The equatorial ionosphere contains imbedded bubbles that rise vertically through a horizontally stratified plasma. When gravity, neutral winds or external electric fields act on the F region plasma, electric fields form in the density perturbations that comprise the bubbles. The second order differential equation that describes the electric fields is a nonseparable elliptic equation. Exact solutions of this nonlinear differential equation for the electric fields and potentials are derived assuming linear or circular symmetry to the density structures imbedded in the background plasma. A wide variety of analytic solutions for electric potentials are found for both density cavities and density enhancements. These solutions may be used for quasi-analytic transport of the bubbles with equations derived from the continuity equation for the plasma. The analytic solutions may also be used to test numerical solvers of the nonseparable elliptic equations that describe the electric fields in the unstable plasma.
 The F region ionosphere can become unstable if a density perturbation becomes electrically polarized by external forces from electric fields, neutral winds, and gravitational acceleration. Near the geomagnetic equator, gravity can act on the plasma attached to the nearly horizontal magnetic field lines to produce unstable conditions. After sunset when the layer is lifted by ambient electric fields, the bottomside steepens and plasma bubbles are formed. These bubbles rise through the layer in response to a Rayleigh-Taylor type instability. Also, winds or electric fields induce electric fields in both density cavities and enhancements that cause distortions in the density structures. These distorted plasma structures are responsible for degradation of radio propagation that lead to navigation errors and outages, communications systems failures and radar clutter. The modeling of ionospheric bubbles or density enhancements uses computer simulations the calculate the effects of self generated electric fields (E) that are driven by gravity, neutral winds, and external electric fields. Numerical computation of the electric potentials requires the most time and effort in the bubble modeling process. The electric fields for these simulations can be found numerically using direct or iterative solvers of the nonseparable potential equations that describe the self-generated electric fields [e.g., Bernhardt and Brackbill, 1984]. The computational time for solving for the disturbed ionosphere is often prohibitive so analytic solutions to both the transport and potential equations are useful. Exact analytic solutions for the electric potential can be used to test the numerical algorithms and to determine errors produced by boundary conditions and numerical roundoff. The analytic solutions also yield insight into the conditions for production of ionospheric bubbles.
 The computational and analytical techniques for simulations of equatorial bubbles are compared in Figure 1. Typically, numerical models of equatorial bubbles follow the procedure illustrated by the block diagram given in Figure 1a. A stratified model of the F layer is perturbed by a small density disturbance. Gravity is allowed to setup an electric potential in this plasma. The electric potential is obtained with a numerical solution of a nonseparable elliptic equation using a direct solver such as described in Appendix A. Once the electric potential is obtained, the plasma transported in response to the electric fields for a small time step. A nondissipative flux corrected transport algorithm [e.g., Zalesak et al., 1982] is then used for incrementally move the plasma disturbance. The process is repeated with the generation of a revised electric potential followed by more incremental plasma transport. All of these processes are numerically intensive and can require several hours of computation.
 A new approach for the quasi-analytic model of the equatorial bubble is proposed using the three steps in Figure 1b. First, the electric potential is defined by an analytic function that gives self-consistent expressions for electron density (or Pedersen conductivity) structures that are obtained in the presence of background electric fields, neutral winds and gravity. This procedure is described in this paper. Second, the plasma transport is determined using incompressible motion from the induced electric fields. The plasma transport is derived with the analytic electric potential distorting the coordinates with out changing the density in each coordinate cell. The third step is to adjust the parameters in the analytic models to that the analytic solution given in step 1 matches the quasi-analytic results from step 2. The application of the transport and normalization processes will be described in a future paper.
2. Exact Solutions for the Electric Potential in a Disturbed Ionosphere
 The equatorial ionosphere is commonly thought of as a uniform layer with the occasional imbedded structure or bubble. The modeling of ionospheric bubbles uses computer simulations that calculate the effects of self generated electric fields (E) that are driven by gravity, neutral winds and external electric fields. The equations for these simulations can be found in a number of papers including Bernhardt . For the analytic solutions considered here, the background ionosphere will be uniform in the horizontal, x- and z-directions. The ambient magnetic field, B, is aligned with the z-axis. The altitude variations of the undisturbed ionosphere will be represented by the function ne0(y) where y is the vertical coordinate.
 The layer becomes distorted when a small perturbation grows as electric fields provide incompressible perpendicular motion at F region altitudes. These internal electric fields move plasma across magnetic field lines with the velocity
where v is the velocity perpendicular to the ambient magnetic field B and the electric field E = −∇Φ can be represented as the gradient of a scalar electrostatic potential Φ. Other perpendicular components of velocity driven by pressure gradients, neutral winds, and gravity can be neglected because the plasma in the F region ionosphere is magnetized. This means that the electron and ion gyro frequencies are much larger than the corresponding collision frequencies with the background neutral gas. The inertial terms, described by McDaniel and Hysell , that affect high altitude plume structures are neglected.
