A number theoretical observation of a resonant interaction of Rossby waves
aa r X i v : . [ m a t h . N T ] S e p A number theoretical observation of aresonant interaction of Rossby waves
Nobu Kishimoto
Research Institute for Mathematical Sciences, Kyoto University, Kyoto, 606-8502,Japan
Tsuyoshi Yoneda
Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo152-8551, Japan
Abstract
Rossby waves are generally expected to dominate the β plane dynamics in geo-physics, and here in this paper we give a number theoretical observation of theresonant interaction with a Diophantine equation. The set of resonant frequenciesdoes not have any frequency on the horizontal axis. We also give several clusters ofresonant frequencies. Key words: β plane, Rossby wave, number theory, a Diophantine equation We consider three-wave interactions of the Rossby waves in a number theoret-ical approach. Such waves are observed in an incompressible two-dimensionalflow on a β plane (in geophysics). The β -plane approximation was first in-troduced by meteorologists (see [1,2]) as a tangent plane of a sphere to ap-proximately describe fluid motion on a rotating sphere, assuming that theColioris parameter is a linear function of the latitude. A formal derivationof the β -plane approximation is given in [5]. It has been known that in theincompressible two-dimensional flow on a β plane, as time goes on, a zonalpattern emerges, consisting of alternating eastward and westward zonal flows, Email addresses: [email protected] (Nobu Kishimoto), [email protected] (Tsuyoshi Yoneda ).
Preprint submitted to Elsevier 18 September 2018 imilar to the zonal band structure observed on Jupiter. From a physical pointof view, one of the most important properties of the flow on a β plane linearwaves called “Rossby waves”. The Rossby waves originate from the followingdispersion relation (see [7] for example), ω = − βk k + k , (1)where ω and ( k , k ) are the angular frequency and the wavenumber vector.The Rossby waves have been considered to play important roles in the dy-namics of geophysical fluids (see [6] for example). In [7], they proved a mathe-matical rigorous theorem which supports the importance of the resonant pairsof Rossby waves. However, none of studies tried to consider such resonantwaves in number theoretical approach, and in this paper we attempt to con-sider it in an elementary number theory. Let us be more precise. We define thewavenumber set consisting of wavenumbers in non-trivial resonance as follows: Definition 1 (Wavenumber set of non-trivial resonance.) Let Λ be a wavenum-ber set such that Λ := ( n ∈ Z with n = 0 : n n + n − xx + y − n − x ( n − x ) + ( n − y ) = 0 , for some ( x, y ) ∈ Z with x = 0 and n − x = 0 ) . The role of the above non-trivial resonance can be found in [7] in PDE sense.Thus we omit to explain how it works to the two-dimensional flow on a β plane (in PDE sense). We would like to figure out the exact elements of Λwithout any numerical computation. The following remark ensures that Λ hasat least infinite elements. Remark 2 (Infinite elements.) At least, n = ( n , n ) = ( m , mℓ ) ( m, ℓ ∈ N , m = ℓ ) is in Λ . In this case, we just take ( x, y ) = ( ℓ , − m ℓ ) . Thus Λ has atleast infinite elements. Λ itself is not only mathematically but also physically interesting. In a turbu-lent flow, every wavenumber component should have nonzero energy. Supposethat the initial energy distribution in a wavenumber space is isotropic. Two-dimensional turbulence is known to transfer the energy from small to largescalemotions (energy inverse cascade). If there is no effect of rotation (no Corioliseffect), then the energy therefore becomes concentrated isotropically aroundthe origin in wavenumber space. However, if the rotation effect (Coriolis