Table 14: How $NO$ $95th^{+}$ percentile events in the third trimester affect
birth outcomes given last-trimester-averaged $NO<78\mu g/m^{3}$

weeks $\leq$: | 0 | 1 | 2 | 3
---|---|---|---|---
A. Average ambient $NO$ level in trimester 3
| 30.1 | 33 | 34.6 | 35.1
B. Birth weight
$NO$ | -0.869 | 5.724 | 0.894 | -4.566∗∗
| (0.906) | (13.204) | (1.227) | (2.213)
$NO_{2}$ | 0.293 | 9.550 | -0.450 | 3.297
| (1.976) | (38.983) | (2.120) | (7.254)
$PM_{10}$ | 1.412 | -24.531 | 0.577 | 9.481
| (1.668) | (38.419) | (1.766) | (6.676)
$r^{2}$ | 0.468 | 0.552 | 0.468 | 0.498
Obs. | 195,790 | 2,673 | 165,227 | 29,473
C. Birth length
$NO$ | -0.009 | 0.509 | 0.105∗ | -0.276∗∗∗
| (0.043) | (0.649) | (0.058) | (0.105)
$NO_{2}$ | -0.004 | -0.488 | -0.056 | 0.331
| (0.091) | (1.886) | (0.101) | (0.319)
$PM_{10}$ | -0.042 | -1.580 | -0.089 | 0.185
| (0.086) | (1.753) | (0.094) | (0.301)
$r^{2}$ | 0.464 | 0.597 | 0.466 | 0.492
Obs. | 185,866 | 2,445 | 156,990 | 27,671

  * •

Notes: (1) Regression is based on the benchmark model in observations whose
average maternal $NO$ exposure in the last trimester is less than the $78\mu
g/m^{3}$. These observations are further classified in to sub-samples
according to the number of “high-level $NO$ pollution events”, which is
defined as weeks with average ambient $NO$ concentration higher than
$99th/95th$ percentile ($170\mu g/m^{3}$ and $110\mu g/m^{3}$ separately) of
the weekly $NO$ concentration in the last trimester. (2) The average ambient
$NO$ level in the last trimester for each sub-group is in Panel A. The
independent variables in Panel B and Panel C are birth weight and birth length
separately. (3) Cluster robust standard errors at grunnkrets level in
parentheses, (4) *** $p<0.01$, ** $p<0.05$, * $p<0.1$. (5) All pollutants are
in $\mu g/m^{3}$, birth-weight in gram, birth length in millimeter.

By combining Table 13 and Table 14, I conclude that for the sample with above-
average prenatal environmental $NO$ exposure levels in the last trimester,
long-term exposure to relatively high ambient $NO$ levels caused more harm
than occasional high ambient $NO$ events, whereas for observations with
relatively low average last trimester prenatal environmental $NO$ exposure
levels, occasional high ambient $NO$ events, if present, are also harmful to
birth outcomes.

##  9 Conclusion

In this paper, by using the variance in prenatal ambient air pollution
exposure levels among infants born within a specific calendar-month in the
same sub-zip-code area, I find that exposure to ambient nitric oxide ($NO$) in
the last trimester of pregnancy can significantly reduce birth weight and
length in Norwegian children born between 2000 and 2016. On average, each
standard deviation increase ($25.4\mu g/m^{3}$) in prenatal exposure to $NO$
resulted in a $1\%$ decrease in birth weight and a $0.3\%$ decrease in birth
length, which is similar to the effects of other studies on the effects of
ambient air pollutants. Pollution levels of other types of ambient air
pollutants, such as $NO_{2}$, $PM_{10}$ and $O_{3}$, appear to be safe for
Norwegian fetuses. Prenatal exposure to $SO_{2}$ in the environment appears to
have a negative effect on birth weight and length, but there are not enough
observations to make a precise estimate. I do not find an effect of prenatal
ambient air pollution exposure on APGAR scores.

A limitation of the study methodology in this paper is that the addresses of
pregnant women are updated annually, and I may not have been able to
accurately determine where the mothers resided during pregnancy. This may be
the reason why I find no significant effect of prenatal exposure to ambient
air pollution in the first two months, although the literature suggests that
most abnormal fetal development occurs in the last trimester. Using the
mother’s workplace address in the year of birth, which is regularly recorded
by social welfare and tax agencies, may be a solution for the future.

The affinity of $NO$ for hemoglobin may be a contributor to this adverse
effect. The diffusion of inhaled $NO$ into the blood of the pregnant woman
through the alveoli and capillaries oxidizes the Fe(II) of red blood cell
hemoglobin (Hb) to the Fe(III) state, forming methemoglobin (MetHb) and
impairing oxygen transport. As a result, the fetus is exposed to methemoglobin
through the placental barrier. Although ambient air pollution is associated
with diabetes, which in turn affects birth weight, I have not found any
evidence that ambient $NO$ has such a mechanism. It would be interesting to
further confirm the mechanisms by which environmental $NO$ pollution affects
the fetus. Though the literature has found a link between birth outcomes and
long-term health outcomes, it is not clear whether reduced birth weight and
length due to ambient air pollution can also affect long-term health outcomes.
Understanding the mechanisms by which pollutants affect birth outcomes can
help assess their long-term effects.

In addition, I find that both average ambient $NO$ in the last trimester and
occasional high ambient $NO$ pollution events can be harmful to the fetus.
Although ambient air quality in Norway is generally high and has been
improving in recent years, there are weeks with high ambient $NO$
concentrations that are harmful to the last trimester fetus. As found in the
literature, reductions in birth weight and length may have a negative impact
on the long-term health status of children. This poses a challenge to
environmental pollution management and policy development: not only to focus
on average pollutant levels, but also to pay attention to the containment of
short-term high pollution events.

Prenatal exposure to ambient $NO$ also has heterogeneous effects on different
groups. Consistent with the literature, I find that male infants are more
susceptible to environmental $NO$ pollution than female infants. Infants from
economically and/or ethnically disadvantaged families are more affected than
children from better-off families. This is not surprising because most
immigrants live in large cities like Oslo, where ambient $NO$ concentrations
are occasionally quite high during some weeks, and as mentioned earlier, the
marginal effect of ambient $NO$ on birth outcomes is greater when air
pollution levels are high. Another possible explanation is that less
privileged mothers are physically more vulnerable to the effects of ambient
air pollution. Due to the nature of their work, they may also engage in more
outdoor activities. Future studies may examine why newborns of poorly
conditioned parents are more vulnerable to ambient air pollution and how to
protect them. If infants’ long-term health is made worse by prenatal exposure,
and thus disadvantaged in the labor market in the future, they may be more
likely to be exposed to the same harmful environment - parental and offspring
air pollution exposure reinforcing each other and create a poor-health (and
also poverty) trap.

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## Appendix

Figure A1: Two babies both born in the 2nd quarter of 2010, but have different
prenatal exposure to $NO$

Figure A2: Given the same trimester-average concentration: relatively high
concentrations in all weeks vs. occasionally very high concentration events

Table A1: Air quality standards in different countries

Pollutant | averaging period |  | EU | USA | China | WHO
---|---|---|---|---|---|---
$NO_{2}$ |  |  | annual |  | 40 | 100 | 40 | 40
|  | daily |  | - | - | 80 | -
|  | hourly |  | 200 | 188 | 200 | 200
$PM_{10}$ |  |  | annual |  | 40 | - | 70 | 20
|  | daily |  | 50 | 150 | 150 | 50
$PM_{2.5}$ |  |  | annual |  | 25 | 15 | 35 | 20
|  | daily |  | - | 35 | 75 | 25
$SO_{2}$ |  |  | daily |  | - | 125 | 150 | 20
|  | hourly |  | 350 | 196 | 500 | -
$CO$ |  |  | daily |  | - | - | 4 | -
|  | 8-hour |  | 10 | 10 | - | -
$O_{3}$ |  |  | 8-hour |  | 120 | 137 | 160 | 100

  * •

Notes: (1) $CO$ in $mg/m^{3}$; all pollutants the other pollutants are
measured in $\mu g/m^{3}$. (2) There is no clear guideline for ambient $NO$ in
all these standards. (3) There are different safety standards in the U.S. and
the one in the is meant to “provide public health protection, including
protecting the health of “sensitive" populations such as asthmatics, children,
and the elderly”. (4) There are 2 classes of air quality standard in China and
I adopt the one for residential zones (“2nd class”). (5) Data source: European
Commission, European Environment Agency, U.S. Environmental Protection Agency,
Ministry of Ecology and Environment People’s Republic of China, World Health
Organization,

Table A2: Cross-validation $R^{2}$: how well the interpolation and fixed
effects predict the real weather condition (%)

p | miles | $humi$ | $precip$ | $press$ | $temp$ | $wind$
---|---|---|---|---|---|---
0.1 | 10 | 91.4 | 88.7 | 100.0 | 99.3 | 89.2
15 | 90.9 | 85.8 | 100.0 | 99.2 | 88.6
20 | 90.5 | 86.1 | 100.0 | 99.2 | 88.8
25 | 88.6 | 85.1 | 99.9 | 99.1 | 89.1
30 | 87.6 | 80.4 | 99.9 | 99.1 | 90.9
1 | 10 | 91.5 | 88.7 | 100.0 | 99.2 | 89.2
15 | 90.8 | 85.8 | 100.0 | 99.2 | 88.6
20 | 90.4 | 86.2 | 100.0 | 99.1 | 88.7
25 | 88.7 | 85.3 | 99.9 | 99.1 | 89.0
30 | 87.7 | 80.6 | 99.9 | 99.1 | 90.8
2 | 10 | 91.5 | 88.5 | 100.0 | 99.2 | 89.2
15 | 90.8 | 85.7 | 100.0 | 99.2 | 88.6
20 | 90.3 | 86.1 | 100.0 | 99.1 | 88.6
25 | 88.6 | 85.2 | 99.9 | 99.1 | 88.9
30 | 87.6 | 80.6 | 99.9 | 99.1 | 90.6
5 | 10 | 91.5 | 88.2 | 100.0 | 99.2 | 89.2
15 | 90.7 | 85.4 | 100.0 | 99.1 | 88.5
20 | 90.2 | 85.6 | 100.0 | 99.1 | 88.4
25 | 88.4 | 84.6 | 99.9 | 99.0 | 88.6
30 | 87.3 | 80.1 | 99.9 | 99.0 | 90.3

  * •

Notes: (1) As defined in Section 4.1, $e$ is the power of distance as defined
in equation; radius determines within which range the monitoring stations are
used for interpolation. (2) Column 3-7 contains the $R^{2}$ of in regression 2
for different meteorological conditions, which shows how well the real
conditions is explained by the interpolation and fixed effects.

