DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS–SERIES B Volume 6, Number 6, November 2006
Website: http://AIMsciences.org pp. 1403–1416
REVISITING THE SLOW MANIFOLD OF THE LORENZ-KRISHNAMURTHY QUINTET
M. Phani Sudheer
1
Department of Mechanical Engineering National Institute of Technology Karnataka, Surathkal, India
Ravi S. Nanjundiah Center for Atmospheric and Oceanic Sciences Indian Institute of Science, Bangalore 560 012, India
A. S. Vasudeva Murthy TIFR Centre Indian Institute of Science, Bangalore 560 012, India
(Communicated by Roger Temam) Abstract. The slow-manifold for the Lorenz-Krishnamurthy model has been studied. By minimizing the evolution rate we find that the analytical functions for the fast variables are devoid of high frequency oscillations. However upon solving this model with initial values of the fast variables obtained from the analytical functions, the LK model exhibits high frequency oscillations. Upon using the time derivatives of the analytic functions for computing the evolution of fast variables, we find a slow-manifold in the neighbourhood of the LK model. Minimization of evolution rate does not guarantee the invariance of the manifold. Using a locally linear approximate reduction scheme, the invariance can be maintained. However, the solutions so obtained do develop high frequency oscillations. The onset of these high frequency oscillations is delayed vis-a-vis other previous studies. These methods have potential to be used in improving the predictions of weather systems.
1. Introduction. In the dynamics of atmosphere as well as oceans, we come across a vast range of spatial and temporal scales. It is extremely difficult, or even impossible, to resolve all the scales. In any case daily observations of field variables are too sparse to allow accurate resolution. If we were to pick one large scale variation and one small scale variation that could be crucial for accurate numerical weather prediction then it would be the low frequency Rossby wave (RW) and the high frequency gravity wave (GW). The solutions to the primitive equations (PE) that model the atmosphere permit both RW and GW. It is extremely expensive to accurately evaluate the GW by solving the PE numerically. Besides the gravity waves have little energy with most of the energy concentrated in the RW. The question therefore boils down to whether it is possible to get the RW accurately without an accurate resolution of the GW. More precisely the question is whether we can find 2000 Mathematics Subject Classification. Primary: 37D10; Secondary: 35B42. Key words and phrases. invariant slow-manifold, minimization of evolution rate, predictability. 1 Undergraduate student, presently student at Indian Institute of Science, Bangalore
1403
1404
PHANI, NANJUNDIAH AND MURTHY
a set of initial conditions for the PE such that the trajectories in the phase space of PE do not contain the GW but only the RW. Such a set of trajectories is called as the “slow manifold”. There are many definitions of the slow manifold in the literature specifically for atmospheric models [8]. In general, the reduced manifold can be defined as the lower-dimensional manifold of the desired dimension that attracts most solution trajectories of the full system. The attracting nature of such a manifold makes it the slowest and the densest [1, 2]. The slow manifold is invariant i.e. an orbit originating from a point on the manifold is completely constrained in the manifold. Also the reduced system with all the required attributes has to be a slowest invariant manifold [2]. Since the dimension of the phase space for PE is infinite and PE’s are non-linear, the question of existence of a slow manifold for the PE is extremely difficult. However we can try to answer the above question on simple finite dimensional models. Toward this end Lorenz [4] introduced a simplified version of the PE model using mode truncation to reduce it to a five dimensional system of ordinary differential equations (ODEs). He then identified the variables representing