6.1 On the nature of compressive Turbulence

Considerable efforts, both theoretical and observational, have been made in order to disclose the nature of compressive fluctuations. It has been proposed (Montgomery et al., 1987

; Matthaeus and Brown, 1988

; Zank et al., 1990 ; Zank and Matthaeus, 1990 ; Matthaeus et al., 1991

; Zank and Matthaeus, 1992

) that most of compressive fluctuations observed in the solar wind could be accounted for by the Nearly Incompressible (NI) model. Within the framework of this model, (Montgomery et al., 1987 ) showed that a spectrum of small scale density fluctuations follows a $k -5/3$ when the spectrum of magnetic field fluctuations follows the same scaling. Moreover, it was showed (Matthaeus and Brown, 1988 ; Zank and Matthaeus, 1992 ) that if compressible MHD equations are expanded in terms of small turbulent sonic Mach numbers, pressure balanced structures, Alfvénic and magnetosonic fluctuations naturally arise as solutions and, in particular, the RMS of small density fluctuations would scale like $M 2$ , being $M = dv/Cs$ the turbulent sonic Mach number, $dv$ the RMS of velocity fluctuations and $Cs$ the sound speed. In addition, if heat conduction is allowed in the approximation, temperature fluctuations dominate over magnetic and density fluctuations, temperature and density are anticorrelated and would scale like $M$ . However, in spite of some examples supporting this theory (Matthaeus et al.

, 1991

reported $13%$ of cases satisfied the requirements of NI-theory), wider statistical studies, conducted by Tu and Marsch (1994

), Bavassano et al. (1995

) and Bavassano and Bruno (1995

), showed that NI theory is not applicable sic et simpliciter to the solar wind. The reason might be in the fact that interplanetary medium is highly inhomogeneous because of the presence of an underlying structure convected by the wind. As a matter of fact, Thieme et al. (1989

) showed evidence for the presence of time intervals characterized by clear anti-correlation between kinetic pressure and magnetic pressure while the total pressure remained fairly constant. These pressure balance structures were for the first time observed by Burlaga and Ogilvie (1970 ) for a time scale of roughly one to two hours. Later on, Vellante and Lazarus (1987 ) reported strong evidence for anti-correlation between field intensity and proton density, and between plasma and field pressure on time scales up to $10 h$ . The anti-correlation between kinetic and magnetic pressure is usually interpreted as indicative of the presence of a pressure balance structure since slow magnetosonic modes are readily damped (Barnes, 1979 ).

These features, observed also in their dataset, were taken by Thieme et al. (1989 ) as evidence of stationary spatial structures which were supposed to be remnants of coronal structures convected by the wind. Different values assumed by plasma and field parameters within each structure were interpreted as a signature characterizing that particular structure and not destroyed during the expansion. These intervals, identifiable in Figure 65 by vertical dashed lines, were characterized by pressure balance and a clear anti-correlation between magnetic field intensity and temperature.

These structures were finally related to the fine ray-like structures or plumes associated with the underlying cromospheric network and interpreted as the signature of interplanetary flow-tubes. The estimated dimension of these structures, back projected onto the Sun, suggested that they over-expand in the solar wind.

Figure 65:

From top to bottom: field intensity $|B |$ ; proton and alpha particle velocity $vp$ and $va$ ; corrected proton velocity $vpc = vp- dvA$ , where $vA$ is the Alfvén speed; proton and alpha number density $np$ and $na$ ; proton and alpha temperature $Tp$ and $Ta$ ; kinetic and magnetic pressure $Pk$ and $Pm$ , which the authors call $Pgas$ and $Pmag$ ; total pressure $Ptot$ and $b = Pgas/Pmag$ (adopted from Thieme et al., 1989

The idea of filamentary structures in the solar wind dates back to Parker (1963 ), followed by other authors like McCracken and Ness (1966 ), Siscoe et al. (1968 ), and more recently re-proposed in literature with new evidences (see Section 9). These interplanetary flow tubes would be of different sizes, ranging from minutes to several hours and would be separated from each other by tangential discontinuities and characterized by different values of plasma parameters and a different magnetic field orientation and intensity. This kind of scenario, because of some similarity to a bunch of tangled, smoking “spaghetti” lifted by a fork, was then named “spaghetti-model”.

