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Revisit of the Thermal Wind Equation: Application to Planetary Atmospheres at Low Latitudes

Liming Li, F. Michael Flasar, Barney J. Conrath, and Peter J. Gierasch

(August 30, 2008) Abstract. The standard thermal wind equation, which is based on the hydrostatic and geostrophic balances, has been widely used in the explorations of planetary atmospheres. The two balances work well for Earth's atmosphere even when approaching the equator. However, the two balances behind the standard thermal wind equation should be used with caution for the equatorial regions of other planets given the large variations of jet velocities, radii, and rotation periods. Here, we examine a general relationship between the wind field and the temperature field without the hydrostatic and geostrophic balances. The standard and general thermal wind equations are both tested by a meteorological dataset of Earth's atmosphere. The comparisons of results between the two equations suggest that the general thermal wind equation is a better relationship between the wind field and the temperature field within the equatorial regions. Finally, we apply the general thermal wind equation to Jupiter and Saturn.

1. Introduction Hydrostatic balance describes a static equilibrium of fluid between the gravitational force and the vertical component of the pressure gradient force (i.e. buoyancy force). Even though the fluid is in motion, the so-called Archimedian principle is also applicable when the vertical accelerations due to the other forces besides the gravity force and the pressure gradient force are negligible. In general, the hydrostatic balance provides a good approximation for the large-scale motion with the nearly horizontal character. Geostrophic balance describes an equilibrium between the Coriolis force and the pressure-gradient force for the large-scale motions of fluid. The reference frame for the geostrophic balance is often set to the horizontal plane, in which the horizontal component of the Coriolis force is balanced by the pressure-gradient force. The two balances have become part of the foundations of present day meteorology, oceanography, and planetary sciences.

Combining the two balances, the standard thermal wind equation offers a relationship between the vertical shears of the geostrophic winds and the horizontal temperature gradients along isobaric surfaces. The physical sense behind the standard thermal wind equation is straightforward, as follows: The spatial distribution of temperature induces titling of constant-pressure surfaces. The magnitude of their tilting will vary in the vertical direction if the horizontal temperature gradients exist, resulting in a vertical shear of the geostrophic winds. The thermal wind equation is extraordinarily useful for the large-scale motions of atmospheres. Once the temperature field is known, the vertical structure of horizontal winds is immediately determined with suitable boundary conditions. Likewise, the temperature field can be derived from the structure of horizontal winds with suitable boundary conditions.

The hydrostatic balance works well for the large-scale motions in which the vertical accelerations can be neglected. In general, the geostrophic balance also works well for the large-scale motions outside of the equatorial regions. It is well known that the geostrophic balance ceases to be valid when approaching the equator. More importantly, the large variations of large-scale horizontal winds, rotation periods, and radii of different planets in our solar system makes

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some forces, which are discarded in the hydrostatic and geostrophic balances, are not negligible when approaching the equator. The concern that the standard thermal wind may not work near the equators of planets has already been noticed in some previous studies (Verdiere and Schopp 1994; Juarez, et al., 2002; Flasar et al., 2005). Beginning from the primitive equations, the first two studies both derive a more general thermal wind relationship from a vorticity viewpoint (Vierdiere and Schopp, 1994) or from geostophic wind balance with a full Coriolis force (Juarez et al., 2002). Flasar et al. (2005) also get a similar relationship from a viewpoint of gradient wind balance when studying Titan’s thermal wind. Here, we revisit the thermal wind equation with a simplified momentum equation. We compare the standard thermal wind equation and the general thermal wind equation with emphasis on the physical sense behind the general thermal wind equation and the applications to other planets.

The following section (Section 2) is a revisit of the thermal wind equation from the viewpoints of mathematics and physics. We begin with simplified momentum equations in order to clarify physics behind the standard and the general thermal wind equation. The two thermal wind equations are both examined with a meteorological dataset of Earth, which is put in the Section 3. Applications to Jupiter and Saturn are put in the Section 4. In section 5, we summarize our discussions in the context of previous sections. For Earth, we use the general longitude and latitude. For other planets, we use the planetographic latitude and System III west longitude. In this study, we focus on the zonal component of horizontal winds in planetary atmospheres mainly because observations of meridional winds in other planets are limited. 2. Thermal Wind Equations 2.1 The Standard Thermal Wind Equation in Rotating Spherical Coordinates Following meteorological traditions, we use the spherical coordinates fixed in rotating planets. The coordinate axes are then ),,( rφλ , where λ is longitude, φ is latitude, and r is radial distance from the center of the planet. The standard thermal wind equation is based on the geostrophic wind, which can be written as the following for the zonal wind u (Batchelor, 1967; Pedlosky, 1987; Holton, 2004):

