PREDICTION OF THE PROTON-TO-TOTAL TURBULENT HEATING IN THE SOLAR WIND

We adopt the same specific model for the high-speed solar wind used by Cranmer et al. (2009) to facilitate the comparison to their empirical turbulent heating estimate. This model is used to specify, as a function of heliocentric radius R, the two key plasma parameters required by the turbulent heating prescription: the proton plasma beta β_p and the proton-to-electron temperature ratio T_p/T_e.

Analytic fits to in situ measurements of the high-speed solar wind (faster than 600 km s⁻¹) from the Helios and Ulysses spacecraft over the range 0.29 AU < R < 5.4  AU were used by Cranmer et al. (2009) to generate equations for the proton and electron temperatures as function of heliocentric radius R,

$$\begin{equation} \ln \left(\frac{T_p}{10^5\mbox{ K}} \right) = 0.9711 - 0.7988 x + 0.07062 x^2 \end{equation} \tag{ 1 }$$

$$\begin{equation} \ln \left(\frac{T_e}{10^5\mbox{ K}} \right) = 0.03460 - 0.4333 x + 0.08383 x^2, \end{equation} \tag{ 2 }$$

where x ≡ ln (R/1 AU).

In addition to the proton temperature, we need to specify the form of the proton density and magnetic field strength to determine the proton plasma beta β_p = 8πn_pT_p/B². Following Cranmer et al. (2009), we take a proton density of the form

$$\begin{equation} n_p(R) = n_0 (R/1\mbox{ AU})^{-2}, \end{equation} \tag{ 3 }$$

where n₀ = 2.5 cm⁻³. The empirical turbulent heating constraints calculated by Cranmer et al. (2009) used a colatitude θ = 15° to model the high-latitude Ulysses measurements. For the heliocentric distances covered by this model, the winding of the magnetic field into the Parker spiral for the high-speed streams at this colatitude is relatively weak, so a simple monopolar model for the magnetic field strength is a reasonable approximation,

$$\begin{equation} B(R) = B_0 (R/1\mbox{ AU})^{-2}, \end{equation} \tag{ 4 }$$

with B₀ = 2.5 × 10⁻⁵ G.

Using these functions for T_p, T_e, n_p, and B in high-speed solar wind streams, we find that the proton plasma beta varies from β_p = 0.92 at 0.29 AU to β_p = 34 at 5.4 AU, and the proton-to-electron temperature ratio varies from T_p/T_e = 3.9 at 0.29 AU to T_p/T_e = 1.3 at 5.4 AU.

The heating prescription presented in Howes (2010) is determined using a model for the turbulent cascade of energy in a magnetized, weakly collisional plasma (Howes et al. 2008a). The cascade model determines the steady state form of the magnetic energy spectrum of Alfvénic fluctuations, based on three primary assumptions: (1) the Kolmogorov hypothesis that the energy cascade is determined by local interactions (Kolmogorov 1941), (2) the turbulence maintains a state of critical balance at all scales (Goldreich & Sridhar 1995), and (3) the linear kinetic damping rates are applicable in the nonlinearly turbulent plasma.

The dependence of the nonlinear energy transfer rate on the local turbulent fluctuations in the cascade model (Howes et al. 2008a) is inspired by the following theoretical picture of low-frequency, anisotropic Alfvénic turbulence in a magnetized, weakly collisional plasma (Howes 2008; Schekochihin et al. 2009). The energy of Alfvénic fluctuations is injected into the turbulence isotropically at a scale much larger than the ion Larmor radius, L₀ ≫ ρ_i, corresponding to an isotropic driving wavenumber k₀ρ_i ≪ 1. Since the damping of Alfvénic fluctuations at this large scale by wave–particle interactions in a weakly collisional plasma is negligible, the turbulent fluctuations rise to sufficient amplitudes that nonlinear interactions between counter-propagating Alfvén wave packets transfer the turbulent fluctuation energy to smaller scales. This sets up a critically balanced, anisotropic cascade of MHD Alfvén waves over all scales down to the perpendicular scale of the ion Larmor radius, k_⊥ρ_i ≲ 1 (Goldreich & Sridhar 1995; Boldyrev 2005).² Even in a weakly collisional plasma, the dynamics of this Alfvén wave cascade is rigorously described by the equations of reduced MHD (Schekochihin et al. 2009). At the perpendicular scale of the ion Larmor radius k_⊥ρ_i ∼ 1, the turbulence transitions to a critically balanced, anisotropic cascade of kinetic Alfvén waves over the perpendicular scales k_⊥ρ_i ≳ 1.

