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Physical Interpretation of the Attractor Dimension for the Primitive Equations of Atmospheric Circulation J. L. LIONS,* O. P. MANLEY,1 R. TEMAM,#,@
AND
S. WANG#
*College de France, Paris, France 1
Department of Energy, Washington, D.C.
Department of Mathematics and Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, Indiana
#
Laboratoire d’Analyse Nume´rique, Universite´ Paris-Sud, Orsay, France
@
(Manuscript received 18 April 1995, in final form 8 October 1996) ABSTRACT In a series of recent papers, some of the authors have addressed with mathematical rigor some aspects of the primitive equations governing the large-scale atmospheric motion. Among other results, they derived without evaluating it an expression for the dimension of the attractor for those equations. It is known that the long-term behavior of the motion and states of the atmosphere can be described by the global attractor. Namely, starting with a given initial value, the solution will tend to the attractor as t goes to infinity. The dimension estimate of the global attractor is evaluated in this article, showing that this global attractor possesses a finite but large number of degrees of freedom. Using some arguments based on the known physical dissipation mechanisms, the bound on the dimension of the attractor in terms of the observable quantities governing the heating and energy dissipation accompanying the motion of the atmosphere is made immediately transparent. Consequently, to the extent that the resolution needed in numerical simulations of the long-term atmospheric motion is related to the dimension of the attractor, the result in this article suggests that the required resolution is quite sensitive to the magnitude of the effective (or eddy) viscosity, while it appears to be less sensitive to the details of the way that the atmosphere is heated.
1. Introduction Following the pioneering works by Bjerknes (1904), Richardson (1922), and Charney et al. (1950), it is recognized that the prediction of the weather and climate essentially amounts to solving an initial and boundary value problem for the partial differential equations governing the motion and states of the atmosphere. From both the mathematical and physical points of view, according to von Neumann (1960), the motion of the atmosphere can be divided into three categories depending on the timescale of the prediction. They are motions corresponding respectively to the short-time, long-term, and medium-range behavior of the atmosphere. For long-term weather prediction and climate, the motion and states of the atmosphere can be described by the global attractor. Namely, starting with a given initial value, the solution will tend to the attractor as t goes to infinity. The dimension estimates of the global at-
Corresponding author address: Roger Temam, Institute for Scientific Computing and Applied Mathematics, Indiana University, Rawles Hall, 618 East Third Street, Bloomington, IN 47405-5701. E-mail:
[email protected]
q 1997 American Meteorological Society
tractor presented in this note show that this global attractor possesses only a finite number of degrees of freedom. In other words, the long-term weather and climate are essentially determined by a finite number of parameters. The several more or less sophisticated attempts to derive low-dimensional climate attractors by analyzing the available time series, summarized by Ghil et al. (1991), have led to some debatable results. At about the same time, prompted by the discovery of the Lorenz attractor (Lorenz 1963a), a number of relatively primitive models with finite but not necessarily low numbers of degrees of freedom have been studied. Typically, they have aimed principally at mimicking selected aspects of atmospheric circulation, for example, blocking and variations in atmospheric predictability (Lorenz 1963b; Charney and De Vore 1979; Legras and Ghil 1985). For the admittedly successful purpose of modeling the selected phenomena, the model systems were assumed to be driven by a prescribed mechanical forcing related to the Coriolis force. That leaves unclear how the number of degrees of freedom constituting the quasigeostrophic model system is related to that in the underlying, more complex primitive equations (PEs). In addition, given the experience with the truncation (or resolution) de-
