Testing the Hill model of transpolar potential ... - Wiley Online Library

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. A12, 1467, doi:10.1029/2003JA010154, 2003

Testing the Hill model of transpolar potential saturation D. M. Ober and N. C. Maynard Mission Research Corporation, Nashua, New Hampshire, USA

W. J. Burke Air Force Research Laboratory, Hanscom Air Force Base, Massachusetts, USA Received 21 July 2003; revised 23 October 2003; accepted 31 October 2003; published 31 December 2003.

[1] The Hill model predicts that the transpolar potential (PC) saturates for strong solar

wind electric fields (ESW) and saturates at higher values for larger solar wind ram pressures (PSW) and/or lower ionospheric Pedersen conductances (P). Three months of Defense Meteorological Satellite Program (DMSP) measurements of polar cap potentials have been compared with predictions of the Hill model. Data from representative days that span a wide range of interplanetary and solar-cycle conditions were selected for presentation. In the comparison we augmented the Hill model by adding a constant term to the magnetospheric potential and expressing P in terms of the 10.7-cm solar radio flux (F10.7). Temporal variations in observed PC are in good agreement with the predictions of the augmented Hill model. Comparison of two events with significantly different average PSW but similar F10.7 and ESW found that DMSP observed higher potentials when PSW was larger. DMSP also observed higher potentials when P was lower, when comparing two events with significantly different F10.7 (solar minimum versus solar maximum) but similar PSW and ESW. Both comparisons agree with the predictions of the INDEX TERMS: 2736 Magnetospheric Physics: Magnetosphere/ionosphere interactions; Hill model. 2411 Ionosphere: Electric fields (2712); 2409 Ionosphere: Current systems (2708); 2475 Ionosphere: Polar cap ionosphere; KEYWORDS: Hill model, transpolar potential, polar cap potential, saturation, MI coupling Citation: Ober, D. M., N. C. Maynard, and W. J. Burke, Testing the Hill model of transpolar potential saturation, J. Geophys. Res., 108(A12), 1467, doi:10.1029/2003JA010154, 2003.

1. Introduction [2] The transpolar potential (PC) is a key parameter for understanding solar wind-magnetosphere-ionosphere (SMI) coupling. Observations show that as the solar wind electric field (ESW) increases PC increases and eventually saturates for large ESW [Reiff and Luhmann, 1986; Russell et al., 2000; Russell et al., 2001]. However, the mechanism by which the solar wind, magnetosphere, and ionosphere interact to produce transpolar potential saturation remains an unanswered question. Following the conjecture of Hill et al. [1976], Siscoe et al. [2002a] developed a theoretical model of SMI coupling that predicts how and when PC saturates. This paper reports on tests of Hill-model predictions using PC measurements by Defense Meteorological Satellite Program (DMSP) satellites, combined with observations of solar-wind drivers from the ACE and WIND spacecraft. [3] Our knowledge of PC behavior mostly derives from empirical models. Boyle et al. [1997] developed a model of the steady state polar cap potential using DMSP measurements of PC acquired between 1987 and 1990. Their analysis provided no empirical evidence for PC saturation for large southward interplanetary magnetic Copyright 2003 by the American Geophysical Union. 0148-0227/03/2003JA010154

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field (IMF), probably due to their data-selection criteria. Their most restrictive selection criterion required 4 consecutive hours of quasi-steady IMF conditions, which eliminated many of the strong IMF events. Nor did they find any dependence of PC on the solar wind ram pressure (PSW). However, they did find that PC strongly depended on the solar wind flow velocity (VSW). In the Boyle et al. [1997] model PC is a function of the IMF orientation, IMF magnitude, and VSW, with the dependence of PC on the IMF magnitude being linear. Boyle et al. [1997] also noted that PC does not go to zero for vanishing IMF. They call this residual potential a ‘‘vis2 . A seasonal cous’’ contribution that depends on VSW dependence was also noted, with slightly higher potentials occurring in the winter than the summer hemisphere. However, the highest overall potentials occurred near the equinoxes. DMSP data acquired during 1987 – 1990 extends from solar minimum into solar maximum, with monthly averaged values of the 10.7-cm solar radio flux (F10.7) increasing from 71.5 to 235.4 during this time. [4] Weimer [1995, 2001] developed an empirical model of PC, based on measurements from the Dynamics Explorer 2 (DE 2) satellite from between August 1981 to March 1983. The best fit to the data occurred for PC proportional to BT2/3, where BT is the magnitude of IMF components in the Y-Z plane. Weimer [2001] attributed the

