On the mathematical fluid dynamics of the atmospheric Walker circulation

Starting from the general, governing equations for a viscous, compressible fluid, with an associated description of its thermodynamics, we outline an asymptotic derivation based on the thin-shell approximation. [The details appear in another publication.] This produces a reduced system of equations which retain all the dynamics and thermodynamics of the steady atmosphere, the thin-shell approximation alone being the basis for the construction of the asymptotic solution. The leading order describes the background state of the atmosphere, and the next order provides a simple set of equations that can be used to investigate, for example, the Walker circulation, a particular atmospheric flow which is restricted to the neighbourhood of the Equator across the Pacific Ocean. Our formulation of this problem shows, explicitly and in detail, how the pressure and temperature gradients in the azimuthal direction drive the circulation; this extends the usual physical arguments used to describe the Walker circulation. An initial investigation highlights the rȏle of the variable eddy viscosity and then, on the basis of these observations, a solution is obtained which describes in detail the velocity and temperature fields in the Walker cell. In particular, we present an example of the temperature profile and of the streamlines for the flow along the Equator and which is bounded above by the tropopause. Further details of the Walker circulation are given, together with an identification of the heat sources that drive the motion. Finally, we comment on the changes to the flow pattern that arise during an El Niño event.


Introduction
main controller of the weather in this region of the Pacific Ocean. Indeed, this motion of the surface, coupled to the Equatorial Undercurrent, is an important element in the dynamics of the equatorial Pacific Ocean. For a general introduction to atmospheric flows, including a discussion of the Walker circulation, see, for example, [9,12]; for a mathematical model of this oceanic flow, with associated wave interactions, see [2,3].
The mathematical description that we explore here is based on the system of equations developed in [6], where a compressible, viscous fluid, with suitable thermodynamic properties, is used to represent the atmosphere. The formulation hinges on the thin-shell asymptotic approximation to describe the atmosphere on a (nearly) spherical, rotating Earth. In this presentation, we will briefly outline how these equations are obtained, and then write down the main system of equations that couple the dynamics and the thermodynamics; the full details are given in [6]. The plan is to take these equations and apply them to the Walker circulation; this will then constitute a special reduction of the general asymptotic formulation. The resulting system is readily analysed, the forcing required to generate the flow in the cell being explicit; it takes a simple form which is easily interpreted and can be related directly to the physical structure of the flow field. A few general observations about the flow structure in the Walker cell were given in [6], but the intention here is to provide far more detail. Indeed, we are able to include the adjustments needed to accommodate the changes associated with El Niño and El Niña events (which are described, for example, in [13]).

Governing equations
The underlying model that we use for our description of the atmosphere is based on the general equations for a compressible, viscous fluid, coupled to an equation of state and a suitable version of the first law of thermodynamics. In mathematical fluid dynamics, this is what constitutes a model; the development then follows the familiar route of non-dimensionalisation, scaling and the construction of an asymptotic solution. (More details about how the atmosphere is modelled, and the general principles that underpin the analysis, can be found in [6].) For this discussion, we choose to work in rotating, spherical coordinates, assuming a spherical Earth; the ellipsoidal approximation of the Earth's geoid-not invoked here-is carefully described in [6]. We allow the eddy viscosity to vary with height above the Earth's surface; indeed, in many models it is taken to be virtually zero beyond about 2 km altitude; see [15].