gravity wave activity as the ones which exhibit fast oscillations, and defined the slow manifold as an invariant manifold in the five-dimensional phase space for which fast oscillations never develop. Later Lorenz and Krishnamurthy [3] (hereafter LK) introduced damping in the model and identified an orbit which by construction was not supposed to have GW. However accurate numerical solution showed that sooner or later fast oscillations developed, implying the nonexistence of a slow manifold. This has triggered a series of papers by several workers questioning the very definition of slow manifold and its existence. We shall not go into this but refer to Camassa and Tin [10] and Fowler and Kember [9] and references therein. Our interest in the LK model was motivated by the work of Girimaji [1] [2] who has proposed an interesting criteria for getting the slow manifold in a system of ODES that contain both fast and slow time scales. This criteria is based on the hypothesis that given the values of slow variables, an arbitrary solution trajectory is most likely to be found at the state with the longest residence time i.e the smallest evolution rate. The precise mathematical statement will be given in the next section. The main aim of this work is to apply this criteria on the LK system and test its efficacy in suppressing the GW. We end this section with the LK quintet dU/dt dV /dt dW/dt dX/dt
= = = =
−V W + bV Z − aU U W − bU Z − aV + aF −U V − aW −Z − aX
(1) (2) (3) (4)
dZ/dt
=
bU V + X − aZ
(5)
In the above equations, a is the damping coefficient, b is the nonlinear coupling coefficient and F is the time-invariant forcing. We denote Rossby amplitudes by (U, V, W ) and the gravity mode amplitudes by X and Z. Although the model is derived from primitive equations (PE), the variables are amplitudes of a generalized Fourier series and U, V and W represent the coefficients of the first mode, second mode, third mode of the streamfunction respectively while X and Z represent the coefficients of the third mode of velocity potential and height of the free surface
APPROXIMATE SLOW MANIFOLD
1405
respectively. For more details of the LK quintet refer to [4] and [3]. Note that U,V and W are the slow variables while X and Z are the fast variables. 2. The Slow Manifold of Girimaji. Consider an autonomous dissipative set of ODEs dz = g(z); dt z(o) = zo , where z(t) = (z1 (t)z2 (t)...zn (t)) and g is a smooth function on Rn that has an attracting fixed point. In [1] Girimaji has argued that if the system has disparate time scales, then the solution exhibits a typical three stage behaviour. 1. Initiation condition dependent transient stage dominated by small-time scales 2. the intermediate slow-manifold in which the solutions “bunch” together in a lower-dimensional space dominated by the slow-time scales, 3. the final steady state. If we partition z(t) into the slow variable y(t) and the fast variable x(t) Girimaji’s proposal is x(y) ≈ minx F (x, y) where F = Σni=1 gi2 Basically this means that the variation in z due to fast variable x is minimized. For higher order, Girimaji proposes to minimize F m (x, y) but we shall restrict to m = 1. Also in the nonlinear Galerkin method of Marion and Temam [11], z(t) is a Hilbert space valued function and is approximated by zm (t) which is split as zm (t) = xm (t) + ym (t) xm (t) = Σm j=1 ajm (t)wj ym (t) = Σ2m j=m+1 bjm (t)wj where {wj } is an orthonormal basis for the Hilbert space. If one can find a function φ satisfying ym (t) = φ(xm (t)) this would then correspond to an inertial manifold or an approximate slow manifold. In general there is no recipe for obtaining this manifold φ. The above criteria of minimization of evolution rate gives a method for obtaining it. Finally, applying the latter scheme to the LK system ( 1)-(5) we obtain · ¸2 · ¸2 · ¸2 · ¸2 · ¸2 dU dV dW dX dZ F = + + + + dt dt dt dt dt Minimizing F with respect to fast variables X and Z gives the values of X and Z as functions of U, V and W that minimizes the evolution rate. Therefore −bU V (6) X = f (U, V, W ) = 1 + a2 bW (V 2 + U 2 ) + abU (V + F ) Z = g(U, V, W ) = (7) b2 (U 2 + V 2 ) + 1 + a2