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OBER ET AL.: POTENTIAL SATURATION

nonlinear response of PC to BT on decreasing geoeffective widths of the solar wind coupling region. Weimer [2001] also found that PSW more strongly influenced PC than VSW. The DE 2 data were acquired during the declining phase of solar maximum with the monthly averaged values of F10.7 decreasing from 224.3 to 118.6 during this time. [5] In the following, we describe a theoretical construct of the Hill model suggested by Siscoe et al. [2002a] and compare predictions with PC values measured by DMSP satellites. After demonstrating general agreement between Hill-model predictions and observations, we explore the dependence of PC saturation on PSW and ionospheric Pedersen conductance P.

2. The Hill Model [6] Hill et al. [1976] proposed the following representation for the transpolar potential H (H being the Hill model prediction of PC), H ¼

M S ; M þ S

ð1Þ

where M is the potential that would result from unhindered dayside merging, and S is the saturation potential achieved when region 1 currents are strong enough to prevent any further increase in the merging rate. When M is large, H = S; conversely, when S is large, H = M. If M = S, then H = M/2 = S/2. [7] Siscoe et al. [2002a] quantified Hill’s concept by developing explicit theoretical formulas showing how the region 1 currents and H scale as functions of PSW, P, and ESW. The reconnection potential M depends on the effective length of the reconnection line, the size of the magnetosphere (scaled with PSW), the effects of magnetosheath compression, the strength of ESW, and the merging efficiency. The ionospheric saturation potential S is derived considering the Northern and Southern Hemisphere region 1 currents as flowing in two circular wire loops aligned with the dawn-dusk terminator plane. The paired current loops produce a magnetic perturbation field B that is southward at the dayside magnetopause and sunward (antisunward) over the northern (southern) polar cap. When B at the dayside magnetopause becomes a significant fraction of the dipole field, saturation occurs. The resulting equations for M and S expressed as functions of PSW, P, and ESW are, 1=6

M ðkVÞ ¼ 30:0ðkVÞ þ 57:6ESW ðmV=mÞPSW ðnPaÞ

ð2Þ

1=3

S ðkVÞ ¼

1600PSW ðnPaÞ ; P ðSiemensÞ

ð3Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ESW is defined as VSW B2Y þ B2Z sinðq=2Þ, BY and BZ are the Y and Z GSM components of the IMF, q is the 2 , n is the IMF clock angle in the Y-Z plane, PSW = nmpVSW solar wind number density, and mp is the proton mass. We have modified the equation for M as expressed by Siscoe et al. [2002a, 2002b] by adding a 30 kV baseline [Burke et al., 1999]. The 30 kV is the part of PC not attributable to ESW, such as viscous contributions. Boyle et al. [1997] found that the ‘‘viscous’’ portion of PC is proportional to 2 . For simplicity we adopt a constant value. VSW

[8] The two major sources of ionospheric Pedersen conductivity are solar extreme ultraviolet (EUV) radiation and particle precipitation. However, variations in the average P between solar minimum and maximum are largely due to variations in the solar EUV radiation. Thus ignoring contributions from particle precipitation, P is proportional to F10.7 [Robinson and Vondrak, 1984] where, P ¼ C0

pffiffiffiffiffiffiffiffiffiffi F10:7 ;

ð4Þ

with the constant of proportionality C0. Since we are only interested in understanding the variability of the potential rather than its absolute value, the exact value of C0 is not critical. Choosing C0 = 0.77 fits the observations well and allows P to vary from about 5 to 12 Siemens between solar minimum and maximum conditions. [9] Combining equations (1) –(4) allows us to compare predictions of the Hill model with observed potentials. Specifically, the dependence of the PC saturation limit on PSW and P can be tested. From equation (3) we expect that PC will attain larger values when PSW is high and/or P is low (low F10.7).