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pendent qualitative behavior of the solutions of convectively driven flows (Treve and Manley 1982; Curry et al. 1984), it is not clear how stable with respect to additional modes the computed features are in such models. These questions suggest that as a point of departure one ought to determine the number of degrees of freedom, or the related dimension of the attractor associated with the underlying, closest to physical reality primitive equations. This has been carried out by Lions et al. (1992a) without relating the attractor dimension to the physical parameters governing atmospheric circulation. The present paper explores that explicit relationship. We evaluate a bound on the dimension of the attractor in terms of observable physical quantities. We will comment on the fact that this bound is very large, but at this point we remark that one possible explanation is that not all modes on this attractor are equally likely (statistically equivalent) so that, possibly, only a few of them are effective. It was mathematically shown that such a situation actually prevails for systems much simpler than the PEs considered here see (Birnir and Grauer 1994). The upper bound on the dimension of the attractor presented here indicates the sufficient number of degrees of freedom needed in any direct numerical simulation of global atmospheric circulation based solely on the primitive equations. Here sufficiency is deemed to be the condition that the time-asymptotic behavior of the simulation is independent of the chosen initial conditions and is free of any artifacts arising from using a smaller number of degrees of freedom. Instances of such misleading artifacts have been discussed in the past (Treve and Manley 1982; Curry et al. 1984). On more familiar ground, it has been shown (Constantin et al. 1985b) that the conventional estimate of the number of degrees of freedom for Navier–Stokes equations, Re9/4, is in fact such an upper bound. Here we report on our establishing the not generally known fact that the dimension of the attractor for the primitive equations is finite, even though the system is governed by a set of partial differential equations. The dimension of the attractor is indicative of the number of degrees of freedom needed to simulate the given dynamical system. We find that the bound on the attractor dimension for physically realistic atmospheric parameters is well beyond the capability of the existing or contemplated computer system. Perhaps that reflects the possibility that even for a relatively nominal Reynolds number, the primitive equations yield an enormous amount of details about the atmosphere, observable far in excess of what is actually necessary for the most important features of atmospheric circulation. The quantitative assessment of this possibility needs further study. Discouraging as that result is, it is encouraging to note that the next conventional physically based approximation beyond that of hydrostatic equilibrium, namely, the geostrophic balance constraining the full primitive equations, leads to a dramatic reduction in the bound on the attractor dimension (Wang
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1992). That constraint puts the computation just within the reach of existing or near-term computing resources. It suggests that other reasonable physically justified constraints on this system of equations may reduce the bound on the attractor dimension to manageable proportions. The method outlined here, based on more extensive mathematical developments presented elsewhere, will provide the necessary tools for calculating such modified dimension estimates. This also suggests that some simplifications of the mathematical models are indispensable. At the present stage of our knowledge we still cannot be sure to what degree the inevitable simplifying assumptions made in constructing those models are strictly justifiable. Inevitably, then, one can question the validity of the longtime numerical solutions of the relevant model equations. With the still shifting details of the key physics in climate modeling it should be comforting to know that the model equations as they stand at any given time make at least mathematical sense; that is, their solutions are stable, in the sense that they do not blow up either in finite time or asymptotically in time. Often such anomalous behavior signals some previously overlooked problem with the physics of the model. Recently, three of the authors published a series of papers addressing the mathematical properties of a class of climate models including the coupling of the atmosphere to the ocean (see Lions et al. 1992a,b; 1993). They showed that the system of the given equations can be put in a form similar to that studied extensively in the context of Navier–Stokes equations in three dimensions, namely, the abstract form du 1 Au 1 R(u) 5 f, dt where A is a linear operator, R is a nonlinear operator with quadratic nonlinearities, involving first-order derivatives and also nonlocal operators, and f is a source function driving the system. For completeness, a sketch of the derivation of the attractor dimension for this model of atmospheric motion is