Energy Conversion Associated with Intermittent Currents in the Magnetosheath Downstream of the Quasi-Parallel Shock

Abstract

Using the data from the Magnetospheric Multiscale (MMS) mission, we studied the energy conversion between electromagnetic fields and particles (ions and electrons) in a spacecraft rest frame inside a turbulent magnetosheath downstream of the quasi-parallel shock. The results show that the energy conversion was highly intermittent in the turbulent magnetosheath, and the perpendicular electric fields dominated the energy conversion process. The energy conversion among the electromagnetic fields, ions, and electrons was related to the current intensity. In the region with weak current, the ions gained energy from electromagnetic fields, while the electron energy was released and transferred into electromagnetic fields. In contrast, in the intense current region, the energy of ions was transferred into the electromagnetic fields, but the electrons gained energy from electromagnetic fields. The results quantitatively established the relationship between energy conversion rate and current density and revealed that the energy conversion among the electromagnetic fields, ions, and electrons was related to the local current intensity inside the shocked turbulence.

1. Introduction

The magnetosheath, which is downstream of Earth’s bow shock, is situated between the bow shock and the magnetopause. The magnetosheath predominantly covers a shocked solar wind, which is a thermalized subsonic plasma flow [1,2]. Based on the angle (${\theta }_{BN}$) between the interplanetary magnetic field (IMF) and the normal direction of the upstream shock, the magnetosheath is usually classified into two different regions: the quasi-perpendicular magnetosheath, located downstream of the quasi-perpendicular shock (${\theta }_{BN}\ge {45}^{\circ }$), and the quasi-parallel magnetosheath, located downstream of the quasi-parallel shock (${\theta }_{BN}\le {45}^{\circ }$) [3]. Inside the quasi-perpendicular magnetosheath, the structures and dynamic processes are governed by the ion–cyclotron waves or mirror waves which are excited by the temperature anisotropy (ion perpendicular temperature was greater than parallel temperature) [4,5,6,7,8]. Compared to the quasi-perpendicular magnetosheath, the structures and dynamic processes are more complex and abundant inside the quasi-parallel magnetosheath [9,10,11,12,13], causing the quasi-parallel magnetosheath to gain more attention.

In addition to the linear waves, a lot of nonlinear current structures could be produced inside the quasi-parallel magnetosheath, such as reconnecting current sheets, magnetic flux ropes, vortexes, etc. [14,15,16,17,18,19,20,21,22,23,24]. The first reconnecting current sheet in the quasi-parallel magnetosheath was observed using Cluster, and the intense energy conversion process and electron heating were observed inside the current sheet [25]. Recently, using the measurements from the Magnetospheric Multiscale (MMS) [26], a lot of reconnecting current sheets have been reported in the quasi-parallel magnetosheath [14,15,16,27,28,29], and most of them are electron-only reconnecting current sheets, constituting a new-type of reconnection without the bursty ion outflows that has also been observed in the magnetotail [30,31,32,33,34]. The simulations suggested that these reconnecting current sheets could form after the upstream waves or discontinuities penetrated through the shock and were then compressed [10,13,35], as proven in a recent observational study [36]. The magnetic flux rope also contributed to the energy conversion process, and it was found that the secondary reconnection could take place inside the magnetic flux rope to facilitate energy conversion [37,38,39]. Moreover, violent energy conversion processes were also found in the vortex arrays and kinetic-scale discontinuities [19,40]. These studies suggested that the current structures should make a significant contribution to the energy conversion inside the magnetosheath.

