SITUACIÓN ACTUAL DE LA TITULIZACIÓN DE ACTIVOS EN EL PERÚ

For compressive viscosity the damping rate is given by Eq. (4.47) withΩidepending onV =kzLalone.

The fundamental and first overtone modes correspond tokz =π/(2L)andkz =π/Lrespectively. Thus,

we may write the damping time of the fundamental mode as

τdamp=

2L πcsΩiV= π

2

4.5 Damping Rates 105

Figure 4.16: Plot showing the damping timeτdamp(= 1/ωi)in seconds as a function of temperatureT0(in

◦_{K) for the fundamental mode of a loop of length2L= 50Mm for the exact solution (4.49) (solid curve)}

and the approximations (4.52) and (4.56) (dashed curves). Damping is due to thermal conduction only.

and the damping time of the first overtone as

τdamp=

L

πcsΩi(V =π). (4.58)

Figure4.17 shows the damping time for the fundamental and first overtone modes as a function of temperatureT0for various loop half-lengthsL. As for thermal conduction, we note that for compressive

viscosity, the damping rate of the fundamental loop with L = L1 is equal to the damping rate of the

first overtone forL = 2L1. As for thermal conduction we convince ourselves of this by considering the

formulae for the damping rate given above by Eqs. (4.57) and (4.58). If we consider the damping time of the fundamental mode whenL=L1and the damping time of the first overtone whenL= 2L1and recall

thatis given by = 0 L, where 0= 4ν0csT 7/2 0 3γp0 .

4.5 Damping Rates 106

first overtone mode of a loop of half-length2L1are both given by

τdamp= 2L1 πcsΩi V= )π 2L1 .

Figure4.17shows that initially the damping time of the first overtone of a loop is less than the damping time of its corresponding fundamental. In the limitT0→0the damping time for both the fundamental mode

and the first overtone tends to infinity (as for thermal conduction). In the case of compressive viscosity, however, an increase in temperature causes an indefinite decrease in damping time until for an infinitely hot loop the damping time is effectively zero, i.e. in the limitT0 → ∞,τdamp → 0: the hotter the loop,

the quicker it is damped by compressive viscosity. As for thermal conduction, the damping time initially increases as the loop length is increased but asT0 → ∞the damping time is independent of loop length

and mode of oscillation.

Figure 4.17: Plot showing the damping timeτdamp(= 1/ωi) in seconds for the fundamental mode (solid

curves) and its first overtone (asterisks) as a function of temperatureT0(in◦K) for various loop half-lengths

(L= 25Mm - black,L= 50Mm - blue ,L= 100Mm - green, andL= 200Mm - red). Damping is due to compressive viscosity only.

Figure4.18shows the damping time for the fundamental and first overtone modes as a function of loop length2L for various temperaturesT0. In agreement with Fig. 4.17, it shows that the damping time is

increased as the loop length is increased but that this increase is counteracted by an increase in temperature. We conclude that, as for thermal conduction, it is the temperature and not the loop length which dominates the rate of damping for compressive viscosity.

4.5 Damping Rates 107

Figure 4.18: Plot showing the damping timeτdamp(= 1/ωi) in seconds for the fundamental mode (solid

curves) and its first overtone (asterisks) as a function of loop length2L(in metres) for various loop temper- atures (T0 = 1MK - black,T0 = 2MK -blue ,T0 = 3MK - green,T0 = 4MK - red, andT0 = 5MK -

pink). Damping is due to compressive viscosity only.

In Tables4.12and4.13we give values of the damping time for specific loop lengths and temperatures for the fundamental mode and the first overtone respectively. The damping time ranges from seconds to months. For example, it takes the first overtone of a loop of length50Mm (L= 25Mm) and temperature

T0 = 8MK∼26seconds to be damped but it takes the fundamental mode of a loop of length400Mm

(L= 200Mm) and temperatureT0= 1MK around16weeks to damp.

Table 4.12: The damping timeτdamp (in seconds) for the fundamental mode for various loop half-lengths

Land temperaturesT0. Damping is due to compressive viscosity only.

L(Mm) T0= 1MK T0= 2MK T0= 6MK T0= 8MK

25 1.51e5 1.34e4 2.86e2 1.04e2 50 6.04e5 5.34e4 1.14e3 4.17e2 100 2.42e6 2.14e5 4.57e3 1.67e3 150 5.44e6 4.81e5 1.03e4 3.76e3 200 9.67e6 8.55e5 1.83e4 6.67e3

At this stage we note that for typical TRACE temperatures (T0= 1−2MK), damping of both the first

overtone and the fundamental is quicker under thermal conduction but that for typical SUMER temperatures (T0 > 6MK), damping of the first overtone and the fundamental is generally quicker under compressive

4.5 Damping Rates 108

Table 4.13: The damping timeτdamp(in seconds) for the first overtone for various loop half-lengthsLand

temperaturesT0. Damping is due to compressive viscosity only.

L(Mm) T0= 1MK T0= 2MK T0= 6MK T0= 8MK

25 3.78e4 3.34e3 7.14e1 2.61e1 50 1.51e5 1.34e4 2.86e2 1.04e2 100 6.04e5 5.34e4 1.14e3 4.17e2 150 1.36e6 1.20e5 2.57e3 9.39e2 200 2.42e6 2.14e5 4.57e3 1.67e3

We can determine the behaviour of the damping time analytically by recalling Eq. (3.74) in Chapter3,

Ω = 1 2Vi± 1−1 4V 2 1/2 . (4.59)

ProvidedV <2, Eq. (4.59) gives

Ωi= 1

2V. (4.60)

We note thatV <2is satisfied for typical coronal conditions (see Chapter3). Using this result we find that the damping time is given by

τdamp= 2 k2 zL2 τvisc= 3 2 ˜ µp0 ν0k2zRT 7/2 0 , V<2, (4.61) since V = τs τvisc kzL, τs= L cs , τvisc= 3 4 ˜ µp0L2 ν0RT 7/2 0 .

As such for smallV the damping time is directly proportional to the compressive viscosity timescale. The conditionV<2is equivalent toT0<(2p0L/V0)1/3orL >V0T03/2p0where

V =V0 T 3 0 p0L , V0= 4 3 ν0 r_γR ˜ µ γ kzL

and so is valid in the limitsT0 → 0 and L → ∞. We note that for all V < 2 the damping time is

directly dependent on the pressurep0, temperatureT0and dependent on the loop half-lengthLand mode

of oscillation through dependence onkz. To relate this to Figs. 4.14and4.15in the limits ofT0 → 0

andL → ∞as we increase the temperature or decrease the loop half length, the damping time decreases (wherep0is fixed).

As an illustration, in Fig. 4.19we show that for a loop of length50Mm the damping time of the first overtone as a function of temperatureT0for compressive viscosity is given exactly by Eq. (4.61) forV<2.

4.5 Damping Rates 109

Figure 4.19: Plot showing the damping timeτdamp (= 1/ωi)in seconds as a function of temperatureT0

(in◦K) for the first overtone of a loop of length2L= 50Mm for the exact solution (4.58) (solid curve) and the approximation (4.61) (dashed curve). Damping is due to compressive viscosity only.