φρ ∂∂−

=p

rfu 1 (1)

where f is Coriolis parameter ( φsin2Ω=f , and Ω is the angular speed of rotation of the planet), ρ is density of the atmosphere, and p is the pressure. Differentiating eq. (1) with respect to r , we obtain

r

prrr

pr

prrr

uf∂∂

∂−⎟⎟

⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂−

∂∂

=∂∂

φρρ

ρφρφρ

211111 (2)

The range of radial distance r is from the bottom to the top of atmospheres. In general, r1

is much smaller than ( )( )r∂∂ρρ1 in the first term of the right hand side of eq.(2) because r is much larger than the scale height of density ( ( )r∂∂− ρρ ), which is on the order of magnitude

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of 110 km. Therefore, combining the hydrostatic balance in radial direction ( gdrdp ρ−= , where g is the gravity of planet), we can simplify eq.(2) as

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−∂∂

∂∂

=∂∂

+∂∂

∂∂

=∂∂

φρρ

φρφρ

ρρ

φρ rp

rp

rrg

rp

rruf 22

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⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂∂∂

−∂∂

∂∂−

=r

rr

grrp

prp

r p

ρφφ

ρρ

ρφφρ

ρ 2

1 (3)

Utilizing the chain rule of partial derivatives and the ideal gas law ( RTp ρ= , where T is temperature and the specific gas constant R is the ratio between the universal gas constant *R and the mean molecule weight of atmosphere), we can re-write eq.(3) as

ppp

TrTf

grf

gr

rrf

gru

φφρ

ρρ

φφρ

ρ ∂∂−

=∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+∂∂

=∂∂ (4)

Likewise, we have the thermal wind equation of meridional wind v as p

TrTfg

rv

λφ ∂∂

=∂∂

cos.

2.2 The Thermal Wind Equation Without Geostrophic and Hydrostatic Balances The whole momentum equations in the meridional (φ ) and radial ( r ) directions can be written as follows (Batchelor, 1967; Pedlosky, 1987; Holton, 2004):

φφφ

φρφ F

rup

rrvw

ru

DtDv

+∂Φ∂

−Ω−∂∂−

=++1sin21tan2

(5)

rFr

urp

rvu

DtDw

+∂Φ∂

−Ω−∂∂−

=+

− φρ

cos2122

(6)

where w is the velocity in the radial direction, Φ is the effective geopotential, defined as the sum of the geopotential due to gravity ( ∫=Φ drrgr ),(),(1 φφ ) and the geopotential due to

rotation of planets ( rr 22 ),( Ω−=Φ φ ). The term

φ∂Φ∂

−r1 of eq. (5) comes from the non-spherical

(oblate) shape of the planet. φF and rF are frictions in the meridional and radial directions,

respectively. The total derivative DtD can be expanded as

rw

rv

ru

tDtD

∂∂

+∂∂

+∂

∂+

∂∂

=φλφcos

in the spherical coordinates. For the large-scale motions, we have wind relations as follows: WVU >>>> , where U , V , and W are the scales of zonal wind u , meridional wind v , and

vertical wind w , respectively. We set that the time-scale of large-scale processes as UL , where L is the length-scale of large-scale motions. Using the scalar analysis for the large-scale steady axisymmetric flow and neglecting friction, we can simplify eqs. (5) and (6) as

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φ

φφρ

φ∂Φ∂−

=Ω+∂∂

+r

uprr

u 1sin21tan2

(7)

r

urp

ru

∂Φ∂

−=Ω−∂∂

+− φ

ρcos212

(8)

Taking rr)( ∂∂ on (7) and )( φ∂∂ on (8), and then subtracting the above two equations to cancel the geopotential Φ , we have

( ) 011cos21tan 2 =∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

−∂∂

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+Ω+⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

+∂∂

rpp

ruru

rr ρφφρφ

φφ (9)

Let unit vector z be along the rotating axis of the planet and λ be the unit vector along the longitudinal direction. Multiplying eq. (9) with r1 and projecting the first term to the direction of z , and the second and third terms to the direction of λ , we have