A spectral analysis performed by Marsch and Tu (1993a ) in the frequency range $6 .10- 3-6 .10- 6$ showed that the nature and intensity of compressive fluctuations systematically vary with the stream structure. They concluded that compressive fluctuations are a complex superposition of magnetoacoustic fluctuations and pressure balance structures whose origin might be local, due to stream dynamical interaction, or of coronal origin related to the flow tube structure. These results are shown in Figure 66 where the correlation coefficient between number density $n$ and total pressure $Ptot$ (indicated with the symbols $pT$ in the figure), and between kinetic pressure $Pk$ and magnetic pressure $Pm$ (indicated with the symbols $p k$ and $p b$ , respectively) is plotted for both Helios s/c relatively to fast wind. Positive values of correlation coefficients $C(n, pT)$ and $C(pk, pb)$ identify magnetosonic waves, while positive values of $C(n, pT)$ and negative values of $C(pk,pb)$ identify pressure balance structures. The purest examples of each category are located at the upper left and right corners.

Figure 66:

Correlation coefficient between number density $n$ and total pressure $pT$ plotted vs. the correlation coefficient between kinetic pressure and magnetic pressure for both Helios relatively to fast wind (adopted from Marsch and Tu, 1993b ).

Following these observations, Tu and Marsch (1994

) proposed a model in which fluctuations in temperature, density, and field directly derive from an ensemble of small amplitude pressure balanced structures and small amplitude fast perpendicular magnetosonic waves. These last ones should be generated by the dynamical interaction between adjacent flow tubes due to the expansion and, eventually, they would experience also a non-linear cascade process to smaller scales. This model was able to reproduce most of the correlations described by Marsch and Tu (1993a ) for fast wind.

Later on, Bavassano et al. (1996a ) tried to characterize compressive fluctuations in terms of their polytropic index, which resulted to be a useful tool to study small scale variations in the solar wind. These authors followed the definition of polytropic fluid given by Chandrasekhar (1967 ): “a polytropic change is a quasi-static change of state carried out in such a way that the specific heat remains constant (at some prescribed value) during the entire process”. For such a variation of state the adiabatic laws are still valid provided that the adiabatic index $g$ is replaced by a new adiabatic index $g'= (cP - c)/(cV - c)$ where $c$ is the specific heat of the polytropic variation, and $cP$ and $cV$ are the specific heat at constant pressure and constant volume, respectively. This similarity is lost if we adopt the definition given by Courant and Friedrichs (1976 ), for whom a fluid is polytropic if its internal energy is proportional to the temperature. Since no restriction applies to the specific heats, relations between temperature, density, and pressure do not have a simple form as in Chandrasekhar approach (Zank and Matthaeus, 1991 ). Bavassano et al. (1996a ) recovered the polytropic index from the relation between density $n$ and temperature $T$ changes for the selected scale $Tn1 -g'= const.$ and used it to determine whether changes in density and temperature were isobaric ( $' g = 0$ ), isothermal ( $' g = 1$ ), adiabatic ( $' g = g$ ), or isochoric ( $' g = oo$ ). Although the role of the magnetic field was neglected, reliable conclusions could be obtained whenever the above relations between temperature and density were strikingly clear. These authors found intervals characterized by variations at constant thermal pressure $P$ . They interpreted these intervals as a subset of total-pressure balanced structures where the equilibrium was assured by the thermal component only, perhaps tiny flow tubes like those described by Thieme et al. (1989 ) and Tu and Marsch (1994 ). Adiabatic changes were probably related to magnetosonic waves excited by contiguous flow tubes (Tu and Marsch, 1994 ). Proton temperature changes at almost constant density were preferentially found in fast wind, close to the Sun. These regions were characterized by values of $B$ and $N$ remarkable stable and by strong Alfvénic fluctuations (Bruno et al., 1985 ). Thus, they suggested that these temperature changes could be remnants of thermal features already established at the base of the corona.

Thus, the polytropic index offers a very simple way to identify basic properties of solar wind fluctuations, provided that the magnetic field does not play a major role.