1406
PHANI, NANJUNDIAH AND MURTHY
Note that the expression for X is the same as given by Camassa and Tin [10] (eqn C4) but the expression for Z is different. The equations in Camassa and Tin are obtained by setting the derivatives in (1) - (5) to zero i.e. the steady state solution for these equations. 2.1. Eliminating the Fast Variables. We are now interested in finding out as to how best the above equations represent the slowness of the manifold, with the dynamics of the model remaining unchanged. X and Z in the expressions for the time derivatives of U,V and W in ( 1) - (3) are replaced by the functions f and g given in 6-7, leading to a set of three ODEs with rhs being only functions of U,V and W. · ¸ bW (V 2 + U 2 ) + abU (V + F ) dU/dt = −V W + bV − aU (8) b2 (U 2 + V 2 ) + 1 + a2 · ¸ bW (V 2 + U 2 ) + abU (V + F ) dV /dt = U W − bU − aV + aF (9) b2 (U 2 + V 2 ) + 1 + a2 dW/dt = −U V − aW . (10) Initial Conditions are 0.1,0,−0.1,0,1 0.11 Ug U 0.1
0.09
U and Ug
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0
100
200
300 t
400
500
600
Figure 1. Graph showing evolution of by solving the original LKmodel eqn( 1-5) and the solution Ug obtained by solving the reduced set of equations ( 8-10) by minimization of evolution rate. The graph is for a = 0.02; F = 0.2; b = 0.5 (same values as used by [3]) We have solved these set of ODEs using a second-order Runge-Kutta method. Figures 1 and 2 clearly show that U and V obtained by solving the system (1)-(5) has slowly varying U and V contaminated with the fast oscillations. This is due to the fact that the fast variable Z is coupled with U and V in the system. On the other hand, the value of U obtained by solving the system ( 8)-( 10) (denoted by Ug in the figure) not only has no noise disturbing its solution but also captures the
APPROXIMATE SLOW MANIFOLD
1407
Initial Conditions are 0.1,0,−0.1,0,1 0.06 Vg V
0.04
V and Vg
0.02
0
−0.02
−0.04
−0.06
0
100
200
300
400
500
600
t
Figure 2. Same as fig 1 but for V complete dynamics of the original system. This elimination of noise could be useful in numerical weather forecasting, where one wants the Rossby phase space to be free of gravity waves. Nevertheless inaccuracies do prevail in capturing the dynamics of the original system, though fast oscillations were eliminated in the Rossby phase space. This was prominent in some cases where the steady state reached by the values of the slow variables obtained by solving the systems (1)-(5) and ( 8) - (10) are different. This could be due to the fact that the LK system is unstable and has three steady states for values of F > Fc .(Fc has the same meaning as represented by [3] and its value is around 0.02 for the data considered here). Hence, the final steady state attained depends on the initial values of the variables. We note that this deviation occurred when the initial values of U and W were equal. Figures 3 and 4 illustrate the fact with initial conditions for U, V, W, X and Z as 0.1,0,0.1,0,1 respectively. 2.2. Orbit from a Point on the Slow Manifold. In continuation with the previous observation, we examined the trajectory emanating from a point on the slow manifold i.e calculate X and Z for a given set of initial values of U, V and W from ( 6) and (7). According to Lorenz [5], if the orbit from the point on the slow manifold does not have any corrugations or high frequency oscillations, it will be the slow manifold for the given system. Unfortunately the obtained slow manifold by minimization of evolution rates could not suppress the oscillations in Z for the above mentioned conditions as indicated in the Figure 5. Initial conditions for U,V,W chosen here were 0.125,0.325,0.125, the same as that of Lorenz [5]. This implies that the minimum evolution rate scheme did not take care of invariance. This was evident from the fact that the original evolution rates of X and Z were quite different from that of the new evolution rates obtained from the functions f and g.