01ˆ2ˆ2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∇×∇⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛ Ω+∇⋅ p

rruu

zc

c

ρλ (10)

where φcosrrc = , which is the cylindrical radius at the latitude φ . Symbols ‘∇ ’and ‘× ’ are the gradient and cross product operators in the spherical coordinates. With the ideal gas law, we can rewrite the second term of eq. (10) as

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂∂∂

−∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∇×∇⋅

rT

rppT

rp

rpRp

rT

rpT

rpRp φ

φφφρλ 1ˆ

⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂∂∂

−∂∂

∂∂

=rTrT

rp

rpR

rT

rppT

rp

rpR

pφφφ

φ (11)

Utilizing the chain rule of partial derivative, we have

=⎟⎟⎠

⎞⎜⎜⎝

⎛∇×∇⋅ p

ρλ 1ˆ

pp

Trp

rpRprT

rp

rpR

φφφφ ∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛

∂∂

∂∂

+∂∂

∂∂ (12)

Combining eqs. (10) and (12), we have

pc

c

r

Trp

rpR

rruu

zc

φ∂∂

∂∂

=⎟⎟⎠

⎞⎜⎜⎝

⎛ Ω+∂∂ 22

(13)

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An alternative derivation of eq. (13) is to write the momentum equations in the cylindrical coordinates (please see Appendix A). Equation (13) offers a general description between the large-scale wind and temperature fields, which also works for the equatorial regions. The general relationship shown in eq. (13) is convenient for numerical evaluation.

With the two assumptions used in the standard thermal wind equation (4) (i.e. hydrostatic balance and geostrophic balance, meaning cru Ω<< except for regions close to the poles) we can simplify eq. (13) into

ppr

TrTgT

rpgR

zu

cφφ

ρ∂∂−

=∂∂−

=∂∂

Ω2 (14)

Using the approximation of φsinrz ∂≈∂ and the definition φsin2Ω=f , equation (14) can

be changed back to eq. (4) as

ppr

TrTf

gruT

rTg

zu

cφφ ∂∂−

=∂∂

⇒∂∂−

=∂∂

Ω2 (15)

In general, the geostrophic balance and hydrostatic balance both work well in the middle and

high latitudes for the large-scale motions. Therefore, the general thermal wind equation (eq. (13)) is equivalent to the standard thermal wind equation in the middle and high latitudes for a thin atmosphere. It should be emphasized that the approximation of φsinrz ∂≈∂ only works for the middle and high latitudes of a thin atmosphere. Therefore, the general thermal wind equation cannot be changed back to the standard thermal wind equation if the approximation does not hold (i.e. a thick atmosphere or the low latitudes). 2.3 Physical Interpretation of the General Thermal Wind Equation. The right hand side of eq. (13) represents that the change of centrifugal force and Coriolis force in the direction of the rotating axis, which comes from the derivatives of centrifugal and Coriolis forces in the meridional (φ ) and radial ( r ) directions. The left hand side of eq. (13) describes the temperature changes in the constant pressure surface, which can be converted back to the change of pressure gradient force in the meridional and radial directions. The process of derivation from equation (7) and (8) to eq. (13) better illuminates the above points. Therefore, equation (13) represents the balances between variations of forces in different directions. From this perspective, equation (13) is similar to the standard thermal wind equation (eq.(4)), which also represents a balance of vertical variations between the Coriolis force and pressure gradient force. Equation (10) further suggests a balance between the twist forces that represent drives for the zonal component of vorticity. One twist is due to the gradients of the centrifugal and Coriolis forces, and the other is the solenoidal twist force arising from the baroclinicity of atmosphere (Holton, 2004).

At the equator, the situation is qualitatively different because there is nothing to balance the pressure gradient in the meridional direction in eq. (7). Rather than balance the pressure gradient force, the roles of centrifugal acceleration and Coriolis force are to modify the pressure field in such a way as to eliminate the meridional pressure gradient. It accomplishes this by modifying the effective gravity to offset the expansion or contraction of atmosphere that temperature

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variations would otherwise cause. In general, the expansion or contraction of atmosphere that temperature variations cause will produce the pressure-gradient in horizontal direction. To show this, consider a hypothetical cylindrical case around equator and let rΔ is the distance between two cylindrical surfaces with constant pressure p and pp δ+ at the equator. Utilizing

gr ~∂Φ∂ , we can rewrite eq. (8) as

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2−

⎟⎟⎠

⎞⎜⎜⎝

⎛Ω−−−=Δ u

rugpr

ρδ (16)

