## Abstract

The effects of the flow of an electrically conducting fluid upon a magnetic field anchored at the boundary of a domain are studied. By taking the resistivity as a small parameter, the first-order approximation of an asymptotic analysis yields a boundary layer for the magnetic potential. This layer is analysed both in general and in three particular cases, showing that while in general its effects decrease exponentially with the distance to the boundary, several additional effects are highly relevant.

## 1. Introduction

The effects of a fixed transversal magnetic field at the boundary of a domain containing a plasma of low magnetic Reynolds number have long been recognized as being important both theoretically (Gerard-Varet 2002; Thess *et al.* 2007) and for engineering applications, notably in the case of the Hartmann layer (Hartmann & Lazarus 1937; Davidson 2001). The inverse effect, that of the flow upon the magnetic field, while lying at the core of highly relevant subjects such as dynamo theory, has been less studied as a source of boundary layers. In particular, for a plasma of high conductivity (the opposite of the Hartmann case), the system is likely to act nearly as ideal magnetohydrodynamics at some distance from the walls of the domain, i.e. magnetic field lines will be transported by the flow as material points. On the other hand, if we fix the normal component of the magnetic field at the boundary, a transitional layer to adapt the passive magnetic field to these conditions will take the form of a boundary layer which may be studied by an appropriate asymptotic analysis, analogously to the classical Prandtl layer in fluid dynamics. There exist some important physical problems such that the normal component of the magnetic field is given *a priori*. Probably the archetypal example consists in a static force-free field in a solar corona (e.g. Low & Flyer 2007 and references therein). The solar corona, as a fully ionized plasma, may be considered in a first approximation as a static perfect conductor, so the Lorentz force vanishes. The boundary magnetic field is obtained e.g. from polarimetric observations at the coronal base, and one must solve the force-free equation (∇×B)×B=0 with these boundary values to obtain a model of the field in the whole corona. Also, in the theory of Parker (1994) of creation of current sheets due to magnetic relaxation, the footpoints of the field lines are fixed. In general, wherever the magnetic field is known in the outside of our domain, Maxwell’s equations yield the normal component of the field at the boundary, which is therefore fixed *a priori*. When the Lorentz force is weak enough that we may ignore the action of the magnetic field upon the flow, we may take the kinematic approximation and consider the induction equation alone. This may occur for several reasons: obviously for force-free configurations, for weak magnetic fields like those occurring at the onset of a dynamo, or when the Lorentz force equals a gradient which is compensated by the kinetic pressure. In particular, the onset of a fast dynamo, when the magnetic field grows exponentially, is a well-studied subject, in which the topology of the flow plays the main role (Childress & Gilbert 1995). This must be chaotic for fast dynamos to develop (Klapper & Young 1995), but even in this case at a certain point exponential growth stops and dynamo saturation sets in. This occurs because the kinematic approximation fails and the magnetic field reacts upon the flow, changing its stretching properties. The period where this occurs has been estimated (Thiffeault & Boozer 2003). A connection with our problem is that the presence of walls modifies substantially the mixing properties of a chaotic flow, and therefore its dynamo generation abilities (Gouillart *et al.* 2007). Anyway, we will consider the velocity **v** of the flow fixed and time independent, and study the evolution of the magnetic field **B** as given by the induction equation
1.1
where *ϵ* represents the magnetic diffusivity (or, when setting the equation as non-dimensional, the magnetic Reynolds number).

From now on, we will consider the two-dimensional case. Since **B** is a plane solenoidal field, it possesses a scalar potential *A*:
1.2
Equation (1.1) may be uncurled to yield
1.3
where the gauge *f* must be in this case just a function of time. Any choice of *f* would yield the same magnetic field lines; only the labelling of them would change. Thus, we choose *f*=0.

We will assume that the domain *Ω* where the field lies possesses a boundary where *A* is fixed. Since the tangential derivative of *A* is the normal component of **B**, this is equivalent to fix **B**⋅**n** at ∂*Ω* and *A* at a single point of every connected component of ∂*Ω*.

The key to the successful analysis of equation (1.1) is the assumption that the magnetic diffusivity is low enough that the zeroth order term of an asymptotic expansion of *A* in powers of *ϵ* describes adequately the solution. Since *ϵ* multiplies the highest order term of equation (1.1), this is a singular perturbative parabolic system. The study of singular perturbations of elliptic systems (to which we will refer ours) was started in Eckhaus (1972), Eckhaus (1979) and Grasman (1968, 1974) and has been extended in several directions (Il’in 1992; de Jager & Furu 1996). We will describe briefly the main lines of this theory.