The development rests on one fundamental parameter: ε = H R , the thin-shell parameter, where H is the maximum thickness of the troposphere (about 16 km) and R is the average radius of the (spherical) Earth. (We use primes to denote physical (dimensional) variables; we will dispense with these as we move to nondimensional variables.) In the construction of the asymptotic version of this problem, we keep all other parameters fixed, as ε → 0, thereby retaining every physical attribute that contributes, at the same order, to both the dynamics and thermodynamics of the atmosphere. The Earth is rotating at the constant angular speed ≈ 7 · 29 × 10 −5 rad s −1 , and using this we non-dimensionalise according to: u = H (u, v, kw), the velocity vector in spherical coordinates (φ, θ, r ), where k measures the strength (in terms of ε) of the vertical velocity component; r = R (1 + εz); p = ρ ( R ) 2 p is the pressure, where ρ is an average density of the atmosphere; ρ = ρ ρ is the density. The coordinates are chosen so that φ is the azimuthal angle, and θ the meridional angle, being zero at the Equator and ±π 2 at the North/South poles, respectively. For a consistent thin-shell approximation, we choose to set k = ε, and then the governing equations for the steady atmosphere, with error terms indicated, can be written as these being the three components of the Navier-Stokes equation. The parameters that have been introduced here are where the former is a Reynolds number (defined using an average value of the dynamic eddy viscosity, μ ), and the latter (which can be interpreted as the ratio of the square of two speeds) typically takes a value of about 0 · 72, for H = 16 km; these are treated as O(1), i.e. fixed, as ε → 0. (This choice of parameter definitions ensures that we have a well-defined background state of the atmosphere, together with a dynamicthermodynamic coupling which describes its motion; an extensive discussion of this formulation is given in [6].) In addition, we have written the dynamic viscosity as μ (r ) = μ m(z) and the equation of mass conservation becomes The thermodynamic elements of the atmosphere are described, firstly, by the equation of state, where we have defined the temperature as with ≈ 287 m 2 s −2 K −1 the gas constant; secondly, the first law becomes where c p = c p ≈ 5 · 25, κ = κ c p H 2 ; c p is the specific heat of air, κ the thermal diffusivity of predominantly dry air and Q is the (non-dimensional) totality of heat sources/sinks. These final two parameters are also held fixed as ε → 0. The normalisation of the temperature, which uses the factor ( R ) 2 (about 800°K), produces a temperature variation of approximately T = 0 · 36 down to T = 0 · 27, from the bottom to the top of the troposphere. Although it transpires that R e is very large, and κ is very small, there is no necessity to incorporate additional assumptions or approximations: any thin viscous or thermal boundary layers, for example, are automatically included in the solutions. Finally, we observe that the second law of thermodynamics, which sets limits on the transformation between heat energy and mechanical energy, plays no direct ro "le in the calculations that we present here.

Asymptotic structure of the solution
To proceed, we seek an asymptotic solution, based on ε, and obtain the first two terms in an asymptotic expansion of the form where q, and correspondingly q n , represent each of u, v, w, p, ρ, T and Q. Further, we assume that the boundary conditions follow this same pattern, so that no terms appear in addition to those in the asymptotic sequence {ε n }. A general discussion of the nature and validity of this asymptotic expansion is given in [6]. The leading order is then obtained directly from Eqs. (1)-(3), (5) and (6), which gives This system has the solution where ς = gz − 1 2 cos 2 θ and p 0 is otherwise an arbitrary function at this stage, with The classical solution which describes the stationary background state of the atmosphere, independent of the velocity field, is given by and then Q 0 ≡ 0. This choice of model for the atmosphere shows that there are no external heat sources; the only heat supplied is that up from the surface of the Earth into the atmosphere. Further, we note that, although the choice (12) removes any direct coupling to the leading-order velocity field, this velocity field is, in general, a contributor to the solution at leading order (and is determined at the next order, as we now demonstrate). At the next order, O(ε), we obtain the system of equations which connects all the dynamics and thermodynamics of the motion: At this stage, we have written down the equations that appear, at O(1) and at O(ε), as the relevant descriptions of the general (steady) motion in the atmosphere, invoking only the thin-shell approximation. These equations can be used to describe many different phenomena: the Ekman spiral, geostrophic balance, the thermal wind, the Hadley-Ferrel-polar cell structure, the appearance of jet streams high in the troposphere and the Walker cell; see [6] for a discussion of all the foregoing applications, but where the last example is covered in only the barest outline. We now restrict the application of these equations to a careful and complete analysis of the Walker circulation.