3. DMSP Observations [10] We have examined DMSP drift meter measurements acquired during the months of October 1995, May 1999, and March 2001. However, only DMSP F12 and F13 data from 18– 19 October 1995, and DMSP F12, F13, F14, and F15 data from 27 –31 March 2001 will be presented. DMSP satellites are three-axis stabilized spacecraft that fly in circular polar orbits at an altitude of 840 km. The orbital planes are sun-synchronous. The ascending nodes of the DMSP F12 and F13 were near 2130 and 1740 geographic local time, respectively, during October 1995. The ascending nodes of the DMSP F12, F13, F14 and F15 were near 1950, 1800, 2030, and 2110 geographic local time, respectively, during March 2001. Since the mid-1980s the scientific payloads of all DMSP satellites have included ion drift meters that face in the ram direction to measure the cross-track components of plasma motion in the vertical and horizontal directions. These measurements are combined with IGRF model values of the local magnetic field to calculate the electric field component along the spacecraft trajectory. Integration of this electric field component yields the potential distribution encountered by the satellite. The best estimate of PC comes from the DMSP F13 satellite whose orbital plane is near the dawn-dusk local time meridian. Owing to the offset between the geographic and the geomagnetic poles, these satellites experience a wide range of magnetic latitude (MLat) and local times (MLT). Depending on the size of the polar cap and the UT time of day, these spacecraft may sample a large or small fraction of the applied PC. [11] Figure 1 shows observations of the solar wind and transpolar potential for 27– 31 March 2001. The top four panels show ACE solar wind observations of GSM BZ, the IMF magnitude, PSW, and ESW. As seen in the figure, IMF conditions varied widely. The IMF BZ ranged between 40 and +40 nT, while PSW varied from a few to >50 nPa. Vertical black lines in the bottom three panels indicate


Figure 1. The 27 –31 March 2001 ACE and DMSP observations. DMSP observations show the minimum and maximum potentials sampled (vertical red lines) with predictions of the transpolar potential B (black) from Boyle et al. [1997], W (black) from Weimer [2001], and H (black) and S (blue) from the Hill [Siscoe et al., 2002a] model (equations (1) – (4)). minimum and maximum potentials measured by the DMSP spacecraft, plotted at the central times of each pass. [12] Overlaid on the observations are values of the transpolar potential predicted by the Boyle et al. [1997] (B), Weimer [2001] (W), and Hill (H) models. The ACE parameters used in these calculations were measured near the L1 point and have been appropriately lagged to their interaction time at the magnetopause. The blue curve in the bottom panel is S from the Hill model. The difference between H and S represents the level of saturation. Estimates of H and S are split evenly between positive and negative values. In the Boyle et al. [1997] model, B 2 (kV) = 104VSW + 11.7 B sin3(q/2) kV, where B is the magnitude of the IMF, and q is the IMF clock angle in the Y-Z plane. The split of B is offset 4.1 kV to the dusk convection cell which was observed on average to be slightly larger than the dawn cell. Our primary purpose

