3.1. Finding the Dispersion Relation
Let us seek solutions to Equation (
2) in the form of global, collective modes of the plasma. In addition, let us assume that the whole plasma oscillates with a common frequency,
. Hence, the temporal dependence is put proportional to
. With this condition, the Alfvén operator becomes
where
is the position-dependent Alfvén frequency squared, namely
with
the Alfvén velocity. After some algebra, Equation (
2) becomes
We note that the second term on the left-hand side of Equation (
8) diverges at the specific position where
. This is the Alfvén resonance and
is the resonance position. Mathematically, the presence of the Alfvén resonance introduces an imaginary part to the quasi-mode frequency,
. Physically, the presence of the Alfvén resonance causes the damping of the collective oscillation of the plasma.
Outside the nonuniform region, i.e., for
, the density is uniform and Equation (
8) simplifies to
whose solutions for perturbations vanishing at
, as consistent with a surface mode, are in the form of exponentials that decay away from the interface, namely
where
and
are constants. The perturbations in the left plasma,
, and those in the right plasma,
, need to be connected through the nonuniform interface,
. To do so, the solution to the full Equation (
8) needs to be found in the nonuniform transition.
There are a number of possible ways to connect the perturbations through the nonuniform layer. A usual approach relies on the so-called thin boundary approximation, i.e., assuming
. In the thin boundary approximation, the jumps of the perturbations at the resonance position are used as connection formulae across the entire nonuniform transition (see, e.g., Refs. [
19,
20]). Actually, this convenient method avoids finding solutions of Equation (
8) in the nonuniform transition, since only the jumps of the perturbations are required. Another method, valid for arbitrary thickness of the nonuniform layer, consists in expressing the solution of Equation (
8) as a Frobenius series around the resonance position and considering enough terms in the series for their convergence radius to cover the whole nonuniform region (see, e.g., Refs. [
14,
21]). A third alternative is trying to find the exact analytic solution of Equation (
8), which is only possible for specific density profiles (see, e.g., Refs. [
15,
17,
22]). Here, this last approach is followed. Although the obtained solution will only be valid for the chosen density profile, it will suffice to perform a rather general discussion of the underlying physics.
Let us consider hereafter a linear variation for the density in the transitional layer,
which allows us to rewrite Equation (
8) as
where the Alfvén resonance position,
, is given by
Here,
is the so-called kink frequency given by
where
and
denote the Alfvén frequencies in the left and right uniform plasmas, respectively.
Equation (
12) is a modified Bessel equation of order 0. The general analytic solution, which is applicable when
, is
where
and
are the modified Bessel functions of order 0 (see [
23]), and
and
are constants. As noted before, the solution should diverge in the nonuniform region where
because of the Alfvén resonance. Indeed, the singularity in
is present due to the
function. A series expansion of
in the vicinity of
reveals that the dominant term is given by a logarithmic singularity, namely
If a different density profile was adopted, the solution would no longer be in the form of modified Bessel functions, but the logarithmic singularity would necessarily remain enclosed somehow in the solution. This logarithmic singularity would explicitly appear if the solution was expressed with the method of Frobenius [
21]. The presence of a logarithmic singularity in
is the key ingredient to describe the resonant absorption of the plasma collective motions into the Alfvén continuous spectrum [
24].
Now, let us consider together the solutions in the uniform plasmas (Equation (
10)) and in the nonuniform transition (Equation (
15)). Then, the continuity of
and
at
are imposed. This gives us a system of four algebraic equations for the constants
,
,
, and
. The dispersion relation is obtained from the condition that there is a nontrivial solution of the system. The intermediate steps are omitted. The final expression of the dispersion relation is
where
and
are the modified Bessel functions of order 1, and
and
are defined as
No restrictions are imposed on the thickness of the nonuniform transition, so that Equation (
17) is valid for arbitrary values of
l. We also note that Equation (
17) is an equation for the quasi-mode frequency,
, which is hidden within the definitions of
and
(see Equations (18) and (19)).
