some simple solutions for heat-induced tropical circulation

Quart. J. R. Met. SOC. (1980). 106, pp. 447-462 551.513.2551.515.51 Some simple solutions for heat-induced tropical circulation By A. E. GILL Department of App lied Mathematics and Theoretical Physics, University of Cambridge (Received 20 July 1979; revised 23 November 1979) SUMMARY A simple analytic model is constructed to elucidate some basic features of the response of the tropical atmosphere to diabatic heating. In particular, there is considerable east-west asymmetry which can be illus- trated by solutions for heating concentrated in an area of finite extent. This is of more than academic interest because heating in practice tends to be concentrated in specific areas. For instance, a model with heating symmetric about the equator at Indonesian longitudes produces low-level easterly flow over the Pacific through propagation of Kelvin waves into the region. It also produces low-level westerly inflow over the Indian Ocean (but in a smaller region) because planetary waves propagate there. In the heating region itself the low-level flow is away from the equator as required by the vorticity equation. The return flow toward the equator is farther west because of planetary wave propagation, and so cyclonic flow is obtained around lows which form on the western margins of the heating zone. Another model solution with the heating dis- placed north of the equator provides a flow similar to the monsoon circulation of July and a simple model solution can also be found for heating concentrated along an inter-tropical convergence line. 1. INTRODUCTION The first theory of tropical circulation can be attributed to Halley (1686), who proposed that heating in the tropics caused air to become ‘less ponderous’ and hence to rise. This explained the equatorward component of the trade winds (which Halley stated would be accompanied by a reverse flow aloft), but not the easterly component. That was explained later by Hadley (1735) in terms of the tendency of the equatorward flowing air to conserve its angular momentum about the earth’s axis. The zonally averaged seasonal pictures of the meridional circulation (e.g. Newel1 et al. 1974) show the motion envisaged by Halley, although the picture is not symmetric about the equator, the sinking part of the circulation being nearly all in the winter hemisphere. However, the tropical circulation is by no means zonally symmetric. In particular, Bjerknes (1969) drew attention to the Walker circulation in the Pacific Ocean. This is a circulation in the plane of the equator with rising air over the warm zone in the West Pacific and descending motion over the cold ocean of the East Pacific. Variations in the U‘alker circulation seem to be an important feature of the Southern Oscillation (Julian and Chervin 1978). Much of the heating in the tropics is found to be over the three continental areas of the tropics, i.e. Africa, South America and the Indonesian region (see e.g. Ramage 1968 and the discussion in Krueger and Winston 1974), each of which is relatively small in extent. That causes speculation on the result of heating a limited area centred on or near the equator at a particular longitude. If the heating is applied to a resting atmosphere, and is small enough to use linear theory, the response can be modelled in terms of equatorially trapped waves. If the heating were switched on at some initial time, Kelvin waves would carry the information rapidly 447 448 A. E. GILL eastward, thereby creating easterly trade winds in that region, the trades providing inflow to the region of heating and so providing a Walker-type circulation with rising over the source region and sinking to the east. If dissipative processes are added, a steady state can be reached with the trade winds gradually dying out with distance from the source region. Thus trade winds over the Pacific are generated in this model by the heating over Indonesia. In addition, the switching on of the heating would generate a planetary wave which would carry information westwards into the Indian Ocean. The fastest planetary wave, however, travels at a third of the Kelvin wave speed, so this effect would be expected to penetrate only one-third as far as the Kelvin wave. This planetary wave response could explain the surface westerlies in the Indian Ocean (Lettau 1973), as being a result of the heating over the Indonesian region. Such qualitative results are sufficiently realistic to suggest it would be useful to produce a simple mathematical model on the above lines, and this is done in subsequent sections. The model is designed to show how the response to heating varies in the horizontal. Although emphasis is given to problems with forcing in limited regions (which nicely illustrate east-west asymmetries in the response), the model can also be applied to more general forms of forcing. 