effect)2s dominant, the energy transfer becomes governed by the resonant interactionof Rossby waves Λ, and the number of resonant triads gives a rough estimateof the strength of the nonlinear energy transfer. Therefore, roughly speaking(in a physical point of view), the wavenumbers not in Λ are then expected togain less energy compared with wavenumbers in Λ. In a numerical computa-tion (see [7]), we can expect that Λ has anisotropic distribution. Thus our aimis to know Λ rigorously, and prove (in a number theoretical approach) thatits distribution is anisotropic (however, it seems so difficult that we need toprogress little by little). For the first step, in this paper, we give nonexistenceof three wave interaction on n -axis by using a Diophantine equation. In Ap-pendix, we give suitable definitions to describe resonant points Λ, and giveseveral specific resonant points. Up to now, the points were found one by one(not theoretically). The main theorem is as follows: Theorem 3 (Nonexistence of the three wave interaction on n -axis.) If n , x, y ∈ Z and n n = xx + y + n − x ( n − x ) + y , (2) then n x ( n − x ) = 0 . Remark 4
In order to consider more general setting, namely, to figure outwhether ( n , n ) ( n , n ∈ Z , n = 0 ) belongs to Λ or not, we need to considerthe following equality (just derived from Definition 1): y − n y − x ( n − x ) y + 2 n x n − x + n n ! y − x ( n − x )( x − n x + n + 2 n ) − n xn = 0 for x, y ∈ Z with x = 0 .This equality might be related to “elliptic curve” more or less. In this point ofview, the ideas of Mordell’s theorem and “infinite descent” might be useful. Assume there is n and x such that n x ( n − x ) = 0. Since Λ is symmetric,we can assume n > x >
0. From (2), we see3 x + y ) { ( n − x ) + y } = n x { ( n − x ) + y } + n ( n − x )( x + y ) ⇔ y + { x + ( n − x ) − n x − n ( n − x ) } y + { x ( n − x ) − n x ( n − x ) − n x ( n − x ) } = 0 ⇔ y − x ( n − x ) y + x ( n − x ) − n x ( n − x ) = 0 ⇔ y = x ( n − x ) ± q x ( n − x ) . Clearly, we do not treat complex numbers in this consideration, thus n − x > < x ( n − x ) < n (we have already assumed that n > x >
0, thus n − x ≤ n ), we have 0 < x ( n − x ) < n q x ( n − x ). Thus y = x ( n − x ) + n q x ( n − x ) . Otherwise, y becomes a complex number. In particular, x ( n − x ) =: p ( p ∈ N )and p + n p are square numbers (if x ( n − x ) is not square number, then y isnot in Z and it is in contradiction). Here, we can assume x and n are relativelyprime. In fact, if the greatest common divisor is d >
1, we set x ′ = x/d ∈ N and n ′ = n /d ∈ N and then( y/d ) = x ′ ( n ′ − x ′ ) + n ′ q x ′ ( n ′ − x ′ ) . Since the left hand side of the above equality is a rational number, then x ′ ( n ′ − x ′ ) is a square number, namely, the right hand side is a natural number: y ′ := y/d ∈ N . Therefore we can regard n ′ , x ′ and y ′ as n , x and y . Since x and x ( n − x ) are relatively prime and x ( n − x ) is a square number, x and n − x are also square numbers. In fact, if either x or n − x is not squarenumber, then (at least) two p j in the following expression x ( n − x ) = p = p p · · · p N ( p , · · · , p N are prime numbers, and some p i and p j ( i = j ) may be the same)must belong to both x and ( n − x ). In this case, x and n − x are not relativelyprime. Therefore x = q , n = q + r , p = qr, q, r ∈ N are relatively prime . (3)We see that q , r and q + qr + r are all relatively prime. For example, if q and q + qr + r are not relatively prime, there is a prime number p such that q = s p and q + qr + r = s p ( s , s ∈ N ). Since q + qr is multiple of q (namely,multiple of p ) then r is also multiple of p . However, if r is multiple of p ,then r itself must be multiple of p . This means that q and r are not relativelyprime. It is in contradiction