On the stability of certain collisional configurations of hydromagnetic flows

Physics Letters A 374 (2010) 4128–4132

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Physics Letters A www.elsevier.com/locate/pla

On the stability of certain collisional configurations of hydromagnetic flows Manuel Núñez Departamento de Análisis Matemático, Universidad de Valladolid, 47005 Valladolid, Spain

a r t i c l e

i n f o

Article history: Received 21 April 2010 Received in revised form 22 July 2010 Accepted 6 August 2010 Available online 13 August 2010 Communicated by A.P. Fordy

a b s t r a c t We study the flow of a hydromagnetic fluid toward an obstacle in two different cases: when this is a rigid wall or when two plasma masses collide with each other. The magnetic field far from the obstacle is assumed to be aligned with the flow. The diffusivity is taken as low, and a boundary layer approach for the stationary MHD system is considered. The relevant equations turn out to be a generalization of the Falkner–Skan ones, and while analytical solutions are impossible to obtain, a qualitative analysis shows that whenever the size of the Alfvén speed far from the interface exceeds the size of the fluid velocity, the system has no nontrivial solutions. The interpretation of this is that in this case disturbances occurring in the boundary layer travel upstream and disturb the boundary conditions at the outer layers. © 2010 Elsevier B.V. All rights reserved.

1. Introduction We wish to study the stability of MHD flows colliding against an obstacle when far away from it the magnetic field and the velocity are aligned. In particular we deal with two different scenarios: the first one is a direct generalization of the Hiemenz flow (see e.g. [1, p. 112]) where an incompressible two-dimensional fluid of low viscosity flows against a rigid wall and escapes laterally. The stream function of the flow, away from the wall, has the form

ψ(x, y ) = axy .

(1)

The MHD stationary equations are studied under the assumption that the low viscosity will mean that the fluid is inviscid except in the immediate vicinity of the wall, where a boundary layer develops. This yields a particular case of the Falkner–Skan equation [2], which also includes the classical case of parallel flow around a sheet [3]. A good account of these problems may be found in [1]. If we admit that the fluid, in common with most astrophysics plasmas, has high conductivity and the magnetic flux function away from the wall has the form

A (x, y ) = bxy

(2)

(which means that magnetic field and velocity are aligned there), the Falkner–Skan equation turns into a system involving ψ and A about which new questions arise. Foremost of them is the very existence of solutions satisfying the MHD system plus the boundary conditions. The second scenario is probably more interesting, as it is related to the classical picture of planar magnetic reconnection. The

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first rigorous studies of this topic [4,5] involved the collision of two masses of plasma with magnetic fields which are parallel and opposite at the current sheet where contact takes place, and where the plasma escapes laterally. Even this simple geometry is too complex to obtain analytical solutions of the MHD system: the most successful studies [6] consider only the evolution of the first Taylor coefficients of the flux and stream functions near the center. The transport of a transverse magnetic field by a conducting flow has great physical importance by its relation with magnetic reconnection, and for certain choosings of flux functions it may be analytically solved [7,8]. However, there exists in many instances a tendency for the flow and the magnetic field to be aligned [9]. The most extreme case is the alfvénic state B = ±v, which yields null magnetic induction, as well as null momentum forcing (except for the gradient of the total pressure). This state is noticeable e.g. in the solar wind [10]. The Falkner–Skan equations are obtained when looking for self-similar solutions of the relevant equations; we will also look for self-similar flux and stream functions satisfying appropriate boundary conditions away from the interface. We will need to make an assumption when looking for self-similar solutions: the quotient between the viscosity ν and resistivity η (or the kinetic and magnetic Reynolds numbers in adimensional formulation), called the magnetic Prandtl number Prm , is equal to one. It would be misleading to state that this is often the case: in many astrophysical totally ionized plasmas, Prm is very small ( 10−2 in the Sun’s convection zone [11–13]) while for rarefied partially ionized plasmas such as the ones in interstellar clouds, Prm ∼ 105 . Prm is close to one in certain turbulent plasmas [14,15], but this is not our case. The reason is purely pragmatic, as this simplifies considerably the system, and as such has been used in many numerical analyses of theoretical models: see e.g. [6,16,17]. In spite of the strength of this assumption, we hope that the result will il-

M. Núñez / Physics Letters A 374 (2010) 4128–4132

luminate the role played by the magnetic field in the stability of collision configurations. The model studied in this Letter is certainly not new; many variations of boundary problems between fluids with different characteristics have been studied from the nineteen sixties. However, classical papers such as [18] study only flat flows (Blasius problem) and consider only linearized analyses of the main equations, with all its obvious limitations. Nonetheless, the physical intuition of this particular paper is remarkable.

