Module 12 Heat-Induced Tropical Circulations in the Atmosphere

Transcription

1 Module 12 Heat-Induced Tropical Circulations in the Atmosphere 12.1 Introduction Oceans mostly dominate the tropical latitudes. The atmospheric circulations in tropics are complex because they are forced and maintained by the convective fluxes of heat and moisture from the sea surface. Large amount of latent heat is released in the precipitating deep cumulonimbus cloud towers that generally occur over the oceanic regions in the tropical atmosphere where warm sea surface temperatures persist; for example, the convection over the warm pool in the western Pacific Ocean. The ocean plays an important role as it supplies moisture, which is picked up by dry warm thermals, and their equivalent potential temperature (θ e ) will rise. Especially over the oceans, only air near the surface has a sufficiently high θ e, which becomes buoyant if it is forcibly lifted by the low level convergence produced by the large-scale circulation in the tropics. This is a plausible triggering mechanism of convection in tropics. As a consequence, convection in the tropics is generally organized meaning that both large-scale circulation and the small-scale processes cooperate to produce longer spells of intense rainfall. Without elaborating further, it is imperative that an understanding of the oceanatmosphere interaction is the key to developing successful mathematical models that explain tropical circulations. Moreover, non-uniform conditions in tropics imply that radiative cooling and latent heat released due to convection do not balance in some oceanic regions. Evidently, the excess heat thus liberated would force deep convection in those regions, though it might be highly intermittent. From observations, H. Riehl (1979) explained the nature of the convection in the tropics with: (i) diluted ascent of convective parcels if they entrain environmental air during their rise; and (ii) undiluted ascent of convective parcels produce the so called undiluted hot towers because such parcels do not entrain environmental air and their buoyancy is increased due to release of latent heat as they rise further up in the atmosphere. The atmosphere undoubtedly responds to such intense heating. A.E. Gill (Quart. J. Royal Met. Soc. vol.102, 1980) has presented solutions of a simple model (T. Matsuno J. Met, Soc. Japan, vol.44, 1966) by adding symmetric and asymmetric heat sources about the equator. These solutions explain large scale circulations that are oriented either zonally (Walker circulation) or meridionally (Monsoon circulation). In the Gill s model, heating is applied to an atmosphere at rest, i.e. an atmosphere with no mean motion. This is an important contribution to the dynamics of tropical atmosphere; therefore, we derive here the solutions of the governing equations of the Gill s model, as it provides a deeper insight into the complex dynamics of largescale tropical circulation due to anomalous heating in the tropics The linear model The response of the tropical atmosphere to a given distribution of heating can be studied with the help of linear shallow water equations. Our aim is to examine the steady forced motion in the tropics, which is obtained as forced solutions of a two-dimensional linear model. In case of a three dimensional model, a normal modes decomposition in the vertical will result in a series of independent two-dimensional shallow water models for a characteristic equivalent depth. Further in tropics, it is well in order to use the equatorial beta plane approximation where the Coriolis parameter f is given by the product of 1

2 β (= df ) and distance y from the equator. The nondimensional form of the governing dy equations of the linear shallow water model, first derived by Matsuno (1966), is as follows: Momentum equations u t 1 2 yv = p (12.1) x v t + 1 p yv = (12.2) 2 y Continuity equation p t + u x + v = Q (12.3) y In the Matsuno-Gill model equations (12.1) (12.3), the independent variables (x, y) represent nondimensional distances with x varying eastward, and y along north with y = 0 at equator (i.e. f = βy). The dependent variables (u,v) are proportional to the components of horizontal velocity; p is proportional to pressure. Taking w proportional to vertical velocity, it can be obtained from (12.3) as w = (u,v) = p t + Q (12.4) In (12.3), Q is proportional to the heating rate; also if Q > 0 (heating) would imply w > 0, that is, upward motion in the vertical. However, if Q is steady, then the amplitude of the motion will continue to grow unless dissipation mechanisms are included in the system of equations (12.1) (12.4). The simplest way to include dissipative processes in a shallow water model is to replace the operator, t + ε and for the sake of t mathematical simplicity the same ε represents friction in the momentum equations, and cooling in the continuity and vertical velocity equations. The parameter ε is the decay rate. Dropping the time-dependent terms from the resultant equations, one obtains the steady-state version of equations (12.1) (12.4) with the simplest form of dissipation viz., Rayleigh friction and Newtonian Cooling, given as 1 p ε u yv = 2 x (12.5) 1 p ε v + yu = 2 y (12.6) ε p + u x + u = Q y (12.7) w = ε p + Q (12.8) The solutions of the above equations will represent the steady forced motion in the tropics. It will be seen that the procedure is so elegant that the final expressions thus obtained, may be regarded as the simple solutions representing heat-induced tropical circulations. 2

3 12.3 The longwave approximation It is assumed that ε << 1; and the forcing function is such that its y-scale is of the order of unity. Also, if k is the east-west wave number of the forcing, then we have 2kε << 1. This relation implies that the east-west scale (k) of the forcing is large as compared to 2ε. Further, with such an assumption, the term εv in the second momentum equation (12.6) can be neglected and it becomes 1 p yu = (12.9) 2 y That is, the eastward flow (u) is in geostrophic balance with the pressure gradient. This simplification is also equivalent to long wave approximation if the time-dependent terms were retained in the model. We now proceed to solve this system of equations with this approximation in the next section The method of solution The model equations are comprised of (12.5), (12.7) (12.9), hence the new system of governing equations is as follows εu 1 2 yv = p x 1 2 yu = p y (12.10) ε p + u x + u y = Q w = ε p + Q The method of solution was first described by H. Lamb (1945) and it has been used by Gill (1980) for the above model. Define two new variables, q = p + u (12.11a) r = p u. (12.11b) On summing (12.5) and (12.7), and using (12.11a) we obtain εq + q x + v y 1 yv = Q. (12.12) 2 By subtracting (12.5) from (12.7) and using (12.11b) we obtain εr r x + v y + 1 yv = Q (12.13) 2 And, equation (12.9) takes the following form q 1 r 1 + yq + yr = 0 y 2 y Free solutions (12.14) The free solutions of (12.12) (12.14) are obtained by setting the heating rate identically zero (Q 0) in these equations. By inspecting these equations one can easily identify two types of operators given below, 3

4 y y ( ) and y 1 2 y ( ). The free solutions of (12.12) (12.14) are the parabolic cylindrical functions D n (y) (M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, 1965, Chapter 19), which satisfy the following properties dd n + 1 dy 2 y D n = n D n 1 (12.15) dd n dy 1 2 y D n = D n+1 (12.16) The parabolic cylindrical functions have the following form ( ) exp 1 4 y2 D o, D 1, D 2, D 3 = 1, y, y 2 1, y 3 3y 12.6 Forced solutions: reduction of governing equations to ODEs (12.17) With Q n (x) 0, the solutions of forced problem (12.12) (12.14) can be obtained by expanding the dependent variables q, r, v and Q in terms of the parabolic cylinder { } which form a complete set of basis functions. When these expansions functions D n (y) are substituted in the governing equations, the partial differential equations will reduce to a sequence of ordinary differential equations, which can be solved by standard methods. The expansions of the dependent variables in terms of the parabolic cylinder functions D n (y) are as follows q(x, y) = q n (x)d n (y) ; r(x, y) = r n (x)d n (y) (12.18 a, b) v(x, y) = v n (x)d n (y); Q(x, y) = Q n (x)d n (y) (12.18 c, d) Substituting (12.18a) in equation (12.12) one obtains dq ε q n (x)d n (y) + n dx D (y) n + v n (x) dd n dy 1 2 yd (y) n = Q n (x)d n (y) For any n, the above equation gives dq n dx + εq n D (y) v D (y) n n n+1 = Q n (x) When the above form is expanded, then we get the following equation dq o dx + εq + Q (x) o o D + dq 1 o dx + εq v + Q (x) 1 o 1 D 1 + dq 2 dx + εq v + Q (x) D + dq 3 2 dx + εq v + Q (x) D +... = 0 3 D n 4

