CALCULATION OF THE Dst INDEX

1
CALCULATION OF THE Dst INDEX
Presentation at LWS CDAW WorkshopFairfax,
VirginiaMarch 14-16, 2005

• R.L. McPherron
• Institute of Geophysics and Planetary Physics
• University of California Los Angeles
• rmcpherron_at_igpp.ucla.edu

2
GEOGRAPHIC COORDINATES USED IN MAGNETIC
MEASUREMENTS

• Dipole is tilted and inverted relative to
  rotation axis
• Dipole field lines are nearly vertical above 60?
  latitude
• Cartesian geographic coordinates are defined in a
  plane tangent to earth at observers location
• X component is towards geographic north pole
• Y component is east along a circle of latitude
• Z component is radially inward or down

3
LOCAL VIEW OF VARIOUS COORDINATE SYSTEMS USED IN
GEOMAGNETISM

• Origin is located at observer
• X points north, Y points east, Z points down in
  the local tangent plane
• F is the total vector field
• H is the horizontal projection of the vector F
• D is the east declination of H from geographic
  north in tangent plane
• I is the inclination of F below the tangent plane
  
• X, Y, Z are the geographic Cartesian components
  of F

4
DISTRIBUTION OF RING CURRENT AND ITS PERTURBATION
IN A MERIDIAN

• Most of the current is concentrated close to the
  equator
• Eastward current inside and westward outside
• Perturbations curl around the volume of current
• The perturbation over the earth is nearly uniform
  and axial

5
Origin of Dst

• Moos, N.A.F., Colaba Magnetic Data, 1846 to 1905,
  2, The Phenomena and its Discussion, Central
  Government Press, Bombay, 1910.
• Figure below taken from the following reference
  to illustrate work by Moos
• Chapman, S., and J. Bartels, Geomagnetism,
  Vol 1, Clarendon Press, Oxford, 1962.

• Use a large set of storms with start time
  uniformly distributed in local time
• For each hour after an ssc (storm time) find the
  average departure of H at a single station from
  its mean value in the corresponding months
  (disturbance) obtaining the disturbance versus
  storm time or Dst
• Separate the storms by the local time at which
  the ssc occurred to illustrate the asymmetry of
  the development as seen by a single station

6
IGY Calculation of Dst
Measured field at ith station
Disturbance Variation
Lunar Variation
Solar Quiet Day Variation
Main Field Variation
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Average Variation over Longitudinal Chain
For each hour average the preceding equation over
8 stations around the world at fixed latitude
Average Disturbance
Average Lunar variation 0
Average Sq at 8 stations
Average secular variation at 8 stations
Round-world average variation at time t
8
Average Sq Variation

• For ith station in month m take the average of
  the five international quiet days (25 hours)
  defined by Greenwich time
• From this average quiet day subtract a linear
  trend connecting midnight at the two ends of the
  Greenwich day
• For each month average the quiet day variations
  over all stations

• Model the residual average Sq variation with a
  double Fourier series in time T and month M.
  Estimate up to 6th harmonic.
• Use this series to estimate the average Sq
  variation at any hour of any day of year.
  Subtract the estimate from DH(t).

9
Average Secular Variation

• Plot the residual variation versus time. Found
  that there was no trend in its baseline, i.e. the
  average secular variation was constant
• Determine the average level of the baseline
  during quiet intervals not affected by magnetic
  storms
• Subtract this constant from the previous residual
  obtaining

• Assume that the last term in is approximately
  zero so that
• Assume that only the ring current contributes to
  the average disturbance so that we have found
  the disturbance as a function of storm time

10
Modern Dst CalculationSugiura, M., and T. Kamei,
Equatorial Dst index 1957-1986, IAGA Bulletin No
40, pp. 7-14, ISGI Publications Office,
Saint-Maur-des-Fosses, France, 1991.

• Use four stations distributed in longitude near
  25? magnetic latitude
• SECULAR VARIATION
• For each station calculate annual means from the
  5 quiet days of each month
• Use current and four preceding annual means to
  determine a polynomial fit to the quiet days. Let
  t be time relative to some reference epoch.
• Use the preceding 5-year fit to predict the
  baseline value on first day of current year.
  Include this value as a data point in the current
  5-year fit
• Create the deviation of H from the secular trend
  for each hour of current year

• QUIET DAY VARIATION
• Use the five local days closest to the Greenwich
  monthly five quiet days plus 1-hour at each end
  of these days
• At each hour calculate monthly averages of the
  local quiet days
• Subtract a linear trend passing through average
  of first and last hours
• Fit a double Fourier transform in hour of day and
  day of year to the 12 sets of 24 hourly values
• Use the fit coefficients to calculate the quiet
  day variation at every hour of year
• Subtract the estimated Sq variation from the
  deviation time series
• Calculate Dst as the latitude weighted average
  disturbance variation

