Interpretation

×

"

340 + 20 sin 2πd 365

!#

, (2.14)

where the seasonal cycle is a function of the day, d, and the amplitude is a function of the eccentricity, e, on year, y. Here, the year is approximated as having an integer number of days. The selected constants in Eq. 2.14 give values roughly corresponding to the seasonal variations in global mean insolation, measured in W/m². The ec-centricity term in Eq 2.14 represents the secular insolation variability, while the sine term represents the seasonal variability.

Continuing with the simple example, P0(d,y)=I(d,y) because insolation is spa-tially uniform. Furthermore, the singular value decomposition of P0 will only require a single pair of singular vectors withU∼(340 + 20 sin(^2πd₃₆₅)),V∼[1/(1−e(y)²)]², and a singular value S=^P_y^P_dI²(d, y); the ’∼’ means proportional to. Thus, insolation at any given latitude (φ), day (d), and year (y) can be calculated as,

I(φ, d, y) = L0(φ) U(d) S V^T(y).

Note that the variability in space, seasonal time, and secular time are divided between the Legendre polynomials, U singular vectors, and V singular vectors respectively.

In this simplified case, the result is trivial, and there is little benefit to representing the insolation in terms of Legendre polynomials and singular vectors. However, for the real insolation variability, the Legendre/singular vector representation proves an efficient descriptions which provides real insight into the modes of insolation variabil-ity.

2.5.4 Interpretation

Returning now to the real earth-sun system, Figure 2-15 shows the fraction of variance explained by each set of singular vectors for each matrix of Legendre polynomials, Pn. The leading three pairs of singular vectors explain over 99.99% percent of the variance in eachPn. After the first, the singular values associated with eachPncome in pairs — that is, the second and third set of singular vectors explain nearly equal amounts of variance, likewise for the fourth and fifth, etc. Closer inspection of these paired singular vectors indicates that they are nearly identical, except for a 90^◦ shift

1 2 3 4 5 6 7 8 9 10⁻¹²

10⁻¹⁰ 10⁻⁸ 10⁻⁶ 10⁻⁴ 10⁻² 10⁰

fraction of variance explained

singular vector

P0 P1 P₂ P3

Figure 2-15: The variance explained by the leading singular vectors for each matrix of Legendre polynomial weightings,P_n. In each case, the leading singular vector explains over 99% of the variability in the weightings — note the logarithmic scaling of the y-axis. The variance explained by higher order singular vectors comes in pairs, where each pair explains roughly 99% of the remaining variance. In this case, the pairings of explained variance indicates the presence of a traveling wave of insolation, due to the effects of precession on climate. Since the anomalistic year, controlled by precession, comes in and out of phase with the tropical year, which is controlled by obliquity, the precessional influence is manifested as a traveling wave, and is represented by two sets of singular vectors which are separated by 90^◦ of phase.

in phase. This result implies the presence of a traveling wave present in the insolation variability. Because its phase shifts relative to the basic seasonal cycle, two sets of singular vectors are required to explain this wave, in direct analogy with adding a sine and cosine together to form a new signal with the same period but different phase.

It is possible to describe this wave using a single set of complex singular vectors, by means of a Hilbert transform of P_n [e.g. von Storch and Zwiers, 1999], but for these purposes it is simpler to interpret the second and third sets of singular vectors together.

To understand the presence of this traveling wave, observe that Eq. 2.1 contains two different annual periods: λs and λs−$. λs has a period of one tropical year (365.2422 days) and measures the mean interval between vernal equinoxes. Following the example of the Gregorian calendar, the insolation calculations used here define the vernal equinox as the 80th day of year, tying the seasonal phase to the vernal equinox and the tropical year. Theλs−$period is associated with the anomalistic year (currently equal to 365.2422 days minus 1/21,000KY, or 365.2596 days) and

measures the time from one perihelion passage to another. Because insolation is calculated with respect to the tropical year, in the time it takes the angle of perigee,

$, to make a 360^◦ rotation the anomalistic year will move in and out of phase with the tropical year, thus manifesting as a 21KY period traveling wave in the insolation calculations. Thomson [1995] has made the somewhat perplexing observation that temperatures in different cities around the world seem to follow one or the other of these annual cycles. Thus, this seemingly small difference in years appears to have real physical effects even on relatively short timescales.

This dual-period year effect is present in all insolation calculations, but only when the full annual cycle is resolved does it become obvious. When only a portion of the year is resolved, a significant fraction of the variability can be due to the anomalistic year coming in and out of phase with the tropical year. If only a portion of the annual cycle or a portion of the meridional variability is resolved, it is difficult to distinguish between a traveling wave, changes in mean annual insolation, and meridional shifts in insolation. Presumably, each of these redistributions of insolation will force differ-ent types of climatic responses, and it appears knowledge of the full insolation field is crucial when attempting to understanding the dynamical response to insolation forcing.