 An analytic approach is derived to solve for the electric potential for localized electron density perturbations driven by external vector fields. Kelley  gives the electric current in the F region as
where σp = is the Pedersen conductivity, ne(x, y, z) is the electron density, νin is the ion neutral collisions frequency, E0 is the external electric field perpendicular to B, U = Ux + Uy gives the normal components of the neutral wind vector, g = −g0 is the gravitational acceleration, and B = B0 is the magnetic field vector. Assume that the normal electric fields are constant along the magnetic field in the z-direction and that there is no z-component of current J. For specific gravitational accelerations, neutral winds and external electric fields, the potential equation is found from (2) using
Substitution of (2) into (3) and integration along the z-direction leaves an equation for the potential in the perpendicular () x- and y- directions.
where Σp = ∫σpdz is the field-line-integrated Pedersen conductivity and E = −⊥Φ is the induced electric field. For simplicity, the driving fields E0, U, and are assumed constant in space and time, then the potential equation simplifies to
where the equivalent electric field vector defined by ET ≡ E0 + (U + ) × B. Given a spatial distribution for the Pedersen conductivity (or electron density), the potential is usually obtained numerically from the nonseparable elliptic equation (5). Often iterative solvers requiring relatively long solution times or direct solvers requiring large memories are required to compute this solution.
 A computational alternate approach assumes that the potential is given and (5) is used to find the associated electron density. For this solution, only Pedersen currents in the horizontal, x-direction will be considered so
where ETx represents the equivalent driving fields from a static electric field in the positive x-direction, a neutral wind in the positive y-direction and gravitational acceleration along the positive y-axis. The sign of ETx is negative for the downward acceleration of gravity. In our notation, the growth rate for the Rayleigh-Taylor instability is γ = (−ETx/B0)/LN where LN is the scale length of the gradient on the bottom side of the ionosphere [Zalesak et al., 1982; Sultan, 1996].
 In normalized Cartesian coordinates, (4) becomes
where often gravitational forcing is the sole contribution to ETx, = is the dimensionless, normalized potential, (, ) = (, ) are the normalized coordinates, and r0 is a constant scale factor for all distances. Note that (7) has many self-similar elements. Multiplying the - and - coordinates as well as by a constant scale factor (i.e., r0) does not change the equation. Multiplying Σp by a constant C0 also yields a solution. Consequently, if (, ) and ΓP(, ) satisfies (7), then so do the pairs of functions r0(, ) and C0 ΓP(, ). Normalized coordinates (, ) will be used to simplify the notation for the analytic solutions.
 The existence of analytic solutions for (7) was discovered by examining numerical solutions. A numerical algorithm for non-separable elliptic equations similar to (7) was written using a block tridiagonal solver of the algebraic equation derived from finite difference approximations to the partial derivatives (Appendix A). When a circularly symmetric function, ΣP where = , was used for the Pedersen conductivity it was found that the integral of the resulting electric potential along was also circularly symmetric. In mathematical terms,
This immediately shows that the form of the potential is the coordinate multiplied by a circularly symmetric function since
Functions of single variables such as can be solved analytically.
 General analytic solutions to (7) can be derived by making simplifying assumptions about the form of and Log(Σp). Numerical simulations for symmetric perturbations in x with x-directed fields given by (6) yield electric potentials that are odd functions of x with the form
where a0 and a1 are constants, q(, ) = is a single variable representing an elliptical perturbation for the potential and “s” determines the polarization of the coordinate ellipse.
 The Pedersen conductivity (or electron density) takes the form of an elliptically shaped perturbation modulating the background conductivity.
where Lp(q,s) is the natural log of the conductivity/density perturbation and Σp0() = ∫ r0 d is the integrated conductivity associated with the horizontally stratified plasma layer. The last term of the left side of (7) is x-direction gradient of the log-Pedersen conductivity which drives the solution for the potential. Consequently, the variation through the function q is required to obtain useful solutions for the potential.
 Substitution of (10) and (11) into (7) and solving for the derivative of Lp(q) yields the equation
where the prime (′) denotes the derivative with respect to q. If the functions of vanish, (12) may be integrated directly. The 2 terms vanish only if s = 0, or 1.
 General solutions for an extended vertical plume imposed on a horizontally stratified ionosphere are considered first. For s = 0 the potential has no variations in the altitude coordinate () and the solutions for (10) and (11) are given by
where C0 is a constant chosen to give the background conductivity as → ∞. Exact solutions for pairs of Pedersen conductivity and electric potentials from (13) are easily found. Table 1 gives several examples these pairs.
Table 1. Electron Density Disturbances and Companion One-Dimensional Potential Functions
Pedersen Conductivity Function, Σp(0) (, )
Electric Potential, (, )
(a0 + a1) exp( −b2)
(a0 + a1) sech[b ]
 Exact solutions for pairs of Pedersen conductivity and electric potentials from (13) are easily found. The last equation in (13) can be inverted to give
where ΣP(0)() = is the normalized one-dimensional density cross section in the horizontal direction. From the second equation in (13), the electric potential is found as
 An example of the second function in Table 1 is illustrated in Figure 2 with the parameters a1 = −0.5 and b = 3. The conductivity trough (Figure 2a) centered along the y-axis has a ridges that increase in amplitude as the parameter “a1” is increased. If a1 > 0, the trough is replaced by a Pedersen conductivity enhancement as the sense of the potential (Figure 2b) is reversed. The parameter “b” simultaneously controls the steepness of the walls on the conductivity irregularity and the spatial decay of the potential function. With separation of variables, even more general solutions to (7) can be found in Cartesian coordinates (Appendix B).