to (3). Recall that p + n p = q r + ( q + r ) qr = pr ( q + qr + r ) is a square number. Since q , r and q + qr + r are all relativelyprime, we can rewrite q = s , r = t , s + s t + t = u , s, t, u ∈ N . Lemma 5 ([3]) The following Diophantine equation X + X Y + Y = Z , X, Y, Z ∈ Z only have a trivial integer solution: X = 0 or Y = 0 . In this section we give several specific resonant points. The points were foundone by one (not theoretically). In order to state the resonant points, “clus-ter” concept is very useful. Note that on a sphere case, Kartashova andL’vov [4] have already tried to classify several Rossby waves into clusters.Let { Ω finitej } j ⊂ Λ be a family of clusters composed by finite elements and { Ω inftyj } j ⊂ Λ be a family of clusters composed by infinite elements defined asfollows:
Definition 6 (Clusters with finite elements) Let { Ω finitej } j ⊂ Λ be a familyof wavenumber clusters satisfying the following properties: • For any n ∈ Ω finitej , there is ( x, y ) ∈ Ω finitej such that n and ( x, y ) satisfythe definition of Λ . • For any n Ω finitej , there is no ( x, y ) ∈ Ω finitej such that n and ( x, y ) satisfythe definition of Λ . • For each j , number of elements in Ω finitej is always finite. • We set λ ,j := inf {| n | : n ∈ Ω finitej } . Then λ ,j ≤ λ ,k ( j < k ). If there is j and k ( j < k ) such that λ ,j = λ ,k , again, we set λ ,j := inf {| n | : n ∈ Ω finitej , | n | = λ ,j } and then λ ,j ≤ λ ,k ( j < k ). If there is j and k ( j < k )such that λ ,j = λ ,k , again, we proceed the same manner. Definition 7 (Clusters with infinite elements) Let { Ω inftyj } j ⊂ Λ be a familyof wavenumber clusters satisfying the following properties: • For any n ∈ Ω inftyj , there is ( x, y ) ∈ Ω inftyj such that n and ( x, y ) satisfy thedefinition of Λ . • For any n Ω inftyj , there is no ( x, y ) ∈ Ω inftyj such that n and ( x, y ) satisfythe definition of Λ . • For each j , number of elements in Ω inftyj is always infinity. • We set λ ,j := inf {| n | : n ∈ Ω inftyj } . Then λ ,j ≤ λ ,k ( j < k ). If there is j and k ( j < k ) such that λ ,j = λ ,k , again, we set λ ,j := inf {| n | : n ∈ Ω inftyj , | n | = λ ,j } and then λ ,j ≤ λ ,k ( j < k ). If there is j and k ( j < k )such that λ ,j = λ ,k , again, we proceed the same manner. emark 8 • We see Ω finitej ∩ Ω finitek = ∅ ( j = k ) , Ω inftyj ∩ Ω inftyk = ∅ ( j = k ) and Ω finitej ∩ Ω inftyk = ∅• Theoretically, each Ω finitej and Ω inftyj ( j = 1 , , · · · ) are uniquely determined. • We see
Λ = (cid:16) ∪ j Ω finitej (cid:17) [ (cid:16) ∪ j Ω inftyj (cid:17) . In order to find specific clusters, the following observation is useful. For fixed n , we only need to see finite combinations of ( x, y ) satisfying the followinginequality: | n || n | ≤ √ x + y + 1 q ( n − x ) + ( n − y ) ≤ √ x + y , q ( n − x ) + ( n − y ) ) . Note that too large x or y will break the above inequality. Now we give severalclusters:Ω finite = { (1 , , (8 , − , ( − , } Ω finite = { (3 , , (32 , − , ( − , , (8 , , (27 , − } and Ω infty = { (1 , − , (3 , − , (5 , , (8 , , (13 , , (15 , , ( − , − , (27 , − , ( − , − · · · } . Since ( − , −
4) is two times ( − , − infty includes ∞ [ j =1 { ( j, − j ) , (3 j, − j ) , (5 j, j ) , (8 j, j ) , (13 j, j ) , (15 j, j ) , ( − j, − j ) , (27 j, − j ) , ( − j, − j ) } . This means that Ω infty has infinite elements. However, we could not figure outthe exact elements in Ω infty so far. Acknowledgements.
The first author was partially supported by Grant-in-Aid for Young Scientists (B), No. 24740086, Japan Society for the Promotion of6cience. The second author was partially supported by Grant-in-Aid for YoungScientists (B), No. 25870004, Japan Society for the Promotion of Science.
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