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Creation of the Secular Variation

• For every calendar month select 10 international
  quiet days
• Determine the monthly median value at local
  midnight (red dots)
• Take 2-year running average of midnight medians
• Fit a cubic smoothing spline to the filtered data
  (black line)

12
Creation of Monthly Quiet Day Curve

• Create ensemble of the 10 international quiet
  days for a month
• Subtract value at local midnight
• Subtract linear trend through left and right
  local midnight
• Calculate median variation as function of time

13
Solar Cycle Effects on Sq Variation

• Calculate monthly median quiet day for each month
  of four solar cycles (44 years)
• For each year of an 11-year solar cycle calculate
  mean of four monthly medians
• Compare all means (all years in lower right
  panel)
• There appears to be little effect of solar cycle
  on the median quiet day
• We can ignore effect of phase of solar cycle in
  Sq

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KakiokaMonthlyQuiet Days1960 to 2004

• For each month in 44 years find median Sq of 10
  quiet days
• Find median of each month for all years
• Arrange as a map of variation as function of
  local time and month
• Use coefficients of two-dimensional Fourier
  transform to calculate Sq

15
Fourier Synthesis of Arbitrary Quiet Day

• Use data from four solar cycles
• Find the median quiet day for each month
• Load data into a 12 month by 24 hour 2-D array
• Perform a double Fourier transform
• Expand array to 366 by 24 and move the Fourier
  coefficients to correct location
• Inverse transform to obtain quiet day for each
  day of year

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Removal of Secular and Quiet Variations

• Select the portion of the secular variation curve
  for the interval
• Synthesize the quiet day variation for the
  appropriate days of year
• Subtract both from the original data to obtain
  the disturbance variation for the given station
  component

17
LONGITUDINAL PROFILE OF ?Bj FROM MAGNETOSPHERIC
CURRENTS

• Symmetric ring should create nearly constant
  longitudinal profile in H component
• Local time average of ?H at equator approximates
  ?B at center of Earth
• But other magnetospheric currents create local
  time dependent deviations from symmetry
• Assume asymmetric component has zero mean when
  averaged over local time
• Define the disturbance storm time index Dst as
  local time average of observed ?H profile

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Relation of Dst to Stream Interface

• The figure shows the relation of several solar
  wind and magnetospheric variables to CIRs The
  main stream interface at leading edge of high
  speed stream is taken as epoch zero in a
  superposed epoch analysis
• The colored patches show upper and lower
  quartiles of the variable as function of epoch
  time
• The heavy red line is the median curve
• Stream interfaces cause weak storms

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COMPARISON OF SEVERAL OBSERVED AND PREDICTED
QUIET DAYS AT GUAM IN 1986
Observed
Quiet
Disturbance (nT)
Residual
40
41
42
43
44
45
46
47
48
49
50
Day in 1986
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CORRECTED H AT GUAM DURING RECOVERY FROM A
MAGNETIC STORM
60
40
20
0
Quiet H
-20
-40
Disturbance (nT)
-60
-80
Corrected H
-100
Observed H
-120
-140
40
41
42
43
44
45
46
47
48
49
50
Day in 1986
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The End!

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SCHEMATIC ILLUSTRATION OF EFFECTS OF RING CURRENT
IN H COMPONENT
Projection of a uniform axial field onto Earths
surface
Magnetic effects of a symmetric equatorial ring
current
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In
Out
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REMOVAL OF SECULAR TREND FROM HOURLY VALUES OF H
AT GUAM DURING STORM
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MINOR MAGNETIC STORM RECORDED AT SAN JUAN -
11/24/96
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THE DESSLER-PARKER-SCKOPKE RELATION
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CONTRIBUTIONS TO THE VARIATION IN THE H COMPONENT
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ESTIMATION OF THE SECULAR TREND INH COMPONENT AT
SAN JUAN
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QUIET VALUES DURING STORM USED IN QUIET DAY (Sq)
ESTIMATION
80
Flagged Point
Quiet Value
70
60
50
40
30
Transient H (nT)
20
10
0
-10
-20
115
120
125
130
135
140
Day in 1986
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QUIET GUAM H TRACE AT EQUINOX AND SOLSTICE 1986
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Sq FOR H AT SAN JUAN IN 1978 AS FUNCTION OF DAY
OF YEAR AND UT
-5
350
-5
0
10
0
300
20
25
15
0
5
250
0
20
200
Day of Year
15
0
0
5
0
0
150
10
15
25
20
30
31.1
100
50
20
25
0
-5
5
0
5
10
15
20
UT Hour
-5
0
5
10
15
20
25
30
Diurnal Variation (nT)
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REMOVAL OF STORM EFFECTS IN QUIET DAY (Sq)
ESTIMATION
COMPARISON OF DETRENDED GUAM H TO MIDNIGHT SPLINE
50
0
-50
Disturbance (nT)
Midnight Spline
-100
H Comp
115
120
125
130
135
140
DETRENDED AND STORM CORRECTED GUAM H IN 1986
80
60
40
Residual H (nT)
20
0
-20
115
120
125
130
135
140
Day in 1986
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(No Transcript)
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CURRENTS CONTRIBUTING TO MIDLATITUDE MAGNETIC
PERTURBATIONS