 General solutions for circular holes in an ionosphere with simple vertical structure are considered next. For s = 1 the density disturbance is circularly symmetric with radius r around the (, ) origin. The solutions from (12) require a0 = 0 and a1 = 1 with the result
To eliminate any dependence on of in (16), the background Pederson conductivity takes the functional form Σp0() = C0m where m and C0 are a constants. With this substitution, (16) becomes
which is identical to (13) with a0 = 0, a1 = 1 and m = −1. The corresponding formula for the spatial variation of Pedersen conductivity is
In Appendix C, using separation of variables, all of the solutions for (7) in cylindrical geometry are derived.
 The one-dimensional (s = 0) and two-dimensional, circularly-symmetric (s = 1) expressions are similar. With a0 = 0 and a1 = 1, the rational polynomial function of the form G(q) = used in (10), and the corresponding potentials from (13) and (14) are
where a and b are constants. Analytic solutions can be obtained from (13) through (16).
 Derivatives of the density perturbations become
The corresponding Pederson conductivity expressions are
where A1 = and B1 = −a(b − 1) − 2 and . The constants of integration are chosen to yield the background density at large distances where x → ∞. The physically acceptable solutions have b > 0.
 The analytic solutions give insight into the relationships between localized density perturbations and the associated electric potentials. Two types of conductivity structures, cavities and enhancements, are described by (21a) and (21b). In the parameter range ≡ aMin > a > 0, the plasma structure is a cavity centered at x = 0. These limits are found by solving for A1 = 0. As parameter “a” approaches the value of “amin” the sides of the density cavity becomes steeper. With a = amin, the wall of the cavity is located at radius is = . For b > m + 2, the cavity has a ridge located at = . With 0 > a > 1, a conductivity enhancement is found at the origin. This enhancement can represent the increased Pedersen conductivity produced by an artificial ion cloud from Barium or similar material released in the sunlit ionosphere. As a → 1, the sides of the density enhancement become steeper. Solutions exist for all values of b > 0 but if b > 1, the potential vanishes at large distances. If a > 1, then the solutions in (21a) and (21b) become complex and are not physically possible. The maximum upward velocity for the potential (1) (x, y) in (17b) is Vy0 = = . If a > 1, then the conductivity would move with a velocity larger than the ETx/B0 velocity of the driving force, which is not possible. For instance if only a vertical wind Uy is considered in the equation (6) for ETx, then the upward velocity is Vy0 = aUy. An unreasonable value of a > 1 would permit the conductivity enhancement to rise faster than the neutral wind driver.
 The restrictions on the ranges for the potential amplitude “a” indicate that not all electric potentials correspond to a physical density or Pedersen conductivity structure. For a given force on the plasma from external electric fields, neutral winds, or gravity, the induced potential is determined by the gradients on the wall of density cavity or enhancement. These gradients are physically limited by infinite steepness and the amplitude of the potential is a maximum at this limit. Thus for the solution to (4), a given physical density structure will always correspond to a potential function. The magnitude of a potential function can be increased to the point that there is no corresponding plasma density function.
 The one-dimensional expression (21a) can represent a horizontal density modulation that uniformly changes the background density of a stratified ionosphere. These may be produced by horizontally traveling acoustic-gravity waves can act as seeds for equatorial bubbles. The elongated shapes of these modulations are illustrated by the example in Figure 2. The elongations can be found in nature as the extensions of an ionospheric plume below its top. The horizontal electric field vectors calculated as gradients of the potential yield vertical plasma transport. This transport is normal to the density gradients and, consequently, no net change in the densities is produced. The electric fields near the top of the bubble are the primary drivers for plasma transport.
 The two-dimensional solutions to the potential equation are more useful than the one-dimensional solutions. The expression for Σp(1) (, ) in (21b) describes a plasma disturbance with two-dimensional structure. The conductivity (or electron density) vanishes in Σp(1)(, ) at y = 0 unless m = 0. Figure 3 illustrates three examples of the analytic density cavities and the associated electric potential for a uniform background using m = 0. By changing the parameters in the analytic model, a wide variety of density structures is obtained.
 The background plasma variation can be approximated using nonzero values of m. The Pedersen conductivity from (21b) vanishes at the = 0 boundary if m > 0. With ΣP = 0 at the lower = 0 boundary, the potential equation (7) reduces to = 0. To satisfy this condition, ∂/∂ = 0 because vertical gradient ∂Σp/∂ is nonzero-positive at the lower boundary. This condition is automatically built into the analytic expression for the electric potential because of the 2 symmetry of (, ) = G().
 Two quantitatively different ionospheres can become polarized with the same potential variation. Figure 4 illustrates two solutions of (21b) with identical parameters for a and b but with (a) m = 1 for a linear background profile and (b) m = 0.25 for a forth-root of y profile. As seen by the solutions in (21a) and (21b), a family of Pedersen conductivity structures can be associated with a single electric potential (or field) distribution. This non-uniqueness property can be exploited for modeling the evolution of the density structures.
3. Formation of Bubble Structures From Analytic Solutions for Troughs and Holes
 The analytic solutions derived in section 2 can for the building blocks for analytic descriptions of equatorial bubble structures. A single plasma bubble or the individual finger of plasma bifurcation can be decomposed into holes and troughs. By adjusting the parameters of the analytic descriptions for the holes and troughs, any finger of a plasma irregularity can be approximated. The procedure for bubble finger formation is based on noting that an analytic potential in (14) with the form (1)(, ) = G1 is shown to have a density structure of the form
where = and m = 0.
