison, we also calculate a traditional zonal-mean field based on 1991–2000 SH January data, and plot its

*γ*ratio in Fig. 6. The GEM field consistently represents more synoptic temperature variance in the SH with

*γ*ratios higher by 10-point percentage.

Despite the fact that *ϕ*_{300–850} is a tropospheric parameter, the GEM field captures a significant portion of stratospheric variability during summer, a sign that the tropospheric EB structure extends into the lower stratosphere. The local *γ* maximum at 100 mb is contributed by the large stratospheric baroclinicity in terms of meridional temperatures range.

Another view of the GEM performance is via a reconstruction of temperature section from the GEM field using *ϕ*_{300–850} values from the observed field. In Fig. 2 such a GEM-derived section replicates the tropospheric frontal structure on 15 January 2003 and agrees quite well with the original daily map. The GEM field itself is based on 1991–2000 data.

GEM function Γ(*ϕ, p*) represents invariant temperatures profile on each baroclinic streamfunction contour. The columnar structure is illustrated in Fig. 7 where *θ* profiles are grouped by *ϕ*_{300–850} values. The reanalysis data between 400 and 850 mb exhibit small variance around the correspondent GEM profile, and the GEM profile itself changes gradually across midlatitudes.

## 4. Circumglobal GEM analysis

### a. Spatiotemporal characteristics

Similar to the oceanic zonal flow, the atmospheric westerlies exhibit zonally asymmetric components due to orography and land–ocean heating contrasts. We expect tropospheric GEM to vary zonally, and proceed to construct sectional GEM every 15° of longitude around the globe. The composite of sectional GEM is EB on the synoptic scale but allows departure from EB on the planetary scale. This enables us to diagnose longitudinally dependent phenomena.

Figure 8 gives a plan view of the circumglobal GEM fields. The asymmetric features near the surface reflect lower boundary conditions, with a wavenumber-2 pattern in the Northern Hemisphere (NH) and a weakwavenumber-1 pattern in the SH. The global meander of the ACC brings cold polar waters northward at the east of Drake Passage and has visible influence on the PBL temperature. In the NH winter, significant asymmetry not only appears in the lower troposphere, but also near the tropopause.

The degree of zonal asymmetry can be measured by the standard deviation of GEM fields along streamfunction contour. As Fig. 9 shows, large zonal asymmetry appears at the poleward side during winter and at the equatorward side during summer. In both seasons, the SH GEM fields appear to be more zonally symmetric.

The zonal-mean GEM fields calculated in streamfunction space (Fig. 10) gives a better view of the seasonal contrast of baroclinic mean field: the stratospheric temperature at the poleward side falls in winter; the midtroposphere has the smallest seasonal variation; in the SH GEM the lower troposphere is warmer in winter because of the equatorward shift of mean streamfunction over the subtropical oceans (the mean latitude for 7.7 km contour shifts from 50°S in summer to 40°S in winter); the NH is influenced by vast landmass and the lower troposphere becomes slightly colder during winter, despite the equatorward shift of *ϕ*_{300–850} contours.

The summer GEM fields contain less baroclinicity than the winter GEM, as the *ϕ*_{300–850} range decreases by about 10% in the SH and 30% in the NH. The buildup of baroclinicity in winter is accompanied by strengthened jet streams, stronger eddy activity and larger GEM rms errors. The summer GEM rms errors in the two hemispheres are of comparable magnitude (Fig. 11). In winter, the NH GEM performs worse because of the presence of NH stationary waves.

Such seasonal and hemispheric differences of GEM performance are also reflected in the percent ratio of variance captured by GEM fields (Fig. 12); the zonally averaged *γ* ratios are lower in the NH than in the SH, and the SH summer has the highest *γ* ratio in the troposphere. The NH summer GEM has the worst performance partly because its baroclinicity signal, as expressed by the thickness range, is small.

The January 500-mb map in Fig. 13 shows the SH GEM rms errors are smaller over the ocean, and the increase of rms error because of perturbation from continents, such as Australia and eastern Antarctic, is apparent. Similar error pattern is found in SH July. While the SH exhibits certain degree of zonal symmetry in GEM rms distribution, the NH does not. In the NH winter, a dipole pattern of GEM rms error appears over the North Atlantic.

So far only January and July data are used. The GEM analysis can be applied to other months and forms a seasonally varying tropospheric mean field. The SH example given in Fig. 14 corroborates the above results on GEM rms error and *γ* ratio. The seasonal variation is well explained if we establish a connection between GEM rms error and westerly strength, that is, the instability process in strong winter westerlies tends to produce larger synoptic departure from EB.

