A three-dimensional numerical simulation in the nighttime midlatitude ionosphere is developed for the first time and applied to the Perkins instability in the midlatitude F region. Growth of the Perkins instability is successfully reproduced under nighttime condition, and the numerical results basically agree with a linear theory and previous two-dimensional numerical studies. Northwest-southeast (NW-SE) alignment of density perturbations is generated from random seeding by applying a southeastward neutral wind. The perturbations are dominant at altitudes of 200–300 km where a steep density gradient exists, which is consistent with the altitude of 630-nm airglow emission that often shows NW-SE alignment. Further simulation in terms of the E-F coupling can be done in the near future.
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 Since the discovery of turbulent upwellings associated with midlatitude spread F with the MU radar [Fukao et al., 1991], the midlatitude ionosphere has been intensively studied. Two-dimensional airglow images and GPS-TEC (Total Electron Content) maps have revealed that banded structures, or medium-scale traveling ionospheric disturbances (TIDs), frequently occur in the nighttime midlatitude ionosphere in summer [e.g., Saito et al., 2001]. They are aligned from northwest to southeast (NW-SE) with a wavelength of a few hundred kilometers and travel southwestward with a velocity of approximately 100 m s−1 [e.g., Miller et al., 1997]. Although the Perkins instability [Perkins, 1973] may be the most likely mechanism to explain the TID structures, the rapid growth of the TIDs cannot be derived from the linear theory of the Perkins instability.
 Progress in this field has taken place in terms of coupled electrodynamics between the ionospheric E and F regions since Tsunoda and Cosgrove  proposed a scenario based on early observations. Recent coordinated observations showed strong indications of the coupling between midlatitude spread F and E-region irregularities [e.g., Otsuka et al., 2007; Saito et al., 2007] and between conjugate hemispheres [Otsuka et al., 2004]. Cosgrove and Tsunoda  studied the E-F coupled instability theoretically and concluded that the growth rate of the coupled instability exceeds that of the Perkins instability.
 Numerical simulations of the Perkins instability have been conducted in a two-dimensional field in which field-line-integrated Pedersen conductivity is considered [Miller, 1996; Zhou et al., 2005, 2006]. Although NW-SE oriented conductivity perturbations are successfully obtained in these simulations, the rapid growth of the structures is not yet clear. Different types of simulation have shown that polarization electric fields in the coupled E region have significant effects on F-region electrodynamics [e.g., Yokoyama et al., 2004a; Shalimov and Haldoupis, 2005; Cosgrove, 2007]. Therefore, an E-F coupled numerical model is required for the study of F-region electrodynamics and the Perkins instability at midlatitudes.
2. Model Description
 We have employed a three-dimensional numerical model of the midlatitude ionosphere that was extended from our previous model applied to E-region electrodynamics [Yokoyama et al., 2004a]. Positive ion consists of only O+ for the F region, and the E-region plasma is not included in the present simulation. The momentum and continuity equations are written as
where the subscript j denotes ion or electron, n (= ni = ne) is the plasma density, v is the velocity, P and L are the production and loss coefficients of O+ in the nighttime [Strobel et al., 1980; Schunk and Nagy, 2000], q is the charge including its sign, m is the mass, E is the electric field, B is the geomagnetic field, g is the gravity acceleration, kB is the Boltzmann constant, T is the temperature, νjn is the collision frequency with neutrals, and u is the neutral wind velocity. The electric field generated by the ambipolar diffusion is explicitly written and subtracted as [Miller, 1996]
where E0 is the background electric field, ϕ is the electrostatic potential of the polarization electric field, Δ = meνen/miνin, and g∥ is the B-parallel component of g. B is uniform in the model with ∣B∣ = 4.6 × 10−5 T and with an inclination angle of 45°. ϕ is calculated to satisfy the following condition
Equations (1)–(4) are solved with a finite difference method in Cartesian coordinates, where the x, y, and z axes are directed eastward, northward, and upward, respectively. The model covers altitudes from 90 km to 470 km, and the horizontal domain is 320 km × 320 km, with a grid spacing of 2 km. Background parameters are given by the MSISE-90 and IRI-95 models at 34.85° Latitude and 136.1° Longitude at 22 LT on 27 June 1996 [Hedin, 1991; Bilitza et al., 1993]. Periodic boundary conditions are applied in the horizontal direction, and the top boundary is set to maintain diffusive equilibrium. Equipotential magnetic field lines except for the ambipolar field are assumed in the present simulation for simplicity. Equations for ϕ derived from (4) are solved with a non-stationary iterative method called Bi-CGSTAB(2), which provides fast and smooth convergence for this problem [Sleijpen and Fokkema, 1993]. The continuity equation (1) is calculated with the constrained interpolation profile (CIP) method [Yabe et al., 2001] for advection terms and with the fully-implicit scheme for diffusion terms. The CIP scheme can suppress numerical diffusion considerably by incorporating the advection of spatial derivatives of a quantity. The time step for solving (1) is 0.5 s.