 Analytic bubbles or fingers are formed by using (22) for the top portion or “tip” of the finger above some altitude y0 and by extending the horizontal, x-axis cross section downward for the plume portion of the bubble for y less than y0. The bubble is then fully described by
 Away from the transition altitude y0, the corresponding electric potential is given by
These equations satisfy the normalized potential equation (7) at all locations near y = y0 where the electric potential from one type solution penetrates into the region of the other type solution.
 The bubble potential can be thought of as a weighted sum of the analytic solutions for the potentials of the trough and hole. If the end of the trough joins the center of the hole (i.e., y0 = 0), then the potential at that point is the average of the two potentials. This is proven by noting at y = 0 that = 0 and therefore = 0. Without the y-derivative, the potential equation (7) becomes
By assuming that 0 = 0, the potential equations for the hole and the trough take similar forms
 Adding (26) and (27) and comparing with (25) immediately yields that the bubble potential as the average of the trough and the hole potentials
Considering asymptotic limits, the bubble potential from a finger structure (23) takes the form
where the transition function g(, ) is bounded by 0 and 1 with the limits
and for 0 = 0, g(, 0) = . From (29), the transition function is given by
which is well behaved as long as the denominator is not zero.
 To illustrate the analytic finger formation, rational polynomial function
is used to generate density and potential structures. The bubble will be formed in a uniform background plasma by letting m = 0. The transition from plume to tip will occur at y = y0 using (21b) to give the analytic result
where = for > 0 and = for < 0.
 The electric potential is found from (24) with the asymptotic forms given as for and dq for . Two examples of ionospheric bubble structures will be considered. First, a bubble is formed by attaching the top half of the hole to the trough so that y0 = 0. This conductivity perturbation is shown in Figure 5a for the model parameters of a = −0.9 and b = 3. A numerical solver (Appendix A) yields the corresponding electric potential of Figure 5b. The potential is primarily one-dimensional for y ≪ y0 and it rapidly decays as for y ≫ y0. For comparison, the analytic potentials for the trough (Figure 5c) and for the hole (Figure 5d) are provided. For this example, the electric potential of the trough is much larger than the electric potential for the hole.
 The transition function is found numerically from (30) using the potentials given by Figures 5b, 5c, and 5d. The computed transition function (Figure 6) is dominated by monotonic variations in the y-direction. The function is also anti-symmetric about the y = 0 line around the value g = . In this case one can define a simplified transition function that does not depend on the horizontal coordinate or the -coordinate direction so g(, − 0) ≈ g( − 0) = 1 − g(0 − ). One function with these properties is
and this function will be used for computing the analytic approximations to the bubble potentials. This approximation is validated using numerically derived potentials.
 An accurate analytic representation of the electric potential is given by (29) with the transition function (33). With this simple transition, the potential associated with the trough leaks into the region of the semihole at the top of the bubble. The analytic potential is given by
The numerical and analytic representations of the bubble potential are similar in both shape and magnitude (Figure 7 especially near the y-axis where the potential has the largest affect on the transport of plasma gradients. The presence of the trough enhances the electric potential (and electric field) at the head of the bubble above potential that would be present with only the circular portion of the bubble. This has a strong effect on the evolution of the bubble by transport processes.
 The bubble potential drives the plasma transport to change the density of the plasma. The incompressible continuity equation is given by
where the plasma velocity is computed from the electric potential using
Normalizations of the potential, derivatives, velocities, and time are given by = , = r0, ≡ , and = t with the negative sign for the electric field used because with downward gravity ETx < 0. With these normalizations, the continuity and plasma drift velocity combined to give the rate of change for the integrated Pederson conductivity
The gradient of the potential (i.e. electric field) must be orthogonal to the gradient in the Pederson conductivity for the densities to change. In the trough portion of the plasma bubble, the horizontal (x-direction) density gradients are primarily aligned with the horizontal gradients in the potential (as in Figure 2) but the transition function provides some vertical gradients in the potential that will couple with the horizontal gradients in the tail of the plasma bubble just below the hole structure. The primary driver of evolution for the plasma bubbles is the horizontal gradients electric potential near the tip which couple to the vertical gradients at the top of the bubble structures.
 The computed time variation in the Pedersen conductivity for both numerical and analytic solutions are illustrated by the example in Figure 8 for a y0 = 0. In all regions, the change in conductivity is less than zero with a maximum change at the top of the bubble representing the rise of the bubble structure. The change at the sides of the bubble that is smaller in magnitude represents widening of the bubble as it rises. The electric potential simultaneously lifts the bubble and causes ballooning of the bubble structure. Because in this particular example, the electric potential associated with the trough is much larger than the potential associated with the hole (see Figure 5), the evolution of the bubble is primarily determined by the fringing fields of the trough that “leak” in to the region of the top of the bubble. With this process, the head of the bubble will be come wider until its potential structure dominates that of the tail portion. This ballooning process is often observed with numerical simulations of rising plasma bubbles in the ionosphere [Zalesak et al., 1982].