The range of scales traversed by the MHD Alfvén wave cascade, between the driving scale and the ion Larmor radius scale, is commonly designated the "inertial range" (Kolmogorov 1941) of MHD turbulence—i.e., the range of scales over which the effects of driving and dissipation are negligible. For a sufficiently large inertial range, the anisotropy of the energy cascade (k_∥∝k^2/3_⊥ in the Goldreich–Sridhar theory or k_∥∝k^1/2_⊥ in the Boldyrev theory) leads to turbulent fluctuations, at the transition to the kinetic Alfvén wave cascade at k_⊥ρ_i ∼ 1, that are highly elongated along the direction of the local magnetic field, k_∥/k_⊥ ≪ 1. Such anisotropic fluctuations are optimally described by a low-frequency expansion of kinetic theory called gyrokinetics (Rutherford & Frieman 1968; Frieman & Chen 1982; Howes et al. 2006; Schekochihin et al. 2009). For Alfvénic fluctuations, the anisotropy implies that, even at scales k_⊥ρ_i ∼ 1, the turbulent fluctuation frequency remains much smaller than the ion cyclotron frequency, ω ≪ Ω_i. This is an important limit for the applicability of gyrokinetic theory and enables one to determine quantitatively the limit of validity of the heating prescription (see Section 2.5). When this limit is satisfied, the ion cyclotron resonance plays a negligible role in the collisionless damping of the turbulent fluctuations (Lehe et al. 2009). Instead, collisionless damping of the fluctuations occurs via both the ion and electron Landau resonances at scales k_⊥ρ_i ≳ 1, implying that the kinetic Alfvén wave cascade comprises the "dissipation range" of Alfvénic turbulence.

It is in the dissipation range that wave–particle interactions transfer the electromagnetic fluctuation energy to the ion and electron particle distribution functions. Ultimately, this free energy in the particle distribution functions is transferred to small scales in velocity space through an entropy cascade (Schekochihin et al. 2009; Tatsuno et al. 2009; Plunk et al. 2010; Plunk & Tatsuno 2011), enabling arbitrarily weak collisions to thermalize this energy, increasing the entropy, and leading to irreversible heating of the plasma. Wave–particle interactions via the Landau resonance typically peak at k_⊥ρ_i ∼ 1 for ions and k_⊥ρ_i > 1 for electrons, so the predicted result for ion-to-electron heating Q_i/Q_e is sensitive to the model for the nonlinear energy transfer rate in the kinetic Alfvén wave cascade.

Although the physical model of the turbulent cascade presented here remains controversial within the heliospheric physics community, there exists significant numerical and observational evidence in support of two of its key features: (1) the turbulent frequency remains low, ω ≪ Ω_i, even for k_⊥ρ_i ≳ 1 and (2) the turbulence transitions to a cascade of kinetic Alfvén waves at k_⊥ρ_i ∼ 1. A gyrokinetic numerical simulation of the transition from the MHD Alfvén wave to the kinetic Alfvén wave cascade at k_⊥ρ_i ≳ 1 (Howes et al. 2008b) produces magnetic and electric energy spectra that are consistent with Cluster measurements of turbulence in the solar wind (Bale et al. 2005). A recent gyrokinetic simulation spanning the entire dissipation range from the ion to the electron Larmor radius (Howes et al. 2011) yields a magnetic energy spectrum that is quantitatively consistent with in situ measurements of the dissipation range turbulence up to 100 Hz (Sahraoui et al. 2009; Kiyani et al. 2009; Alexandrova et al. 2009; Chen et al. 2010; Sahraoui et al. 2010). The striking agreement between the predictions for a kinetic Alfvén wave cascade and the observed magnetic and electric power spectra found by Sahraoui et al. (2009) provide observational support for this model. Finally, a k-filtering analysis of multi-spacecraft Cluster measurements demonstrates that the wavevectors of the turbulent fluctuations at scales k_⊥ρ_i ∼ 1 are aligned nearly perpendicular to the local magnetic field (Sahraoui et al. 2010); for Alfvénic turbulent fluctuations, this implies low turbulent frequencies, ω ≪ Ω_i, in support of the turbulent model employed in this study.

