THIRD MOMENTS AND THE ROLE OF ANISOTROPY FROM VELOCITY SHEAR IN THE SOLAR WIND

The Navier–Stokes equations for incompressible HD and constant density can be re-expressed under assumptions of homogeneous fluctuations to yield a relation between the mean rate of energy dissipation per unit mass and correlation functions of velocity v. This approach is used to describe turbulent fluctuations with a large number of degrees of freedom such that the fluctuations become well behaved in a statistical sense. In these situations, velocity shears associated with turbulent eddies are presumed to exist over a large range of directions and to vary in speed up and down with equal repetition so that the average shear is zero. More explicitly, v is a random variable whose average is centered at zero and has the same statistical properties everywhere so as to satisfy homogeneity. The energy cascade is forward from larger to smaller scales, and this corresponds to a positive value of .

von Kármán & Howarth (1938) re-expressed the Navier– Stokes equations in the above manner for homogeneous isotropic turbulence. Their resulting equation and its generalizations will be referred to as the KH equation. Kolmogorov (1941a, 1941b) introduced increments of velocity δv(≡ v(x + r) − v(x), where r is the spatial separation lag). Like differential quantities in the Navier–Stokes equations, these increments are sensitive to the flow dynamics and for the given flow are also homogeneous. Kolmogorov recast the KH equation in terms of δv, and in doing so, he derived the four-fifths law of isotropic and fully developed HD turbulence in the limit of small but inertial range r. His findings state that 〈(δv_r)³〉 = −(4/5)r, where δv_r are increments of the longitudinal velocity (i.e., components of velocity along the lag direction r) and 〈⋅⋅⋅〉 is the ensemble average of the enclosed quantity. Here, the third power of the increment, called a third-order structure function, varies linearly with lag r, has the opposite sign to r, and is proportional to the . Monin (1959; see also Monin & Yaglom 1975) showed that (in the inertial range where viscous terms may be neglected) the KH equation is generalizable to anisotropic turbulence as ∇ · 〈(δv)²δv〉 = −4, where (δv)² ≡ δv · δv. This vector form may be simplified to one dimension by integrating it over the volume of the sphere of radius r, and applying Gauss' theorem to the divergence operator to get a surface integral. In this paper, we use the KH in spherical average form

$$\begin{equation} \frac{1}{4 \pi } \int \langle (\delta v)^2 \delta {\bf v} \rangle \cdot {\hat{\bf r}} \, d \Omega = - \frac{4}{3} \epsilon r, \end{equation} \tag{ 1 }$$

where dΩ(≡ sin θdθdϕ) is the element of solid angle, θ and ϕ are spherical polar coordinates, and ${\hat{\bf r}}$ is the unit radial vector normal to the sphere (see Nie & Tanveer 1999). The integral in Equation (1) is carried out over all angles and normalized by 4π, and so calculates the scale-to-scale flux of energy through the sphere. In Equation (1) and throughout the remainder of the paper, we assume statistically stationary flow. Owing to this, also equals the energy cascade rate. Equation (1) is valid regardless of the anisotropy of the δv. Note that the integral of the third-order structure function is negative, linear in radius (r is always nonnegative in spherical coordinates), and proportional to . The sense in which v is a homogeneous field is critical to the derivation of the KH equation. Here, v refers to a fluctuating velocity with a mean value that is zero everywhere.

Casciola et al. (2003) obtained a different third-moment equation for examining the role of velocity shear on turbulence in the absence of boundary effects. They set

$$\begin{equation} {\bf v} = {\bf u} +({\bf x} \cdot {\bf \nabla }) {\bf U}. \end{equation} \tag{ 2 }$$

Equation (2) separates total velocity into a fluctuation component u and a varying mean part U. Here the mean velocity is incompressible and is restricted to a plane-parallel shear flow that varies linearly with position. This case is referred to as a homogeneous shear flow (e.g., Hinze 1975; Townsend 1976; Davidson 2004). As such, increments of U for any lag are the same no matter the position of the origin. Casciola et al. call v in Equation (2) a homogeneous field, but we do not since the average of v varies with position. As such, the velocity used in the KH equation given by Equation (1) will not prove to be interchangeable with the one given by Equation (2). The decomposition of velocities into fluctuating and mean parts also follows the Reynolds analysis of the Navier–Stokes equations and usefully employs results from this approach. The Navier–Stokes equation is then rewritten in a similar manner as the KH equation. The resulting equation will have structure functions that are separated into parts showing the role of the background shear. Numerous experiments have explored quasi-homogeneous shear flows. These include physical experiments (e.g., Champagne et al. 1970; Harris et al. 1977; Shen & Warhaft 2000, 2002) and numerical simulations (e.g., Rogers & Moin 1987; Lee et al. 1990; Kida & Tanaka 1994; Pumir & Shraiman 1995; Pumir 1996; Schumacher & Eckhardt 2000; Schumacher 2004; Gualtieri et al. 2002; Casciola et al. 2003, 2007).

For the homogeneous velocity shear and without loss of generality, we select a coordinate system so that the background flow magnitude varies linearly with position in a single direction across the flow. Taking the background velocity U to be $\alpha y {\hat{\bf x}}$, its gradient is $\alpha {\hat{\bf y}}$, where α is the magnitude of speed change with position and is a constant because the solution of the mean momentum equation has U independent of time. The mean kinetic energy is then fixed as its production and dissipation are balanced by its convection in the mean gradient direction (e.g., Champagne et al. 1970). Fluctuations on the background shear have velocities u in all three directions and vary with position in all three directions. Casciola et al. (2003) derive an equation for this case in terms of velocity increments and averaged over of a sphere of radius r. The resulting equation is referred to here as the Casciola–Gualtieri–Benzi–Piva (CGBP) equation. We neglect a viscous term for r values greater than the dissipative scales, and in terms of structure functions, the CGBP equation is given by

$$\begin{equation} S^{{\rm HD}}_3 + S^{{\rm HD}}_U + S^{{\rm HD}}_P = - \frac{4}{3} \epsilon r, \end{equation} \tag{ 3 }$$

where

$$\begin{equation} S^{{\rm HD}}_3 = \frac{1}{4 \pi } \int \langle (\delta u)^2 \delta {\bf u} \rangle \cdot {\hat{\bf r}} \, d \Omega, \end{equation} \tag{ 4 }$$

$$\begin{equation} S^{{\rm HD}}_U = \frac{ \alpha r}{4 \pi } \int \sin ^2 \theta \sin \phi \cos \phi \langle (\delta u)^2 \rangle d \Omega, \end{equation} \tag{ 5 }$$

and

$$\begin{equation} S^{{\rm HD}}_P = \frac{2 \alpha }{4 \pi r^2} \int \!\!\int \langle \delta u_x \delta u_y \rangle r^2 {d} r {d} \Omega. \end{equation} \tag{ 6 }$$

Here, the superscript HD refers to the hydrodynamic case, δu(≡ u(x + r) − u(x)) is the increment of u along the lag, and (δu)² ≡ δu · δu. Integrals are definite. In Equations (4) and (5), they are evaluated over all angles, and in Equation (6) they are evaluated over the spherical volume with radius r. S^HD₃ is the third moment due to fluctuations alone and has the same form as the left-hand side of Equation (1). The terms S^HD_U and S^HD_P are proportional to the shear, and their values are reliant on the particulars of the fluctuation anisotropy in the (trace and off-diagonal x–y element of the tensor) second moment in the presence of shear. The term S^HD_U calculates the flux of energy through the sphere due to shear, while S^HD_P is the shear production of energy in the volume.

