How the Ionosphere Responds Dynamically to Magnetospheric Forcing

Ground magnetic field variations have been used to investigate ionospheric dynamics for more than a century. They are usually explained in terms of an electric circuit in the ionosphere driven by an electric field, but this is insufficient to explain how magnetic field disturbances are dynamically established. Here we explain and simulate how the ionosphere dynamically responds to magnetospheric forcing and how it leads to magnetic field deformation via Faraday's law. Our approach underscores the causal relationships, treating the magnetic field and velocity as primary variables (the B, v paradigm), whereas the electric field and current are derived, in contrast to the E, j paradigm commonly used in ionospheric physics. The simulation approach presented here could be used as an alternative to existing circuit‐based numerical models of magnetosphere‐ionosphere coupling.


Introduction
Magnetic field perturbations observed on ground are often regarded to be produced by ionospheric currents, modeled as a horizontal current j h on a spherical shell with density where E is the electric field, u is the neutral wind velocity, assumed to be constant in the height interval represented by the spherical shell, and B 0 is the main magnetic field of the Earth.Σ P and Σ H are height-integrated Pedersen and Hall conductivities, respectively, referred to as conductances.
Equation 1 is the (height-integrated) ionospheric Ohm's law.It is a steady-state solution to the momentum equation for the average ionospheric ion species (Brekke, 1997;Dreher, 1997), expressed as a relationship between electric currents and the electric field in the frame of the neutral fluid.Very often the electric field is also described as a potential field, which implies that it has no rotational part and therefore that Faraday's law can be set to zero.While it has been shown that the rotational part of the electric field can be strong (Madelaire et al., 2024;Vanhamäki et al., 2007), it is usually small compared to the potential part of the field.It nevertheless plays a critical role in the dynamics, as we will return to below.
Magnetic field perturbations ΔB, where B = B 0 + ΔB, can be calculated from the ionospheric Ohm's law by a Biot-Savart integral over the current.Since it is often a good approximation, and since it relates so many different quantities in the ionosphere, the ionospheric Ohm's law is extremely useful for describing ionospheric electrodynamics (K.M. Laundal et al., 2022).However, since it is an equilibrium equation it is not suitable for explaining causal chains of events.Vasyliunas (2001), Vasyliunas (2005aVasyliunas ( , 2005bVasyliunas ( , 2012)), and Parker (1996Parker ( , 2007) ) argued that this so-called E, j paradigm, which attempts to explain plasma dynamics using circuit theory, is severely limited.Instead, the correct starting point for a dynamic description of space plasma dynamics is the plasma velocity v and magnetic field B. E and j should be derived from B and v, and not vice versa.
As discussed in a review by Leake et al. (2014), the B, v paradigm has been successfully applied in almost all areas of space plasma physics, except for in the ionosphere where the E, j perspective remains prevalent.For example, despite a handful of illuminating simulations that take the B, v perspective (Dreher, 1997;Otto & Zhu, 2003;P. Song et al., 2009;Tu & Song, 2019;Tu et al., 2014;Yano & Ebihara, 2021), gaining an intuitive understanding of the sequences of events within the B, v-framework that lead to ground magnetic field variations proves challenging.Some of the models do not include horizontal gradients, or they neglect dynamic variations at the base of the ionosphere, both of which are critical for understanding ionospheric magnetic field variations on the ground.In the next section we show how ionospheric dynamics can be modeled in two dimensions, and how it leads to magnetic field disturbances seen on the ground.The approach is an adaptation of the concepts presented in Vasyliunas (2012) and Parker (2007, Chapter 9.5), tailored for application to a 2D sheet ionosphere, simplified to accentuate key processes, and converted to SI units.In Section 3 we present results of a numerical simulation, and Section 4 concludes the paper.

Model
In the description and simulation presented below, we consider the ionosphere to be 2-dimensional and horizontal.We use this simplification to elucidate the conceptual differences between a dynamic description and the conventional circuit description, Equation 1.Since we model the ionosphere as a two-dimensional infinitesimally thin surface, magnetic field lines have singular points there; , where the superscripts indicate positions just above (+) or below ( ) the ionosphere at r = r 0 .Inside the sheet ionosphere, we only consider the radial magnetic field component.