Table A3: Month of childbirth in the sample and the rest part of the population  | Baseline sample |  | Population uncovered
---|---|---|---
Month | Freq. | Percent |  | Freq. | Percent
1 | 38,134 | 8.21 |  | 45,213 | 8.30
2 | 35,973 | 7.75 |  | 42,481 | 7.80
3 | 39,380 | 8.48 |  | 46,224 | 8.48
4 | 39,699 | 8.55 |  | 46,769 | 8.58
5 | 42,001 | 9.05 |  | 47,633 | 8.74
6 | 41,450 | 8.93 |  | 46,587 | 8.55
7 | 43,027 | 9.27 |  | 49,247 | 9.04
8 | 40,780 | 8.78 |  | 48,254 | 8.86
9 | 37,867 | 8.16 |  | 46,993 | 8.63
10 | 37,745 | 8.13 |  | 44,271 | 8.13
11 | 34,500 | 7.43 |  | 40,722 | 7.47
12 | 33,748 | 7.27 |  | 40,418 | 7.42
Total | 464,304 |  | 544,812

Table A3 shows the distribution of birth months in my sample. The month of
delivery is evenly distributed, with slightly more babies being born in the
summer months. Some parents may prefer to have their babies in early summer
(May-July), either to avoid the summer heat or to take parental leave between
summer vacations (so they can enjoy a "double summer vacation"). Information
such as the fact that babies born in the summer get more vitamin D and are
therefore healthier (which is essential in high latitude countries such as
Norway) may also be at play here. I also compared the demographics of parents
with “summer babies” and those without “summer babies” and found no
significant differences.

Table A4: Benchmark regression with standard errors clustered at different
levels

| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9)
---|---|---|---|---|---|---|---|---|---
cluster: | p | muni | family | q-g | m-p | q-p | s | m-s | q-s
A.Birth weight |  |  |
$NO$ | -1.387∗ | -1.387∗∗∗ | -1.387∗∗ | -1.387∗∗ | -1.387∗∗ | -1.387∗∗ | -1.387∗∗ | -1.387∗∗ | -1.387∗∗
| (0.068) | (0.008) | (0.035) | (0.027) | (0.026) | (0.033) | (0.011) | (0.011) | (0.017)
$NO_{2}$ | -0.259 | -0.259 | -0.259 | -0.259 | -0.259 | -0.259 | -0.259 | -0.259 | -0.259
| (0.902) | (0.841) | (0.881) | (0.891) | (0.881) | (0.890) | (0.876) | (0.876) | (0.887)
$PM_{10}$ | 1.329 | 1.329 | 1.329 | 1.329 | 1.329 | 1.329 | 1.329 | 1.329 | 1.329
| (0.455) | (0.244) | (0.412) | (0.318) | (0.413) | (0.315) | (0.364) | (0.421) | (0.331)
B.Birth length |  |  |
$NO$ | -0.052 | -0.052∗∗∗ | -0.052∗ | -0.052∗ | -0.052∗ | -0.052∗ | -0.052∗∗ | -0.052∗∗ | -0.052∗∗
| (0.141) | (0.000) | (0.087) | (0.065) | (0.089) | (0.069) | (0.035) | (0.041) | (0.023)
$NO_{2}$ | -0.023 | -0.023 | -0.023 | -0.023 | -0.023 | -0.023 | -0.023 | -0.023 | -0.023
| (0.816) | (0.710) | (0.784) | (0.770) | (0.779) | (0.769) | (0.804) | (0.800) | (0.792)
$PM_{10}$ | -0.004 | -0.004 | -0.004 | -0.004 | -0.004 | -0.004 | -0.004 | -0.004 | -0.004
| (0.967) | (0.944) | (0.962) | (0.956) | (0.961) | (0.956) | (0.952) | (0.954) | (0.943)

  * •

Notes: (1) The standard errors are clustered at level: p(postcode),
muni(municipality), family(children of the same mother), q-g(calendar quarter
and grunnkrets), m-p(calendar month and postcode), q-p(calendar quarter and
postcode), s(nearest monitoring station), m-s(calendar month and nearest
monitoring station) and q-s(calendar quarter and nearest monitoring station).
(2) The independent variables in Panel A and Panel B are birth weight and
birth length separately. (3) *** $p<0.01$, ** $p<0.05$, * $p<0.1$. (4) All
pollutants are in $\mu g/m^{3}$, birth-weight in gram, birth length in
millimeter.

Table A5: Air pollution exposure during the whole pregnancy

|  | birth weight |  | birth length
---|---|---|---|---
trimester | all | same | stay |  | all | same | stay
$1st$ | $NO$ | 0.958 | 1.181 | 0.884 |  | 0.022 | 0.017 | 0.049
| (0.773) | (0.970) | (1.093) |  | (0.036) | (0.046) | (0.052)
$NO_{2}$ | -1.568 | -2.520 | -2.226 |  | -0.065 | -0.072 | -0.133
| (2.058) | (2.582) | (2.765) |  | (0.097) | (0.127) | (0.125)
$PM_{10}$ | 1.151 | 1.826 | -0.396 |  | 0.105 | 0.088 | 0.108
| (1.882) | (2.192) | (2.600) |  | (0.091) | (0.103) | (0.120)
$2nd$ | $NO$ | 0.699 | 0.803 | 0.063 |  | -0.046 | -0.091 | -0.001
| (1.268) | (1.585) | (1.676) |  | (0.060) | (0.073) | (0.078)
$NO_{2}$ | 2.587 | 1.747 | 0.331 |  | 0.238∗ | 0.203 | 0.158
| (2.866) | (3.571) | (3.673) |  | (0.138) | (0.166) | (0.169)
$PM_{10}$ | 1.225 | 4.367 | 2.223 |  | -0.048 | 0.154 | -0.083
| (3.182) | (3.731) | (4.267) |  | (0.146) | (0.171) | (0.189)
$3rd$ | $NO$ | -1.010 | -0.777 | -0.692 |  | -0.063 | -0.066 | -0.048
| (0.858) | (1.052) | (1.102) |  | (0.040) | (0.049) | (0.052)
$NO_{2}$ | 0.786 | -0.003 | -0.566 |  | 0.071 | -0.024 | 0.023
| (2.168) | (2.718) | (2.801) |  | (0.102) | (0.133) | (0.127)
$PM_{10}$ | 1.610 | 3.475 | 3.984 |  | -0.019 | 0.101 | 0.077
| (1.952) | (2.402) | (2.657) |  | (0.104) | (0.136) | (0.125)
$r^{2}$ | 0.464 | 0.469 | 0.502 |  | 0.462 | 0.461 | 0.501
Obs. | 211,935 | 135,535 | 135,714 |  | 200,292 | 128,855 | 127,816

  * •

Notes: (1)The three trimesters are regressed in one model, not separately. (2)
$all$ means all observations in the working data is used; $same$ means mothers
lived in the same municipalities in the delivery year as where the children
were born, $stay$ keeps only those whose mothers resided in the same
grunnkrets in the delivery year and the year before delivery. (3) Cluster
robust standard errors at the grunnkrets level in parentheses. (4) ***
$p<0.001$, ** $p<0.01$, * $p<0.05$. (5) All pollutants are in $\mu g/m^{3}$,
birth-weight in gram, birth length in millimeter.

In Table A5, I include average ambient air pollution levels and weather
conditions throughout the third trimester of pregnancy in the baseline
regression model. The samples in columns (3) and (5) are the baseline sample
(“all”), while the infants in columns (4) and (6) (“same”) are born in the
same city as their mother’s residence in the year of delivery, as a robustness
check (as discussed in Section 5.).

When air pollution and weather conditions in the first two quarters are
included, the $NO$ coefficients in the last three months remain very similar
to the baseline regressions of birth weight and birth length. Although most of
the coefficients in Table A5 are not significant at the $10\%$ level, the
coefficients on $NO_{2}$ and $PM_{10}$ are less significant, i.e., the
magnitude of the effect is smaller and the standard error is larger. Based on
the findings in Table A5, I included only the last three months of air
pollution levels and weather conditions in the baseline analysis.