From Equation (3), the sum of the individual terms on the left-hand side is linear with r. Casciola et al. (2003) used three-dimensional (3D) incompressible HD numerical simulations to examine a linearly varying plane shear flow with turbulent fluctuations (see also Gualtieri et al. 2002). From the simulation results, they found that the fluctuations satisfy Equation (3) well in the inertial and energy-containing ranges. The individual terms themselves are not, however, linear with r. The sum of transfer terms S^HD₃ and S^HD_U is most significant in the inertial range, corresponding to the cascade of energy from larger to smaller scales. In the energy-containing scales, both of these transfer terms become small at large r. The term S^HD_P can be insignificant at the smallest scales of the inertial range but becomes significant in the energy-containing range, and the dominant term at large r. It is referred to as the shear production term because it accounts for the energization of the cascade. The sum S^HD₃ + S^HD_U is comparable to S^HD_P at the scale r_s = (/|α|³)^1/2 (Toschi et al. 1999).

The importance of the terms is certainly affected by the magnitude of α. In the limit α = 0, Equation (3) reduces to the KH equation given by Equation (1). In the opposite limit of large |α|, we can consider |α|r ≫ [δu(r)]_rms, where [δu(r)]_rms is the root-mean-squared value of δu averaged over all angles at fixed r. For this condition, |S^HD₃| can be small compared to either |S^HD_U| or |S^HD_P| or to both. If only one term proportional to α remains in this limit in Equation (3), then the corresponding fluctuation correlation is independent of r. This situation is likely in the energy-containing range where |S^HD_U| ≪ |S^HD_P| when r ≫ r_s. It might also occur within a large inertial range if |S^HD_U| ≫ |S^HD_P| and [δu(r)]_rms/|α| ≪ r ≪ r_s.

The S^HD₃ term can contribute significantly to the cascade even when the fluctuations are isotropic or nearly so. For smaller r, fluctuations tend to become more isotropic (e.g., Hinze 1975) due in part to the effects of pressure strains that cancel relative to energy scale transfer, but not for energizing fluctuation velocity components of the same scale. As a result, S^HD₃ is likely to be dominant at the small-scale end of the inertial range, and isotropy can be used to estimate this term.

The shear term S^HD_U is sensitive to the fluctuation anisotropy and so is more difficult to estimate. In Equation (5), the angular sensitivity comes from the projection of δU onto the normal to the sphere $\hat{{\bf r}}$. The factor sin θcos ϕ is from the projection along the flow direction $\hat{{\bf x}}$ onto $\hat{{\bf r}}$, and the sin θsin ϕ part is from the projection along the flow gradient direction $\hat{{\bf y}}$ onto $\hat{{\bf r}}$. With this projection, an isotropic fluctuation energy dependence, or one where the fluctuation energy is isotropic in the x–y plane, causes S^HD_U to vanish. This velocity shear in 3D is, however, associated with the nonlinear vortex stretching of eddies. It is known to yield a strain along two principal axes in the x–y plane located half-way between ${\hat{\bf x}}$ and ${\hat{\bf y}}$ (see Tennekes & Lumley 1972; Kundu 1990). Along one axis, the vorticity and energy of eddies are maximally amplified at the expense of these same quantities along the other axis. The principal axis on which eddies are amplified or not reverses with the sign of α. This contributes to fluctuation anisotropy and gives S^HD_U a negative value.

The strain associated with vortex stretching also causes the correlation between δu_x and δu_y to become nonzero. This is a condition indicative of anisotropic fluctuations. For α > 0, this correlation is negative, and for α < 0 this correlation is positive. As a result, S^HD_P will have a negative value. Lumley (1967) used dimensional analysis to determine the value of the 〈u_xu_y〉 omni-directional spectrum in the inertial range of a homogeneous shear flow turbulence. He showed that the Kolmogorov energy cascade would be associated with a k^−7/3 wavenumber spectrum for 〈u_xu_y〉. This indicates that the corresponding second-order structure function 〈δu_xδu_y〉 should vary as r^4/3.

To evaluate cases for a homogeneous velocity shear, we need to use the CGBP equation and not the KH equation. To illustrate this, we replace δv in Equation (1) with δu + δU as would occur when the velocity shear is not detrended. We then obtain

$$\begin{equation} S^{{\rm HD}}_3 + S^{{\rm HD}}_U + S^{{\rm HD}}_{{\rm extra}} = - \frac{4}{3} \epsilon r, \end{equation} \tag{ 7 }$$

where S^HD_extra is given by

$$\begin{eqnarray} S^{{\rm HD}}_{{\rm extra}} &=& \frac{1}{4 \pi } \int \langle (\delta U)^2 \delta {\bf u} + 2(\delta {\bf u} \cdot \delta {\bf U}) \\ &&\,\times\, (\delta {\bf u} + \delta {\bf U}) \rangle \cdot {\hat{\bf r}} \, d \Omega, \nonumber \end{eqnarray} \tag{ 8 }$$

and the cubic term involving δU vanishes upon integration. These extra terms do not equate with S^HD_P in Equation (3), and thus the value of obtained from Equation (7) would differ from that in Equation (3).

When solar wind velocity shear intervals of particular magnitude and sign are combined, the background velocity is not spatially invariant. This situation requires using the generalization of the CGBP equation to MHD. Moreover, we will need a method for representing the shear-induced anisotropy to evaluate a term analogous to S^HD_U, which will rely heavily on the above discussed vortex stretching process. We now turn to consider the above approach to MHD.