We also make another very useful simplification in our description: That the electron fluid and magnetic field are frozen-in throughout the ionosphere.This assumption holds as long as the electron gyro-frequency is much larger than the collision frequency, which is generally assumed to be the case above 100 km (Brekke, 1997).This assumption is helpful because the electron momentum equation reduces to the following relationship between the horizontal component of the electron velocity v eh and the horizontal component of the electric field: where r is a unit vector in the radial direction.This equation is also referred to as the generalized Ohm's law, not to be confused with the ionospheric Ohm's law that is Equation 1. Equation 2 implies that electrons behave as an infinitely light fluid that immediately neutralizes any electric field in their own frame of reference; and that the electric field is non-zero in all but the electrons' frame of reference.For a 3D version of this equation that includes electron collisions, electron inertia, gravity, and pressure, see Leake et al. (2014, Equation 26).
Our assumption about the electrons also implies that when B and the horizontal ion velocity v ih are known, the electron velocity can be found from Ampere's law, where n is the plasma density (particles per m 2 ), e is the elementary charge, and we have used that the horizontal current density j h = ne(v ih v eh ).The horizontal magnetic field difference represents ∇ × B in the 2D ionosphere.We neglect the displacement current in line with our assumption that electrons are infinitely light.In reality, it is the electric field of the displacement current that accelerates electrons to cancel any charge accumulation implied by an imbalance between ∇ × B and j (Y.Song & Lysak, 2006), but this happens on plasma frequency time scales, much faster than the processes we consider here.
In our description, the electric field and electron velocity are equivalent quantities that are in dynamic equilibrium, derived from the time-dependent primary variables in our model: v ih , B, and n.The time-variation in n is described by the continuity equation, where P is the plasma production rate and αn 2 is the decay rate.The n 2 dependence is appropriate for dissociative recombination, which is the most important loss process in the lower ionosphere where the ionospheric currents are strongest (Ieda et al., 2014).
The time-variation in ion velocity is dictated by the momentum equation (see e.g., Parker, 2007, Equation 9.13), where u h is the horizontal neutral wind velocity, m i is the average ion mass and ν in is the ion-neutral collision frequency for two-dimensional momentum transfer.In this equation we neglect the divergence of momentum density flux and momentum changes due to loss and production of plasma, for simplicity and since these are assumed to be small compared to the two dominating terms: The Lorentz force and ion-neutral collisions.Notice that by using Equation 2to replace E, the Lorentz force can also be expressed as j × B. Both Equations 4 and 5 can be seen as height-integrated versions of their three-dimensional counterparts.The relationships between the conductances from Equation 1, and the parameters in this equation, P, α, and ν in , and their 3D counterparts, is described in Appendix A.
The time development of B r is described by Faraday's law as where Equation 2was used to write Faraday's law as a conservation equation for magnetic flux.That is, B r v eh can be thought of as the flux of magnetic field lines, and ∇ ⋅ (B r v eh ) as the rate at which magnetic field lines are squeezed together or brought apart by the electron motion.Equation 6shows that B r can only change if B r v eh is nonuniform.An idealized situation where uniform electric and magnetic fields produce currents according to Equation 1 is therefore not possible to establish dynamically.
Equations 2-6 constitute a complete set of equations for describing the processes that lead to changes in the magnetic field, as long as we know the magnetic field above and below the ionosphere, B + h (r 0 ) and B h (r 0 ) , at all times.The magnetic field above the ionosphere, B + h , depends on the magnetospheric dynamics and can be calculated with ideal MHD.The magnetic field below the ionosphere, B h , must also be calculated.In this region there is no current due to the insulating atmosphere, so the magnetic field can be described as B = ∇V where V is the magnetic potential.The magnetic field below can be calculated by solving Laplace's equation for V with B r from Equation 6as an upper boundary condition (R. Lysak & Song, 2001;Ozeke & Mann, 2004;Sciffer & Waters, 2002;Yoshikawa & Itonaga, 1996, 2000).The magnetic field below the ionosphere thus adapts to changes in B r at the speed of light, while B r is controlled by ionospheric dynamics according to Equations 2-6.An ionospheric current that satisfies Ampere's law will automatically be present, but it is not what causes the magnetic field disturbances.This description of how the ground magnetic field varies in response to space plasma dynamics is fundamentally different from conventional explanations, which first calculates the current with Equation 1 and then the magnetic field on ground using the Biot-Savart law.