presented in the appendix. The abstract form has well-known properties from which the qualitative attributes of the climate model equation can be readily derived. One such property is the existence of an attractor and its finite dimension. Another useful outcome of the analysis is the secure knowledge of the functional setting of the given equations, and hence the potential for selecting the most efficient and most accurate algorithm for the numerical solution of the model equations. In this brief article, we shall discuss how the dimension of the attractor for this model climate system depends on the physical parameters. Readers less familiar with the relevant literature dealing with attractor dimensions of dynamical systems may find it useful to refer to Constantin et al. (1985a) illustrating the methodology for Navier–Stokes equations in
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two dimensions and the more extensive and rigorous treatment in Hale (1988), Ladyzhenskaya (1991), Temam (1988), and Babin and Vishik (1992). Expository material on the various kinds of dimensions, notably the fractal and Hausdorf dimensions, is treated in Mandelbrot (1977). For the sake of brevity, we do not repeat here the precise definitions of those dimensions, since by now they are commonly available. However, it is important to stress that the usefulness of the concept of attractor dimension lies in the fact that the number of degrees of freedom in a dynamical system is effectively no more than about twice the dimension of the attractor (Man˜e´ 1981; Takens 1981). 2. An upper bound of the dimension of the attractor To make some progress, we consider a particular case of the atmosphere model offered in Lions et al. (1992a). Namely, we disregard the fact that the eddy transport coefficients in the horizontal direction differ from those along the vertical. The mathematical arguments presented there remain unchanged. Throughout we take the eddy transport coefficient to be of the order of 2.5 3 105 m2 s21 (Washington and Parkinson 1986). Again, in the interest of simplicity, we restrict our discussion of the attractor dimension to the case of unity Prandtl number, that is, thermal diffusion coefficient being equal to the kinematic viscosity coefficient. The model assumes that the heating rate of the atmosphere E (net solar energy heating rate per unit density) is given, thus avoiding here the complications arising from the need to solve the equation of radiative transfer. Then using (A11)–(A12) in the appendix 2 2 dim(Attractor) # c5 Re3 (gk)3/2 5 c5 Re9 [z f2z ND sup\u\ ND ]3/2 , (1)
where g [ c5Re sup\u\2;c5 is a natural constant; Re 5 2 2 aU/y,\u\2ND 5 (\y \ND 1\T\ND ); f2 5 aR2T0e/CU3Cp; a is the earth’s radius; U is a representative air velocity; T0 is a representative temperature; y 5 ya/U the horizontal air velocity; ya normalized by U; T 5 Ta/T0 the air temperature Ta normalized by T0;R is the gas constant; Cp is the heat capacity for air at constant pressure, and ¯ RT(p)
1
C[R
Cp
scale defined following (5). Similarly, z f z designates the L2 2 norm of f,z f z2 5 ∫Vˆ z f(x)z2dV and z f zND its nondimensional form z f z2ND 5 z f z2/a2H. From a mathematically rigorous point of view, we are not able to estimate g in terms of the physically relevant quantities such as the Reynolds number. Finally, k is defined in the appendix where the following rigorous estimate is proved [see (A.11), (A.14)]: 2 k # c6Re3 zf2zND ,
where Re k 1 and c6 is a natural constant. Thus, physically speaking, according to (1) the upper bound on the attractor dimension is determined by the enstrophy, namely, the mean square vorticity and the mean square temperature gradient. Also consistent with the conjecture that atmospheric circulation is only weakly dependent on the details of the field of heating (Lorenz 1967), the attractor dimension depends only on the mean square value of the energy source f2. Note that unlike the case of two-dimensional flows, in three dimensions the enstrophy is not guaranteed to be finite. Hence the finiteness of the attractor dimension for atmospheric circulation depends on the empirical evidence of the absence of singularities in the observed flows (Constantin et al. 1985b). The finiteness of the attractor dimension does not necessarily preclude the presence of an infinite number of modes in the flow field. Rather, it appears that in fluid flows beyond some large wavenumber, the finer structure is determined by or slaved to the modes constituting the degrees of freedom (Constantin et al. 1985b). We need to estimate on physical grounds Sup\u\2ND . Sup\y \2ND 1 Sup\T\2ND,
where all the quantities are nondimensional. Consider first the velocity term, that is, the kinetic energy dissipation rate E5
y zVz
E
dV(=y) 2;
V
all variables dimensional, and V , R3 .