The Walker cell: formulation
We start with a few salient features of the Walker cell. Its strength is attributed to the variation of sea-surface temperature across the Pacific along the equator-a difference of about 5°C, the warmer water to the West and the colder to the East. The cold water is present by virtue of advection northward along the coast of South America by the Humboldt current; see the discussions in [1,10]. This describes the general situation, but it can be disrupted every few years. In particular, the Walker cell weakens during an El Niño year, but it is strengthened in an El Niña year. So, for example, during an El Niño event, the displacement of the west-Pacific warm-pool to the mid-Pacific triggers the appearance of a double-cell, with an ascending component in the central Pacific (see [13]). In order to provide a mathematical description of these various phenomena, we must introduce some appropriate simplifications, aimed at producing a suitable set of (correctly asymptotic) equations that represent the type of flow field which supports the Walker cell in the neighbourhood of the Equator. Thus we set θ = 0 and assume no dependence on θ , together with no motion in the meridional direction, so v 0 ≡ 0; this describes flow in the (φ, z)-plane at the Equator (and we interpret the solution as being appropriate to a neighbourhood of θ = 0); see Equation (14) is identically satisfied, Eq. (15) is now and (16) simplifies to give The general approach that we adopt here is that developed in [6]. So, rather than input the heat sources that drive the motion-the obvious manoeuvre based on the physical nature of the problem-we aim to input a suitable temperature profile (as we have already done at O(1); see (12)), deduce the associated velocity field and then we may identify the heat sources required to drive and maintain the motion. This is the best way to proceed, we argue, and on two counts: (1) the precise nature of the heat sources, and how to model them, are notoriously difficult problems (see [7]); (2) the temperature field throughout the atmosphere is well known, using ground-based and satellite measurements (see [8]). So following this philosophy, and the more general development given in [6] (where all the details appear), we write which leads to where which allows us to find where G is an arbitrary function. Using (25) in (23), and evaluating on z = 0, yields and so from (19) we obtain with (from (21)) These two equations are our main results, defining the two-dimensional velocity field, given the pressure and temperature gradients; when coupled with (18), which simplifies to we are then able to identify the heat sources associated with this motion. We note, in particular, that Eq. (26) shows that the horizontal component of the velocity field is driven by the pressure gradient in the azimuthal direction evaluated on z = 0, and by the corresponding temperature gradient, but this also possesses a suitable z-structure. These observations are consistent with the accepted mechanism for the maintenance of the Walker cell (see [8,11]), but our version is quite explicit in providing the details of this forcing. These two gradients which drive the motion are, in our formulation, regarded as given forcing terms, independently assigned. A study of the oceans, however, suggests that the surface wind-stress generates upwelling, bringing colder water to the surface in the East, and so we have cooler water to the East, and warmer to the West, thereby producing the temperature gradient at the surface. A model for this upwelling, in the presence of the westward surface flow and the Equatorial Undercurrent, can be found in [4]. All the above relates to that region of the flow where the eddy viscosity plays a ro "le, most particularly in the lower regions of the troposphere. The upper regions, which are traditionally regarded as inviscid, we treat by using a suitable model for the eddy viscosity. Some of the evidence-see [15,16] for example-indicates that the eddy viscosity decays rapidly in the upper reaches of the atmosphere, and is virtually zero above about 2 km, although many alternative models for the viscosity appear in the literature; see the overview included in [5]. In our formulation, in conjunction with a variable viscosity, we impose the speed in the azimuthal direction at the bottom of the cell, u 0B (φ) (which is zero if the no-slip condition is invoked). Thus the twodimensional velocity field describing the Walker circulation is given by the solution of (26) and (27) which satisfies Note that the solution that we seek is bounded below by the ocean's surface and above by the tropopause (z = z 0 = 1) or a boundary close to this (z = z 0 < 1), and so we must satisfy w 0 = 0 on these two boundaries, and satisfying this condition at the top fixes u 0T (φ).
example, the main purpose here is to emphasise the choices that are available. This will indicate, in particular, how it is possible to introduce suitable adjustments and additions based on observational data, opening the door to further investigations. As we mentioned earlier, we would, in the best of all possible worlds, aim to input the heat sources, determine the temperature field and then obtain the associated velocity field. This, however, is virtually impossible because neither the background knowledge nor the specific detailed data are available that would enable us to proceed. So we opt for a development based on, in principle, a choice for the temperature field as the starting point. Given this (i.e. T 1 (φ, z) or τ 1 (φ, z)) we obtain the velocity field directly (by integration) and then we may identify the heat sources. But even this sequence is not completely straightforward or useful, because it is far from clear what precise form of perturbation temperature profile, T 1 (φ, z), will generate the flow associated with a cell. Rather, it is better to guide the choice of T 1 (φ, z) (or τ 1 (φ, z)) by noting the type of velocity field that we need in order to recover a Walker cell. In addition, we must also choose a model for the vertical behaviour of the dynamic eddy viscosity (i.e. m(z)). It is reasonable to adopt this approach since our overall aim is to show that suitable solutions do exist, which provide a description of the Walker cell, and that will also enable us to include the adjustments needed to recover the cell structure that is observed in El Niño years. Equation (26) describes, in some detail, the vertical structure of the solution and this is to be consistent with the existence of a cell. However, the horizontal structure must be imposed in order to generate cells, the physical property that guides this being the obvious one: the Pacific Ocean is bounded by land masses to the East and to the West. The simplest choice which describes the required property is to set where is a constant and the cell sits in φ 0 < φ < φ 1 , expressed in degrees. (Clearly many other choices are possible, but we will limit our discussion here to just this one: we are aiming to confirm the existence of suitable solutions.) The horizontal velocity component then takes the form and so Eq. (26) becomes Before we proceed with a more general analysis of Eq. (32), we carry out a simple check to see if this formulation captures some of the important features of the cell structure: we choose T (z) = TT 2 0 (z), with T constant, take m = constant = 1 and which satisfies the no-slip condition on z = 0, and also U (z m ) = 0 where we take 0 < z m < z 0 ≤ 1, and α > z 0 with α + z m = −3ˆ T, the choice of α fixing the speed at the top of the cell; an example of this profile is shown in Fig. 3. Necessarilŷ < 0 for T > 0, this latter condition ensuring that the flow is westwards low down in the troposphere and eastwards higher up. Thus, from (30), we see that.