Table A6: The effects of ambient $O_{3}$ in the 3rd. trimester on birth
outcomes

| (1) | (2) | (3) | (4)
---|---|---|---|---
A. Birth weight |
$O_{3}$ | 0.911 | -0.141 | -0.121 | -0.335
| (0.955) | (1.143) | (1.370) | (1.383)
$NO$ |  | -0.985∗ | -1.006 | -1.796∗
|  | (0.529) | (0.769) | (1.038)
$NO_{2}$ |  |  |  | 2.808
|  |  |  | (2.528)
$r^{2}$ | 0.439 | 0.423 | 0.417 | 0.417
Obs. | 172,352 | 150,629 | 144,298 | 144,298
B.Birth length |
$O_{3}$ | 0.052 | 0.010 | 0.032 | 0.024
| (0.048) | (0.056) | (0.067) | (0.068)
$NO$ |  | -0.038 | -0.021 | -0.050
|  | (0.024) | (0.034) | (0.047)
$NO_{2}$ |  |  |  | 0.104
|  |  |  | (0.115)
$r^{2}$ | 0.445 | 0.432 | 0.424 | 0.424
Obs. | 162,243 | 142,259 | 136,416 | 136,416

  * •

Notes: (1) Regressions are based on the benchmark model but include only
$O_{3}$, $NO$ and $NO_{2}$ in order to examine the correlation between the
pollutants (2) The independent variables in Panel A and Panel B are birth
weight and birth length separately. (3) Cluster robust standard errors at
grunnkrets level in parentheses. (4) *** $p<0.01$, ** $p<0.05$, * $p<0.1$. (5)
All pollutants are in $\mu g/m^{3}$, birth-weight in gram, birth length in
millimeter.

Table A7: Excluding mothers exposed to high-level ambient air pollution in
3rd. trimester

| (1) | (2) |  | (3) | (4) |  | (5) | (6) |  | (7) | (8)
---|---|---|---|---|---|---|---|---|---|---|---
| $NO$ |  | $NO_{2}$ |  | $PM_{10}$ |  |  $NO_{2}$ & $PM_{10}$
| $<99p.$ | $<95p.$ |  | $<99p.$ | $<95p.$ |  | $<99p.$ | $<95p.$ |  | $<99p.$ | $<95p.$
A.Birth weight |  |  |  |
$NO$ | -1.896∗∗ | 0.756 |  | -2.342∗∗∗ | -2.717∗∗∗ |  | -1.240∗ | -1.448 |  | -2.298∗∗∗ | -2.012
| (0.784) | (1.151) |  | (0.693) | (0.954) |  | (0.707) | (0.920) |  | (0.785) | (1.224)
$NO_{2}$ | 0.765 | -0.445 |  | 1.750 | 0.693 |  | 0.472 | 1.626 |  | 2.933 | 1.139
| (1.886) | (2.118) |  | (1.906) | (2.551) |  | (2.017) | (2.545) |  | (2.207) | (3.336)
$PM_{10}$ | 1.140 | 0.537 |  | 1.716 | 0.745 |  | 0.871 | -0.287 |  | 0.592 | 1.429
| (1.620) | (1.763) |  | (2.079) | (2.995) |  | (1.620) | (1.863) |  | (2.274) | (3.583)
$r^{2}$ | 0.467 | 0.468 |  | 0.464 | 0.468 |  | 0.467 | 0.470 |  | 0.466 | 0.470
Obs. | 205,170 | 165,555 |  | 206,770 | 156,122 |  | 206,417 | 163,289 |  | 190,339 | 127,154
B.Birth length |  |  |  |
$NO$ | -0.053 | 0.082 |  | -0.085∗∗ | -0.099∗∗ |  | -0.053 | -0.012 |  | -0.082∗∗ | -0.055
| (0.038) | (0.055) |  | (0.033) | (0.045) |  | (0.034) | (0.043) |  | (0.037) | (0.054)
$NO_{2}$ | 0.007 | -0.056 |  | 0.068 | 0.008 |  | 0.014 | 0.020 |  | 0.124 | 0.006
| (0.087) | (0.101) |  | (0.084) | (0.111) |  | (0.092) | (0.117) |  | (0.097) | (0.145)
$PM_{10}$ | -0.054 | -0.096 |  | -0.046 | -0.117 |  | -0.017 | -0.128 |  | -0.078 | -0.086
| (0.083) | (0.094) |  | (0.099) | (0.137) |  | (0.083) | (0.097) |  | (0.107) | (0.163)
$r^{2}$ | 0.463 | 0.466 |  | 0.462 | 0.466 |  | 0.464 | 0.467 |  | 0.465 | 0.470
Obs. | 194,486 | 157,291 |  | 195,830 | 148,232 |  | 195,498 | 155,014 |  | 180,669 | 121,101

  * •

Notes: (1) Regressions are based on the benchmark regression model in babies
whose mother were not exposed to high-level ambient air pollution. Here $99p.$
and $95p.$ means that the weekly maternal exposure to certain pollutants in
the third trimester is all below 99th and 95th percentile. (2) Cluster robust
standard errors at grunnkrets level in parentheses, (3) *** $p<0.01$, **
$p<0.05$, * $p<0.1$. (5) All pollutants in $\mu g/m^{3}$, birth-weight in
gram, birth length in millimeter.

The regressions in Table A7 are for certain environmental contaminants in the
subsample where mothers are not exposed to any high levels of environmental
contamination events (weekly mean concentrations >$99th/95th$ percentile) in
the last trimester. For example, weekly prenatal environmental $NO$ exposures
in the last trimester ($11$ weeks total) in column (1) are below the $99th$
percentile. Columns (1) and (2) are copied from columns (1) and (5) in Table
12 for comparison purposes. The coefficient on $NO$ is no longer negative when
all $11$ weeks of prenatal exposure to $NO$ in the last trimester are below
the $95th$ percentile, and the exclusion of high levels of environmental
$NO_{2}$ and $PM_{10}$ pollution events does not affect the sign of the
coefficient on $NO$. This means that it is the high levels of $NO$, not
$NO_{2}$ or $PM_{10}$, that reduces the birth outcome.


# Theory of incompressible MHD turbulence with scale-dependent alignment and
cross-helicity

J. J. Podesta and A. Bhattacharjee  Center for Integrated Computation and
Analysis of Reconnection and Turbulence,University of New Hampshire, Durham,
New Hampshire, 03824

###### Abstract

A phenomenological anisotropic theory of MHD turbulence with nonvanishing
cross-helicity is constructed based on Boldyrev’s phenomenology and
probabilities $p$ and $q$ for fluctuations $\delta\bm{v}_{\perp}$ and
$\delta\bm{b}_{\perp}$ to be positively or negatively aligned. The positively
aligned fluctuations occupy a fractional volume $p$ and the negatively aligned
fluctuations occupy a fractional volume $q$. Guided by observations suggesting
that the normalized cross-helicity $\sigma_{c}$ and the probabilities $p$ and
$q$ are approximately scale invariant in the inertial range, a generalization
of Boldyrev’s theory is derived that depends on the three ratios
$w^{+}/w^{-}$, $\epsilon^{+}/\epsilon^{-}$, and $p/q$. It is assumed that the
cascade process for positively and negatively aligned fluctuations are both in
a state of critical balance and that the eddy geometries are scale invariant.
The theory reduces to Boldyrev’s original theory when $\sigma_{c}=0$,
$\epsilon^{+}=\epsilon^{-}$, and $p=q$ and extends the theory of Perez and
Boldyrev when $\sigma_{c}\neq 0$. The theory is also an anisotropic
generalization of the theory of Dobrowolny, Mangeney, and Veltri.

##  I Introduction

Phenomenological theories of incompressible MHD turbulence that take into
account the anisotropy of the fluctuations with respect to the direction of
the mean magnetic field $\bm{B}_{0}$ were pioneered by Goldreich & Sridhar and
others in the 1990s. The influencial and somewhat controversial theory of
Goldreich & Sridhar (Goldreich and Sridhar, 1995, 1997) established the idea
that the timescale or coherence time for motions of a turbulent eddy parallel
and perpendicular to $\bm{B}_{0}$ must be equal to each other and this unique
timescale then defines the energy cascade time. This concept, called critical
balance, leads immediately to the perpendicular energy spectrum
$E(k_{\perp})\propto k_{\perp}^{-5/3}$ and to the scaling relation
$k_{\parallel}\propto k_{\perp}^{2/3}$ describing the anisotropy of the
turbulent eddies.

The decade following the publication of the paper by Goldreich & Sridhar in
1995 was a time when significant advances in computing power were brought to
bear on computational studies of MHD turbulence. Simulations of incompressible
MHD turbulence during this time showed that when the mean magnetic field is
strong, $B_{0}^{2}\gg(\delta b)^{2}$, the perpendicular energy spectrum
exhibits a power-law scaling closer to $k_{\perp}^{-3/2}$ than
$k_{\perp}^{-5/3}$ (Maron and Goldreich, 2001; Müller et al., 2003; Müller and
Grappin, 2005). Motivated by this result, Boldyrev modified the Goldreich &
Sridhar theory to explain the $k_{\perp}^{-3/2}$ power-law seen in simulations
(Boldyrev, 2005, 2006). A new concept that emerged in Boldyrev’s theory is the
scale dependent alignment of velocity and magnetic field fluctuations whereby
the angle $\theta$ formed by $\delta\bm{v}_{\perp}$ and $\delta\bm{b}_{\perp}$
scales like $\lambda_{\perp}^{1/4}$ in the inertial range. This alignment
effect weakens the nonlinear interactions and yields the perpendicular energy
spectrum $E(k_{\perp})\propto k_{\perp}^{-3/2}$. Evidence for this alignment
effect has been found in numerical simulations of incompressible MHD
turbulence (Mason et al., 2006, 2008) and in studies of solar wind data
(Podesta et al., 2008, 2009).