• View is from behind and aabove earth looking
  toward Sun
• Current systems illustrated
• Symmetric ring current
• Dayside magnetopause current
• Partial ring current
• Tail current
• Substorm current wedge
• Region 1 current
• Region 2 current
• Current systems not shown
• Solar quiet day ionospheric current
• Secular variation within earth
• Main field of Earth

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EFFECTS OF MAGNETOPAUSE ON THE Dst INDEX

• Balance magnetic pressure against dynamic pressure

10
8
6
4
Neutral
Point
2
Z (Re)
0
-2
Solar
-4
Wind
-6
-8
-10
0
5
10
15
X (Re)
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A SHEET CURRENT MODEL OF EFFECT OF TAIL CURRENT
ON Dst

• Magnetic Effects

• Tail Current Model

xxx
x
x
x
Ri
Ro
Bz
xxx
x
x
x
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MAGNETIC EFFECTS OF A SUBSTORM CURRENT WEDGE

• Transverse currents in the magnetosphere are
  diverted along field lines to the ionosphere
• Viewed from above north pole the projection of
  the current system has a wedge shape
• Midlatitude stations are primarily affected by
  field-aligned currents and the equatorial closure
  (an equivalent eastward current)
• The local time profile of H component is
  symmetric with respect to the central meridian of
  wedge
• The D component is asymmetric with respect to
  center of wedge

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STEPS IN THE CALCULATIONOF Dst INDEX

• Define the reference level for H component on a
  monthly basis
• Fit a polynomial to reference H values (secular
  variation)
• Adjust H observed on a given day by subtracting
  secular variation
• Identify quiet days from same season and phase of
  solar cycle
• Remove storm effects in quiet values and offset
  traces so that there is zero magnetic
  perturbation at station midnight
• Flag all values recorded during disturbed times
  and interpolate from adjacent quiet intervals
• Create some type of smoothed ensemble average of
  all quiet days
• Subtract average quiet day from adjusted daily
  variation to obtain disturbance daily variation
  for station
• Repeat for a number of stations distributed
  around the world at midlatitudes
• Project the local H variations to obtain axial
  field from ring current and average over all
  stations

39
Magnetograms from several midlatitude stations
during storm
40
MAJOR SUBSTORMS DURING MAGNETIC STORM OF APRIL
3-5, 1979
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CONCLUSIONS

• The Dst index is defined to be linearly
  proportional to the total energy of particles
  drifting in the radiation belts (symmetric ring
  current)
• Dst must be estimated from surface measurements
  of the horizontal component of the magnetic field
• Surface field measurements include effects of
  many electrical currents other than the symmetric
  ring current
• These effects must be estimated or eliminated by
  the algorithm that calculates the Dst index
• Extraneous currents include secular variation,
  Sq, magnetopause, tail, Region 12, partial ring
  current, substorm current wedge, magnetic
  induction
• There are numerous assumptions and errors
  involved in Dst calculations and the index
  contains systematic and random errors as a
  consequence
• Be aware of these problems and take them into
  account in interpreting Dst!

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EXAMPLE OF MIDLATITUDE MAGNETIC DATA DURING
MAGNETIC STORM
43
INTERPLANETARY MAGNETIC FIELD, AE AND Dst INDICES
DURING STORM

• Coronal mass ejection produce intervals of strong
  southward Bz at the earth
• Magnetic reconnection drives magnetospheric
  convection
• Convection drives currents along field lines and
  through ionosphere
• Ground magnetometers record effects of
  ionospheric currents in H and other components
• H traces are used to construct the AE and Dst
  index

44
MAGNETIC EFFECT OF A RING CURRENT AT EARTHS
CENTER

• Axial field from a circular ring current

Z

• Field at center of ring

X
LRRe
Westward RingCurrent

• Convenient units

45
THE SOLENOIDAL EFFECT OF THE RADIATION BELT
CURRENTS

• A more realistic model of the ring current
• Shows the magnetic perturbations
• Shows the distortion of dipole current contours
• Perturbation field from ring current

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DESSLER-PARKER-SCKOPKE DERIVATION