The incompressible (single fluid) MHD equations that are normally written in terms of the Elsässer variables Z^±(=V  ±  B/(4πρ)^1/2, where V is the velocity, B is the magnetic field, and ρ is a constant density) can also be re-expressed using the same assumptions for HD as relations between and third-order structure functions. In this paper, the magnetic Prandtl number is assumed to be unity so that viscosity and resistivity are made equal. The two-dimensional (2D) and 3D cases without shear were derived by Politano & Pouquet (1998a, 1998b), and we will refer to these relations as the PP equations. For this paper, the 3D PP equations in spherical average form are given by

$$\begin{equation} \frac{1}{4 \pi } \int \langle (\delta Z^{\pm })^2 \delta {\bf Z}^{\mp } \rangle \cdot {\hat{\bf r}} \, d \Omega = - \frac{4}{3} \epsilon ^{\pm } r, \end{equation} \tag{ 9 }$$

where δZ^± ≡ Z^±(x + r) − Z^±(x) and (δZ^±)² ≡ δZ^± · δZ^±. The integral represents the flux of the energy from scale to scale through the sphere. Equation (9) is quite general and valid for both anisotropic turbulence and for flow with zero average shear.

The total rate ^T is given by ^T = (⁺ + ⁻)/2 and is also the mean dissipation rate. The (twice) total fluctuation energy per unit mass is given by the sum ((δZ⁺)² + (δZ⁻)²)/2 which is why the sum of rates leads to the rate for total energy. The difference corresponds to the cross-helicity σ_c which is normalized by total energy and is given by ((δZ⁺)² − (δZ⁻)²)/((δZ⁺)² + (δZ⁻)²).

Wan et al. (2009, 2010a, 2010b) examined the resulting relations for MHD when the velocity is separated into fluctuations and background linear shear flow in both two and three dimensions. Wan et al. (2010b) particularly focus on the 2D case where the velocity and magnetic field fluctuations are confined to a plane, and where there is no background magnetic field. In this 2D case, the fluid vorticity is nonlinearly strained via electrodynamic vortex amplification associated with a differential Lorentz force (e.g., Spangler 1999). They undertook incompressible 2D MHD numerical simulations with periodic boundaries of driven turbulence amidst a velocity shear. The prescribed time constant background velocity varied up and down spanwise, and sharply between two regions of nearly linear shear. In the linear shear region, the contributions from fluctuations and shear terms were evaluated and found to be in good agreement with theory. They showed that the power spectrum of the fluctuations exhibits anisotropic behaviors at lower wavenumbers. This is induced by the velocity shear which, judging from their Figure 10, is oriented approximately with principal axes that are streamwise and spanwise. Power is greater in the spanwise direction. Moreover, they considered the total velocity without any detrending for shear, and the 2D PP equations were found to give good estimate of the cascade rate when the entire spatial domain was considered so that the net shear vanished.

In the solar wind, the effects of the velocity shear influence fluctuations with vectors and variations in all three spatial directions. Moreover, there is a background interplanetary magnetic field B₀ that must also be considered, and here we assume that B₀ is a constant. We set

$$\begin{equation} {\bf Z}^{\pm } = {\bf z}^{\pm } + \alpha y {\hat{\bf x}} \pm {\bf B}_0/(4 \pi \rho)^{1/2}, \end{equation} \tag{ 10 }$$

where z^± = u ± b/(4πρ)^1/2 is the fluctuating part, b is the fluctuating magnetic field, and the mean velocity is as in Section 2.1. Wan et al. (2009) derive the shear expressions from the 3D MHD equations that we refer to here as the Wan–Servidio–Oughton–Matthaeus (WSOM) equations. The WSOM equations are obtained using the fluctuating and mean parts and integrating over a spherical volume. We recast the equations in a form analogous to Equation (3) as

$$\begin{equation} S^{{\rm MHD},\pm }_3 + S^{{\rm MHD},\pm }_U + S^{{\rm MHD},\pm }_P = - \frac{4}{3} \epsilon ^{\pm } r, \end{equation} \tag{ 11 }$$

where

$$\begin{equation} S^{{\rm MHD},\pm }_3 = \frac{1}{4 \pi } \int \langle (\delta z^{\pm })^2 \delta {\bf z}^{\mp } \rangle \cdot {\hat{\bf r}} \, d \Omega, \end{equation} \tag{ 12 }$$

$$\begin{eqnarray} S^{{\rm MHD},\pm }_U &=& \frac{ \alpha r}{4 \pi } \int \sin ^2 \theta \sin \phi \cos \phi \\ &&\,\times\, \langle (\delta z^{\pm })^2 \rangle d \Omega, \nonumber \end{eqnarray} \tag{ 13 }$$

and

$$\begin{equation} S^{{\rm MHD},\pm }_P = \frac{2 \alpha }{4 \pi r^2} \int\!\! \int \langle \delta z^{\pm }_x \delta z^{\mp }_y \rangle r^2 d r d \Omega. \end{equation} \tag{ 14 }$$

Again we have neglected dissipative terms since the application will be for lags longer than the dissipation scale. The nature of the terms and limits with respect to α are in accord with the above discussion for HD. Summing over the ± terms in Equations (11)–(14) and then dividing by 2 gives the corresponding dissipation rate for total energy. The WSOM equations are valid for any form of anisotropic turbulence which arises in association with the velocity shear.

When σ_c = 0, all corresponding ± terms in Equations (12)–(14) are equal to each other (S^{MHD, +}₃ = S^{MHD, −}₃, etc.) and contribute to the forward energy cascade so that each term is nonpositive. When σ_c = ±1, nonlinearity in the incompressible MHD equations vanishes and so should all terms. For intermediate σ_c, there is no known constraint on the sign of the terms. The expectation is that for σ_c not near ±1, the cascade will be forward and all terms in Equations (12)–(14) will be nonpositive.

Equation (11) provides what is needed to examine simulation results from incompressible 3D MHD quasi-homogeneous shear flow. Yet, the solar wind is a flow where the ratio of the plasma to magnetic pressure is typically below unity, and thus is far from an incompressible medium. Here, B₀ and its attendant anisotropy with respect to direction become vitally important. Turbulent velocity and magnetic field fluctuations in the presence of B₀ nonlinearly interact most strongly across B₀ (e.g., Shebalin et al. 1983; Matthaeus et al. 1996a). With highly oblique wave vectors, the fluctuations are found to approach a nearly incompressible (e.g., Zank & Matthaeus 1993) or weakly compressive (e.g., Bhattacharjee et al. 1998) state. For this case, the incompressible MHD equations contain the dominant nonlinear effects needed to describe the turbulent fluctuations. Observations show that interplanetary fluctuations are relatively noncompressive in the inertial range. Moreover, lags across B₀ are readily available at 1 AU. MacBride et al. (2005, 2008) showed from Equation (9) with B₀-directed anisotropy models that the cross-field cascade in the solar wind is indeed stronger than along B₀. The amount of anisotropy is modest enough that an isotropic approximation for Equation (9) is still useful in cascade rate determination (see also Stawarz et al. 2010). We will henceforth ignore anisotropy arising from B₀ in order to focus on velocity shear.

