Theoretical and Mathematical Physics, 139(1): 500–512 (2004)
CALCULATION OF INTEGRALS OF THE HUGONIOT–MASLOV CHAIN FOR SINGULAR VORTICAL SOLUTIONS OF THE SHALLOW-WATER EQUATION S. Yu. Dobrokhotov,∗ E. S. Semenov,∗ and B. Tirozzi† We discuss the problems of the Hugoniot–Maslov chain integrability for singular vortical solutions of the shallow-water equations on the β plane. We show that the complex variables used to derive the chain automatically give most of the integrals of the complete and the truncated chains. We also study how some of these integrals are related to the Lagrangian invariant (potential vorticity). We discuss how to choose solutions of the chain that can be used to describe the actual trajectories of tropical cyclones.
Keywords: singular vortical solutions, trajectories, shallow water, integrals, chains
1. Introduction According to Maslov’s concept [1], several types of singular solutions of systems of quasilinear hyperbolic equations have the same mathematical nature and can be analyzed from the same standpoint, although they have different physical interpretations. All of them (in both the vector and the scalar cases) can be written as w(x, t) = f (x, t) + g(x, t)F S(x, t) , x = t (x1 , x2 , . . . , xn ) ∈ Rn . (1) Here, the scalar “phase” S(x, t), the vector (or scalar) “background” f (x, t), and the “amplitude” g(x, t) are smooth (infinitely differentiable) functions. The scalar function F (τ ) smoothly depends on τ for τ = 0 and has a singularity at τ = 0. The singularities of the function w(x, t) are thus determined by the zeros of the phase S. Hereafter, the left superscript t denotes transposition. These solutions also include shock waves; in this case, F = Θ(τ ), where Θ(τ ) is the Heaviside function (Θ = 0 for τ < 0 and Θ = 1 for τ ≥ 0). These solutions also include infinitely narrow solitons F = Sol(τ ), where Sol(τ ) = 0 for τ = 0 and Sol(τ ) = 1 for τ = 0. In the one-dimensional problems, the phase S has the form S = x− X(t) in both cases. Another example is given by weak singular solutions whose singularity has √ the type of the square root of a quadratic form. In this case, F = τ , and in the first approximation, S is a nondegenerate positive quadratic form with respect to x centered at the points x = X(t) that determine the trajectory Γ = x = X(t) of the singularity motion. Some properties of singular solutions of such systems of shallow-water equations on the β plane are studied in this paper. In this case, x = t (x1 , x2 ) ∈ R2 , the vectors w, f , and g have three components, and we have the formulas ρ(x, t) η R(x, t) u1 = u1 (x, t) + U1 (x, t) S(x, t) , (2) u2 u2 (x, t) U2 (x, t)
3
x − X(t), G(t) x − X(t) S= + O x − X(t) . 2 ∗ Institute
for Problems in Mechanics, RAS, Moscow, Russia, e-mail:
[email protected],
[email protected]. of Physics, University “La Sapienza,” Rome, Italy, e-mail:
[email protected].
† Department
Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 139, No. 1, pp. 62–76, April, 2004. 500
c 2004 Plenum Publishing Corporation 0040-5779/04/1391-0500
Here, u(x, t) = t u1 (x, t), u2 (x, t) is the two-dimensional velocity vector, the function η(x, t) is the excess over the free surface in the shallow-water theory or the so-called geopotential in atmospheric physics, and G(t) is a positive 2×2 matrix with distinct eigenvalues. The system of shallow-water equations is written later. We now once again stress that any solution of form (2) has a weak singularity: its components are continuous functions, and only some of its derivatives are discontinuous. In addition, we note that the singular component of solution (2) has a vortex character. A solution of form (2) thus describes a solitary vortex “running” over a smooth “background.” Various general properties of solutions (1) are discussed in [2]–[8]. In particular, these are the properties of “structural” self-similarity and stability and the following algebraic property: any power or any product of solutions (1) has a structure similar to (1) and can be decomposed into a smooth component and a singular part determined by the same function F (i.e., solutions with the above functions F form an algebra with two generators). For many hyperbolic systems, the properties of “structural” self-similarity and stability, in some sense, permit proving the uniqueness of singularities of the above types in a wide class of singular solutions. For example, among functions of form (1) with a continuous function F such that F (0) = 0, F → ∞ as τ → +0, and F is smooth for τ > 0,1 only functions (2) having a singularity of the type of the square root of a quadratic form can be solutions of the system of shallow-water equations. The system of shallow-water equations on the β plane has the form ∂η + (∇, ηu) = 0, ∂t where
T=
∂u + (u, ∇)u − ωTu + ∇η = 0, ∂t
0 1 −1 0
,
∇=t
(3)
∂ ∂ , . ∂x1 ∂x2
This system arises in many fields of physics and mechanics. It is important for us that this system is also used as the simplest model without viscosity and without dispersion to describe the motion (but not the creation and dissipation) of mesoscale vortices in the atmosphere (the tropical cyclones) [9]. In this case, ω =ω
+ βx2 is the doubled Coriolis frequency on the β plane, and ω
and β are the respective values of this frequency and of its derivative along the meridian x2 at the point where the β plane is tangent to the Earth’s sphere. The coordinate system is oriented such that the x1 axis is directed along the parallel to the East and the x2 axis is directed along the meridian to the North. We consider only vortices in the northern hemisphere and therefore assume that ω
> 0 and β ≥ 0. System (3) has several useful properties, in particular, a Hamiltonian structure and conservation laws (see, e.g., [9], [10]). It is important for us that this system has the Lagrangian invariant (the Rossby invariant of the potential vorticity) Π=
u2x1 − u1x2 + ω , η
which satisfies the relation Πt + (u, ∇)Π = 0.
