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

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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

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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

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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

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

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IGY Calculation of Dst Disturbance Variation Main Field Variation Solar Quiet Day Variation Lunar Variation Measured field at i th station Main field and its secular variation Secular Variation from average IGY Average )()()()()( )()()()( 0 )( tDtLtSqtHtH iiiii

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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

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Average Sq Variation 1.For i th station in month m take the average of the five international quiet days (25 hours) defined by Greenwich time 2.From this average quiet day subtract a linear trend connecting midnight at the two ends of the Greenwich day 3.For each month average the quiet day variations over all stations 4.Model the residual average Sq variation with a double Fourier series in time T and month M. Estimate up to 6 th harmonic. 5.Use this series to estimate the average Sq variation at any hour of any day of year. Subtract the estimate from H(t).

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Average Secular Variation 1.Plot the residual variation versus time. Found that there was no trend in its baseline, i.e. the average secular variation was constant 2.Determine the average level of the baseline during quiet intervals not affected by magnetic storms 3.Subtract this constant from the previous residual obtaining 4.Assume that the last term in {} is approximately zero so that 5.Assume that only the ring current contributes to the average disturbance so that we have found the disturbance as a function of storm time

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Modern Dst Calculation Sugiura, M., and T. Kamei, Equatorial Dst index , IAGA Bulletin No 40, pp , ISGI Publications Office, Saint-Maur-des-Fosses, France, 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 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)

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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

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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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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

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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

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LONGITUDINAL PROFILE OF B j 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 Day in 1986 Disturbance (nT) Observed Quiet Residual

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CORRECTED H AT GUAM DURING RECOVERY FROM A MAGNETIC STORM Day in 1986 Disturbance (nT) Quiet H Observed H Corrected H

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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 Earth’s 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 IN H COMPONENT AT SAN JUAN x 10 4 Year H (nT) Fourth Order Trend Daily Average 80% Point

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QUIET VALUES DURING STORM USED IN QUIET DAY (Sq) ESTIMATION Day in 1986 Transient H (nT) Flagged Point Quiet Value

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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 UT Hour Day of Year Diurnal Variation (nT)

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REMOVAL OF STORM EFFECTS IN QUIET DAY (Sq) ESTIMATION Day in Disturbance (nT) COMPARISON OF DETRENDED GUAM H TO MIDNIGHT SPLINE Residual H (nT) DETRENDED AND STORM CORRECTED GUAM H IN 1986 Midnight Spline H Comp

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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 X (Re) Z (Re) Solar Wind Neutral Point

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A SHEET CURRENT MODEL OF EFFECT OF TAIL CURRENT ON Dst Xgsm (Re) Bz (nT) Normal Tail Inner Edge Total Earth Tail Current Model Magnetic Effects BzBz xxx xxx xxx RiRi RoRo

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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 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

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Magnetograms from several midlatitude stations during storm

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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 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!

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EXAMPLE OF MIDLATITUDE MAGNETIC DATA DURING MAGNETIC STORM

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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

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

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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

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