Planet Mercury has a global magnetic field of internal origin. Its low intensity, 340 nT at the equatorial surface, indicates that the Hermean dynamo is special as much stronger fields are to be expected. Here, we suggest a feedback dynamo model where Mercury is embedded in an external field which is generated by the Chapman-Ferraro current of the small magnetosphere of Mercury. This self-generated ambient field has a pronounced influence on the dynamo action. Based on a kinematic α-Ω-dynamo model we show that the feedback dynamo has two solutions, a Hermean-type solution with a weak magnetic field and an Earth-like one with a strong field. The Hermean solution is that one where the external field opposes dynamo action, while an Earth-like solution results as the Chapman-Ferraro field is negligible in the dynamo region.
 In 2008 NASA's MESSENGER spacecraft will reach Mercury. One of the scientific interests of this mission [Solomon et al., 2006] is a study on the nature of the Hermean magnetic field. Dynamo action is thought to be the reason for global magnetic fields observed at Mercury as well as its companions Earth, Jupiter, Saturn, Uranus, and Neptune. Rapid convective motion in an electrically conducting core is sufficient to overcome magnetic field diffusion out of the field generating region. In its simplest form differential rotation transforms poloidal fields into toroidal ones, the Ω-effect. Other convective motions cause an α-effect, transforming toroidal fields into poloidal ones, closing the dynamo cycle [e.g., Stevenson, 2003].
 Assuming magnetostrophic balance a surface field strength thirty times stronger than measured is expected [Christensen, 2006] for Mercury. Several attempts have been made to explain this weak magnetic field. Measurements made during the flyby of Mariner 10 at Mercury in 1974 only allow to estimate the dipole component; any higher-order field contributions remain unknown [e.g., Ness et al., 1975]. A weak surface field does not necessarily imply that dynamo action is absent, but can indicate that magnetic field is primarily generated at small-scales which are effectively attenuated toward the surface [e.g., Glassmeier et al., 2000a]. Stanley et al.  suggested a thin-shell dynamo to understand the nature of the Hermean magnetic field. Using numerical experiments on the dynamo process they show that a thin-shell dynamo for Mercury's field is possible with toroidal fields more efficiently produced through differential rotation than poloidal fields generated via interaction of the convection flow with the toroidal fields. As the poloidal fields generated are also of smaller scale the surface field is weak.
 Another solution for the weak Hermean field situation was offered by Christensen  who argues that the outer regions of Mercury's liquid core are stably stratified. This implies a deep seated dynamo generating a strong magnetic field dominated by rapidly fluctuating small-scale contributions, attenuated by the stably stratified outer core regions. At the surface only the weaker dipole and quadrupole contributions can be detected by a spacecraft. Further possibilities to explain the weak Hermean field are also discussed by Heimpel et al.  and Takahashi and Matsushima .
 Here, we like to suggest another possibility. Embedded in a large mass density solar wind environment and equipped with a magnetic field of minor strength, planet Mercury exhibits a small magnetosphere [Ness et al., 1975]. The mean magnetopause distance is about 1.7 RM. Chapman-Ferraro currents flowing in the magnetopause cause magnetic fields at the planetary surface of the order of 50 nT [e.g., Glassmeier, 2000; Grosser et al., 2004]. At Earth these fields are of the order of 20 nT, that is much less than the internal geomagnetic field strength. Thus, Mercury is embedded in a significant ambient magnetic field. This field may cause planetary dynamo amplification and modification much as discussed for the Galilean moons Io and Ganymede [e.g., Sarson et al., 1997, 1999], but with the ambient field self-generated by the dynamo and its magnetic field-solar wind interaction. We shall explore the characteristics of such a feedback dynamo using a kinematic approach following Levy .
 The physical situation we envisage is schematically shown in Figure 1: Chapman-Ferraro currents generate a magnetic field which enhances the magnetospheric field and ideally cancels the field outside the magnetosphere. In the dynamo region the Chapman-Ferraro field is always opposing the dynamo generated field; the dynamo is embedded in an ambient field of opposite polarity. It is this situation which needs further detailed consideration.
2. Ambient External Field
 Magnetic field measurements taken on March 29, 1974 by the Mariner 10 spacecraft indicate a clear jump of the field strength of about 24 nT across the Hermean magnetopause [Ness et al., 1975]. Such a jump is caused by a Chapman-Ferraro sheet current density of jCF = 1.9 · 10−2 A/m. This current sheet covers all of the dayside magnetopause and its extension in meridional direction is of the order of πRMP, where RMP = 1.7 RM is the average radial magnetopause distance. To simplify the discussion we model the magnetic effect of the Chapman-Ferraro current sheet as that one of a circular current loop in the equatorial plane of the planet. The strength of this ring current is approximated as ICF = π · RMP · jCF; with this the total magnetopause current amounts to about ICF = 2.5 · 105 A.
