#### 2.1. The Magnetic String Equation

[4] The shape of the magnetic field lines around the magnetosphere is described by the magnetic string equations, which are second order, hyperbolic partial differential equations. They read as follows (see further details in Paper 1; and by *Farrugia et al.* [1995])

Here **r** is the position vector, *ρ* is the mass density, *M*_{A}, the Alfvén Mach number, and Π is the total pressure, i.e., the sum of the magnetic and plasma pressures. All quantities are normalized to their solar wind values. The coordinate system (*α*, *τ*, *ξ*) is defined such that coordinate *α* varies along the magnetic field lines, *τ* varies along the flow streamlines, and *ξ* varies along the electric field lines (**E** = −**v** × **B**). So these coordinates are related to the physical quantities (**V**, **B**, **E**). The right hand side of (1) contains two force terms and, as explained in Paper 1, which concentrated on to the equatorial plane, it is the cooperation followed by the competition between these two forces which first accelerates the plasma and then causes it to reach a terminal speed equal to the solar wind speed.

[5] To visualize this, we show in Figure 1 (top) the sum of the two forces (*F*_{res}) as a function of distance X (in *R*_{E}) from the Earth in the equatorial plane (Z = 0). As in Paper 1, the calculation is based on an IMF pointing due north. The shape of the magnetosphere is that of *Shue et al.* [1998]. The four curves are parametrized by the Alfvén Mach number *M*_{A}. All curves have a maximum in *F*_{res}, whose size depends on *M*_{A} and which is reached on the dayside, followed by a gradual decrease on the nightside. The two forces cancel each other at X ∼ −9 *R*_{E} and the maximum speed is reached there. After that the resultant force is negative, i.e., points sunward, and the flows are decelerated until they reach the solar wind speed far downtail.

[6] For completeness, we show in Figure 1 (middle) the Lorentz force **J** × **B** tangential to the surface. The difference between the Lorentz force and the resultant force is just the gradient of the plasma pressure (Figure 1, bottom; multiplied by 10), which is small in the magnetic barrier. With increasing *M*_{A}, the contribution of the pressure gradient increases while that of the Lorentz force decreases.

#### 2.2. Development of a Two-Humped Latitude Flow Profile

[7] We work with two *M*_{A}'s (3, 8) and start with *M*_{A}= 3. This would generally correspond to passage at Earth of magnetically-dominated configurations, the most common of which are interplanetary coronal mass ejections (ICMEs) [*Neugebauer and Goldstein*, 1997, and references therein] and their subset, magnetic clouds [*Burlaga et al.*, 1981].

[8] Figure 2 shows the draped configuration of magnetic field lines (blue) and the flow streamlines (green) for *M*_{A}= 3. The vertical axis labeled Z is along the external magnetic field direction and points north, and the horizontal axis is the X axis directed from Earth to Sun, positive sunward. The field lines at Z = 0 are initially convex sunward (to the right) and thus produce a tailward-pointing force due to field line tension. Tailward of X ≤ −9*R*_{E} they become convex tailward and then produce a sunward force, decelerating the plasma.

[9] Away from the Z = 0 plane the situation is, however, different. Far from Z = 0 (the *XY*plane), the field line has developed two points of inflection because the bend at Z = 0 has propagated symmetrically north and south and thereby deformed the field line. At X-distances where the field is convex tailward at low latitudes it is still convex sunward at higher Z values. Therefore forces develop at higher latitudes which are now pointing tailward and thus adding to the total pressure gradient force. This is different from what is happening at Z = 0 where the forces are still opposed and decelerate the plasma. Thus the peak accelerated flows migrate north/south of the*XY* plane. This migration is with respect to the magnetic field line. Since magnetic field lines are being dragged tailward by the flow, between two adjacent field lines the high speeds have moved northward and southward. The red blobs refer to the location of maximum speeds.

[10] Figure 3shows the flow profiles along the magnetic field lines at 8 different X-values in the range [= 3*R*_{E} (brown), −24 *R*_{E} (red)]. The vertical axis points along the external field (north), and the horizontal axis is the ratio *V*/*V*_{SW}. Tailward of X = −9 *R*_{E}, the maximum speeds start to move away from the equatorial plane to higher northern and southern latitudes, while still remaining, of course, close to the magnetopause. The peak speeds continue to increase with increasingly negative X in the range shown.

[11] As we see, the profile changes from being single-peaked in latitude (for lines 1–4) to being double-peaked, symmetric about the equatorial plane. Further, the peaks separate from each other such that the further downstream one goes the further north/south that they lie. Importantly, the ratio of the maximum magnetosheath speed (*V*) to that of the solar wind *V*_{SW} increases with downtail distance. Thus V/Vsw = 1.52 (X = −11 *R*_{E}) and 1.58 (X = −24 *R*_{E}). The lines approach V = *V*_{SW} as the field lines approach the bow shock at both ends of the figure.

[12] This acceleration mechanism is to be contrasted with that of the equatorial situation shown in Figure 1. As explained above, at Z = 0 the acceleration resulted from cooperation followed by competition of the two forces in the magnetic string equation. By contrast, north or south of the points of inflection in the field line, both the total pressure gradient force as well as the magnetic tension force point in the same (tailward) direction. For that reason the asymptotic behavior is different (see further below).

[13] Figure 4 shows the corresponding figure for *M*_{A} = 8, in a format similar to that of Figure 3. Again, we note a pronounced two - humped Z speed profile developing after*X* ≈ −9*R*_{E}. The maximum values of *V*_{SW} at all downtail distances is smaller than for *M*_{A} = 3, reaching up to only 1.25 *V*_{SW}. This is a combined effect of weaker magnetic tension and weaker pressure gradient force. We also note that, contrary to the case for *M*_{A} = 3, these maximum accelerated flows at higher north/south latitudes barely exceed values at Z = 0.

[14] What happens to these accelerated flows as we go even further downstream than X = −30 *R*_{E}? Figure 5 shows the result for the two Alfvén Mach numbers (dashed: *M*_{A} = 8, solid trace: *M*_{A} = 3). In both cases the flow speeds approach an asymptotic value, which is higher for *M*_{A} = 3 (= 1.58 *V*_{SW} for *M*_{A} = 3 versus 1.25 *V*_{SW} for *M*_{A} = 8). Asymptotic values are reached nearer Earth for *M*_{A} = 8 than for *M*_{A} = 3. The asymptotic behavior comes about because at large negative X the bends in the field lines straighten out, reducing the magnetic tension force, while simultaneously the pressure gradient force decreases.