3.2. Thin Transition: Analytic Approximations
Equation (
17) can be solved numerically for arbitrary values of
l. However, analytical approximations can be obtained by considering the case of a sharp, but still continuous, transition in density, so that the nonuniform interface is thin. The transitional layer is assumed to be thin when
, which means that the width of the interface is a small fraction of the wavelength. By performing a first-order expansion of the modified Bessel functions in Equation (
17) with respect to the small parameter
, we obtain the dispersion relation in the thin boundary (TB) limit, namely
In the case of an abrupt jump in density,
, and Equation (
20) simplifies to
which can be solved exactly, namely
The resulting frequency for the incompressible surface MHD wave is a sort of weighted average of the Alfvén frequencies at both sides of the interface, where the respective densities are the weights of the average. This is also the frequency of the compressible surface MHD waves propagating nearly perpendicularly to the magnetic field (see, e.g., Ref. [
4]) and the frequency of transverse kink and fluting waves in thin magnetic cylinders (see, e.g., Refs. [
25,
26]).
Returning to Equation (
20), let us note the presence of a logarithmic function in the term proportional to
. This logarithmic term is the remnant of the
and
functions of the general dispersion relation (Equation (
17)). Due to the presence of the logarithmic term, the dispersion relation is a multivalued function with branch points at
and
. To make the dispersion relation univalued, the branch points can be connected in the complex
-plane with branch cuts, as explained in, e.g., Refs. [
7,
17,
22].
One expects quasi-modes to have complex frequencies, namely
, where
and
are the real and imaginary parts of
, respectively. When
, the frequency is real and so the surface waves are undamped (Equation (
22)). Hence, if the nonuniform transition is thin, one may assume that the damping is weak, i.e.,
, and use Equation (10.87) from [
22] to express the logarithm in Equation (
20) as
where
n denotes the order of the Riemann sheet. Due to resonant damping,
, and one shall take
. The first term on the right-hand side of Equation (
23) complicates matters if we aim to find an analytic expression for
. To approximate this term, the following reasonable assumption is made. The term proportional to
l in Equation (
20) is assumed to be small when the transitional layer is thin. Hence, the result for
(Equation (
21)) is used to approximate
in the first term on the right-hand side of Equation (
23), and therefore,
Using this last result in Equation (
20), one obtains:
As anticipated, Equation (
26) has no physically acceptable solutions on the principal Riemann sheet, i.e., when
, because complex eigenvalues do not exist in ideal MHD (see, e.g., Refs. [
6,
7,
22]). To find the damped quasi-mode, we take
. According to [
6], where the Laplace transform is used. to analytically solve the initial-value problem, the zero of the dispersion relation found on the
sheet is the one that has the dominant contribution at intermediate times after the initial stages dominated by the excitation but before the collective oscillation has already decayed. On the other hand, the isolated eigenmode found in the dissipative MHD spectrum in the limit of small dissipation [
27], which corresponds to the global mode, has the same real and imaginary parts of the frequency as the ideal quasi-mode found on the
sheet. Additionally, the results from full numerical, time-dependent simulations [
28] show that the period and decay rate of the global oscillation obtained from the simulations match those predicted by the quasi-mode on the
sheet. Equation (
26) becomes
Now, let us write the frequency as
. This expression is used in Equation (
27). Since weak damping is assumed, terms with
and higher powers are neglected. After some algebraic manipulations, one finally obtains the approximate expressions for
and
, namely
The real part of the frequency,
, is the same as in the
case (Equation (
22)). In turn, Equation (
29) gives the same damping term found previously by, e.g., Refs. [
6,
11,
13]. Remarkably, this is also the same expression as Equation (79b) of [
20] for surface waves in a cylinder if in their expression
and
is replaced by
k. Equation (
29) predicts a linear dependence of
with
, so that
monotonically decreases when
increases. So, the results already obtained in the past with different methods are consistently recovered. One must keep in mind that Equations (
28) and (
29) are only strictly valid when
. The generalization of these approximations to the case of thick transitions may be misleading, as shown later.
3.3. Transition of Arbitrary Thickness: Numerical Results
For arbitrary
, one has to solve the general dispersion relation (Equation (
17)). This must be carried out numerically. As in the case with
, no solutions exist on the principal Riemann sheet, so the
Riemann sheet is considered to find the quasi-mode complex frequency. Specifically, this is accomplished by considering how the logarithmic terms enclosed in the series expansions of
and
jump when crossing the resonance position.