(4)
We can say that the use of system (3) and of its solutions of form (2) to describe mesoscale vortices in the atmosphere imposes very strong restrictions on both the trajectory Γ and the behavior of the phase S and the values of the geopotential ρ0 = ρ|Γ , as well as on the velocities u and their derivatives on the trajectory Γ. In particular, the appropriate trajectories Γ must be sufficiently smooth in the sense that they cannot have loops during the lifetime of tropical cyclones (5–10 days). The geopotential can vary along 1 For example, such functions are τ α , τ log τ , and their linear combinations.
501
the trajectory rather slowly. In what follows, we keep these considerations in mind. We also note that for mesoscale vortices in the atmosphere, the potential vorticity takes sufficiently large values and at least does not vanish. We point out another important property of solutions (1) and (2): they generate (infinite) chains of ordinary equations relating the Taylor coefficients of the functions f , g, and S in a neighborhood of a singularity. For shock waves, the first equations in this chain are the well-known Hugoniot conditions. The other equations can be considered corrections to them. These chains first appeared in [1]; we call them Hugoniot–Maslov chains. Their closure (or truncation) leads to finite systems (truncated chains), which can be used to construct approximate descriptions of the evolution of solutions (1). The integrability properties of the truncated Hugoniot–Maslov chains corresponding to solutions (2) of system (4) were found in [3] and [4]. It turned out that these properties are related to a passage to new variables in which the chain has a relatively simple form. But the meaning of these transformations was not clear. In this paper, we propose an approach that makes the meaning of the new variables sufficiently transparent. This approach also simplifies the rather cumbersome calculations performed earlier. Moreover, we study the behavior of solutions of the truncated chain in a neighborhood of special invariant submanifolds.
2. Equations for smooth and singular components of the solution: The Cauchy–Riemann conditions Substituting a solution of form (2) in system (3) and equating the smooth and nonsmooth components separately to zero, we obtain the following two subsystems: for the regular part (which does not contain √ S), ∂ ρ + (∇, ρu + RU S) = 0, ∂t
(5)
∂ 1 u + (u, ∇)u + ∇ρ − ωTu + S(U, ∇)U + (U, ∇S)U = 0, ∂t 2 √ √ and for the irregular part (containing S ) after appropriate differentiations and eliminating S, St + (u, ∇S) R + ρ(P, U ) + 2Sf = 0, St + (u, ∇S) U + P R + 2SF = 0,
(6)
where f = Rt + (∇, uR + ρU ) and F = Ut + (u, ∇)U + (U, ∇)u − ωTU + ∇R. We thus obtain six equations for the seven unknowns ρ, u1 , u2 , R, U1 , U2 , and S. It was shown in [6] that, without loss of generality, it can be assumed that the function S satisfies the eikonal (Hamilton–Jacobi) equation corresponding to the “hydrodynamic or slow mode” of system (3): St + (u, ∇S) = 0.
(7)
It follows from Eq. (7) that system (6) can be rewritten as 1 S Rt + (∇, uR + ρU ) + ρ(U, ∇S) = 0, 2 1 S Ut + (u, ∇)U + (U, ∇)u − ωTU + ∇R + R∇S = 0. 2 Equations (5), (7), and (8) form a closed system for the seven functions ρ, u1 , u2 , R, U1 , U2 , and S. 502
(8)
Equation (4) can also be divided into the smooth and nonsmooth parts. As a result, in addition to (5), (7), and (8), we obtain two more equations: ∂ 1˘ + (u, ∇) P + (U, ∇)P˘ − P(U, ∇)S = 0, (9) S ∂t 2 ∂ (10) + (u, ∇) P˘ + S(U, ∇)P = 0. ∂t Here, P=
ρ(curl u + ω) − SR curl U + R(T∇S, U )/2 , ρ2 − SR2
S curl U − (T∇S, U )/2 − SR(curl u + ω) P˘ = . ρ2 − SR2 Because (4) follows from (3), system (5), (7)–(10) is not overdetermined: Eqs. (9) and (10) can be derived from (5), (7), and (8). But, as we see below, precisely Eqs. (9) and (10) are responsible for obtaining convenient new variables, conservation laws, and chain integrals corresponding to solution (2). We note that Eq. (7) and the condition that the phase S “in the first approximation” is a quadratic form in x − X(t) imply the condition that the trajectory Γ is “frozen” in the velocity field u: def ˙ X(t) = u X(t), t = V (t).