5 Next, the coefficients of D 0, D 1, D 2, D 3 etc. are equated to zero in the above equation to arrive at a system of ordinary differential equations for n = 0, n = 1, n = 2, etc. which must be solved for obtaining the complete solution. This system of equations reads dq 0 dx + εq = Q 0 0 (x), n = 0 (12.19) dq n+1 dx + εq v = Q n+1 n n+1 (x), n 1 (12.20) On substituting (12.18b) in (12.13) we get ε dr r n (x)d n (y) n dx D dd n + v n n dy 1 2 yv D n n = Q n (n)d n (y) The above equation takes the following form εr n dr n dx D (y) dd n + n dy y D (y) n v n = Q n (x)d n (y) Using the property of parabolic cylinder functions, the above equation can be written as ε r n d r n dx D (y) n + nv n D n 1 (y) = Q n D n (y). On writing the summations in the expanded form and grouping the terms, we get ε r 0 d r 0 dx + Q 0(x) D 0(y) + ε r 1 d r 1 dx + v 0 + Q 1 (x) D 1(y) + ε r 2 d r 2 dx + v + Q (x) 1 2 D (y) +... = 0 2 On equating the coefficients of D 0, D 1, D 2, D 3 etc. to zero, we obtain n = 0, coefficient of D 0 gives : ε r 0 d r 0 dx + v 1 = Q 0 (x) n = 1, coefficient of D 1 gives : ε r 1 dr 1 dx + 2v 2 = Q 1 (x) etc. for increasing n. This sequence of equations can be precisely written as ε r n 1 dr n 1 dx + nv n = Q n 1 (x) n 1 (12.21) Next, substituting the expansions of q and r from (12.18a) and (12.18b) respectively in the eq. (12.14), we get dd n dy yd n q dd n + n dy 1 2 yd n r n = 0. Using the relations (12.15) and (12.16), the above equation reduces to 5

6 n D n 1 q n D n+1 r n = 0, which when written in the expanded form, gives q 1 D 0 (y) + (2q 2 r 0 )D 1 + (3q 3 r 1 )D = 0. If the coefficients of D 0, D 1, D 2 etc. are all set to zero in the above equation, we get q 1 = 0 (12.22) (n +1)q n+1 r n 1 = 0 n 1 (12.23) Note that the PDEs (12.12) (12.14) have been reduced to a system of ordinary differential equations, which can be solved easily for a prescribed form of heating rate Forcing function The solutions of the governing equations are to be found with two types of forcing functions: (i) symmetric about the equator (y = 0), and (ii) asymmetric about the equator. It is assumed that the forcing is localized about x = 0. Further, the forcing Q(x, y) has been expanded in terms of the parabolic cylinder functions as given in the expression (12.18d). Therefore the y dependence of the forcing can be expressed in terms of these parabolic functions, and the x dependence can be expressed in terms of an analytical function. With such an analytical formulation of the heating rate Q(x, y), analytical solutions will be amenable to solutions, which provide a deep insight in to the tropical dynamics. Now, we give the mathematical form of such forcing functions used by Gill (1980) in what follows. (i) Symmetric forcing about the equator: The parabolic function D 0 (y) = exp( 1 4 y2 ) is symmetric about the equator; hence a symmetric heating rate can be formulated as Q(x, y) = F(x)D 0 (y) = F(x)exp( 1 4 y2 ) (12.24) (i) Asymmetric forcing about the equator: The parabolic function D 1 (y) = y exp( 1 4 y2 ) is asymmetric about equator; hence an asymmetric heating rate in the tropics can be formulated in terms of D 1 (y) as Q(x, y) = F(x)D 1 (y) = F(x) y exp( 1 4 y2 ) (12.25) It now remains to prescribe an appropriate form of function F(x), which is localized about x = 0, hence it could be of the following form cos kx x < L F(x) = 0 x > L The wavenumber k is defined as k = π 2L (12.26a) (12.26b) 6

7 Because the forcing function has been defined for the given problem, it is now possible to find the complete response of the atmosphere to this forcing by calculating the solutions of the system of linear equations representing (i) the response that propagates eastward, and (ii) the response that propagates westward in the tropics. These two parts of the response have been formulated as Problem P1 and Problem P2 as given below Problem P1: Eastward propagating response of the heat source ( n = 0 ) The solution of Problem P1 gives the eastward propagating response. Hence, by putting Q 0 (x) = cos kx in (12.19), the corresponding boundary value problem becomes dq 0 dx + ε q 0 = cos kx L < x < L (12.27) dq 0 dx + ε q = 0 0 x L (12.28) From (12.28), it can be inferred that eastward propagating response decays exponentially in the region east to the heat source as shown in the Fig below. Since the response does not propagate westward, therefore, we have immediately q 0 = 0 x L (12.29) The solution (12.29) thus becomes a boundary condition for the solution of (12.27) at the left extremity x = L of the heat source. This condition will be used to evaluate the constant of integration. Further Q 1 (x) = 0; Q 2 (x) = 0 etc. Kelvin wave response x L x = L x = 0 x = L x L q 0 = 0 q 0 e εx No westward propagation Heat source region Response decays exponentially Spatial decay rate = ε Match solution at boundaries (x = ±L) Fig Schematic of calculating eastward propagating response of the heat source on the equator. The equation (12.27) can be solved using an integrating factor e εx ; hence multiply (12.27) with e εx on both sides, the result of this step can be written as d(e εx q 0 ) dx = e εx cos kx The integration of the above form gives the formal solution of (12.27) as q 0 (x) = e εx e εx cos kx dx + c 1 e εx, (12.30) 7