 Using cylindrical coordinates the magnetic field = (Br, Bϕ, BCF) of such a circular current loop in the plane of the loop is given by [e.g., Smythe, 1968, p. 291]
where RL = RMP denotes the radius of the loop, i.e. the magnetopause distance, and r the distance of the receiving point to the center of the current loop; K and E are elliptic integrals of the first and second kind.
 The precise radius of the Hermean core is as yet unknown. Adopting the value RC = 1,860 km suggested by Spohn et al. , using the estimate ICF = 2.5 · 105 A, putting r = RC, and assuming a value RMP = 1.7 RM we have BCF ≈ 47 nT at the Hermean core-mantle boundary. Apparently the magnetic field of the Chapman-Ferraro current significantly contributes to the magnetic field in Mercury's dynamo region.
 Let us assume that the interplanetary magnetic field is negligible. In this case the Chapman-Ferraro current is determined by the magnetic field strength BMP at the magnetopause:
where BC is the magnetic field strength at the core surface. RMP depends on the solar wind dynamic pressure pSW and the planetary magnetic field strength at the core surface via
With this and equation (3) the Chapman-Ferraro field at the core surface reads
This gives the Chapman-Ferraro magnetic field strength in which the Hermean dynamo is embedded as a function of the Hermean field itself with the solar wind dynamic pressure playing the role of an external parameter.
 The circular current model used here to calculate the magnetic field of the Chapman-Ferraro currents is only an approximation. It underestimates the actual field by, for example, not taking into account the drag coefficient or the actual shape of the magnetopause (see Holzer and Slavin  for a discussion of these effects). Furthermore, the actual Chapman-Ferraro current system has a pronounced day-night asymmetry which causes a diurnal field variation on the time scale of the Hermean rotation period. Temporal variations of the solar wind dynamic pressure will result in additional variations on smaller and comparable time scales. Such temporal variations are not of importance here as the magnetic diffusion time scale of the Hermean core is of the order of 100,000 years.
3. Levy Dynamo
Parker  suggested a kinematic dynamo considering magnetic field generation by fluid flows consisting of a differential rotation and cyclonic motion. This model is also suitable to tackle the problem of a dynamo embedded in an ambient magnetic field [Levy, 1979]. Sarson et al. [1997, 1999] used a much more elaborated numerical model, simulated the dynamo process under the influence of an ambient field, and basically confirmed earlier results by Levy . We use this as a justification for our kinematic approach to demonstrate key features of the feedback dynamo.
 In the presence of an external field the induction equation reads
where denotes the magnetic field generated by the dynamo process and ext the external field; η is the magnetic diffusivity. Two consequences of this external field immediately emerge: the dynamo process is no more symmetric with respect to the sign of as this symmetry is broken by the presence of the external field. Furthermore, fluid motion against the already existing external field gives rise to magnetic field generation which can be used to counteract the consequences of magnetic diffusion. The external field can lower the threshold for dynamo field generation [Levy, 1979; Sarson et al., 1997, 1999].
 Following Parker  the magnetic field can be decomposed into its poloidal and toroidal part. For a spherical body and axial symmetry the induction equation then reads
where r, ϕ, and θ denote spherical coordinates; Bϕ and Br are corresponding components of the magnetic field. The external field ext is approximated as a uniform field aligned with the rotation axis. Here, we assume Bext = BCF, that is we approximate the external field strength by the equatorial value of the Chapman-Ferraro circular current field at the core-mantle boundary. The toroidal magnetic vector potential component is denoted as Aϕ. The two functions Ω(r, t) and α(r, t) describe differential rotation and cyclonic motion. Following Levy  the nonuniform rotation is confined to a thin layer near the surface of the fluid core and resulting from a rigid rotation of a thin shell of outer radius R past the inner sphere of radius l:
R is the radius of the dynamo region.
 Cyclonic fluid motion is also assumed to be confined to thin axially symmetric rings placed symmetrically on either side of the equatorial plane, which gives one [Levy, 1979]
Here, ξ and ψ give the positions of the rings of cyclonic motion; ψ is the angle between the rotation axis and the vectors of length ξ connecting the origin with the outer circumference of the rings (see Levy  for further details). Functions α0(r, θ) and Ω0(r, θ) give the strength of the cyclonic motion and differential rotation. With (8) and (9), defining BC as the field strength of the dipole component at the polar surface of the dynamo region, and assuming a stationary dynamo solution equation, (6) can be reduced to the expression [Levy, 1979]
where Γ contains information about the rings of cyclonic motion causing the α-effect. The dynamo number N is defined as N = αΩL3η−2, NC is a critical dynamo number which is of the order of 100 for a self-regenerative dynamo [e.g., Parker, 1970], and L is a typical spatial scale of the dynamo. The interpretation of relation (10) is straightforward: without any external field N = NC, and for any finite value of Bext N can take values different from NC as the ambient field supports or opposes dynamo generation as mentioned above.
4. A Feedback Dynamo
 As a feedback dynamo we define a situation where an ambient field in which the dynamo is embedded is generated by the dynamo action itself (cf. Figure 1). The origin of such a self-generated ambient field can be the interaction of the planetary magnetic field with the solar wind plasma. As the dynamo is influenced by the ambient field and this field depends on the dynamo generated field a feedback situation emerges.