(11)
If we assume that the background functions ρ and u are given, then Eq. (7) can be integrated, and we obtain a linear system for R and U . If the function S did not vanish on the trajectory, then according to the well-known facts in the theory of linear hyperbolic systems, we could construct a smooth solution of the Cauchy problem for Eq. (8) with the initial data from a wide class of smooth functions. But the function S has a second-order zero on the trajectory x = X(t), which generally cannot be compensated by the first-order derivatives of S in the last terms in Eqs. (8), because these derivatives have only a first-order zero. We thus obtain a system with degeneration (this system is singularly perturbed), and this system can have smooth solutions only under certain additional conditions on its coefficients, i.e., on the functions u and ρ. An unexpected fact discovered in [1] is that the first of these equations are the Cauchy–Riemann conditions for the complex velocities u˜ = u1 + iu2 on the trajectory Γ (also see [8]):
∂u2
def ∂u1
= = q(t), ∂x1 Γ ∂x2 Γ
∂u1
∂u2
def =− = p(t). ∂x2 Γ ∂x1 Γ
(12)
In relations (12), we introduce new unknown functions q(t) and p(t). Deriving eikonal equation (7) and Cauchy–Riemann conditions (12), we simultaneously determine the first terms of the Taylor expansion of the functions R and U [3],
2
U = AT∇S + O x − X(t) ,
R=
3
2 A(2p − ω)S + O x − X(t) , 3
(13)
where A is a constant. Conditions (11) and (12) can be considered analogues of the Hugoniot conditions for square-root-type solutions (2). These conditions, which play an important role in describing the properties of singular solutions and of their trajectories, are obtained by equating the Taylor coefficients up to the third order inclusively in system (8) to zero. Equating the higher-order coefficients, we obtain sets of additional relations. 503
To derive them explicitly, we must perform a large amount of nontrivial calculations and transformations; we have recently obtained such relations only for the first- and second-order coefficients [6]. We note that we do not assume a priori that the complex velocity u(x, t) is analytic. Moreover, the analyticity property does not hold for all t (x1 , x2 ) ∈ R2 : it holds only on the trajectory Γ. It turns out [3] that this property is in some sense a consequence of the existence of the conservation law for potential vorticity (4). On the other hand, it is not yet completely clear why the Cauchy–Riemann conditions arise here. In further calculations, we use only the equation for the regular part. Moreover, we consider the Cauchy–Riemann conditions the coupling equations for the derivatives of the velocity.
3. Derivation of the Hugoniot–Maslov chain in complex variables Equating the Taylor coefficients in Eqs. (5) for the smooth component, we obtain a nonclosed chain of ordinary differential equations with a very complicated structure. To use this chain in problems of finding the trajectories of the singularity (of a solitary vortex), we must close this chain. We do this as in [11], preserving the equations for the Taylor coefficients up to the second order inclusively, i.e., preserving the coefficients of the same order as in the nonsmooth component. This “truncation” gives a system already consisting of 17 nonlinear ordinary differential equations. As previously noted, this system has curious properties, such as the integrability [3], based on this system having several integrals that can be easily found if we pass to new variables in the original chain [3], [8]. One of the goals in this paper is to show that these variables and the equations related to them arise naturally if instead of the real variables x1 and x2 in the original shallow-water equations and in the description of the functions ρ and u, we use the complex variables z = x1 − X1 (t) + i x2 − X2 (t) , z¯ = x1 − X1 (t) − i x2 − X2 (t) . We pass from the variables u to the variables u : u = u + V . This transformation is equivalent to passing to a coordinate system with the singularity center as origin and then introducing the complex variables z and z¯. To simplify the notation in what follows, we omit the primes on the new variables. To avoid complicated notation, we set ∂ ∂ ∂ ∂u ∂ +u ¯ , div u = u+ u¯ = 2 Re , ∂z ∂ z¯ ∂z ∂ z¯ ∂z 1 ∂u ∂ u ¯ ∂u curl u = − . = 2 Im i ∂z ∂ z¯ ∂z
(u, ∇) = u
We study Eqs. (5) and (9) simultaneously. In the new variables, Eqs. (5) can be written as ∂ ρ + (u, ∇)ρ + ρ div u = F (R, U, S), ∂t
(14)
∂ ∂ u + (u, ∇)u + V˙ + iω(u + V ) + 2 ρ = G(U, S), ∂t ∂ z¯ where F = − div(SRU ) and G = −S(U, ∇)U − U (U, ∇)S/2. Equation (9) does not change its form in the new variables and in the new notation. We let g (k) denote the homogeneous polynomials of degree k in the Taylor expansion of the functions g in powers of z and z¯: g
(k)
=
k i=0
504
1 gi,k−i z i z¯k−i . i! (k − i)!
