If you wish to contribute or participate in the discussions about articles you are invited to contact the Editor

Pole Tide

From Navipedia
Jump to navigation Jump to search

Title Pole Tide
Author(s) J. Sanz Subirana, J.M. Juan Zornoza and M. Hernández-Pajares, Technical University of Catalonia, Spain.
Level Intermediate
Year of Publication 2011

The instantaneous earth rotation axis shifts inside a square of about [math]\displaystyle{ 20 }[/math] meters in relation to a point with fixed coordinates on the earth (i.e., Chandler wobble with a period of [math]\displaystyle{ 14 }[/math] months). This entails a varying elastic response of the earth's crust. This has an effect smaller than [math]\displaystyle{ 2.5 }[/math] centimetres in vertical and [math]\displaystyle{ 0.7 }[/math] centimetres in horizontal, but must be taken into account if the observations are carried out over periods longer than two months.

From the IERS Conventions [Denis et al., 2004] [1], pages 83-84, the following expression [footnotes 1] can be derived for the displacement at a point of geocentric latitude [math]\displaystyle{ \displaystyle \varphi }[/math] and longitude [math]\displaystyle{ \displaystyle \lambda }[/math]:

[math]\displaystyle{ \begin{array}{ll} \delta_{\hat{\mathbf r}}=&-\frac{\omega^2_E \, R_e}{2\,g}\,h \, \sin 2\varphi \left ( m_1 \cos \lambda + m_2 \sin \lambda \right ) \,\hat {\mathbf r}\qquad \\[0.3cm] \delta_{\hat{\mathbf \lambda}}= & -\frac{\omega^2_E \, R_e}{g}\, l \, \sin \varphi \left (-m_1 \sin \lambda + m_2 \cos \lambda \right ) \, \hat{\mathbf \lambda} \qquad \\[0.3cm] \delta_{\hat{\mathbf \varphi}}= & -\frac{\omega^2_E \, R_e}{g}\, l\, \cos 2\varphi \left (\, m_1 \cos \lambda + m_2 \sin \lambda \right )\, \hat{\mathbf \varphi}\qquad \end{array} \qquad\mbox{(1)} }[/math]

where ([math]\displaystyle{ \displaystyle m_1 }[/math],[math]\displaystyle{ \displaystyle m_2 }[/math]) are the displacements (in meters) from the 1903.0 CIO, pole position, and [math]\displaystyle{ \displaystyle h= 0.6027 }[/math], [math]\displaystyle{ \displaystyle l=0.0836 }[/math] are the Love numbers.

Taking the earth's angular rotation [math]\displaystyle{ \omega_E= 7.29\cdot 10^{-5} rad/s }[/math], the earth's equatorial radius [math]\displaystyle{ R_e=6378\cdot 10^3\, m }[/math] and the gravitational acceleration [math]\displaystyle{ g=9.8\, m/sec^2 }[/math], it follows:

[math]\displaystyle{ \begin{array}{ll} \delta_{\hat{\mathbf r}}=&-32 \, \sin 2\varphi \left ( x_1 \cos \lambda + x_2 \sin \lambda \right ) \, \hat {\mathbf r}\qquad \mbox{(mm)} \\[0.3cm] \delta_{\hat{\mathbf \lambda}}= & -9 \, \sin \varphi \left (-x_1 \sin \lambda + x_2 \cos \lambda \right ) \, \hat{\mathbf \lambda} \qquad \mbox{(mm)} \\[0.3cm] \delta_{\hat{\mathbf \varphi}}= & -9 \, \cos 2\varphi \left (\, x_1 \cos \lambda + x_2 \sin \lambda \right )\, \hat{\mathbf \varphi}\qquad \mbox{(mm)} \end{array} \qquad\mbox{(2)} }[/math]

where ([math]\displaystyle{ \displaystyle x_1 }[/math],[math]\displaystyle{ \displaystyle x_2 }[/math]) are the displacements (given in seconds of arc). Pole displacements can be found at ftp://hpiers.obspm.fr/iers/eop/eop.others.

The displacement [math]\displaystyle{ \displaystyle \delta }[/math] is given in the radial, longitude and latitude [math]\displaystyle{ (\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi}) }[/math] vectors (positive upwards, eastwards and northwards, respectively). Thus, the displacement vector in the (x, y, z) ECEF Cartesian coordinates is given by

[math]\displaystyle{ \left[ \begin{array}{c} \Delta x \\ \Delta y \\ \Delta z \end{array} \right]= \underbrace{\left( \begin{array}{ccc} \cos \lambda \cos \varphi & -\sin \lambda & -\cos \lambda \sin \varphi \\ \sin \lambda \cos \varphi & \cos \lambda & -\sin \lambda \sin \varphi \\ \sin \varphi & 0 & \cos \Phi \end{array} \right)}_{{\mathbf R}_3(-\lambda )\cdot {\mathbf R}_2(\varphi )} \left[ \begin{array}{l} \delta_{\hat{\mathbf r}}\\ \delta_{\hat{\mathbf \lambda}}\\ \delta_{\hat{\mathbf \varphi}} \end{array} \right] \qquad\mbox{(2)} }[/math]

Figure 1: Transformation from UEN [math]\displaystyle{ (\hat{\mathbf r}, \hat{\boldsymbol \lambda}, \hat{\boldsymbol \varphi}) }[/math] to TRS [math]\displaystyle{ (x,y,x) }[/math] coordinates.

where [math]\displaystyle{ {{\mathbf R}_3(-\lambda )\cdot {\mathbf R}_2(\varphi )} }[/math] are the rotations in latitude (1) and longitude (2) indicated in figure 1 (see Reference Frames in GNSS).


  1. ^ Notice the use of latitude [math]\displaystyle{ \displaystyle \varphi }[/math] in equations (1 and 2), instead of the co-latitude [math]\displaystyle{ \displaystyle \theta }[/math] used in the IERS equations.


  1. ^ [Denis et al., 2004] Denis, D., McCarthy and Petit, G., 2004. IERS Conventions (2003). IERS Technical Note 32.. IERS Convention Center., Frankfurt am Main.