The phenomenological theory of Galtier et al. (2005) can also be used to
explain the observed $k_{\perp}^{-3/2}$ energy spectrum. Using a slightly
modified critical balance relation that retains the $k_{\parallel}\propto
k_{\perp}^{2/3}$ scaling of the Goldreich & Sridhar theory, their model admits
the $k_{\perp}^{-3/2}$ energy spectrum (as well as other solutions). However,
the theory of Galtier et al. (2005) does not include the scale dependent
alignment that arises in Boldyrev’s theory and, more importantly, is seen in
the solar wind (Podesta et al., 2008, 2009).

The theories discussed so far (Goldreich and Sridhar, 1995, 1997; Boldyrev,
2005, 2006; Galtier et al., 2005) all assume that the cross-helicity vanishes
and, therefore, these theories cannot be applied to solar wind turbulence.
When the cross-helicity of the turbulence is nonzero it is necessary to take
into account the cascades of both energy and cross-helicity. A generalization
of the Goldreich & Sridhar theory to turbulence with nonvanishing cross-
helicity, also called imbalanced turbulence, has been developed by Lithwick,
Goldreich, & Sridhar (Lithwick et al., 2007). Other theories of imbalanced
turbulence have been derived by Beresnyak & Lazarian (Beresnyak and Lazarian,
2008) and Chandran (Chandran, 2008; Chandran et al., 2009). However, none of
these theories contains the scale dependent alignment of velocity and magnetic
field fluctuations seen in the solar wind. Therefore, to develop a theory that
may be applicable to solar wind turbulence it is of interest to generalize
Boldyrev’s theory to incompressible MHD turbulence with non-vanishing cross-
helicity. An extension of Boldyrev’s theory to imbalanced turbulence has been
discussed by Perez & Boldyrev (Perez and Boldyrev, 2009). The purpose of the
present paper is to develop a theoretical framework which generalizes the
results of Perez & Boldyrev and is consistent with solar wind observations.

Our theory is founded, in part, on two new solar wind observations presented
in this paper. The first is the observation that the normalized cross-helicity
$\sigma_{c}$, the ratio of cross-helicity to energy, is _scale invariant_ in
the inertial range. The second is the observation that the probabilities $p$
and $q$ that fluctuations are positively or negatively aligned, respectively,
are also _scale invariant_ , that is, these quantities are approximately
constant in the inertial range. Experimental evidence for the scale invariance
of $\sigma_{c}$ comes from solar wind observations by Marsch and Tu (1990) and
also Figure 1 below, and from numerical simulations (Verma et al., 1996; Perez
and Boldyrev, 2009). Evidence for the scale invariance of $p$ and $q$ is shown
in Figure 2 below. Assuming these quantities are all scale invariant we deduce
expressions for the energy cascade rates and the rms fluctuations that
generalize the results in (Boldyrev, 2006) and (Perez and Boldyrev, 2009) and
are consistent with the concept of scale dependent alignment of velocity and
magnetic field fluctuations, a concept neglected in other phenomenological
theories (Galtier et al., 2005; Lithwick et al., 2007; Beresnyak and Lazarian,
2008; Chandran, 2008). The resulting theory, which is founded on the concept
of scale-invariance and grounded in solar wind observations, contains the
theories of Boldyrev (2006) and Perez and Boldyrev (2009) as special cases,
but opens up a broader range of physical possibilities.

Consistent with numerical simulations and solar wind observations, in our
approach the fluctuations at a given point may assume one of two possible
states referred to as positively aligned
$\delta\bm{v}_{\perp}\cdot\delta\bm{b}_{\perp}>0$ and negatively aligned
$\delta\bm{v}_{\perp}\cdot\delta\bm{b}_{\perp}<0$. Each state is characterized
by its own rms energy $v^{2}$, alignment angle $\theta$, and nonlinear
timescale $\tau$. Positively aligned fluctuations have a characteristic
spatial gradient which determines their nonlinear timescale and negatively
aligned fluctuations have a different spatial gradient which determines their
nonlinear timescale. These timescales are estimated from the nonlinear terms
in the MHD equations as described in sections 2 and 3.

Section 2 describes the geometries of velocity and magnetic field fluctuations
that are either aligned ‘$\uparrow$’ or anti-aligned ‘$\downarrow$’ and these
are used to form estimates of the nonlinear terms in the MHD equations. From
this foundation, estimates of the energy cascade times are constructed in
section 3 and the theory of the energy cascade process is developed in section
4. The summary and conclusions are presented in section 5.

##  II Fluctuations in imbalanced turbulence

Consider velocity and magnetic field fluctuations measured between two points
separated by a distance $\lambda_{\perp}$ in the field perpendicular plane.
Let $\bm{v}$ and $\bm{b}$ denote the fluctuations in the plane perpendicular
to the local mean magnetic field, where $\bm{v}$ and $\bm{b}$ are both
measured in velocity units. Suppose that $\bm{v}$ and $\bm{b}$ are aligned
with some small angle $\theta>0$ and assume, as for Alfvén waves, that
$|\bm{v}|=|\bm{b}|$. Then $\bm{w}^{+}=\bm{v}+\bm{b}$ is nearly aligned with
$\bm{v}$ and $\bm{w}^{-}=\bm{v}-\bm{b}$ is nearly perpendicular to $\bm{v}$ as
sketched in Figure 1.

Figure 1: Geometry of the fluctuation vectors $\bm{v}$ and $\bm{b}$ for
positively aligned fluctuations (a) denoted by ‘$\uparrow$’ and negatively
aligned fluctuations (b) denoted by ‘$\downarrow$’. The gradient is
perpendicular to the velocity fluctuation $\bm{v}$. The magnitude of $\bm{v}$
for positively and negatively aligned fluctuations are $v_{\uparrow}$ and
$v_{\downarrow}$, respectively. The angles formed by $\bm{v}$ and $\bm{b}$,
$\theta_{\uparrow}$ and $\theta_{\downarrow}$, are both assumed to be small.

It follows from the identity $\bm{w}^{\pm}\times\bm{v}=\mp\bm{v}\times\bm{b}$
that

$w^{+}\sin\theta^{+}=v\sin\theta=w^{-}\sin\theta^{-},$ (1)

where $\theta^{+}$ is the angle formed by $\bm{w}^{+}$ and $\bm{v}$,
$\theta^{-}$ is the angle formed by $\bm{w}^{-}$ and $\bm{v}$, and $\theta$ is
the angle formed by $\bm{v}$ and $\bm{b}$. In addition,
$\theta^{+}+\theta^{-}=\pi/2$. Following Boldyrev (2006), suppose that the
gradient of the fluctuations is in the direction perpendicular to $\bm{v}$. In
this case,

$(\bm{w}^{-}\cdot\nabla)\simeq\frac{w^{-}\sin\theta^{-}}{\lambda_{\perp}}=\frac{v\sin\theta}{\lambda_{\perp}}$
(2)

and

$(\bm{w}^{+}\cdot\nabla)\simeq\frac{w^{+}\sin\theta^{+}}{\lambda_{\perp}}=\frac{v\sin\theta}{\lambda_{\perp}}.$
(3)

The time rate of change caused by nonlinear interactions is estimated from the
relations

$\frac{\partial}{\partial
t}\frac{|\bm{w}^{+}|^{2}}{2}\simeq\bm{w}^{+}\cdot(\bm{w}^{-}\cdot\nabla)\bm{w}^{+}$
(4)

and

$\frac{\partial}{\partial
t}\frac{|\bm{w}^{-}|^{2}}{2}\simeq\bm{w}^{-}\cdot(\bm{w}^{+}\cdot\nabla)\bm{w}^{-}.$
(5)

If $\bm{v}$ and $\bm{b}$ are aligned with some small angle $\theta$, then the
fluctuations are called “positively aligned” and denoted by ‘$\uparrow$’
(Figure 1a). Similarly, if $\bm{v}$ and $-\bm{b}$ are aligned with some small
angle $\theta$, then the fluctuations are called “negatively aligned” or
“anti-aligned” and denoted by ‘$\downarrow$’ (Figure 1b). For positively
aligned fluctuations, equations (2)–(5) imply

$\frac{\partial}{\partial
t}\frac{(w^{+}_{\uparrow})^{2}}{2}\simeq\frac{(w^{+}_{\uparrow})^{2}w^{-}_{\uparrow}\sin\theta^{-}_{\uparrow}}{\lambda_{\perp}}=\frac{(w^{+}_{\uparrow})^{2}v_{\uparrow}\sin\theta_{\uparrow}}{\lambda_{\perp}}$
(6)

and

$\frac{\partial}{\partial
t}\frac{(w^{-}_{\uparrow})^{2}}{2}\simeq\frac{(w^{-}_{\uparrow})^{2}w^{+}_{\uparrow}\sin\theta^{+}_{\uparrow}}{\lambda_{\perp}}=\frac{(w^{-}_{\uparrow})^{2}v_{\uparrow}\sin\theta_{\uparrow}}{\lambda_{\perp}},$
(7)

where $\theta_{\uparrow}$ is the angle formed by $\bm{v}$ and $\bm{b}$ and
quantities with the subscript $\uparrow$ describe positively aligned
fluctuations. It is clear from the middle term in equation (6) that the time
rate of change of $w^{+}_{\uparrow}$ depends on $w^{-}_{\uparrow}$, consistent
with the nonlinear terms in the MHD equations, although this dependence is not
immediately apparent in the last term in (6). For negatively aligned
fluctuations, equations (2)–(5) imply