The distance depends on temperature, through density ρ , as well as on the zonal wind u ,

through its deduction of effective gravity by centrifugal force and Coriolis force. In order to eliminate the pressure gradient along the rotating axis z (the same as the meridional direction φ at the equator) it requires that the distance rΔ between two constant pressure surfaces does not change along the direction z . Therefore, we have

020)(12

=⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛Ω−−

∂∂

⇒=∂Δ∂

−

ur

ugpzz

rρδ (17)

Using the ideal gas law to eliminate ρ and assuming that centrifugal force and Coriolis force are much smaller than gravity at the equator, we can rewrite eq. (17) as

zT

rp

pR

zT

Tgu

ru

z c ∂∂

∂∂

=∂∂

−=⎟⎟⎠

⎞⎜⎜⎝

⎛Ω+

∂∂ 2

2

(18)

Equation (18) is actually the thermal wind equation in the cylindrical coordinate (please see

Appendix A), which is same as eq. (13) at the equator when 1cos =φ and φ∂=∂ rz . In summary, the general thermal wind equations (eqs. (13) and (18)) suggest a unique balance at the equator in which the centrifugal and Coriolis forces modify the gravity field so that the thickness of atmosphere does not change between the two constant pressure surfaces. The constant thickness of atmosphere helps to eliminate the pressure gradient force in the meridional direction.

The unique physical balance at the equator leads to a different scaling for the relationship between temperature contrast and thermal wind between the middle latitudes and the equatorial regions. At the middle latitudes, we have the approximations of ( ) Hprp 1−=∂∂ and

φsinHz ≈∂ (where H is the scale height). Assuming Lr ≈∂φ in the meridional direction, we have the scaling relationship in eq. (13) as

H

TRLHU

rU Δ

⎟⎠⎞

⎜⎝⎛=ΔΩ−

Δ 22

(19)

At the equatorial regions, we use eq.(18) to do the scale analysis. Therefore, we have

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H

TRUrU Δ

=ΔΩ−Δ 2

2

(20)

A factor LH exists between the middle latitudes and the equatorial regions. In general, the

horizontal length scale L is much larger than the scale height H . Therefore, winds are much more sensitive to temperature contrasts in the equatorial regions than in the middle latitudes. If there is a dynamical constraint on the magnitude of the wind, this will imply that only relatively small temperature contrasts can exist near the equator. The implication helps to explain why the temperature gradient is relatively small around the equator in some planets of our solar system. 2.4 Differences between the Standard and General Thermal Wind Equations An obvious difference between the standard thermal wind equation and the general thermal wind equation is that the latter does not need the two assumptions (geostrophic balance and hydrostatic balance), assumed by the standard thermal wind equation. In principle, the general thermal wind equation is a more precise description of the relationship between the large-scale dynamic and thermal fields. The assumption of geostrophic balance breaks down around the equator, where the Coriolis parameter f approaches zero. The assumption of hydrostatic balance should also be used with caution because the centrifugal force due to the relative motion ( ru 2 ) can be large with strong equatorial jets in giant planets. We will discuss more about the two assumptions in the section of applications.

Figure 1. Sketch of the cylindrical routine for the general thermal wind equation. Latitude excursion δφ between the radial integration routine of standard thermal wind equation (AO) and the cylindrical integration routine of the general thermal wind equation (AB) is illustrated in the sketch. The critical latitude 1Cφ , in which the cylindrical routine will be tangential with the reference bottom pressure level, is also shown.

The other important difference between the two thermal wind equations is the different integration routines used when applying the thermal wind equation to estimate zonal winds at different pressure levels. To evaluate the zonal winds at different pressure levels, the standard

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equation is integrated in the radial direction ( r∂ ) and the general equation is integrated in the cylindrical path along the rotation axis ( z∂ ). An approximation ( φsinrz ∂≈∂ ) exists between the two integration paths in the middle and high latitudes. But the approximation breaks down when approaching a specific latitude where the cylindrical path originating at the top of the atmosphere becomes tangent to the equatorial surface (Fig. 1). For a spherical body, geometric construction shows that the critical latitude 1Cφ is given by

⎟⎟⎠

⎞⎜⎜⎝

⎛+

=⇒+

=ra

aara

aCC δ

φδ

φ coscos 11 (21)

where a is the reference radius of a planet and rδ is the total thickness of the atmosphere.