## 2. General asymptotic analysis

For an elliptic equation of the type
2.1
the regular expansion has the form
2.2
obtained by substituting formally in and equating powers of *ϵ*, i.e.
2.3
and
2.4
The problem lies in making this solution conform to previously given Dirichlet boundary conditions,
2.5
To this end, the boundary ∂*Ω* is divided into three parts: the one where **v** is transversal to the boundary and points to the interior of *Ω*, the one where it points outwards from *Ω*, and the one where **v** is parallel to the boundary. Obviously, the partition may become extremely complex unless both domain and flows are simple: moreover, the points of contact between different portions of the boundary yield additional types of layers which complicate what is already a difficult problem. Therefore, the literature deals with simple domains, such as circles and rectangles, and vectors **v** which have a single direction (parallel to two of the sides in the case of a rectangle). Let us take this instance for concretion: *Ω*=(0,*a*)×(0,*b*), **v**=(*v*,0),*v*>0. Then all the terms *A*_{n} in equation (2.4) are determined by the values of *A*_{n} in {0}×[0,*b*]. Specifically, we set
2.6
Next, the boundary condition at the parallel boundary must be dealt with. There is no reason to think that the functions *A*_{n} will yield something like equation (2.5) in (0,*a*)×{0} and (0,*a*)×{*b*}; the way to adapt them is to add a parabolic boundary layer to the regular solution. This is obtained by assuming that the solution will vary more rapidly transversally than in parallel to the boundary, and choosing as a new variable ( in the upper side). We will concentrate on (0,*a*)×{0} to present the calculations. The idea is to add a perturbed field
2.7
where
2.8
and satisfies the boundary conditions
2.9
For *n*>0,
2.10
Finally, the ordinary boundary layer at {*a*}×(0,*b*) is intended to adapt the regular expansion to the value *g*. It is obtained by defining a local variable *θ*=(*a*−*x*)/*ϵ* so that the ordinary layer expansion
2.11
satisfies the following equations:
2.12
The solution may be easily found and decreases exponentially away from *x*=*b*. The rest of the terms satisfy
2.13
At the points (0,0) and (0,*b*) the phenomenon of birth of boundary layers exist, whereas corner layers occur at (*a*,0) and (*b*,0). These play a transitional role between different boundary layers. An analysis of these phenomena, plus the proof that
2.14
really represent an asymptotic expansion of the solution valid in the whole domain may be found in the above references. For the zero-order terms the corner layers do not represent a problem: it is for higher values of *n* that singularities appear. Since for our problem, we will consider only these terms and there will be no ordinary boundary layer (that is, we impose no restrictions on the potential for the exit flow), we will study *A*_{0} and . These are in fact the most important terms: for the analogous problem in Hydrodynamics would represent the Prandtl boundary layer. Equation (2.8) may be studied as follows: assuming *v*(*x*,0)≥*δ*>0, define a new variable *ξ* by
2.15
so that
2.16
Then equation (2.8) may be written as
2.17
where *c*(*ξ*)=*f*(*x*(*ξ*)). Let *P*′(*ξ*)=*c*(*ξ*). Equation (2.17) becomes
2.18
with the boundary conditions
2.19
Let us denote by *K*(*s*,*η*) the heat kernel for these boundary conditions:
2.20
Then
2.21
i.e.
2.22
Analogous results may be obtained for parabolic equations, changing *f* in equation (2.1) by ∂/∂*t*. However, it will be simpler in our case to refer to the resulting elliptic system after performing a Laplace transform in time. We will use this method to analyse the equation
2.23
where *F*,*v*≥*δ*>0. By defining *ξ* as in equation (2.15), equation (2.23) may be written as
2.24
where *c*(*ξ*)=*F*(*x*(*ξ*))≥*δ*>0. Assume (as will be always the case in our examples) that for all *ξ*,*η*. Then the Laplace transform of equation (2.24) yields for
2.25
plus the boundary conditions
2.26
According to equation (2.22) the solution to this problem is
2.27
where *P*′(*ξ*)=*c*(*ξ*). Since the Laplace transform of the translated *τ*_{a}*f* of a function *f*, *τ*_{a}*f*(*s*)=*f*(*s*−*a*), is , equation (2.27) means
2.28
The integrand vanishes whenever *t*−*P*(*ξ*)−*P*(*ξ*−*s*)<0, which, since *P* is strictly increasing, means that the interval of integration reduces to 0≤*s*≤*ξ*−*P*^{−1}(*P*(*ξ*)−*t*). Hence
2.29
Recall that in our case *h*=*A*−*A*_{0} at *η*=0. In fact, the use of the Laplace transform should be limited to problems where , but we use it only as a method to obtain the final formula (2.29), which is valid for any interval of *ξ*.