(3)
Here, in the three-dimensional domain V, the vertical coordinate is replaced by a pressure coordinate in virtue of the hydrostatic relation
]T¯ 2p 5 const, ]p
2
while T¯(p) is climate-averaged temperature on an isobaric surface, p. The supremum in (1) is taken over the attractor. More precisely, Sup\u\ is the supremum of the H1 norm of u, for u on the attractor (included in the phase space). The notation \u\ designates the H1 2 norm of u, namely the root-mean-square of the gradient of u: \u\2 5 ∫Vˆ z,uz2dV, and \u\ND its nondimensional form, \u\2ND 5 \u\2/H, where H is a specific vertical length
(2)
]p 5 2rg. ]z Hence
E
E E E P
dV 5
p0
5
dp rg
P 2 p0 r0 g
(4)
a 2 dS 2 1
0
r0 dh r
E
a 2 dS2 .
(5)
Here, S2 denotes the unit sphere surface. Note that (P
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2 p0)/r0g has the dimension of length, say H. Thus in our coordinate system, and in terms of nondimensional variables (x → x/a,y/U → y), we have
E
1
E 5 nU 2 H
0
5n
12 U a
E1
r0 dh r
S2
dy dx
2
2
dS 2 /a 2 H
2 2 \y \ ND ,
(6)
the subscript ND denoting the nondimensional nature of this H1 2 norm; \y \2ND is the quantity appearing in (1). We have taken zVz 5 a2H, that is, the volume of the atmosphere. Note that E is independent of the precise choice of H. Since at this time a reliable a priori estimate of \y \2ND is not available and the conventional estimates applicable to strong turbulence do not apply (Ghil et al. 1985), we make the natural assumption that on average the kinetic energy dissipation rate E is balanced by the fraction of the heat input driving the atmospheric circulation. This has been observed to be about 1% of the total solar heating available (Lorenz 1967). Thus 2 \y \ ND 5
12
0.01zE z a n U
2
a 5 0.01zE zRe 3 . U
(6a)
We turn now to the consideration of \T\2. We assume that its supremum is of the same order of magnitude as its time average. First we find the H1 2 norm of T: form the inner product of T and the fully dimensional form of (A4)1; then, after taking a time average, we have, on using the Poincare´ inequality, ˜ 2 & 5 ^( ˜f2 , T)& ˜ # z ˜f2z^zTz& ˜ m ^\T\ ˜ 2 &1/2 ^\T\ # z ˜f2z . K0
(7)
Here m is the thermal diffusivity and K0 is the lowest wavenumber characteristic of the flow geometry; we can take as K0 ; 1/a. From which, with ˜f2 5 e, ˜ 2& # ^\T\
1 2 1 2 z ˜f 2z mK0
2
5
zE z mK0
2
a 2 H.
2 ^\T\ ND &#
1
2
zE z NDa . mK0 T0 C p
1
1
2
2
2
12
1 2
2 \u\ ND # 0.01zez ND Re
sup u∈A
(9)
The nondimensional form of \T\ND in (9) can be cast in terms of the large-scale atmospheric motion and energy exchange processes: zE z NDa 2 ^\T\ ND &# mK0 T0 C p
where we have set m ø n (unity Prandtl number). Thus in (10), apart from the Reynolds number, we have the ratio of the atmospheric kinetic energy to its thermal energy, and the ratio of the net thermal energy input to the rate of kinetic energy dissipation. Note that here, dealing as we are with the primitive equations, the forcing of the circulation is assumed to be of thermal origin, essentially powered by the flux of solar radiation. The flow of energy in this system goes from sunlight heating the lower part of the atmosphere, leading to upward motion of the air, storing the energy in the form of available potential energy (Lorenz 1967), which is converted to kinetic energy, and eventually dissipated by friction with the earth’s surface and in the very small scale, high wavenumber motion of the air. That is unlike the simpler geostrophic model due to Legras and Ghil (1985) in which the forcing is mechanical. Another distinction from the work of Legras and Ghil (1985), who followed Charney and DeVore (1979), is that, unlike their work emphasizing the role of energy in governing various aspects of the modeled phenomena, namely, anomalies, blocking, and the like, we find that the dimension of the attractor depends critically on the gradients of the velocity field (shear) and the temperature gradients. This evokes ideas going back to Helmholtz (1888), who recognized how important to atmospheric circulation are rapid changes in the direction of the flow field. Further, that suggests a possible connection with the second law of thermodynamics, which is not readily evident in models emphasizing the dominant role of energy. We recall here that Wang (1992) estimated the dimension of the attractor for the quasigeostrophic equations of the atmosphere introduced by Charney (1951). The upper bound on that dimension was found to be of the order of Re 1 Re5/3 (Rozfz)2/3, where Ro is the Rossby number and f is the mechanical forcing in the quasigeostrophic model. Here we will not pursue further this point concerning the reduction of the attractor dimension, leaving it to future studies. Use of (6a) and (10) in (2) yields
(8)
Hence, in nondimensional terms
2 a zE z ND 5 Re 2 , U C 2p T02
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2
(11)
in which for Re . 10, the second term arising from the temperature gradients dominates the contribution due to velocity shear. From which, on substituting in (A15) and neglecting the small contribution due to vorticity, dim(Attractor) ø m # cRe12
2
Here, T˜ and f˜2 are the fully dimensional forms of T and f2.