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for using the Boyle model here is to highlight the difference between a linear and nonlinear response of PC to BT. The Weimer [2001] calculations of W are based on a sphericalharmonic technique of pattern fitting. The Weimer [2001] model is available upon request from Dr. Daniel R. Weimer ([email protected]). Lagged ACE measurements served as inputs to the three models. Daily values of F10.7 were used in the calculation of H and S. [13] Observations from 27 – 31 March 2001 represent equinox and solar maximum conditions (F10.7 260). For most of the interval, all three model-predictions agree fairly well with the DMSP PC observations. The greatest disparity between models occurs during the strongly southward IMF period on 31 March 2001 when the Boyle model predicts larger potentials than either DMSP observes or the Hill and Weimer models predict. Note also that this is where the Hill models predict H S. [14] Two cautions should be kept in mind while evaluating DMSP observations of PC. First, the measurements take about 30 min during which interval the convection pattern may change. Second, DMSP satellites rarely sample the potential minima and maxima, and thus must be viewed as lower-limit estimates of actual PC values. [15] Figure 2 shows a second example of solar wind and ionospheric observations for 18– 19 October 1995 in the same format as Figure 1. In this cases the top four panels show observations from the WIND spacecraft at (176, 3, 13) RE GSE, again lagged to their interaction at the magnetopause when used in the model calculations. These observations are representative of equinox and solar-minimum conditions (F10.7 = 80.5). Notice the larger values of S due to the lower ionospheric Pedersen conductivity during this interval. When the IMF turns strongly southward at 2300 UT on 18 October, the Boyle model again predicts larger potentials than what was observed or predicted by the other models. In both Figures 1 and 2 the largest measured/ predicted PC values occur when the IMF is most strongly southward. [16] Figure 3 contains a scatter plot of the measured PC versus the predicted H potentials. Data from 27 and 31 March 2001 and 18 – 19 October 1995 periods are shown. Overlaid on the scatter plot is a unit line indicating where H = PC. If DMSP always measured the full PC and the data and theory were in perfect agreement, all of the plotted points would lay perfectly along this unit line. Instead, Figure 3 shows that all of the data points lie close to or below the unit line. [17] The Hill model predicts that the saturation potential is directly proportional to the cube root of PSW and inversely proportional to the P. Simply put, higher PSW and lower P leads to higher saturation potentials. To illustrate these dependencies in DMSP observations, Figures 4 – 6 plot measured PC as a function of ESW, for varying PSW and F10.7 conditions. The scatter plot in Figure 4 shows measured DMSP potentials versus ESW for 31 March 2001. Overlaid on the measurements are lines representing B (dotted), W (dashed), and H (solid) as functions of ESW for constant PSW, n, and VSW. Average values of F10.7 and PSW used in the calculations of predicted potentials are indicated at the top of the plot. For larger ESW the linear Boyle model predicts much higher PC values than were observed by DMSP or predicted by the other models.

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Figure 2. The 18– 19 October 1995 WIND and DMSP observations (same format as Figure 1).

functions of PSW, P, and ESW. For comparison with the DMSP observations we augmented the formulation of Siscoe et al. [2002a] by adding a constant term to the magnetospheric potential and expressing P in terms of F10.7. [20] The Boyle et al. [1997], Weimer [2001], and Hill [Siscoe et al., 2002a] models all compared well with the DMSP observations for typical solar wind conditions. But there are significant differences between the models during extreme solar wind conditions. In the empirical Boyle et al. [1997] model PC increases linearly with increasingly southward IMF. For moderate IMF conditions, the steadystate Boyle et al. [1997] model compares favorably with the observed potentials even for nonsteady IMF conditions, as Boyle et al. [1997] suggested. However, for strongly southward IMF conditions our comparison with DMSP shows that the Boyle model predicts larger than observed potentials. This effect is most evident in the 31 March 2001 observations (Figure 1). While DMSP seldom samples the full potential drop, it would be difficult to attribute all of the disagreement to DMSP measurement uncertainties. [21] The empirical Weimer [2001] model contains a nonlinear response to the strength of the IMF, with PC proportional to BT2/3. As a result, potentials predicted during strongly southward IMF intervals on 31 March 2001 are much smaller than those predicted by the Boyle model and in general fit the observations much better. The Boyle model likely failed to identify a nonlinear response of PC due to their restrictive data-selection criteria. Comparison of the Boyle and Weimer models with the observations serves to demonstrate clearly the nonlinear response of the transpolar potential to large southward IMF conditions. Hairston et al. [2003] also demonstrated the nonlinear response of the transpolar potential observed by DMSP during this time consistent with the predictions of the Hill model. [22] A visual inspection of Figures 1 and 2 indicates that the Hill and Weimer models predict just slightly different

[18] Figures 5 and 6, respectively, compare DMSP observations of 27 March 2001, and 18– 19 October 1995 with model predictions, in the format of Figure 4. Observations on 27 and 31 March occurred during intervals of similar F10.7 (solar maximum) but very different average PSW. Conversely, the observations of 27 March 2001 and 18– 19 October 1995 occurred during times of similar PSW but different F10.7 (solar maximum versus solar minimum). The lowest PC values were measured under low PSW and high P (Figure 5) conditions versus the opposing conditions of high PSW (Figure 4) or low P (Figure 6). We note that under solar minimum conditions the Weimer [2001] model predicts PC values that are too low (Figure 6).