$\frac{\partial}{\partial
t}\frac{(w^{+}_{\downarrow})^{2}}{2}\simeq\frac{(w^{+}_{\downarrow})^{2}w^{-}_{\downarrow}\sin\theta^{-}_{\downarrow}}{\lambda_{\perp}}=\frac{(w^{+}_{\downarrow})^{2}v_{\downarrow}\sin\theta_{\downarrow}}{\lambda_{\perp}}$
(8)

and

$\frac{\partial}{\partial
t}\frac{(w^{-}_{\downarrow})^{2}}{2}\simeq\frac{(w^{-}_{\downarrow})^{2}w^{+}_{\downarrow}\sin\theta^{+}_{\downarrow}}{\lambda_{\perp}}=\frac{(w^{-}_{\downarrow})^{2}v_{\downarrow}\sin\theta_{\downarrow}}{\lambda_{\perp}},$
(9)

where $\theta_{\downarrow}$ is the angle formed by $\bm{v}$ and $-\bm{b}$ and
quantities with the subscript $\downarrow$ describe negatively aligned
fluctuations. Here, $0<\theta_{\uparrow}<\pi/2$ and
$0<\theta_{\downarrow}<\pi/2$.

In general, the fluctuations $\bm{v}$ and $\bm{b}$ observed at any point
$(\bm{x},t)$ are either positively or negatively aligned. For a point
$(\bm{x},t)$ picked at random, let $p$ and $q$ be the probabilities the
alignment is positive or negative, respectively ($p+q=1$). Then, on average,

$\frac{\partial}{\partial
t}\frac{(\tilde{w}^{+})^{2}}{2}\simeq\frac{1}{\lambda_{\perp}}\big{[}p(w^{+}_{\uparrow})^{2}v_{\uparrow}\sin\theta_{\uparrow}+q(w^{+}_{\downarrow})^{2}v_{\downarrow}\sin\theta_{\downarrow}\big{]}$
(10)

and

$\frac{\partial}{\partial
t}\frac{(\tilde{w}^{-})^{2}}{2}\simeq\frac{1}{\lambda_{\perp}}\big{[}p(w^{-}_{\uparrow})^{2}v_{\uparrow}\sin\theta_{\uparrow}+q(w^{-}_{\downarrow})^{2}v_{\downarrow}\sin\theta_{\downarrow}\big{]},$
(11)

where the rms values $\tilde{w}^{\pm}$ are defined by

$\displaystyle(\tilde{w}^{+})^{2}$
$\displaystyle=p(w_{\uparrow}^{+})^{2}+q(w_{\downarrow}^{+})^{2},$ (12)
$\displaystyle(\tilde{w}^{-})^{2}$
$\displaystyle=p(w_{\uparrow}^{-})^{2}+q(w_{\downarrow}^{-})^{2}.$ (13)

The following relations also hold. For a positively aligned fluctuation,
assuming $|\bm{v}|=|\bm{b}|$,

$\displaystyle\bm{w}_{\uparrow}^{+}\cdot\bm{w}_{\uparrow}^{+}$
$\displaystyle=2v_{\uparrow}^{2}(1+\cos\theta_{\uparrow}),$ (14)
$\displaystyle\bm{w}_{\uparrow}^{-}\cdot\bm{w}_{\uparrow}^{-}$
$\displaystyle=2v_{\uparrow}^{2}(1-\cos\theta_{\uparrow}),$ (15)

and $w^{+}_{\uparrow}w^{-}_{\uparrow}=2v_{\uparrow}^{2}\sin\theta_{\uparrow}$.
The energy of a positively aligned fluctuation is $v_{\uparrow}^{2}$. For a
negatively aligned fluctuation

$\displaystyle\bm{w}_{\downarrow}^{+}\cdot\bm{w}_{\downarrow}^{+}$
$\displaystyle=2v_{\downarrow}^{2}(1-\cos\theta_{\downarrow}),$ (16)
$\displaystyle\bm{w}_{\downarrow}^{-}\cdot\bm{w}_{\downarrow}^{-}$
$\displaystyle=2v_{\downarrow}^{2}(1+\cos\theta_{\downarrow}),$ (17)

and
$w^{+}_{\downarrow}w^{-}_{\downarrow}=2v_{\downarrow}^{2}\sin\theta_{\downarrow}$.
The energy of a negatively aligned fluctuation is $v_{\downarrow}^{2}$. Thus,
the rms values (12) and (13) are

$\displaystyle(\tilde{w}^{+})^{2}$
$\displaystyle=2\big{[}pv_{\uparrow}^{2}(1+\cos\theta_{\uparrow})+qv_{\downarrow}^{2}(1-\cos\theta_{\downarrow})\big{]},$
(18) $\displaystyle(\tilde{w}^{-})^{2}$
$\displaystyle=2\big{[}pv_{\uparrow}^{2}(1-\cos\theta_{\uparrow})+qv_{\downarrow}^{2}(1+\cos\theta_{\downarrow})\big{]}.$
(19)

If the angles are small, $\theta_{\uparrow}\ll 1$ and $\theta_{\downarrow}\ll
1$, then the small parameter $\theta$ can be used to order the terms in
equations (18) and (19) so that to leading order

$(\tilde{w}^{+})^{2}\simeq
4v_{\uparrow}^{2}p\qquad\mbox{and}\qquad(\tilde{w}^{-})^{2}\simeq
4v_{\downarrow}^{2}q,$ (20)

where $p+q=1$. This may be derived as follows. In equations (18) and (19)
assume that the angles are both small and then substitute $1+\cos\theta\simeq
2$ and $1-\cos\theta=2\sin^{2}(\theta/2)$ to obtain

$(\tilde{w}^{+})^{2}\simeq
4\big{[}pv_{\uparrow}^{2}+qv_{\downarrow}^{2}\sin^{2}(\theta_{\downarrow}/2)\big{]}\\\
$ (21)

and

$(\tilde{w}^{-})^{2}\simeq
4\big{[}pv_{\uparrow}^{2}\sin^{2}(\theta_{\uparrow}/2)+qv_{\downarrow}^{2}\big{]}.$
(22)

As $\lambda_{\perp}\rightarrow 0$, both $\theta_{\uparrow}\rightarrow 0$ and
$\theta_{\downarrow}\rightarrow 0$ and, therefore, to first order, the terms
proportional to $\sin^{2}(\theta)$ may be neglected. Alternatively, note that

$\bigg{(}\frac{\tilde{w}^{+}}{\tilde{w}^{-}}\bigg{)}^{2}\simeq\frac{(pv_{\uparrow}^{2}/qv_{\downarrow}^{2})+\sin^{2}(\theta_{\downarrow}/2)}{(pv_{\uparrow}^{2}/qv_{\downarrow}^{2})\sin^{2}(\theta_{\uparrow}/2)+1}.$
(23)

As discussed below, solar wind observations show that this quantity is
approximately constant in the inertial range. Now, as
$\theta_{\uparrow}\rightarrow 0$ and $\theta_{\downarrow}\rightarrow 0$ the
only way that this can remain constant is if
$pv_{\uparrow}^{2}/qv_{\downarrow}^{2}$ is bounded away from zero and

$\bigg{(}\frac{\tilde{w}^{+}}{\tilde{w}^{-}}\bigg{)}^{2}\simeq\frac{pv_{\uparrow}^{2}}{qv_{\downarrow}^{2}}.$
(24)

This justifies the approximation in Eqn (20).

Equation (20) shows that at a given scale $\lambda_{\perp}$ the total energy
$[(\tilde{w}^{+})^{2}+(\tilde{w}^{-})^{2}]/4$ is partitioned into two parts,
the energy $(\tilde{w}^{+})^{2}/4$ associated with positive alignment and the
energy $(\tilde{w}^{-})^{2}/4$ associated with negative alignment. The
normalized cross-helicity $\sigma_{c}$ is defined as the ratio of the cross-
helicity to the energy at a given scale and can be written

$\sigma_{c}=\frac{(\tilde{w}^{+})^{2}-(\tilde{w}^{-})^{2}}{(\tilde{w}^{+})^{2}+(\tilde{w}^{-})^{2}}.$
(25)

For small angles, equations (10) and (11) become, to leading order,

$\displaystyle\frac{\partial}{\partial t}\frac{(\tilde{w}^{+})^{2}}{2}$
$\displaystyle\simeq\frac{4pv_{\uparrow}^{3}\theta_{\uparrow}}{\lambda_{\perp}},$
(26) $\displaystyle\frac{\partial}{\partial t}\frac{(\tilde{w}^{-})^{2}}{2}$
$\displaystyle\simeq\frac{4qv_{\downarrow}^{3}\theta_{\downarrow}}{\lambda_{\perp}}.$
(27)

To express these in terms of the rms values $\tilde{w}^{\pm}$, eliminate
$v_{\uparrow}$ and $v_{\downarrow}$ using equation (20). This yields

$\displaystyle\frac{\partial}{\partial t}\frac{(\tilde{w}^{+})^{2}}{2}$
$\displaystyle\simeq\frac{(\tilde{w}^{+})^{3}\theta_{\uparrow}}{2\lambda_{\perp}p^{1/2}},$
(28) $\displaystyle\frac{\partial}{\partial t}\frac{(\tilde{w}^{-})^{2}}{2}$
$\displaystyle\simeq\frac{(\tilde{w}^{-})^{3}\theta_{\downarrow}}{2\lambda_{\perp}q^{1/2}}.$
(29)

These estimates shall be used to derive the cascade times.