12
2 a a zE z ND 1 Re 2 , 3 2 2 U U C p T0
(10)
1
ø aR12
1 21 2 a U
3
zE z ND C p T0
2
azE z , UaH1/2 C p T0
3
3
(12)
where c is a natural constant of the order of unity; the factor multiplying Re12 can be shown, after some re-
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arrangement, to be the cube of the energy absorbed in the atmosphere during a characteristic period of time; and a/U ø 6.4 3 106 m/10 m s21 ø 7 days, divided by the sensible heat stored in the atmosphere, which is equal to several days worth of solar energy absorbed by the atmosphere. Hence, that factor is essentially time independent and of the order of unity. Thus finally we see that the upper bound on the dimension of the attractor for the primitive equations is simply about Re12. Using the previously mentioned effective kinematic viscosity of 2.5 3 105 m2 s21, a characteristic length of 1000 km, and a representative velocity of 10 m s21, we find that the effective Reynolds number is about 40. Therefore, in order to ensure trustworthy qualitative behavior of the numerical solutions of the primitive equations, we should have adequate computational resources to follow about 1019 degrees of freedom, be it in terms of a spectral decomposition (eigenfunction expansion) or in terms of mesh points (finite spatial resolution elements) clearly beyond present day capabilities. This large bound on the attractor dimension reflects the possibility of a ‘‘worst case,’’ as well as the existing limitations on the power inherent in the methods of functional analysis used to prove the results presented here. Here, worst case means the case of primitive equations including initial conditions that make mathematical sense but may not be physically accessible, and hence leading to unrealizable or unlikely circulation patterns. As we mentioned in the introduction, it might also happen that not all modes are equally likely on the attractor; this is another problem beyond the scope of this article. We feel that it is important to understand first how the attractor dimension depends on the given parameters of the most fundamental model, the primitive equation, which although perhaps impractical for specific calculations, is after all a reflection of most of the a priori relevant physics governing the atmospheric circulation. The estimate of the attractor dimension discussed here may then be regarded as a benchmark against which future improvements in atmospheric circulation models can be judged. It is expected that as the empirically gained knowledge of atmospheric circulation is incorporated in simplifying modifications of the primitive equations, for example, geostrophic flow, zonal averaging, multicell meridional circulation, and the like, the estimates of the attractor dimension will be correspondingly decreased, if only because each of these simplifications restricts portions of the accessible phase space and hence reduces the number of possible degrees of freedom. That has been actually demonstrated by Wang (1992) in the case of the quasi-geostrophic equations [cf. discussion following Eq. (10)]. These ideas will be pursued further in the future. It is perhaps of some interest to remark that the rigorous analysis underlying the result in (12) is consistent with the conjecture (Lorenz 1967) that the details of the actual distribution of the forcing heat flux from the sun