In order to proceed further with solar wind data analyses, we need to consider approximations for the individual terms in Equation (11) since single-spacecraft analyses lack direct measurement of fluctuation anisotropy with respect to the velocity shear. Again, because we are going to combine intervals with a particular varying background velocity, we must use the WSOM equations to obtain and not the PP equations, for the same reasons discussed earlier concerning the HD case. In Section 3, we show that the PP equations, using only the observed lag measurements and assuming isotropy to evaluate the integral in Equation (9), yield incorrect results for such cases.

In Equation (11), the terms S^{MHD, ±}_U require anisotropy, and thus we treat first the estimation of these terms. The S^{MHD, ±}_U terms from Equation (13) involve an integral over the fluctuation energy that vanishes for isotropic fluctuations in the x–y plane. With the assumed form of the velocity shear and with fluctuations varying in all three spatial dimensions, the most straightforward expectation for fluctuation energy anisotropy induced by the shear is the alignment of this anisotropy with the mean strain principal axes $(\hat{{\bf y}} \pm \hat{{\bf x}})/ 2^{1/2}$. This is due to the aforementioned mechanism of vortex stretching. We can evaluate the integral in Equation (13) by assuming that the fluctuation energy [δz^±(r, ϕ, θ)]² varies with θ and ϕ in the simplest possible manner: θ and ϕ vary according to an ellipsoidal model with coefficients that are independent of r so that the anisotropy is constant with r. This neglects the expected attenuation of anisotropy for smaller r in the inertial range. In the solar wind, however, the inertial range is moderately short, and moreover the estimates for will be based on averages taken over the observed linear range of r. Thereby, the model coefficients can approximate an average anisotropy over the extent of this range. In this approach, the range of coefficients is limited to values where mean dissipation and expected plasma heating rates are in acceptable agreement.

We replace [δz^±(r, ϕ, θ)]² with f(ϕ, θ)[δz^±(r)]² where

$$\begin{eqnarray} f(\phi,\theta) &=& \frac{3}{1/a^2+ 1/b^2 + 1/c^2} \hspace{36.135pt}\nonumber \\ &&\,\times \left[ \frac{\sin ^2 \theta \,(1/2 + \cos \phi \sin \phi)}{a^2} \right. \nonumber \\ &&\,\left. + \frac{\sin ^2 \theta \,(1/2 - \cos \phi \sin \phi)}{b^2} + \frac{\cos ^2 \theta }{c^2} \right]\quad \end{eqnarray} \tag{ 15 }$$

is the ellipsoid of anisotropy with principal axes radii a and b in the x–y plane and the out of plane axis with radius c. (Note that a larger radius along its axis corresponds to smaller amplitude along that axis for fixed radius r.) The value of f is normalized so that 4π results when f is integrated over all solid angle. The mean values 〈[δz^±(r)]²〉 over all angles can be related to the values along the streamwise lag 〈[δz^±(r, 0, 90°)]²〉. The values along the streamwise lag are hereafter denoted by 〈[δz^±(r)]²_s〉, that in application will be replaced by measured values from solar wind data. Dividing through with the value of f in the x direction, we obtain

$$\begin{equation} \langle [\delta z^{\pm }(r)]^2 \rangle = A_s \big\langle [\delta z^{\pm }(r)]^2_{s} \big\rangle, \end{equation} \tag{ 16 }$$

where the factor A_s ≡ 1/f(0, 90°) is given by

$$\begin{eqnarray} &&A_s = \left[ \frac{3}{1/a^2 + 1/b^2 + 1/c^2} \right]^{-1}\,{\times}\, \left[ \frac{1}{2} \left(\frac{1}{a^2} + \frac{1}{b^2} \right) \right]^{-1}, \nonumber\\ \end{eqnarray} \tag{ 17 }$$

or more compactly by

$$\begin{equation} A_s = \frac{2}{3} \frac{c^2(a^2 + b^2) + a^2 b^2}{c^2(a^2 + b^2)}. \end{equation} \tag{ 18 }$$

With these assumptions, the resulting form for Equation (13) is

$$\begin{equation} S^{{\rm MHD},\pm }_U = -\frac{2 | \alpha | r }{15} \, \frac{| b^2 - a^2 |}{a^2 + b^2} \big\langle [ \delta z^{\pm } (r) ]^2_{s} \big\rangle, \end{equation} \tag{ 19 }$$

which is independent of c by virtue of the cancellation of factors in Equations (15) and (17). We assume that the forward energy cascade is enhanced by both S^{MHD, +}_U and S^{MHD, −}_U when they are nonzero because the data used in Section 3 combine intervals without regard for σ_c. The average |σ_c| is not so large as to significantly alter the inertial range cascade (Smith et al. 2009; Stawarz et al. 2009). With this assumption, we take absolute values in Equation (19) so that S^{MHD, ±}_U is always nonpositive. This then means that we can also take b² ⩾ a² without loss of generality and not be concerned with the sign of α when evaluating Equation (19). Note that isotropy in the x–y plane a² = b² gives S^{MHD, ±}_U = 0. Moreover, the choice of principal axes in the x–y plane in Equation (15) maximizes |S^{MHD, ±}_U| for fixed x–y anisotropy b²/a², whereas if the axes were along $\hat{{\bf x}}$ and $\hat{{\bf y}}$, S^{MHD, ±}_U are zero. Alternatively, for fixed values of S^{MHD, ±}_U, b²/a² is minimized because a different orientation for the principal axes in the x–y plane requires larger b²/a² to obtain the same values of S^{MHD, ±}_U.

The model Equation (19) will be essential in the analyses below. The values of a and b are to be taken as free parameters to vary anisotropy in the x–y plane and to examine the resulting values of ^± to compare with independently determined dissipation and heating rates. We neglect the effect of σ_c in the present analysis and use the same parameters a, b, and c for the Elsässer amplitudes |z⁺| and |z⁻|. Taking a case from Section 3, one sets a = 1, and b = 2, then S^{MHD, ±}_U = −0.08|α|r〈(δz^±)²〉 with a proportionality constant of 0.08. The actual range of the proportionality constant is quite limited. The largest magnitude is 2/15 and occurs when b²/a² → ∞. As a result, the magnitudes of S^{MHD, ±}_U are sensitive to small departures from isotropy but insensitive to large departures.

