Earth system models and mathematical remarks

Computer Methods in Applied Mechanics and Engineering 89 (1991) l-9 North-Holland

Earth system models and mathematical

remarks

J.L. Lions Colkge

de France,

3, rue d’Ulm,

7.5005 Paris,

France, and C.N.E.S.,

France

Received 25 October 1990

1. Introduqtion Already in 1824, Fourier raised the question which is now known as the ‘global change’ question. Indeed, in his paper [l] he wrote: “The development and the progress of Human Societies, the action of natural forces can significantly change in large areas the composition of the soil surface, the distribution of water and the main flows of the atmosphere. Such modifications could modify, in the perspective of several centuries, the mean temperature!” During more than one century, it was impossible to go very far along the lines of this question raised by Fourier, by lack of data and by lack of adequate computing facilities. A large amount of data - in particular showing the increase of CO, in the atmosphere - has been obtained now for more than 30 years and has convinced the best experts in several fields, from atmospheric sciences to biological sciences, that indeed Fourier was right in m&sing the question. Moreover they have been convinced by data and by numerical simulations that the problem is global. Many phenomena react on one another, by mechanisms called ‘feedbacks’. The most famous of these feedbacks is due to Charney: the desert feeds back itself (although the more simulations are getting precise, the more this feedbacks raises open questions). It is with this global aspect that ICSU (International Council of Scientific Unions) started in 1986 a program called IGBP (International Geosphere Biosphere Program) which is now running in parallel and in collaboration with WMO (World Meteorological Organization). These are global programs. They eventually rest on two key points: (i) obtain mathematical models; (ii) exploit these mathematical models with the help of super computers (computers being the second essential tool which did not exist at the time of Fourier). Of course we do not, by any means, underestimate the difficulties which are met in order to obtain adequate models. World wide efforts are devoted to these questions, and, in particular along the following lines: (i) obtain more data - and, among other things, obtain data by using dedicated satellites, in order to study the energetic balance of the tropical system, or the ice caps, etc.; (ii) make these data accessible to research laboratories (and, at the same time, to see to it that these data are properly calibrated); (iii) study the feedbacks which connect the ‘subsystems’ between themselves - this being one of the very fundamental questions.

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J. L. Lions,

Earth system models

and mathematical

remarks

The point of view which will be taken here is the following: assuming that models do exist and are available, what can be done with them? Is a ‘brute force’ approach, using ever more powerful supercomputers, going to be sufficient (it is obvious that super computers are necessary here)? Before we proceed, let us make an explicit assumption - or an article of faith . . . - which is implicit in most of the publications devoted to these questions: There exist models of planet earth, or of some of its subsystems, which consist of sets of non-linear partial differential equations of dissipative type. (1.1) A few remarks are in order. 1.1. We speak of models and not of a model. Not only will models depend on the subsystem we particularly want to study, but these models will depend on the time horizon where the problem is considered.

REMARK

1.2. There is an apparent contradiction when referring to ‘subsystems’ in (1.1) and in Remark 1.l, since indeed the IGPB (global change program) is based on the recognition that ‘all’ subsystems are connected. . . . We will return to that in Section 5.

REMARK

1.3. Actually the non-linear PDE’s (partial differential equations) referred to in (1.1) can also contain delays (hysteresis) making the problem non-local and raising the most preoccupying questions, cf. [2].

REMARK

REMARK attractor,

1.4. If (1.1) is admitted, then it follows - at least formally’ -that there exists an i.e. a subset J& of the (infinite dimensional) phase space such that & is ‘much smaller’ than the whole phase space and could even be of finite (l-2) dimension (Hausdorff or fractal dimension); all solutions of the model(s) converge towards ti as the time t goes to +m . (l-3)

This is the so called climatic attractor. But the mere existence of a finite dimensional climatic attractor is somewhat controversial, without speaking of its dimension. One can consult [4, 5]‘*. 1.5. The structure of some of the main ‘building blocks’ of the models referred to in (1.1) is well-known. This is the case for the atmosphere and for the oceans. This remark does not mean that the problem is simple! But the main structure of all the models is based on the equations of fluid mechanics and on various diffusion equations. Most of the problems considered in applied mathematics are met in the global change models!

REMARK

I For situations where this is actually proven, cf. [3], 2nd edition, and the references therein. I* (Added in proof.) Cf. in this direction [ll].

J.L.