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are not too important in determining the observed atmospheric circulation. 3. Conclusions Previous rigorous efforts by some of the present authors (Lions et al. 1992a,b; 1993) have yielded a finite bound on the dimension of the attractor for a model of atmospheric circulation. That bound was not immediately transparent with respect to its sensitivity to the given physical aspects of atmospheric circulation. In this note, by using some arguments based on the known physical properties of energy flow in the atmosphere, we have been able to cast the bound on the dimension of the attractor in terms of the observable quantities governing the heating and energy dissipation accompanying the motion of the atmosphere. To arrive at this result, we have made some simplifying, but physically realistic assumptions, such as taking the Prandtl number of the air to be of the order of unity, while the distribution of the heating rate in the atmosphere was assumed to be given, say from direct observations. To the extent that the resolution needed in numerical simulations of the long-term atmospheric motion is related to the dimension of the attractor, our results suggest that the required resolution is quite sensitive to the magnitude of the effective (or eddy) viscosity, while it appears to be less sensitive to the details of the way that the atmosphere is heated. These conclusions rest on the dependence of ‘‘dim A’’ on a rather high power of the Reynolds number, while dim A depends only on the L2 2 norm, that is, mean square of the atmospheric heating rate. It is perhaps of some interest to remark that the rigorous analysis underlying the result in (12) is consistent with the conjecture (Lorenz 1967) that the details of the actual distribution of the forcing heat flux from the sun are not too important in determining the observed atmospheric circulation. Finally, we have reported on the finding that the dimension of the attractor for the primitive equations is finite, albeit large, supporting the belief that a finite but large number of degrees of freedom is needed for a trustworthy numerical simulation of atmospheric circulation phenomena. In the context of atmospheric circulation, this summary of past relevant results in the mathematical literature may serve as a caveat against the risk of carrying out simplified calculations based on purely physical arguments but leading to less than fully trustworthy numerical results. The point is that, unlike the case of linear systems, the qualitative aspects (e.g., oscillatory behavior, chaotic behavior, steady state, etc.), of the solutions of nonlinear partial differential equations may change spuriously because of injudicious suppression of some modes, merely to accommodate limitations on computational resources.
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Acknowledgments. We are grateful to an anonymous referee for drawing our attention to some relevant earlier literature. Two of the authors (RT, SW) were partially supported by the National Science Foundation under Grant NSF-DMS-9400615, by the Office of Naval Research under Grants NAVY-N00014-91-J1140 and NAVY-N00014-96-1-0425, and by the Research Fund of Indiana University.
5
]f 2 eK3 (h)T 1 eL4v 5 0, ]h
(A2)
1 ]v 5 0, e ]h
(A3)
1 ]t + = T 2 e v ]h2 2 K (h)v + L T 5 f ,
(A4)
div v 2
k0
]T
1 ]T
v
3
2
2
where
5
1 2 1 1 ] ] L 52 D2 K (h) 2 R e R ]h 1 ]h d d ] ] L 52 D2 K (h) 2 , 1 R e R ]h ]h L1 5 2
1 1 ] ] D2 2 K1 (h) R1h e R1v ]h ]h
2
1
2
2h
5
Tr(AoQm) 5 qm,
(A8)
where the index m is determined by the Kaplan–Yorke conjecture Kaplan and Yorke (1979), as extended and proved in Constantin et al. (1985a,b). Thus, in words, qm is derived from the trace of the Jacobian of the system linearized about a solution of the primitive equations, and it is a partial sum of the Lyapunov exponents, which in turn is related to the dimension of the attractor. It can be shown that qm # 2 #2
(A6)
(A7)
where V can be approximated in terms of a suitably chosen set of orthonormal functions, say, Cj(j 5 1, 2, · · · , m). For a given m, let the trace of A projected onto the subspace formed by {Cj}mj51 be