 At Mercury the Chapman-Ferraro field BCF takes the role of the external field Bext. With this we have:
Equations (5) and (11), define relations between the field strengths BCF and BC and a further relation for the dynamo field strength BC results:
From this we derive the following relation for the dynamo number as a function of the dynamo generated field:
For three different values of Γ (25 · 10−2, 3 · 10−2, 3 · 10−2), Figure 2 displays this dependence. The situations described correspond to the (ψ, ξ/R) combinations (ψ = 45°, ξ/R = 1.0), (ψ = 45°, ξ/R = 0.5), and (ψ = 30°, ξ/R = 0.5). A ratio ξ/R = 0.5 indicates cyclonic motion deeper in the core. A dynamic pressure pSW = 13.4 nPa is assumed, corresponding to a proton number density of 5 · 107 m−3 and a solar wind speed of 400 km/s.
 Of particular interest are those situations where the dynamo number is close to the critical dynamo number. At this critical dynamo number we have nominal convection conditions, that is convection with magnetic field generation in the absence of any ambient magnetic field. For such nominal conditions a self-regenerative kinematic dynamo can exist [e.g., Parker, 1970].
 As Figure 2 indicates there are apparently always two situations where one can meet such nominal convection conditions. One occurs at a relatively weak core magnetic field, the other at large magnetic fields. The feedback dynamo discussed here has two stationary solutions, a weak field and a strong field one. In both cases the convection system does not need to be altered significantly and is described by its critical dynamo number, determined by differential rotation and cyclonic motions.
 The two situations differ in the following way. The strong planetary field situation causes, according to equation (4), a large magnetopause distance. In turn, this implies a weak Chapman-Ferraro field in the planetary core much as observed at Earth. The dynamo operates in the absence of an ambient field. The terrestrial dynamo is a realization of such a solution.
 The other solution, that one with a weak planetary field, can be explained as follows (Figure 3): differential rotation causes toroidal field generation out of the primary poloidal field with this field being regenerated via the α-effect. The primary poloidal field causes the generation of a magnetosphere whose Chapman-Ferraro currents generate a secondary poloidal field in the differentially rotating liquid core. As this secondary poloidal field is oriented in the opposite direction of the primary poloidal field the secondary toroidal field opposes the primary toroidal magnetic field. This in turn leads to a smaller regenerative poloidal field. We conclude that the proposed feedback dynamo offers two stationary solutions, a Hermean-type with a weak planetary magnetic field, and an Earth-like solution with a significantly stronger field.
 Planet Mercury is a suitable candidate for the operation of a Hermean-type dynamo. From our kinematic dynamo model we infer that its field strength should be of the order of a few hundred nT. This corresponds remarkable well with the actually measured field at Mercury.
5. Summary and Outlook
Levy's  kinematic dynamo model has been modified and used to outline the possibility of a feedback dynamo which we suggest to operate in the fluid core of planet Mercury. Though the model is a kinematic one it strongly indicates the importance of taking into account any self-generated ambient magnetic field, caused by magnetospheric Chapman-Ferraro currents or any other magnetospheric current system [e.g., Potemra, 1984; Glassmeier, 1984, 2000].
 The way the feedback dynamo explains the weak planetary field is also different from the dynamo models suggested by Stanley et al. , Heimpel et al. , Takahashi and Matsushima , or Christensen . The feedback dynamo does not necessarily require dominating small-scale field components. It can well explain a weak surface field dominated by low-order moments. It is the interaction of the ambient field with the differentially rotating fluid and its counter-acting influence on the toroidal field which causes the weak field.
 The stationary solutions outlined do not explain the path the dynamo reaches a particular solution. We merely show that two significantly different solutions are possible. Future numerical simulations are required to understand the nature of the highly nonlinear feedback dynamo system suggested and the conditions necessary to realize either solution. Also, their stability needs to be studied. This, however, is beyond the scope of the current study and subject to future analysis.
 Any experimental test to prove the validity of either model requires a spherical harmonic analysis of the Hermean magnetic field. The MESSENGER mission will contribute to this [e.g., Korth et al., 2004]. But a more definite answer is expected from the European-Japanese BepiColombo mission [e.g., Schulz and Benkhoff, 2006; Mukai et al., 2006]. Two spacecraft, the Mercury Magnetospheric Orbiter and the Mercury Planetary Orbiter, will orbit planet Mercury starting in 2019. The combined magnetic field measurements from both spacecraft will allow a highly accurate determination of the topology of Mercury's magnetic field.
 We are grateful to Stephan Stellmach, Johannes Wicht, Gerhard Haerendel, Rudolf Treumann, and Dragos Constantinescu for illuminating discussions. This work was financially supported by the German Ministerium für Wirtschaft und Technologie and the German Zentrum für Luft- und Raumfahrt under contract 50 QW 0602.