Figure 1a shows the real part of the frequency as a function of
. When
, one finds a solution with
, as expected according to the analytic approximations and to be called the `q-mode’ after the `quasi-mode’. Unexpectedly, there is another solution that appears only when
. The presence of this other solution was not predicted by the analytic approximations. The real part of the frequency of this additional solution grows from zero as
l increases and, for this reason, it is labelled as the `
l-mode’. Certainly, the
l-mode owes its existence to the presence of the nonuniform transition. As
increases, the q-mode and the
l-mode eventually converge, and two different solutions or branches emerge afterwards. For the set of parameters used in
Figure 1a, this happens around
. The behaviour of
of the two new branches is strikingly different. In the case of the lower branch, its
tends to
when
. For this reason, the lower branch is labelled as the `i-mode’. Conversely, the real part of the frequency of the upper branch tends to
when
, so the upper branch is labelled as the `e-mode’. The q-mode, which is the descendent of the surface wave of the abrupt interface, ceases to exist as such when it collides with the
l-mode and the i- and e-branches subsequently emerge. This suggests that an interface with a sufficiently thick transitional layer is not able to support a global collective mode. Below, more arguments in support of this idea are given.
To shed more light on the behaviour of the solutions, let us turn now to
.
Figure 1b displays
as a function of
. For the q-mode, the behaviour of
when
is again correctly described by the analytical formula for a thin transitional layer. Before the q-mode and the
l-mode merge, it is found that the q-mode
increases (in absolute value) with
. The agreement between the approximate linear dependence with
and the full result is good enough when
.
On the other hand, the nature of the
l-mode becomes even more puzzling when one realizes that it is a strongly attenuated solution. This fact can be better visualized in
Figure 1c, which displays the quality factor,
Q, of the solutions as a function of
. The quality factor measures the efficiency of the damping and is defined as
Modes are overdamped when
. An overdamped mode decays in a shorter timescale than its own period, which means that overdamped modes do not represent actual oscillations in the plasma but very short-lived motions. If a single period cannot be completed, the plasma motion can hardly be called an oscillation. Initially, the
l-mode is heavily overdamped, and so it does not represent an actual oscillation. As
increases, the
l-mode quality factor increases until it crosses the critical value
. It turns out that this happens, precisely, when the real part of the frequency of
l-mode coincides with the lowest Alfvén frequency, i.e.,
, so that
l-mode enters inside the Alfvén continuum (see the blue circles in panels a and c of
Figure 1). From there on, and until it merges with the q-mode, the
l-mode is another oscillatory solution that lives in the Alfvén continuum.
After q- and l-modes collide and the i- and e-modes subsequently emerge, the situation is as follows. The imaginary part of the frequencies of both the i- and e-modes decreases (in absolute value) and then tends to zero for . The quality factor of the e-mode is always smaller than that of the i-mode. Essentially, this is a consequence of the e-mode having a larger .
Although the physical interpretation of the q-mode is straightforward, it is the quasi-mode that descends from the undamped surface mode, and understanding the nature of the
l-, i- and e-modes is more challenging. To further explore the nature of the solutions,
Figure 2 displays the real part of their corresponding Lagrangian displacement,
, in two cases, when
and when
. These two values of
are chosen as representatives to the scenarios before and after the merging of the q- and
l-modes. The solutions when
are the q- and
l-modes, while the solutions when
are the i- and e-modes.
The perturbations of the q-mode when retain the shape expected from a surface wave, with the additional presence of the jump in due to the logarithmic Alfvénic singularity. The q-mode is an even (symmetric) mode as the surface wave for should necessarily satisfy the physical requirement that must be continuous at the abrupt interface. However, when , the hard requirement that must necessarily be an even function no longer applies. The two uniform plasmas may equally be connected through the nonuniform transition with either an even or an odd function. In principle, the two symmetries should be possible. The q-mode, being the descendant of the surface wave, retains the even symmetry. In turn, a new solution is introduced, the l-mode, whose is an odd (antisymmetric) function that jumps and changes sign at . Let us recall that l-mode is an overdamped solution for small and does not physically represent an oscillation. Only the q-mode represents an actual oscillation supported by a thin nonuniform interface.
Concerning the perturbations of the i- and e-modes when
, the jumps in
due to the resonances of these two modes in the Alfvén continuum are evident in
Figure 2. The i-mode has a larger amplitude near the left boundary of the interface, while the e-mode has a larger amplitude near the right boundary. This result supports the idea that the two modes that split after the coalescence of the q- and
l-modes are actually related with the uniform plasmas at each side of the interface and do not represent true collective modes of the whole interface. One can also see that the i-mode has inherited the even symmetry of the q-mode in the sense that the sign of
is the same in the two plasmas at both sides of the interface, while the e-mode retains the odd symmetry of the
l-mode; i.e., the sign of
is the opposite in the two uniform plasmas.