According to the Vlasov–Maxwell equations, the energy conversion rate between electromagnetic fields and plasma species $\mathsf{\alpha }$ is:

$${\partial }_t{E}_{\alpha }=n_{\alpha }{q}_{\alpha }(E\cdot V_{\alpha })$$

(1)

where $E_{\alpha }$ is the total energy of the plasma species; $n_{\alpha }$, and $q_{\alpha }$ are the number density and charge; $V_{\mathsf{\alpha }}$ is the bulk flow velocity; and $E$ is the electric field. For the magnetosheath plasma, where the ions are dominated by the protons, the energy conversion rate between electromagnetic fields and ions (electrons) can be computed by ${\partial }_t{E}_i=n_{i}q(E\cdot V_i)=J_i\cdot E$ (${\partial }_t{E}_e=-n_{i}q(E\cdot V_e)=J_e\cdot E$), where $n_i$ ($n_e$) and $q$ are the number density of protons (electrons) and elementary charge, $V_{\mathrm{i}}$ ($V_{\mathrm{e}}$) is the bulk flow velocity of protons (electrons), and $J_i=n_{i}q{V}_i$ ($J_e=-n_{e}q{V}_e$) is the current density supported by ions (electrons). The total energy conversion rate between electromagnetic fields and plasma can be obtained from the following:

$${\partial }_{t}E={\partial }_t{E}_i+{\partial }_t{E}_e=n_{i}e(E\cdot V_i)-n_{i}e(E\cdot V_e)=J\cdot E$$

(2)

where $J=n_{i}q(V_i-V_e)$ is the current density. It is worth noting that the energy conversion rate $J\cdot E$ is dependent on the reference frame, and our analyses in this paper were carried out in the spacecraft rest frame.

In the present study, using the accurate measurements of plasma moments and electromagnetic fields from MMS, the aforementioned energy conversion rates were effectively calculated. On this basis, we studied the energy conversion related to the current density in a quasi-parallel magnetosheath, and tried to reveal the rules of the energy conversion between electromagnetic fields and particles (ions and electrons) inside the magnetosheath.

2. Instruments and Event Overview

In this paper, the data were obtained from the MMS spacecraft. Electric field data were measured using the electric field double probe (EDP) with a time resolution of 128 Hz (8192 Hz) in survey (burst) mode [41,42]. The Fast Plasma Investigation (FPI) provides data for the electron and ion moments [43]. The time resolution was 30 ms (4.5 s) for electrons and 150 ms (4.5 s) for ions in burst (survey) mode. The survey (16 Hz) and burst (128 Hz) magnetic field data were taken from the Flux Gate Magnetometer (FGM) [44].

We analyzed the data of MMS1 during 22:30–23:50 UT on 29 November 2018, when the spacecraft moved from [10.7, 7.3, 5.0] Re to [11.2, 8.3, 5.4] Re in the Geocentric Solar Ecliptic (GSE) coordinates. Figure 1 shows the overview of the event. During 22:30–22:43 UT, the spacecraft was in the magnetosheath downstream of the quasi-perpendicular shock, characterized by the dense and hot plasma (Figure 1a,b), magnetic fields with larger magnitude and weak fluctuations (Figure 1c) and stable plasma flow (Figure 1d). Around 22:43 UT, the shock changed from quasi-perpendicular to quasi-parallel. Inside the downstream magnetosheath, the fluctuations of plasma number density (Figure 1b), magnetic fields (Figure 1c), and ion bulk flow velocity (Figure 1d) became larger, and the flux of the higher-energy (>3 keV) ions was clearly enhanced (Figure 1a), consistent with previous observations of the quasi-parallel magnetosheath [12,16,25], which was a highly turbulent region. Moreover, the presence of the high-energy (>3 keV) ions indicated that the ion heating or acceleration was more evident in the quasi-parallel magnetosheath than in the quasi-perpendicular magnetosheath. Around 23:36 UT, the spacecraft crossed the shock and entered into the solar wind region, which was characterized by the smaller magnitude of the magnetic field (Figure 1c), tenuous and cold plasma (Figure 1a,b), and super-Alfven ion bulk flow velocity (Figure 1d). Moreover, during 22:43:00–23:31:00 UT, MMS briefly left the magnetosheath several times at 22:54:00–22:56:00 UT, 23:04:00–23:06:00 UT, 23:14:00–23:15:30 UT and 23:28:00–23:29:30 UT (the shadowed region in Figure 1), where the ion energy spectrogram (Figure 1a) changed clearly. In this paper, we mainly focus on the energy conversion process in the magnetosheath, and the influence of the bow shock and solar wind should be avoided. Therefore, in the following analyses, we just used data for 22:43:00–23:31:00 UT, except for 22:54:00–22:56:00 UT, 23:04:00–23:06:00 UT, 23:14:00–23:15:30 UT and 23:28:00–23:29:30 UT.

