Numerical Modelling of the Earth's magnetic
field induced by ocean tides
Jan Dostal1, Zdeněk Martinec2, Maik Thomas1
(1)
Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences
(2)
Dublin Institute for Advanced Studies, Dublin 2, Ireland
09.10.2013
Geopotential satellite missions
CHAMP
2000 - 2010
accuracy ~1 nT
SWARM
November 2013
accuracy ~0.1 nT
Magnetic field
Motional induction
GRACE
Gravity field
Earth system
dynamics
Ocean tides
Observed M2-tide signal in nT
(Tyler et al. 2003)
GOCE
Mass redistribution
Electromagnetic induction in the ocean
atmosphere
BE
ocean
u
sediment
Main magnetic
Field BE
x
ocean velocity u
Electromagnetic induction in the ocean
atmosphere
BE
ocean
+++
+++
---
+
u
-
sediment
Main magnetic
Field BE
x
ocean velocity u
Lorentz force:
el. charge deflection
el. field
Electromagnetic induction in the ocean
atmosphere
BE
ocean
+++
+++
---
horizontal j
u
+
-
Main magnetic
Field BE
x
ocean velocity u
Lorentz force:
el. charge deflection
el. field
electric current
density j
vertical j
sediment
Electromagnetic induction in the ocean
poloidal mag. Field B
atmosphere
BE
ocean
+++
+++
---
horizontal j
u
+
-
toroidal mag. Field B
Main magnetic
Field BE
x
ocean velocity u
Lorentz force:
el. charge deflection
el. field
electric current
density j
vertical j
sediment
ocean-induced
magnetic field B
Ocean-induced
magnetic field constituents
poloidal magnetic field
due to conductivity contrast
atmosphere
poloidal magnetic field
due to ocean flow variability
continent
ocean
sediment, crust
toroidal
magnetic field
el. currents
upper mantle
lower mantle
Symmetrically layered model
For the ocean (source layer)
- inhomogeneous differential equation
Electric conductive layers (sediment, mantle)
- homogeneous differential equation
u
BE
Solution and parmetrization
●
●
The induction equation has the form of the Helmholtz equation
→ analytical solution in frequency domain possible
Parametrization in lateral direction
- vector spherical harmonics
●
Description of the source term in vector spherical harmonics
●
Prarmatrization in radial direction:
Numerical solution
Analytical solution
Matrix-propagator technique
Spherical Bessel functions
(Dostal et al. 2012)
validation
Weak formulation
Finite elements
(time domain solution)
Background magnetic field
POMME 6 (Maus et al. 2010)
●
●
toroidal field
– cut-off degre jmax= 6
poloidal field
– dipol term only
in 10 4 nT
Water transport of M2-tide
Ocean velocities:
●
●
●
●
●
●
●
●
OMCT model
tidal dynamic M2-tide (T=12h 42min )
ocean flow with well known period
allows solution in frequency domain
average velocities weighted by
bathymetry (transport)
cut-off degree jmax = 48
only horizontal flow
barotropic flow
Water transport of M2-tide
components
imag. part (sin-part)
real part (cos-part)
in cm/s
Radial profile of electrical conductivity
beneath the ocean
in the ocean layer:
σ = 3.2 – 4.7 S/m
for uniform ocean:
σ = 3.5 S/m
ocean layer depth
h = 4 km
in S/m
Ocean-induced
magnetic field constituents
poloidal magnetic field
due to conductivity contrast
atmosphere
poloidal magnetic field
due to ocean flow variability
continent
ocean
sediment, crust
toroidal
magnetic field
el. currents
upper mantle
lower mantle
Verification of the amplitude
of primary poloidal magnetic
field induction
thin sheet approach
(verticaly integrated values)
finite layer approach
(continent not considerd)
OMCT
in nT
Dostal (2009)
Profile of the poloidal magnetic field
mantle
atmosphere
in nT
Ocean-induced
magnetic field constituents
poloidal magnetic field
due to conductivity contrast
atmosphere
poloidal magnetic field
due to ocean flow variability
continent
ocean
sediment, crust
toroidal
magnetic field
el. currents
upper mantle
lower mantle
Amplitude of the toroidal
magnetic field
at ocean bottom
in 10 -3 nT
Radial profile of the toroidal magnetic field
beneath the ocean
in 10 -3 nT
detail ocean
Amplitude of the secondary
poloidal magnetic field
ocean layer:
σ = 3.5 S/m
continent
σ = 10-3 S/m
in nT
Summary
●