Similar to the reconstruction of longitudinal section in Fig. 2, we can reconstruct zonal temperature section from circumglobal GEM fields. Figure 15 shows such an exercise at 50°S. Except for synoptic features near the PBL that has phase lag and resemble unstable boundary waves, the two fields agree quite well through the troposphere. The fact that daily three-dimensional fields can be retrieved from a two-dimensional baroclinic proxy (*ϕ*_{300–850} as plotted in Fig. 1) demonstrates the equivalent-barotropic approximation of the tropospheric thermal field across a wide range of spatiotemporal scales. The baroclinic mean field defined by GEM thus persists through synoptic variability and has more dynamic relevance than the traditional Eulerian mean description.

To study GEM geometry we look at inverse thickness

which unlike potential vorticity is entirely determined by GEM *θ* field. Figure 16 shows the isobaric and isentropic thickness distributions in the SH GEM fields. The isentropic Π gradients are large near the tropopause and considerably smaller in the troposphere between 400 and 850 mb. Uniform thickness distribution characterizes the summer isentropes. In winter the Π gradients become negative (decrease poleward) above the PBL. The seasonal progress of thickness distribution is qualitatively similar in the NH GEM.

### b. Planetary deviation from EB

While two-dimensional GEM represents synoptic thermal structure that is strictly EB, a circumglobal composite of sectional GEM may represent planetary thermal structures that are zonally asymmetric and deviate from EB.

In NH winter, the wavelike perturbation of GEM isotherms penetrates through the entire troposphere (Fig. 17). We can derive meridional thermal winds from the circumglobal GEM fields using a mean hemispheric distribution of *ϕ*_{300–850}. Since thermal winds at 300 mb relative to 850 mb follow the *ϕ*_{300–850} contour, the appearance of meridional winds on a zonal *ϕ*_{300–850} section represents vertical turning of thermal winds relative to 300 mb (Fig. 17). The influence from the NH land–ocean heating contrast is apparent: tropospheric thermal winds veer with height over the North Pacific and the North Atlantic (turns in a clockwise direction), and back with height over the continents. Figure 18 shows the planetary deviation from EB is much less in SH summer, consistent with the previous analysis of GEM rms errors.

The most common turning of geostrophic wind is at synoptic scales and is due to crossing between baroclinic–barotropic components. Little attention has been paid to the vertical turning of geostrophic wind shear (thermal wind), which involves at least three mechanisms: baroclinic instability (e.g., isotherm phase lag in the Eady instability solution) contributes to the most of the GEM rms errors; planetary *β*-effect (so-called beta spiral in oceanic meridional currents); along-stream thermal evolution in westerly associated with its global stationary wave, as revealed here.

Deviation from EB in the circumglobal GEM fields is associated with planetary stationary eddies. Meanwhile the rms errors in sectional GEM represent synoptic deviation from EB and are associated with transient eddies. The magnitude of planetary thermal-wind turning is about 1–3 degrees, much smaller than the synoptic turning induced by temperature advection.

### c. Global correlation analysis

The vertical coherence of tropospheric thermal variations can be examined by correlation analysis. A mean annual cycle, calculated by averaging the ten-year data for each calendar day, is removed from the 1991–2000 time series. The temperature correlation between 400 and 850 mb is quite high in extratropical regions where westerlies are located (>0.7), and minimal in tropical regions (figure not shown). Calculation of lag correlation by shifting one time series forward or backward gives lower correlation values everywhere on the globe, suggesting no prominent vertical phase difference (on a time scale of days) in tropospheric temperature.

We then look at the phase relation between baroclinic and barotropic fields, choosing 300–850-mb thickness as the baroclinic variable and 850-mb geopotential as the barotropic variable. Figure 19 shows deseasonalized *ϕ*_{300–850} and *Z*_{850} have relatively high correlation over the Southern Ocean and the eastern parts of the North Pacific and North Atlantic. The two variables are weakly anticorrelated in tropical regions. Calculation of lag correlation shows that *Z*_{850} leads *ϕ*_{300–850} by 1 day in extratropical regions. The phase lag is most robust at places with large increase of correlation value, such as the two NH storm tracks (Fig. 20).

Similar results are obtained if warm season and cold season data are treated separately. However, correlation analysis of geopotential field, such as geopotential heights at 300 and 850 mb, does not reveal globally systematic phase shift. This is because the upper tropospheric geopotential, following our definition in section 2, is a mixture of barotropic and baroclinic components.

In the eastern parts of the NH oceans where baroclinic and barotropic fields are more correlated, their phase differences are minimal since Fig. 20 shows little increase of correlation value. In fact, the traditional EB concept is related to the state of baroclinic–barotropic fields being in phase. An EB temperature field, when combined with a barotropic field leading in phase, would give a westward tilt with height in geopotential fluctuations.