Inside the highly turbulent magnetosheath, a lot of current structures were observed (current spikes in Figure 1e). The total energy conversion rates ($J\cdot E$) between electromagnetic fields and plasma, with their parallel components ($J\cdot E_{\parallel }$, where $E_{\parallel }$ represents the parallel electric field with respect to the background magnetic fields) and perpendicular components ($J\cdot E_{\perp }$, where $E_{\perp }$ represents the components of the electric field perpendicular to the background magnetic fields), are shown in Figure 1f. The values of the energy conversion rate were large, and the peak value was up to 40 nW/m³, which was on the same scale as the typical value of the energy conversion rate inside the electron diffusion region at the magnetopause [45,46,47]. However, the values of the energy conversion rate, along with its components, tended to alternate between positive and negative values, which means that the energy conversions from electromagnetic field to particles and from particles to electromagnetic field simultaneously took place inside the different areas of the magnetosheath. The energy conversion rates between ions and electromagnetic fields ($J_i\cdot E$, Figure 1g) and the energy conversion rates between electrons and electromagnetic fields ($J_e\cdot E$, Figure 1h) were similar to the total energy conversion rate (Figure 1f). From the timing diagram of the different types of the energy conversion rates (Figure 1f–h), it could be concluded that the intense energy conversion processes in random directions were taking place inside the turbulent magnetosheath, but it is difficult to obtain the general rules for the energy conversion between electromagnetic fields and particles (ions and electrons). In the following section, we describe how we calculated the cumulative distribution functions (CDFs) of the energy conversion rate with its components, and tried to reveal the rules of the energy conversion in this time interval (22:43:00–23:31:00 UT).

3. Energy Conversion between Electromagnetic Fields and Particles

Figure 2a shows the CDFs of the current density ($J$) and energy conversion rate ($J\cdot E$), and the independent variable for both of them was the magnitude of the current density ($\left|J\right|$). The CDF of current density was computed by ${\displaystyle {\sum }_{\left|J\right|}^{\infty }f}$, where $f$ is the probability density function of current density. The current density changed from 0 µA/m² to 2.6 µA/m² (Figure 1e), but most density values were less than 1.05 µA/m², and they accounted for 99.9% of all data points (see the CDF of current density, the black trace in Figure 2a). The CDF of $J\cdot E$ was computed by ${\displaystyle {\sum }_0^{|J|}J\cdot E}$. When the independent variable $\left|J\right|=J_0$, the value of ${\displaystyle {\sum }_0^{|J|}J\cdot E}$ represents the sum of energy conversion rate $J\cdot E$ of the data points with a current density magnitude of less than $J_0$.

The CDF of the energy conversion rate $J\cdot E$ is shown in Figure 2a by the red trace. When the current density $\left|J\right|$ < 0.08 µA/m², the ${\displaystyle {\sum }_0^{|J|}J\cdot E}$ remained about 0 nW/m³, which means that the net energy conversion with current density magnitude $\left|J\right|$ < 0.08 µA/m² was negligible, although these data points accounted for more than 90% of all data points in this interval. At 0.08 < $\left|J\right|$ < 0.95 µA/m², the ${\displaystyle {\sum }_0^{|J|}J\cdot E}$ continued to increase, indicating that the net energy conversion rate $J\cdot E$ was positive. Namely, the net energy was transferred from the electromagnetic fields to particles when 0.08 < $\left|J\right|$ < 0.95 µA/m². When the current density $\left|J\right|$ > 0.95 µA/m², the variation of the ${\displaystyle {\sum }_0^{|J|}J\cdot E}$ was small, which suggested that the contribution from data points with $\left|J\right|$ > 0.95 µA/m² to the global energy conversion was negligible, which might be caused by the tiny percentage of these data points (less than 0.1%).