ρ,

(3) (4) (5)

u denotes the flow velocity, B the magnetic field, p the pressure and J = ∇ × B the current density. When both lie in the plane (x, y ), a streamfunction ψ and magnetic potential A may be found: they satisfy



∂ψ ∂ψ , ,− ∂y ∂x   ∂A ∂A . ,− B= ∂y ∂x u=

(6) (7)

For static problems, ψ and A are unique except by the addition of constants, so it is enough to label a given streamline and magnetic field line to fix ψ and A. We will take ψ = 0 and A = 0 at Γ . The induction equation (4) may be uncurled to yield (see e.g. [11, p. 366]):

∂ A ∂( A , ψ) + = η A . ∂t ∂(x, y )

y

ν xf √

(9)

ν

We will from now on take straightforward to find





u = xf  , − f ,



ν = 1 for simplicity of notation. It is 



B = xg  , − g ,

  u = xf  , − f  ,

The MHD equations for an incompressible plasma of density viscosity ν and resistivity η may be written as



√



u · ∇ u = xf  2 − xf f  , f f  ,

2. Mathematical formulation

∂u 1 = ν u − u · ∇ u + J × B − ∇ p, ∂t μ0 ρ ∂B = ηB − u · ∇ B + B · ∇ u, ∂t ∇ · u = ∇ · B = 0.

 , ν   √ y . A (x, y ) = ν xg √

ψ(x, y ) =



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  B = xg  , − g  .

As stated, we will assume η = ν and define a new A (denoted √ by the same symbol) as the old one divided by μ0 ρ , to avoid dragging this coefficient. We see that A is really the Alfvén speed; nonetheless we will often use the words magnetic potential to avoid confusion with real flow velocity, and because in most cases where the Alfvén speed is used it is a constant datum (given e.g. by a background fixed magnetic field) and not a variable. We will study (3)–(4) in the stationary case and assuming that ν is low enough to be ignored except in the regions where the gradients of ψ and A are large; that is, near the collision region. (Notice that (1) and (2) satisfy trivially (8) when ν = η = 0.) This is the rigid wall in the first scenario and the interface between the plasma masses in the second; we will take both to be at the x-axis y = 0. In the vicinity of this line we will take the boundary layer approximation, which corresponds to the zeroth order approximation in an expansion of ψ and A in powers of ν ; for a rigorous mathematical justification see e.g. [19]. For practical purposes this means that since all the magnitudes are assumed to vary much more rapidly in the y-direction than the x-direction, ∂ f /∂ x  ∂ f /∂ y for all relevant f and the x-derivatives are ignored whenever they are added to an analogous y-derivative. Thus e.g. ψ becomes ∂ 2 ψ/∂ y 2 . On the other hand, for y → ∞, ψ should behave as axy. Analogous results hold for A, and in the second case we also assume that for y → −∞, ψ ∼ cxy, A ∼ dxy. As stated, we will look for self-similar √ solutions. In this case the self-similar variable is simply s = y / ν [1]; we therefore assume

(10)

The horizontal component of the momentum equation yields

−xf  = −xf  2 + xf f  + xg  2 −

∂p . ∂x

(11)

For the vertical component we use the fact that at the boundary layer ψ and A (i.e. f and g) are vanishingly small. This follows from assumptions on the nature of the wall which will be explained later. Thus the equation reduces to the vanishing of the forcing given by the total pressure gradient,

  ∂ 1 2  p + x g = 0. ∂y 2

(12)

This implies

p=−

x2  2 g + h(x), 2

(13)

at the boundary layer. On the other hand, for y 1,

u · ∇ u = a2 (x, y ),

B · ∇ B = b2 (x, y ),

(14)

which means that the momentum equation there is

∇ p = −a2 (x, y ). Comparing (13) with (15) and making y →

(b2 − a2 )x, so that at the layer (8)





B · ∇ B = xg  2 − xgg  , gg  ,

(15)

∞, we find h (x)

  ∂p = −xg  g  + b2 − a2 x, ∂x

=

(16)

and after dividing by x, the momentum equation reduces to

f  = − f f  + f  2 + gg  − g  2 + b2 − a2 .