which correspond to the observed properties of the Walker circulation: the pressure at the bottom of the troposphere is higher to the East, and the temperature is higher to the West. However, this simplistic observation ignores many of the detailed ingredients that make up Eq. (32), most notably the ro "le of a variable viscosity. We now take this initial examination a little further. The perturbation-temperature profile used in the preceding calculation, with T constant, is not likely to be relevant to any realistic description of the Walker circulation, although the general form of the velocity profile is what we expect (when we use the no-slip condition). To proceed, and in order to make more transparent the important properties of this flow, we now use a simplified velocity profile which excludes the no-slip condition at the bottom and therefore admits a wind blowing directly over the surface of the ocean. The simplest such profile is linear: where U 0 > 0 and 0 < z m < z 0 are constants. The viscosity that we work with, and the one for which most of the data is available (as mentioned earlier), is the kinematic eddy viscosity; thus we introduce and then we shall make choices for n(z). Equation (32) now becomes and an important observation follows directly. Evaluation on z = 0 gives and so a model for the viscosity in which n(0) = 0 and dn dz(0) > 0 (see [15], for example) gives < 0, which recovers the result ∂ p 1 ∂φ > 0, consistent with the observations. On the other hand, if n(z) = constant (> 0), then > 0; if n(0) > 0 and dn dz(0) < 0, then again > 0. We conclude that the choice of model for eddy viscosity has a profound effect on the underlying pressure gradient, even though the airflow near the surface of the ocean is always westwards in our formulation. But of course a critical feature of any solution that describes the Walker circulation is the temperature variation and the resulting gradient in the azimuthal direction.
The construction of the temperature perturbation, T 1 (z) (see (30)), is obtained directly when we choose the velocity profile, U (z) (see (31)), together with the model for the kinematic eddy viscosity. In this first stage of the investigation, we have used n(z) = 1, n(z) = e −νz , n(z) = νze 1−νz , where ν > 0 is a constant, these describing a constant viscosity, exponentially decreasing viscosity, and a viscosity which increases from zero followed by exponential Table 1 The results of the calculations using the cubic and the linear velocity profiles, and the choice of viscosity model: constant, exponentially decreasing, increasing followed by exponentially decreasing. The signs of the gradients in the azimuthal direction on z = 0 are listed Profile → cubic cubic linear linear respectively. In each model, we have set the maximum value of n to be 1, which is consistent with a suitable choice of the non-dimensionalisation based on μ .
We have chosen two velocity profiles for the calculations: the cubic polynomial drawn in Fig. 3 (and see (34)) and the linear model profile in (35). In both cases we have a flow which is westwards in the lower region of the troposphere and eastwards higher up. We tabulate (see the Table 1) the resulting signs of ∂ p 1 ∂φ and ∂ T 1 ∂φ on z = 0 and then, to correspond to the observed properties of the Walker cell, we should expect to use (as mentioned earlier) (The properties listed in the Table 1 are the same for all ν > 0.) If we use the conditions in (38) as the guiding principle for selecting suitable solutions, then the Table shows that we may use either a constant viscosity or a decreasing viscosity, in conjunction with the no-slip condition at the surface of the ocean, or the increasing-decreasing viscosity for the linear profile. In the light of these observations, we now examine in detail a simple profile which accommodates both a linear variation and a no-slip condition (but we produce the results for only one of these).