The relationship between the magnitude of the energy conversion rate and the current density is shown in Figure 2b. It is clear that the magnitude of the energy conversion rate has a linear dependence relation with the current density, consistent with the previous study in the turbulent outflow region of magnetotail reconnection [48]. Moreover, the relationship between the average energy conversion rate $〈J·E〉$ and current density is shown in Figure 2c, where a dashed line representing $\propto J^2$ is shown simultaneously. The average energy conversion rate is proportional to the $J^2$, which is also consistent with previous simulations and observations in the turbulent plasma [49,50,51,52].

Distinguishing the energy contributions from parallel or perpendicular electric fields is important for understanding energy conversion mechanisms. Thus, similar to the CDF of the total energy conversion rate ${\displaystyle {\sum }_0^{|J|}J\cdot E}$, the CDFs of energy conversion rates caused by parallel electric field ($J\cdot E_{\parallel }$) and perpendicular electric field ($J\cdot E_{\perp }$) were calculated using ${{\displaystyle {\sum }_0^{|J|}J\cdot E}}_{\parallel }$ and ${{\displaystyle {\sum }_0^{|J|}J\cdot E}}_{\perp }$, respectively. As shown in Figure 3a, when the current density $\left|J\right|$ < 0.3 µA/m², most of the increase in the CDF of $J\cdot E$ (${\displaystyle {\sum }_0^{|J|}J\cdot E}$, black trace) is derived from the increase in the CDF of $J\cdot E_{\perp }$ (${{\displaystyle {\sum }_0^{|J|}J\cdot E}}_{\perp }$, red trace), suggesting that the perpendicular electric fields dominated the energy conversion process in this range of current density. With the enhancement of current density, the contribution from parallel electric fields became important. At current density $\left|J\right|$ > 0.6 µA/m², the slope of ${{\displaystyle {\sum }_0^{|J|}J\cdot E}}_{\parallel }$ was almost equal to that of ${{\displaystyle {\sum }_0^{|J|}J\cdot E}}_{\perp }$, which indicated that the contribution from ${{\displaystyle {\sum }_0^{|J|}J\cdot E}}_{\parallel }$ to the increase in ${\displaystyle {\sum }_0^{|J|}J\cdot E}$ was nearly equal to that from ${{\displaystyle {\sum }_0^{|J|}J\cdot E}}_{\perp }$. As a whole, the perpendicular electric field still dominated the energy conversion inside the turbulent magnetosheath, and it contributed about 80% of the total energy conversion.

Another angle to reveal the energy conversion mechanisms is to understand how the released magnetic energy is divided between electrons and ions or vice versa. Therefore, the CDFs of the energy conversion rate between electromagnetic field and ions ($J_i\cdot E$), and between electromagnetic field and electrons ($J_e\cdot E$), were calculated using ${\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}$ and ${\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}$. Meanwhile, the CDFs of $J_i\cdot E_{\perp }$, $J_e\cdot E_{\perp }$, $J_i\cdot E_{\parallel }$ and $J_e\cdot E_{\parallel }$ were calculated using ${{\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}}_{\perp }$, ${{\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}}_{\perp }$, ${{\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}}_{\parallel }$, and ${{\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}}_{\parallel }$, and are shown in Figure 3c,d. It is worth noting that the independent variables for all of them were the magnitude of the current density ($\left|J\right|$).

The CDFs of total energy conversion rate (${\displaystyle {\sum }_0^{|J|}J\cdot E}$, black trace), energy conversion rate for electrons (${\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}$, magenta trace), and energy conversion rate for ions (${\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}$, cyan trace) are shown in Figure 3b simultaneously. When the current density 0 < $\left|J\right|$ < 0.16 µA/m², ${\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}$ increased monotonically with the increase in current density $\left|J\right|$, while ${\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}$ decreased monotonically. The increase in ${\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}$ is mainly supported by the increase in ${{\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}}_{\perp }$ (cyan trace in Figure 3c), and the contribution from ${{\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}}_{\parallel }$ was nearly negligible (cyan trace in Figure 3d). Similar to the ${\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}$, the decrease in ${\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}$ was supported by the decrease in ${{\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}}_{\perp }$ (magenta trace in Figure 3c), and the contribution from ${{\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}}_{\parallel }$ was also nearly negligible (magenta trace in Figure 3d). This indicates that when the current was weak (0 < $\left|J\right|$ < 0.16 µA/m²), the net $J_i\cdot E$ was positive, while $J_e\cdot E$ was negative. The ions received energy from the electromagnetic fields, but the energy of electrons was transferred to the electromagnetic fields. Both the energy conversion processes of ions and electrons were dominated by the perpendicular electric fields.