2. THEMODEL The aim is to study the response of tropical atmosphere to a given distribution of heating, using as simple a model as possible. It is natural, therefore, to use the linear theory for small perturbations a resting atmosphere, i.e. the heating rate will be assumed small enough for linear theory to apply. Other effects may well play a significant role in practice but will not be considered here. The unperturbed state, therefore, consists of an atmosphere at rest with properties a function of height z only. In the absence of dissipative processes and forcing, an effective means of studying motions with large horizontal scale (i.e. large compared with the vertical scale) has been separation of the solution into a height-dependent part and a part which depends on the horizontal coordinates and time. The idea was already implied in the discussion by Laplace (1778/9) of thermal oscillations in the atmosphere where he effec- tively stated that there exists a mode which satisfies his tidal equations with an equivalent depth equal to the scale height. (This is now called the Lamb mode.) Taylor (1936) showed how the technique can be applied to a compressible atmosphere with undisturbed tempera- ture an arbitrary function of height. Details are discussed, e.g. by Holton (1975). In the case of an isothermal atmosphere, for instance, there is a continuous spectrum of modes for each of which the square root of density times the vertical displacement varies sinusoidally with altitude and one additional mode (the Lamb mode) for which the vertical displacement is zero but the pressure perturbation decays exponentially with height. For each of these modes, variations with horizontal position and with time are governed by the ‘shallow water’ equations but with a different ‘equivalent depth‘ of water for each mode. A technique for dealing with forced problems is to expand the forcing function in terms of normal modes. This method has been applied, for instance, by Lighthill (1969) to an oceanographic problem. The equivalent in the present problem is to express the diabatic heating rate as a Fourier-type integral over the complete set of modes. Important contribu- tions to the solution will be expected to come from modes with inverse vertical wavenumber m-l of the same order as the scale of the forcing function. Since, in the tropical atmosphere, the diabatic heating tends on average (see e.g. Hantel and Baader 1978) to be a maximum at 5 km, this can be taken as a representative value form-’. The wave speed for such a mode HEAT-INDUCED TROPICAL CIRCULATIONS 449 in the absence of rotation is about 60m s- ' (it is approximately given by N/m where N is a mean value for the buoyancy frequency, compressibility effects being minor at this scale) corresponding to an equivalent depth of about 400 m. The above remarks are intended merely to put the present calculation in perspective, for attention here is concentrated on structure in the horizontal and solutions will be found for one mode only. Since the solutions are all for forcing in a limited region, solutions for different modes with different (but similar magnitude) equivalent depths would have similar structure but with slightly different scales. There are other approaches which lead to the same shallow-water equations, e.g. a two-layer atmospheric model with small perturbations about a rest state would give the same set. In this model, heating is equivalent to increasing the amount of high potential temperature fluid, i.e. transferring mass from the lower layer to the upper layer (cf. Gill 1979; Gill et al. 1979). Another approach is to consider an incompressible atmosphere with constant buoyancy frequency N and with a rigid lid at height z = D. The gravest mode in this case has pressure perturbations and horizontal velocity components which vary with height as cos(lrz/D), a vertical velocity which varies like sin(az/D), and an equivalent depth H given by c = (gH)* = ND/lr, (2.1) where g is the acceleration due to gravity and c is a separation constant which is equal to the speed of long waves in the absence of rotation. If the diabatic heating rate is chosen to vary like sin(nz/D), then only the gravest mode will be stimulated and hence the shallow-water equations for a single mode describe the complete solution. It is convenient when pictorial representations of the vertical structure are required to use this mode. It should be remem- bered, however, that the solution has wider significance as indicated above. The problem has now been reduced to solving the forced shallow-water equations in the tropics. Because the motion is confined to the tropics, it is convenient to use the equatorial beta-plane approximation in which the Coriolis parameter is approximated as B times