4. Discussion [19] Hill et al. [1976, 1984] proposed a heuristic model of SMI coupling where the maximum convection potential achievable is limited by the smaller of either the ionospheric convection potential or the potential that would result from the dayside merging process. Siscoe et al. [2002a] extended Hill’s work by adding explicit, theoretically based formulas that indicate how the region 1 currents and PC scale as

Figure 3. The 27 and 31 March 2001 and 18– 19 October 1995 scatter plot of the observed versus predicted Hill potentials.


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transpolar potentials. The Hill model has been compared with 3 months of DMSP observations with results that are similar to those shown here. Mainly, the Hill model does a good job of predicting the variations in the envelope of the DMSP observed minimum and maximum potentials. Figure 3 further illustrates the strength of this comparison. While the constant C0 in equation (4) was selected to fit the observations, it is encouraging that the same value of C0

works for both solar minimum and solar maximum conditions. The combined results of Figures 1 – 3 strongly indicate that the PC does not scale linearly with increasingly larger southward IMF and that the Hill model closely approximates the variations in PC over a wide range of solar wind conditions. [23] The Hill model not only appears to predict the variations in PC well (Figures 1– 3), but it also suggests specific reasons as to why PC saturates. In the Hill model, PC is limited by feedback effects of associated region 1 currents on force-balance with PSW [Siscoe et al., 2002b]. This leads to a prediction that PC asymptotes at larger values for larger PSW and lower P. [24] A comparison of data in Figures 4 and 5 illustrates the PSW dependence of PC. On 31 March 2001 PSW was over twice as large as on 27 March 2001. The F10.7 index varied only slightly between the two days. For the same ESW strength, PC clearly reached higher values on 31 March than it did on 27 March. On 27 March, PC < 130 kV, while on 31 March, PC > 180 kV. This indicates that S was higher on 31 March when PSW was higher than it was on 27 March when PSW was lower. The Hill model closely depicts these differences. The Weimer model predicts nearly the same potentials for both days. (The minimal response in the Weimer model to PSW is due to the way that the multiple linear regression used to derive the coefficients behaves. Weimer is working on improving this aspect of the model along with several other things (D. Weimer, Mission Research Corporation, personal communication, 2003)). [25] Comparing data in Figures 5 and 6 demonstrates the dependence of PC with variations in P. The F10.7 index was 80.5 on 18 –19 October 1995 and 273.4 on 27 March 2001. This indicates over a factor of 3 difference in the solar UV intensities on the 2 days. P on 18– 19 October 1995 should have been nearly half of what it was on 27 March 2001. PSW varied only slightly between the two events. For the same ESW, PC clearly reached higher values on 18–

Figure 5. The 27 March 2001 observations in the same format as Figure 4.

Figure 6. The 18– 19 October 1995 observations in the same format as Figure 4.

Figure 4. The 31 March 2001 scatter plot of measured DMSP potentials versus ESW with the Boyle et al. [1997] (dotted), Weimer [2001] (dashed), and Hill [Siscoe et al., 2002a, 2002b] (solid) model potentials overlaid. The interval average solar wind ram pressure and F10.7 values are labeled at the top of the figure.