Assuming a fully ionized plasma of protons and electrons with isotropic Maxwellian equilibrium velocity distributions, Howes (2010) used the turbulent cascade model (Howes et al. 2008a) to calculate the total proton and electron heating resulting from collisionless damping of the electromagnetic fluctuations of the Alfvénic turbulent cascade. The model employs the linear collisionless gyrokinetic dispersion relation (Howes et al. 2006) to determine both the linear kinetic damping rate via the Landau resonances and the nonlinear energy cascade rate, so the resulting heating prescription is only valid in the gyrokinetic limit, ω ≪ Ω_p.

The resulting prescription for the ratio of proton-to-electron heating for T_p/T_e > 1 is given by

$$\begin{equation} Q_p/Q_e = c_1\frac{c_2^2 + \beta _p^\alpha }{c_3^2 + \beta _p^\alpha } \sqrt{\frac{m_p T_p}{m_e T_e}} e^{-1/\beta _p}, \end{equation} \tag{ 5 }$$

where c₁ = 0.92, c₂ = 1.6/(T_p/T_e), c₃ = 18 + 5log (T_p/T_e), and α = 2 − 0.2log (T_p/T_e). The requirement that the proton cyclotron resonance plays a negligible role in the dynamics and dissipation of the turbulence enables the regime of validity of this turbulent heating prescription to be quantified. The limit of the regime of validity can be expressed as a constraint on the minimum width of the inertial range (k₀ρ_p)⁻¹_min, shown in panel (a) of Figure 2 as a contour plot in the (β_p, T_p/T_e) parameter space.

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Substituting the values of β_p(R) and T_p/T_e(R) specified in Section 2.1 into Equation (5) enables the calculation of the predicted ratio of proton-to-electron turbulent heating Q_p/Q_e(R) as a function of heliocentric radius. In Figure 1, we plot the predicted ratio of proton-to-total turbulent heating Q_p/(Q_p + Q_e) versus heliocentric radius R.

We compare this theoretical prediction to the empirical estimate by Cranmer et al. (2009) (dashed) for 0.8 AU ≲ R ⩽ 5.4 AU in Figure 1. The error estimates (dotted) in this figure are derived by attempting to account for the error arising from both modeling and observational uncertainties. Modeling uncertainties are derived by taking the curves for outflow speeds of 650, 700, 750, and 800 km s⁻¹ in Figure 3(a) and colatitudes of 0°, 15°, and 30° in Figure 4(b) of Cranmer et al. (2009). The observational uncertainties in the calculated turbulent heating due to observed variations in the proton and electron temperatures and electron heat flux are estimated roughly by taking ±10% of the proton-to-total turbulent heating ratio. The error estimates (dotted) plotted in Figure 1 are determined by taking the outer envelope of these modeling and observational uncertainties.

We find generally good agreement between the prediction of the turbulent heating prescription (solid) and the empirical estimate by Cranmer et al. (2009) (dashed) for 0.8 AU ≲  R  ⩽  5.4 AU. The disagreement at R ≲ 0.8 AU, as we shall see in Section 2.5, is attributed to the violation of the gyrokinetic approximation, and, therefore, to exceeding the limits of validity of the turbulent heating prescription. The downturn in the empirical estimate of Q_p/(Q_p + Q_e) (dashed) seen at R > 3 AU may be an artifact of the bifurcation of the electron temperatures measured by Ulysses, as seen in Figure 1(a) of Cranmer et al. (2009), and may not represent an actual decrease in the proton-to-total turbulent heating ratio for the high-speed wind. Cranmer et al. (2009) noted that this appeared to be a solar cycle effect, but further work will required to ascertain the significance of this downturn.

We can compare our prediction of the turbulent heating based on Equation (5) with predictions based on simple theoretical models of the turbulent heating by Cranmer et al. (2009), presented in Figure 5 of their paper. Their approach used a quasilinear framework to estimate the proton-to-electron heating rates for three particular models of the distribution of turbulent energy in wavevector space: an isotropic distribution, a slab distribution of only parallel wavevectors, and a "two-dimensional" (2D) distribution of nearly perpendicular wavevectors. All three of the models showed significant disagreement with the empirically determined heating ratio. The slab model predicted 100% proton heating, the isotropic model overestimated the proton heating for R ≲ 2 AU, and the 2D model significantly underestimated the proton heating at R ≳ 1 AU. In light of these results, the agreement between the proton-to-total turbulent heating ratio predicted by Equation (5) and the empirical estimate in Figure 1 is quite good. This result suggests that the turbulent cascade model (Howes et al. 2008a) captures the dominant physical mechanisms (described qualitatively in Section 2.2) that play a role in the dissipation of solar wind turbulence at ≳ 0.8 AU.