1408
PHANI, NANJUNDIAH AND MURTHY Initial Conditions are 0.1,0,0.1,0,1 0.2
0.15
0.1
U and Ug
Ug 0.05
0
U
−0.05
−0.1
−0.15
0
100
200
300
400
500
600
t
Figure 3. Evolution of U and Ug for initial conditions U0 = W0 Initial Conditions are 0.1,0,0.1,0,1
W and Wg
W
0
Wg
0
100
200
300
400
500
600
t
Figure 4. Evolution of W and Wg when U0 = W0 2.3. A Neighborhood System with an Exact Slow Manifold. The manifold obtained by minimizing the evolution rate was not able to eliminate oscillations in the orbit emanating from a point on the assumed slow manifold. We now try to obtain a new system in which the evolution rates for X and Z of the LK model (1)-(5) are replaced with those obtained from f and g. dX dt dZ dt
= =
∂f ∂U ∂g ∂U
dU ∂f dV ∂f + + dt ∂V dt ∂W dU ∂g dV ∂g + + dt ∂V dt ∂W
dW dt dW dt
(11) (12)
APPROXIMATE SLOW MANIFOLD
1409
Initial conditions are 0.125;0.325;0.125;−0.02030438;0.00798901 0.04
0.03
0.02 Z
Z
0.01
0
−0.01
−0.02
−0.03
−0.04
0
5
10
15
20
25
30
35
40
45
50
t
Figure 5. Rapid oscillations in Z,the orbit emanating from assumed slow manifold Figure 6 illustrates the graph of Z vs t evaluated from the new system for the same initial conditions of U,V and W used in Figure 5. Initial X and Z were calculated from f and g. Initial conditions are 0.125;0.325;0.125;−0.02030438;0.00798901 0.04
0.03
0.02
0.01
Z
Z 0
−0.01
−0.02
−0.03
−0.04
0
5
10
15
20
25
30
35
40
45
50
t
Figure 6. Evolution of Z for the new system, the orbit from the assumed slow manifold The new system of equations is devoid of high frequency oscillations and the solution converges to original Z in the steady state (Figure 6). This clearly indicates that f and g is the exact slow manifold for this system. Hence a new system which has a exact slow manifold has been deduced from the previous system.
1410
PHANI, NANJUNDIAH AND MURTHY
We say that this new system is in the neighbourhood of the original system in terms of its total energy. By the term ’total energy’ we mean the sum of the squares of the evolution rates of all the variables. ¸2 · ¸2 · ¸2 · ¸2 · ¸2 · dU dV dW dX dZ E= + + + + (13) dt dt dt dt dt Figure 7 shows the evolution of total energy for the original system and the new system for the same initial values of U,V and W (0.125,0.325,0.125) with initial X and Y calculated from f and g. −3
7
0.125;0.325;0.125;−0.020304;0.007989012
x 10
New E Old E 6
5
E
4
3
Total Energy
2
1
0
0
100
200
300
400
500
600
t
Figure 7. Evolution of Total Energy for the new(En )and old system(Eo ) A careful observation of the Figure 7 is that the total energy of the original system (1)-(5) and the new system of equations are comparable except for the peak values which are slightly different. Thus we conclude that the new system lies in the neighbourhood of the original system in terms of energy, and more importantly it has the slow invariant manifold given by f and g. Figures 8 and 9 show the variation of the energy of slow and fast variables respectively both for the new (En ) and the old (Eo ) systems. 3. Obtaining the Slow Manifold with the Slowest Invariant Approach. In our previous discussion, we have studied the slow manifold for the LK-model by minimization of the evolution rate and find that it suffered from shortcomings such as the non-elimination of gravity waves in the orbit emanating from a point on the slow-manifold. The main reason was that in the latter scheme, the slowest portion of phase space was recognized to be the appropriate reduced system but it was not required to be an invariant manifold [2]. Here we attempt to include invariance and predict the slow-manifold using local reduction techniques, as global reduction techniques may not be viable for complex systems such as numerical weather prediction models. We apply Girimaji’s technique of local reduction [2] that uses the properties of invariant manifold to the LK-model (eqns 1-5).