The conditions of being frozen and conditions (12) imply the following relation for u: u = u(1) +u(2) +O |z|3 , u(1) = (q − ip)z. For ω, we obtain β ω = ω0 (t) − i (z − z¯), 2
ω0 (t) = ω|Γ .
We note that ρm,n = ρ¯n,m and Pm,n = P n,m because ρ and P are real-valued functions. We write ρ0 instead of ρ(0) . The problem of deriving the chain now consists in equating the coefficients of the same order in Eqs. (14). Differentiating with respect to t does not change the order of polynomials in z and z¯ in the Taylor expansion. Differentiating with respect to z and z¯ decreases the order of the corresponding polynomial by unity. Multiplying the polynomial by z m z¯n increases the order by m + n. We can therefore rewrite system (9), (14) as ∂ρ0 ∂ρ0 + div u(1) ρ0 ≡ + 2qρ0 = 0, ∂t ∂t
(15)
dV ∂ + iω0 V + 2 ρ(1) = 0, dt ∂ z¯
(16)
∂ (0) P = 0, ∂t
(17)
and as ∂ρ(k) (1) (k) + u , ∇ ρ + ρ(k) div u(1) + (u(k) , ∇)ρ(1) + ρ(1) div u(k) = Mk + F (k) , ∂t ∂u(k) + (u(1) , ∇)u(k) + (u(k) , ∇)u(1) + iω0 u(k) = Nk + G (k) , ∂t ∂ (k) P + (u(1) , ∇)P (k) = Lk S (2) ∂t
(18) (19) (20)
for k ≥ 1. Here, the functions Mk and Nk are homogeneous polynomials of degree k depending on homogeneous polynomials of lower degrees in the Taylor expansion of ρ and u. They also depend on ρ(k+1) and u(k+1) and their derivatives with respect to z and z¯. The function Lk contains neither polynomials P(k) nor their derivatives. It follows from formulas (13) that L0 = L1 = L2 = L3 = 0 and F (k) = G (k) = 0 for k = 0, 1, 2. Equation (17) readily implies one of the integrals of the chain: def
C =
ω0 − 2p = P (0) ≡ Π(0) = const . ρ0
(21)
We have the following statement based on Cauchy–Riemann conditions (12). Although obvious, it is important for further considerations. Statement. The monomials z m z¯n are the eigenfunctions of the operator u(1) , ∇ with the eigenvalues q(m + n) − ip(m − n) : (1) m n u , ∇ z z¯ = q(m + n) − ip(m − n) z m z¯n . (22) Differentiating Eqs. (19) successively with respect to z and z¯ and then using (22), we now obtain the system of equations for the Taylor coefficients: ∂ ∂ m+n um,n + (1 + m + n)q − i(1 + m − n)p + iω0 um,n = m n (Nm+n + G (m+n) ). ∂t ∂z ∂ z¯ 505
We note that this system of equations is diagonal with respect to um,n . Unfortunately, similar equations for the Taylor coefficients obtained from Eq. (18) do not have this structure. But a similar structure is inherent in the equations for the Taylor coefficients Pm,n , as follows from Eq. (20): ∂ ∂ m+n Pm,n + q(m + n) − ip(m − n) Pm,n = m n Lk . ∂t ∂z ∂ z¯ Therefore, in deriving equations for the Taylor coefficients of the functions determining singular solution (2), along with the continuity equation in system (3), it is convenient to consider Eq. (4) for the potential vorticity. By relation (22), “complex” frequencies of the form N q(m + n) + ipM + iω0 K appear in the equations for um,n and Pm,n . Using Eq. (15), we can “kill” the real parts of these frequencies by the change of (1+m+n)/2 (m+n)/2 variables um,n = ρ0 Pm,n , which leads to the equations u˜m,n , Pm,n = ρ0 m+n ∂ −(1+m+n)/2 ∂ u ˜m,n + i ω0 − (1 + m − n)p u˜m,n = ρ0 Nk + G (k) , m n ∂t ∂z ∂ z¯
(23)
m+n ∂
m,n = ρ−(m+n)/2 ∂ Pm,n − ip(m − n)P Lk , 0 ∂t ∂z m ∂ z¯n
(24)
which already have purely real frequencies. We consider the chain equations with k = 1, 2 in more detail. Expanding the functions determining the background in the Taylor series, substituting them in Eqs. (14), and collecting the terms of the same degree in z and z¯ up to the second order inclusively, we obtain the set of equations ∂ρ(1) (1) (1) + u , ∇ ρ + ρ(1) div u(1) + ρ0 div u(2) = 0, ∂t
(25)
∂ρ(2) (1) (2) + u , ∇ ρ + ρ(2) div u(1) + u(2) , ∇ ρ(1) + ρ(1) div u(2) + ρ0 div u(3) = 0, ∂t