While equation (2.29) is a closed solution for , we desire a simpler analytic expression, not involving integrals. A good approximation may be obtained if we assume that *η*≫0, i.e. the approximation will be accurate for *y*>0 fixed and small *ϵ*. In practice, the approximation is extremely good even for moderate *η*. It is obtained by integration by parts of equation (2.29), and it corresponds to the first term of an asymptotic expansion related to the well known one of the error function. In order to abbreviate the notation, let us define for fixed *t* and *ξ*
2.30
Since
2.31
equation (2.29) may be written as
2.32
Since *f*(*α*)=*h*(0,*P*^{−1}(*P*(*ξ*)−*t*)) and we have assumed for all *ξ* and *η*, we have in particular for all *ξ*. Hence *f*(*α*)=0 and the first term in the last line of equation (2.32) vanishes; so we are left with the last integral. A new integration by parts yields, calling *g*(*s*)=d(*s*^{1/2}*f*(*s*))/d*s*,
2.33
If the first term does not vanish and *h* is smooth enough for d(*s*^{2}*g*(*s*))/d*s* to be e.g. bounded, the quotient of the last integral by the first term tends to zero when ; this is a simple consequence of the behaviour of the exponentials and Lebesgue’s dominated convergence theorem. This, however, is guaranteed for bounded intervals of *t* and *ξ*; it is not necessarily uniformly valid for all time, as we will see later. To evaluate the first term, let us write
2.34
Since *h*(0,*ξ*)=0 for all *ξ*, the only term which does not necessarily vanish at *s*=*α* is the middle one, whose value is *α*^{1/2}(*D*_{1}*h*)(0,*P*^{−1}(*P*(*ξ*)−*t*)). Assuming this is different from zero and remembering that *P*′=*c*, we are left with
2.35
For the case where *c*=1, *P*(*ξ*)=*ξ*, *α*=*t* and equation (2.35) simplifies to
2.36
The reason why for all *ξ* and *η* is that we will always choose the same initial condition for *A* and the ideal solution *A*_{0}. This will become clear in the examples developed in the next section. Recall that we need *h* smooth enough for equations (2.35) and (2.36) to hold: we will also see that this does not always happen.

## 3. Examples of boundary layers

In the following instances we will always take a time-independent incompressible flow parallel to the boundary. We will not present ordinary boundary layers, which are anyway unusual in realistic physical problems.

### (a) Infinite wall

Let us consider that the domain is a half-space (*x*,*y*):*y*>0. A horizontal incompressible flow has the form **v**(*x*,*y*)=(*v*(*y*),0), where we must at least assume *v*(*y*)≥*δ*>0, and the associated induction equation (1.3) becomes
3.1
We will assume as in the classical Hartmann case a transversal magnetic field; only now we cannot ensure its permanence beyond the boundary. Let us therefore pose the boundary condition
3.2
The first step is to find the zeroth-order term of the regular expansion. This satisfies
3.3
The general solution is any smooth enough function of the form
3.4
Since we wish *A*(0,*x*,*y*)=*A*_{0}(0,*x*,*y*), the solution is determined by its value at *t*=0:
3.5
It is natural to choose the initial condition for *A* to conform with the given value at the boundary: *A*(0,*x*,*y*)=*x*. Then equation (3.5) yields *A*_{0}(*x*,*y*,*t*)=*x*−*v*(*y*)*t*. Therefore
3.6
Equation (2.29) becomes in this case
3.7
while the approximation for *η*≫0 of equation (2.36) is
3.8
Note that is independent of *x*, which is logical as it is independent of *x* at *y*=0. The zeroth order approximation to the magnetic potential is therefore for *y*>0, *ϵ*≪1,
3.9
Note the rapid (exponential) decrease of as we move away from the boundary; nonetheless, this decrease slows in time, due to the gradual diffusion of the boundary value of as time progresses. The independence of this perturbed potential from *x* means that the associated magnetic field is horizontal. In this example, there is no place for an ordinary boundary layer, because there is no boundary where the flow exits the domain.

### (b) Semi-infinite wall

Consider now that the domain is the first quadrant *x*>0, *y*>0 and again **v**=(*v*(*y*),0). Boundary conditions must now include the values of *A*(*t*,*x*,0) and *A*(*t*,0,*y*) for all time. Requesting as before a transversal field, we again set
3.10
and to obtain a potential continuous in the boundary,
3.11
It is intuitive that the transport by the flow of this null magnetic potential will quench the magnetic field progressively, leaving only the perturbative one due to the parabolic boundary layer. As in equation (3.4), the regular potential *A*_{0} is a function of (*x*−*v*(*y*)*t*,*y*); in this case, the characteristic curves do not intersect all the plane *t*=0, so that the value of *A*_{0} is given by
3.12
Anyway *A*(0,*x*,*y*)=*A*_{0}(0,*x*,*y*). Thus,
3.13
Therefore,
3.14
and
3.15

Therefore, the integral in equation (2.29) may be written as
3.16
Since the function *h* is not differentiable at *t*=*ξ*, the approximations (2.35)–(2.36) cannot be applied. However, by analogy with equation (3.7) we obtain directly
3.17
where . Writing , and recovering *y* from *η*, we get
3.18
By adding this to the expression in equation (3.12), we obtain the zeroth-order approximation to *A*. Note that the regular term *A*_{0} vanishes for fixed *x* and large *t*, meaning that the zero value of *A* at *x*=0 quenches gradually the potential; only is left, which depends only on (*x*,*y*) from a certain time on.

