L19 Geomagnetic Field Part I


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1 Intoduction to Geophysics L191 L19 Geomagnetic Field Pat I 1. Intoduction We now stat the last majo topic o this class which is magnetic ields and measuing the magnetic popeties o mateials. As a way o motivating this topic let s ecall a little about the histoy o the idea o continental dit. Remembe it was way back in 191 that Aled Wegene suggested continental dit. Wegene was a meteoologist and was especially inteested in paleoclimatology, and ound that by plotting his paleoclimatic data on his supecontinent econstuction that these data omed climatic belts just like seen today. E.g., he could see the equatoial topical ain belt, two adjacent dy belts, two tempeate ain belts, and two pola ice caps. But, the biggest stumbling block o continental dit was its inability to explain the mechanism by which dit took place. Wegene basically thought subcustal mateial was capable o viscous yield ove long peiods o time which allowed the continents to dit though the ocean cust like ships though wate. This idea eceived a lot o skepticism. Fo example, one o ou heo s om seismology Si Haold Jeeys agued that it was physically impossible o a lage mass o solid ock to plow though the ocean loo without beaking up. Jeeys held a lot o sway at the time and people believed him besides that he had the distinction o being ight. But, in the 195s people wee stating to look at paleomagnetism which is the ecod o the Eath s ancient magnetic ield. Thee wee eally main lines o evidence om paleomagnetic wok that ultimately paved the way o plate tectonics: 1) Appaent pola wande the appeaance that the pole has shited with time. ) Sea loo speading. So, we will next show how studying the Eath s magnetic ield led to the discovey o these two items.
2 Intoduction to Geophysics L19. Magnetic Field and Potential To a ist appoximation the magnetic ield o the Eath (the geomagnetic ield) is a dipole ield. That is, we can epesent the geomagnetic ield by a magnetic dipole situated at the cente o the Eath (like a ba magnet at Eath s cente). One thing to note in this pictue is that the geomagnetic noth pole is called so because the Nend o a compass needle points to it (it is theeoe actually a magnetic Spole). To ist ode, a easonable it with the Eath s magnetic ield can be made by a magnetic dipole appoximation. The best itting dipole is aligned ~11.5 with espect to the Eath s spin axis (i.e., the geogaphic nothsouth axis).
3 Intoduction to Geophysics L19 Keeping that in mind, let s deine some things in the above pictue: geomagnetic poles: the two points at which the axis o the bestitting dipole intesects the Eath s suace (geomagnetic noth and south poles). magnetic poles: the two points on the Eath s suace at which the magnetic ield is vetical (and has no hoizontal component). geomagnetic equato: the equato o the best itting dipole axis. magnetic equato: the line along which the magnetic ield is hoizontal and has no vetical component. An impotant point to emembe hee is that i the ield was exactly a dipole ield the magnetic poles and equato would align coincidentally with the geomagnetic poles and equato. Howeve,
4 Intoduction to Geophysics L194 the magnetic ield o the Eath is not exactly a dipole ield. The dieence between the bestit dipole ield and the Eath s magnetic ield is called the nondipole ield. Instead o just using a dipole; the Eath s magnetic ield can be descibed in geate detail using spheical hamonic analysis. Spheical hamonics descibe the vaiation o magnetic ield on a spheical suace. This is simila to a Fouie seies. With a Fouie seies we can explain a seies (e.g., a time seies) as an ininite sum o sine and cosine unctions. Recall, the Fouie seies epesentation has a om that looks like this: a ( x) an cos n sin n1 Similaly, spheical hamonics ae expansions like: nx b nx ( ) ( ) ( ) ( ) m whee P cos n ae Associated Legende Polynomials (with Schmidt nomalization). The point is that similaly to Fouie seies, we can expess the magnetic potential as an ininite sum o weighted sine and cosine unctions but on a spheical suace. What this ultimately means is that instead o descibing Eath s magnetic ield as a simple dipole we can make a bette desciption by adding moe tems:
5 Intoduction to Geophysics L195 But, o simplicity we will just conside the magnetic ield associated with a dipole. So, the magnetic potential o a dipole is witten as: ( ) This is just the ist tem, in the ininite seies o spheical hamonics. This gives the magnetic scala potential (V) due to a dipole situated at the cente o the Eath at any point given by the position vecto. Recall back to ou lectues on gavity: in ode to move a test mass away om the attacting mass wok has to be done against the attactive oce, which is equal to the gain o potential enegy o the test mass. When the test mass was a unit mass, the attactive oce was called the gavitational ield and the gain in potential enegy was called the change in gavitational potential. Magnetic potential is simila but depends on how much wok must be done to move a magnetic monopole away om anothe monopole. The value m is the dipole moment. This is a vecto aligned with the dipole axis. Basically, this gives the stength o the systems magnetic souce. Speciically it is popotional to the toque exeted by a magnetic ield to tun a magnetic dipole paallel to the ield diection. Fo the Eath, the magnitude o the dipole moment is: Fom the magnetic scala potential we can detemine the magnetic ield by: ( ) ( ) Whee, μ magnetic pemeability o ee space (o moe oten just called the magnetic constant):
6 Intoduction to Geophysics L kg m A s So, we know we can to ist ode assume a dipole ield o the Eath, but what is the magnetic ield? Hee we assume we can model the geomagnetic ield as the ield o a dipole aligned along the geogaphic NS axis. Although this is not quite tue at pesent, on aveage the dipole axis has been aligned with the geogaphic axis. I not, all paleomagnetic estimates o past positions o ock samples would be meaningless, as they would be elative to the position o the geomagnetic pole at the time each sample acquied its magnetization. So, we ind the magnetic ield due to a dipole in the ollowing coniguation: Whee, B = the adial component o the magnetic ield, and B θ is the hoizontal component. This poblem demonstates adial symmety which should immediately make us think that the easiest way to solve it is in spheical coodinates.
7 Intoduction to Geophysics L197 Review: Spheical Coodinates In Catesian coodinates we deine ou position in space by its location in x, y, and z. But in spheical coodinates we deine ou position by, θ, φ. In paticula: as the distance om the oigin to ou point. θ the angle between the positive zaxis and ou adial line ( θ 18 ). φ the angle between the positive xaxis and the pojection o ou adial line onto the xy plane. With these deinitions we can detemine the spheical coodinate position om the Catesian coodinate position om the ollowing omulas: x tan cos y y x z O, i we have the position in spheical coodinates we can ind the Catesian coodinate position with the ollowing omulas: z cos z x sin cos y sin sin
8 Intoduction to Geophysics L198 Retuning to ou poblem: Assuming that ou dipole is aligned along the negative zaxis: ( ) Recall that we can wite ou vecto dot poduct as: a b a b cos. Hence, ( ) ( ) To get the magnetic ield we use the omula: ( ) ( ) ( ) [ ] So, we can easily calculate the values o the vaious components o the magnetic ield: Radial component: B,, V mcos 4 mcos 1 4 mcos 4 mcos 4 Hoizontal component: 1 mcos B,, 4 m cos 4 m sin 4 msin 4
9 Intoduction to Geophysics L199 φ  component: mcos B,, sin 4 (deivative o a constant = ) Magnitude: ( ) 4 m cos m sin m 4cos sin 4 So, by way o example and to see i these equations make sense, what ae the components o the magnetic ield at the equato (θ = 9 ): mcos B,9, 9 4 msin 9 B,9, 4 m 4 Does this answe make sense? Hee s a quick diagam, whee ou measuement point at the equato is indicated by the light blue cicle. So, as we know the magnetic ield at the equato should point due up. And as we can see, the adialcomponent = implying thee is no outwad component o the Bield. Remembe, ou θ
10 Intoduction to Geophysics L191 component is measued as an angle with espect to the positive zaxis. Hence the negative θ component hee indicates the Bield points in the upwad diection. Just as expected. Calculating the components o the magnetic ield at the Noth Pole: B, mcos, 4 m B,, msin 4 Does this make sense? Daw a pictue to conim this. So, the above equations ae valid inside and above the suace o the Eath. But, i we ae just concened with the Eath s suace we can deine a constant B : B m R Whee R is the adius o the Eath. Now, plugging this into ou equations o the components o the magnetic ield: Next time, we will begin ou discussion on how these equations apply to ock magnetization.
11 Intoduction to Geophysics L1911 Review: Gadient The gadient is sometimes symbolized by eithe gad o (i.e., an upside down delta symbol). Fo example: gad o, Whee is a unction o scala ield. The gadient is a vecto ield that which points in the geatest ate o incease in the scala ield. An example is a hill. In this example, think o elevation as the scala ield. Then the gadient at any point on the hill is a vecto that points in the diection o the steepest slope at that point. The magnitude o that vecto will tell you how steep the slope is at that point. Mathematically we calculate the gadient as: Catesian Coodinates: z z y y x x z y x,, Spheical Coodinates: sin 1 1,,
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