(17)

The induction equation is found simply by substituting in (8) the functional form of ψ and A:

g  = f  g − f g  .

(18)

Often this problem is studied by applying the curl operator to the momentum equations to get rid of the pressure gradient and then integrating the resulting equation (of fourth order in f ). However, in order to find the appropriate integration constant one needs to assume

lim f  (s) = 0,

s→∞

(19)

which in principle overdetermines the system. We only need

lim f  (s) = a,

s→∞

lim g  (s) = b.

s→∞

(20)

In the plasma collision case, the limits when y → −∞ must also be taken into account: if we set ψ(x, y ) ∼ cxy, A (x, y ) ∼ dxy when y  0, (20) must be repeated replacing a by c and b by d when s → −∞. Let us consider now each of the two cases. If a rigid wall

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M. Núñez / Physics Letters A 374 (2010) 4128–4132

is set at y = 0, the presence of viscosity means that the velocity vanishes at the wall. Since



 v(x, 0) = xf (0), f (0) , 

(21)

we must have f (0) = f  (0) = 0. The condition for g depends on our assumptions on the nature of the wall. If it is perfectly conducting, the normal component of the field vanishes, which means g (0) = 0; the field does not penetrate the wall. Any other assumption would need to account for diffusion of the field into the wall, which in practice would mean solving an additional boundary value problem coupled with the first one. Summing everything up, the case of a rigid perfectly conducting wall satisfies

f  = − f f  + f  2 + gg  − g  2 + b2 − a2 , f (0) = f  (0) = g (0) = 0, lim f  (s) = a, lim g  (s) = b.

(22)

s→∞

The second scenario is somewhat different. To impose boundary conditions for f  , f  and g  at ±∞ plus initial conditions at s = 0 would overdetermine the system. In practice this means that we cannot guarantee the axis y = 0 to be the collision front, which is logical since we cannot guarantee it even to be a stream (and field) line. There is one clear exception: when a = c, b = d, the symmetry of the equations means that f (−s) = − f (s), g (−s) = − g (s), i.e. ψ(x, − y ) = −ψ(x, y ), A (x, − y ) = − A (x, y ); thus f (0) = f  (0) = g (0) = 0 holds. The best we can do is to pose a transmission problem: 

g = f g− fg , f  = − f f  + f  2 + gg  − g  2 + b2 − a2

for s > 0,

f  = − f f  + f  2 + gg  − g  2 + d2 − c 2

for s < 0,

lim f  (s) = a,

[f]= f



= f

(30)

F  G  + b2 − a2 tends to zero when s → ∞, it is a bounded

Since h = function: say |h|  M. Let us write (26) as

F  = −G F  + h,

(31)

and let

s



 

= [ g ] = g = 0,

(32)

s0

Since (31) is equivalent to



 e Γ F  = e Γ h,

F (s) = e

(33)

−Γ (s)



s



F (s0 ) +

e

Γ (r )

h(r ) dr .

(34)

s0

lim g  (s) = d,

 



G (r ) dr < −α s2 − s20 .

Γ (s) =



The estimate (32), together with the fact that h is bounded, means that the function

s→−∞

g  = f  g − f g  ,

(23)

(24)

r → e Γ (r ) h(r ), is integrable in (s0 , ∞). Hence there exists the limit

λ(s0 ) = lim

s→∞



s



F (s0 ) +

e

Γ (r )

h(r ) dr

s0

∞



e Γ (r ) h(r ) dr .

= F (s0 ) +

(35)

s0

If λ(s0 ) = 0, F  (s) behaves for large s as

F  (s) ∼ λ(s0 )e −Γ (s) .

and define the new (Elsässer-like) variables

(36)

Since by (32)

F = f + g,

e −Γ (s) > e −s0 e s , 2

G = f − g.