The disagreement between the prediction of the proton-to-total turbulent heating ratio (solid) and the empirical estimate (dashed) in Figure 1 can be understood if we evaluate the limit of the regime of validity of the turbulent heating prescription (Equation (5)) using the plasma parameters specified for the solar wind model in Section 2.1. The violation of the gyrokinetic approximation, or, equivalently, the point at which the proton cyclotron resonance begins to play a non-negligible role in the dynamics and dissipation of the turbulence, is cast as a requirement for the minimum dynamic range spanned by the inertial range, given by the driving scale divided by the proton Larmor radius, L₀/ρ_p ∼ (k₀ρ_p)⁻¹. We denote this as the minimum "width" of the inertial range, (k₀ρ_p)⁻¹_min, plotted in panel (a) of Figure 2 as a contour plot in the (β_p, T_p/T_e) parameter space. The proton Larmor radius ρ_p = v_tp/Ω_p = c(2T_pm_p)^1/2/(q_pB) is easily determined as a function of heliocentric distance R given the models for T_p(R) and B(R) given in Section 2.1. Estimating isotropic driving wavenumber of the turbulence k₀(R), however, requires the incorporation of additional empirical constraints.

We interpret the isotropic driving scale L₀ = 2π/k₀ to be the outer scale of the inertial range, and we identify this scale observationally as the break in the solar wind magnetic energy spectrum from the f⁻¹ energy containing range to the f^−5/3 inertial range. To estimate k₀, we employ measurements of magnetic energy spectrum from Helios 2 data published in Figure 23 of Bruno & Carbone (2005). The frequency f₀ of the spectral break marking the outer scale of the inertial range measured from this figure is given in Table 1. Each of these spectra are measured in the same corotating fast stream with a velocity v_sw ≃ 700 km s⁻¹ (taken from Figure 17 of Bruno & Carbone 2005), so we may calculate the corresponding inertial range outer scale length L₀ = v_sw/f₀ or wavenumber k₀ = 2π/L₀.

The function

$$\begin{equation} k_0 = K_{0} \left(\frac{R}{1\mbox{ AU}}\right)^{-2} \end{equation} \tag{ 6 }$$

provides a reasonable fit for the evolution of k₀ as a function of heliocentric radius R with the value K₀ = 5 × 10⁻⁶ rad km⁻¹. Although this fit is based solely on measurements in the inner heliosphere and may not be an accurate representation of the evolution of k₀(R) for R > 0.9 AU, we will see that error in the estimation of k₀ at R ≳ 1 AU does not strongly impact the applicability of the turbulent heating prescription for the plasma parameters derived from this solar wind model.

Using the values of ρ_p(R) and k₀(R) derived here, we calculate the width of the inertial range k₀ρ_p as a function of heliocentric radius and compare it to the constraint on (k₀ρ_p)⁻¹_min, as shown in Figure 2. Panel (a) presents a logarithmic contour plot of the constraint (k₀ρ_p)⁻¹_min over the (β_p, T_p/T_e) parameter space. Also shown is the path though this parameter space (red) traversed by the solar wind model from 0.29 AU to 5.4 AU. In panel (b), the value of the width of the inertial range (k₀ρ_p)⁻¹ for our solar wind model (solid) is plotted against the constraint on the minimum width of the inertial range (k₀ρ_p)⁻¹_min for the validity of our heating model (dashed). (For the heating prescription to be valid, the dashed line must fall below the solid line in this plot.) As previously mentioned, for the plasma parameters β_p and T_p/T_e specified by the solar wind model in Section 2.1, the constraint on (k₀ρ_p)⁻¹_min does not strongly restrict the applicability of the heating prescription for R > 1 AU. It is clear, however, that this constraint is violated for heliocentric distances R ≲ 0.7 AU, signaling that the gyrokinetic representation of the turbulent dissipation mechanisms is no longer valid because the proton cyclotron resonance has begun to play a non-negligible role. Significantly, this limit to the validity of our prediction for the proton-to-total turbulent heating ratio coincides with the point where our prediction (solid) begins to fall significantly below the empirical estimate (dashed) in Figure 1. If the contribution to proton heating from the proton cyclotron resonance becomes non-negligible for heliocentric distances R ≲ 0.8 AU, a physical process is not represented in the gyrokinetic turbulent heating prescription, then the result would be an empirically estimated proton-to-total heating rate that exceeds the gyrokinetic predictions at these radii, in agreement with the behavior shown in Figure 1.