Simulation Results
We now present simulation results to illustrate the processes described above.A detailed description of the numerical implementation is given in Appendix B. We use a flat ionosphere in Cartesian coordinates (x and y), at an altitude r = r 0 = 100 km.The driver of the ionospheric dynamics is a flow channel in the magnetosphere at r = r 0 + 500 km, which is confined in the x direction.Gradients in the y-direction are zero.The neutral wind is zero and the main magnetic field is downward, with a magnitude of 50,000 nT.This geometry is illustrated in Figure 1.It is arguably the simplest possible geometry in which the dynamic explanation of magnetic field variations is meaningful, and is designed to elucidate the process.
We simplify the description of the time-development in space at r > r 0 by assuming that the Alfven speed there is infinite and that the magnetic field between the ionosphere and the top of the simulation domain is force-free.The resulting magnetic field in the y-direction is uniform but tilted from the vertical direction by an angle that depends on the differential motion between the electron fluid in the ionosphere and the plasma at the top boundary.In the x-direction the magnetic field is assumed to mirror the field below the ionosphere, described by a Laplacian potential, consistent with an assumption that j y = 0 in the region between the ionosphere and the top boundary.Thus, we ignore the propagation of Alfven waves back and forth between the magnetosphere and ionosphere (Wright, 1996), and assume that the magnetic field immediately adapts to the boundary conditions.Because of these simplifications, the timescales of the variations in our simulation may be unrealistic.However, the main purpose of this simulation is to show the sequence of events that establishes magnetic field disturbances in and below the ionosphere.
The sequence of events is illustrated in the simulation output displayed in Figure 2. Starting from a stationary ionosphere, the differential motion between ionospheric electrons and the plasma at the top boundary causes the magnetic field lines in space to bend in the y-direction.The resulting kink in the magnetic field lines at r = r 0 corresponds to a curl in the magnetic field in the x-direction, causing a current.The current is at first carried by electrons, which, since the flow channel is confined in the x-direction, compress the magnetic field lines at one side and bring them apart at the other side of the flow channel (Figure 2a2).As the magnetic field above and below the ionosphere adapts to the resulting B r , a curl in the perpendicular y-direction is produced in the ionosphere.The electron motion required to carry the resulting current changes the differential motion between ionosphere and magnetosphere, changing the rate at which the field-lines bend in the y-direction.This process continues until the system reaches equilibrium.In equilibrium the electron motion in the y-direction equals the speed of the flow channel, while ions lag behind.This is the steady-state Hall current.The steady-state Pedersen current, which flows in the x-direction, is entirely carried by ions, forced through the neutral fluid by the Lorentz force or j × B.
Figure 3 shows the time-development of B x on ground (panel a), the ionospheric ion velocity in the y direction (b), and of B y above the ionosphere (c), all evaluated at the center of the flow channel.The first 120 s of six different model runs are shown in different colors and line styles.The model runs have different combinations of plasma production P and ion-neutral collision frequency ν in (both kept uniform in space), corresponding to different combinations of Hall and Pedersen conductances (see Appendix A).We see that the steady-state value of the magnetic field on ground depends on the Hall conductance, and in space depends on the Pedersen conductance.The ion velocity in the y direction depends on the ratio between Hall and Pedersen conductances.While these results are expected from steady-state theory (Brekke, 1997), the time variations are not possible to predict without a dynamic treatment.Figure 3 shows that the rate of change of the magnetic field on ground depends on the ratio of Hall to Pedersen conductance.For small ratios the magnetic field slowly converges toward its equilibrium value, while large ratios give rapid variations and oscillations.A large ratio corresponds to more ionization near the base of the ionosphere where the ion-neutral collision frequency is high, which can happen during energetic particle precipitation.