Remaining terms on the left-hand side of Equation (11) give nonzero values even for isotropic fluctuations, but the underlying fluctuations will be anisotropic for finite α. To lowest order, we expect the anisotropy of these terms to align with the mean strain principal axes. We approximate the anisotropy as one equivalent to the fluctuation energy and separate integrands as quantities whose angular dependence is given by Equation (15) and whose amplitude dependence is only on r. The integration over all angles of f(ϕ, θ) only results in the cancellation of the 4π factor in Equations (12) and (14). The mean value of the amplitude dependence ratioed to that measured in the streamwise direction is proportional to A_s. This introduces a dependence upon the free parameter c, as well as a and b. From Equation (12), the terms dependent only on fluctuations are estimated by

$$\begin{equation} S^{{\rm MHD},\pm }_3 = A^{3/2}_s \, \langle [(\delta z^{\pm })^2 \delta z^{\mp }_x ]_{s} \rangle, \end{equation} \tag{ 20 }$$

which retain the signs of the third-order structure functions along the streamwise lag. The shear production terms S^{MHD, ±}_P involve an evaluation over the volume. With this approach, we use the streamwise values of 〈δz^±_xδz^∓_y〉 so that

$$\begin{equation} S^{{\rm MHD},\pm }_P = \frac{2 \alpha A_s}{ r^2} \int ^r_0 \langle [ \delta z^{\pm }_x \delta z^{\mp }_y ]_{s} \rangle r^2 d r. \end{equation} \tag{ 21 }$$

With increasing r, the values of S^{MHD, ±}_P accumulate and generally increase in magnitude. Note that absolute values are not introduced in Equation (21) because the signs and magnitudes of the integrands are obtained with measurements. Their relation with the sign of α will give a consistency check on the assumption that velocity shear driving of turbulence is preeminent.

The dependence of A_s on c can vary the magnitude of the estimates for S^{MHD, ±}₃ and S^{MHD, ±}_P. For the typical case in Section 3, a = c = 1 and b = 2, we find from Equation (18) that A_s = 1.2 which differs little from complete isotropy where a = b = c = 1 and A_s = 1. In the limit that c²/a² → ∞, which corresponds to fluctuation wave vectors confined increasingly to the x–y plane, A_s = 2/3. In the opposite limit where c²/a² → 0 and b²/a² is finite, A_s → ∞, so that large anisotropy in the z direction can greatly affect the values of S^{MHD, ±}₃ and S^{MHD, ±}_P.

The final expressions used in the present analyses change some notation and this notation is consistent with that used in MacBride et al. (2005, 2008) and Stawarz et al. (2009, 2010). The signs of Z^± and z^± are redefined with respect to the background magnetic field so as to correspond to propagating Alfvén waves traveling inward to (denoted by superscript "I") or outward from (denoted by "O") the Sun. (Note that propagation here can be a formality since in directions where k · B₀ = 0, Alfvén waves do not propagate and yet eddies with mixed Elsässer signs do interact nonlinearly.) The background velocity is mainly in the heliocentric radial direction $\hat{{\bf R}}$ away from the Sun. In Cartesian coordinates, this corresponds to the $\hat{{\bf x}}$ direction used above. We take $\hat{{\bf y}}$ to be in the direction of the Earth's revolution about the Sun which is the $\hat{{\bf T}}$ direction of the Radial–Tangential–Normal (RTN) coordinate system. The RTN coordinate system is used in the present analyses of ACE spacecraft measurements. The measured lag is taken from V_SWτ, where V_SW is the solar wind speed and τ is the time lag of increments. Since the wind carries the fluctuations past the spacecraft, the increasing time lag corresponds to positions going back to the Sun, so that the spherical radius lag r is replaced by the Cartesian lag −V_SWτ. We hereafter also drop the subscript denoting streamwise measurements, and the superscript MHD since that is the only case we will consider. Equation (11) with all approximations included is rewritten as

$$\begin{equation} D^{I/O}_{3} + D^{I/O}_{U} + D^{I/O}_P = \frac{4}{3} \epsilon ^{I/O}_{{\rm SH}} V_{{\rm SW}} \tau, \end{equation} \tag{ 22 }$$

where the functions of V_SWτ are

$$\begin{equation} D^{I/O}_{3} = A^{3/2}_s \, \big\langle (\delta z^{I/O})^2 \delta z^{O/I}_R \big\rangle, \end{equation} \tag{ 23 }$$

$$\begin{equation} D^{I/O}_{U} = \frac{2 | \alpha | \, V_{{\rm SW}} \tau }{15} \, \frac{| b^2 - a^2 |}{a^2 + b^2} \langle (\delta z^{I/O})^2 \rangle, \end{equation} \tag{ 24 }$$

and

$$\begin{equation} D^{I/O}_P = - \frac{2 \alpha A_s V_{{\rm SW}}}{ \tau ^2 } \int ^{\tau }_0 \big\langle \delta z^{I/O}_R \delta z^{O/I}_T \big\rangle \tau ^2 d \tau. \end{equation} \tag{ 25 }$$

We also define the sum of structure functions in Equation (22) by

$$\begin{equation} D^{I/O}_{3,{\rm SH}} \equiv D^{I/O}_{3} + D^{I/O}_{U} + D^{I/O}_{U}, \end{equation} \tag{ 26 }$$

and total structure functions by

$$\begin{equation} D^{T}_{3} \equiv \big(D^{O}_{3} + D^{I}_{3}\big)/2, \end{equation} \tag{ 27 }$$

$$\begin{equation} D^{T}_{U} \equiv \big(D^{O}_{U} + D^{I}_{U}\big)/2, \end{equation} \tag{ 28 }$$

$$\begin{equation} D^{T}_{P} \equiv \big(D^{O}_{P} + D^{I}_{P}\big)/2, \end{equation} \tag{ 29 }$$

and

$$\begin{equation} D^{T}_{3,{\rm SH}} \equiv \big(D^{O}_{3,{\rm SH}} + D^{I}_{3,{\rm SH}}\big)/2. \end{equation} \tag{ 30 }$$

The total and mean dissipation rate is given by

$$\begin{equation} \epsilon ^{T}_{{\rm SH}} =(\epsilon ^{O}_{{\rm SH}} + \epsilon ^{I}_{{\rm SH}})/2. \end{equation} \tag{ 31 }$$

The subscript "SH" denotes that the expressions correspond to the WSOM theory for the case of homogeneous shear, and use detrended velocities to evaluate fluctuations and their moments. Note that the sign of the structure functions as most naturally sampled with the spacecraft is positive when the energy cascade is forward.