### (c) Flow towards a wall

Let us consider now a fluid flowing towards a wall and escaping laterally. We will model this flow by the velocity **v**(*x*,*y*)=(*x*,−*y*), so that the induction equation becomes
3.19
and let the domain be the upper half plane *y*>0. The first term of the regular expansion therefore satisfies
3.20
whose general solution is any function of the form
3.21
If we take as usual *A*_{0}(0,*x*,*y*)=*A*(0,*x*,*y*), this means
3.22
The flow is symmetric with respect to the vertical axis. If we take symmetric initial and boundary conditions, the potential *A* will also be symmetric. Thus it is enough to choose boundary conditions on *x*≥0,*y*=0. To analyse equation (3.19), we take parabolic coordinates
3.23
so that
3.24
It is well known that
3.25
and
3.26
To avoid dragging the constant , let us redefine a new time as the old one times . Then the induction equation (3.19) becomes
3.27
and the mapping (*x*,*y*)→(*u*,*v*) transforms the upper half plane *y*>0 into the whole plane minus the positive real axis *v*=0,*u*≥0, which becomes the new boundary. By our symmetry assumption, it is enough to consider *v*≥0, but in contrast to our first example, boundary conditions are only set in the positive portion of the real axis. Let us take the same initial and boundary condition as in equation (3.1),
3.28
Then
3.29
In parabolic coordinates, this becomes
3.30
defined for *u*≥0. At the point (0,0) a birth of boundary layer must be taken into consideration (Eckhaus 1979), but it matters only for higher order approximations. The equation satisfied by *A*_{0} is obtained as before by taking . This is defined for *u*≥0:
3.31
with boundary conditions
3.32
With the notations of §2,
3.33
Let us rewrite equation (2.29) as
3.34
Although we could apply formula (2.35), it is worth repeating the argument in this particular case to avoid errors in the time dependence of the approximate solution. Let *α*=*u*(1−*e*^{−t}). Then, after one integration by parts,
3.35
which, after a further integration by parts, becomes
3.36
The first term in equation (3.36) becomes the approximation of equation (2.35),
3.37
Note the presence of the exponentially growing term *e*^{t/2}. Before we rush to believe that the perturbative potential grows exponentially in time, let us recall that equation (3.37) is a good approximation of equation (3.36) for fixed *u*,*t* and large *η*; it is not uniformly valid for all times. This is clear from the fact that in the integral in the second line of equation (3.36) occur terms of the form *e*^{3t/2}, so it is not true that this remainder is smaller than the first term for large times. We must consider equation (3.37) a valid approximation for *v*>0 fixed, fixed intervals of *u* and *t*, and small *ϵ*.

The next step would be to substitute *u* and *η* by its values in terms of *x*, *y* and *ϵ*. The final expression is prohibitively complex, but equation (3.37) is enough to note a behaviour roughly similar to the infinite wall case, substituting the coordinate lines *x*,*y*=const. by the hyperbolas *u*,*v*=const. This is not totally unexpected, since the hyperbolas *v*=const. are the streamlines of the flow, although the fact that it is *v* which measures the rate of decrease of as we leave the boundary is not obvious *a priori*.

## 4. Conclusions

While the effects of a transversal magnetic field upon a flow of low magnetic Reynolds number lead to the well-known Hartmann layer and have been widely studied, the opposite case about the action of a flow of large conductivity upon a magnetic field fixed at some boundary have remained largely unexplored, in spite of possible applications to problems like the evolution of solar coronas. The mathematical problem is simplified assuming the Lorentz force weak enough not to interfere with the flow; in that case the induction equation alone will determine the magnetic field evolution. For two-dimensional problems and steady incompressible flows, an asymptotic analysis, considering the resistivity of the fluid as a small parameter, provides good approximations to otherwise very complex solutions. The field is assumed to have infinite conductivity in the body of the fluid, and therefore is transported passively by the flow, while boundary layers connect this solution to previously fixed values at the boundaries. For boundaries parallel to the flow, the problem may be reduced in the first approximation to the solution of a diffusion equation. While this approximation may be obtained exactly by an integral formula involving the heat kernel, further work is necessary in order to get analytically simpler expressions of the perturbed field. A general theory plus three different examples are studied, showing quite diverse phenomenologies in different settings.

- Received March 4, 2010.
- Accepted April 28, 2010.

- © 2010 The Royal Society