More importantly, because the standard thermal wind equation and the general thermal wind equation integrate along different paths (radial path vs. cylindrical path), the meridional temperature gradient φ∂∂T is different and will lead to error in the standard approach. Figure 1 shows that the angle δφ between the radial integration path and the cylindrical integration path. The error in the standard thermal wind equation due to the different integration path is largest at the equatorial regions because the latitude excursion δφ increases when approaching the equator. To estimate the critical latitude 2Cφ at which the standard approach becomes unacceptable, something must be assumed about the nature of the horizontal temperature gradient. Let us assume that the standard thermal wind equation is sufficiently accurate as long as the horizontal excursion of the radial integration path from the correct cylindrical integration path is less than a specified amount ( )Cφδ . The simple geometry (Fig.1) gives the critical latitude of acceptability 2Cφ , corresponding to the specified small ( )Cφδ , as

( ) ( ) ⎥⎦

⎤⎢⎣

⎡>⇒<≈⇒≈ −

CCC a

ra

rraφδ

δφφδφ

δφδφ

δφδ 12 tan

tantan (22)

Equation (22) suggests that the critical latitude of acceptability 2Cφ is determined by the ratio

between the thickness of atmosphere and the reference radius of planet arδ and the specified ( )Cφδ . The error in the standard thermal equation will become unacceptable when latitudes are smaller than the critical latitude 2Cφ . It should be mentioned that eq. (22) only works for small φδ so that the approximation used in the equation holds. The latitude excursion φδ is not very small when approaching the equator even for a thin atmosphere. Therefore, a

more precise expression should be used to estimate the latitude excursion in the equatorial regions.

( )ra

ara

aaraδ

φδφδφδφ

φδφφφδφδ+

=−⇒+

=+

⇒=++ )sin(tan)cos(cos

)cos(cos)cos( (23)

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It is not easy to get the analytical solution of the excursion φδ from equation (23), but equation (23) is suitable for the numerical evaluation of φδ for a known rδ . 3. Validation of the Two Thermal Wind Equations The atmosphere of Earth has been observed and studied extremely extensively by comparison with other planets in our solar system. The plentiful observations offer a unique opportunity to verify the thermal wind equations. Here, we pick up daily-mean variables from a random day (January 1 of 2006) with spatial resolution 2.5º in the latitudinal and longitudinal directions from the Reanalysis 2 meteorological dataset produced by the National Centers for Environmental Prediction ― National Center for Atmospheric Research (NCEP-NCAR) (Kalnay et al., 1996; Kanamitsu et al. 2002). The above dataset is generally referred as the NCEP2. The NCEP2 is a modern meteorological dataset, which is mainly constructed of satellite-based observations. We use the zonal wind u , temperature T , and geopotential height Z from the NCEP2. These variables are “type A variables”, which are of the highest data quality in the NCEP2 reanalysis dataset.

Firstly, we examine the two assumptions behind the standard thermal wind equation. The hydrostatic balance is an easy one to test. For Earth, the centrifugal force due to the relative motion ru 2 is much smaller than the effective gravity ( rgge

2Ω+= ). In addition, the Coriolis force ( V×Ω ), whose radial component is in the same direction as gravity, is also much smaller than the effective gravity by simple scalar analysis. Therefore, the hydrostatic balance works well for the large-scale motions on Earth.

Figure 2. Comparison between the observed zonal winds and the geostrophic zonal winds calculated from the geospotential height field. Panel (A) shows the geostrophic zonal winds constructed from observational geopotential heights in the NCEP2. The gap at the equator in panel (A) is due to the breaking of geostrophic balance at the equator. Panel (B) is the observed zonal winds from the NCEP2 at the same time of panel (A).

The zonal geostrophic wind (eq.(1)), which is a balance between pressure gradient force and Coriolis force, can be rewritten in the p coordinate as ( ) φ∂∂−= Zrgfu . Therefore, we can use the geopotential height Z to derive the geostrophic winds, and compare with the observed

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winds. Figure 2 is the comparison, which shows a good agreement between the geostrophic winds (upper panel) and observed winds (bottom panel) even though for the equatorial regions. The scalar analysis of large-scale motion suggests that geostrophic balance holds only when the Rossby number ( LfURo = ) is much smaller than 1. We have 3.0≈oR when latitude

o15=φ with assumption smU /10≈ and kmL 1000≈ . The Rossby number is not much less than one around latitude 15º, which suggests that the geostrophic balance does not work for the equatorial regions with latitudes less than 15º by the traditional perspective. However, Figure 2 shows that the geostrophic balance still holds eventhe Rossby number is not much less than one, which suggests that the Rossby number is not a perfect criterion for determining whether the geostrophic balance or the scales of variables in the Rossby number should be re-chosen.