For comparison purposes, the case with total velocity given by Equation (9) without detrending and ignoring shear uses

$$\begin{equation} D^{I/O}_{3,{\rm NOSH}} = \big\langle (\delta Z^{I/O})^2 \delta Z^{O/I}_R \big\rangle, \end{equation} \tag{ 32 }$$

assuming isotropy. (Note that the D_U and D_P terms are not relevant here since total velocity is used.) The total rate ignoring shear is taken from

$$\begin{equation} D^{I/O}_{3,{\rm NOSH}} = \frac{4}{3} \epsilon ^{I/O}_{{\rm NOSH}} V_{{\rm SW}} \tau, \end{equation} \tag{ 33 }$$

$$\begin{equation} D^{T}_{3,{\rm NOSH}} \equiv \big(D^{O}_{3,{\rm NOSH}} + D^{I}_{3,{\rm NOSH}}\big)/2, \end{equation} \tag{ 34 }$$

and

$$\begin{equation} \epsilon ^{T}_{{\rm NOSH}} =\big(\epsilon ^{O}_{{\rm NOSH}} + \epsilon ^{I}_{{\rm NOSH}}\big)/2. \end{equation} \tag{ 35 }$$

Here the subscript "NOSH" denotes no shear.

We now turn to ACE observations.

In order to apply Equations (22)–(31), intervals are sorted into bins according to ΔV_SW in a specified range. The ΔV_SW bins that we use range from −100 to +100 km s⁻¹ with equal widths of 25 km s⁻¹. Note that negative values of ΔV_SW correspond to regions where faster wind is followed by slower wind so that these are regions of rarefaction. Positive values of ΔV_SW correspond to regions where a slower wind is being overtaken by a faster wind, and thus these are regions of compression.

The present analysis bins compression and rarefaction intervals for equivalent levels of shear without regard for the duration of the event. With equivalent levels of shear, the theory has dissipation rates that are the same in the rarefaction and compression intervals. Important differences in these intervals when analyzed, however, will be shown below. This might arise due to assumptions in our analysis such as the neglect of latitudinal gradients. Our theory, in general, is symmetric regardless of whether the shear is positive or negative. Results are shown without additional comment on these assumptions. Section 4 then discusses how the assumptions can impact the results.

Averages over the 12 hr intervals having ΔV_SW within the appropriate bins are performed to find the average value of D^T_{3, SH} as a function of spatial lag. Then the uncertainty-weighted average of 3D^T_{3, SH}/(4 r) is calculated for each spatial lag r which is less than the corresponding time lag τ of 8000 s based on a reference speed of 400 km s⁻¹. The long-lag limit is further constrained so that D^T_{3, SH} varies almost linearly from the shortest lag to the long-lag limit. The linear scaling of D^T_{3, SH} is required by the theory. This constraint is further shown below to be important when ΔV_SW > 0. This calculation gives the average ^T_SH for each bin. A similar approach is used to obtain ^T_NOSH.

To compute uncertainties we use the same method described in Stawarz et al. (2009, 2010). Take X to be some quantity, e.g., D^I/O₃, at a particular spatial lag, which is to be averaged over the 12 hr intervals analyzed. Each value of X computed from an individual 12 hr interval is taken to be a statistically independent estimate with no intrinsic uncertainty because 12 hr is greater than the correlation length and the propagation of measurement error is likely the same for all samples. As such, estimates from the 12 hr intervals will follow a Gaussian distribution, and Gaussian statistics can be used to compute the mean standard deviation and error of the mean for the ensemble-averaged quantities. From there this uncertainty can be propagated through further calculations using standard methods.

The average value of _heat is also obtained from all 12 hr intervals per ΔV_SW bin. The expression for _heat in Equation (37) is determined by an analysis based on the spherically symmetric proton equation of state. Hence, the heating rate is the average over the spherical polar angles and so does not correspond to rates in separate rarefaction and compression regions. Based on the work of Burlaga & Ogilvie (1973) who found that the net heating in compressions over rarefaction is ∼15% on average, we would expect that the ΔV_SW > 0 bins have somewhat more heating than predicted by _heat, while ΔV_SW < 0 bins would have less. The amount of relative deviation is, however, likely to be only ≲ 5%. Thus, _heat is deemed to be a sufficiently good guide to the required dissipation rate from the turbulence in the present analyses. Uncertainty for the average value of _heat is determined as above.

Figure 1 plots the value of ^T_SH, ^T_NOSH, and _heat as a function of ΔV_SW (see also Table 1). Here the value of ^T_SH is obtained by assuming an anisotropy of 2:1, corresponding to the values a = c = 1 and b = 2 in Equations (18) and (24). Errors of the mean are plotted but are typically smaller than the symbols used, and horizontal bars indicate the range of ΔV_SW analyzed for each data point. Except for ^T_NOSH, the plotted rates increase with |ΔV_SW|, as would be expected of a turbulent energy cascade driven by velocity shear. The values of ^T_SH and _heat approximately follow each other, and ^T_SH is always above zero, consistent with a forward cascade dissipating and heating the plasma. The dissipation rate ^T_NOSH ignoring shear appears to be dominated by the sign of the shear and has no correspondence with _heat. The absolute value of ^T_NOSH is smaller for the bin with ΔV_SW < 0 than the one for the same but positive signed ΔV_SW bin.

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Table 1. Rates for Figure 1

ΔVSW Range	Number of	TSH	TNOSH	Theat
(km s−1)	Samples	(× 103 J kg−1 s−1)
−100 < ΔVSW < −75	159	3.94 ± 0.11	−4.59 ± 0.09	2.39 ± 0.14
−75 < ΔVSW < −50	404	3.20 ± 0.05	−2.75 ± 0.04	1.94 ± 0.07
−50 < ΔVSW < −25	915	1.56 ± 0.03	−1.89 ± 0.03	1.63 ± 0.05
−25 < ΔVSW < 0	1334	0.62 ± 0.06	−0.85 ± 0.02	1.54 ± 0.04
0 < ΔVSW < +25	782	0.32 ± 0.11	1.41 ± 0.04	1.72 ± 0.05
+25 < ΔVSW < +50	410	1.51 ± 0.29	5.18 ± 0.09	2.36 ± 0.09
+50 < ΔVSW < +75	204	1.18 ± 0.34	9.66 ± 0.12	2.70 ± 0.13
+75 < ΔVSW < +100	136	6.59 ± 0.38	16.94 ± 0.26	3.02 ± 0.17

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The values of ^T_SH for the bins closest to zero, i.e., −25 to 0 km s⁻¹ and 0 to +25 km s⁻¹, in Figure 1 are anomalously small compared to _heat and are not a good match. The poor match probably occurs because cases of weak shear are less likely to be characterized well by a linear trend of velocity shear. In these cases, the intervals tend to be located between intervals of positive and negative shear and may be at local maximum or minimum V_SW. Thus, we will focus on the outlying bins where the shear is more significant and better characterized.