8 with c 1 as the constant of integration to be evaluated by matching the solutions at the western boundary of the heat source. On evaluating the integral, the solution (12.30) takes the form (k 2 + ε 2 )q 0 (x) = ε cos kx k sin kx + c 1 e εx (12.31) The constant of integration c 1 can be evaluated using (12.29) which implies that the solution (12.31) vanishes at x = L. This step evaluates c 1 = ke εl and (12.31) takes the following form (k 2 + ε 2 )q 0 (x) = ε cos kx k [sin kx + e ε (x+l) ] (12.32) The solution that decays in the region x L is the solution of (12.28), which reads as q 0 (x) = c 2 e εx for x L The constant c 2 is to be obtained by matching the decaying solution with (12.32) at the right extremity x = L and c 2 evaluates to 1 c 2 = k 2 + ε 1+ e 2εL 2 e εl Hence the solution in the region east to the heat source is given by (k 2 + ε 2 )q 0 (x) = k [1+ e 2εL ]e ε ( L x) (12.33) Therefore, the complete solution of the Problem P1 is as follows (k 2 + ε 2 )q 0 (x) = 0 x < L (k 2 + ε 2 )q 0 (x) = ε cos kx k [sin kx + e ε (x+l) ] L < x < L (k 2 + ε 2 )q 0 (x) = k [1+ e 2εL ] e ε(l x) x < L (12.34) It is now required to obtain the expressions for u, v, w and p from the above solution that corresponds to n = 0. Now (12.20) gives dq 1 dx + εq 1 v 0 = Q 1 (x) = 0 Also, from (12.22) q 1 = 0 and Q 1 (x) = 0, therefore we have v 0 = 0 (12.35) For the Kelvin wave response, it is evident from (12.20) that both q 2 = 0 and v 1 = 0. Then from (12.23), we have From (12.11), we have r 0 = 2q 2 r 0 = 0 (12.36) q 0 = p 0 + u 0 and r 0 = p 0 u 0 We obtain the following important relation p 0 = u 0 (since r 0 = 0) and u 0 = 1 2 q 0 (12.37) 8

9 Thus, the Kelvin wave part of the response of the atmosphere to heat source in the tropics is given by u = p = 1 2 q 0 exp( 1 4 y2 ) v = 0 (12.38) w = 1 2 {εq + 2F(x)}exp( y2 ) If the heat source is thought to be the one situated over the Indonesian region, then solution (12.38) shows easterlies (u < 0, as q 0 < 0) over the Pacific Ocean, which is the region east to the forcing region; w > 0 is maximum at x = 0 (the centre of the heating) as q 0 ~ ε and F(x) 1. The easterly trades are parallel to equator because, there is no meridional component to the wind (v = 0). This is the Walker circulation over the Pacific with easterlies parallel to equator flowing into the heat source region where they rise and then flow eastward aloft. Also note that pressure p, as calculated from (12.38), is minimum at the centre of the forcing region. It means a trough is created on the equator and easterlies flowing parallel to equator, imply a flow down the pressure gradient along the equator Problem P2: Westward propagating response of the heat source ( n = 1) To derive the relevant equations for n = 1, we begin with the following equations obtained from (12.20), (12.21) and (12.23) to get dq 2 dx + εq 2 v 1 = Q 2 (x) Note that q 2 (x) = 0 for x > L. Putting Q 2 (x) = 0 in the above equation, we get v 1 = dq 2 dx + εq 2 ε r 0 dr 0 dx + v 1 = Q 0 (x) 2q 2 r 0 = 0 The above three equations can be combined to give the following equation 3ε q 2 dq 2 dx = Q 0 (x) (12.39) The solution of the Problem P2 gives the westward propagating response of heating located over the equator with heat source functions Q 0 (x) = cos kx in (12.39) and Q 2 (x) = 0, Q 3 (x) = 0 etc. in (12.21). Next, the formulation of Problem P2 is presented which is based on the fact that Rossby wave response of the atmosphere to a prescribed heat source on the equator does not propagate in the region east to the heating region, therefore q 2 = 0. The relevant equations are the following 9

10 dq 2 dx 3ε q 2 = 0 x < L dq 2 dx 3ε q 2 = cos kx L < x < L q 2 (x) = 0 x > L (12.40) r 0 = 2q 2 (12.41) v 1 = dq 2 dx + εq 2 (12.42) Note that the Rossby response to the heat source dissipates three times faster than the spatial decay rate of the Kelvin wave response. As shown in Fig. 12.2, the Rossby response to the heat source is to be computed separately in the three distinct regions, viz., x < L (region west to the heat source), L < x < L (heat source region) and x > L (region east to the heat source). A formal solution of (12.40) over the heat source region ( L < x < L ) can be easily obtained as q 2 (x) = e 3εx e 3εx cos kx dx + c 3 e 3εx, or q 2 (x) = e 3εx e 3εx cos kx dx + c 3 e 3εx = 1 {(3ε) 2 + k 2 } [k sin kx (3ε)cos kx]+ c 3 e3εx. Rossby wave response x L x = L x = 0 x = L x L q 0 e 3εx Response decays exponentially Spatial decay rate = 3ε Heat source region Match solution at boundaries (x = ±L) q 0 = 0 No eastward propagation Fig Schematic of computing westward propagating response of heat source on the equator. The solution on the east extremity ( x = L ) of the heat source vanishes because the response does not propagate in the region east to the heat source. Hence in the above solution, c 3 evaluates to c 3 = k e 3εL ; and the solution over the heating region reads {(3ε) 2 + k 2 }q 2 (x) = (3ε)cos kx + k[sin kx e 3ε (x L) ] L < x < L (12.43) The westward propagating Rossby response to the heat source dissipates in the region west to the source region with the decay rate of 3ε (compare it with the Kelvin wave decay rate of ε ), the solution of (12.40) in this region x < L is given by 10

11 q 2 (x) = q 2 e 3εx, q 2 = constant x < L (12.44) The unknown constant q 2 can be evaluated by matching the solutions (12.43) and (12.44) at x = L, the west extremity of the heat source region; and q 2 (x) evaluates to q 2 = k[1+ e 2(3ε )L ] {(3ε) 2 + k 2 } e3εl Therefore the complete solution for n = 1 which represents the westward propagating Rossby response of the atmosphere to the heat source prescribed on the equator is as follows, {(3ε) 2 + k 2 }q 2 (x) = k[1+ e 2(3ε )L 3ε (x+l) ]e x {(3ε) 2 + k 2 }q 2 (x) = (3ε)cos kx + k[sin kx e 3ε (x L) ] {(3ε) 2 + k 2 }q 2 (x) = 0. < L L < x < L x > L (12.45) The above solution corresponds to long planetary wave propagating westward at speed 1 3 that gives a spatial decay rate of 3ε. The general solution of equations (12.20), (12.21) and (12.23) for an arbitrary n may be written from (12.45) as follows {(2n +1) 2 ε 2 + k 2 }q n+1 (x) = k[1+ e 2(2n+1)εL (2n+1)ε (x+l) ]e x < L {(2n +1) 2 ε 2 + k 2 }q n+1 (x) = (2n +1)ε cos kx + k[sin kx e (2n+1)ε (x L) (12.46) ] L < x < L {(2n +1) 2 ε 2 + k 2 }q n+1 (x) = 0, x > L. To understand the response of the atmosphere, we next calculate expressions for the dependent variables u, v, w and p. The long planetary wave is constituted from q and r given by q = p + u = q 2 D 2 and r = p u = r 0 D 0. Now, q 2 (x) is given by (12.45) and r 0 = 2q 2 (x) from (12.41), hence we have u = 1 2 (q 2D 2 2q 2 D 0 ) and p = 1 2 (q 2D 2 + 2q 2 D 0 ). On substituting expressions of D 0 and D 2 from (12.17) in the above relations, we have u = 1 2 q 2(x)(y 2 3)e 1 4 y2. (12.47) p = 1 2 q 2(x)(y 2 +1)e 1 4 y2 (12.48) The expression for v is obtained from v 1 given by (12.42) and D 1 which reads v = v 1 D 1 (y) = ( dq 2 dx + εq )D (y) = (dq dx 3εq + 4εq ) 2 2 ye Hence, the final expression for v is given by v = (F(x) + 4εq 2 )ye 1 4 y2 (12.49) 1 4 y2 11