 Consider a second bubble which has the tail attached away from the center of the hole to emphasize the head. Figure 9 illustrates the bubble structures for the parameters a = −0.9, b = 3, and the normalized y0 = −1. The hybrid structure of a trough attached to a hole yields a realistic description for a plasma bubble that has ballooned at the top (Figure 9). The numerical solver provides the electric potential for the analytic bubble structure is computed and is shown in Figure 9. The hybrid potential based on (32) agrees very well with the computation in all regions (Figure 9). For reference, the analytic potential associated with just the tip of the bubble is given in Figure 9. It turns out that this bubble tip potential is the primary driver of the transport for this plasma bubble.
 Examination of Figures 8b and 8c indicates that the trough portion of the plasma bubble has the density gradients that are aligned with the gradients in the potential so there will be no change in the plasma density in this region where y < y0. Consequently, the primary driver of evolution for the plasma bubbles is the electric potential near the tip of the bubble structures.
Figure 10 provides a comparison of the Pedersen conductive time derivative (29) for the analytic model bubble shown in Figure 9 for both the numerical potential (Appendix A) and the hybrid analytic potential model (24). These two descriptions of the potential yield a density reduction at the top of the bubble because as the bubble rises, the low density portion of the bubble is transported upward. The potential provided by the tail portion of the bubble yields negligible density fluctuations.
 From this study it is concluded that the transport potential from a plasma bubble can be estimated with a weighted sum of the analytic potentials of the circular portion and the linear portions of the plume structure. The potential of the circular portion of the bubble is most easily found by fitting a plasma density holed such as given by (21b) and then using the fit parameters to describe the potential with the analytic representation of (19b). The potential of the tail is found directly from the integral equation (15). After estimating the transition plane between the tail and the head of the bubble, (33) is used to provide the full expression for the analytic electric potential. This procedure provides the basis for the plasma transport algorithm developed a companion paper. This discussion justifies the quasi-analytic approach illustrated in Figure 1.
 The analytic formula for the electric provides insight into the evolution of an equatorial plume. If the head of the bubble is about the same diameter as the tail portion as shown by Figure 5, the electric potential of the tail or trough portion is much larger than the potential contribution from the head portion. In this case, the tail-portion potential leaks into the head region and causes ballooning at the top of the plume. The processes continue until the head potential grows to be larger than the potential contribution by the tail. At this time the expansion of the head stops and the bubble only continues to rise through this plasma.
4. Application of Exact Potential Solutions to Triggering and Bifurcations of Equatorial Bubbles
 The triggering of the Rayleigh-Taylor instability in the equatorial ionosphere is influenced by (1) the vertical density gradient at the bottom of the F layer, and (2) the magnitude and shape of the density perturbation that seeds the instability. The analytic formulas for bottomside growth rates only consider the first influence where a small perturbation excites the Rayleigh Taylor instability. Ionospheric disturbances that trigger bubbles may be comparable in size to the background value of integrated Pedersen conductivity. Consequently, the effect of bubble triggers will depend on the plasma gradients produced by the trigger disturbance to the background plasma.
 One application of the analytic solutions of the potential equation (7) is to determine the optimum structure for artificial excitation of bubbles. In a program called Colored Bubbles, Haerendel et al.  proposed that a chemical release that produced two density enhancements could be used to artificially trigger equatorial bubbles. Two barium releases were released over Brazil near the equator from a rocket traveling in the zonal direction perpendicular to the magnetic meridian. The releases were placed at the bottom of the F layer with a separation chosen produce an upwelling plasma drift between the two plasma clouds. Details on the photoionization of barium to yield enhanced plasma clouds is described by Bernhardt et al. .
 The Colored Bubbles concept will be tested using two density enhancements described by (21b) with parameter a > 0. Each enhancement is described by
The centers of the two enhancements are separated by distance 20 using the form
where the function g is given by (33). This analytic form was chosen because the log of the Pedersen conductivity is the weighted sum of the log for the each enhancement centered at ±0. The nonlinear equation for the electric potential (7) asymptotically satisfies (39) for 0 1 or 0 1 with the electric potential given as
where 2 (, ) = . Numerical simulations validate this asymptotic formula for the electric potential for all values of spacing parameter 0. The minimum and maximum for the electric potential function occur where = 0 at ≡ M = ±(b − and = 0. Figure 3c illustrates a single enhancement with a = and b = 4 where the maximum and minimum potentials are located at M = ± ≅ 0.76. The analytic derivates of (40) are used for the electric field. As the parameter 0 is varied, the electric field between the two clouds changes.
 The optimum location for the two barium cloud plasma enhancements occurs when the separation yields the maximum potential gradient or electric field. If the two clouds are separated using 0 ≈ M, the positive potential from one cloud will cancel the negative potential of the other cloud so that the electric field at the center of the two clouds is expected to be near zero. If the clouds are separated by a greater distance (0 > M), the electric potentials from the two clouds will not cancel and the central electric field will be large.
 The effects of separation of the clouds are illustrated in Figures 11 and 12 for two model enhancements with parameters a = , b = 4, and m = to represent barium clouds released on the bottom side of the F layer. Zero central electric field is obtained for a separation parameter 0 = 0.88M. The electric potential (Figure 11b) is flat in the region between the two plasma clouds (Figure 10a). Using (37), the rate of change for the integrated Pedersen conductivity is computed for the plasma clouds on the bottom of the ionosphere. Figure 11c shows that the density grows near the centers of the clouds indicating that the plasma enhancements are falling and that the rate of change is zero in the region below the clouds. As predicted from the analytic potential model, the electric field between the clouds is zero because the electric potential from the right could cancel the electric field from the left cloud.
