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Recent observation of oscillating the two stream instability (TSI) in a solar type III radio bursts and spatial damping of Langmuir oscillations has made this instability as an important candidate to understand the coronal heating problem. This instability has been studied by several authors for cold plasma found to be stable for high frequencies (greater than plasma frequency
ω_{p}). In this paper, we prove that this instability is unstable for warm plasma for higher frequencies (greater than plasma frequency
ω_{p}) and much suitable to study the solar coronal heating problem. We have derived a general dispersion relation for warm plasma and discussed the various methods analyzing the instability conditions. Also, we derived an expression for the growth rate of TSI and analyzed the growth rate for photospheric and coronal plasmas. A very promising result is that the ion temperature is the source of this instability and shifts the growth rate to high frequency region, while the electron temperature does the reverse. TSI shows a high growth rate for a wide frequency range for photosphere plasma, suggesting that the electron precipitation by magnetic reconnection current, acceleration by flares, may be source of TSI in the photosphere. But for corona, these waves are damped to accelerate the ions and further growing of such instability is prohibited due to the high conductivity in coronal plasma. The TSI is a common instability; the theory can be easily modifiable for multi-ion plasmas and will be a useful tool to analyze all the astrophysical problems and industrial devices, too.

Solar corona is hot to the level of million degree kelvin; several mechanisms have been put forward for the last 60 years which is continued as the problem exist. In this paper, we try to add one more to the list of possible mechanisms as a source of coronal heating, which is the streaming instability. Streaming instabilities arise when there is relative velocity between ions and electrons in a plasma. The simplest type of streaming instability is the two stream instability (TSI). This arises in an electron-proton plasma with electrons in relative motion with ions. This type of streaming instability arises in stellar atmospheres, since the stellar plasmas are predominantly elec- tron-proton type. A specific example for this instability is an electron-precipitation related phenomenon in solar chromosphere [

The electromagnetic wave propagation and instabilities for the counter streaming astrophysical situations for cold plasma have been discussed by several authors [_{p}), but TSI is a high frequency wave and also the growth rate is negligible for cold plasma. But for warm plasma, we find it is unstable for a very wide range of frequencies and even for the frequencies much greater than ω_{p} and that the growth rate is very large and sensitively depends on the ion (electron) temperatures which is ideal for solar atmosphere.

An interesting result is that the growth rate shifts to the higher frequency region as the ion temperature in- creases, while the electron temperature shifts it to the lower frequency region. It is the ion temperature that becomes the source of TSI as the high frequency oscillations, for it is a necessary condition presumed for this instability and hence the damping of this instability will heat the ions. Also, this instability depends on the electron temperature, but stabilises the plasma. It is found that the growth rate region shifts towards the low frequency regions as the electron temperature increases in reverse of the shift due to ion temperature. As the kinetic energy of electrons shifts the growth rate to the low frequency region which is nothing, damping of high frequency oscillation results in the heating of ions. The high frequency oscillations are induced by the electric field due to the electron drift which accelerates the ions at the expense of kinetic energy of electrons, and this is true because the increase in drift velocity decreases the instability. This instability is a common one and hence this theory can be applied to any hot electron-proton plasma and can be better for fusion studies too.

For the theoretical study of streaming instability in a hot plasma, we consider an electron-proton plasma, with ions assumed stationary and electrons moving with a velocity

The linearised equation of motion for protons and electrons are respectively:

We consider electrostatic waves of the form,

where

Under these conditions Equation (1) gives

i.e.,

Similarly, Equation (2) gives

The same results hold good for parallel electrostatic plasma oscillations (i.e., wave propagation parallel to B_{0}.)

The ion continuity equation for our case is

Linearising this and noting that

Similarly the electron continuity equation is,

Linearising this we get

Simplifying Equation (7) and Equation (9) we get,

and

The plasma waves leading to TSI are high frequency plasma oscillations. To deal with these type of waves it is well known that we should use Poisson’s equations

i.e.,

Here the electric field induced is due to the perturbation in the density. Assuming the perturbation as

Equation (14) is a fourth-order equation in

represents a possible oscillation

occur in complex conjugate pairs, and real part represents the propagation modes and imaginary part represents the instability. This could be analyzed by putting

Positive

As a check for knowing whether there is any unstable modes in the plasma, we follow the procedure in [

where

of ion and electron respectively. The dispersion relation shown in Equation (15) is a function of four variables