Figure 3. Comparison between the observed zonal winds and zonal winds constructed from the standard thermal wind equation. Panel (B) is the observed zonal winds from the NCEP2. Panels (A) and (C) are derived zonal winds, which are integrated by the standard thermal wind equation from the top layer and the bottom layer, respectively.

The exploration of planetary atmospheres often requires us to derive information about the

deep atmosphere from above or derive information about the upper atmosphere from below. Therefore, we test the two thermal wind equations by integrating them from the top of atmosphere and from bottom of atmosphere. Figure 3 shows the comparison between the observed winds (panel B) and the winds derived from the standard thermal wind by assuming that zonal winds are known at the top layer (panel A) or bottom layer (panel C). The figure shows that the standard thermal wind equation works well not only for middle and high latitudes but also for most of the equatorial regions. The good agreement between the observed winds and the winds derived from the thermal wind equation suggests that the two hydrostatic and geostrophic approximations hold for the global atmosphere except for a small fraction near the equator. The small fraction near the equator with the biggest discrepancy is probably due to the accumulation of the error associated with the approximation of geostrophic balance along the integration routines.

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Figure 4. Same as Fig.3 except for zonal winds (A and C) constructed from the general thermal wind equation. The integration from bottom (panel (C)) has a latitude excursion ~ 5 degrees when the integration reaches to the top of atmosphere. The integration from top (panel (A)) almost covers all grid points in the equatorial regions. The few missing grid points in panel (A) are filled with a linear interpolation from the neighboring grid points.

Figure 4 is the same as Fig. 3 except for the general thermal wind integrations. Figure 4 shows that the cylindrical integrations based on the general thermal wind equation also do a good job for estimating of zonal winds. Figure 5 is a close comparison of winds derived from the general thermal wind equation and the standard thermal wind equation for the equatorial regions. The comparison shows that the general thermal wind equation (panels D and F) does a better job than the standard thermal wind equation (panels A and C) for the equatorial regions. The comparison between C and F shows that the latitude excursion of integration path is ~ 5º around the equator, which is the main reason why the relationship between the temperature field and the wind field is better described in the general thermal wind equation than in the standard thermal wind equation.

Figure 5. Comparison of winds between the standard thermal wind equation and the general thermal wind equation for the equatorial regions. The left column (panels (A), (B), and (C)) is same as figure 3 except for the equatorial regions. The right column (panel (D), (E), and (F)) is the same as figure 4 except for the equatorial regions.

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The superiority of the general thermal wind equation in the equatorial regions does not mean that the general thermal wind equation will replace the standard thermal wind equation. Figure 4 shows that there is no noticeable difference between the general thermal wind equation and the standard thermal wind equation for the middle and high latitudes, but the standard thermal wind equation has the following two advantages: 1) it is simpler than the general thermal wind equation; and 2) its integration path is consistent with the general observational grids in the radial direction. The above two advantages make the standard thermal wind equation a better choice for the exploration in the middle and high latitudes. In summary, we suggest that the standard thermal wind equation and the general thermal wind equation should be combined to explore the global atmospheres of planets. In the combination, the standard thermal wind equation is recommended for the high and middle latitudes and the general thermal wind equation is recommended for the equatorial regions.

Figure 6. Comparison between the observed temperatures and the derived temperatures. Panel (A) is the observed temperatures from the NCEP2. Panel (B) is the integration of the standard thermal wind equation in a reverse direction by utilizing the known zonal winds.

We also test the reverse direction of the relationship between the temperature field and zonal wind field. We derive the temperature field by integrating the zonal wind field based on the standard thermal wind equation (the general thermal wind equation presents almost same results). The results are shown the Fig.6. Figure 6 shows an agreement of temperature patterns at the global-scale between the observations (panel A) and the estimates from the thermal wind equation (panel B). The figure also shows some discrepancy between observations and derivations for the middle and high latitudes probably due to the accumulation of the error related to approximations along the integration routines.

