Modeling the time and energy behavior of the GCR intensity in the periods of low activity around the last three solar minima
M. B. Krainev, G. A. Bazilevskaya, M. S. Kalinin, A. K. Svirzhevskaya, N. S. Svirzhevsky
aa r X i v : . [ a s t r o - ph . S R ] N ov ND I NTERNATIONAL C OSMIC R AY C ONFERENCE , R
IO DE J ANEIRO T HE A STROPARTICLE P HYSICS C ONFERENCE
Modeling the time and energy behavior of the GCR intensityin the periods of low activity around the last three solar minima K RAINEV
M.B., B
AZILEVSKAYA
G.A., K
ALININ
M.S., S
VIRZHEVSKAYA
A.K., S
VIRZHEVSKY
N.S.
Lebedev Physical Institute, Moscow, Russia [email protected]
Abstract:
Using the simple model for the description of the GCR modulation in the heliosphere and the setsof parameters discussed in the accompanying paper we model some features of the time and energy behavior ofthe GCR intensity near the Earth observed during periods of low solar activity around three last solar minima.In order to understand the mechanisms underlying these features in the GCR behavior, we use the suggestedearlier decomposition of the calculated intensity into the partial intensities corresponding to the main processes(diffusion, adiabatic losses, convection and drifts).
Keywords: modeling GCR intensity, GCR intensity around solar minima, unusual solar minimum 23/24
The period of low solar activity of the last solar cycle(SC) 23 was rather strange not only because of the record–setting heliospheric and GCR characteristics in the mini-mum 23/24 between SC 23 and 24 (see references in ouraccompanying papers [1, 2]). Some other time and energydetails were also unusual [3, 4, 5, 6, 7]: the abrupt changeof the energy dependence of the GCR modulation sometime before the moments of the maximum of the inten-sity; the unusual sequence of these moments for the lowand high energy particles; the unusual correlation betweenchanges of the different heliospheric parameters. A fewworks were devoted to the interpretation of these details[8].In this paper using the simple model for the descrip-tion of the GCR modulation and the sets of parameters dis-cussed in [2] we try to reproduce some of the above timeand energy features in the GCR intensity near the Eartharound three last solar minima. Besides, in order to un-derstand the mechanisms underlying these features in theGCR behavior, we discuss the behavior of some other GCRcharacteristics, the radial gradients and the partial intensi-ties corresponding to the main processes (diffusion, adia-batic losses, convection and drifts), which we calculate us-ing the method of decomposition of the calculated inten-sity suggested in [9, 10].
In [2] we discuss the differential boundary–value problemfor the distribution function U ( ~ r , p , t ) = J ( ~ r , T , t ) / p [11,12, 13]: − (cid:209) · ( K (cid:209) U ) | {z } diff. + ~ V sw (cid:209) U | {z } convect. − (cid:209) · ~ V sw p ¶ U ¶ p | {z } adiab.loss + ~ V dr (cid:209) U | {z } drift = r = r min , r max andpoles (without termination shock and heliosheath) and the”initial” condition U | p = p max = U um ( p max ) , where J , p and T are the intensity, momentum and kinetic energy of theparticles, p max =
150 GeV/c and U um is the distribution function of the unmodulated GCRs. The set of constantmodel parameters { h i } c is chosen in [2] as well as the setsof the main { B r , E , a t , V sw , E } m and additional { r max , a R } a parameters necessary to describe the GCR intensity in theminima of the last solar cycles. Here B r , E , V sw , E , a t are themagnitude of the HMF radial component, the solar windvelocity (both near the Earth) and the heliospheric currentsheet (HCS) tilt angle, respectively, and r min , r max and a R are the minimum and maximum radii of the modulationregion and the index in the rigidity dependence of the GCRdiffusion coefficients, respectively.So using the same { h i } c , the measured values of { B r , E , V sw , E , a t } m and the additional factors { r max , a R } a one can calculate the GCR intensity for any time in thewhole heliosphere for any energy. Here we are interestedin comparison of the time and energy behavior of the cal-culated and observed GCR proton intensity near the Earth( r = r E = J = J E =
90 deg ) in three periodsof low solar activity around solar activity minima 21/22,22/23 and 23/24 for the low T low =
200 MeV and high T high = a qt ,the half of the heliolatitude HCS range calculated usingthe Wilcox Solar Observatory (WSO) model [16], can beused as a HCS tilt angle a t in the models of drift velocity[17, 18, 19]. In the process of solving boundary–value problem (1) foreach time run (the Carrington rotation) the finite differenceapproximation of the radial gradient of the relative inten-sity u ( r E , J E , T ) ≡ J ( r E , J E , T ) / J um ( T ) is saved for each odeling GCR intensity around the last three solar minima33 ND I NTERNATIONAL C OSMIC R AY C ONFERENCE , R
