## 1. Introduction

[2] One of the controversial issues regarding magnetic reconnection at the magnetopause is the location of the reconnection line on the dayside magnetopause during periods of a large interplanetary magnetic field *B*_{y}. The antiparallel merging model [*Crooker*, 1979; *Luhmann et al.*, 1984] predicts that reconnection would occur where the magnetic shear across the magnetopause is largest and predicts no reconnection in the subsolar region when the IMF *B*_{y} (the so-called guide field) is large. In contrast, the component merging model [*Sonnerup*, 1974; *Gonzalez and Mozer*, 1974] predicts that the reconnection line passes through the subsolar point and has an orientation that is controlled by the IMF. As we discussed in an accompanying paper [*Daughton and Karimabadi*, 2005, hereinafter referred to as Paper I], the results from observations of magnetopause as well as theories of reconnection have been inconclusive. In order to address the issue of reconnection onset at the magnetopause, we have used a combination of linear theory and high-resolution simulations to compare and contrast the efficiency of collisionless reconnection in the antiparallel (zero guide field) and component merging (finite guide field) geometries. Paper I examined the detailed properties of the tearing mode as a function of guide field strength. It was found that the properties of tearing mode can be categorized into three regimes depending on whether the guide field *B*_{y} is smaller or larger than a characteristic guide field given by

Here

is the ion gyroradius, *v*_{ti} is the ion thermal velocity (2T_{i}/m_{i})^{1/2}, and L is the current sheet half-thickness. The three parameter regions are (1) the weak guide field regime *B*_{yo} < *B**_{y} which includes the zero guide field as a special case, (2) the strong guide field limit *B*_{yo} > 3*B**_{y}, and (3) in between these two limits, there exists a previously unknown regime that we refer to as the intermediate regime 3*B**_{y} ≳ *B*_{yo} > *B**_{y} where tearing has unusual properties such as maximum growth at oblique angles. Finally, we showed that changing the guide field from 0 to a value equal to the main field reduces the growth rate by a factor of ∼3.75. Thus we concluded that component merging cannot be ruled out based on linear theory of tearing.

[3] Given this result, the next obvious question is whether there are significant differences in the nonlinear evolution of the tearing mode between the zero guide field and finite guide field configurations. However, a theoretical study of saturation for these two limits has not been done systematically and previous works have yielded contradictory results. In the second paper in this series [*Karimabadi et al.*, 2005, hereinafter referred to as Paper II], we considered the nonlinear evolution of the tearing mode in the presence of one unstable mode. We found that a single tearing mode saturates at too small of an amplitude to be of relevance at the magnetopause. Here we extend this calculation to the case of multiple unstable modes.

[4] There exist several previous works that dealt with the nonlinear evolution of tearing in the presence of multimodes but in the absence of a guide field. In the collisionless limit, *Biskamp et al.* [1970] used quasilinear theory to conclude that tearing would saturate at amplitudes much smaller than the singular layer thickness. *Malara et al.* [1992], using incompressible MHD, found that unlike the single mode case, the saturation occurs due to inverse energy cascade (i.e., coalescence in the physical space). The coalescence process takes away energy from the tearing unstable modes, resulting in their saturation. However, the longest wavelengths continue to grow and achieve amplitudes much larger than those predicted by *Biskamp et al.* [1970]. A one-species simulation of the tearing mode in the presence of multiple unstable modes by *Pritchett* [1992] also revealed an explosive growth, following the coalescence phase (see also earlier papers such as *Leboeuf et al.* [1982]). Pritchett attributed the explosive growth to the vacuum effect. It is interesting to note, however, that the explosive growth phase has not observed in the fluid simulations of tearing [e.g., *Malara et al.*, 1992].

[5] The saturation mechanism of the tearing mode in the presence of a guide field is even less understood. Early theoretical studies ruled out component-merging as a possibility. The physics behind the stabilization of guide field tearing is the modification of resonant particle orbits by the growing island. Taking into account the influence of the perturbed electron orbits, it was shown [e.g., *Drake and Lee*, 1977; *Coroniti and Quest*, 1984] that the single tearing mode saturates at minute amplitudes (∼50 m), much smaller than the magnetopause current layer thickness of 50–200 km. In Paper II, we showed that these theories underestimate the saturation amplitude, but even with the corrected estimates, single mode tearing saturates at small amplitudes (∼9–36 km). Magnetic field line stochasticity was considered as a way to obtain larger saturation amplitudes [e.g., *Galeev et al.*, 1986, and references therein]. The overlap of neighboring magnetic islands can lead to destruction of magnetic surfaces, allowing the field lines to percolate through the magnetopause boundary layer. However, this diffusion process is too slow and may lead to saturation amplitude that is at most 3–4 times larger than the single tearing mode case [*Wang and Ashour-Abdalla*, 1994].

[6] In this paper, we use two-dimensional full particle and hybrid (electron fluid, kinetic ions) simulations to examine in detail the nonlinear evolution of multimode tearing instability as a function of the guide field. The remainder of this paper is organized as follows. Section 2 includes the simulations model. Section 3 presents the simulation results and our new theory for the saturation mechanism for the antiparallel geometry. Section 4 includes a description of nonlinear theories of the guide field tearing, followed by our simulation results. Section 5 pools the results from sections 3–4 and uses them to addresses the issue of antiparallel versus component merging at the magnetopause. The reader interested only in the application to the magnetotail can skip the first three sections and go directly to section 5. Summary and conclusion follow in section 6.