4. Applications to outer giant planets 4.1 Jupiter Firstly, we discuss the hydrostatic and geostrophic balances in the equatorial regions of Jupiter. The equatorial jet is ~ 100 m/s on Jupiter, so that the centrifugal force due to the relative motion

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is 24722 104.1107100 −−×≈×≈ msaV , which is much smaller than the effective gravity of Jupiter (roughly the same as the gavity of Jupiter 224 −= msg J ). The Coriolis force ( 224 107.1100107.1 −−− ×=××≈×Ω msV ) is also much smaller than the gravity. The geostrophic balance cannot be tested directly because the geopotential height Z is difficult to define and measure for Jupiter. The banding phenomenon (Ingersoll, 1990; Gierasch, 1996), large-scale vortices including the Great Red Spot, and planetary waves (Allison, 1990; Li, et al., 2007) suggests that the length-scale of the large-scale motions on Jupiter are probably on the order of

km410 , so the scale analysis shows that Rossby number at latitude 10º is ~ 0.2 ( 2.0)1010sin2(100 7 ≈×Ω≈= oLfURo ), which is not much smaller than 1. Therefore, it seems that the geostrophic balance probably does not work for the equatorial regions of Jupiter.

Figure 7. Comparison of derived zonal winds of between the two thermal wind equations for Jupiter. Panel (A) shows the derived zonal winds by integrating the standard equation with the Cassini CIRS temperatures (spatial resolution 3º in the meridional diection). Panel (B) is the same as panel (A) except for integrating the general equation. The two equations are integrated from a known zonal wind at bottom layer ~ 500 mbar from Cassini measurements (Porco et al., 2003). The region between 3ºS and 3ºN in panel (A) is neglected because the standard thermal wind equation does not work when approaching the equator. The latitude excursion between the two different integration routines is ~ 6º when reaching to top of atmosphere around the equator.

Figure 7 shows the comparison of winds derived from the two thermal wind equations, which are based on the temperatures from the Cassini composite infrared spectrometer (CIRS). In the middle and high latitudes, the two thermal wind equations present similar results, which are basically consistent with the previous results (Flasar, et al., 2004b; Simon-Miller, et al., 2006). However, there is some discrepancy between the standard thermal wind equation and our general thermal wind equation in the equatorial regions due to the different integration routines and the possible breaking of geostrophic balance. Firstly, the strong high-altitude (between 1-mbar and 10-mbar) equatorial jets have larger area in the altitude-altitude cross section in the winds derived from the standard thermal wind equation (panel A) than in the winds derived from the general thermal wind equation (panel B). Particularly, the strong high-altitude equatorial jets in the southern hemisphere predicted by the standard thermal wind equation (panel A) almost disappear in the winds derived from the general thermal wind equation (panel B). In addition, the

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strong negative jet (eastly jet) above the tropopause (between 10-mbar and 100-mbar) in the southern hemisphere is stronger in the winds derived from the general thermal wind equation than in the winds derived from the standard thermal wind equation. The strong eastly jet has roughly the same magnitude as the westly equatorial jet below it. The strong equatorial jet changes directions with altitude in Jupiter is interesting, and should be explored further. 4.2 Saturn Likewise, we discuss the two balances in the equatorial region first. The equatorial jet is ~ 400 m/s on Saturn. The corresponding centrifugal force due to the relative motion is

23722 107.2106400 −−×≈×≈ msaV , which is much smaller than the effective gravity of Saturn (the effective gravity is estimated as

27242 7.8)106()107.1(4.10 −− =×××−≈Ω−= msagge ). The Coriolis force ( 224 108.6400107.1 −−− ×=××≈×Ω msV ) is also much smaller than the effective gravity by simple scalar analysis. Likewise, the banding phenomenon (Ingersoll, 1990; Giegrash, 1996) and planetary waves (Achterberg and Flasar, 1996) suggests that the length-scale of large-scale motions on Saturn probably is in the order of km410 , so the simple scale analysis shows that Rossby number at latitude 30º is 0.2 ( 2.0)1030sin2(400 7 ≈×Ω≈= oLfURo ), which is not much less than one.

The CIRS high-spatial-resolution ( o1 in the meridional direction) temperatures are used to derive the zonal winds based on the thermal wind equations. One example of the zonal-mean CIRS temperature maps is shown in Figure 8. Flasar et al., (2005) apply the standard thermal wind equation for the low and middle latitudes (8ºS-30ºS) of the southern hemisphere of Saturn based on a relatively low-spatial-resolution (8º) CIRS temperatures. Fletcher et al. (2007) also use the standard thermal wind equation to estimate the zonal winds in the polar regions of Saturn.