##  III Energy cascade time

When nonlinear interactions are strong and a large number of Fourier modes are
excited, fluctuations occur continuously in time and space. During a time
$\tau$ the fractional change in the quantity $(\tilde{w}^{+})^{2}$ is, from
(28),

$\chi^{+}(\tau)\simeq\frac{(w^{+})^{3}\theta_{\uparrow}}{2\lambda_{\perp}p^{1/2}}\cdot\frac{2\tau}{(w^{+})^{2}}=\frac{w^{+}\theta_{\uparrow}\tau}{\lambda_{\perp}p^{1/2}},\qquad\tau\leq\tau^{+},$
(30)

where $\tau^{+}$ is the cascade time at the lengthscale $\lambda_{\perp}$ and
the tildes have been dropped. Similarly, the fractional change in the quantity
$(\tilde{w}^{-})^{2}$ is, from (29),

$\chi^{-}(\tau)\simeq\frac{(w^{-})^{3}\theta_{\downarrow}}{2\lambda_{\perp}q^{1/2}}\cdot\frac{2\tau}{(w^{-})^{2}}=\frac{w^{-}\theta_{\downarrow}\tau}{\lambda_{\perp}q^{1/2}},\qquad\tau\leq\tau^{-},$
(31)

where $\tau^{-}$ is the cascade time of $\tilde{w}^{-}$ and the tildes have
been dropped for brevity. Hereafter, the tildes will be omitted and $w^{+}$
and $w^{-}$ will always represent the rms values.

According to the definition of the energy cascade time, the fractional change
$\chi^{+}$ is of order unity when the interaction time $\tau$ is equal to the
cascade time $\tau^{+}$. Therefore, the relations (30) and (31) imply

$\tau^{+}\simeq\frac{\lambda_{\perp}p^{1/2}}{w^{+}\theta_{\uparrow}},\qquad\tau^{-}\simeq\frac{\lambda_{\perp}q^{1/2}}{w^{-}\theta_{\downarrow}}.$
(32)

By similar reasoning, equations (6) and (9) imply

$\tau_{\uparrow}\simeq\frac{\lambda_{\perp}}{2v_{\uparrow}\theta_{\uparrow}},\qquad\tau_{\downarrow}\simeq\frac{\lambda_{\perp}}{2v_{\downarrow}\theta_{\downarrow}}.$
(33)

Moreover, equations (32), (33), and (20) imply $\tau^{+}=\tau_{\uparrow}$ and
$\tau^{-}=\tau_{\downarrow}$. Thus, the energy cascade times for the rms
Elsasser amplitudes are equal to the energy cascade times for the positively
and negatively aligned fluctuations.

For balanced turbulence, $\sigma_{c}\rightarrow 0$, $w^{+}/w^{-}\rightarrow
1$, $p=q$, $\theta_{\uparrow}=\theta_{\downarrow}$, and the energy cascade
times (32) reduce to the cascade time in Boldyrev’s original theory (Boldyrev,
2006). For imbalanced turbulence, $\sigma_{c}\neq 0$, the cascade times (32)
are different from the cascade times
$\tau^{\pm}\sim\lambda_{\perp}/w^{\mp}\theta^{\mp}$ in the theory of Perez &
Boldyrev (Perez and Boldyrev, 2009). The theory presented here is different
from the theory of Perez & Boldyrev (Perez and Boldyrev, 2009) because the
latter theory does not take into account the existence of two separate types
of fluctuations, positively and negatively aligned, with separate
probabilities of occurrence $p$ and $q$. Taking this into account and also the
definitions of the rms amplitudes (12) and (13), it follows from the preceding
analysis that the timescales for the rms amplitudes take the form (32).

As pointed out by Kraichnan (Kraichnan, 1965), Dobrowolny, Mangeney, and
Veltri (Dobrowolny et al., 1980), and others, the energy cascade in MHD
turbulence occurs through collisions between Alfvén wavepackets propagating in
opposite directions along the mean magnetic field. In other words, it is the
interaction between $w^{+}$ and $w^{-}$ waves that causes the energy to
cascade to smaller scales in MHD turbulence. Consequently, the cascade time
for $w^{+}$, say, should depend on $w^{-}$. While it may appear from equations
(32)–(33) that the timescale for $w^{+}$ fluctuations depends only on $w^{+}$
and, therefore, the interaction with the $w^{-}$ waves is absent, this is not
true. The interactions are still present in the expressions (32) and (33)
through the dependence on the angles and other parameters as will be shown in
the next section.

##  IV Theory of the energy cascade process

Assuming there is no direct injection of energy or cross-helicity within the
inertial range and there is no dissipation of energy or cross-helicity within
the inertial range, the energy cascade rate $\varepsilon$ and the cross-
helicity cascade rate $\varepsilon_{c}$ are scale-invariant in the inertial
range. It follows that the energy cascade rates for the two Elsasser variables
$\varepsilon^{\pm}=\varepsilon\pm\varepsilon_{c}$ are also scale-invariant.
The theory of the energy cascade process is based on Kolmogorov’s relations

$\frac{(w^{+})^{2}}{2\tau^{+}}=\varepsilon^{+}\qquad\mbox{and}\qquad\frac{(w^{-})^{2}}{2\tau^{-}}=\varepsilon^{-},$
(34)

where the non-zero constants $\varepsilon^{+}$ and $\varepsilon^{-}$ are the
energy cascade rates per unit mass for the two Elsasser variables $w^{+}$ and
$w^{-}$, respectively. These equations describe the conservation of energy
flux in $\bm{k}$-space (Fourier space). In addition to Kolmogorov’s relations,
there are two observational constraints that must be taken into account.

Solar wind observations show that the energy and cross-helicity spectra of the
turbulence follow approximately the same power law in the inertial range (Fig.
2)

Figure 2: Typical energy $E$ and cross-helicity spectra $C$ (trace spectra)
obtained using 3-second plasma velocity and magnetic field data from the Wind
spacecraft near the orbit of the earth at 1 AU. (a) An interval of highly
Alfvénic high-speed wind from 3 Jan 1995 09:00 to 8 Jan 1995 00:00, 4.625
days. (b) A weak high-speed stream embedded in low-speed wind; 24 Jul 1996
12:00 to 7 Aug 1996 00:00, 14 days. (c) The normalized cross-helicity
$\sigma_{c}=C/E$ as a function of frequency. The rapid change in $\sigma_{c}$
near the Nyquist frequency is at least partly caused by the FFT processing
techniques and may not be a real physical effect.

which implies that the normalized cross-helicity $\sigma_{c}$ is approximately
constant. In other words, the quantity $\sigma_{c}$ is approximately scale
invariant. Similar results have been found in simulations of incompressible
MHD turbulence (Verma et al., 1996; Perez and Boldyrev, 2009; Beresnyak and
Lazarian, 2008). In particular, the 3D simulations of Perez & Boldyrev (Perez
and Boldyrev, 2009) indicate that the perpendicular Elsasser spectra are
proportional to each other in Fourier space. Solar wind observations also
suggest that the probabilities $p$ and $q$ are approximately scale invariant
as shown in Fig. 3. These observations will now be taken into account in the
theory.

Assuming $\sigma_{c}$ and $p$ are both scale invariant quantities, then
$w^{+}/w^{-}$, $v_{\uparrow}/v_{\downarrow}$, $\tau^{+}/\tau^{-}$, and
$\theta_{\downarrow}/\theta_{\uparrow}$ are scale invariant by equations (25),
(24), (34), and (32), respectively. In all, there are six different scale
invariant ratios in the theory

$\frac{w^{+}}{w^{-}},\quad\frac{\varepsilon^{+}}{\varepsilon^{-}},\quad\frac{p}{q},\quad\frac{v_{\uparrow}}{v_{\downarrow}},\quad\frac{\tau^{+}}{\tau^{-}},\quad\frac{\theta_{\uparrow}}{\theta_{\downarrow}}.$
(35)

At most, only three of these are independent, say, the first three. Equations
(24), (34), and (32) imply

$\displaystyle\frac{v_{\uparrow}}{v_{\downarrow}}=\sqrt{\frac{q}{p}}\cdot\frac{w^{+}}{w^{-}},$
(36)
$\displaystyle\frac{\tau^{+}}{\tau^{-}}=\left(\frac{w^{+}}{w^{-}}\right)^{2}\frac{\varepsilon^{-}}{\varepsilon^{+}},$
(37)
$\displaystyle\frac{\theta_{\downarrow}}{\theta_{\uparrow}}=\sqrt{\frac{q}{p}}\left(\frac{w^{+}}{w^{-}}\right)^{3}\frac{\varepsilon^{-}}{\varepsilon^{+}}.$
(38)

Therefore, the six scale invariant ratios (35) can all be expressed in terms
of the first three $w^{+}/w^{-}$, $\varepsilon^{+}/\varepsilon^{-}$, and
$p/q$.

Figure 3: The probabilities $p$ and $q$ obtained from solar wind data by
integrating the observed probability density function for the angle $\theta$
from 0 to $\pi/2$ and from $\pi/2$ to $\pi$, respectively. The data was
acquired by the Wind spacecraft between 8 Jan 1997 and 9 June 1997 and
analyzed using the techniques described in (Podesta et al., 2009). Examples of
the probability density functions can be found in (Podesta et al., 2009).