For any given

We could see four intersection in the first graph, which means the plasma is stable. But for in the second

graph, there is only two intersection means the plasma is unstable, for

when

Now we can look into the dispersion relation of warm plasma, as it is to compare with the cold plasma, the plot is done for

We see the plasma is unstable with finite ion thermal velocity even without any thermal velocity of electron. For any higher values of b the plot is same and hence the plasma shows instability. This is quite interesting that the ion temperature becomes a source of instability without the factor of considering the electron temperature. The additional mobility given to the ions by the temperature, negate the deficit of the density due to the high drift speed of electrons or the moving ions could easily find the sufficient electrons in the new spatial situation which sustains the field.

Also this could be verified by looking in to the Poissons equation that the electric field induced is depends on the perturbation density of ions and electrons. If we look into Equation (10)

we see the second term in the denominator is with negative sign, which shows that the density of ions is increased by the temperature ions and hence the ion temperature becomes the source of instability. This could also viewed as the electric field induced by the drift of steaming electrons accelerate the ions to maintain the instability or the damping waves heats the ions.

Then we wish to analyze the effect of electron temperature, we set the ion temperature zero

Figures 3(a)-(c) show the plot for the value of

plasma is unstable for lower values of drift speed (

depends on the sustainability of the electric field, which requires the charge neutrality conditions should be maintained for the oscillations. When the drift velocity of electrons increases, the plasma loses the charge neutrality conditions or local energy is reduced and hence the oscillations easily die out for larger drift speed. When the thermal electrons are added, it gives sufficient background density to support the electric field or it gives an additional local energy to maintain the oscillations. This could be verified from the equation, i.e., density of perturbation of electrons given as

If we look into the equation, we see the second term in the denominator contains the square of drift speed which decreases the electron density, but the last term in the denominator is the kinetic pressure term it also reduces the density of electrons. So, it justifies the result that drift velocity of electrons and electron temperature reduces the instability.

As we have seen that the instability is sensitive to the drift speeds and thermal speeds, we have to analyze the growth rate for various temperatures. For deducing the growth rate, expand Equation 16, we arrive at the following fourth power equation. We know that for sufficiently small y, that is the plasma is unstable for

i.e.,

Since

After some simplifications and, the dispersion relation can be brought to the form

This is the general dispersion relation for warm plasma for two stream instability. This is a forth power equ- ation in x

To check the effect these waves over the solar atmosphere, we plot the growth rate for photosphere and corona and found the results are as expected shown in

Streaming instabilities arise when there is relative velocity between ions and electrons in a plasma. The simplest type of streaming instability is the two stream instability which arises in an electron-proton plasma with electrons moving faster relative to ions. This instability has been studied by several authors for cold plasma, but for warm plasma, it is the first time that a study has been done by deriving a general dispersion relation. The dispersion relation for TSI is a fourth power equation in the angular frequency and wave vector, which has been studied for different conditions. For cold plasma, this instability arises only for lower values

This instability sensitively depends on the ion temperature, the growth rate shifts to the higher, and higher frequency region as the ion temperature increases. The high frequency oscillations are necessary conditions or pre-assumptions for this instability, that is the ion temperature become the source of this instability or the reverse can be more sensible that the damping of this instability will heat the ions.

Also, this instability depends on the electron temperature; the thermal velocity of electrons must be greater than a critical value which is double of the drift velocity, which means that the kinetic energy of electrons must be double times to negate the loss of the energy due to the drift velocity. It is found that the growth rate shifts towards the low frequency regions when the electron temperature increases. This is also quite exciting as it is a high frequency oscillation, if the kinetic energy of electrons shifts the growth rate to the low frequency region, to conserve the energy; the ions must be heated at the expense of the kinetic drift of electrons. This is quite true from the study when the drift velocity increases the instability decreases.

We have studied this instability for the coronal plasma and photospheric plasma, and found the result as predicted that there is no growth rate for coronal plasmas; the oscillations are damped out to heat the ions but for photosphere there is a good growth rate. This instability is a common one and hence this result can be applied to any hot electron-proton plasma and can be more useful to fusion devices.

The authors acknowledge the financial help made by the UGC for this work as a minor project on “Role of Macro Instabilities in Solar Coronal Heating”.