First, we normalise Eq. (32) by writing , which takes a maximum value of 1 on z = 0 (consistent with our non-dimensionalisation). Thus we obtain where the prime denotes the derivative with respect to z. We see directly, by evaluating (40) on z = 0, that which determines P, and hence the pressure gradient in the azimuthal direction at the surface of the ocean. Here, we choose to work with the simple velocity profile where z 1 , z m , β and γ are constants. This profile is linear if βγ = −1 and it satisfies the no-slip condition at the ocean's surface if z 1 = 0; two examples are shown in Fig. 4 where the upper boundary of the flow is fixed at the tropopause (z = 1). Although we investigated the effects of a number of different profiles based on (42), of all the choices that we might make, we opt for a model which describes a wind that blows (westwards) over the ocean, with a profile which is linear. This choice is the one which, in a direct and natural way, captures the important properties of the wind structure in the atmosphere; other profiles are accessible by suitably choosing the values of z 1 , z m , β and γ . Further, we invoke the most reasonable model for the kinematic eddy viscosity: where ν > 0 is a constant. The particular velocity profile that we use for the calculations is shown Fig. 5, and we set ν = 10 throughout. We find that P ≈ −1 · 568 associated temperature perturbation, D(z), is depicted in Fig. 6 and, because D(0) > 0, we see that ∂ T 1 ∂φ < 0 on z = 0. Furthermore, the temperature profile shows that the perturbation temperature decreases rapidly at higher altitude. This is the type Fig. 4 Two examples of the profile given in (40); red curve: where ψ(φ, z) is the stream function for the flow; thus we have which ensures that w 0 = 0 on z = 0. However, in addition, we must choose the various parameters that describe the velocity profile, (42), so that we also satisfy w 0 = 0 on z = 1: the flow is bounded below by the surface of the ocean and above by the tropopause. The choices given above (see Fig. 5) satisfy this requirement. We are now able to produce the streamline pattern for this Walker cell, defined by lines ψ = constant, which is shown in Fig. 7 and plotted for the flow along the streamlines is in the clockwise direction when viewed northward. Finally, we may use our solution to provide a representation of the special flow configuration which arises during an El Niño event. This is most easily accomplished by simply adjusting the periodicity in the azimuthal direction, so plotting the streamlines associated with.
gives the streamline pattern shown in Fig. 8 (again, viewed northward). This solution can be analysed to extract the properties associated with this flow, such as the pressure and temperature distributions that are required to maintain this structure. The details of the solution that we have obtained can now be used in the version of the first law of thermodynamics which is appropriate at this order, namely Eq. (28). We have obtained, explicitly, the background temperature, T 0 (z), and its perturbation, T 1 (φ, z), numerically; we can then determine both ρ 1 (φ, z) and p 1 (φ, z). All this enables us to find an expression for the heat sources/sinks, Q 1 , that are required to Fig. 8 The streamline pattern that corresponds to the flow during an El Niño event; the flow direction in the right-hand cell is clockwise, and anticlockwise in the left-hand cell. Thus there is an ascending flow in the central Pacific maintain the Walker cell. In particular we note that there is a contribution to the background (i.e. solar) heating from the terms.
and the other terms: are heat sources that move with the fluid and so are associated with latent heat. Although we have not produced the (numerical) details here, they are readily available and, we suggest, are worth exploring if reliable data can be used to produce further properties of the Walker circulation.

Discussion
The development presented here is based on the Navier-Stokes equation for a compressible fluid, with variable viscosity, coupled to an equation of state and a suitable version of the first law of thermodynamics. These general governing equations have been non-dimensionalised and then an asymptotic solution is constructed which uses only the thin-shell approximation to describe the atmosphere; all other parameters are held fixed in the limiting process. Although the details are not developed here-they are available in [6]-we have presented the main results to aid the reader. These comprise the equations describing the background state of the atmosphere, together with its perturbation which incorporates all the dynamics and thermodynamics of the steady atmosphere. We have chosen the background state to be that which exists independently of the underlying velocity field; the issue then is to construct suitable solutions which describe the superimposed steady motion. The particular exercise undertaken here is to find a solution which represents and describes the Walker circulation that sits over the Pacific Ocean and along the Equator. The main drivers for this motion are well known; here, we aim to provide a careful mathematical treatment of this phenomenon. This can then be used to investigate, in detail, various properties of the flow and how it might change according to the ambient conditions. The resulting formulation, Eqs. (26) and (27) with (28), is the main theoretical conclusions of the work. In particular, we can be precise about the mechanisms that drive the motion in the cell: the pressure gradient in the azimuthal direction at the surface and a corresponding temperature gradient. One immediate consequence of our development is the form taken by this temperature-gradient term; this involves a combination of both the background temperature and the gradient of the perturbation temperature, via an integral-possibly not an obvious combination. Furthermore, this element of the forcing vanishes at the ocean's surface, leaving only the pressure gradient there. The nett result is to produce a simple, very specific differential system which couples the velocity field to the temperature and pressure gradients in the azimuthal direction, together with the opportunity to include any suitable, variable viscosity.