(26)
∂ β(z − z¯)V ∂u(1) (1) (1) + u , ∇ u + iω0 u(1) + 2 ρ(2) + = 0, ∂t ∂ z¯ 2
(27)
∂u(2) (1) (2) ∂ ∂ β(z − z¯)u(1) + u , ∇ u + u(2) u(1) + iω0 u(2) + 2 ρ(3) + = 0. ∂t ∂z ∂ z¯ 2
(28)
Performing a similar procedure for Eq. (9), but collecting the third-order terms in it and dividing by S (2) , we obtain ∂ (1) (1) (1) P + u , ∇ P = 0. (29) ∂t Substituting u(1) = (q − ip)z in (27), we have (q˙ − ip)z ˙ + (q − ip)2 z + 2ρ11 z + 2ρ02 z¯ + iω0 (q − ip)z +
β(z − z¯)V = 0. 2
Collecting the Taylor coefficients of z¯ and equating them to zero, we obtain the relations ρ¯20 = ρ02 = βV /4, which can be considered coupling equations similar to (12). Further, equating the real and imaginary parts of the coefficients of z to zero, we obtain the equations q˙ − p2 + q 2 + ω0 p + 2ρ11 + p˙ + 2pq − ω0 q − 506
βV2 = 0. 2
βV1 = 0, 2
(30) (31)
We note that Eq. (21) can also be derived from Eqs. (15) and (31) and the equation for ω0 , ω˙ 0 = β Im V,
(32)
which is obtained by differentiating the relation ω0 = ω
+ βX2 with respect to t. Introducing a constant C, we can thus eliminate the function p(t) and Eq. (31) from the chain. We recall that we consider the case C = 0. We can easily deduce from Eqs. (15) and (30) that the function µ = 1/ |C|ρ0 satisfies the equation ω02 + 2β Re V −3 1 4 µ ¨+ µ=µ − 2ρ11 µ . (33) 4 4 Instead of ordinary first-order differential equations (15) and (30), we can therefore use a second-order equation for µ. Differentiating (28) successively with respect to z and z¯, we obtain Eqs. (23) for m + n = 2: 2ρ12 β(q − ip) d u ˜11 + i(ω0 − p)˜ u11 = − 3/2 + , 3/2 dt ρ0 2ρ0 2ρ21 β(q − ip) d u ˜20 + i(ω0 − 3p)˜ u20 = − 3/2 − , 3/2 dt ρ0 ρ0 d 2ρ03 u ˜02 + i(ω0 + p)˜ u02 = − 3/2 . dt ρ 0
To eliminate the variable q from these equations, it is convenient to introduce the variables 1 β Y = 2˜ u11 − 3/2 , |C|5/2 3ρ0
β 1 Z= u ˜20 + 3/2 , |C|5/2 3ρ0
U=
1 u ˜02 . |C|5/2
This implies the equations ρ12 µ3 iβ(2p + ω0 )µ3 Y˙ = i(p − ω0 )Y − −4 , 3|C| |C|
(34)
ρ21 µ3 iω0 βµ3 Z˙ = i(3p − ω0 )Z + −2 , 3|C| |C|
(35)
3
ρ03 µ U˙ = −i(p + ω0 )U − 2 . |C|
(36)
√ Differentiating (29) with respect to z¯ and multiplying the result by 1/ ρ0 , we obtain Eqs. (24) for m = 0
01 = 0. Instead of P 01 , it is more convenient to use the function and n = 1: dP 01 /dt + ipP µ3 β 2i W = − 5/2 P01 = 2 iCρ01 − u11 + u20 + |C| 2 |C| satisfying the equation ˙ = −ipW. W
(37)
This readily implies that |W | is an integral of the complete chain. Of course, this equation can be obtained by combining Eqs. (25) and (28). But it can be derived from the equation for the potential vorticity by straightforward calculations. 507
Expressing ∂ρ(1) /∂ z¯ = ρ01 in Eq. (16) in terms of W , Y , Z, and µ, we obtain the equation iσ V˙ + iω0 V − 3 (Y + W − 2Z) = 0, µ
(38)
where σ = sgn C. Applying ∂ 2 /∂z∂ z¯ to (26) and replacing the corresponding Taylor coefficients with the functions Y , Z, U , and W , we obtain the equation ρ˙ 11 + 4qρ11 + σC 4 ρ30 Im (Z + Y )W + 3Y Z + ρ0 u21 + ρ0 u21 = 0. To eliminate the variable q from this equation, it is convenient to introduce the variable r = ρ11 µ4 , for which we have the equation r˙ +
C Im (Z + Y )W + 3Y Z + µ2 (ρ0 u21 + ρ0 u12 ) = 0. 2 µ
(39)
Equation (33) can be rewritten as
µ ¨+
ω02 + 2β Re V 1 µ = µ−3 − 2r . 4 4
(40)
Equations (32)–(40), where p = (ω0 − σµ−2 )/2, can be considered the first equations in the Hugoniot– Maslov chain for singular solutions (2) of Eqs. (3). We truncate the chain by setting ρ21 = ρ12 = ρ03 = 0 and u21 = u12 = 0. We note that the function U enters only Eq. (36), which is always solvable for U , and we can hence neglect this equation while seeking the trajectory x = X(T ).