12 Finally w is obtained from the expression, w = ε p + Q 0 (x)d 0 (y) ; which gives w = 1 2 [2F(x) + εq 2(x)(y 2 +1)] e 1 4 y2 (12.50) This completes the derivation of the westward propagating n = 1 long planetary wave solution of the equations (12.1) (12.3) which reads p = 1 2 q 2 (x)(y2 +1)e 1 4 y2 u = 1 2 q 2 (x)(y2 3)e 1 4 y2 v = (F(x) + 4εq 2 )ye 1 4 y2 w = 1 2 [2F(x) + εq 2 (x)(y2 +1)]e 1 4 y2 An immediate inference about zonal flow can be made from (12.51). (12.51) (i) u = 3 2 q 2(x) at y = 0 (equator) and q 2 (x) is negative; hence the winds are westerly along the equator and the flow is down the pressure gradient. A trough (low pressure) is produced along the equator and the pressure decreases to the east on the equator. (ii) u = 0 at y = ± 3 ; that is, the westerly winds vanish at on the either side of the equator at a distance 3 from the equator. The above inferences imply that air is descending everywhere in the region x < L and directed toward equator. (iii) The vertical velocity is upward over the heating region which is maximum at equator and reduces exponentially in the meridional direction Complete solution for steady motions due a symmetric heat source The system of governing equations is linear so the principle of superimposition of the solutions can be applied. Therefore, the complete response of the atmosphere to a prescribed symmetric heat source on the equator is the sum of the solutions of the Problem P1 and Problem P2. Accordingly, the steady motion of the tropical atmosphere forced by heating, is obtained by summing up solutions (12.38) and (12.51) which combines the Kelvin and Rossby responses forced by the symmetric heat source on the equator. The Gill s solution depicting total response of the tropical atmosphere to a symmetric heat source has been graphically shown in Fig to have a clear picture of the response. In Fig. 12.3a, the winds are westerly in the region west to the heating zone and easterly in the region east to the forcing region. The vertical velocity in Fig. 12.3a is superimposed on the wind field. One may notice upward motion over the heating zone and descending motion away from the equator in the region west to the forcing region. Moreover, the shape of the region of upward motion is similar to the forcing function. The perturbation pressure contours are also superimposed on the wind fields as shown in Fig.12.3b. The perturbation pressure is everywhere negative and a trough forms over the equator due to heating. However, the pressure in the region west to the forcing region is relatively high at points away from the equator in comparison to those on the equator. 12

13 Along the equator pressure decreases towards east so the flow is down the pressure gradient along the equator and attains speed of a jet in the atmosphere. 4 y (a) w < 0 0 w < 0 4 w > x 15 4 y (b) L 0 Westerly jet L 4 z x 15 (c) Outflow aloft Outflow aloft x= 10 Stream function x= 5 Walker Circulation Surface inflow Surface inflow x= 0 x=5 x=10 x=15 Perturbation pressure p Fig Heat-induced tropical circulation with ε = 0.1 in the lower troposphere: (a) vertical velocity and wind field forced by heating located in the region x < 2 ; (b) the perturbation pressure with induced circulation showing strong inflow on the west and relatively weak inflow on east along with outflows in the meridional direction; (c) the meridionally integrated flow showing stream function with inflow at the surface and outflow aloft over the heating zone. (Gill, 1980) The westerly jet produces strong inflow in the region west to the heating zone. The Kelvin wave response of the heat source also produces a weak inflow on the eastern flank of the heating zone. Under the action of two lows, the flow at the surface is directed poleward from the heat source region. Further, the meridionally integrated flow is 13

14 presented Fig. 12.3c with the stream function, which shows surface inflow in the region of heating and outflow aloft. The heating produces the Walker circulation in the region east to the heating zone and Hadley circulation in the region west to it. Under the action of Hadley cell air rises on the equator, moves aloft poleward with gradual sinking away from equator in the upper troposphere and then it again moves equatorward at the surface. Let us examine the meridional flow over the heating region in the limit ε 0 from the complete solution obtained by summing up (12.38) and (12.51). Note that v = 0 in (12.38), so the expression for v from (12.51) in the limit ε 0 gives v = y F(x) e 1 4 y2 (12.52) Clearly v decays exponentially away from the equator. But the most interesting feature revealed by (12.52) is that for y > 0, v > 0 i.e. a northward flow; and for y < 0, v < 0 i.e. a southward flow. This implies that there is poleward outflow from the heating region x < L, which can be seen in Fig. 12.3b. Also, w > 0 which in conjunction with the direction of v implies that flow over the heating zone is upward and outward from the heat source and not towards the heat source from meridional direction. It may however be noted that in the nonrotating case, flow would be upward and towards the heat source as, for example, the microcirculation forced by a bonfire. The rotation of the earth thus produces outflows in the meridional direction. Moreover, the outflow forced by heating can also be explained from the vorticity equation since we have ε( u y v x ) 1 2 y( u x + v y ) 1 2 v = 0. The above equation one can derive by differentiating (12.5) with respect to y and (12.6) with respect to x and subtracting the resultant equations. In the limit ε 0 it reduces to y( u x + v y ) + v = 0 (12.53) Equation (12.53) states that divergence is balanced by the βv term. Using (12.53) to replace the divergence term in (12.7) and taking the limit ε 0, we get v = yq (12.54) Also note that when ε 0, the vertical velocity w = Q ; that is, heating produces vertical motion which causes stretching of the vortex tube (the bundle of vortex lines) producing positive (cyclonic) relative vorticity. As a result, the absolute vorticity is increased. However, in order to conserve absolute vorticity, the air rising over the heating zone will move poleward to latitudes where its vorticity is same as the background (planetary) vorticity. Hence in the limit ε 0, a heat source placed on the equator induces poleward air motion in the lower troposphere and equatorward motion in the upper troposphere. To summarize, the response of an anomalous symmetric heat source positioned on the equator produces a strong inflow on the west flank and a relatively weak inflow on the east flank into the heating region along with poleward outflows in the meridional direction. Such a heat source induces Hadley circulation in the region west to its location and Walker circulation in the region east to it. 14

