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

Challenges in Data Assimilation

In my research what I'm trying to do is to prove that data assimilation would be an effective way to gain better results in ocean numerical modelling specifically in the sea ice. However today I read an article about the errors arise from data assimilation. It was an eye opener.

There are three types of errors


1. Temporal error:

This is the kind of error that arise from mismatch of time and space.  For an example the value of temperature that is being measured might be a point data while the data used by the computation model depends upon the time step, usually in my case it's one day. This also applies for the position of data obtained. Matching the grid points and actual position of data obtaining points is quite challenging.

2. Instrumental error:

This is the error rising from the error in instruments. The observational data that we use in assimilation might be wrong and the error can get accumulated with cycles.

3. Assimilation Residuals:

The above errors might be easy to avoid with proper handling of data measurement but residuals are inherent to the concept of data assimilation. Data assimilation tends to nudge the results towards observation results. This might lead to avoid the extreme weather predictions.

Other than these errors, it must also be considered that obtaining observational data is extremely difficult in oceanography.

Ocean Stratification

Stratification refers to the layered nature of the ocean and the atmosphere. Ocean is consisting of several layers. This is a picture of the several layers lies in the ocean.

The mixed layer between the atmosphere and the ocean is called Marine boundary (mixed) layer. The layer from the surface until 100m deep is called the Mixed layer. Most action takes place in the mixed region since this is the layer that gets heated up by the sun.
from 100m 1000m is the isothermal layer. The layer below the isothermal layer is called Thermocline. The temperature decreases in this region with depth. There's very little vertical motion in this region

Ref
http://stream2.cma.gov.cn/pub/comet/MarineMeteorologyOceans/ocean_models/comet/oceans/ocean_models/navmenu.htm

Sigma coordinates.

Sigma coordinates are widely used in oceanography due to the differences in the elevation of the seabed.  The elevation can vary from several meters deep near the coast and several kilo-meters in the deep basins. The regular z coordinates are mostly compatible only with uniform basins.

z is the vertical length from water surface to any point. H is the depth of the total water column. η is the sea surface height that is affected by the tidal force. Sigma varies from 0 to -1 where -1 is at the  bottom earth surface where z=-H and 0 is at the top of water surface z=η.

From Wikipedia

 In sigma coordinates system the number of vertical levels in the water column is the same everywhere in the domain though the depth of water column is different from place to place. The Navier Stokes equation will then be represented in sigma coordinates.

Zonal and meridional

They are used to describe the directions on earth

Zonal refers to the direction from east to west

Meridional refers to the directions from North to South of earth

Zonal temperature or meridional wind flows are some commonly used terms

Reference:

http://en.wikipedia.org/wiki/Zonal_and_meridional