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Do we need various assumptions to get a good FCN?

-- A new multiple layer spectral method

Cheng-li HuangMian Zhang

Shanghai Astron. Obs., CAS, China
Journees 2014, 21-24 Sept., @St. Petersburg, Russia


Contents
· ·

Motivation: FCN Methodology:
·

Multiple layer spectral method: Finite Element Method Linear Operator Method

· ·

Our FCN results


Study of Free-core-nutation (FCN)

·

·

FCN is a normal mode of the earth as the rotating axes of the FOC and of Mantle the mantle don't coincide a key parameter & key question to be answered: No fluid OC? No FCN ! · The calculated period of FCN from VLBI 430±1 traditional theory differs largely from the high-precision obs. SG(GGP) 430±5 · FCN reflects (depends on) the theoretical 458470 physics of the FOC, mantle & CMB; calculated · FCN influences strongly the -1yr. nutation due to its resonance.


FCN: Assumptions:

·

extra flattening at CMB: +5%: too big to be consistent with the overall near-HE shape of the Earth as a whole. magnetic/ viscous/topographic couplings @CMB,

·

·
·

2nd order (e2) terms effect?
etc.


Contributions of EMC @CMB to nutation (mas)

(Huang et al., 2011) (Buffett,1992)

(Buffett,2002)
MBH2000

Mathews & Guo (2005); Buffett(2011): + viscous


Do /can we really need these various unproved assumptions to

get a good FCN ?

FCN,FCN,FCN...


Multiple layer spectral method (MLSM) + Linear Operator Method (LOM)


Finite Element Method (FEM)
· ·

Traditional approach solves one order ellipsoid only. FEM can solve more complex models.


Main Idea of FEM
Boundary Surface could be described as:

Let's consider how to solve the dynamic equation:


Traditional Approach: Equivalent Spherical Domain
. P . r


Traditional Approach: Equations of Simple Example
Governing Equations:

Variables:


Traditional Approach: Direct numerical integration approach
Variables
Ordinary Differential Equations


Traditional Approach: Direct numerical integration approach
·

Give a possible period and ODEs' initial values

·

Integrate ODEs from center to surface
Check the integrated values at surface so as to determine whether the possible period is veritable

·


Traditional Approach Problem: the More Complex Figures ? â


FEM: The Whole Domain is separated into several subdomain
·

eg, Earth could be separated into 3 subdomains:
· ·

Solid inner core Fluid outer core

·

Solid mantle (crust)


FEM: Express the Equation in Each Subdomain
The inner and the outer Boundary Surfaces could be described as:

Let's consider how to solve the dynamic equation:


FEM multiple layer spectral method (MLSM)

Outer Boundary Surface

Inner Boundary Surface


Integrate governing eqs. in each subdomain

multiple layer spectral method (MLSM):

Trial vectors(functions)

Dynamic equation

We use MLSM instead of direct numerical integration approach. Variables are expanded in basis function series


MLSM: Galerkin Method
eg, the (2,0) spheroidal displacement field is represented as:

coefficients

Basis function

eg., trial vector(function) of (2,0) spheroidal is:


MLSM: Boundary Conditions
Boundary surface could be described as:

The Normal vector of boundary surface is:


MLSM: Boundary Conditions turn into surface integral


MLSM: Combine All Equations
·

Combine these equations so as to build a matrix:
· ·

Volume integral of governing equations in each subdomain Surface integral of boundary conditions between 2 adjoint subdomain Free surface boundary condition

· ·

No need for the initial value at center. As center condition just require parameters to be reasonable which is the absence of r or r2 basis terms.


MLSM: Search the Period

·

Pick an period and compute the condition number of the matrix.


Linear Operator Method (LOM)

·

Why Use Linear Operator method?
·

Generalized Spherical Harmonics(GSH) are a little bit abstruse. It needs knowledge of group theory and representation theory. Boundary conditions could be easily solved.

·


Linear Operator Method (LOM)
·

Equations is based on spherical harmonics (SH) with unified form:

·

Each SH can be built up by 3 atoms:

·

If we know 3 atoms' actions on each other, all computation about SH are obtained.


LOM Example: Product of two Spherical Harmonics

·

As we know:

·

By

·

We can get


LOM Example: Product of a SH and a VSH
As we know:

·

·

We can get


LOM Example: Dot-product of two VSH
·

Vector Spheroidal harmonics VSH) can be written in
following form:

·

H0 and H1 are combination of SH:


LOM Example: Dot-product of two VSH

·

If we get:

·

We can finally get:


LOM
·

Use this method, we get
·

product of
·
·

2 SH
a SH & a VSH

· · · · ·

dot product of 2 VSH
cross product of 2 VSH gradient of a SH curl of a VSH divergence of a VSH


LOM: Tensors

·

It is difficult to represent the stress tensor in a stand-alone form. But in the equation it only needs the divergence of the tensor, while in boundary conditions it only needs the dotproduct of the normal vector and the tensor, and these two terms can be represented by the LOM.


Our Preliminary Earth Model
·
· ·

PREM
1 order ellipticity 3 layers
· · · ·

without ocean 1 layer for solid inner core 1 layer for fluid outer core 1 layer for mantle and crust (while 10 layers of parameters)


Validation: Tilt-Over-Mode
·

The displacement field is truncated as:
Each term is expanded in polynomial series, in each subdomain

imax
2 3 5 8

TOM
1+1.8e-2 1+1.8e-3 1+1.5e-4 1+2.0e-5

·

Order of Polynomial in trial function


FCN Result
·

The displacement field is truncated as:

VLBI SG(GGP)
Theory calculated

430±1 430±5 458470

·

FCN is very sensitive to the ellipticity at CMB. Our value is equivalent to the most authors': 2.54656*10-3. Result converges when polynomial order imax >= 4.

·

Our Approach

435±3


DiscussionWhy MLSM is Better?
·

avoiding derivatives of some parameters which are not precise in earth model. r ( , , m)


·

r

r ( , , m )dr ( , , m )

MLSM focuses on global characteristics


Discussion & Next Step?
· · ·

Clairaut coordinates (Rochester et al. ,2014): =458 2nd order (e2) terms effect ? truncated coupling chain: +... ?

·

Rotational modes of Jupiter, Saturn & exoplanet

Thanks!



Study of Free-core-nutation (FCN)
·

FCN is a normal mode of the earth as the rotating axes of the FOC and of the mantle don't coincident

·

FCN depends on the physics of the FOC, mantle & core-mantle-boundary. It influences strongly the retro-annual nutation due to its resonance, so it is a key parameter & interesting topic.

Mantle

·

The calculated period of FCN from traditional theory No fluid OC? No FCN ! differs largely from the high-precision observation.
We developed an integrated Galerkin method and spectral element method that can study any antisymmetric earth without GSH. Obs.(VLBI+S 430±1 G) Theoretically 458-465 435

·

·

These methods are applied on the computation of Calculated FCN period from PREM earth without any assumption (eg., extra flattening at CMB, magnetic/ This work viscous/topographic couplings at CMB, etc.). Our result is 435 sid. day !

·

(Zhang & Huang, 2014a,b, c)