15 12.11 Problem P3: Response of the tropical atmosphere to asymmetric heating The asymmetric heat source will be defined from (12.18d) as Q(x, y) = Q 1 (x)d 1 (y) ; Q 1 (x) = F(x) (12.55) The form of the function F(x) is same as given in (12.26). There are two parts of the response that corresponds to: (i) n = 0 (mixed planetary-gravity wave); and (ii) n = 2, the planetary wave. (i) The long mixed waves do not propagate outside the forcing region; hence the first part of the response is confined to the forcing region only. The response of the heating for n = 0 is computed from (12.20) and (12.22) which give the following system dq 1 dx + εq 1 v 0 = Q 1 (x) q 1 = 0 From above two equations, the response corresponding to n = 0 is given by q 1 = 0, v 0 = Q 1 (x) (12.56) (ii) The second part of the response corresponds to n = 2 computed from (12.20), (12.21) and (12.23) which give the following system of equations dq 3 dx + εq 3 v 2 = Q 3 (x) = 0 ε r 1 dr 1 dx + 2v 2 = Q 1 (x) 3q 3 r 1 = 0 The above equations are combined to obtain the following set of governing equations for the heat source region x < L v 2 = dq 3 dx + εq 3 (12.57) r 1 = 3q 3 (12.58) dq 3 dx 5ε q 3 = Q 1 (x) (12.59) In the above set of equations, it is only required to solve (12.59) for q 3 (x), as the other variables can be determined from q 3 (x). Boundary conditions: The response of an asymmetric heat source does not propagate in the region extending on the east to the heat source. This means, one can set q 3 (x) = 0, x > L. But the solution has limited propagation in the westward region of the heating as the decay rate is 5ε for the planetary wave n = 2. This argument helps in setting up the initial condition for the first order ordinary differential equation (12.59). The schematic of finding the solution of (12.59) is given in Fig The solution of the Problem P3 in essence reduces to solving eq. (12.59) subject to the following initial condition, q 3 (L) = 0 (12.60) 15

16 The complete solution of (12.59) subject to condition (12.60), can easily be found out as {(5ε) 2 + k 2 }q 3 (x) = 5ε cos kx + k[sin kx e 5ε (x L) ] < x < L (12.61) over the heat source region and the region west to the heating zone. In the eastern region x > L, the solution of (12.59) continues to grow exponentially and becomes unbounded ( q 3 ); hence, for the solution to be realistic, it is set to zero for the region x > L, and we have {(5ε) 2 + k 2 }q 3 (x) = 0 x > L (12.62) That is, the response does not propagate eastward. From the above form, one can write a general form of the complete solution for any n as {(2n +1) 2 ε 2 + k 2 }q n+1 (x) = (2n +1)ε cos kx + k[sin kx e (2n+1)ε (x L) ] < x < L (12.63) {(2n +1) 2 ε 2 + k 2 }q n+1 (x) = 0 x > L Asymmetric heat source response x L x = L x = 0 Heating Cooling Limited westward propagation Heat source region x = L x L q 0 = 0 No eastward propagation Match solution at boundary (x = L) Fig Schematic of computing response of the asymmetric heat source about equator It now remains to find out the expressions for p, u, v and w in terms of the solution q 3 (x) given by (12.61) over the heat source region. These quantities are obtained as p = 1 2 [q + r] = 1 2 [q 3 (x)d 3 (y) + r 1 (x)d 1 (y)] u = 1 2 [q r] = 1 2 [q 3 (x)d 3 (y) r 1 (x)d 1 (y)] v = v 0 (x)d 0 (y) + v 2 (x)d 2 (y) w = ε p + Q 1 (x)d 1 (y) = ε p + F(x)D 1 (y) Substituting for r 1, v 0 and v 2 we obtain the final expressions which read as follows 16

17 p = 1 2 q 2 (x)(y2 +1)e 1 4 y2 u = 1 2 q 2 (x)(y2 3)e 1 4 y2. (12.64) v = (F(x) + 4εq 2 )ye 1 4 y2 w = 1 2 [2F(x) + εq 2 (x)(y2 +1)]e 1 4 y2 The solution (12.64) is plotted in Fig for ε = 0.1 and L = 2. The asymmetric heat source produces cross-equatorial flow much like that is established during the summer monsoon over India. The asymmetric heating is confined to x < 2 ; further, there is no air motion in the region x > 2. The decay rate in the region west to the heat source is 5ε for the planetary wave n = 2, therefore the response propagation is limited westward General solutions The general forcing can be constituted with non-zero Q n (x) in the expansion (12.18d) of heating function Q(x, y). The solution technique remains the same and it is required to solve a series of problems for each n to get the contribution of heating. Further it is to be noted that as n increases, the decay rate for the planetary waves will also increase limiting their influence zonally. For the case of generalized heat sources, Zhang and Krishnamurti (J. Atmos. Sci., vol.53, 1996) have done these calculations. However, the major response though would be contributed from Q 0 (x)d 0 (y) and Q 1 (x)d 1 (y), yet all solutions could be summed up to obtain the complete response of a combined symmetric-asymmetric heat source about the equator, as done by Zhang and Krishnamurti. If a line of symmetric heat sources is placed close to equator, then also one can find the general solution as the response of the linear model. Gill (1980) has shown that the complete solution is the sum of the solutions of the linear model corresponding to each heat source. The model will thus simulate the Intertropical Convergence Zone ITCZ). Similarly, the solution obtained by adding the solutions corresponding to a symmetric heat source on the equator and an asymmetric heat source with heating mainly concentrated to the north gives easterlies in the region east to the heating. A strong inflow from west to the heating will develop on the north to the equator but it would be much weaker on the south (Fig. 3 in Gill s paper in Quarterly Journal of Royal Met Soc., 1980). From an understanding of such circulations, the role of dynamically evolving heat sources in the tropical atmosphere can be easily assessed in numerical simulations from more complete atmospheric global models. The Gill s contribution is important to tropical meteorology because different kinds of tropical circulations can be explained as the response of the atmosphere to diabatic heat sources, which constantly, evolve, propagate and dissipate in this region. As a matter of fact, Gill s solutions are very helpful in interpreting the sensitivity studies of general circulation models to prescribed heating, such as anomalous sea surface temperatures, diabatic heating and radiative forcing. For example, while studying the sensitivity of Indian summer monsoon to radiative forcing, Sharma et al. (J. Climate, vol.11, 1998) 17

18 obtained the response of imposed cloudiness derived from satellite data (ISCCP) in a global circulation model (LMD GCM), which shows a strong countercurrent (Fig. 12.6) over the Indian subcontinent coming from the east. y 4 0 (a) Fig Circulation induced by an asymmetric heat source. (a) vertical velocity (w) superimposed over wind field with w > 0 over the heating region in the northern hemisphere and descending motion ( w < 0 ) in the southern hemisphere; (b) y L 0 H x x (b) the perturbation pressure shows a low pressure over warmer region with cyclonic circulation whereas a high over the cold area with an anticyclone. Note the cross equatorial flow due to this disposition of low and high pressure centres about the equator at the surface. The flow is much like the summer monsoon circulation over India. One may notice in Fig.12.6, a strong inflow on the west and a relatively weaker inflow on the east below the equator that has been simulated by the GCM. The most dominant feature of this circulation that model simulated is the counter monsoon current which was first demonstrated by Krishnamurti et al. (J. Climate, vol.2, 1989) for the failed monsoon of 1987 producing droughts in various parts of India. The different features of this GCM simulation suggest that the model has simulated an anomalous heating on the equator in the Indian Ocean as shown in Fig due to imposed cloudiness. Essentially as illustrated by Gill (1980), such a result in the Matsuno-Gill steady linear model is produced by the combined effect of symmetric and asymmetric heat sources about the equator, which undoubtedly has helped us in analyzing and understanding the response of the GCM for the monsoon circulation. The key impact of the countercurrent (Fig. 12.6) is that the global model simulated drastically reduced 18