3. Numerical Results
 Before starting the simulation of the Perkins instability, the equilibrium state of plasma density is obtained as an initial background profile by applying a uniform neutral wind u0 of 200 m s−1 toward 37° south of east for 1800 s to the density profile from IRI. The corresponding direction of u0 × B is 53° north of east. The wave vector k direction with the maximum growth rate is determined by the condition α = θ/2 where α is the angle between k and the east and θ is the angle between E0 + u0 × B and the east [Perkins, 1973]. In the above case (θ = 53°), α should be 26.5°, which is satisfied if kx = 2ky. Initial perturbations are then applied by raising the density profile along the geomagnetic field. In the first run, random perturbations are applied with a range of 0–100 m, which provides a 0.26% variation of the integrated Pedersen conductivity along the geomagnetic field. Figure 1 shows plasma density perturbation on a horizontal plane ((n − )/ where is the averaged density on the plane) at an altitude of 280 km at t = 0, 1200, 2400 and 3600 s. The random structure is filtered into NW-SE patterns growing with time. However, they do not propagate in any directions because a background electric field is not imposed. The power spectral density at t = 3600 s in a wave vector domain is shown in Figure 2. Large power areas lie in the first and third quadrants divided by two solid lines which represent the theoretical cutoff of the Perkins instability. (The figure is symmetric about the origin.) The spectral peak appears at kp = (kpx, kpy) = (2π/53.33 km−1, 2π/106.67 km−1), which corresponds to a wavelength of 47.7 km, and short-scale perturbations are filtered out. Although the dominant wavelength is relatively short, the direction of the wave vector agrees with that which maximizes the linear growth rate of the Perkins instability. Figure 3 shows the profiles of plasma density and perturbation averaged on each horizontal plane. The perturbation is dominant between 200 and 300 km altitude, where the upward density gradient is steep because the perturbation grows by the vertical motion of plasma. This region matches the 630-nm airglow emission altitudes where NW-SE alignment is frequently observed [e.g., Shiokawa et al., 2003].
 In order to examine the wavelength dependence of the instability, sinusoidal perturbations with an amplitude of 100 m are introduced with varying initial wavelengths. The wave vector of the initial perturbation is directed 26.5° north of east which maximizes the growth rate. Figure 4 shows the time variation of the perturbations of the integrated Pedersen conductivity for four wavelengths, (kx, ky) = (2π/80 km−1, 2π/160 km−1), (2π/40 km−1, 2π/80 km−1), (2π/30 km−1, 2π/60 km−1) and (2π/20 km−1, 2π/40 km−1), which are labeled Cases 1, 2, 3 and 4, respectively. All k vectors satisfy the maximum growth rate condition (kx = 2ky) and smooth variation on the periodic boundary. The dotted line represents the linear growth rate of 6.25 × 10−4 s−1 derived from the Perkins' theory. The perturbation grows following the theoretical estimate approximately except for the shortest wavelength case (Case 4), then begins to saturate when the amplitude reaches 3–4%. Perturbation with a shorter wavelength has a smaller growth rate and saturates at a smaller amplitude, which is consistent with the previous result that short-scale perturbations are filtered out. It is also consistent with the previous studies; Huang et al.  pointed out that spatial harmonics generated from the second-order nonlinear terms cause the damping of the instability which makes perturbation with a shorter wavelength saturate at a smaller amplitude, and Zhou et al.  showed similar saturation with their two-dimensional simulation.
