6.3 Zonally asymmetric circulations in the tropics: the

Gill model

[Gill, Quart. J. R. Meteor. Soc., 106, 447-462 (1980).]We consider the zonally varying circulation forced by steady thermal forc-

ing on the equatorial β-plane. We linearize about a stably stratified state ofno motion. The perturbation thermodynamic equation is

∂T ′

∂t+ w′S =

J

ρcp,

where S =(dT0dz+ g

cp

). To relate this to a shallow water model of depth D,

we replace

T ′ → −h′Sw′ → −D

(∂u′

∂x+∂v′

∂y

)

J → DρcpS Q

to give a set of perturbation shallow water equations

∂u′

∂t− βyv′ = −g∂h

′

∂x∂v′

∂t+ βyu′ = −g∂h

′

∂y

∂h′

∂t+D

(∂u′

∂x+∂v′

∂y

)= −DQ

which are the same as the unforced set but with the addition of the ”thermal”forcing. In order to incorporate dissipation (we’ll see that this is importantto the solutions) we include both linear drag and “Newtonian” cooling, eachwith the same rate ε̃, to get

∂u′

∂t− βyv′ = −g∂h

′

∂x− ε̃u′

∂v′

∂t+ βyu′ = −g∂h

′

∂y− ε̃v′

∂h′

∂t+D

(∂u′

∂x+∂v′

∂y

)= −DQ− ε̃h′

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Now we non-dimensionalize1, using the deformation radius LE = (c/β)1/2,

and the time scale τE = LE/c = (cβ)−1/2, where c = (gD)1/2 is the gravity

wave speed. Note that, with c = 55.7 ms−1, and β = 2.29 × 10−11m−1s−1,we have LE = 1560 km, and τE = 28000 s = 7.78 hr. Then, with u′ = cu,v′ = cv, y → LEy, x→ LEx, h

′ = Dh, we get

∂u

∂t− yv = −∂h

∂x− εu

∂v

∂t+ yu = −∂h

∂y− εv

∂h

∂t+∂u

∂x+∂v

∂y= −Q− εb

where ε = τE ε̃. Finally, we make the“long wave” approximation that thezonal length scale � LE; this implies that |u| � |v| and so, provided ε � 1,we can neglect εv in the second equation. Then, in steady state, we have

εu− yv = −∂h∂x

(29)

yu = −∂h∂y

εh+∂u

∂x+∂v

∂y= −Q .

Now, if we associate ε with −iω, we know that the homogeneous (un-forced) version of (29) have eigenfunctions that are Hermite functions, so weanticipate that expanding in these functions will be a good approach to theproblem. First, however, we follow Gill by writing

u =1

2(q − r)

h =1

2(q + r)

1Our defintion for LE follows a more conventional approach than that of Gill, whosedefinition differs from ours by a factor of

√2. Accordingly, differences of factors of

√2

appear between our equations and those of Gill.

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when (29) can be rewritten and reorganized to

εq +∂q

∂x+∂v

∂y− yv = −Q

εr − ∂r

∂x+∂v

∂y+ yv = −Q (30)

∂q

∂y+ yq +

∂r

∂y− yq = 0 .

Now we expand

qrvQ

=

∞∑

n=0

qn(x)rn(x)vn(x)Qn(x)

Vn(y) (31)

where Vn are the Hermite functions we encountered previously. They havethe recurrence relations

dVndy

+ yVn = 2nVn−1

dVndy

− yVn = −Vn+1 (32)

Vn+1 − 2yVn + 2nVn−1 = 0

Substituting into (30) and using (32), we get∞∑

n=0

[(εqn +

∂qn∂x

+Qn

)Vn − vnVn+1

]= 0

∞∑

n=0

[(εrn −

∂r

∂x+Qn

)Vn + 2nvnVn−1

]= 0 (33)

∞∑

n=0

[2nqnVn−1 − rnVn+1] = 0

Given the forcing Q(x, y) =∑

Qn(x)Vn(y), equations (33) can be usedto build the solution. Note that the Hermite functions are normalized suchthat ∫

∞

−∞

Vm(y) Vn(y) dy = n!2n√πδnm , (34)

so that

Qn(x) =1

n!2n√π

∫∞

−∞

Q(x, y) Vn(y) dy . (35)

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6.3.1 Localized forcing on the equator

Suppose that Q is localized in x, being zero except in −L < x < L, say, andthat in latitude it is simply described by V0(y) = exp

(−12y2), so that only

Q0(x) is nonzero. Then (33) gives us

εq0 +∂q0∂x

= −Q0

εq2 +∂q2∂x

− v1 = 0

εr0 −∂r0∂x

+ 2v1 = −Q0

4q2 − r0 = 0

and all other coefficients are zero. We can use the 2nd, 3rd and 4th of theseto get

3εq2 −∂q2∂x

= −12Q0 .

Taking the first equation first, then outside the forcing region,

εq0 +∂q0∂x

= 0

i.e., q0 ∼ e−εx. Demanding a bounded solution leads us to conclude that

q0(x) =

0 , x < −L−∫ x−L

Q0(x′) eε(x

′−x)dx′ , −L < x < 0

−e−εx∫ L−L

Q0(x′) eεx

′

dx′ , x > L

(36)

This part of the solution, which has u = h ∼ exp(−12y2)and v = 0 is the

Kelvin wave part. There is no Kelvin wave response to the west of the forcingsimply because the Kelvin wave group velocity is eastward. To the east thesolution decays as e−x/λ where

λ = ε−1 =group velocity

disspation rate

(recall that in dimensional terms ω = k for the Kelvin wave, whence thegroup velocity is unity).