distance northwards from the equator. It is also convenient to write the equations in a non- dimensional form (Gill and Clarke 1974), using as length scale the equatorial Rossby radius (c/2j)*, which is about 10" of latitude when the equivalent depth is 400m, and the time scale (2Pc) -*, which is about one-quarter of a day. Justification of the beta-approximation comes from the small size of the Rossby radius compared with the 90" span from equator to poles. The equations have the form (cf. Matsuno 1966): (2.2) a p au a U at ax ay -+- +- = - Q . In these equations (x, y ) is non-dimensional distance with x eastwards and y measured northwards from the equator, (u, v ) is proportional to horizontal velocity and p is propor- tional to the pressure perturbation. Q is proportional to the heating rate, the signs being such that if Q is positive (positive heating), the signs of u, u, p correspond to those at the surface. Equations (2.2) and (2.3) are the momentum equations, while (2.4) is the modal version of the continuity equation, the vertical velocity being proportional to w = - + Q , aP . at 450 A. E. GILL the latter being derived from the buoyancy equation. In order to study the response to steady forcing, dissipative processes must be included somehow. The most convenient is the so-called ‘Rayleigh friction’ and ‘Newtonian cooling’ forms which replace the operator a/& by a/at+e. The mathematics is simplest when the epsilon for friction is the same as the one for cooling, so the steady-state versions of Eqs. (2.2) to (2.5) are aP eu-+yv = -- ax, au av ax ay ep+-+-= -Q , w = &p+Q. (2.9) This model was used by Matsuno (1966). The above set of equations can be reduced to a single equation for v namely a2u azv av aQ aQ ax ay ax ay +YG e 3 v + ~ e y 2 v - e - 2 - ~ 7 - + - = E-- (2.10) (Note: McCreary (1980) has shown, with suitable assumptions, this method can be extended to include all vertical modes. Vertical eddy transfer can be allowed for by making E a function of m and hence of H.) One further approximation will now be made. The forcing will be chosen to have a y-scale of order unity and E will be assumed small. Hence the term e3v at the beginning of Eq. (2.10) can be neglected. Secondly, if the forcing has east-west wavenumber k, the term ea2u/ax2 is small compared with the term +au/ax provided 2ek < 1. . (2.11) This will be assumed to be true also, i.e. that the forcing has east-west scale large compared with 28. This is no real restriction because E is assumed to be small. These approximations are equivalent to neglecting the term ev in Eq. (2.7), which now becomes (2.12) i.e. the eastward flow is in geostrophic balance with the pressure gradient. This is also equivalent to making the ‘long wave’ approximation in the transient problem (cf. Gill and Clarke 1974). 3. METHOD OF SOLUTION In order to solve the three Eqs. (2.6), (2.8) and (2.12), it is convenient first to introduce two new variables q and r to replacep and u. These are defined (Gill 1975) by q = p + u . (3.1) r = p - u . . (3.2) The sum and difference of (2.8) and (2.6) then yield HEAT-INDUCED TROPICAL CIRCULATIONS 45 1 while Eq. (2.12) can be rewritten in the form 84 dr - + + y q + - - + y r = 0. . aY aY (3.5) The free solutions of (3.3), (3.4) and (3.5) have the form of parabolic cylinder functions Dn(y) (Abramowitz and Stegun 1965, Ch. 19) and solutions of the forced problem can be f6und by expanding the variables q, r, u and Q in terms of these functions, namely etc. These have the properties So substituting. the expressions of the form (3.6) into (3.3), (3.4) and (3.5) gives +nun = - Q n - , Ern-, - - drn- I d x n > 1, (3.6) (3.7) (3.8) (3.9) (3.10) (3.11) In the following sections’solutions will be found for two special cases where the forcing has a particularly simple form. One is the case where the heating rate Q is symmetric about the equator and has the form Q ( ~ , Y ) = F(x)Do(y) = F ( x ) e x ~ ( - b ~ ) . - (3.12) The second has heating antisymmetric about the equator and of the form Q(x,Y) = F(x)D~(Y) = F(x)Y~xP(-~Y’)- . (3.13) The advantage of these forms is that the response only involves parabolic cylinder functions up to order 3, and these are given by Do, D l , D,, D , = (~,Y,Y2--,y3-3y)exp(-aYZ). (3.14) The forcing will be assumed localized in the neighbourhood of x = 0 and to have the form coskx x < L F(x) = (0 IxI > L . (3.15) where k = 7~/2L . . (3.16) 452 A. E. GILL 4. SYMMETRIC FORCING This is the case where the heating rate has the form (3.12), that is, the only non-zero coefficient Q,, corresponds to n = 0 and is given by QO = F(x) - (4.1) For this form of forcing, there are two parts of the response. The first part involves qo only, and this coefficient satisfies (3.9) with n = 0. This part represents a Kelvin wave which is damped out as it progresses eastwards. The wave moves at unit speed with decay rate E, so has spatial decay rate E as well. Since no information is carried westwards, the solution is zero for x < -L and so the solution of (3.9) is given by (4.2) I (&'+k2)q0 = 0 f o r x < -L (6, + k2)qo = - E cos kx - k[sin kx + exp{ - E(X + L)}] for 1x1 < L (6' + k2)qo = - k{ 1 + exp( - EL)} exp{E(L - x)} for x > L Consider now what this Kelvin-wave part of the solution looks like. Using (3. I), (3.2), (3.9) and (2.5) it follows that this part of the solution is given by (4.3) 1. . . 