Figures 2-16 through 2-19 shows the leading three singular vectors associated with each matrix of Legendre polynomial loadings, P_n. The singular vectors of the Pn describe how the spatial patterns of insolation vary at both seasonal and secular timescales. The singular vectors associated with theUdescribe the daily variations in insolation and are referred to as the seasonal vectors; while theV describe long-term changes in insolation and are called the secular vectors. For instance, variations in eccentricity cause changes in net annual insolation and are therefor associated with the secular vectors of P₀ [see Figure 2-16]. The leading secular vector associated with P0, as originally computed, had a small component of precessional variability which accounted for less than 0.0001% of the spatially averaged insolation variance.

To make the mean annual insolation variability easier to interpret [see Figure 2-16], this precessional variability was removed and placed in the precessional vectors. This was the only change made. For the decomposition of the otherPn,n >0, the secular variability in the leading set of singular vectors is determined by obliquity, and these singular vectors are therefor referred to as theobliquity vectors. The secular variability in the second and third sets of singular vectors are controlled by the climatic precession

and its 90^◦ phase shift, ecos($), and are referred to as the precession vectors.

The seasonal obliquity vectors associated with even (symmetric) Legendre poly-nomials [see Figures 2-16 and 2-18] have a non-zero mean value indicative of changes in obliquity causing meridional redistributions of insolation. Conversely, changes in precession are independent of any annual mean variations at any latitude and thus the seasonal precession vectors are always zero-mean. It appears that the net insolation received by each hemisphere is equal as the seasonal vectors associated with all the odd (asymmetric) Legendre polynomials [e.g. Figures 2-17 and 2-19] are zero mean.

The precession vectors associated with odd (asymmetric) Legendre polynomials [see Figures 2-17 and 2-19] are doubly-periodic. To understand this, consider the case when aphelion occurs at the summer solstice. Then the argument of perigee is 90^◦, climatic precession is positive, and cos(90^◦) = 0: thus, the only precessional contri-bution will come from the seasonal vector associated with the precession parameters.

By Kepler’s second law, being closer to the sun during Northern Hemisphere summer means the Earth will move more quickly, therefor reaching summer solstice prior to the average date of June 19th (depending on the magnitude of eccentricity). Summer solstice is a maximum in the inter-hemispheric gradient, and approaching it quickly results in an anomalously large insolation gradient for that period of the year. In ad-dition, being closer to the sun during summer solstice increases the inter-hemispheric gradient in insolation. Because Earth reached the summer solstice quicker than usual, it also leaves more quickly, and by mid-July the precessional effect is negative. This negative trend continues, reaching a minima at the autumnal equinox. Note that adding this precessional seasonal vector to the obliquity vector yields seasonal vari-ations which have both the summer solstice maximum and the autumnal equinox zero-crossing occurring earlier in the year. The cycle then repeats for the Northern hemisphere winter solstice, but now with a relatively slow Earth velocity, an anoma-lously large distance to the sun, and a change in sign of the insolation-gradient. These effects combine to create another oscillation with a maximum during the winter sol-stice, giving the doubly-periodic signal. The larger response during the autumnal equinox occurs because its date has the most variability, being furthest away from the fixed vernal equinox.

Paralleling the simple example given earlier, the insolation, I, is represented as I(φ, d, y) =

N

X

n=0

L(φ, n)

M

X

m=1

U_n(d, m)S_n(m, m)V_n(y, m)

!

, (2.15)

100 200 300

Figure 2-16: Singular value decomposition of P0 showing spatial average insolation variability over the last 1000 KY. The seasonal singular vectors (left) are scaled to units of W/m²; the secular singular vectors (right) indicate the modulation of the mean and annual variability. a,b: The leading pair of singular vectors explain 99.8%

of the spatial average insolation variance. a, the leading seasonal singular vectors represents the spatial and time average insolation over the last million years, 341.4 W/m²; the solar constant is 1365W/m². There is no seasonality associated with this mean value, thus the flat line. b, the secular variations in total annual insolation are only a function of eccentricity, scaling as 1/(1−e²)², and increasing by 0.65%

from a minimum eccentricity, 0.005, to a maximum, 0.057. As discussed in the text, a slight modification was made to these singular vectors. c,d: the second and third sets of singular vectors, termed the precession vectors, show the anomalistic year coming in and out of phase with the tropical year and each account for 0.1% of the variance, almost all the variance not explained by the leading set of singular vectors.

c, the seasonal precession vectors are a sine (solid) and cosine (dashed) pair with zero phase at the vernal equinox (the vertical dotted line at day 80, or March 20th) and can together be interpreted as a traveling wave. d, variations in eccentricity and argument of perigee control the amplitude and phase of this annual period wave in spatial average insolation. One secular precession vector is proportional to the precession parameter (solid); the other is phase shifted by 90^◦, and is proportional to ecos$(dashed).