 When the spacing between the clouds is increased to 0 = 1.44M the central potential gradient is a maximum yielding an electric field that lifts the plasma. Figure 12 shows plasma enhancements that are optimally spaced for maximum lifting of the bottomside ionosphere for triggering of bubbles. The positive and negative rates of Pedersen conductivity reduction (Figure 12c) shows that the plasma will rise in the between the clouds and to fall at the near the centers of each cloud. These results use analytic derivates of the potential equation (40) to determine the optimum spacing between the plasma clouds for the strongest effect. Similar results were obtained by Çakır et al.  using a full numerical solution for the electric fields between the two barium clouds.
 The final application for the exact solutions of the electric potential associated with a plasma density structure is the formation of bubble bifurcations. Bifurcations are formed when the top of the bubble flattens to the point that the center does not rise as a fast as the regions to the left and right sides. To illustrate this process, the analytic model given by (38), (39), and (40) is used with parameters representing a plasma depression (a = −1/3, b = 3) in a uniform plasma (m = 0). The spacing between the Pedersen conductivity holes is adjusted using the parameter 0 until a split in the rate of density reduction is observed. This occurs when 0 = 0.6 M as shown in Figure 13. With this spacing, the electric potential is flat and the horizontal electric field is monotonic over the center region. The bifurcation only forms because of a slight increase in the Pedersen conductivity at the center near the top edge of the bubble. This yields a dip in the vertical transport flux at that location and the maximum rise of the structures is in two regions on either side of the bubble center. At this point, a bifurcation is initiated.
 As the spacing between the conductivity holes is increased, a splitting forms near the centers of both the electric potential and the conductivity. When the spacing is increased to 0 = M, the bifurcation process is fully developed (Figure 14).
 In summary, exact analytic solutions have been found for the nonlinear potential equation commonly used for determination of electric fields for ionospheric plasma irregularities. These solutions provide easy means to calculate the distributions of plasma conductivity associated with analytic models for the electric potential. These solutions can represent plasma depletions or enhancements depending on the model parameters. The analytic examples demonstrate that a large family of density structures can correspond to identical electric potentials. Also, if the amplitude of a potential is too large, there will not be any corresponding electron density structure. One use of the analytic solutions is to test numerical techniques to solve for the electric potential associated with arbitrary distributions of electron density under the influence of gravity, winds, and ambient electric fields. This is described in the next section.
5. Testing of Numerical Solvers With Analytic Formulations
 The numerical solution of nonseparable elliptic equations illustrated by (7) has been discussed by many authors (see review by Englund and Helsing ). Numerical solutions to this type of equation are influenced by (1) the finite difference approximations to the derivatives, (2) the boundary conditions, and (3) the convergence error for iterative solvers. Closed form solutions are often used for validation of the numerical solvers for both function and accuracy. These closed form solutions are difficult to find because nontrivial solutions of (7) contains both first and second order derivatives for either dependent variable Σp or . In absence of closed form solutions, numerical computations must be tried over a wide range of grid spacings and with many convergence criteria to determine the utility of the numerical solution algorithms.
 The analytic solutions (19b) and (21b) to the potential equation (7) have provided provide test cases for the numerical solutions of this type of equation. The parameters a, b, and m can be adjusted to give a wide range of plasma conductivity structures to look at (1) the effects of gradient scale lengths that approach grid resolution distance or (2) the boundary specifications. A direct numerical solver described in Appendix A was used to calculate the electric potential from (7) and the results were compared with the exact solution. The Pedersen conductivity was specified using the second, two-dimensional analytic model (21b) with the parameters a = −0.53, b = 4, and m = 1. The analytic potential (1)(, ) = from (19b) is compared in Figure 15 with the numerical solutions for the potential with a variety of boundary conditions.
 For uniform boundary densities as illustrated in Figure 3, doubly periodic, doubly fixed (or Dirichlet) (with the boundary potential set to = 0), or doubly derivative (or Neumann) (with ∂/∂ = 0 and ∂/∂ = 0 normal to each boundary) specifications work well. Because of the nonuniformities for the Pedersen conductivities at the boundaries in Figures 4 and 14, care is required in the selection of the boundary conditions for numerical solutions. The Pedersen conductivity and corresponding analytic potential are shown in Figures 14a and 14b, respectively. As illustrated by Figures 14c and 14d, inaccurate solutions are obtained for the potential with boundaries specified as doubly periodic and doubly fixed/Dirichlet ( = 0), respectively. These two boundary conditions force equal potentials at the top and bottom where the real densities are different. Neumann (i.e., zero derivative) boundary conditions yield a useful solution (Figure 15e). Another accurate solution is obtained by using a Neumann (i.e., zero derivative) boundary at the bottom, a Dirichlet (i.e., zero potential) boundary at the top and periodic boundaries at the sides of the solution space (Figure 15f). Table 2 lists the maximum potential for each solution using a 64 × 32 grid. All of the numerical solutions with the correct shape (Figures 14e and 14f) yield a computed potential that is about 11% less than the actual values. This reduction is the result of the finite difference approximations for the derivatives in (7).