To be able to solve Kolmogorov’s relations (34) for $w^{\pm}$ it is necessary
to express the alignment angle $\theta_{\uparrow}$ in terms of $w^{\pm}$. In
general, $\theta_{\uparrow}$ can depend on $w^{+}$, $w^{-}$, the Alfvén speed
$v_{A}$, the lengthscale $\lambda_{\perp}$, the cascade rates
$\varepsilon^{+}$ and $\varepsilon^{-}$, and the probabilities $p$ and $q$. By
dimensional analysis, $\theta_{\uparrow}$ must be a function of the following
six dimensionless quantities

$\frac{w^{+}}{v_{A}},\quad\frac{w^{-}}{v_{A}},\quad\frac{\varepsilon^{+}\lambda_{\perp}}{v_{A}^{3}},\quad\frac{\varepsilon^{-}\lambda_{\perp}}{v_{A}^{3}},\quad
p,\quad q.$ (39)

Moreover, $\theta_{\uparrow}$ must change into $\theta_{\downarrow}$ when
$w^{+}$, $\varepsilon^{+}$, and $p$ are interchanged with $w^{-}$,
$\varepsilon^{-}$, and $q$, respectively, to be consistent with the nonlinear
terms (28) and (29). For a theory composed of power law functions, the only
forms that satisfy all these requirements are

$\displaystyle\theta_{\uparrow}$
$\displaystyle\propto\bigg{(}\frac{w^{+}}{v_{A}}\bigg{)}^{\\!\alpha}\bigg{(}\frac{w^{-}}{v_{A}}\bigg{)}^{\\!\beta}\bigg{(}\frac{\varepsilon^{+}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!\gamma}\bigg{(}\frac{\varepsilon^{-}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!\delta}p^{\mu}q^{\nu},$
(40) $\displaystyle\theta_{\downarrow}$
$\displaystyle\propto\bigg{(}\frac{w^{-}}{v_{A}}\bigg{)}^{\\!\alpha}\bigg{(}\frac{w^{+}}{v_{A}}\bigg{)}^{\\!\beta}\bigg{(}\frac{\varepsilon^{-}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!\gamma}\bigg{(}\frac{\varepsilon^{+}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!\delta}q^{\mu}p^{\nu},$
(41)

where $\alpha$, $\beta$, $\gamma$, $\delta$, $\mu$, and $\nu$ are constants
that must be determined by the theory. In addition, there is a leading
coefficient which is omitted.

The substitution of (40) and (41) into equation (38) yields $\beta=\alpha+3$,
$\delta=\gamma-1$, and $\nu=\mu-1/2$. The parameters are further constrained
by considering the geometry of the “turbulent eddies” associated with the
fluctuations $v_{\uparrow}$ and $v_{\downarrow}$. The parallel correlation
length is defined by $\lambda_{\parallel}^{\uparrow}=v_{A}\tau_{\uparrow}$ and
the correlation length in the direction of the velocity fluctuation is
$\bm{\xi}_{\uparrow}=\bm{v}_{\uparrow}\tau_{\uparrow}$. Similarly, the
correlation lengths for negatively aligned fluctuations are
$\lambda_{\parallel}^{\downarrow}=v_{A}\tau_{\downarrow}$ and
$\xi_{\downarrow}=v_{\downarrow}\tau_{\downarrow}$. In the plane perpendicular
to the local mean magnetic field $\bm{\xi}$ is parallel to $\bm{v}$, the
gradient direction is perpendicular to $\bm{v}$ with lengthscale
$\lambda_{\perp}$, and the eddy dimensions are $\xi\times\lambda_{\perp}$. The
dimension parallel to the mean magnetic field is $\lambda_{\parallel}$. Hence,
in physical space the turbulent eddies can be visualized as three-dimensional
structures with dimensions
$\lambda_{\perp}\times\xi\times\lambda_{\parallel}$.

The coherence times for longitudinal and transverse motions of the eddy must
be equal to each other and also to the cascade time. This is the critical
balance condition of Goldreich and Sridhar which is also implicit in the work
of Higdon Higdon (1984). Equation (33) and the definitions of the correlation
lengths in the last paragraph immediately yield the critical balance condition

$\tau_{\uparrow}=\frac{\lambda_{\parallel}^{\uparrow}}{v_{A}}=\frac{\xi_{\uparrow}}{v_{\uparrow}}\simeq\frac{\lambda_{\perp}}{2v_{\uparrow}\theta_{\uparrow}}$
(42)

with a similar condition for the negatively aligned fluctuations

$\tau_{\downarrow}=\frac{\lambda_{\parallel}^{\downarrow}}{v_{A}}=\frac{\xi_{\downarrow}}{v_{\downarrow}}\simeq\frac{\lambda_{\perp}}{2v_{\downarrow}\theta_{\downarrow}}.$
(43)

Now consider the eddy geometry. When the mean magnetic field is strong enough
that $w^{\pm}/v_{A}<1$, then $\lambda_{\parallel}>\xi>\lambda_{\perp}$ and the
eddies are elongated in the parallel direction. The condition
$w^{\pm}/v_{A}<1$ is assumed hereafter. Equation (42) shows that the aspect
ratio in the field perpendicular plane is
$\phi_{\uparrow}=\lambda_{\perp}/\xi_{\uparrow}=2\theta_{\uparrow}$ and the
aspect ratio in the parallel direction is, from equations (42) and (20),

$\psi_{\uparrow}=\frac{\xi_{\uparrow}}{\lambda_{\parallel}^{\uparrow}}=\frac{v_{\uparrow}}{v_{A}}=\frac{w^{+}}{2v_{A}p^{1/2}}.$
(44)

The two aspect ratios will scale in the same way if
$\phi_{\uparrow}/\psi_{\uparrow}$ is scale invariant. This implies that
$\alpha=-1$ and $\gamma=1/2$. The assumption that the ratio
$\phi_{\uparrow}/\psi_{\uparrow}$ is scale invariant is different from
Boldyrev’s original approach in which he assumed that the alignment angles in
and out of the field perpendicular plane are simultaneously minimized.
Nevertheless, our assumption retains the spirit of Boldyrev’s original theory
which implies the geometry of turbulent fluctuations are scale-invariant.

Solving Kolmogorov’s relation (34) using (32), (40), (41), and the parameter
values obtained so far, one finds

$\frac{w^{\pm}}{v_{A}}\simeq\bigg{(}\frac{w^{+}}{w^{-}}\bigg{)}^{\\!\pm
1/2}\bigg{(}\frac{\varepsilon^{-}}{\varepsilon^{+}}\bigg{)}^{\\!\pm
1/8}\bigg{(}\frac{\varepsilon^{\pm}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!1/4}(pq)^{-\nu/4}$
(45)

and the total energy cascade rate
$\varepsilon=(\varepsilon^{+}+\varepsilon^{-})/2$ is

$\varepsilon=\frac{(w^{+}w^{-})^{2}}{4v_{A}\lambda_{\perp}}\Bigg{(}\sqrt{\frac{\varepsilon^{+}}{\varepsilon^{-}}}+\sqrt{\frac{\varepsilon^{-}}{\varepsilon^{+}}}\Bigg{)}(pq)^{\nu}.$
(46)

The total energy at scale $\lambda_{\perp}$ is

$\frac{(w^{+})^{2}+(w^{-})^{2}}{4}=\frac{w^{+}w^{-}}{4}\bigg{(}\frac{w^{+}}{w^{-}}+\frac{w^{-}}{w^{+}}\bigg{)}\equiv
v^{2}.$ (47)

Therefore, the energy cascade rate can be written

$\epsilon=\frac{4v^{4}}{v_{A}\lambda_{\perp}}\Bigg{(}\sqrt{\frac{\varepsilon^{+}}{\varepsilon^{-}}}+\sqrt{\frac{\varepsilon^{-}}{\varepsilon^{+}}}\Bigg{)}\bigg{(}\frac{w^{+}}{w^{-}}+\frac{w^{-}}{w^{+}}\bigg{)}^{\\!-2}(pq)^{\nu}.$
(48)

Assuming the rms energy $v^{2}$ at scale $\lambda_{\perp}$ is held constant,
the terms on the right-hand side describe the dependence of the energy cascade
rate on the ratios $\varepsilon^{+}/\varepsilon^{-}$ and $w^{+}/w^{-}$. The
value of $\nu$ may be determined by comparison with experiment or possibly by
further physical considerations. This parameter does not affect the inertial
range scaling laws and is left undetermined for the moment.