100 200 300

Figure 2-17: Similar to Figure 2-16 but for P1 — the inter-hemispheric gradient in insolation. a, b: the leading set of singular vectors, termed the obliquity vectors, account for 99.4% of the variance in the insolation gradient. a, The leading seasonal obliquity vector is a sine wave with an annual period, amplitude of 110 W/m², and zero phase at the vernal equinox (vertical dotted line) — during the equinoxes there is no inter-hemispheric gradient in insolation. The seasonal obliquity vector is zero mean because changes in obliquity redistributes insolation between the hemispheres, but does not change global insolation values. b, the annual variability in insolation gradient is modulated by up to±4% due to changes in obliquity. Because obliquity is always positive, the associated secular singular vector is also a positive function. c,d:

The precessional vectors together explain 0.5% of the inter-hemispheric gradient. c The seasonal variability is mostly controlled by changes in the timing of the inter-hemispheric insolation gradient. As the vernal equinox is fixed to March 20th and, by definition, has no inter-hemispheric gradient in insolation its value must be zero.

The doubly-periodic nature of the seasonal vectors and the large excursion near the autumnal equinox is explained in the text. d Changes in climatic precession can add or subtract up to 20 W/m² from the inter-hemispheric insolation gradient. Note the precession singular vectors are zero mean because precession variations do not affect annual mean insolation at any latitude.

and is a function of latitude (φ), the day (d), and the year (y). The inner summation is over the leading M singular vectors to obtain the weighting for the nth Legendre

100 200 300

Figure 2-18: P2 measures symmetric shifts of insolation from low-latitudes towards the high-latitudes (>52^◦). a,b the leading set of singular vectors accounts for 99.8%

of the variability inP2 (0.0012% due to obliquity variability), and always contributes a negative value (note the secular obliquity vector is negative), primarily accounting for the increased angle of incidence at high-latitudes. a, The seasonal singular vector is proportional to a doubly-periodic cosine wave with zero phase at the vernal equinox plus a mean value of 215W/m². During the equinoxes insolation is at a maximum over the equator, thus the obliquity vectors make a maximal negative contribution.

The double-period reflects the suns twice-annual zenith in the tropics. b, the secular obliquity vector is negative because an increase in obliquity causes greater inter-hemispheric asymmetry and less variance to be explained by the symmetric Legendre polynomial,P2. c,dThe precession vectors together explain 0.2% of the variance inP2

and are nearly the negative of those associated with P0. c When the Earth is closest to the sun during vernal equinox, that is cos($) = 1), the earth receives roughly 10 W/m² more insolation at low-latitudes than at high latitudes. As is generally the case for the Legendre polynomials greater than zero, the seasonal precession vectors are not perfect harmonics because of the change in the timing of the seasons (i.e. frequency modulation) caused by precession and changes in eccentricity. dPrecession can cause symmetric shifts in insolation from high to low latitudes with magnitudes as high as 20W/m². These effects are purely seasonal, however, and average out in the annual mean.

100 200 300

Figure 2-19: P3 primarily describes shifts of insolation from the Northern mid-latitudes to the Southern mid-mid-latitudes. This seasonal vectors are nearly the negative of those forP1, suggesting that when the gradient in insolation is energized, so are the Northern mid-latitudes at the expense of the Southern mid-latitudes. a, bThe obliq-uity vectors describe 99.4% of the variance. During Northern hemisphere summer this mode indicates there are 160W/m² more insolation at Northern than Southern mid-latitudes. Changes in obliquity will modulated this value by up to ±4%. c,d The precessional vectors explain 0.5% of the insolation variability, and again have a doubly-periodic structure due to changes in the timing and amplitude of the seasons.

polynomial as a function of time. TheU_nmatrix represents the seasonal variability in the Legendre polynomials weights. Whennis odd, the annual averageUnweights are zero mean, demonstrating that the net annual insolation received by each hemisphere is equal. The Vn matrix represents the secular variability in Legendre polynomial weights. The secular variations inV are composed of either eccentricity, obliquity, or precession signals depending on then andm. The precession variability inV is zero-mean as there are no net annual changes in I resulting from precession. The leading four singular vectors (M=4) and four Legendre polynomials (N=4) are generally sufficient to reconstruct over 99% of the variability in I.