4. Integrals of the truncated chain and smooth trajectories for β = 0 As we previously noted, the variables C and |W | are integrals of the complete chain. Obviously, closing the chain for β = 0, we obtain three more integrals: |Y |, |Z|, and |U |. One more integral for β = 0, found in [3], is given by 1 3 4 λ = − 2rµ − 2|C| Re (Z − Y )W + Y Z . 4 2 In contrast to the above integrals, we still do not understand the origin of this integral. If we now introduce the variables λ instead of the variable r and the variable s = µ, ˙ then we finally obtain the desired truncated chain in the form σ V˙ + iω0 V − 3 (Y + W − 2Z) = 0, µ
X˙ = V,
iβ(2p + ω0 )µ3 , Y˙ = i(p − ω0 )Y − 3|C|
iω0 βµ3 Z˙ = i(3p − ω0 )Z + , 3|C|
(42)
˙ = −ipW, W
(43)
µ˙ = s,
(44)
ω˙ 0 = β Im V,
3ZY ω02 + 2β Re V −3 µ=µ s˙ + λ + 2|C| Re (Z − Y )W + , 4 2 4p Im W λ˙ = βµ3 ω0 Im Y − (2p + ω0 ) Im Z − , 3 508
(41)
(45) (46)
where p = (ω0 − σµ−2 )/2. Instead of the last two equations, we can use the equation r˙ +
C Im (Z + Y )W + 3Y Z =0 µ2
and Eq. (40). If we use the function ψ = 1/µ instead of µ, set b2 = (ω02 + 2β Re V )/ 4|C|2 , and introduce t t the new variable Φ = 0 (1/µ2 ) dt = |C| 0 ρ0 dt instead of t, then the equation for µ can be rewritten as the Ermakov equation d2 ψ b2 + Qψ = (47) dΦ2 ψ3 with the potential Q = λ + 2|C| Re (Z − Y )W + 3Y Z/2 . For β = 0, because of the special form of the “frequencies” in the equations for Y , Z, and W , the potential Q has the form Q = λ + Re(α1 eiΦ + α2 e2iΦ ), where α1 and α2 are complex constants. According to the well-known properties of the Ermakov equation [12], Eq. (47) for β = 0 is therefore equivalent to the Hill equation zΦΦ + Q(Φ)z = 0.
(48)
The existence of integrals of the truncated chain thus permits reducing the integration of it for β = 0 to the integration of the well-known scalar equation with a periodic coefficient. This also permits finding several properties of the trajectories Γ and relating them to the properties of Hill equation (48) (such as stability and the existence of zones and gaps in the spectrum; see [3], [4], [8]). It is interesting that the Hill equation is obtained in finding some exact solutions of the shallow-water equations with β = 0 [13]. For β = 0, some simple trajectories Γ can be easily obtained for system (41)–(44). In particular, by successively setting Z = 0, W = 0 and Y = 0, W = 0 and Z = 0, Y = 0, we obtain trajectories in the form of circles on which the direction of motion depends on the stability zones of Eq. (48) (see [3], [4]). If we assume that the parameter β is small, then the terms with β in (41)–(44) play the role of an adiabatic perturbation, which, in particular, “deforms” the above trajectories. One of the effects caused by these terms is that the motion along the “deformed” circles can change its direction and trajectories Γ in the form of zigzags thus appear. It also turns out that such “deformed” trajectories can be described using Hamiltonian systems with a single degree of freedom [7], [8].