Table 2. Error Analysis for Numerically Derived Potentials
 As the number of mesh points is increased, the numerical solution becomes more accurate. Table 3 shows the effect of the grid size on the maxima of the computed potentials. The numerical solutions use an nx by ny grid in the - and -directions, respectively. In all cases, the doubly derivative or Neumann boundaries yield slightly better solutions than the mixed boundary solution with doubly periodic in the -direction and Neumann/Dirichlet (derivative/fixed) boundary values in the -direction. These examples illustrate that the analytic solution pair (19b) and (21b) provides an easy way for testing (1) the utility of the numerical solutions with various boundary conditions and mesh sizes and (2) that the error in the numerical solution vanishes as the grid becomes denser.
Table 3. Mesh Size Affects on Numerically Derived Potentials
Number of Cells in -Direction: nx
Number of Cells in -Direction: ny
Maximum Potential, % Error
 The analytic solutions provide a quantitative test of numerical algorithms for the nonseparable PDE's that describe electric potentials in the ionosphere. All numerical solvers use the approximations of (1) finitely spaced solution samples (2) boundaries at finite distances. The analytic solution provides infinite spatial resolution with no boundary limits. Test cases can be set up to show the effects on the numerical solutions of boundary placement and boundary types as well as discrete sampling of the coordinate system.
 Exact analytic solutions have been found for the nonlinear potential equation commonly used for determination of electric fields for ionospheric plasma irregularities. These solutions provide an easy means to calculate the distributions of plasma conductivity associated with analytic models for the electric potential. These symbolic solutions can represent plasma depletions or density enhancements depending on the model parameters. With these solutions, it is not necessary to have a numerical potential solver to understand the physics of gravity driven polarization of representative plasma irregularities.
 The exact solutions have been found for a nonlinear potential equation without resorting to small amplitude assumptions or to numerical treatments. The analytic examples demonstrate that a large family of density structures can correspond to identical electric potentials. Also, if the amplitude of a potential is too large, there will not be any corresponding electron density structure.
 The exact solutions have been applied to bubble structures to show the origin of the transport from various parts of the bubble. The electric potential from the head of the bubble is the primarily driver of the vertical transport. The leakage of the electric potential from the tail of the bubble into the region of the head is responsible for the horizontal expansion or “ballooning” of the head. This ballooning continues causing the size of the head to grow until the tail electric fields no longer affect the head. At this point, the ballooning stops and the bubble rises without any widening of the head.
 Artificial triggering of equatorial bubbles was examined with two density enhancements that represent barium cloud releases on the bottom of the ionosphere. The analytic formulations yield the critical spacing of two clouds for maximum induction of upward rise of the plasma between the clouds. The same computation for two density reductions yields the critical spacing for the onset of equatorial bubble bifurcation. There are many more applications of the exact solutions that can yield insight into the dynamics of equatorial bubbles. These will be considered in a follow-on paper where transport is combined with the electric potential solutions as shown in Figure 1.
 The last use of the analytic solutions is to test numerical techniques to solve for the electric potential associated with arbitrary distributions of electron density under the influence of gravity, winds, and ambient electric fields. With realistic analytic solutions for plasma density structures, the influence of grid spacing on numerical accuracy for a controlled range of density gradients and boundary conditions can be studied.
 The next step to complete the quasi-analytic bubble model is to use the exact forms of the electric potential derived in this paper to provide transport of the plasma structures. This will be described in a future paper where the potential given by (19a) and (19b) is used to produce plasma bubbles that will be formed consistently with the Pedersen conductivity relation (21a) and (21b).
Appendix A:: Numerical Solution of the Potential Equation Using a Direct Solver
 Numerical solutions are required when conditions of simplified geometries or uniform flows yield a complicated, nonlinear partial differential equation for the electric potential. In Cartesian coordinates, the nonseparable elliptic equation that describes the electric potential is given by (7) and is repeated in equivalent form here
The conductivity function ΣP(, ) is known and the potential function (, ) is found as a numerical solution to the equation (A1).
 This equation is converted into a set of linear equations using the usual finite difference approximations to the derivatives given by
The Pedersen conductivity is sampled in the uniform solution grid spaced by Δx and Δy to form the array variables ΣPi,j with (i = 1, 2, …, M) and (j = 1, 2,…, N). To complete the solution, boundary conditions of periodic, fixed/Dirichlet, derivative/Neumann or mixed form are provided.
 The unknowns i,j for (i = 1, 2, …, M) and (j = 1, 2, …, N) are grouped into linear arrays given by
The resulting linear system can be written as an extended tri-diagonal matrix
where the M × M block matrices Aj, Bj, and Cj are functions of the finite difference parameters and the Pedersen conductivity samples ΣP i,j. The matrices A1 and D1 are needed for periodic boundaries in the y-direction. The string of matrices [D1, D2, D3, …, DN-1, DN] allow inclusion of an additional condition on the potential such as
This condition arises when the addition of a constant to a solution also yields solution. The nonuniqueness occurs if the boundary conditions are completely periodic and/or specified by constant derivatives (i.e., Neumann). With these types of boundary conditions, (A6) prevents the square matrix in (A5) from being singular and a numerical solution can be obtained. Finally, the right side of (A1) and boundary conditions are contained in the linear arrays Yj.
