## Online material

### Appendix A: Derivation of kink mode equation

We give a short derivation of the basic equations, starting from Eqs. (3)–(5). The solutions are obtained in the three regions, inside the flux tube, outside the flux tube and in the transition layer.

#### A.1. Internal solution

Using the thin tube (or long wavelength) limit, the internal solutions can be
expressed as (A.1)Hence, at
*r* = *R* − *l*/2 = *R*(1 − *ϵ*/2),
we have (A.2)

#### A.2. External solution

Again using the thin tube limit, we have
(A.3)Hence, at
*r* = *R* + *l*/2 = *R*(1 + *ϵ*/2),
we have (A.4)Since both
*ξ*_{r} and *P* are
continuous across the transition layer as *ϵ* → 0, we can state
*ξ*_{e} = *η* + *δ**ξ*_{r}, *P*_{e} = *P*_{i} + *δP,*
where both *δ**ξ*_{r} and
*δP* tend to zero as *ϵ* → 0. Using (A.4), we have, correct to
*O*(*ϵ*^{2}), (A.5)Rearranging
Eq. (A.5), the final equation for
the propagating kink mode is (A.6)Note that the
right hand side of (A.6) is of
*O*(*ϵ*). It is the leading order expressions for
*δ**ξ*_{r} and
*δP* that we now need to calculate and this is done from the
transition layer solutions.

#### A.3. Transition layer solution

Integrating Eq. (3) across the thin
transition layer, we have (A.7)Integrating
(4) we have
(A.8)Remembering that
*η* is independent of *r*,
is *l* times
the average value of the operator ℒ and, for the linear density profile, the average
is ℒ_{k}, thus, (A.9)since
ℒ_{k}*η* = *O*(*ϵ*). Hence, for
the linear density profile (A.10)

We substitute Eq. (A.7) into
Eq. (A.6) and, using both Eq. (A.10),
ℒ_{i} + ℒ_{e} = 2ℒ_{k} and that again
ℒ_{k}*η* = *O*(*ϵ*), this
results in the propagating kink mode equation (A.11)

####
A.4. Integration of *ξ*_{ϕ} across the
transition layer

In this section we evaluate
(A.12)Using the solution
for *ξ*_{ϕ} given by (26), the integral across the transition
layer is made up of four terms. These are evaluated in turn.

##### A.4.1. Term 1

Now the integral of the first term on the RHS of Eq. (26), due to the radial profile of the driving boundary condition
of *ξ*_{ϕ}, is
For
large *kz*, this is proportional to
(*kz*)^{-1}. The influence of the choice of boundary
condition does becomes less important after several wavelengths.

For small values of *kz*, we can expand the result in a series to
show that the first term is For small
*κ*, Term 1 can be expressed as
where we have
defined (A.13)

##### A.4.2. Term 2

The second term on the RHS integrates to give where
Ci(*x*) and Si(*x*) are the *Cosine integral
*and *Sine integral *respectively, defined by

where *γ* = 0.57721*...* is Euler’s constant and
Again this term is
proportional to (*kz*)^{-1} for large *kz*.

For small values of *kz*, it is easier to start from the integral
expression. Hence, the first two terms in the Taylor series are
For
small values of *κ*, term 2 can be shown to reduce to
where
*Z* = *κkz*/2, as above.

##### A.4.3. Term 3

The third term is The
expansion of the coefficient of
for small *kz* gives to leading order
The expansion for
small *κ* gives, where
*Z* = *κkz*/2,

##### A.4.4. Term 4

Consider the final term, The
expansion for small *κ* gives

#### A.5. Final expression

We can now bring together the expressions for all four terms to rewrite the kink mode
equation, (A.11), as
Expressing
*η* as , our
final equation is (A.14)where
,
,
ℒ_{1} = d^{2}/d*z*^{2} − 2i*k*d/d*z*
and
ℒ_{e} = −*k*^{2}*κ* + ℒ_{k}.
In Eq. (A.14), the operator,
ℒ_{1}, acting on the final terms in the curly brackets on the right hand
side, results in terms that are small for *κ* ≪ 1. In fact, the terms
remain small even for *κ* ≤ 1/2. Hence, we will
neglect them and the comparison with the numerical results confirms this is a valid
assumption (see Sect. 5).

Equation (A.14) is an inhomogeneous, integro-differential equation for , the slowly varying amplitude function that describes the damping of the kink mode.

*© ESO, 2013*