Lions,

Earth system models and mathematical

3

remarks

We would like to proceed with some brief remarks concerning specific problems, which are not met (or rarely considered) in the ‘classical’ applied mathematics questions. The topics we want to consider in this respect are systems with incomplete data (Section 2), sentinels (Section 3), connections with controllability (Section 4) and “Are there subsystems?” (Section 5). 2. Systems with incomplete data Think of any possible model concerning meteorology or oceanography. We deal with a system of non linear PDEs. If Y = {Yl,. . * 7YNI

(2.1)

denotes the state of the system, the equations can be written in the form aylat

+ A(y)

=f

,

(2.2)

where A is a system of non-linear partial differential operators and where f denotes the source terms. In (2.2) the space variable x spans a set 0. Depending on the situations fl can be a 2-dimensional sphere, or a subset of it, or an open set in R3 (for instance any time we cannot use shallow water theory). To (2.2) one should add boundary conditions (except in the case when 0 is a sphere), that we assume known (which is of course not always the case!) and we should add initial conditions, i.e., y(x, 0) = given function y”(x) , x E R ,

Y0 =

{Y:,

* * * , y;>

.

(2.3)

But y0 is not known completely, once the ‘initial time’ t = 0 has been chosen. We do have that we can write in the form

informations,

YO=L

K = given subset of the phase space .

(2.4)

But we have also many other informations, obtained at different times. We can write these in the form (2.5) where pi refers to a measurement and where kj is the result of this measurement. In order to proceed with a computation (theoretical and numerical) of the solution2 of (2.2), we should know y”, or, at least, the ‘best possible’ element y” in K such that (2.5) holds true. ’ Actually there are more and more very powerful tools to prove existence of (weak) solutions of more and more classesof nonlinear PDEs, cf. [6, 71 and the references therein. But progress is much slower as far as uniqueness is concerned. As an example, we recall that the question of uniqueness of weak (global) solutions of Navier-Stokes equations in 3 space dimensions is open [g].

4

J. L. Lions,

Earth system models and mathematical

remarks

The 1st idea goes back to Gauss and Legendre. We set Ye7 0) = se4 *

(2.6)

Given 5 let y( 5) be the solution of (2.2), (2.6) (and subject to the appropriate boundary conditions). Then we are looking for 5 such that 5 E K and such that 5 ol,[(Y(OY j=l

kj) - kj12

(2.7)

is minimized (over 5). In (2.7) the weights CX~ > 0 can be chosen according to various rules. All this leads to classical conditions for 5, involving the adjoint equations (or the Lagrange multipliers). But we face here a second dificulty. The solution y( 6) may be very sensitive to 5 as observed by Lorenz [9] in his classical paper. The idea is then to divide y in 2 components, a possibly rapidly oscillating part that we want to drop, and a slowly varying part of the state, which is the most important for our object. Let us set Y = Y, + Ys .

(2-8)

We then proceed by (essentially) using the same method as in classical least square methods but with the additional step of a ‘projection’ on the SZOWvariety (‘the’ variety where y, lies). A variant of this approach has been recently introduced in the mathematical literature under the name of inertial varieties, cf. [lo] and the references therein. The notion of slow manifold for climatology problems has been introduced in [ll]. That the two approaches (i.e. slow varieties and inertial varieties) are intimately connected has been conjectured in [2] and will be proven in [12]. The ‘projection’ on ‘the’ inertial variety can be made, from a practical view point, in very many different ways. When least squares methods and some kind of projection on a slow variety are used simultaneously, the corresponding method is called an assimilation method. For a recent quite exhaustive overview of work being conducted in these directions, we refer to [13]. 2.1. Another way of saying things, avoiding technicalities, is that in assimilation methods one tries to use the state equations and other scattered informations in order to compute the missing data. REMARK

We now proceed by examining another possibility. 3. Sentinels Let us consider again the state equations (2.2) and let us assume now that y(x, 0) is ‘given’ in the following form:

.I. L. Lions, Earth system models and mathematical remarks

y(x, 0) belongs to the ball (in a suitable Hilbertian space) of center y” and radius T, where T is ‘small’ .

5

(3-I)

We assume that the boundary conditions are known. Contrary to what has been said in Section 2, we do not want now to ‘compute the missing data’. We rather address the following question: what are the functionals, or generalized averages, which can be computed and which are not sensitive to the uncertainty following from (3.1)? Sentinels (as introduced in [14, 151) are (hopefully) giving a (very partial) preliminary answer to this question. Let w be a (possibly small) open subset of fl and let ho be a given function in L2(w X (0, 2)). We consider here the scalar case, i.e. N = 1 in (2.2), only in order to simplify the exposition. The method explained here is completely general. Let us set y(x, 0) = y" + TjO )

(3.2)

where E” belongs to the unit ball of, say, L2(0). Let y(x, t; T) = (2.2) subject to (3.2) and to the given boundary conditions. We introduce w E L2(o X (0, T)) to be defined later on

Y(T)

denote ‘the’ solution of

(3.3)

and we define (ho +

W))‘(T)

dx dt .