19 precipitation in comparison to the mean rainfall. Earlier, T.N. Krishnamurti et al. (1989) found that reduced rainfall of 1987 (drought year) was linked to the monsoon counter current. Fig.12.6 Sensitivity of Indian monsoon circulation to radiative heating (OP Sharma, H Le Treut, G Sèze, L Fairhead and R Sadourny, J. Climate, vol.11, 1998) as analysed in the light of the response of the Matsuno-Gill model to concurrent symmetric and asymmetric heat sources about the equator Baroclinic response of the tropical heat sources To understand the baroclinic response of the atmosphere to heating, let us conceptually analyse the circulation in the lower and upper atmosphere induced by a heat source placed on the equator. If the earth were not rotating, the circulation induced by the heat source is much like that one observes around a bonfire as depicted in Fig. 12.7a, with convergence into the heat source and the rising air diverging out in the upper levels to descend slowly to surface to complete the closed cell as shown in this figure. However, the situation will be different when the rotation is considered. The Gill s solution, as shown conceptually in Fig. 12.7b, gives easterlies in the region east and strong westerlies in the region west to the heat source in the lower atmosphere. Further in the north and south directions, the Gill s solution shows outflows in the lower atmosphere. As a result, 19

20 Ascending Descending two cyclones form in the western region and two weak anticyclones in the region east to the heating about the equator (Fig. 12.7b). 200 hpa Upper atmosphere Divergence 850 hpa Lower atmosphere Convergence Heat source (a) NO ROTATION CASE Upper atmosphere LONGITUDE Equator LATITUDE 200 hpa level strong anticyclonic Upper atmosphere A A C C A = anticyclonic circulation C = cyclonic circulation weak cyclonic Convection 850 hpa level strong cyclonic Lower atmosphere C C Heat source (b) ROTATION CASE A A weak anticyclonic LONGITUDE Equator LATITUDE Fig Baroclinic response of the heat source placed on the equator: (a) no rotation case, and (b) rotation case (following Gill s solution) Meteorologically, the easterlies and westerlies advect cold air into the warm area, hence wind will back in accordance to the thermal wind relation. That is, the strong westerly flow in association with two cyclones in the lower atmosphere (850 hpa) will be replaced by strong easterly flow associated with two strong anticyclones (Fig. 12.7b) in the upper atmosphere (200 hpa level). Similarly, in region east to heating zone, the easterlies will be replaced by westerlies in the upper atmosphere in association with the two weak cyclonic circulations as one may observe in Fig. 12.7b. In the light of this response, it is much easier now to understand the response of the upper atmosphere to heat sources forming dynamically in the lower atmosphere Circulations forced by land heating and sea-surface buoyancy fluxes This section deals with circulations that are forced by land-sea contrast in the heating and sea-surface buoyancy fluxes due to air-sea interaction. It is important to recognize that the upward flux of buoyancy mainly contributes towards raising the available potential energy (APE), which is converted to kinetic energy of moving air parcels. In 20

21 this manner atmospheric circulation is strengthened. But circulation redistributes APE so that a steady state could be established again; that is, the constant potential temperature (θ ) surfaces become almost parallel when steady state is re-established. The derivation of APE has been included as an exercise in the Problems of this chapter with proper steps that one could follow to obtain the final expression. Or, one may take the help of the textbooks to derive the expression for APE. How APE is generated in the atmosphere? In this context, it is well known that APE is generated in the atmosphere by (i) uneven radiative heating arising from absorption and emission of radiation, (ii) latent heat release in clouds and (iii) air-sea exchange processes that induce changes in sea surface temperatures. In the ocean, APE is mainly generated by salinity changes and conduction of heat from the boundaries. In the preceding sections, the heat-induced tropical circulations have been discussed, and it is now amply clear that tropical circulations are intimately related to the distribution of heat sources. Deep convection occurs in the atmosphere over the locations of heat sources, which may also move laterally to produces changes in the circulation on different time and spatial scales. Though we have only learnt the linear model based explanations of circulations forced by heat sources, the true dynamics is nonlinear which become important in order to account for scale interaction. Without going into much details, we describe the low-frequency oscillations of the tropical circulation that occur at interannual, annual and intraseasonal scales, which are respectively the El Niño and Southern Oscillation (ENSO), the Southwest Indian summer Monsoon (ISM), and the Madden-Julian Oscillation (MJO). The ENSO (interannual event) has the strongest signal during winters in the central and eastern Pacific; the monsoon (annual event) has the strongest signal in the summer over India and North Australia; while MJO (seasonal phenomenon) characterized by a day oscillation, has the dominant signal in tropical Indian Ocean and Pacific. They primarily arise from complex interaction of atmospheric and oceanic processes and all these phenomena are associated with deep convection in the tropics. Further, they not only influence tropical circulations but also directly impact the climate of tropical countries. Besides, systematic studies on their fluctuations could help in identifying even signatures of an ensuing climate change. The ENSO and ISM are the best examples of atmosphere-ocean interaction in the tropical atmosphere. It is now aimed below to explain them in terms of the heat induced tropical circulations. (i) The El Niño and Southern Oscillation (ENSO): The time scales of ENSO are interannual with a frequency of its occurrence as 2-3 years in the Pacific during northern hemisphere winter. The tropical heat source over the warm pool which moves from west to the east Pacific, forces the circulation in the atmosphere, which, in turn, drives the circulation in the ocean by the action of wind stress on surface waters. Thus ENSO is an important example of air-sea interaction. From the theory of heat-induced circulation described above, the Walker circulation is obtained as a response of the symmetrical heat source produced by the maritime warm pool over Indonesia. The solution (12.38) of Problem P1 formulated with the Matsuno-Gill model shows rising motion over the heat source with air moving east aloft and sinking slowly over the eastern Pacific and again moving westward at the surface towards the heat source to complete the cell (Fig. 12.3c) in the xz-plane (x - east; z - vertical). There are different phases of the circulation over the Pacific which are directly linked to the location of the heat source; these are 21

22 characterized as: (i) normal conditions with west Pacific waters warm and east Pacific waters cold; (ii) the El Niño conditions when warm waters have moved eastward to the Date Line or further east to produced SSTs warmer than normal in central/east Pacific; and (iii) the La Niña conditions when eastern Pacific water are much colder than normal, a situation that is opposite of El Niño. A widely acceptable model of ENSO is a simple Cane-Zebiak model, which uses the Matsuno-Gill type model equations for coupling the atmosphere and ocean in its formulation. This model successfully reproduces the increase in the heat content of equatorial waters before the onset of ENSO, which actually reduces after the arrival of the event. Warm Pool Thermocline Cold Warm Pool Thermocline Cold Warm Pool Thermocline Cold E Pacific Ocean 80 0 W E Pacific Ocean 80 0 W E Pacific Ocean (a) Normal conditions (b) El Niño conditions (c) La Niña conditions Fig : Oscillations in the Pacific Ocean during ENSO and non-enso years. The atmospheric circulation loops are shown in association with the location of deep convection. The response in the ocean as depicted in Fig could be understood in the backdrop of Fig. 2.4 that explains the Ekman layer dynamics its mathematical treatment has been given in Module 8. The Ekman drift will produce upwelling under the action of large-scale easterlies along the equator causing cold SST anomaly. Stronger equatorial cold SST anomalies in the eastern Pacific on the Peru coast are caused by upwelling of deeper waters to the surface and the Southern Ocean cold waters brought by the northward flowing Humboldt Current. As a result, the thermocline slopes westward as shown in Fig. 12.8a; that is, it is deeper in the western Pacific and relatively shallower in the eastern Pacific. This sort of thermal structure in the ocean and convective circulation in the atmosphere is referred to as the normal conditions that prevail generally over the Pacific Ocean equatorial region. However, this kind of situation is disrupted in some years when warm waters from western Pacific warm pool spread towards east. As a result, warm SST anomalies in the eastern Pacific begin to grow displacing the otherwise cold SST anomalies in the region. When the warm SST anomalies appear near the South American west coast, the descending branch of the Walker circulation is reversed into the rising motion over the eastern Pacific, and the thermocline in the ocean deepens causing a barrier for the nutrient-rich water below the thermocline to reach surface. This event is known as the El Niño (the Christ Child) as it occurs in December or January. The El Niño conditions refer to the warming of equatorial Pacific waters by C which persists for several seasons. Sir Gilbert Walker (Director of India Meteorological Department), while working on the long-range forecasting of Indian summer monsoon, noticed an oscillation in the sea level air pressure difference of Tahiti (Southern Pacific) and Darwin (Western Pacific) as 80 0 W 22