 The algorithm for solving (A5) is a generalization of the Thomas Algorithm for scalar tridiagonal systems [Dahlquist and Bjorck, 1974]. Initialize with new matrix variables
Continue with the equations
for the index in the range i = 2, …, N − 1. The next operations define a new set of variables starting with S′N = 0, b′N = I, where I is the M × M identity matrix, and continue with
for the index “i” in the range i = 1, …, N − 1. The solution for j = N is given by
The remaining solution vectors are found from
where the index “i” is given by the range i = 1, …, N − 1. This algorithm was used to provide the numerical potential solutions illustrated in Figure 5 and the data given in Tables 2 and 3.
Appendix B:: Exact Solutions to the Elliptic Potential Equation in Cartesian Coordinates
 An analytic solution to the non-separable elliptic equation that relates the Pedersen conductivity to the electric potential in the ionosphere was given by (16). A localized plasma structure may become polarized in a gravitational, neutral wind or electric field flow field. The equation describing this polarization has been given by (4) as
In Cartesian coordinates, (B1) was written as (7) assuming that only a horizontal equivalent electric field ETx was present. Here, the total electric field that drives the potential in the general case is assumed to have the from
where and are the unit vectors in the x- and y-directions and α is the angle between the equivalent driving field and the x-axis. In Cartesian coordinates, this equation becomes
where the normalizations = Φ/(r0 ET), = x/r0, and = y/r0 have been used. A general solution can be found using separation of variables with
With these substitutions into (B2), the solution for the derivative of the Lx is found to be
The functions Gy() and H() are chosen so that only a function of remains. Examination of the denominator of (B5) requires that
to eliminate one of the functions of . With this substitution, (B5) becomes
The -dependence in H() is eliminated by forcing H′(y)/H(y) ≡ m with the substitution
This result (B9) yields the second equation in (13) if F() = (a0 + a1)G(∣∣) and α = 0. To summarize, the one dimensional potential function
is found from is found from (B3) and (B6) for an external electric field ET making an angle α with the horizontal axis. The corresponding Pedersen conductivity function is found by substituting (B8) and the integral of (B9) into (B4) with the result
Rotation of the coordinate axes (, ) by α so that ET is horizontal will yield solutions for obliquely oriented potential functions.
 The solution in (B11) is only valid if α ≠ π/2. For the case that cos α = 0, (B5) becomes
The only substitutions that eliminate the -variations in (B12) are
where C1 and C2 are constants. The resulting log-density function is
To summarize, for the special case that the ET vector is aligned with the -axis (α = π/2), a potential function of the form
is produced by a Pedersen conductivity variation with the form
Rotation of the solutions in (B14) and (B15) by π/2 yield solutions for a horizontal electric field ET of the form
These solutions co-exist with the horizontal electric field ET (α = 0) solutions of (B10)
Thus with a horizontal external electric field (or equivalently a vertical gravity or neutral wind force), two separate families exact solutions for potentials and densities are available with (B17) and (B18). With rotations the equations (B10) and (B11), a large number of exact oblique solutions can be generated. The example illustrated in Figure 2 is only a small of sample of the analytic results for Cartesian coordinate geometry.
Appendix C:: Exact Solutions to the Elliptic Potential Equation in Cylindrical Coordinates
 A second general solution to the potential equation can be derived by using cylindrical coordinates by selecting either radial or axial symmetric for the function representations. As defined previously in (4) and (B1), the elliptic equation describing the relation between the Pedersen conductivity (ΣP) and the electric potential (Φ) is given as
Assume that ambient forces yield motion in the y-direction by an equivalent electric field ETx in the x-direction given by
The cylindrical (r, , z) coordinates are related to Cartesian coordinates by
and the unit vector in the x- direction is
In the cylindrical coordinate system, the potential equation becomes
After normalization this equation is written as
where the previously defined normalizations = Φ/ETx and = r/r0 have been used.
 To obtain one class of radial-dependent solutions, apply separation of variables with the substitutions
Then, solve for Lr () in terms of functions that only depend on r. With the substitution of (C6) and (C7) into (C5), the solution for the first derivative of Lr() is
The first step in elimination of the dependences on the right side of (C8) is to let F() = cos with the result
Next, the dependent relation tan H′()/H() in (C9) is set equal to a constant “m.” The solution of tan θH′(θ)/H(θ) ≡ m is H(θ) = C1 sinmθ where C1 is a constant. With these substitutions, (C6) and (C7) become
 To obtain a complementary set of angular-dependent solutions, use separation of variables with the functions
and solve for () in terms of functions that only depend on . With (C13) and (C14), (C5) is rewritten as
The first step for elimination of functions of r from (C15) is to let G() = so (C15) simplifies to
The r dependence is removed from (C16) by letting H() = m so H′()/H() ≡ m where “m” is a constant. With this substitution, (C16) is further reduced to
In summary, if the potential has the form
and the Pedersen conductivity has the form
This is written out as the integral equation
This solution is given for completeness. There have been no useful applications of (C19) to the ionosphere because the initial potential grows with radius and the radial electric field extends indefinitely.
 This research was sponsored at the Naval Research Laboratory by the Office of Naval Research. The author thanks Steven T. Zalesak for critical comments on this paper.
 Amitava Bhattacharjee thanks the reviewers for their assistance in evaluating this paper.