If w = 0 and ho Z=0, JJoX(o,r) ho dx dt = 1, then modify this process in such a way that

(3.4) S(T)

is an averaging process. We want to

; s(T)(,=o= 0,

(3.5)

i.e. the functional (3.4) is not sensitive, to the 1st order, to changes in the initial conditions. We also want S(T) to be as close as possible to the operation ~~ox(o,r)hoy(~) dx dt, i.e., we want = minimum bll L2(OX(0,T))

(3.6)

among all possible choices of w’s such that (3.4) holds true. Of course we also want that

Actually this last condition generically follows from (3.9, (3.6). If all conditions above are satisfied, one says that S(T) is a sentinel.

6

J. L. Lions,

Earth system models

and mathematical

remarks

3.1. The terminology ‘sentinel’ is explained by the fact (cf. [14]) that one sentinel can detect some perturbations. One needs an infinite number of sentinels to detect ‘all’ possible perturbations (cf. [El), an interesting fact being that in the (parabolic) dissipative cases, these complete sets of sentinels can be constructed on the same set w. We can think of o as being an observatory. With a complete set of sentinels, there are no stealthy perturbations. REMARK

3.2. Sentinels can be computed (cf. [16]) if the solution of (2.2) subject to the given boundary conditions and to y(x, 0) = y” can be computed. One could replace here y by its slow component y,, but theory is lacking and no computational experiments have been conducted in this respect.

REMARK

3.3. Applications of sentinents are presented in [17-191 for situations (such as those arising in river basins) where the difficulty hinted at in Remark 3.2 does not arise.

REMARK

4. Connections with controllability Are we experimenting irreversible changes in the climate of planet earth? Taken from the very restrictive and very particular view point of a mathematician, a (slightly) related question is as follows: let us again consider (2.2) and let us assume that the boundary conditions are known and that y(x, 0) is given, ye,

(4-l)

0) = Y0 .

Under the actions off, which contain ‘natural forces’ and also ‘human actions’, the system will be, at the horizon time T, in a state yk

T)

=

Y’

.

(4.2)

Let us assume that we consider y’ as highly non-desirable and that we would like to return as soon as possible at the initial state y” or, even, that we would like to reach at time S (> T) an ‘ideal’ state y*. 3 Let us make this question more precise. We consider now aylat+A(y)=f+v,

(4.3)

where u is a control variable, subject to some constraints such as: u is concentrated in given regions, etc. (4.4) 3 Of course this is an oversimplified and an entirely unrealistic situation, since there are obvious conflicts of interest between nations and even between regions in a nation as far as an ‘ideal’ climate is concerned. But even so, we are lead to apparently interesting and open questions.

J. L. Lions,

Earth

system models and mathematical

7

remarks

We are given two states z” and z’. We solve (4.4) with the initial conditions

W)

y(0) = z" .

We are given a time horizon to (to = S - T). Is it possible

to find

u subject

to the constraints

y(to) is as close as we wish to z’?

such that

(4.6)

This question is directly connected with the following: are there connections between controllability and chaos? We conjecture (cf. [20] for some preliminary indications in favor of this conjecture) that, at least generally, the answer to (4.6) is ‘yes’. A number of recent results (unpublished) point also towards a positive answer. We refer to [21-241. Of course we emphasize once again that this is a seemingly very interesting research topic but one that is at a very early stage.

5. Are there subsystems? It is quite clear that actions (i.e. implementation of controls in the system sciences terminology) are needed ZocaZZy to prevent hopefully undesirable effects which are already observed. Water pollution in closed seas, river basins, estuaries, lakes, etc. offers very many cases. Other actions are also needed in order to prevent (hopefully) global undesirable effects - such as the increase of the CO, proportion in the atmosphere (actually decisions have already been taken along these lines). But, of course, decisions will be implemented in a local fashion and therefore they will be connected with subsystems. But the ‘global change’ program is (essentially) based on the observation that subsystems do not exist, that ‘all’ phenomena are connected. To take into account on a global scale of ‘all’ the local actions is known as parameterization theory. It is connected with very many interesting mathematical questions related to the large number of very different time scales and spatial scales which come into play in these problems. This does not change the fact that actions are taken locally. This means that here the subsystem is defined geographically. More precisely, we are given a subset 0 of 0 and in 0 we consider a state equation azlat + B(Z) = g ,

(5.1)

where z = {z,, . . . , zNS} denotes the state we are interested in, in the region 0 C a, where B is a system of non-linear partial differential operators and where g denotes the source and therefore also contains the control variables. As an example 6 can be an estuary or a harbour. The boundary a6’ of 0 consists in ‘natural’ parts r. - such as the interface liquid/solid - and also of artificial parts such as, in the above example, the ‘open sea’ r, C do.

