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1. Radially variable spherical viscoelastic Maxwell Earth

The aim of this problem is to generate a set of relaxation times and load and tidal Love numbers for a layered, spherical, Maxwell-viscoelastic Earth for three radial viscosity distributions. All earth models consist of a 70 km thick elastic lithosphere, a layered viscoelastic mantle, and an inviscid core. The viscosity functions are,

Viscosity-depth functions [Pa s]
Radius [km]	earth-1-y	earth-2-y	earth-3-y
6371.	infinity	infinity	infinity
6301.	1.0E+21	7.0E+20	4.0E+20
5951.	1.0E+21	7.0E+20	4.0E+20
5701.	2.0E+21	7.0E+21	1.0E+22
3480.	0.0	0.0	0.0

The elastic parameters are grouped into three different subgroups.

1a. Volume-averaged PREM, incompressible

These three earth models are termed earth-x-a, where x is the number for the viscosity structure given above. The elastic properties are,

Radius [km]	Density [kg/m^3]	Shear modulus [Pa]
6371.	3037.	5.0605E+10
6301.	3438.	7.0363E+10
5951.	3871.	1.0549E+11
5701.	4978.	2.2834E+11
3480.	10750.	0.0

and the bulk modulus is infinite.

1b. Volume-averaged PREM, compressible

These three earth models are termed earth-x-b, where x is the number for the viscosity structure given above. The elastic properties are,

Radius [km]	Density [kg/m^3]	Shear modulus [Pa]	Bulk modulus [Pa]
6371.	3037.	5.0605E+10	5.7437E+10
6301.	3438.	7.0363E+10	9.9633E+10
5951.	3871.	1.0549E+11	1.5352E+11
5701.	4978.	2.2834E+11	3.2210E+11
3480.	10750.	0.0	1.1018E+12

1c. Realistic PREM, compressible

For the third set of calculations a realistical variation of the elastic properties derived from PREM is assumed. These three earth models are termed earth-x-c, where x is the number for the viscosity structure given above.

Output format

Some preliminaries. Years should be 365.25 days * 24 hours * 3600 seconds. G (gravitational constant), moments of inertia C,A and rate of rotation Omega, should be:

G = 6.6732E-11 m^3 kg^-1 s^-2
C = 8.0394E+37 kg m^2
A = 8.0131E+37 kg m^2
Omega = 7.292115E-05 rad/s

Results should be calculated for Legendre degrees 1 to 200, the output format should be, for each Legendre degree n, first line degree, dummy value, elastic load Love numbers (h^e_n, l^e_n, and k^e_n), subsequent lines degree, logarithm of negative inverse relaxation times in 1/ka (log10(-sⁱ_n)) and the modal amplitudes (-hⁱ_n/sⁱ_n, -lⁱ_n/sⁱ_n, and -kⁱ_n/sⁱ_n), e.g.

n	dummy	h^e_n	l^e_n	k^e_n
n	log10(-s¹_n)	-h¹_n/s¹_n	-l¹_n/s¹_n	-k¹_n/s¹_n
n	log10(-s²_n)	-h²_n/s²_n	-l²_n/s²_n	-k²_n/s²_n
...	...	...	...	...

and possibly formatted using "format(i4,f10.5,3(e13.5))". Append the degree 2 tidal Love numbers in the same way at the end of the file.

This format is suitable for both pure and mixed collocation results. For any other numerical method please contact the authors directly.

Polar wander can also be calculated from a normal mode form. However the form will differ slightly depending on whether the Chandler wobble term is removed from the outset of the derivation (eg Peltier & Wu, 1983, GRL, 10, 181-184) or afterwards (eg Vermeersen & Sabadini, 1996, GJI, 127, F5-F9). In the latter case, there are the same number of modes as for the load and tidal Love numbers, but with complex strengths and relaxation times. In the former case, all relaxation modes have real relaxation times and strengths, but the M0 mode is replaced by an elastic term. If the pure collocation method is employed, the weights can be determined numerically as for the calculation of load Love numbers.

First, we define the factor beta = k^T_f /( k^T_f-k^T_e) where k^T_f is the fluid limit of the tidal potential Love number of degree 2 and k^T_e is the elastic tidal potential Love number of degree 2. beta relates the Chandler wobble frequency for a fluid Earth to that of a rigid Earth. Using the nomenclature of Peltier & Wu (1983), please give the elastic term (beta D₁), the secular term (beta D₂) and the log inverse relaxation times log10(-lambda_i) with their strengths (beta E_i)/(-lambda_i). The lambda_i should be in units of (kyr)^-1 and also the secular term D₂, while the strengths (beta E_i)/(-lambda_i) and the elastic term (beta D₁) are dimensionless. As a check that you have the correct normalisation, beta D₁ = beta(1+k^L_e) where k^L_e is the elastic potential load Love number. The suggested form of output is:

elastic	beta*D₁
secular	beta*D₂
log10(-lambda₁)	-beta*E₁/lambda₁
log10(-lambda₂)	-beta*E₂/lambda₂
...	...

using the format(f10.5,e13.5).

If you used the method described by Vermeersen & Sabadini, there will be no elastic term D₁. The suggested form is then:

secular	beta*D₂	.	.
Re(lambda₁)	Im(lambda₁)	-Re(beta*E₁/lambda₁)	-Im(beta*E₁/lambda₁)
Re(lambda₂)	Im(lambda₂)	-Re(beta*E₂/lambda₂)	-Im(beta*E₂/lambda₂)
...	...	...	...

using the format(4e13.5)

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Georg Kaufmann (georg@rses.anu.edu.au)
Paul Johnston (paul@rses.anu.edu.au)
Research School of Earth Sciences
Australian National University
Canberra, ACT 0200
Australia