finite layer model for ocean-induced magnetic field
●
considering real radial conductivity profile
●
identical source term (M2-tide):
–
primary poloidal magnetic component
~ 4-6 nT
–
secondary poloidal magnetic component ~ 4-5 nT
(coastal effect)
–
toroidal magnetic field ~ 10-3 nT (barotropic flow)
Summary
●
finite layer model for ocean-induced magnetic field
●
considering real radial conductivity profile
●
identical source term (M2-tide):
–
primary poloidal magnetic component
~ 4-6 nT
–
secondary poloidal magnetic component ~ 4-5 nT
(coastal effect)
–
toroidal magnetic field ~ 10-3 nT (barotropic flow)
=> The secondary poloidal magnetic field achieves the same
magnitudes like the primary poloidal part
Summary
●
finite layer model for ocean-induced magnetic field
●
considering real radial conductivity profile
●
identical source term (M2-tide):
–
primary poloidal magnetic component
~ 4-6 nT
–
secondary poloidal magnetic component ~ 4-5 nT
(coastal effect)
–
toroidal magnetic field (barotropic flow)
~ 10-3 nT
=> The secondary poloidal magnetic field achieves the same
magnitudes like the primary poloidal part
=> The thin layer approximation is not sufficient for sophisticated
prediction of the ocean- induced magnetic field
Thank you for your attention.
Governing Equations
Ansatz for poloidal magnetic field
●
Arbitrary vector field decomposition in spheroidal and toroidal
part
●
Poloidal field is divergence free – spheroidal field in general not
●
Condition for magnetic field
●
Solution
●
Induction equation for the poloidal mode
Toroidal magnetic field
real part (cos part)
imag part (sin part)
+
+
+
+
in 10 -3 nT
Radial profile of the toroidal magnetic field
beneath the ocean
in 10 -3 nT
detail ocean
Poloidal magnetic field
+
+
+
in nT
Magnetic field decomposition
Primary induced
poloidal magnetic field
horizontal electric currents
(vertical integrated value -plane)
Toroidal mag. field
vertical electric currents
ocean layer with finite thickness
(calculation with radial
integrated values not possible)
Secondary induced
poloidal magnetic field
horizontal electric currents
Motivation
●
●
global model for ocean-induced magnetic
field constituent (SWARM)
motionally induced magnetic field as
additional indicator for observation
the ocean dynamic
• single layer modelling
– prediction of the magnetic signal
(Tyler et. al. ,2003 , Kuvshinov & Olsen, 2005)
– CHAMP observation
(Tyler et. al. ,2003)
• 3D model
– surrounding electric conductive space
– secondary effect of poloidal magnetic field
Predicted and observed
M2-tide signal in nT
(Tyler et al. 2003)
Average velocities in shalow regions
ocean volume transport by 4km
ocean depth layer (max 3 cm/s)
average velocities by 1km ocean depth layer;
high velocities in coastal region (max. 6 cm/s)
in 10 -3 nT
Ocean flow with radial decay
Study case
1km ocean depth layer
average velocities
Considering the derivative term
100% linear decay of the ocean
Only nonderivative term
velocities between ocean surface
Tidal dynamic
and bottom
Barotropic velocities
dpt
in nT
in 10 -3 nT
Non barotropic ocean flow
Derivative term – dominant factor for toroidal magnetic field
Numerical Modelling of the Earth's magnetic
field induced by ocean tides
Jan Dostal1, Zdeněk Martinec2,3, Maik Thomas1,4
(1)
Helmholtz Centre Potsdam, GFZ German Research Centre for Geosciences,
Section 1.3: Earth System Modelling, Telegrafenberg, D-14473, Potsdam,
Germany
(2)
Dublin Institute for Advanced Studies, 5 Merrion Square, Dublin 2, Ireland
(3)
Department of Geophysics, Faculty of Mathematics and Physics, Charles
University, V Holesovickach 2, 180 00 Prague 8, Czech Republic
(4)
Institut fuer Meteorologie, Freie Universitaet Berlin, D-12165, Germany
Electromagnetic induction in the ocean
poloidal mag. Field B
atmosphere
BE
ocean
+++
+++
---
horizontal j
u
+
-
toroidal mag. Field B
Main magnetic
Field BE
x
ocean velocity u
Lorentz force:
el. charge deflection
el. field
electric current
density j
vertical j
sediment
ocean-induced
magnetic field B