It was quite different when the current density $\left|J\right|$ > 0.16 µA/m². The ${\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}$ decreased monotonically with the increase in current density $\left|J\right|$, while the ${\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}$ increased monotonically. Both the parallel and perpendicular electric fields contributed to the energy conversion in this range of current density. For the ions, the values of ${\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}$ dropped by ~2850 nW/m³ (from 2400 to −450 nW/m³), of which about 75% derived from ${{\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}}_{\perp }$ (the cyan trace in Figure 3c changed from 2700 to 550 nW/m³) and 30% derived from ${{\displaystyle {\sum }_0^{|J|}{J}_i\cdot E}}_{\parallel }$ (the cyan trace in Figure 3d changed from ~0 to −900 nW/m³). For electrons, the values of ${\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}$ increased by ~7000 nW/m³ (from −1900 to 5100 nW/m³), of which about 74% derived from ${{\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}}_{\perp }$ (the magenta trace in Figure 3c changed from −1900 to 3300 nW/m³) and 26% derived from ${{\displaystyle {\sum }_0^{|J|}{J}_e\cdot E}}_{\parallel }$ (the magenta trace in Figure 3d changed from 0 to 1800 nW/m³). The observations suggest that when the current density was large ($\left|J\right|$ > 0.16 µA/m²), the electrons received energy from the electromagnetic field, but the energy of ions was transferred to the electromagnetic field. Both the parallel and perpendicular electric fields contributed to the energy conversion process.

4. Discussion and Conclusions

In this study, by analyzing the CDFs of energy conversion rates between electromagnetic field and plasma particles, the rules of the energy conversion among the electromagnetic field, ions, and electrons at different current intensities were revealed in the turbulent magnetosheath downstream of the quasi-parallel shock.

The energy conversion was highly intermittent in the turbulent magnetosheath, and 80% of the net energy conversion from the electromagnetic field to particles occurred at areas with J > 0.17 µA/m², which just accounted for about 40% of the total measured regions (Figure 2). The observation results were consistent with the previous simulation of the turbulent plasma [49,50] and the observations of the turbulent plasma in different environments [48,51,52,53,54]. The highly intermittent energy conversion suggests that the intermittent current structures might play an important role in the energy conversion process in the turbulent magnetosheath downstream of the quasi-parallel shock.

Overall, the perpendicular electric field dominated the energy conversion process. Based on the guiding center approximation, the energy gain of the adiabatic particles could be divided into three mechanisms: Betatron acceleration, Fermi acceleration, and parallel electric field acceleration, where the Betatron and Fermi accelerations were supported by the perpendicular electric field [55,56]. Therefore, if the energy conversion processes were dominated by the adiabatic processes, our observations suggested that the Betatron or Fermi acceleration should be more important than parallel electric field acceleration in the turbulent magnetosheath. Moreover, various plasma waves were always observed in the turbulent magnetosheath, which indicated that the non-adiabatic wave–particle interaction should also prominently contributed to energy conversion processes in the turbulent magnetosheath, such as Landau-resonant, cyclotron-resonant, and non-resonant interactions [57,58,59,60,61,62].