4. Discussion and Conclusions
 We have simulated the Perkins instability in the midlatitude ionospheric F region by using the three-dimensional numerical model for the first time. The NW-SE density perturbations are produced under a suitable background condition. The altitude range where the perturbations are dominant (200–300 km) is consistent with that of 630-nm airglow emission which often shows NW-SE alignment [e.g., Shiokawa et al., 2003]. It can be concluded that our model have reproduced the Perkins instability successfully, but there are still discrepancies between simulations and observations.
 The major problem with the Perkins instability is that the growth rate is too small to explain observational results as mentioned in the previous reports [e.g., Kelley and Makela, 2001]. The maximum amplitude of the polarization electric field of Case 1 at t = 9000 s in Figure 4 is 0.56 mV m−1, which is much smaller than that expected on the basis of observations [Shiokawa et al., 2003] even though the very large neutral wind (200 m s−1) is applied. Although 10% conductivity variation is large enough to make airglow perturbation, it takes too long time to reach this amplitude from the initial seeding. It is necessary to consider mechanisms to intensify the instability such as a seeding of the F region or the coupled E-region effect.
 The dominant wavelength that grows from the random perturbation is relatively shorter than typical observational results [e.g., Saito et al., 2001]. The primary reasons are that the horizontal domain size (320 km × 320 km) is not large enough to compare with observations and that kp should satisfy the periodic boundary condition, that is, kpx = l × 2π/320 km−1 and kpy = m × 2π/320 km−1 (l = 6 and m = 3 in the above case). Perturbation with a few hundred kilometer-scale structure could be simulated by using a larger simulation domain with more computational resources.
 The southwestward propagation of the banded structure is not reproduced in the present simulation. When only the neutral wind is applied as a background condition, the perturbation structure does not drift in the earth-fixed frame as shown in Figure 1. Although the polarization electric field along the wave front was proposed by Kelley and Makela , it is too small to make the structure propagate southwestward as suggested by Zhou et al. . Zhou et al.  proposed that a northwestward drift of the structure by a northeastward background electric field can produce the southwestward component of phase velocity. However, the source of the large background electric field of more than 4 mV m−1 in the earth-fixed frame used by Zhou et al.  is not clear. Furthermore, although the southeastward neutral wind in the post-sunset period is suitable for generating the NW-SE structure through the Perkins instability, the dynamo electric field induced by the neutral wind is southwestward, which is opposite to the assumption by Zhou et al. .
 Now that we have developed the three-dimensional model of the midlatitude ionosphere, we can proceed further simulation in terms of the E-F coupling. Cosgrove and Tsunoda  proposed that the sporadic-E (Es) layer instability in the E region and the Perkins instability in the F region enhance the growth rate each other. Yokoyama et al. [2004b] showed with the numerical simulation that gravity waves propagating from the lower atmosphere modulate the upper E region through the polarization electric field generated in the Es layer. The electric field can also modulate the F region and seed the Perkins instability. Both are suitable topics of investigation for the model developed in the present paper and will be undertaken in the near future.
 Computation in the present study was performed with the KDK system of Research Institute for Sustainable Humanosphere (RISH) at Kyoto University as a collaborative research project and the super computer at Nagoya University through the joint research program of the Solar-Terrestrial Environment Laboratory, Nagoya University. TY is supported as Research Fellow of the Japan Society for the Promotion of Science (JSPS). The present study was supported by Grant-in-Aid for JSPS Fellows 17·7518.