IO DE J ANEIRO step in energy and then its time behavior can be consideredfor the low and high energy particles.The decomposition of the calculated intensity into thepartial “intensities” connected with the main processes ofthe GCR modulation is discussed in more details in [9, 10].Here we only mention that just as the radial gradient of therelative intensity in the process of solving boundary–valueproblem for each step in energy we also save the finite dif-ference approximation of each term of (1) for r E , J E . Thenwe reconsider the usual partial transport equation (1) asthe ordinary differential equation with respect to momen-tum which can be easily integrated as the approximation ofall terms have been already memorized for all energies. Insuch a way (and converting from the distribution functionto intensity) we get the partial “intensities” correspondingto the diffusion J di f fp , convection J convp and drift J dri ftp , andtheir sum J dcd . As we also know the total calculated inten-sity J , we can calculate the difference J − J dcd and call itthe partial “intensity”, corresponding to the adiabatic loss, J adiabp . So we get J = J di f fp + J convp + J adiabp + J dri ftp . As thisdecomposition is made for each time run we can study thetime behavior of each partial ”intensity” along with the to-tal calculated intensity for the low and high energy parti-cles.The inverted commas in the “intensities” emphasize theconventionality of these terms, which can be negative ifthe process leads to the reduction of the total intensity. Ofcourse, the meaning of, for example, the partial diffusionintensity is not simply the contribution of this process inthe total intensity, as the diffusion term in (1) depends onthe intensity gradient which is the product of all processes. In Fig.1 the time profiles of some time–dependent parame-ters of the model and of the GCR intensity near the Earth(both calculated and observed) are shown for the periodsaround three last solar minima. The 27d averaged HMFcharacteristics near the Earth [20] and the HCS quasi–tilt a qt ([16], classic) are yearly smoothed. As the observedlow energy proton GCR intensity J (
200 MeV) we use for1973–2006 J p ( −
230 MeV) (IMP8/GME, reported in[21]) and for 2006–2009 J p ( −
240 MeV) (PAMELA,constructed in [1] from the data reported in [22]). The timebehavior of the calculated high energy proton GCR inten-sity J ( T e f f ≈ B r , E and a qt which was discussed in[2]. The time profile of V sw , E is not shown in Fig.1 but itdemonstrates the same gradual decrease for the last threeminima. For the calculations in this paper we decided notto change the additional parameters { r max , a R } a aroundeach solar minimum, while their change between minima21/22–22/23 and 23/24, also discussed in [2] is easily seenin Fig.1 (b).The calculated low energy GCR intensity (red line inFig.1(c)) is very close to the observed one for almost thewhole period around 21/22 solar minimum (except for1981–1982) and also for 2008–2009 before the 23/24 so-lar minimum. Note that the HMF polarity A is negative forboth of these periods. In contrast, for the A-positive period Figure 1 : The time–dependent parameters of the modeland the GCR intensity in 1980–2013. In the panels: (a)the absolute value of the HMF radial component (red) andthe HCS quasi–tilt (blue); (b) the radius of the modulationregion (red) and the index of the rigidity dependence ofthe parallel diffusion coefficient (blue); (c) the observed(black) and calculated (red) low energy GCR intensity andalso the calculated high energy intensity (blue) normalizedto the calculated low energy intensity by the linear regres-sion for the period 1983–1986, shown by the horizontalblue line near the time axis. The observed J low ( t ) is shiftedby half a year back in time.around the 22/23 solar minimum the time profile of the cal-culated GCR intensity is more narrow than that of the ob-served intensity although the well–known tendency of thealternating peak-like (for A <
0) and more flat (for A > A –positive period around 22/23 solarminimum and for the first half of A –negative period beforesolar minimum 23/24 getting even greater for 2008–2009.So the results of the calculation support the conclusion of[1] on the gradual softening of the GCR variation spectrumwhen one goes from 21/22 to 22/23 solar minima gettingexceptional near 23/24 minimum. Note, however, that soft-ening of the GCR spectrum during the A–positive period(1990–2000) with respect to the A–negative one (1980–1990) is to some extend just the manifestation of the mag-netic cycle, as the cross–over of the differential spectra forthese two periods occurs at T co ≈