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19 March than it did on 27 March. PC was 170 kV on 18– 19 October. W clearly underestimates the observed PC on 18 –19 October 1995. The Weimer model is derived from observations acquired during the declining phase of a solar maximum which appears to be the reason why it is under predicting the potentials for the solar minimum conditions at this time. [26] It is remarkable that PC should depend on P at all. This indicates a strong feedback mechanism between the ionosphere and the magnetosphere and that PC is not determined by solar wind-magnetosphere coupling alone. A more detailed analysis of the observations is needed to completely understand PSW and P dependencies of PC. [27] Several other observations indicate that transpolar potential saturation occurs. Tsutomu [2002] has found that the Polar Cap Index (a proxy for PC) [Troshichev et al., 1988] also saturates for large values of ESW, in agreement with the predictions of the Hill model. Recently, results from Super Dual Auroral Radar Network (SuperDARN) found that the PC responded nonlinearly and saturated at larger values of ESW [Shepard et al., 2002]. Russell et al. [2000, 2001] using the AMIE procedure to estimate PC during several large magnetic storms, and found that it appeared to saturate during periods of greatly enhanced ESW. [28] Commenting on the October 1995 event, Reiff [1999] finds good agreement between observed DMSP measured potentials and the Boyle model except immediately following the southward IMF turning. The level of disagreement decreases with time as the observed potential slowly increases during the period of strongly southward IMF with a 2 hour time constant. Reiff [1999] attributes this increase to a speedup of the neutral winds. Assuming that the polar cap potential is current limited then as neutral-wind speeds increase the Pedersen conductivity decreases allowing the polar cap potential to further increase. Ridley et al. [2003] used a coupled magnetosphere-ionosphere-thermosphere model to quantify the influence of thermospheric neutral winds on the polar cap potential. They compare two simulations with and without the neutral winds for strongly southward IMF conditions lasting about 1 hour. The simulation shows that the inclusion of the neutral winds increased the polar cap potential at the end of the hour by approximately 6%. Had the IMF been held constant for a longer period the potential increase may have been greater. In most cases the driving conditions vary on shorter time scales than 2 hours.

5. Conclusion [29] The Hill model predicts that PC saturates for conditions of strongly southward IMF and asymptotes to larger saturation potentials for larger PSW and lower P [Hill et al., 1976; Siscoe et al., 2002a, 2002b]. To test these predictions of the Hill model, 3 months of DMSP observations have been analyzed. We find that observed variations in PC agreed well with theoretical predictions of the Hill model when a constant offset term was added to the magnetospheric potential. In particular, during times of strongly southward IMF observed PC did not scale linearly with increasingly southward IMF and appeared to saturate at levels that agree with predictions of the Hill model. DMSP

observations are also consistent with Hill-model predictions of saturation potentials that depend on both PSW and P. Comparing observations on 27 and 31 March 2001 during intervals of similar F10.7 conditions (solar maximum) but different average PSW it was shown that PC and S was larger on the day with the larger PSW, for the same ESW strength. Comparing observations on 27 March 2001 and 18– 19 October 1995 during times of similar average PSW but significantly different F10.7 (solar maximum versus solar minimum) it was shown that PC and S was larger for lower P, for the same ESW strength. A PC dependence with P indicates a strong SMI coupling and feed back mechanism during PC saturation. Future empirical models of PC need to use observations spanning a solar cycle in order to take into account a varying ionospheric Pedersen conductivity. [30] Acknowledgments. This work was performed under funding from the NASA GGS Program (part of the International Solar Terrestrial Physics Program). The authors thank N. F. Ness, Chuck Smith, and the Ace Science Center for making the ACE MFI data available and Dave McComas for providing the high resolution ACE SWEPAM data. We thank Ron Lepping for making the Wind magnetometer data available. [31] Arthur Richmond thanks George Siscoe and another reviewer for their assistance in evaluating this paper.

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OBER ET AL.: POTENTIAL SATURATION Troshichev, O. A., V. G. Andrezen, S. Vennerstrom, and E. Friis-Christensen, Magnetic activity in the polar cap—A new index, Planet. Sci., 36, 1095, 1988. Tsutomu, N., Saturation of polar cap potential by intense solar wind electric fields, Geophys. Res. Lett., 29(10), 1422, doi:10.1029/2001GL014202, 2002. Weimer, D. R., Models of the high-latitude electric potentials derived with a least error fit of spherical harmonics coefficients, J. Geophys. Res., 100, 19,595, 1995. Weimer, D. R., An Improved model of ionospheric electric potentials including substorm pertubations and application to the Geospace Environ-

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ment Modeling November 24, 1996, event, J. Geophys. Res., 106, 407, 2001.

W. J. Burke, Air Force Research Laboratory, 29 Randolph Road, Hanscom AFB, MA 01731-3010, USA. ([email protected]) N. C. Maynard and D. M. Ober, Mission Research Corporation, 589 West Hollis Street, Suite 201, Nashua, NH 03062-1323, USA. (nmaynard@ mrcnh.com; [email protected])