(25)

F  = −G F  + F  G  + b2 − a2 , 

 

2

2

= −F G + F G + b − a ,

plus the boundary conditions

2

(37)

and

By adding and subtracting (18) and (24) we obtain for s > 0

G

3. Unstable configurations



where [h] represents the jump of h at the interface s = 0. This is not a single differential system in the whole space and therefore there is no guarantee that f and g will be totally smooth across s = 0, except in the case where the forcings are identical: b2 − a2 = d2 − c 2 . Then the first three equations in (23) ensure that f  , g  as well as all subsequent differentials are continuous across s = 0, so that the two solutions seam smoothly and form a legitimate global solution. Hence we will consider only this case. Our first step to analyze if solutions to (22) and (23) exist is to differentiate (18) to obtain



We may notice that if ( F , G ) is a solution of (26)–(27), for any γ > 0, the functions P (s) = γ F (γ s), Q (s) = γ G (γ s) form also a solution. This fact is used in the hydrodynamic case to normalize a to ±1. In our case this will not be important for the problem of existence of solutions, and therefore we leave the functions as they are.

s→∞

lim f  (s) = c ,

 

(29)

(26) is equivalent to the integro-differential equation

lim g  (s) = b,

s→∞

s→−∞

lim G  (s) = a − b.

s→∞

G (s) < −2α s.

s→∞



(28)

Let us see what happens when b2 > a2 , i.e. |b| > |a|. Then at least one of the numbers a + b or a − b is less than zero; since both cases are treated in the same way, for concreteness assume a − b < 0. (29) implies that if we call α = (b − a)/4 > 0, there exists s0 > 0 such that for all s > s0 ,

g  = f  g − f g  ,



lim F  (s) = a + b,

s→∞

(26)





s

F (s) = F (s0 ) +

(27)

s0

obviously

F  (r ) dr ,

(38)

M. Núñez / Physics Letters A 374 (2010) 4128–4132





lim F  (s) = sign of λ(s0 ) ∞,

s→∞

(39)

which contradicts (28). The only way to avoid this is that λ(s0 ) = 0. However, since any number greater than s0 may be taken as well, we would need λ(s) = 0 for all s  s0 . In that case, differentiating (35) with respect to s0 , we find

F  (s) = e Γ (s) h(s),

∀s  s0 .

(40)

4131

This means that it takes f  longer to reach its limit a: in other words, the boundary layer thickens, which may be visualized as the consequence of one of the Elsässer variables trying to go upstream and being barely contained by the flow. It is curious to notice that this thickening is interpreted in several classical papers such as [18] as a proof that the magnetic field tends to stabilize the layer. This terminology is doubtful, since the end term of much stabilization seems to be the boundary layer filling the space and effectively blowing up.

Comparing with (33), we find 4. Conclusions

  F  = e Γ F  ,

(41)

from s0 on. Since both F  and e Γ tend to zero when s → ∞, we may integrate (41) to

F  = e Γ F  .

(42)

Unless Γ (and therefore G) vanish for s > s1 , then F  (s) = 0 for s > s1 . Since lims→∞ G  (s) = b − a = 0, the first possibility cannot hold, which means that F is a linear function from some point on. However, since (26) and (27) are analytic equations, there cannot exist a first real number s1 > 0 such that F is linear starting precisely from s1 ; if it is linear to the right of s1 , it is linear to the left. Thus s1 = 0 and F is always linear. Now (26) is reduced to

F  G  + b 2 − a 2 = 0,

(43)

which means that G  is constant as well and therefore G is also linear; thus both f and g are linear for s > 0. Let us consider now the rigid wall case (22). The fact that f (0) = f  (0) = 0 means that f = 0 and therefore a = 0. If g  (0) = 0, also b = 0, which contradicts our hypothesis. If g (0) = 0, g (s) = bs and A (x, y ) = bxy. Hence the only possible solution occurs when the wall is perfectly conducting, there is no flow and B(x, y ) = b(x, − y ). In the collision case, if |b| > |a|, both f and g are linear; given the behavior at infinity of ψ and A (1)–(2), necessarily f (s) = as, g (s) = bs for s > 0. Thus

f (0) = 0,

f  (0) = a,

g (0) = 0,

g  (0) = b.

f  (0) = 0, (44)