Figure 8. Cassini CIRS high-spatial-resolution temperatures of Saturn. The spatial resolution is 1º in the meridional direction. The content information (kernel function) between 5-mbar and 50-mbar is relatively limited so that the temperature retrievals there are essentially interpolations, which suggests that the temperatures between 5-mbar and 50-mbar should be used with caution.

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Figure 9. Saturn’s zonal winds derived from the two thermal equations. The integrations of the two thermal equations are based on the high spatial resolution (1º) temperatures from the Cassini CIRS. Panel (A) is the derived zonal winds by the standard thermal wind equation. Panel (B) is the same as panel (A) except for the general thermal wind equation. The region between 4ºS and 4ºN in panel (A) is neglected because the standard thermal wind equation does not work when approaching the equator. The latitude excursion between the two different integration routines is ~ 9º when reaching the top of atmosphere around the equator.

Figure 9 displays the winds derived from the two thermal equations with the high-spatial-resolution CISR temperatures. The zonal winds at the bottom layer (~ 500-mbar) mainly come from Voyager observations (Gehrels and Matthews, 1984). We interpolate the relatively low-resolution winds (~ 2º) to the high-resolution zonal winds (1º) in order to use the high-spatial-resolution CIRS temperatures. The gap (between 1ºS to 27ºS) of Voyager measurements is filled by combining the Cassini measurement (Porco, et al, 2005). Figure 9 shows that the zonal jet at ~15ºS decays with latitude from the bottom layer to the level ~ 1mbar, which is consistent with the previous estimate (Flasar et al., 2005). Fig. 9 (panel B) further shows that the 15º jet decays with latitude from the bottom layer to the ~ 1mbar level in the northern hemisphere too. The strong equatorial jet (~ 600 m/s) in the stratosphere derived from the general equation (panel B) raises the possibility that intense jets are common in the equatorial stratospheres of giant planets.

Figure 10. Sketch of different integration routines used by the two thermal wind equations. The straight lines with arrows represent one integration path by the standard thermal wind equation, and the curve lines with arrows represent one integration path by the general standard thermal wind equation. The background color map is the temperature gradients based on the temperatures form the Cassini CIRS (Fig. 8).

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Figure 9 also displays obvious discrepancy in the equatorial regions between the winds derived from the standard equation and the winds derived from the general equation. The discrepancy between panel A and panel B is contributed to two possible reasons: 1) the geostrophic balance used by the standard thermal wind equation does not work in the equatorial regions of Saturn; and 2) the different integration routines in the two equations result in different winds. Figure 10 illustrates the two integration routines for the same location (10ºN) at the top layer of atmosphere (0.001mbar). The different integration routines used by the two equations pass different temperature-gradient regions so that different thermal winds are derived. The latitude excursion between the two different integration routines can reach ~ 9º in the equatorial region of Saturn, which suggests that the general thermal wind equation is the better choice for the thermal winds in the equatorial regions of the planets with thick atmospheres. 4. Conclusions. In this paper, we derive a general thermal wind equation from simplified momentum equations with emphasizing the physics and applications. Based on the comparison of winds derived from the standard equation and the general equation, we recommend a combination of the two thermal wind equations, in which the standard equation is used for the middle and high latitudes and the general equation is used for the equatorial regions. The applications of the general thermal wind equation to Jupiter and Saturn reveal some new features in the equatorial regions of the two giant planets.

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Appendix. Following the same process of scalar analysis discussed in the text, we get the simplified governing equation in the cylindrical coordinates fixed in rotating planets ),,( crzλ (λ is still east longitude, z is along the rotating axis of planets, and cr is cylindrical radius) as below

ccc r

urp

ru

∂Φ∂

−=Ω+∂∂

+ 212

ρ (A1)

zz

p∂Φ∂

−=∂∂

ρ1 (A2)

Likewise, taking )( z∂∂ on (A1) and )( cr∂∂ on (A2), then subtracting the above two equations to cancel the geopotential Φ and using ideal gas law, we have

⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

∂∂

−∂∂

∂∂

=∂∂

zp

rT

rp

zT

pR

ru

z ccc

2

(A3)

Using the same reasoning in eqs. (11) and (12), we can rewrite eq. (A3) as

pcc z

Trp

pR

ru

z ∂∂

∂∂

=∂∂ 2

(A4)

The equation (A4) is the general thermal wind equation in the cylindrical coordinates. When

approaching the equator, we have pp

Trz

Tφ∂

∂≈

∂∂ 1 and

rp

rp

c ∂∂

≈∂∂ . Therefore, eq. (A4) is same as

eq. (13) around the equator.

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