1
]T 5 0. ]h The relationships with the dimensional variables 1 m 1 y 5 v, 5 v, R1h aU R 1n aU 1 m T¯ 2 1 n T¯ 2 5 T 30 , 5 T 30 , R2h aU R2v aU U P 2 p 0 RT¯ 0 1 Ro 5 , e5 5 aV P g a m n dm 5 w ø 1, dn 5 w ø 1 mu nv
h 5 1: (y, v) 5 0,
[ ]
dV 5 AV, dt
1n
with nondimensional boundary conditions: ]T h 5 0: (y, v) 5 0, 5 a¯ S (T 2 T¯ S ), ]h
f 5 f9U2
Note that =vv and =vT in (A1)–(A4) are the covariant derivatives of v and T with respect to v, while D in (A5) is the Laplace–Beltrami operator for scalar and vector functions on the surface of a sphere. The system (A1)– (A4) can be linearized about a given orbit in its phase space yielding
n
2
1h
w 5 w9r0 gU
2 pT¯ 0 , ¯ PT(p) ag P ag P K3 (h) 5 2 5 2 , U p U P 2 (P 2 p0 )h R 2 T¯ 20 aT R 2 E k0 5 , f2 5 30 . 2 CU U C Cp
2v
m
4
(A5)
y 5 y9U, T 5 T¯ 0 T9,
K1 (h) 5
APPENDIX
Dimension Estimates of the Global Attractor We summarize here the details of the derivation of the attractor dimension for a simple model of atmospheric circulation. This derivation differs slightly from the previously published one in Lions et al. (1992a). We consider the primitive equations of the atmosphere with vertical viscosity, which in nondimensional form are ]y 1 ]v f 1 = v v 2 v 1 k 3 v 1 =f 1 L1 v 5 0, (A1) ]t e ]h Ro
5
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c1 5/3 2 2 2 m 1 c2 mRe max g(\y\ ND 1 \T\ ND ) Re max c3 5/3 m 1 c4 Re4maxg5/2k5/2 , Re max
(A9)
where Re 5 Remax [ max(R1h, R1v, R2h, R2v). In the main text we assume for simplicity that the transport coefficients in the horizontal direction are the same as those in the vertical direction and that the Prandtl number is approximately one. Hence R1h 5 e2 R1v 5 R2h 5 e2 R2v 5 Re, and
are
2 2 g [ c5 Re sup[ \y\ ND 1 \T\ ND ],
(A10)
u ∈A
and
k 5 Re lim sup H a
t→`
u∈A
1 t
E
t 2 2 dt(\y\ ND 1 \T\ ND ).
(A11)
0
Here and elsewhere, c9i are natural constants of the order of unity. In general, qm is related to the Hausdorff dimension dH and the fractal dimension dF of the attractor for the system (A1)–(A4) as follows:
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for q m # 2amu 1 b, d H (attractor) # m,
u $ 1,
d F (attractor) # 2m,
where m21#
1/u
1 2 2b a
# m.
(A12)
Thus we have dim(attractor) # c5 Re3 g3/2 k3/2.
(A13)
An upper bound on k is found to be
k # c6 Re3 zf2z2
(A14)
and dim(attractor)
[
]
3/2
2 2 # c7 Re9z f2z3 sup(\y\ ND ; 1 \T\ ND ) . u ∈A
(A15)
REFERENCES Babin, A. V., and M. I. Vishik, 1992: Attractors of Evolution Equations. North-Holland, 532 pp. Birnir, B., and R. Grauer, 1994: An explicit description of the global attractor of the damped and driven Sine-Gordon equation. Commun. Math. Phys., 162, 539–590. Bjerknes, V., 1904: Das Problem von der Wettervorhersage, betrachtet vom Standpunkt der Mechanik und der Physik. Meteor. Z., 21, 1–7. Charney, J. G., 1951: Dynamic forecasting by numerical process. Compendium of Meteorology, T. F. Malone, Ed., Amer. Meteor. Soc., 470–482. , and J. G. DeVore, 1979: Multiple flow equilibria in the atmosphere and blocking. J. Atmos. Sci., 36, 1205–1216. , R. Fjørtaft, and J. von Neumann, 1950: Numerical integration of the barotropic vorticity equation. Tellus, 2, 237–254. Constantin, P., C. Foias, and R. Temam, 1985a: Attractors representing turbulent flows. Mem. Amer. Math. Soc., 53, 114, 67 1 vii pp. , , O. P. Manley, and R. Temam, 1985b: Determining modes and fractal dimension of turbulent flows. J. Fluid Mech., 150, 427–440. Curry, J. H., J. R. Herring, J. Loncaric, and S. A. Orszag, 1984: Order and disorder in two- and three-dimensional Benard convection. J. Fluid Mech., 147, 1. Ghil, M., R. Benzi, and G. Parisi, Eds., 1985: Turbulence and Predictability in Geophysical Fluid Dynamics and Climate Dynamics. North-Holland Publishing, 449 pp.