APPROXIMATE SLOW MANIFOLD −3
0.125;0.325;0.125;−0.020304;0.007989012
x 10
7
1411
New E Old E 6
5
E
4
3
2
Energy of Slow Variables
1
0
0
100
200
300
t
400
500
600
Figure 8. Evolution of energy of slow variables for the new(En ) and old (Eo ) systems −4
5
0.125;0.325;0.125;−0.020304;0.007989012
x 10
New E Old E
4
E
3
2
Energy of fast Variables 1
0
0
100
200
300
400
500
600
t
Figure 9. Evolution of Energy of fast variables for the new(En ) and old (Eo ) systems 3.1. The Local Reduction Scheme. The present reduction scheme is based on the fact that the phase velocity of the full system must be locally tangential to the invariant manifold. Assuming x˙ = x(x, ˙ y), y˙ = y(x, ˙ y) is given to us and x = x(y) is the reduced form one is interested in. In the above equations, x and y are the vector representations of the fast and the slow system
1412
PHANI, NANJUNDIAH AND MURTHY
respectively. Now, using the property of invariant manifold, one can write x(x, ˙ y) =
∂x ∂x ∂x ∂x(x, y) y(x, ˙ y) = y˙1 + y˙2 + · · · + y˙n ∂y ∂y1 ∂y2 ∂yn
(14)
The physical meaning of the above equation is simple: the evolution rate of the fast variables computed from the full system will exactly match that computed from the reduced system only if the latter is an invariant manifold of the full system [2]. 3.2. Locally Linear Approximation. The matching of evolution rate implies that on the invariant manifold, Jacobian of the discarded variables obtained from the full system must match that obtained from the reduced system. This concept forms the basis of the locally-linear approximation [2]. If the dynamical system has n fast variables and N slow variables, we differentiate the left hand side and the right hand side of (14) with respect to a slow variable and equate them to get, for i = 1..n and j = 1..N , µ ¶ ¶ N µ n X X ∂ x˙i d ∂xi ∂xi dy˙k ∂ x˙i ∂xk + = y˙k + . ∂xk ∂yj ∂yj dyj ∂yk ∂yk dyj
(15)
k=1
k=1
If the manifold is assumed to be locally linear,we get n N X X ∂ x˙i ∂xk ∂ x˙i ∂xi dy˙k + = . ∂xk ∂yj ∂yj ∂yk dyj
k=1
(16)
k=1
∂xi from which Now (16) can be solved numerically for a locally-linear estimate of ∂y j the needed reduced-manifold relationship x(y) can be determined.
4. Reduction of the LK-model. Now, applying the above-mentioned reduction scheme to the LK-model by using the condition as specified by (14) we obtian dX ∂X dU ∂X dV ∂X dW = + + dt ∂U dt ∂V dt ∂W dt
(17)
dZ ∂Z dU ∂Z dV ∂Z dW = + + (18) dt ∂U dt ∂V dt ∂W dt The left hand side indicates the evolution rate of the fast variables as computed from the full system. The right hand side indicates the evolution rate of fast variables as computed from the reduced system. They should exactly match if the latter is an invariant manifold of the full system. Here we note that Jacobs [6] too started with the same assumption and attempted to find X and Z in terms of power series of U,V and W (the slow variable). But as shown by Lorenz [5] that the manifold emanating from Jacobs’ point on the slow manifold had oscillations after certain period of time when U, V and W fall out of the range of convergence of his power series. The aim of our investigation is also to evaluate X and Z in terms of U , V and W through the local reduction technique. Applying condition (16) and after some mathematical manipulations we arrive at the following set of six partial differential equations.
APPROXIMATE SLOW MANIFOLD
∂Z ∂U ∂X ∂U ∂Z − ∂V ∂X ∂V ∂Z − ∂W ∂X ∂W −
=
bV
=
bV
=
bV
=
bV
=
bV
=
bV
∂Z ∂X ∂X ∂Z ∂X ∂X ∂X − bU + W− V − bZ ∂U ∂U ∂V ∂U ∂V ∂W ∂V ¶2 µ ∂Z ∂Z ∂Z ∂Z ∂Z ∂Z + W − bU − bZ − V − bV ∂U ∂V ∂U ∂V ∂V ∂W ∂Z ∂X ∂X ∂X ∂Z ∂X ∂X + Zb −W − bU −U ∂V ∂U ∂U ∂U ∂V ∂V ∂W µ ¶2 ∂Z ∂Z ∂Z ∂Z ∂Z ∂Z + bZ −W − bU −U − bU ∂V ∂U ∂U ∂U ∂V ∂W ∂Z ∂X ∂X ∂X ∂Z ∂X − V +U − bU ∂W ∂U ∂U ∂V ∂W ∂V ∂Z ∂Z ∂Z ∂Z ∂Z ∂Z −V +U − bU ∂W ∂U ∂U ∂V ∂W ∂V