10 GeV (see [2]). odeling GCR intensity around the last three solar minima33 ND I NTERNATIONAL C OSMIC R AY C ONFERENCE , R
IO DE J ANEIRO
In Fig.2 beside the total GCR intensity, the time profilesof the additional GCR characteristics (the radial gradientsof the relative intensity and partial “intensities” connectedwith the main processes of the modulation) are shown forthe low and high energy particles along with time profilesof B r , E and a qt . Figure 2 : The detailed GCR characteristics in 1980–2013.In the panels: (a) the absolute value of the HMF radial com-ponent (red) and the HCS tilt (blue); (b) the calculated ra-dial gradient of the relative intensity near the Earth for thelow energy (red) and high energy (blue) particles; (c) thetotal calculated GCR intensity (black) and the partial “in-tensities” for the low energy: connected with the diffusion(blue), convection (green), adiabatic loss (violet) and drift(red); (d) the same as in panel (c) but for the high energyparticles. Note that the intensities in panels (c) and (d) areshown in the LOG–scale but taking into account their sign.The time behavior of the calculated local relative inten-sity gradients is rather interesting and unexpected. First,why does the radial gradient increase when one approachesthe moment of the intensity’s maximum for the low en-ergy particle for both types of HMF polarity, while for thehigh energy particles the situation is opposite? Its behav-ior for the high energies looks as more expected as we areaccustomed to the idea that the radial gradients decreaseas one goes from maximum to minimum of the solar cy-cle. Second, for the “normal” pair of solar minima, 21/22and 22/23, the calculated radial gradients of the intensityare higher for A-positive than for A-negative period, al-though we are accustomed to the opposite behavior of theradial gradient. However, in this case one should rememberthat this usual behavior concerns the radial gradients in thefree heliosphere far from both its outer and inner regionswhile the gradients and intensities shown in Fig.2 (b) arefor the inner heliosphere, rather near the Sun with its strongmagnetic fields. And third, why does the radial gradientof the high energy intensity manifest such a strong and abrupt quasi–periodical variations (something like quasi–biannual ones) while for the low energy particles (and evenfor the heliospheric modulating factors) these variationsare much less evident? Now we cannot answer all thesequestions, but the answers are very important for under-standing the GCR intensity behavior and especially for thepartial “intensities” as the latter are directly connected tothe intensity gradients by their definition.
Figure 3 : The detailed GCR characteristics in 1984–1988(left) and 2007–2011 (right). In the panels: (a, e) the ab-solute value of the HMF radial component (red) and theHCS quasi–tilt (blue). The moments of their minima areshown by the vertical dashed lines of the correspondingcolor; (b, f) the calculated radial gradient of the relative in-tensity near the Earth for the low energy (red) and high en-ergy (blue) particles; (c, g) the total calculated GCR inten-sity (black) and the main partial “intensities” for the lowenergy: the sum of the partial “intensities” connected withthe diffusion, convection and adiabatic loss (blue) and drift(multiplied by 5, red); (d, h) the same as in panel (c) butfor the high energy particles.The behavior of the calculated partial “intensities” isalso interesting. First, it can be seen that the diffusion par-tial intensity is always positive while the convective andadiabatic ones are always negative. It means that for thecase considered (the inner heliosphere near the equatoraround solar minima) the diffusion always increases the in-tensity while the convection and adiabatic losses decreaseit. The drift partial intensity is positive for A–negative andnegative for A–positive periods. As to their magnitude forthe case considered the diffusion term is the greatest whilethe convection term is the smallest (their ratio is about 100).The magnitude of the drift term is intermediate and its rel-ative contribution increases as the tilt diminishes. It is in-teresting that near the solar minimum the magnitude of thedrift term for A–negative period is of the same magnitudeas for A–positive periods and, as can be easily shown, thedrift term almost entirely consists of the current–sheet driftcomponent. It means that the flat form of the GCR inten- odeling GCR intensity around the last three solar minima33 ND I NTERNATIONAL C OSMIC R AY C ONFERENCE , R