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J. L. Lions,

Earth

system models and mathematical

remarks

Boundary conditions are generally known on r, but they are generally not known on J”. If z(x, 0) is known, the problem reduces now to finding appropriate boundary conditions orz r,*

There are 3 methods (and combinations of those!).

METHOD 1. One uses a ‘global’ model to compute y in 0. One then deduces z on r, from the numerical results obtained for y. 2. One thinks of the values of z on r, x (0, T) as control variables and one these values by least square methods, assuming of course that we have other information. METHOD computes

3. Transparent boundary conditions can be used. One looks for approximate differential relations satisfied by y on r,.

METHOD

One deduces for z somewhat similar relations and we think of them as boundary conditions. This method has been initiated by [25] and has given rise to a vast literature (cf. [26] and the references therein).

6. Conclusion It goes without saying that not only have we scratched the surface of the problems considered here, but also have we not touched other very interesting questions such as: inverse problems, boundary layers, bifurcations (are abrupt climatic changes of the past some kind of bifurcations), structure of the attractors (and of the ‘blocking’ phenomena in meteorology), etc. We hope to have convinced the reader that a mathematical approach to these questions of obvious general interest can have its own merit.

References [l] J. Fourier, Rapport sur la temperature du globe terrestre et sur les espaces planttaires, Mtmoires de 1’AcadCmie Royale des Sciences de I’Institut de France t. VII (1824) 590-604. [2] J.L. Lions, La Plantte Terre. Lectures at the Instituto d’Espana. (1990), Publication by the Spanish Academy (1991) (in Spanish, translated by I. Diaz et M. Artola). [3] R. Temam, Infinite dimensional Dynamical Systems in Mechanics and Physics (Springer, Berlin, 1988). [4] P. Grassberger, Do climatic attractors exist?, Nature 323 (1986) 609-612. [S] 0. Nicolis and G. Nicolis, Is there a climatic attractor?, Nature 311 (1984) 529-532. [6] P.L. Lions, Lecture at International Congress of Mathematicians, Kyoto (1990). [7] L. Tartar, Lecture at International Congress of Mathematicians, Kyoto (1990). [8] J. Leray, J.M.P.A. 12 (1933) l-82. [9] E.N. Lorenz, Deterministic non periodic flow, J. Atmospheric Sci. 20 (1963) 130-141. lo] C. Foias, O.P. Manley and R. Temam, Non Linear Anal., TMA 11 (1987) 939-967. 111 J.L. Lions, R. Temam and S. Wang, Attractors for the primitive equations of atmosphere, to appear.

J.L. Lions, Earth system models and mathematical remarks

9

[12] R. Temam, Lectures at the ESCURIAL Summer Course, August 1991. [13] W.M.O., World Meteorological Organization, International Symposium on Assimilation of observations in Meteorology and Oceanography, Clermont Ferrand (1990). [14] J.L. Lions, Sur les sentinelles des systtmes distribds. C.R.A.S. 307 (1988) 819-823, 865-870. [15] J.L. Lions, Sentinelles et furtivite. C.R.A.S. (1990). [16] J.L. Lions, Lectures at the ESCURIAC Summer Course, August 1991. [17] 0. Bodart and J.P. Kernevez, to appear. [18] J.P. Kernevez, F.X. Le Dimet and A. Pave, to appear. [19] J. Grassman, to appear. [20] J.L. Lions, Are there connections between turbulence and controllability?, INRIA Symposium, Antibes (1990). [21] F. Colonius and W. Kliemann, Some aspects of control systems as dynamical systems, to appear. [22] D. Gabay, to appear. [23] J.P. Kernevez and R. Lute, to appear. [24] E.D. Sontag and H.J. Sussman, to appear. [25] B. Enquist and A. Majda, Radiation boundary conditions for acoustic and elastic wave calculations, C.P.A.M. xxx11 (1979) 313-357. [26] L. Halpern and M. Schatzman, Artificial boundary conditions for incompressible viscous flows. SIAM J. Math. Anal. 20 (1989) 308-353.