The energy partition between ions and electrons was related to the current intensity. At the region with weak current density, the ions received energy from the electromagnetic field, while electrons lost energy and transferred them into the electromagnetic field. However, the situations reversed at the intense current density region, where the energy of ions was transferred into the electromagnetic field, but electrons gained energy from the electromagnetic field. As a whole, the energy was transferred from electromagnetic fields to particles, and most of it was transferred into electrons. The energy conversion supported by the perpendicular electric fields followed the rules described above. The parallel electric fields always energized the electrons but decelerated the ions, independent of the current intensity.

Magnetic reconnection, which is an effective energy conversion process, is supposed to play an important role in the energy cascade and conversion in turbulent plasma [63,64,65,66]. In turn, plasma turbulence could also develop inside the reconnecting current sheet, changing the structure of the diffusion region, and facilitating the particle acceleration and energy conversion process during the reconnection [48,67,68,69,70,71,72,73]. Here, in the studied turbulent magnetosheath, a lot of possible reconnecting current sheets were observed at 22:47:30 UT, 22:53:52 UT, 23:00:35 UT, 23:10:46 UT, 23:13:23 UT, and 23:21:45 UT. Intense energy conversion rates were observed inside these potential reconnecting current sheets, indicating that reconnection should play an important role in the energy conversion inside the turbulent magnetosheath.

Recently, energy conversion has been studied in the turbulent outflow region of magnetotail reconnection [48]. In this turbulent plasma driven by magnetic reconnection, the energy conversion was dominated by the perpendicular electric field, and the energy conversion rate was directly proportional to the current intensity, which were consistent with that in the turbulent magnetosheath investigated in the present paper. The energy conversion among the electromagnetic field, ions, and electrons was related to the current intensity both in the turbulent magnetosheath and turbulent outflow region of magnetotail reconnection. However, there are still a lot of differences between them. The ions released magnetic energy in the weak currents both in the magnetosheath and outflow region of the magnetotail. However, in the strong currents, ions gained energy in the outflow region of the magnetotail but lost energy in the magnetosheath. Qualitatively, the energy conversion of electrons was similar in these different plasma environments. However, they were still quantitatively different. In the turbulent outflow region of magnetotail reconnection, the energy of the electrons gained in the strong currents was nearly equal to that lost in the weak currents. Differently, in the turbulent magnetosheath, the energy of electrons gained in the strong currents was much greater than that lost in the weak currents. Different background parameters (such as background magnetic field, plasma beta, etc.) might be responsible for these differences, and further study is necessary.

The relationship between current density and energy conversion rates caused by the non-ideal electric field has been studied in the magnetosheath with a time interval of ~4 min [52]. They found that the average energy conversion rates caused by non-ideal electric field ($J\cdot E^{\prime }$, where $E^{\prime }=E+V_e\times B$) were dominated by the parallel components ($J_{\parallel }\cdot {E^{\prime }}_{\parallel }$). The average parallel components ($J_{\parallel }\cdot {E^{\prime }}_{\parallel }$) were always negative, while the perpendicular components ($J_{\perp }\cdot {E^{\prime }}_{\perp }$) were always negative with different current intensities. We studied the energy conversion rates caused by the total electric field and found that both the parallel and perpendicular components were positive with different current intensities (Figure 2a, where the CDFs of $J_{\parallel }\cdot E_{\parallel }$ and $J_{\perp }\cdot E_{\perp }$ increased with the enhancement of current intensities). The difference between these two studies indicates that the electric field supported by $V_e\times B$ might be important for the energy conversion processes.

In summary, we studied the energy conversion among the electromagnetic field, ions and, electrons at different current intensities in the turbulent magnetosheath downstream of the quasi-parallel shock. The energy conversion was highly intermittent, and 80% of the net energy conversion occurred in areas that accounted for only about 40% of the total measured region. The perpendicular electric field dominated the energy conversion process. The energy conversion among the electromagnetic field, ions, and electrons was related to that of the current intensity. In the region with weak current, the ions gained energy from the electromagnetic field, while electrons lost energy and transferred them into the electromagnetic field. Differently, at the intense current region, the energy of ions was transferred into the electromagnetic field, but electrons gained energy from the electromagnetic field. The observations revealed that the energy conversion among the electromagnetic field, ions, and electrons is related to the local current intensity.