1413
(19) (20) (21) (22) (23) (24)
∂X The system of equations represented by (19)-(24) have unknowns in ∂X ∂U , ∂V , denoted by XU , XV , XW , ZU , ZV , ZW respectively hereafter. It is important to note that in the above system the partial derivative represents the derivative of the fast variables computed from the reduced manifold. The above equations have been rewritten in matrix form by splitting the linear and non linear part separately. such manipulation of the system results in the following matrix equation 0 W −V 1 0 0 XU N1 0 −1 0 0 0 W −V XV N2 bV −W XW N3 0 0 −U 0 1 0 + = (25) 0 −1 0 −W 0 −U ZU N4 bU −V U 0 0 0 1 ZV N5 0 0 0 −1 −V U 0 ZW N6 0 ∂X ∂Z ∂Z ∂Z ∂W , ∂U , ∂V , ∂W
where N1 N2 N3 N4 N5 N6
= = = = = =
bV ZU XU − bU XV ZU − bZXV bV ZU2 − bU ZU ZV − bZZV bV ZV XU + ZbXU − bU ZV XV bV ZU ZV + bZZU − bU ZV2 bV ZW XU − bU ZW XV bV ZW ZU − bU ZW ZV
So the matrix form of ( 25) can now be represented as [L][K] + [N ] = [C]
(26)
where L represents the linear part, K the unknown values of partial derivatives, N the non linear part while C represents the constant part of the (19)-(24). As a first approximation, we neglect the non-linear part and find [K] for a given set of values of U ,V and W . We determine X and Z from the property that the evolution rates are equal for the full and the reduced system i.e from (18). We now compute the value of [N] using the obtained numerical values of [K]. The value of [N] so computed is very small for U,V and W less than 1. Figure 10 shows the temporal variation of Z (Zg ) with the same initial conditions of U,V and W (0.125;0.325;0.125) used in the previous graphs while X and Z were calculated from
1414
PHANI, NANJUNDIAH AND MURTHY
the procedure as explained above. Also shown is the variation of Z as obtained by Jacobs’ [6] algorithm of finding X and Z. Note that in Figure 10 trajectory of Z obtained using the above local reduction scheme has been pushed upwards by 0.02 units for better comparison with (Zg ). G:0.125;0.325;0.125;−0.0191377;0.005687,J:0.125;0.325;0.125;−0.01884;−0.00613 0.04
0.03 Zg 0.02
0.01
Z
Zj 0
−0.01
−0.02
−0.03
−0.04
0
5
10
15
20
25 t
30
35
40
45
50
Figure 10. Evolution of Z comparing Jacobs (Zj ) and new trajectory (Zg ) Examining Figure 10 we find that our solution suppresses the oscillations only up to a certain time and corrugations develop later on for the initial values as given in [5]. However there are cases where our results are superior as shown in Figures 11 - 12. Figure 11 show a plot Zg vs t for the initial conditions U0 = 0.1, V0 = 0, W0 = −0.1 while X0 , Z0 are obtained from the procedure described above. Similar graph is plotted (Zj vs t) on the same figure with X0 , Z0 calculated from Jacobs’ algorithm using the same values of U0 , V0 , W0 . The values of Z emanating from a point on our slow-manifold has minimal corrugations when compared to Jacobs’ trajectory. The trajectory obtained by the local reduction scheme has been pushed upwards to have a better comparison between the two graphs. A similar case of superiority is shown in the Figure 12 plotted for the initial conditions U0 = 0.15, V0 = 0, W0 = 0.15. Values of X0 and Z0 have been computed just as above. 5. Concluding Remarks. We have studied the slow-manifold using the LK-model with forcing and dissipation. The analytical functions, obtained by minimization of evolution rates for the fast variables, X and Z (f and g) gave solutions that were devoid of high frequency oscillations. The behaviour of the solution was quite satisfactory in approximating the original dynamics of the system (i.e the LKmodel). However upon solving the original set of equations of the LK-model with values of f and g as initial conditions for X and Z, we notice that fast oscillations develop.