5. Critical manifolds and slow motions in the case of nonadiabatic influence of the β effect In the derivation of the Hamiltonian systems mentioned at the end of the preceding section, it is essential to assume that the parameter β/C is relatively small. If this parameter is not small, then it is clear that the terms with β cannot be considered an adiabatic perturbation; moreover, they can play an essential role in describing the trajectories Γ. System (41)–(44) has not yet been studied from this standpoint. On the other hand, the set of its solutions is very large and depends on 14 arbitrary parameters (because the equation ω˙ = βV2 is obtained as the result of differentiating the equation ω = ω
+ βX2 and one of the 15 constants of integration is not free; we also exclude the initial data for the function U from the set of parameters). We recall that we want to study system (41)–(44) to describe the trajectories of tropical cyclones. As already mentioned, far from all solutions of this system can be used for this purpose, and to choose appropriate solutions, we must use additional arguments. The velocities of their centers are relatively small (∼ 4–20 km/hour) and vary sufficiently slowly during the characteristic lifetime of tropical cyclones (5–10 days). The value of the geopotential on the trajectory varies equally slowly. Therefore, the components V 509
and µ of truncated chain (41)–(44) must also vary sufficiently slowly, and the values of the components V1 and V2 must be relatively small. Recalling the explicit expressions for the right-hand sides in system (41)– (44), we can assume that to obtain the desired solutions in the first approximation, we can omit all the derivatives with respect to t, which leads to an equation for the critical points of the system. An elementary analysis shows that these critical points form two two-dimensional (intersecting) smooth “critical” manifolds characterized by the conditions: (a) W = 0 and (b) p = 0. Such manifolds contain trajectories of uniform motion parallel to the x1 axis (i.e., of motion along the parallel). We can naturally assume that the desired solutions must be in a neighborhood of such critical manifolds and that all the components of these solutions must also vary sufficiently slowly. Clearly, in the first approximation, these solutions are determined by linearized equations and are therefore characterized by the eigenvalues of the corresponding Jacobi matrix. Based on this, we write system (41)–(44) in a somewhat different form. To the complex unknown functions V , Y , Z, and W , we add the complex conjugate functions V , Y , Z, and W and distinguish the variables V , V , Y , Y , Z, Z, W , W , λ, and s in the complete set of unknowns. We let ξ denote the column vector composed of these functions. Then system (41)–(44) can be rewritten as ξ˙ = F 0 (ω0 , µ) + F 1 (ω0 , µ, ξ) + F 2 (ω0 , µ, ξ), ω˙ 0 =
β (V − V ), 2i
(49)
µ˙ = s,
X˙ = V. Here, F 1 (ω0 , µ, ξ) = Q(ω0 , µ)ξ, where Q is a 10×10 matrix; the nonzero components of the vector F 0 independent of ξ are FY0 = −FY0 = −iβ(2p + ω0)µ3 /(3|C|), FZ0 = −FZ0 = iβω0 µ3 /(3|C|), and Fs0 = −ω02 µ/4; and the only nonzero component of the vector F 2 quadratically depending on ξ is determined by the relation Fs2 = µ−3 2|C| Re (Z − Y )W + 3ZY /2 . The vector ξ 0 (ω0 , µ) determining the critical points of system (49) (without the equation X˙ = V ) has the components V0 =V
0
Y0 =Y
0
0
4
λ =µ
=
β(2p − ω0 )(p + ω0 ) , |C|ω0 (p − ω0 )(3p − ω0 )
=
βµ3 (2p + ω0 ) , 3|C|(p − ω0 )
0
Z0 = Z =
ω02 kβ ω02 − 12pω0 − 12p2 − , 4 2ω0 (ω0 − 3p)(ω0 − 2p)
βµ3 ω0 , 3|C|(ω0 − 3p)
0
W 0 = W = 0,
s = 0.
If we add the equation V = 0 to the equations for critical points, then we obtain an equation relating p or µ to ω0 . The following two situations are possible here:
a1.
p = −ω0
a2.
p=
ω0 2
1 ⇐⇒ µ = √ , 3ω0 ⇐⇒ C = 0.
The motion along the trajectories far from a neighborhood of these sets is rather fast, and it is unlikely that the corresponding solutions of the truncated chain can be used in the problem of calculating the tropical cyclones. Hence, we do not consider the neighborhood of critical manifold b (p = 0) here. We thus have 510
one-dimensional invariant manifolds parameterized by ω0 . We now replace the variable ξ with ξ according to the formula ξ = ξ + ξ 0 (ω0 , µ) and rewrite system (49) as ξ˙ = Λ(ω0 , µ)ξ + F 2 (ω0 , µ, ξ ), β (V − V ), µ˙ = s , 2i Here, Λ(ω0 , µ) is a 10×10 matrix acting on the vector ξ as ω˙ 0 =
Λξ = Q(ω0 , µ)ξ −
(50) X˙ = V + V 0 .