The cross-spectrum analysis of NH geopotential height fields by Lau (1979) has shown that the vertical phase shift decreases as transient fluctuations move eastward across the oceans, consistent with our finding. Hess and Wagner (1948) documented the land–sea contrast in the vertical structure of wintertime disturbances passing the northwestern United States: fluctuations in SLP and tropospheric temperature are in phase at the Pacific coast, but a westward tilt with height develops inland and results in low pressure-temperature correlation. This is reflected in Figs. 19 and 20.

## 5. Summary

The study shows that parallel isotherms, unidirectional thermal winds, and GEM temperature field are kinematically equivalent (see the appendices). In contrast with traditional equivalent-barotropic concept and quasigeostrophic models, GEM provides a streamfunction definition of mean field for baroclinic fluids that persists through synoptic processes.

The tropospheric mean field is diagnosed from the NCEP–NCAR reanalysis data with an emphasis on the seasonal variation. Monthly GEM fields account for more than 90% tropospheric thermal variance. Reconstruction of daily section from the GEM field further demonstrates that the tropospheric temperature field at leading order is EB. There exist significant seasonal, zonal and hemispheric asymmetries: the circumglobal GEM performance, in terms of rms error and percent variance ratio, is better in summer than in winter, and also better in SH than in NH. GEM isentropes in summer are characterized by uniform thickness distribution.

In association with the NH stationary waves, the NH winter GEM displays planetary deviation from EB with thermal winds veering over oceans and backing over continents. The zonal asymmetry is seen as a response to the NH land–ocean heating contrast. Meanwhile the synoptic deviation from EB as represented by GEM rms errors is associated with transient eddies. Other mechanisms producing vertical turning of geostrophic wind and thermal wind are also identified.

Correlation analysis shows that tropospheric thermal variations at different heights are generally in phase, and barotropic variations tends to lead the EB baroclinic field by one day in westerly regions. In the NH, this phase difference between barotropic–baroclinic fields are robust at the two storm tracks as well as the northwestern United States, but minimal in the eastern part of the NH oceans. It verifies that temperature and thermal wind, rather than geostrophic winds, hold approximate EB structure in baroclinic fluids.

## Acknowledgments

The NCEP–NCAR reanalysis data were provided by the NOAA–CIRES Climate Diagnostics Center. The work was initially completed during my postdoctoral stay at Princeton University. I would like to thank two reviewers, as well as Bob Hallberg and Gabriel Lau who spent time discussing climate diagnosis with me. Support from NSFC Grant 40476002 is appreciated.

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### APPENDIX A

#### Sanders’ Analytical Model

Sanders (1971) constructed an idealized temperature field

where 1000-mb geopotential is given by

which follows a checkerboard distribution and has no zonal-mean component.

Geopotential above the 1000-mb level is then obtained by hydrostatic integration,

We choose thickness between (*p*_{1}, *p*_{2}) as the baroclinic parameter. It can be derived from the hypsometric equation as

where the two constants are

It shows that the temperature field in the Sanders’ model follows a linear GEM distribution.

### APPENDIX B

#### Parallel Isotherms

We now prove that if isotherms are parallel at all heights, as assumed in Charney et al. (1950), the temperature field can be described by a GEM function. The kinematic analysis is performed at a given time *t* = *t*_{1}.

At each point of the reference isobaric surface *p* = *p*_{0}, a unique isotherm *T*_{0} passes through and is represented by *C*(*x*, *y*, *p*_{0}). There is a parallel isotherm curve *C*(*x*, *y*, *p*) on every other isobaric surface, probably with different temperature value. The map between parallel isotherms signifies a dimensional reduction, establishing a function that relates the three-dimensional thermal field to the temperature on the reference isobaric surface as

we find the two parameters *T*_{0} and *ϕ* are functionally related. If *F* is also bijective, it admits an inverse function

### APPENDIX C

#### Unidirectional Thermal Wind

A unidirectional thermal wind field argued by Sutcliffe and Forsdyke (1950) also leads to a GEM temperature distribution.

The thermal wind field in polar coordinates is

If there is no vertical change of velocity direction, we have

and

Substituting the thermal wind relation into the equality, we obtain

Like in (A2), we choose thickness between (*p*_{1}, *p*_{2}) as the baroclinic parameter. The thermal wind equation is

A relation can be derived from (C3) as

Using (C5) we obtain

Its general solution is in GEM form

## Footnotes

*Corresponding author address:* Che Sun, Institute of Oceanology, 7 Nanhai Road, Qingdao, China. Email: csun@ms.qdio.ac.cn