23 Surface pressure anomalies warm waters over eastern Pacific cause sea level pressure (SLP) to fall over there and its rise over the western Pacific (Fig. 12.9). Sir Gilbert Walker (1928) called this oscillation in sea level pressures as the Southern Oscillation and the difference in Tahiti and Darwin sea level pressures, ΔP TD = (SLP Tahiti SLP Darwin ) is used in defining the Southern Oscillation Index (SOI) as follows, SOI = monthly value of ΔP TD long-term average of ΔP TD standard deviation of ΔP TD 10. There is a kind of seesaw discovered by Sir Gilbert Walker in the difference of pressures of Darwin and Tahiti; this means, when Darwin pressure increases, that of Tahiti decreases, and vice-versa (Fig. 12.9). The worldwide impact of the El Niño is now well documented as it causes drought in Africa and floods in Peru and Ecuador due to torrential rains. Δp s < 0 0 Δp s > 0 Western Pacific Ocean Normal / La Niña Eastern Pacific Ocean El Niño Fig The seesaw in the surface pressure anomalies at Darwin and Tahiti during El Niño conditions and normal conditions (La Niña years included). Darwin Tahiti Fig depicts the pressure signal associated with the major event in the Pacific that is linked to east Pacific SST and the tropospheric circulation, which has been referred to as the ENSO event. Thus, in triggering an ENSO event, interaction of ocean and atmospheric processes plays a key role. In the initial stage, transfer of heat takes place from west to east Pacific due to eastward flowing equatorial current. Weakening of southeast trade winds in the east Pacific then follows. As a consequence, the upwelling of cooler thermocline waters on the Peru and Ecuador coasts will also weaken. Furthermore, in a situation of weaker south-east trade winds, the eastward flowing warm waters along the equator will continue to move further east raising the SST of the central and east Pacific; consequently, the associated deepening of the thermocline follows. That is, the wind circulation plays a crucial role in changing the ocean circulation with sinking waters on the east Pacific and upward moving water masses in the west Pacific. This means that thermocline shoals on the west and deepens in the east Pacific. From the Gill s solution, it has been seen that convergence occurs over the heat source. This implies that the movement of convergence zones in the atmosphere is intimately linked the position of the warm SST in the ocean, which is referred to as the positive feedback between the SST and the wind circulation during the ENSO event. The negative feedback loop will be associated with the La Niña or normal conditions over the Pacific. (ii) The Southwest Indian Summer Monsoon (ISM): The ENSO is an interannual phenomenon in the Pacific Ocean, which begins from December-January with a 23

24 frequency of 2-3 years. On the contrary, the southwest monsoon over India is an annual phenomenon, which arrives on 1st June (its climatological onset date) and remains active during summer each year. The Nature keeps its date with impeccable accuracy insofar as it concerns to the arrival of the southwest monsoon over India because the standard deviation of its onset dates is ±7 days. This means that the numerical models that reproduce the annual cycle of the monsoons correctly within the observed standard deviation shall be successful in predicting the global weather and climate. Monsoons over India have been discussed in Module 2, and their main cause is attributed to land-sea thermal contrast, which essentially forms an asymmetrical heat source with heating mainly concentrated over the landmass of the Indian subcontinent. The Tibetan Plateau acts as an elevated heat source during summer months while ocean temperatures remain cold in comparison to land temperatures. The heating forms a low pressure over the landmass and together with a high-pressure centre over the Indian Ocean. As a result, a strong low-level airflow engenders that heralds the arrival of copious rains over India. This cross-equatorial flow maintains the moisture supply for the monsoon to remain active on the seasonal scale. The tropical circulation forced by such a kind of heat source has been referred to as the Problem P3 for the Matsuno-Gill model and its solution is given by (12.64) and illustrated in Fig In the context of southwest monsoon which is a manifestation of the response of the asymmetric heat source one may notice a low pressure over the heating region and a high pressure over the cold temperatures over Indian Ocean in accordance to the Gill s solution. Further, as discussed in the preceding paragraph, the response in the wind field shows a clear cross equatorial flow in the lower troposphere that subsequently becomes a strong southwesterly flow reaching into the heating region. In the upper troposphere, the baroclinic response of the asymmetric heat source over the Indian subcontinent and Indian Ocean, will result in a strong northeasterly flow from northern hemisphere to southern hemisphere. It may therefore be noted that low pressure in northern hemisphere and high pressure in southern hemisphere together produce a strong interhemispheric exchange that persists for more than 120 days of the monsoon season over India. At time of onset of monsoon, significant changes occur in the positions of the centres of the action in the northern hemisphere, and triggering of monsoon is truly an episodic event. For an elaborate account of these changes, one may consult the book entitled Monsoons by P.K. Das (1998) and a recent book entitled Introduction to Tropical Meteorology by T.N. Krishnamurti, L. Stefanova and V. Mishra (2013) for a more comprehensive and extensive exposition. It is also equally rewarding to read a paper by H. Luo and M. Yanai (Monthly Wea. Review, vol.111, 1983) for the Tibetan Plateau acting as an elevated heat source, the circulation and the associated precipitation based on observations of 1979 summer monsoon. Various researchers have explained the annual event of the monsoon phenomenon differently but distinctly, which are described below. Synoptic meteorologists, physical meteorologists, radiation meteorologists and geophysical fluid dynamicists, all describe the establishment of southwest monsoon over India with a flavour characteristic to their domain of research; and so distinctly that each definition complements each other and brings out the very important aspect of this phenomenon. An enlightened synoptic meteorologist would describe the event: As the southwest monsoon establishes over India, the level of westerlies go down (implying a 24