Since also |d| > |c | the same argument may be applied to s < 0, which implies a = c and b = d; that it, the system is perfectly symmetric: ψ(x, y ) = −ψ(x, − y ) = axy, A (x, y ) = − A (x, − y ) = bxy. The reason for this phenomenon is clear: when the size of the Alfvén speed exceeds the fluid velocity, one of the Elsässer variables u ± B travels backwards along the streamlines. Thus the behavior of the magnitudes at the boundary layer is transported upstream towards the outer layers and it is impossible to set stationary boundary conditions there. In our case, this effect is enhanced because Alfvén waves travel along the hyperbolic paths defined by the ambient field, which has a converging configuration that amplifies the outgoing disturbances. It would be interesting to analyze with precision the evolution of the boundary layer when |b| approaches a to refine this description. Unfortunately (26) and (27) are extremely difficult to integrate with the standard numerical integrators, growing extremely rapidly for small s and surpassing the tolerance of common Runge–Kutta methods, although further study is in progress. Numerical integration of these equations seems to be a considerable undertaking and we omit it from this qualitative study. Anyway, from the fact that F and G behave for large s as 2 e −(a±b)s /2 ,

we deduce that for |b| near a, one of the terms adding to f decreases as exp( s2 ), with small, and the other decreases rapidly.

The modelling of an hydromagnetic fluid flowing toward an obstacle may cover several interesting physical situations. The most straightforward one is to take the obstacle as a rigid wall, in analogy to the classical hydrodynamic Hiemenz flow. Potentially more relevant, due to its relation with the geometry of magnetic reconnection, is the case of the collision of two masses of plasma. We will consider the case where the low diffusivity allows us to treat the collision front as a boundary layer. When the magnetic field and the flow are aligned away from this front, self-similar solutions of the stationary MHD system may be studied, obtaining a system generalizing the Falkner–Skan equation. While for the Hiemenz flow the problem may be set with classical initial and boundary conditions, for the collision model transmission conditions are necessary to close the system. The nonlinear character of the equations precludes obtaining general analytical solutions, but a qualitative study shows that in the rigid wall case, an Alfvén speed larger than the velocity at infinity only allows for the trivial solution when there is no flow at all. In the collision scenario, the same nonexistence result holds when the Alfvén speed is larger than the velocity at both sides, except when they are precisely symmetric, and even in this case only trivial solutions exist. The explanation of this is that when the Alfvén speed exceeds the flow velocity, disturbances occurring at the boundary layer travel upstream making impossible to establish stationary boundary conditions there. This occurs because the group velocity of the Alfvén waves is always directed along the magnetic field, which in our case is collinear with the flow. Moreover, these waves travelling against the flow are channeled by the hyperbolic paths defined by the ambient field, which yields a converging effect that amplifies the outgoing disturbances. In fact when the Alfvén speed nears the flow velocity, the boundary layer thickens tending to fill the whole space and making meaningless the Prandtl approach. Acknowledgements The author wishes to thank the two reviewers of this Letter for their demand of a solid physical and mathematical foundation of it, which led to the explanation of the importance of the Alfvén/flow speed ratio. References [1] H. Schlichting, K. Gersten, Boundary Layer Theory, 8th edition, Springer-Verlag, Berlin–Heidelberg, 2003. [2] V.M. Falkner, S.W. Skan, Phil. Mag. 12 (1931) 865. [3] H. Blasius, Z. Math. Phys. 56 (1908) 1. [4] P.A. Sweet, Nuovo Cimento Ser. X (Suppl. 8) (1958) 188. [5] E.N. Parker, Astrophys. J. Suppl. Ser. 8 (1963) 177. [6] H.K. Moffat, R.E. Hunt, A model for magnetic reconnection, in: K. Bajer, H.K. Moffat (Eds.), Tubes, Sheets and Singularities in Fluid Dynamics, Kluwer Academic Publishers, 2002, pp. 125–138. [7] B.U.O. Sonnerup, E.R. Priest, J. Plasma Phys. 14 (1975) 283. [8] I.J.D. Craig, S.M. Henton, Astrophys. J. 450 (1995) 280. [9] M. Dobrowolny, A. Mangeney, P.-L. Veltri, Phys. Rev. Lett. 45 (1980) 144.

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