The farther outlying bins on the left and negative side of ΔV_SW = −25 km s⁻¹ have ^T_SH that matches or exceeds _heat. These can then provide sufficient energization for the plasma including electron and alpha heating, as well as proton heating. Taking electron heating to be about 50% of proton heating provides a good upper bound to the expected total plasma heating (Vasquez et al. 2007; Stawarz et al. 2010), and the values obtained for ^T_SH can satisfy these bounds. On the other hand, farther outlying bins on the right and positive side of ΔV_SW = +25 km s⁻¹ have a more irregular up and down trend with respect to _heat and are less in agreement with _heat than is the case on the opposite side. This is not a trend to be expected for an ensemble of true homogeneous shear-driven turbulence, where the sign of α or ΔV_SW does not affect the mean dissipation rate. Section 4 discusses how compressional regions in the solar wind can deviate from the theory and contribute to the observed trends.

The analysis based on the PP equations giving ^T_NOSH clearly does not provide the mean dissipation rate for cases with persistent shear. This was explained in Section 2, and these results are in agreement with conclusions reached there. The signal from the velocity shear itself contaminates the measurements to such an extent that the sign of the calculated ^T_NOSH is the same as that of ΔV_SW.

We now consider the relative contribution of fluctuation-only and shear-dependent terms to the mean dissipation rates. The theory requires D^T_{3, SH} to vary linearly. The individual terms that sum to D^T_{3, SH} can vary differently from a linear scaling. Figure 2 plots the values of D^T₃, D^T_U, D^T_P, and the sum of these three terms D^T_{3, SH} as a function of lag for the ΔV_SW bin from −75 to −50 km s⁻¹. The value of D^T_{3, NOSH} as a function of lag within this bin is also plotted for completeness sake, but we do not discuss this quantity further. Figure 3 plots the same but for the ΔV_SW bin from +50 to +75 km s⁻¹. The trends in each figure are typical of the bins with the same sign of ΔV_SW.

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In Figure 2, all quantities vary approximately linearly with lag over the plotted range. By far the largest contribution to D^T_{3, SH} comes from the detrended third-moment term D^T₃. The values of D^T_U and D^T_P are smaller, with D^T_U larger than D^T_P. Though small, D^T_P has a value that is more than one standard deviation from zero. The relative error for D^T_P is less than 1%, and the error bars for D^T_P in Figure 2 are smaller than the plotted symbol size. Its magnitude and sign can be compared to the predictions of shear-driven turbulence. Here, D^T_P has a positive sign as expected from the generation of energy from the shear. Its relatively small value is consistent with the plotted range of lags being in the inertial range. Since D^T_P according to Equations (25) and (30) depends on 〈δz^I/O_Rδz_T^O/I(τ)〉 which we measure to be nonzero at all lag times τ, fluctuations are anisotropic, and this too is consistent with shear-driven turbulence.

In Figure 3 where ΔV_SW > 0, we find complicated behavior which is beyond our expectations for homogeneous shear-driven turbulence. In Section 4, we will discuss how compressive effects can contribution to this behavior. The term D^T_{3, SH} varies linearly with lag starting from near zero out to about 2000 s, which is in accordance with theory, but then decreases and follows a nonmonotonic course. The linear range is only a quarter of the plotted range and smaller than the expected inertial range. The computed value of ^T_SH is taken only from this range. The term D^T₃ changes from positive values to negative ones beyond 2000 s. Both D^T_{3, SH} and D^T₃ have values with much larger uncertainty than corresponding points in Figure 2. The term D^T_U varies nearly monotonically over the whole range which corresponds to increasing fluctuation energy with increasing lag. It even exceeds D^T₃ beyond lags of 1000 s. The term D^T_P remains close to zero in the plotted range. The irregular behavior of D^T_{3, SH} with lag which is found for all bins with ΔV_SW > 0 and its relatively large uncertainty undoubtedly contributes to the scatter about _heat in ^T_SH.

In the theory we expect symmetrical results for rarefaction and compression intervals. In Figure 2, we find that the plotted quantities for rarefactions are well behaved with consistent behavior throughout the entire range and measurement uncertainties are relatively small. By contrast, Figure 3 shows for compressions that the plotted quantities do not follow a consistent trend and do not conform to expectations of linear scaling. Therefore, we fail to find the expected symmetry and conclude based on the observed nonlinear scaling and larger uncertainties that the analysis provides a poorer assessment of compression regions.

Because the terms D^I/O_P provide the most direct measurement of the effects of shear-driven turbulence, we examine separately the underlying fluctuation structure functions. These are the second-order structure functions given by 〈δz^I/O_Rδz^O/I_T〉. In the theory the sign of 〈δz^I/O_Rδz^O/I_T〉 matches that needed to produce energy for the cascade. In addition, the absolute value of 〈δz^I/O_Rδz^O/I_T〉 increases with increasing scale since production is strongest at the large, energy-containing scales. Figure 4 plots 〈δz^I_Rδz^O_T〉 and 〈δz^O_Rδz^I_T〉 as a function of lag for the ΔV_SW bin from −100 to −75 km s⁻¹. Figure 5 plots the same but for the ΔV_SW bin from +75 to +100 km s⁻¹.

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In Figure 4, both 〈δz^I_Rδz^O_T〉 and 〈δz^O_Rδz^I_T〉 have a negative sign consistent with the production of shear-driven turbulence and the magnitude increases with increasing scale, as is expected for driving by shear. Other bins with ΔV_SW < −25 km s⁻¹ show the same behavior.

On the other hand, for compressive flow, Figure 5 shows 〈δz^I_Rδz^O_T〉 and 〈δz^O_Rδz^I_T〉 that are increasing for lags shorter than 2000 s but then are decreasing for larger lags. The sign is positive for shorter lags, consistent with shear-driven turbulence, but can become negative at a large enough lag. This behavior is not found in all bins with ΔV_SW > +25 km s⁻¹ wherein 〈δz^I_Rδz^O_T〉 and 〈δz^O_Rδz^I_T〉 are sometimes found to be negative for all lags. Thereby, results for positive ΔV_SW bins continue to show inconsistencies with shear-driven turbulent predictions.

In Figures 4 and 5, power-law fits have been made to quantities as a function of lag. Fit parameters and lag ranges are given in Table 2. The power-law index is roughly 3/4. This is smaller than the expected value of 4/3 based on Lumley's (1967) dimensional analysis of Kolmogorov-like cascade with linear strain (see Section 2.1). Departures from the Kolmogorov prediction at 1 AU are also found by Tessein et al. (2009) for the second-order velocity fluctuation structure function, whereas the corresponding structure function for the magnetic field fluctuation does satisfy the prediction. Thus, the results for 〈δz^I/O_Rδz^O/I_T〉 may be another manifestation of the velocity fluctuation departure. Potentially, the lack of correspondence with a Kolmogorov cascade indicates that the turbulent state at 1 AU has not yet reached an asymptotic statistically steady state.