(51)
∂ξ 0 β ∂ξ 0 ∂F 2 (V − V ) − s + ω0 , µ, ξ 0 (ω0 , µ) ξ . ∂ω0 2i ∂µ ∂ξ
The critical points are now determined by the relation ξ = 0, and the slow motions in a neighborhood of the critical manifold are primarily determined by the eigenvalues of Λ(ω0 , µ). In the usual procedure for studying systems in a neighborhood of invariant submanifolds, the stability of the system must first be examined in the linear approximation. But we are primarily interested in the existence of slow motions in system (50). If all the eigenvalues of Λ are large in absolute value, then system (50) does not have slow motions independently of whether these values are real or complex, i.e., of whether the linear part of the system gives rise to a stable or unstable situation. Therefore, before we continue our study of system (50) in a neighborhood of critical manifolds, which is generally not a trivial task, we must first understand whether the system can have slow trajectories at all and if so for what values of the parameters. Preliminary calculations show that in case a2, all eigenvalues are relatively large and it is hardly possible to use the corresponding solutions in the problem of calculating tropical cyclones. Hence, we restrict ourself √ here to case a1, where 1/ µ ∼ 3ω0 ⇔ p ∼ −ω0 , and write only the final result obtained using computers. Namely, the eigenvalues λ1 , λ2 , . . . , λ10 of the matrix Λ for the first subsystem at the critical points (where W = 0 and p = −ω0 ) are λ9 = ip, λ10 = −ip, and the other eigenvalues determined by the equation 9δ 171δ 243δ 2 8 2 6 4 4 6 2 λi + 22ω0 λi + 105 + ω0 λi + 148 + ω0 λi + 64 − 81δ − ω08 = 0, 2 2 2 where δ = β 2 /(3|C|ω03 ) is a dimensionless parameter. The slow motions in a neighborhood of the critical manifold are possible if this polynomial has “small” roots λ. One of the eigenvalues is zero if and only if √ the term (64 − 81δ − 243δ 2 /2)ω08 in the last equation is zero. This implies δ = 465 − 9 /27 ∼ 0.465. As an example, we consider the case where the β plane is tangent to the Earth’s surface at a point at the latitude 20◦ ( ω ≈ 0.18 hour−1 ) and ask for what values of the geopotential is one of the eigenvalues zero
in a neighborhood of this point. The corresponding calculations based on the equalities 3ω0 = µ12 = |C|η Γ and η = gH, where g is the acceleration of gravity and H is the conventional height of the atmosphere, show that this situation can be realized if we assume that the value of the geopotential on the trajectory Γ is equal to the value of the geopotential corresponding to the height of an atmosphere layer of ∼ 10 km (η|Γ ≈ 860000 km2 /hour2 ). In other words, slow trajectories are possible in a neighborhood of the latitude 20◦ if we assume that the height of the atmosphere layer is equal to ∼ 10 km. We note that in problems on mesoscale vortices in the atmosphere (in contrast to the ocean), the height of the atmosphere layer is a conventional quantity often assigned a value of 7–12 km.
6. Conclusion We propose a new method for obtaining the Hugoniot–Maslov chain. This method explains the meaning of several integrals of the truncated and complete chains and simplifies the calculations. We obtain necessary conditions for the conventional height of the atmosphere layer under which the truncated chain can have smooth trajectories under the assumption that the influence of the β effect is not adiabatic. 511
Acknowledgments. This study is supported by the Russian Foundation for Basic Research (Grant No. 01-02-00850). REFERENCES 1. V. P. Maslov, Sov. Math. Surv., 35, 252–253 (1980). 2. V. G. Danilov, V. P. Maslov, and V. M. Shelkovich, Theor. Math. Phys., 114, 1–42 (1998); V. V. Bulatov, Yu. V. Vladimirov, V. G. Danilov, and S. Yu. Dobrokhotov, Dokl. Rossiiskoi Akad. Nauk, 338, 102–105 (1994). 3. S. Yu. Dobrokhotov, Russ. J. Math. Phys., 6, 137–173, 282–313 (1999). 4. S. Yu. Dobrokhotov, Theor. Math. Phys., 125, 1721–1741 (2000). 5. S. Yu. Dobrokhotov, K. V. Pankrashkin, and E. S. Semenov, Russ. J. Math. Phys., 8, 25–54 (2001). 6. E. S. Semenov, Math. Notes, 71, 902–913 (2002). 7. S. Yu. Dobrokhotov and B. Tirozzi, Dokl. Rossiiskoi Akad. Nauk, 65, 453–458 (2002). 8. S. Yu. Dobrokhotov, E. S. Semenov, and B. Tirozzi, “Hug´ oniot–Maslov chains for singular vortical solutions of quasi linear hyperbolic systems and typhoon trajectory,” Contemp. Math. (2003 to appear). 9. J. Pedlosky, Geophysical Fluid Dynamics, Springer, Berlin (1982); F. V. Dolzhanskii, V. A. Krymov, and D. Yu. Manin, Sov. Phys. Usp., 33, 495–520 (1990). 10. V. E. Zakharov and E. A. Kuznetsov, Phys. Usp., 40, 1087–1116 (1997). 11. R. Ravindran and P. Prasad, Appl. Math. Lett., 3, 107–109 (1990). 12. V. F. Zaitsev and A. D. Polyanin, Handbook of Exact Solutions for Ordinary Differential Equations [in Russian], Fizmatlit, Moscow (1995); English transl.: A. D. Polyanin and V. F. Zaitsev, CRC, Boca Raton, FL (1995). 13. C. Rogers and W. K. Schief, J. Math. Anal. Appl., 198, 194–220 (1996); C. Rogers, Phys. Lett. A, 138, 267–273 (1989).
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