IO DE J ANEIRO sity time profile during A–positive solar minima can bedue to the fact that the pointed in magnitude drift contribu-tion reduces the intensity making the profile V–like (or atleast more flat), while for A–negative periods it enhancesthe intensity making the profile L –like. Note that this con-clusion is opposite to the usual view that the GCR inten-sity time profile is flat around the qA > J dacp . Then the totalintensity can be considered as the sum of two main partial“intensities”, J = J dacp + J dri ftp . In Fig.3 along with the mainheliospheric modulating factors and the radial gradients ofthe relative intensity the time profiles of the above mainGCR partial intensities are compared for 4–year periodsaround two last solar minima with the same HMF polarity A <
0. Note that now to see the details better we used thelinear scale for the intensities and the drift partial intensityis multiplied by 5.The first interesting detail one can see from Fig.3 isthat for the low energy particles the magnitudes of thedrift term for 21/22 (1987) and 23/24 (2009) solar minimaare about the same (the lower HMF strength in 2009 iscompensated by the higher tilt angle) while the dac–termand total intensity are significantly (50–70 %) higher forthe second minimum. The situation with the high energieslooks much more symmetric, that is the small excess of thetotal intensity in 2009 consists of comparable excesses inboth dac– and drift terms. It means that according to ourmodel the additional flux of low energy particles in 2009with respect to 1987 which is discussed in [3, 6, 2] is duemainly to enhanced diffusion rather than drift. The secondinteresting detail is that the time profile of the calculatedintensity is much more pointed in 2009 than in 1987 andthe moment of its maximum in 2009 coincides with themoment of minimum of tilt, not of HMF strength, as in1987. Finally the third feature seen in the relative behaviorof different main partial intensities is the change of the driftcontribution when compared with the diffusion (or dac–term), especially sudden for 23/24 period. In the middle of2008 the drift term started to grow faster while the dac–term almost stopped growing. Probably this feature canbe relevant to the change in 2008 of the observed GCRintensity variation spectrum reported in [6].Note that in this paper we have not used all the poten-tials of the model. We have not tried to imitate the changeof B / d B in correlation with the HCS tilt reported in [7], B and d B being the HMF strength and its variation, re-spectively. Besides, the gradual component of the changeof r max (cid:181) p N sw V sw in 1980–2000 could result in the grad-ual softening of the GCR intensity variation in accordancewith the observations [1]. At least this explanation of thegradual softening of the GCR intensity variation wouldlook more substantiated by the observations than the grad-ual change of the energy dependence of the diffusion coef-ficient which was used in [14] to reproduce the PAMELAenergy spectrum in 2006–2009 [22].
1. Using rather simple model of the GCR modulation inthe heliosphere it is possible to reproduce to some extendthe important features of the time and energy behavior ofthe GCR intensity in the periods around the last three solaractivity minima: the general form of the time profiles ofthe low energy GCR intensity; the great excess of the lowenergy GCR intensity during the last solar minimum; thegradual softening of the GCR spectrum when one goesfrom one period around solar minimum to the next one.2. To understand the mechanisms underlying the observedfeatures of the time and energy behavior of the GCR inten-sity in the periods around the last three solar activity min-ima the use of some additional calculated characteristics ofthe GCR intensity (the local intensity radial gradients andpartial “intensities” connected with the main processes ofthe modulation) can be very useful.3. The relative changes of the different partial “intensities”,probably, indicate to the causes of some peculiarities in theGCR intensity observations (what mechanisms are behindthe different forms of the intensity–time profiles during pe-riods of opposite HMF polarity; what mechanism is mainlyresponsible for the energy dependent excess of the GCR in-tensity during the last solar minimum; how the change ofthe different heliospheric parameters influences the differ-ent components of the GCR intensity etc). Some featuresof the behavior of the radial gradients of the GCR intensityare still intriguing and need further consideration.
Acknowledgment:
We thank the Russian Foundation forBasic Research (grants 11-02-00095a, 12-02-00215a, 13-02-00585a, 13-02-10006k) and the Program “Fundamental Proper-ties of Matter and Astrophysics” of the Presidium of the RussianAcademy of Sciences.
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