The oscillations also give insight into the dynamic relationships between the three quantities: We see that the ion velocity (d2) varies in step with B y (d3).This can be understood from Equation 5: The ion inertial term is usually small compared to the two force terms, which means that the ion velocity varies approximately in proportion to the Lorentz force, which is proportional to B y in space.Figure 3 also shows that extrema in the B y and v iy time series align with periods when B x changes most rapidly.This can be explained by considering Equations 3 and 6.Since B x on ground depends on B r in space, its time derivative (Equation 6) varies as v ex , which according to Equation 3 is a linear combination of v i and B y in space.  .Simulation output at t = 4.5 s for a simulation where plasma production P and the ion-neutral collision frequency for momentum transfer ν in were chosen to give Σ P = Σ H = 5 mho in steady state.The top two panels show magnetic field lines in the y, r and x, r planes, respectively, with the horizontal component of the magnetic field exaggerated by a factor of 1,000 for visualization purposes.The blue arrows represent electron velocity, with the velocity in the x-direction enhanced by a factor of 5 compared to the vectors in y direction.The vector scale can be read from the third panel (a3) which shows the electron (blue) and ion (green) velocities in the 2D plane, together with the magnetospheric flow channel, as a function of x.The bottom panel (a4) shows the magnetic field perturbations on ground and the radial current density, also as a function of x.With our approach j r can be calculated as the negative divergence of the horizontal current, j r = ∂j x ∂x , or as the curl of the magnetic field in space, j r = 1

Discussion and Conclusions
We have presented an example simulation that elucidates the process by which the ionosphere dynamically responds to magnetospheric forces, causing deformation of the magnetic field.The description is different from conventional descriptions in ionospheric physics because the electric current is not a primary variable, but instead derived from the deformation of the magnetic field.
In our example the magnetosphere, as defined by everything above the 2D sheet ionosphere, was highly simplified.In principle this could be replaced by a more sophisticated MHD code.In most MHD simulations of the global magnetosphere the coupling to the ionosphere is handled by solving the Poisson equation that arises from taking the divergence of Equation 1 for a potential electric field, given distributions of field-aligned currents and conductances (Merkin & Lyon, 2010).The dynamic 2D description presented here could in principle replace this coupling scheme, with the same input (FACs) and output (E).With the parallel current given by the MHD simulation and the assumption that perturbations are weak compared to the main magnetic field, Ampere's law can be used to calculate the corresponding magnetic field just above the ionosphere, which drives the ionospheric dynamics.The output electric field should be calculated using Equation 2. The electric field calculated in this way is not a potential field since induction (Faraday's law) is included.Vanhamäki (2011) proposed and demonstrated a version of this procedure that was based on spherical elementary current systems (Amm, 1997;Vanhamäki & Juusola, 2020).
Even more work has been done on inductive MI coupling in the context of Alfvén wave reflection.This was pioneered by Yoshikawa and Itonaga (1996) and further developed by for example, Yoshikawa and Itonaga (2000), R. Lysak and Song (2001), Sciffer and Waters (2002), R. L. Lysak (2004).These studies use a similar boundary condition for the magnetic field as in this paper, but use the steady-state ionospheric Ohm's law (Equation 1) neglecting ion inertia (Equation 5) and time-dependent plasma transport effects (Equation 4).While this may be a good approximation in many cases, a key benefit of including these effects is, arguably, that it gives a clearer picture of the causal chain of events in the ionospheric response to magnetospheric forcing.It is, also arguably, easier to extend our approach to include neutral dynamics and higher-order fluid moments.For example, it would be valuable to keep track of temperatures by including energy equations, since the temperature influences collision frequencies (Schunk & Nagy, 2009).A more complete description of the dynamic magnetosphere-ionosphere coupling should also include Alfvén wave physics, like the studies cited above.
In the causal chain of events outlined above, the Hall current is produced because of a rotational electric field that is associated with electron motion; and the electron motion serves to exactly maintain charge neutrality from the point of view of the electron fluid (Equation 2).We stress that this is also a simplification, since it assumes the electrons to be fully magnetized, but we find that it is a useful perspective to get intuition; and it is easy to generalize to include more effects in Equation 2. For a discussion of this, see the section about the generalized Ohm's law in Leake et al. (2014), or the section about partially ionized gases in Parker (2007).