The anisotropy of the fluctuation energy is set by the free parameters b and c where a = 1 without loss of generality, and we now consider how this impacts the mean dissipation rate. Here, we limit the discussion to the two bins with ΔV_SW ⩽ −50 km s⁻¹ which are the ones that agree best with homogeneous shear-driven turbulence. Moreover, we only consider varying b, keeping c = 1. We do this because b gives the anisotropy in the plane of the background velocity and gradient, which is the important factor to consider regarding linear shear.

Figure 6 plots ^T_SH as a function of the free parameter b for the ΔV_SW bin from −100 to −75 km s⁻¹. A value of b = 1 corresponds to isotropy, and a value of b = 2 corresponds to the 2:1 anisotropy used in the previous plots. The lower dashed line corresponds to _heat and the upper dashed line corresponds to 50% above _heat, which gives an approximate bound for the additional energy required to heat electrons. For this bin, values of b between 1.06 and 1.69 yield mean dissipation rates in the expected range of plasma heating. For the next adjacent ΔV_SW bin which ranges from −75 to −50 km s⁻¹ (not shown), b = 1 has ^T_SH slightly greater than _heat, while 50% excess is reached for b = 1.69. Thereby, for these bins modest amounts of anisotropy are consistent with plasma heating.

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In the present analysis based on the WSOM equations, we have modeled the anisotropy induced by the velocity shear on the fluctuations. Anisotropy associated with the direction of B₀ has not been considered here.

Previous studies (MacBride et al. 2005, 2008; Stawarz et al. 2009, 2010) employed data that were selected without regard to shear and therefore averaged over approximately equal amounts of increasing and decreasing shear regions, as is required for this approach. These studies found that the PP equations give mean dissipation rates that could account for expected proton heating rates. In order to determine how the results using the WSOM equations compare with these previous studies, we perform the Stawarz et al. (2010) analysis for bins of V_SWT_P using Equations (32)–(35) which correspond to the isotropic fluctuation formalism in their paper. Stawarz et al. (2010) also obtained rates parallel and perpendicular to B₀ whose sums are in better agreement with the expected total plasma heating than the rates based on the isotropic case. Here, we only compare with the isotropic case since the shear formalism does not yet include additional anisotropy effects associated with B₀.

In accordance with Stawarz et al. (2010), 12 hr intervals are placed into seven overlapping V_SWT_P bins based on the average product of the solar wind velocity and the proton temperature within the interval. For each of the V_SWT_P bins, we again compute ^T_SH with a 2:1 anisotropy, ^T_NOSH, and _heat using the methods described above.

Figure 7 plots these three quantities against the average value of V_SWT_P within each bin, and values are given in Table 3. Although the shear is considered in calculating ^T_SH for each 12 hr interval, values are only averaged in Figure 7 by V_SWT_P bins. Compressions and rarefactions occur in all bins. Note that due to a small error in the selection of 12 hr intervals in the Stawarz et al. (2010) analysis and new selection criteria in applying the shear formalism, the values plotted for ^T_NOSH are slightly different from those shown in Stawarz et al. (2010). The same conclusions that were drawn in Stawarz et al. (2010) can, however, be inferred from the slightly revised data. All quantities in Figure 7 tend toward larger rates for increasing V_SWT_P. The value of ^T_SH trends from above _heat in the two lowest V_SWT_P bins to below _heat in the highest bins. The value of ^T_NOSH deviates from _heat in the opposite sense.

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Table 3. Rates for Figure 7

VSWTP Range	Number of	TSH	TNOSH	Theat
(× 107 km s−1 K)	Samples	(× 103 J kg−1 s−1)
0.2 < VSWTP < 2.0	978	0.78 ± 0.03	0.20 ± 0.02	0.49 ± 0.00
1.1 < VSWTP < 3.0	1536	1.13 ± 0.04	0.60 ± 0.03	0.73 ± 0.01
2.0 < VSWTP < 4.0	1446	1.12 ± 0.05	0.86 ± 0.04	1.05 ± 0.01
3.0 < VSWTP < 8.0	1851	1.78 ± 0.04	1.89 ± 0.03	1.79 ± 0.01
4.0 < VSWTP < 12.0	1814	2.44 ± 0.04	3.13 ± 0.04	2.52 ± 0.02
8.0 < VSWTP < 26.0	991	3.06 ± 0.07	5.71 ± 0.07	4.39 ± 0.04
12.0 < VSWTP < 40.0	413	3.65 ± 0.15	7.63 ± 0.15	5.88 ± 0.07

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Stawarz et al. (2010) noted that the two lowest bins of V_SWT_P contain intervals mainly from the lower temperature extremes of the solar wind. Following Vasquez et al. (2007), they suggested that _heat may be overestimated in these lower bins because the analyses for _heat only considered V_SW selection to obtain the radial proton temperature gradient. Freeman & Lopez (1985) found that cold intervals at 1 AU are more consistent with nearly adiabatic behavior. This differs significantly from the average behavior based on V_SW. Thereby, the deviations in the lower bins for ^T_NOSH were not considered detrimental to the analysis of ^T_NOSH, at least with the current knowledge of heating rates.

The two highest bins of V_SWT_P have ^T_NOSH greater than _heat. Stawarz et al. (2010) found that these intervals mostly come from fast winds that always have high T_P. Electron heating has been noted by Pilipp et al. (1990) for fast winds and there is no discernible heating in slower wind intervals. The trend from ^T_NOSH near _heat for the lower bins of V_SWT_P to ^T_NOSH greater than _heat at the highest bins is then consistent with the expectations for total plasma heating.

Two concerns are then evident about the values of ^T_SH as a function of V_SWT_P. For the two lowest V_SWT_P bins, ^T_SH certainly provides sufficient energization for proton heating. Are these values too high even for total plasma heating? Additionally, in the two highest bins, ^T_SH is too small to explain even proton heating.

Figure 8 plots D^T₃, D^T_U, D^T_P, and D^T_{3, SH} as a function of lag for the middle V_SWT_P bin. As expected, D^T_{3, SH} is approximately linear with respect to lag. The term D^T₃ provides the dominant contribution to the total D^T_{3, SH}. These results are typical of all V_SWT_P bins except for the two largest bins where D^T_U begins to dominate the sum at long lags. Recall that D^T_U dominance at longer lags is a feature of ΔV_SW > 0 intervals (see Figure 3). Combine this with the fact that ^T_SH is not well determined for ΔV_SW > 0 and that intervals with ΔV_SW of both signs are mixed together in V_SWT_P bins, and we surmise that intervals with ΔV_SW > 0 are a likely contributor to the lack of agreement between ^T_SH and _heat. Moreover, the highest bins of V_SWT_P contain relatively more intervals of large positive ΔV_SW and thus strong compression regions.

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