APPROXIMATE SLOW MANIFOLD
1415
G:0.1;0;−0.1;1.42956e−05;−3.056287e−04,J:0.1;0;−0.1;−6.488e−04;−0.0016652 0.04
0.03 Zg 0.02
0.01
Z
Zj 0
−0.01
−0.02
−0.03
−0.04
0
5
10
15
20
25
30
35
40
45
50
t
Figure 11. Evolution of Z comparing Jacobs(Zj ) and new trajectory (Zg ) for U0 = 0.1, V0 = 0, W0 = −0.1 G:0.15;0;0.15;−2.9289e−05; 0.0020695,J:0.15;0;0.15;−0.0012943;0.002969 0.04
0.03 Zg 0.02
0.01
Z
Zj 0
−0.01
−0.02
−0.03
−0.04
0
5
10
15
20
25
30
35
40
45
50
t
Figure 12. Evolution of Zj and new trajectory Zg for U0 = 0.15, V0 = 0, W0 = 0.15 Hence the manifold so obtained (by minimization of evolution rate) does not represent a slow manifold for the original LK-model. However, since the solutions obtained with the reduced set (i.e. with analytical expressions for X and Z) and those with evolution equations obtained as derivatives of f and g mimic the slow-response of the original LK-model, we can say that we have found a slow-manifold in the neighbourhood of the LK-system. Minimization of evolution rate did not guarantee the invariance of the manifold. Hence we chose the locally linear approximation reduction scheme in which the property of invariant manifold was used. While this gave solutions that initially did not have corrugations, the solutions developed fast
1416
PHANI, NANJUNDIAH AND MURTHY
oscillations later but were in good agreement with (and sometimes superior to) the solutions of Jacobs. Thus it appears that while we have an exact slow manifold in the neighbourhood of the LK-model, the LK-model itself does not appear to have an exact slowmanifold. However, the use of minimization of evolution rates and locally linear approximation appear to be promising candidates to delay the occurrence of high frequency oscillations and thus could help in improving predictability of weather systems. Acknowledgement. The authors thank Professor J. Srinivasan (IISc) and Professor V. Krishnamurthy (COLA) for their suggestions and comments. Phani thanks Indian Academy of Sciences, Bangalore, for providing the summer research fellowship. Nanjundiah and Murthy thank the Council of Scientific and Industrial Research, New Delhi for their financial support through the NMITLI project “Mesoscale modeling for monsoon related predictions”. REFERENCES [1] S. Girimaji, Reduction of large dynamical systems by minimization of evoluton rate, Physical Review Letters, 82 (1999) 2282–2285. [2] S. Girimaji, Fundamental constraints on kinetics reduction. Tech Report. Aerospace Engineering Dept, Texas A & M University (2004). [3] E.N. Lorenz and V. Krishnamurthy, On the non-existence of a slow manifold, Journal of Atmospheric Sciences, 44 (1987), 2940-2950. [4] E.N. Lorenz, On the existence of a slow manifold, Journal of Atmospheric Sciences, 43 (1986), 1547-1557. [5] E.N. Lorenz, The slow manifold - what is it? Journal of Atmospheric Sciences, 49 (1992), 2249-2451 [6] S.J. Jacobs, On the existence of a slow manifold in a model system of equations, Journal of Atmospheric Sciences, 48 (1991), 793-801. [7] J.P. Boyd, The slow manifold of a five-mode model, Journal of Atmospheric Sciences, 51 (1994), 1057-1063. [8] J.P. Boyd, Eight definitions of the slow manifold: seiches, pseudoseiches and exponential smallness, Dynamics of Atmosphere and Oceans, 22 (1995), 49-75. [9] A.C. Fowler and G. Kember, The Lorenz-Krishnamurthy slow manifold, Journal of Atmospheric Sciences, 53 (1996), 1433-1437. [10] R. Camassa and S-K Tin, The global geometry of the slow manifold in the Lorenz-Krishnamurthy slow manifold, Journal of Atmospheric Sciences. 53 (1996), 3251-3264. [11] M. Marion and R. Temam, Nonlinear Galerkin Methods, SIAM Journal of Numerical Analysis, 26 (1989), 1139-1157.
Received August 2005; revised March 2006. E-mail address:
[email protected] E-mail address:
[email protected] E-mail address:
[email protected]
