# CALCULATION OF THE Dst INDEX

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CALCULATION OF THE Dst INDEX
Presentation at LWS CDAW Workshop Fairfax, Virginia March 14-16, 2005 R.L. McPherron Institute of Geophysics and Planetary Physics University of California Los Angeles

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 observer’s location X component is towards geographic north pole Y component is east along a circle of latitude Z component is radially inward or down

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

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

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

IGY Calculation of Dst Measured field at ith station ) ( t D L Sq H +
t D L Sq H i + = Disturbance Variation Lunar Variation Solar Quiet Day Variation Main Field Variation Main field and its secular variation Secular Variation from average IGY Average

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

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).

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

Modern Dst Calculation Sugiura, M. , and T
Modern Dst Calculation Sugiura, M., and T. Kamei, Equatorial Dst index , 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

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)

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

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

Kakioka Monthly Quiet Days 1960 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

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

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

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

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

COMPARISON OF SEVERAL OBSERVED AND PREDICTED QUIET DAYS AT GUAM IN 1986
-30 -20 -10 10 20 30 40 50 60 70 Observed Quiet Disturbance (nT) Residual 40 41 42 43 44 45 46 47 48 49 50 Day in 1986

CORRECTED H AT GUAM DURING RECOVERY FROM A MAGNETIC STORM
60 40 20 Quiet H -20 -40 Disturbance (nT) -60 -80 Corrected H -100 -120 Observed H -140 40 41 42 43 44 45 46 47 48 49 50 Day in 1986

The End!

SCHEMATIC ILLUSTRATION OF EFFECTS OF RING CURRENT IN H COMPONENT
Magnetic effects of a symmetric equatorial ring current Projection of a uniform axial field onto Earth’s surface

In Out

REMOVAL OF SECULAR TREND FROM HOURLY VALUES OF H AT GUAM DURING STORM
115 120 125 130 135 140 3.575 3.58 3.585 3.59 395.5 x 10 4 Observed H (nT) COMPARISON OF GUAM H WITH SECULAR TREND IN 1986 -100 -50 50 Day in 1986 Transient H (nT) DEVIATION OF GUAM H FROM SECULAR TREND IN 1986 Secular Trend

MINOR MAGNETIC STORM RECORDED AT SAN JUAN - 11/24/96

THE DESSLER-PARKER-SCKOPKE RELATION

CONTRIBUTIONS TO THE VARIATION IN THE H COMPONENT

ESTIMATION OF THE SECULAR TREND IN H COMPONENT AT SAN JUAN
1978 1983 1988 2.7 2.705 2.71 2.715 2.72 2.725 2.73 2.735 2.74 x 10 4 Year H (nT) Fourth Order Trend Daily Average 80% Point

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 -10 -20 115 120 125 130 135 140 Day in 1986

QUIET GUAM H TRACE AT EQUINOX AND SOLSTICE 1986
Spring Summer Fall Winter 5 10 15 20 -10 30 40 50 60 Local Time Pred Quiet H (nT)

Sq FOR H AT SAN JUAN IN 1978 AS FUNCTION OF DAY OF YEAR AND UT
-5 350 -5 10 300 20 25 15 5 250 200 20 Day of Year 15 5 150 15 10 20 25 31.1 30 100 50 20 25 -5 5 5 10 15 20 UT Hour -5 5 10 15 20 25 30 Diurnal Variation (nT)

REMOVAL OF STORM EFFECTS IN QUIET DAY (Sq) ESTIMATION
COMPARISON OF DETRENDED GUAM H TO MIDNIGHT SPLINE 50 Disturbance (nT) -50 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 -20 115 120 125 130 135 140 Day in 1986

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

EFFECTS OF MAGNETOPAUSE ON THE Dst INDEX
Balance magnetic pressure against dynamic pressure 10 8 6 Neutral 4 Point 2 Z (Re) -2 Solar Wind -4 -6 -8 -10 15 10 5 X (Re)

A SHEET CURRENT MODEL OF EFFECT OF TAIL CURRENT ON Dst
Magnetic Effects Tail Current Model xxx x x x -6 -4 -2 2 4 6 -35 -30 -25 -20 -15 -10 -5 -Xgsm (Re) Bz (nT) Normal Tail Inner Edge Total Earth Ri Ro Bz xxx x x x

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

STEPS IN THE CALCULATION OF 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

Magnetograms from several midlatitude stations during storm

MAJOR SUBSTORMS DURING MAGNETIC STORM OF APRIL 3-5, 1979
12 24 36 48 60 72 -2000 -1000 1000 AU and AL index (nT) 125 627 1610 1803 2118 2351 252 757 1143 1702 2200 700 900 1032 1641 2147 -200 -100 100 Time from 0000 UT on April 3 Dst Index (nT)

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 1&2, 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!

EXAMPLE OF MIDLATITUDE MAGNETIC DATA DURING MAGNETIC STORM

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

MAGNETIC EFFECT OF A RING CURRENT AT EARTH’S CENTER
Axial field from a circular ring current Z Field at center of ring X LRRe Westward Ring Current Convenient units

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

DESSLER-PARKER-SCKOPKE DERIVATION

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