24 = P = 440(x)exP(-4YZ) w = H E q o ( X ) + F(xN exp( - 4Y2> v = o If the forcing region is taken to represent the Indonesian area, this solution represents the Walker circulation over the Pacific with easterly trades flowing parallel to the equator into the forcing region, rising there and then flowing eastward aloft. A corresponding pressure trough is found on the equator. Pressure decreases towards the west and the flow along the equator is down the pressure gradient. The second part of the forcing corresponds to putting n = 1 in (3.9), (3.1 1) and (3.10) which then give (4.4) ro = 2q, . (4.5) &-3Eq2 d x = Q o . . (4.6) This Corresponds to the n = 1 longplanetary wave which propagates westwards at speed 113 so giving a spatial decay rate of 3.5. As no information is carried eastwards, the solution of (4.6) is given by {(2n + 1) E + for x < - L {(2n+1)2~Z+k2}q,,+1 = -(2n+l)ccos kx+k[s inkx-exp((2n+l)E(x-L)} ] for 1x1 < L ((2n + 1 ) 2 ~ 2 + k2}q,,+ = O 2 2 k2 }q,,+ = - k[ 1 + exp{ - 2(2n + EL}] exp((2n + ~ ) E ( x + L)} for x > L (4.7) withn = 1. (3.2), (4.4), (4.5), (4.6), (2.5) and (3.14), namely The detailed solution for the pressure and velocity components follows from (3.1), HEAT-INDUCED TROPICAL CIRCULATIONS 453 (4.8) Consider the properties of this solution, beginning with the region x < -L to the west of the forcing zone. Here q2 is negative, so there is a net eastward flow in the lower layer given by J - m The winds are westerly along the equator and fall to zero at y = 43, with weak easterlies at higher latitudes. The air is descending and directed equatorwards throughout the entire region and there is a trough of low pressure along the equator. The pressure decreases to the east and flow along the equator is down the pressure gradient. It remains now to consider the flow in the forcing region 1x1 < L, which requires summing the contributions from the part (4.3) and the part (4.8). Thep and u fields are much as expected, but the meridional flow shows a feature which is somewhat surprising. This feature is most marked in the limit as tends to zero when (4.8) gives (v , w) -+ (Y, 1)F(x)exP(-iY2). (4.10) Thus the flow in the heating region is upwards but away from and not towards the heat source. The explanation comes from the vorticity equation obtained by taking the curl of (2.6) and (2.7). This gives in the limit as E + 0 y (a,: -+- :;) + v = o , (4.11) i.e. the divergence is balanced by the ‘pv’ term. Substitution from (2.8) with E = 0 gives v = Y Q , . (4.12) which is the equivalent of the Sverdrup equation in oceanography. (Note that this gives the opposite sign to that obtained by putting u = 0 in (2.8).) Now the explanation follows easily. Heating, in the limit as E + 0, causes upward motion by (2.9). This upward motion causes vortex lines in the lower layer to be stretched, and so increase their absolute vorticity (cyclonic positive). In the zero-epsilon limit, how- ever, particles must achieve this by changing to a latitude where their vorticity is the same as that of the background. (This is required by (4.14), i.e. by moving polewards.) Thus heating causes poleward motion in the lower layer and equatorward motion aloft. If E is allowed to become non-zero, the rotational constraint is less strong and so the region where heating causes poleward motion becomes smaller in extent, but (4.7) and (4.8) show that v is always positive near x = L. Despite the above ‘peculiarity’ of the forcing region, the zonally integrated flow has the expected form of a double Hadley cell with symmetry about the equator. The integrated flow is most easily obtained by integrating (3.9), (4.4), (4.5) and (4.6) direct, which gives 1 2 & ( 4 0 , 4 2 , ‘ o , V 1 ) d X = -1 - - -- - - ) I / & !Im ( ’ 3’ 3’ 3 (4.13) where 4 Y (a ) 0 -4 4 Y (b ) 0 -4 -5 X . Fi gu re 1 . So lu tio n fo r he at in g sy m m et ri c ab ou t t he e qu at or in th e re gi on 1 x1 < 2 fo r de ca y fa ct or E = 0 .1 . (a ) C on to ur s of v er tic al v el oc ity w (s ol id c on to ur s ar e 0. 0 .3 , 0 .6 , br ok en c on to ur is - 0. 1) s up er - im po se d on t he v el oc ity fi el d fo r t he lo w er la ye r. T he fi el d is d om in at ed by th e up w ar d m ot io n in th e he at in g re gi on w he re it h as a pp ro xi m at el y th e sa m e sh ap e as th e he at in g fu nc tio n. E ls ew