At this point it is of interest to return to the expressions (32) for the
cascade times and ask: How do the cascade times depend on the rms Elsasser
amplitudes? Using the parameter values obtained previously, equation (40)
becomes

$\theta_{\uparrow}\sim\bigg{(}\frac{w^{+}}{v_{A}}\bigg{)}^{\\!-1}\bigg{(}\frac{w^{-}}{v_{A}}\bigg{)}^{\\!2}\bigg{(}\frac{\varepsilon^{+}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!1/2}\bigg{(}\frac{\varepsilon^{-}\lambda_{\perp}}{v_{A}^{3}}\bigg{)}^{\\!-1/2}p^{\nu+1/2}q^{\nu}$
(49)

and the substitution of this result into equation (32) yields

$\tau^{+}\simeq\frac{\lambda_{\perp}}{v_{A}}\bigg{(}\frac{v_{A}}{w^{-}}\bigg{)}^{\\!2}\bigg{(}\frac{\varepsilon^{-}}{\varepsilon^{+}}\bigg{)}^{\\!1/2}(pq)^{-\nu}.$
(50)

A similar expression holds for $\tau^{-}$ so that the ratio
$\tau^{+}/\tau^{-}$ satisfies (37). Ignoring scale invariant factors, the
preceding equation shows that

$\tau^{+}\propto\frac{\lambda_{\perp}v_{A}}{(w^{-})^{2}}\qquad\mbox{and}\qquad\tau^{-}\propto\frac{\lambda_{\perp}v_{A}}{(w^{+})^{2}}.$
(51)

In this form, the angle dependence has been eliminated. Note that the simple
estimate $\tau^{+}\sim\lambda_{\perp}/w^{-}$ suggested by the nonlinear term
in the MHD equations is modified by the factor $v_{A}/w^{-}$ which accounts
for the weakening of nonlinear interactions caused by scale dependent
alignment. The presence of this algebraic factor is one of the hallmarks of
Boldyrev’s original (2006) theory which is generalized here to imbalanced
turbulence. Remarkably, the relations (51) are identical to those in the
isotropic theory of imbalanced turbulence developed by Dobrowolny, Mangeney,
and Veltri; see equation (10) in (Dobrowolny et al., 1980). Recall that
Dobrowolny, Mangeney, and Veltri concluded from their expressions for the
cascade times that steady state turbulence with nonvanishing cross-helicity is
impossible. On the contrary, the theory presented here allows such a steady
state because the additional coefficients shown in (50) but not (51) maintain
the relation (37) even when $\varepsilon^{+}\neq\varepsilon^{-}$. Thus, the
theory presented here is also a generalization of the theory of Dobrowolny,
Mangeney, and Veltri (Dobrowolny et al., 1980).

A remark about the timescales in the theory should be mentioned. If
$w^{+}>w^{-}$, then equation (37) implies it is possible that
$\tau^{+}<\tau^{-}$ since there is nothing in the theory that prevents this.
That is, the energy of the more energetic Elsasser species may be transferred
to smaller scales in less time than the energy of the less energetic Elsasser
species. This is not inconsistent with dynamic alignment, a well known effect
seen in simulations of decaying incompressible MHD turbulence where the
minority species usually decays more rapidly than the dominant species causing
the magnitude of the normalized cross-helicity to increase with time
(Dobrowolny et al., 1980; Matthaeus et al., 1983; Matthaeus and Montgomery,
1984; Pouquet et al., 1986).

In freely decaying turbulence, dynamic alignment occurs whenever the total
energy decays more rapidly than the cross-helicity, that is,
$\varepsilon>|\varepsilon_{c}|$, where the cascade rate of cross-helicity
$\varepsilon_{c}$ may be positive or negative. From the relations
$\varepsilon>0$ and $\varepsilon^{\pm}=\varepsilon\pm\varepsilon_{c}$, it
follows that dynamic alignment occurs if and only if $\varepsilon^{+}>0$ and
$\varepsilon^{-}>0$. If $w^{+}>w^{-}$, it is not necessary that
$\tau^{+}>\tau^{-}$, only that

$\frac{\tau^{+}}{\tau^{-}}>\frac{\varepsilon^{-}}{\varepsilon^{+}},$ (52)

as can be seen from equation (37). Therefore, even though the relation
$\tau^{+}<\tau^{-}$ may seem counter-intuitive, it is not inconsistent with
dynamic alignment.

##  V Summary and Conclusions

Observations of scale dependent alignment of velocity and magnetic field
fluctuations $\delta\bm{v}_{\perp}$ and $\delta\bm{b}_{\perp}$ in the solar
wind suggest that this effect must be included in any theory of solar wind
turbulence (Podesta et al., 2008, 2009). Perez and Boldyrev (2009) have
recently discussed a theory of imbalanced turbulence that includes scale
dependent alignment of the fluctuations $\delta\bm{v}_{\perp}$ and
$\delta\bm{b}_{\perp}$ in the inertial range. We have extended the Perez-
Boldyrev theory by including the probabilities $p$ and $q$ which solar wind
observations indicate are not necessarily equal. Operationally, the
probabilities $p$ and $q$ may be defined as follows. Suppose space is covered
by a uniform cartesian grid or three dimensional mesh. At each grid-point one
may compute the fluctuations $\delta\bm{v}_{\perp}$ and $\delta\bm{b}_{\perp}$
and the angle between them $\theta$. If the angle lies in the range
$0<\theta<\pi/2$, then the fluctuation is positively aligned and if
$\pi/2<\theta<\pi$, then the fluctuation is negatively aligned. By counting
the number of positively and negatively aligned fluctuations in a large volume
$V$, much larger than the lengthscales of the turbulent eddies, the
probabilities $p$ and $q$ may be defined as the fractional numbers of
positively and negatively aligned fluctuations in the volume $V$.

The phenomenological theory developed in this paper was guided primarily by
two new solar wind observations. It should be noted that both of these solar
wind observations are necessary for the development of the theory. At first
glance, it may seem that the condition $\sigma_{c}=$ const implies that $p$
and $q$ are both constant. Or that these two conditions are somehow
equivalent. However, the relation $(w^{+}/w^{-})^{2}\simeq
pv_{\uparrow}^{2}/qv_{\downarrow}^{2}$, equation (24), shows that $p/q$ can
vary with the lengthscale even if $w^{+}/w^{-}$ is constant. Therefore, it is
essential to have separate observations of the scale invariance of
$\sigma_{c}$ and the scale invariance of $p$ and $q$ to support the
theoretical framework developed here.

In summary, using estimates of the cascade times derived from the nonlinear
terms in the incompressible MHD equations and two new observational
constraints derived from studies of solar wind data, we have constructed a
generalization of Boldyrev’s theory (Boldyrev, 2006) that depends on the three
parameters $w^{+}/w^{-}$, $\varepsilon^{+}/\varepsilon^{-}$, and $p/q$. The
theory reduces to the original theory of Boldyrev (2006) when $w^{+}=w^{-}$,
$\epsilon^{+}=\epsilon^{-}$, and $p=q$ since in this limit
$\theta_{\uparrow}=\theta_{\downarrow}$ and the cascade times (32) become
equal to those of Boldyrev (2006). For imbalanced turbulence $w^{+}\neq
w^{-}$, $p\neq q$, and the theory predicts the scaling laws
$w^{\pm}\propto\lambda_{\perp}^{1/4}$,
$\theta_{\uparrow\downarrow}\propto\lambda_{\perp}^{1/4}$, and
$\lambda_{\parallel}^{\pm}\propto\lambda_{\perp}^{1/2}$. Interestingly, the
scaling laws for balanced and imbalanced turbulence are the same. The
perpendicular energy spectrum defined by $k_{\perp}E^{\pm}\sim|w^{\pm}|^{2}$
has the inertial range scaling $E^{\pm}\propto k_{\perp}^{-3/2}$ with

$\frac{E^{+}}{E^{-}}=\bigg{(}\frac{w^{+}}{w^{-}}\bigg{)}^{\\!2}=\frac{1+\sigma_{c}}{1-\sigma_{c}}=\mbox{const}.$
(53)

The theory assumes that the cascades for positively and negatively aligned
fluctuations are both in a state of critical balance (42), although they are
governed by different timescales, and that the eddy geometry is scale
invariant. The positively aligned fluctuations occupy a fractional volume $p$
and the negatively aligned fluctuations occupy a fractional volume $q$ so that
the energy cascade rate is

$\varepsilon=p\frac{v_{\uparrow}^{2}}{\tau_{\uparrow}}+q\frac{v_{\downarrow}^{2}}{\tau_{\downarrow}}$
(54)

or, equivalently,

$\varepsilon=\frac{(w^{+})^{2}}{4\tau^{+}}+\frac{(w^{-})^{2}}{4\tau^{-}}.$
(55)

In the discussion following equation (35) it was shown that at most three of
the ratios $w^{+}/w^{-}$, $\varepsilon^{+}/\varepsilon^{-}$, and $p/q$ can be
independent. However, the two ratios $w^{+}/w^{-}$ and
$\varepsilon^{+}/\varepsilon^{-}$ cannot be independent since in the case of
homogeneous steady-state turbulence $w^{+}=w^{-}$ implies
$\varepsilon^{+}=\varepsilon^{-}$ and vice versa. This is because the
injection of cross-helicity into the system, $\varepsilon_{c}\neq 0$ or
$\varepsilon^{+}\neq\varepsilon^{-}$, will create a nonzero cross-helicity
spectrum and a cascade of cross-helicity from large to small scales which
implies a net accumulation of cross-helicity within the volume
($\sigma_{c}\neq 0$). Hence, at most two of the ratios and $w^{+}/w^{-}$ and
$p/q$ are independent. Whether $p/q$ can be expressed in terms of
$w^{+}/w^{-}$ and $\varepsilon^{+}/\varepsilon^{-}$ is an open question.

###### Acknowledgements.

We are grateful to S. Boldyrev for valuable comments on an earlier version of
the manuscript and to Pablo Mininni and Jean Perez for helpful discussions.
This research is supported by DOE grant number DE-FG02-07ER46372, NASA grant
number NNX06AC19G, and NSF. Additional support for John Podesta comes from the
NASA Solar and Heliospheric Physics Program and the NSF SHINE Program.

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