The intuition of electrons serving as a frozen-in neutralizing fluid can also be used to explain wind-driven ionospheric currents, which dominate at lower latitudes.Qualitatively, if the neutral wind changes the ion motion, the electron motion also changes to correct any imbalance in ∇ × B (Equation 3).The deformation of the magnetic field by the electron motion in turn modifies the current.One way that this can happen is if electrons move in different directions at the two ionospheric footpoints of a magnetic field line.This is how the Sq current is established (Buchert, 2020).The equatorial electrojet, an intense electric current at the magnetic dip equator, can also be qualitatively understood with the dynamic approach: Electrons can flow freely along the dip equator, carrying an intense current without deforming the magnetic field, since the geometry of the magnetic field does not change along the dip equator.This physical explanation of the equatorial electrojet is different from the conventional description (Kelley, 2009) which relies on primary and secondary electric fields in a Cowling channel.
The simulation presented in this paper deliberately employs a basic geometry to simulate the dynamic ionospheric response to magnetospheric forcing.However, it excludes several critical elements that should be addressed in future studies.This includes the dynamic impact of more complex magnetospheric forces and effects of horizontal variations in plasma production, or conductance.These aspects can still be explored within a two-dimensional setup; yet, many other dynamic/inductive aspects of magnetosphere-ionosphere coupling, as evidenced in studies by Lotko (2004), Tu et al. (2014), P. Song et al. (2009), and Tu and Song (2019), demand a transition to where all integrals are over the height of the ionosphere.We see that the 2D velocity is the average 3D velocity weighted by the density; the 2D recombination coefficient is the average 3D recombination rate weighted by the density squared; and the 2D ion-neutral collision frequency for momentum transfer is the average 3D ion-neutral collision frequency for momentum transfer weighted by the momentum.We have assumed isotropy such that ν in is a scalar.where the coefficients c n can be found by using boundary condition Equation B7.We truncate the sum at N = 1,000, the number of grid points where B r is defined.We solve for the N coefficients in each time step in order to find B x = B x (r = r 0 ) .Above the ionosphere boundary condition Equation B10 is applied at r = +∞ instead of r = ∞.This leads to similar solutions, except for a sign change so that B + x = B x .
We now have a complete set of equations that can be integrated in time.We use a time-step Δt = 0.1 ms.Decreasing the time-step further did not change the model results.

Figure 1 .
Figure1.The geometry of the simulations, with a flat ionospheric sheet at r = 100 km.500 km above the ionosphere the plasma flow is held fixed in the y-direction with a peak speed of 500 m/s.

Figure 2
Figure2.Simulation output at t = 4.5 s for a simulation where plasma production P and the ion-neutral collision frequency for momentum transfer ν in were chosen to give Σ P = Σ H = 5 mho in steady state.The top two panels show magnetic field lines in the y, r and x, r planes, respectively, with the horizontal component of the magnetic field exaggerated by a factor of 1,000 for visualization purposes.The blue arrows represent electron velocity, with the velocity in the x-direction enhanced by a factor of 5 compared to the vectors in y direction.The vector scale can be read from the third panel (a3) which shows the electron (blue) and ion (green) velocities in the 2D plane, together with the magnetospheric flow channel, as a function of x.The bottom panel (a4) shows the magnetic field perturbations on ground and the radial current density, also as a function of x.With our approach j r can be calculated as the negative divergence of the horizontal current, j r = ∂j x ∂x , or as the curl of the magnetic field in space, j r = 1 . (b) Same as column (a), but evaluated after the simulation reached steady state.

Figure 3 .
Figure 3.The time-development of B x on ground (a), the ionospheric ion velocity in the y direction (b), and B y above the ionosphere (c), all evaluated at the center of the flow channel.The first 120 s of six different model runs are shown, with different combinations of P and ν in to give different Hall and Pedersen conductances.The thick orange lines correspond to the simulation used in Figure 2.
Writing V = B 0 r + X(x)R(r) and applying boundary conditions Equations B8-B10 we find solutions of the form V(x,r) = B 0 r ∑