An initial investigation, based on cubic and linear velocity profiles, for various models for the kinematic eddy viscosity, was undertaken. This showed that a realistic azimuthal velocity profile-westward lower down and eastward higher up-could be generated for other than the observed signs of the temperature and pressure gradients. Nevertheless, we used these observations to guide a more comprehensive examination of a solution which used a simple but general velocity profile, which could accommodate both a linear variation and also a quadratic variant, allowing for a no-slip condition at the surface. The choice which most closely accords with the existence of trade winds (crucial to early sailors) is a flow which does not satisfy the (classical and technically correct) no-slip condition at the surface. (Of course, we would certainly allow a non-zero speed at the surface if we are to model wind-driven waves.) Thus we have used parameter values that produce a westward flow down to the surface of the ocean and an eastward flow at higher altitude. In addition, the vertical velocity component, w 0 , must satisfy the requirement that the flow is bounded by the ocean surface below and the tropopause (or something close to this) above. Although we can always impose w 0 = 0 on z = 0, for any horizontal velocity profile, we must choose this profile with care so that we also have w 0 = 0 on z = 1 in order to satisfy mass-flow conservation. This describes the vertical structure of the solution; the horizontal structure required to produce a cell was imposed by invoking a simple trigonometric function that gave u 0 = 0 at the furthest extremities of the cell. With all these ingredients in place, a solution was computed (using Maple) which produced a vertical perturbation-temperature profile through the depth of the troposphere and a streamline pattern for the flow in the Walker cell.
This temperature profile, shown in Fig. 6, is quite specific for this solution; on the other hand, given this profile and the maximum speeds at the top and bottom of the cell, the solution for U(z) can be recovered. The speed at the top, we note, is fixed by ensuring that w 0 = 0 along the top of the cell. So the details that we have presented confirm the existence of a suitable solution-and this was the main aim of the work-and, more significantly, the formulation provides an opportunity for further detailed investigation. Thus any number of choices for the variable viscosity, the surface pressure gradient and the temperature profile can be made, and solutions representing the Walker cell constructed; in addition, adjustments to the velocity profile can also be made and the consequences investigated. (We also record that the maximum value attained by the temperature perturbation in our calculations-which always occurs on z = 0-is significantly affected by the choice of ν in the viscosity model.) All this is particularly useful if reliable data are available to guide the choice of viscosity model, the temperature profile and the horizontal structure of the cell. Furthermore, the effects of changes in the ambient conditions (perhaps driven by climate change) can be tested using this system of equations. With this in mind, we introduced a simple modification which models the situation that arises during an El Niño event: we have seen that two cells are easily accommodated, mirroring what happens when the region of warmer water penetrates further to the East.
There is no virtue in including an examination of the El Niña, because this is simply an enhancement-larger gradients and higher wind speeds-of the Walker circulation that we have described. However, there is one final observation that we can make which suggests, albeit in a numerical sense, that we have captured some important attributes of the Walker circulation. Using the values obtained in our numerical example, together with the parameter values and non-dimensionalisation given earlier, taking R e = 5 × 10 5 and assuming that the Walker cell extends over 80°of longitude, we find that a surface wind speed of 5 ms −1 produces a temperature change, εT 1 ( R ) 2 , along the extent of the cell at the ocean's surface of about 5°C. Thus our leading term in the perturbation of the background state produces a result that is altogether reasonable, although there is clearly no certainty about the values that should be used in this rudimentary calculation. The higher-order terms are proportional to ε n , n = 2, 3, … and, since there is no suggestion of non-uniformities in the asymptotic expansions, we appear to have captured the main contributor to the temperature perturbation which drives the Walker circulation.
In conclusion, we have shown that a careful (asymptotic) approach to the general, governing equations that represent the atmosphere have led to a simplified system of equations. These produce a perturbation of a background state, the perturbation combining all the dynamics and thermodynamics of the steady atmosphere. Furthermore, these equations can be used to give a detailed description of the Walker circulation, providing a simple test-bed for the effects of pressure gradient, temperature profile, velocity profile and variable eddy viscosity to be investigated. Associated with this general flow structure, the disruption caused by El Niño and El Niña events can also be examined. There is much, we submit, that can be explored using the equations presented here, particularly if extensive and reliable data are available.
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