25 northward shift of this regime) and that of easterlies go up. As a result, the easterly flow attains a jet speed in the upper troposphere, which is supported by strong southwesterly monsoon current in the lower troposphere. This describes a simultaneous happening which asserts (i) the shifting of the subtropical westerly jet stream suddenly on the northern boundary of the Tibetan Plateau, (ii) the development of easterly jet on the southern periphery of the Tibetan Plateau in the upper troposphere and (iii) the establishment of southwest monsoon current in the lower troposphere. Indeed, as described in the book by T.N. Krishnamurti et al., both (i) and (ii) are an integral part of the subtropical high over the Tibetan Plateau. To a physical meteorologist, the Tibetan Plateau, which acts as an elevated heat source, is important for the onset and maintenance of the southwest monsoon over India. In effect, due to excessive heating of Tibetan Plateau during the month of May, isobars are packed in the upper troposphere creating a high pressure region. With the formation of this subtropical high, a strong equatorward flow is triggered in the upper troposphere with descending branch over the Indian Ocean. This strong equatorward flow becomes the upper tropospheric easterly jet as it moves southward. Subsequently, the southwesterly winds are established in the lower troposphere. The thermally direct monsoon cell is formed in this manner, with rising motion over land and descending motion over the sea. The moisture laden southwesterly winds from ocean on reaching over the Indian subcontinent bring copious rains; and with that, it begins the wet season in the region. A salient point of this description is the formation of subtropical high in the upper troposphere due to elevated heat source over Tibetan Plateau that triggers and maintains the southwest monsoon over India. To a radiation meteorologist, the gain and loss of the radiant energy is important before and after the onset of the SW monsoon. In this context, the excessive heating of Indian subcontinent landmass and Tibetan Plateau produces a net deficit in the radiant energy; that is, due to higher land temperatures, the emitted infrared flux is greater than the net shortwave radiation flux reaching the ground during May and June months. From the radiation balance point of view, the atmospheric column emits excess infrared radiation than it receives. This results in net cooling of an atmospheric column. Consequently, a downward motion sets over this region and descending parcels maintain their energy through adiabatic compression. The evidence of such a net deficit in the radiant energy over the region is the appearance of a heat low over the Western India / Pakistan region. In order to make up this deficit, moisture laden southwest monsoonal winds are established which produce copious rains releasing enormous amount of latent heat. The latent heat released from monsoon clouds offsets the energy deficit to turn the region into a net gainer of energy. However, due to scarce rains over the heat low region, the radiant energy deficit persists and the monsoon current continues to remain active. Accordingly, it is the net deficit of radiant energy in summer months that causes a heat low to develop in the lower atmosphere, which forces subsequent rearrangement of meteorological systems in the region. A heat low is very distinct from a low pressure system with low level convergence and upper level divergence. In a heat low, there is convergence both at the surface and in the upper troposphere but divergence at 850 hpa (the top of the boundary layer). The reader is invited to draw the vertical cross-section of the atmospheric column showing levels of convergence and divergence over a heat low as well as over a low pressure centre. 25

26 To a geophysical fluid dynamicist, the monsoon phenomenon arises from low pressure on land in NH and high pressure over the Indian Ocean in the SH. The lower tropospheric air is forced to move from south to north under the action of the pressure gradient force. However, due to the action of Coriolis, the south-to-north flow experiences Ekman drift on the left in the SH so it is constantly deflected westward. But on crossing the equator, the Ekman drift is on the right so the flow is continuously deflected southward during its northward journey. Thus, a cold southwesterly geostrophic flow arrives over the warm Indian subcontinent, and in accordance with the thermal wind relation these winds will back in the upper troposphere generating the easterly jet. The strong northeasterly upper level flow crosses equator and descends in to SH to complete the thermally direct monsoon cell. In this manner, the existence of southwesterly flow in lower troposphere and the tropical easterly jet in the upper troposphere could be immediately explained. The above definitions of summer monsoon are also applicable for the other monsoons over the globe. They invariably emphasize the role of tropical heating, which contributes to intraseasonal and interannual variability of the monsoons. The Gill s solution gives a very comprehensive view of the response of the tropical atmosphere that is seen in the onset and maintenance of the Indian summer monsoon. It must be remarked that the latent heat release is dynamically distributed and the organized convection in the tropics produces slowly propagating transients that produce intraseasonal variability in the otherwise steady circulations such as the monsoon. Prominent among them is the day intraseasonal oscillation in tropics, also known as the Madden-Julian Oscillation (MJO), which propagates eastward and directly interact with meridionally propagating intraseasonal oscillations to impact the climate of the region. We shall give the solution that is pertinent to MJO from the theory already developed with the Matsuno-Gill model. Since MJO is a propagating phenomenon, the time dependent terms in the Matsuno-Gill model are therefore to be retained in the governing equations. (iii) The Madden-Julian Oscillation (MJO): Several types of intraseasonal oscillations (ISOs) in the tropical atmosphere propagate both meridionally and zonally. The variability introduced by such oscillations range from monthly to biweekly; or, in other words, the variability associated with them dominates the tropics at different time scales. Further, when the eastward propagating Madden-Julian oscillation interacts with northward propagating ISOs, it may considerably spin up or spin down precipitation. The MJO is named after R.A. Madden and P.R. Julian (Description of a day oscillation in zonal wind in the tropical Pacific. J. Atmos. Sci., vol.28, 1971), who discovered this phenomenon by studying 10-year zonal winds and sea-level pressure in the tropics. Some studies have highlighted the role of MJO in the onset and demise of El Niño, but its mechanism is still an open challenge because it in not very well simulated by the numerical models. The MJO dynamics essentially evolves as a result of interaction between the planetary-scale long waves and the mesoscale convection in the tropics. It is therefore pertinent to present the MJO mechanisms that have been proposed. The MJO is primarily a slow eastward (average speed ~ 5ms 1 ) propagating phenomenon where heating plays the dominant role in triggering it. There are two types of hypotheses for MJO initiation. The first type of hypothesis regards MJO as the response of the tropical atmosphere to a stationary but pulsating heat source (T. Yamagata and Y. Hayashi, J. Met. Soc. Japan, vol.62, 1984; A. Anderson and D.E. 26

27 Stephens, J. Atmos. Sci., vol.44, 1987) or a travelling (either of constant or oscillating strength) heat source (Hayashi and Miyahara, J. Met. Soc. Japan, vol.65, 1987). The second type hypothesis is founded on the interaction of tropical diabatic heating and the large-scale flow. The diabatic heating in the atmosphere could arise: (i) due to evaporation heating, which is proportional to surface zonal wind anomalies (evaporationwind feedback) proposed by J.D. Neelin, I.M. Held and K.H. Cook (J. Atmos. Sci., vol.44, 1987) and K.A. Emanual (J. Atmos. Sci., vol.44, 1987); and (ii) condensational heating forced by low-level moisture convergence as proposed in the CISK mechanism (condensational-convergence feedback). It has been mentioned above that MJO is most distinctly observed in the Indian Ocean and the Pacific. Then an important question arises. Does MJO activity occur in one or both the regions simultaneously? The answer is: yes, MJO could be active simultaneously in any one or both the regions but the circulation would be weaker. Because, it is quite likely that one MJO event is in the initiation stage (in Indian Ocean) and the other in the decaying phase (over Pacific Ocean). However, an example of MJO activity on May 9, 2003 in the Indian Ocean (refer Chapter 7 in the book by TN Krishnamurti et al. 2013) is worth highlighting though that kind of MJO occurrence in the Indian Ocean is a very rare event. Therefore, in general, one fully developed MJO event exists at a time in the tropics. Fig The structure of Madden-Julian Oscillations (Adapted from Rui and Wang, Journal of Atmospheric Sciences, Vol. 47, p ) 27