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The second part of the solution has

q2(x) =

−12e3εx

∫ L−L

Q0(x′) e−3εx

′

dx′ , x < −L−12

∫ LxQ0(x

′)e3(x−x′) dx′ , −L < x < L

0 , x > L

. (37)

Here boundedness demands that there is no component like e3εx in x > L,to the east of the forcing. This part of the solution also includes v1(x) andr0(x), where

v1 = εq2 +∂q2∂x

r0 = 4q2

Note that this component is zero to the east of the forcing and, to the west,varies as e3εx. This is the n = 1 Rossby wave component. Recall that,in the long wave limit under consideration here, the n = 1 Rossby wave hasω = −k/3, so that its group velocity is −1/3. Hence there is no contributionto the east of the forcing and to the west the solution decays as ex/λ, wherenow

λ = (3ε)−1 =|group velocity|disspation rate

.

Since the group velocity of the n = 1 Rossby wave is 1/3 of that of the Kelvinwave, it attenuates a factor of 3 more rapidly than the Kelvin wave.

All other components of the expansion (31) are zero, a consequence of thefact that we chose a forcing structure (in y) that does not project onto thehigher modes. The full solution is shown in Fig. 1; the central frame, showinglow-level pressure and wind, clearly shows the predicted characteristics: zonalflow to the east, with a Gaussian latitudinal profile of u and p (≡ h), andtwin cyclones straddling the equator to the west, as expected for the n = 1Rossby wave,which has h structure

h =1

2(q − r)

=1

2[q2(x)V2(y) + r0(x)V0(y)]

=1

2[q2(x) [V2(y) + 4V0(y)]]

= q2(x)(2y2 + 1

)e−y

2/2 .

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Figure 1: Gill’s solution for forcing symmetric about the equator.

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This function has extrema in y at y = 0 and y = ±√3/2 = ±1. 22, consistent

with the locations of the twin low level cyclones on the figure. [Note thatGill’s values for x and y must be divided by

√2 to convert to our notation.]

With our numbers, LE 1600 km, so these values of y correspond to dimen-sional distance of 1950 km, or about ±17.5o latitude. Since we are assuming“first baroclinic mode” structure, we also expect to see twin anticyclones atthese locations in the upper troposphere.

The latitudinally averaged streamfunction in the x−z plane, shown in thebottom frame of Fig. 33, shows the twin circulation cells in this plane. Themore extensive and (slightly) stronger cell is located to the east, and appearsto correspond to the Walker circulation in the atmosphere. It extends adistance of order 7LE 11100 km, about enough to straddle the equatorialPacific Ocean. Note, however, that this depends entirely on the value chosenfor ε, the dissipation rate. Gill used ε = 0.1

√2 (in our notation), so the

predicted e-folding distance is just LE/ε = 11200 km. Given our time scaleof 7.78 hr, Gill’s value for ε corresponds to a dissipation time, for both windand temperature, of 55 hr, rather a short time for both.

6.3.2 Antisymmetric forcing

If the forcing is antisymmetric about the equator, such that Q(x, y) has noprojection onto the even Hermite functions, the the response is qualitativelydifferent. Gill’s solution for the response to such forcing is shown in Fig. 2.(The imposed distribution ofQ is approximately indicated by the distributionin the top frame.) Since there is no projection of Q onto V0(y), there is noKelvin wave component to the solution and, since this component is the onlyone with eastward group velocity, there can be no response to the east of theforcing. Near and to the west of the forcing, the response (dominated by then = 2 Rossby wave) takes the form of a cyclone/anticyclone pair straddlingthe equator near y = ±2 ≈ ±3200 km ±29o latitude, with cross-equatorialflow around and out of the anticyclone and around and into the cyclone.

6.3.3 Forcing centered off the equator

Fig. 3 shows the response to forcing centered in the northern subtropics –in fact, it is simply the sum of the first two cases. (Again, Q is approximatelyshown by the distribution of w.) The response shows many of the featurescharacteristic of the Indian Ocean region in northern summer: low level

21

Figure 2: Gill’s solution to forcing localized in x and antisymmetric abouty = 0.

22

Figure 3: Gill’s solution for response to forcing centered in the northernsubtropics.

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cyclone/upper level anticyclone slightly NW of the upwelling, and cross-equatorial flow to the west of the forcing, becoming a southwesterly inflowinto the region of low-level convergence.

6.3.4 What does the Gill model mean?

The Gill model is remarkably successful at reproducing major characteristicsof the tropical flow, yet it is built on a demonstrably false premise, thatone can treat the dynamics a priori as a first baroclinic mode structure.Further, it appears to rely in unreasonably strong dissipation (in order toget reasonable zonal length scales). So is the model’s success a fluke, oris it somehow doing the right things for the wrong reason? One possibleexplanation (Lindzen & Nigam, 1987; Neelin 1989) is that the single-nodevertical structure of the 1st baroclinic mode is really representing flow inthe boundary layer, and flow of the opposite sign in the free atmosphere;moreover, the importance of the boundary layer then makes Gill’s dissipationseem less unreasonable. More recently (e.g., Neelin & Zeng 2000) it seemsbetter to think of it as a consequence of the quasi-equilibrium nature ofthe tropical atmosphere, where the vertical stratification is constrained bydeep moist convection, in which case “1st baroclinic mode-like” dynamicalstructures are imposed by the vertical structure of the convection, ratherthan by real modal constraints.

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