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, M. Kimoto, and J. D. Neelin, 1991: Nonlinear dynamics and predictability in the atmospheric sciences. Rev. Geophys., 36 (Suppl.), 46–55. Hale, J., 1988: Asymptotic Behavior of Dissipative Systems. Mathematical Surveys and Monogr., No. 25, Amer. Math. Soc., 198 pp. Helmholtz, H., 1888: Ueber Atmosphaerische Bewegung. Sitzungsber. Akad. Wiss. Berlin, 647. Kaplan, J., and J. Yorke, 1979: Chaotic behavior of multidimensional difference equations. Functional Equations and Approximations of Fixed Points, H. O. Peitgen and H. O. Walther, Eds., Lecture Notes in Mathematics, Vol. 730, Springer-Verlag, 204–227. Ladyzhenskaya, O. A., 1991: Attractors for Semigroups and Evolution Equations. Academia Nazionale dei Lincer Series, Vol. 1, Cambridge University Press, 173 pp. Legras, B., and M. Ghil, 1985: Persistent anomalies, blocking, and variations in atmospheric predictability. J. Atmos. Sci., 42, 433– 471. Lions, J. L., R. Temam, and S. Wang, 1992a: New formulations of the primitive equations of the atmosphere and applications. Nonlinearity, 5, 237–288. , , and , 1992b: On the equations of the large-scale ocean. Nonlinearity, 5, 1007–1053. , , and , 1993: Models of the coupled atmosphere and ocean (CAO I). Computational Mechanics Advances, 1, 5–54. Lorenz, E. N., 1963a: Deterministic nonperiodic flow. J. Atmos. Sci., 20, 130–141. , 1963b: The mechanics of vacillation. J. Atmos. Sci., 20, 448– 464. , 1967: The Nature and Theory of the General Circulation of the Atmosphere. World Meteorological Organization, 161 pp. Mandelbrot, B., 1977: Fractals: Form, Chance, and Dimension. Freeman, 468 pp. Man˜e´, B., 1981: On the dimension of the compact invariant sets of certain nonlinear maps. Lect. Notes Math., 898, 230–242. Richardson, L. F., 1922: Weather Prediction by Numerical Process. Dover, 235 pp. Takens, F., 1981: Detecting strange attractors. Turbulence, Dynamical Systems and Turbulence, Warwick, 1980, D. A. Rand and L. S. Young, Eds., Springer, 366–381. Temam, R., 1988: Infinite Dimensional Dynamical Systems in Mechanics and Physics. Applied Mathematical Sciences, Vol. 68, Springer, 1500 pp. Treve, Y. M., and O. P. Manley, 1982: Energy conserving galerkin approximations for 2-D hydrodynamic and MHD Benard convection. Physica D, 4, 319–342. von Neumann, J., 1960: Some remarks on the problem of forecasting climatic fluctuations. Dynamics of Climate, R. L. Pfeffer, Ed., Pergamon Press, 9–12. Wang, S., 1992: Attractors for the 3-D baroclinic quasi-geostrophic equations of large scale atmosphere. J. Math. Anal. Appl., 165, 266. Washington, W. M., and C. L. Parkinson, 1986: An Introduction to Three-dimensional Climate Modeling. Oxford University Press, 422 pp.
