Longitudes, latitudes and the real length between two points on earth

Earth is a sphere. It’s not a perfect sphere but to make the life of most of the engineers and scientists easy we usually make the assumption that it’s a perfect sphere.  At a glance it might seems that calculating the length between two points on a sphere is a very easy task, but ladies and gentlemen, it’s not.  It took me several hours to figure it out.

First we have the great circle formula.

In the figure GCA is the great circle angle. It is the angle that is created by the two axis that can be drawn from the two points to the center of the earth.

The great circle Formula to find out the great circle angle

Cos(great circle angle) = Sin(latA) Sin (latB) + Cos (latA) Cos (latB) Cos (longA-longB)

The derivation of which is given in the  site below

http://faculty.chicagobooth.edu/alwyn.young/research/Papers/MinimumSphericalDistances.pdf

The great circle formula is getting simplified for the points in the same longitude and the same latitude

For the same longitudes it becomes

Cos(great circle angle) = Sin(latA) Sin (latB)

For the same latitudes it becomes

Cos(great circle angle) = Sin^2(latA)  + Cos^2 (latA)  Cos (longA-longB)

Then we can measure the  distance between any two points using the formula below, where Rearth is the radius of earth

L=Rearth X  great circle angle in radians

Avoiding polar singularity in ocean modelling.

 In general singularity is  a point at which a given mathematical function is not defined. When numerically modelling oceans we come across with the two poles that can be considered as singularity points. 

There are several methods to deal with the singularity points. However in my work we are only concerned about the Arctic pole. We deal with it by rotating the North pole in to the equator. It should be mentioned that this rotation affects the Coriolis force.  I will discuss later how to rectify this affect.

To rotate the coordinates -90 degrees about the y axis we use Euler rotation Matrix. Given the rotation angle is tita the matrix can be represented as below.

Rotation Matrix

Congelation ice formation direction

I took a bit of time to understand this phenomena. Once I understood it, it became really interesting to me since I was working on boundary layer computations in my masters.  Before describing the phenomena let me describe some keywords that would make it easier to understand it.

Congelation ice: This is the ice that grows by freezing in to an existing ice bottom that has already formed in a calm sea.

c-axis: It's the reference axis perpendicular to the plane of a movement of rock or minerals

boundary layer: the fluid layer adjacent to a solid.

corrugated:  a series of parallel ridges and furrows- The free dictionary

Corrugated ice

According to the observations of Langhorne, there seems to be a correlation between the ice formarion direction and the direction of the current. Let's consider the following two figures.

(a) current direction is perpendicular to the grooves parallel to C axis

(b) current direction is perpendicular to the c-axis and parallel to grooves

In the figure (a) current direction is perpendicular to the grooves. This creates mixing in the boundary layer  between the ice and the water. The boundary layer is slightly turbulent. We can also assume that the salinity of this boundary layer is quite high since freezing emits the brine in to the water. The mixing in the boundary layer changes the salinity in figure (a) that makes it more favorable in ice growth. In figure (b) since the groves are aligned with the current direction, we could assume that the boundary layer is laminar and therefore no mixing takes place. It isn't favorable for ice growing. Therefore we  can conclude that ice growing has a tendency to align the c axis parallel to that of the current direction.

Reference: 

Ice in the ocean, Peter Wadhams