Scientific Studies On Sea Ice Engineering Essay
Sea ice is nothing more than frozen ocean water, formed at the polar and subpolar regions in both hemispheres. Sea ice is different from ice that forms from freshwater, with regards to its properties and its formation. A significant difference between freshwater and sea water is that the freezing point of sea water is lower than the freezing point of fresh water (0oC) due to the presence of salt. The temperature at which sea water freezes depends strongly on its salinity, the more saline the water the lower the freezing temperature.
A lot of scientific studies focus on sea ice because of its influence on both climate and living beings. The presence of sea ice, can affect humans through navigation in the polar and subpolar seas and activities like hunting of seals or fishing. Also wildlife is affected by sea ice changes since some animals, like polar bears, live in Polar Regions and their survival depends on the presence of sea ice. Sea ice can affect global climatic conditions through its interaction with the atmosphere and the ocean. In order to understand how sea ice interacts and affects the climate, first we have to understand the physic of sea ice.
During autumn and winter, where the temperature of the air is lower than the temperature of the ocean surface, heat transfer from the ocean to the atmosphere occurs (conductive heat transfer). Thus the temperature of the ocean’s surface water decreases due heat loss. Since the surface sea water becomes colder, its density increases and the denser water starts to sinks being replace by the warmer underlying water, which will then cool and the cycle repeats. For typical ocean salinity percentages there is no threshold temperature where further cooling will cause the sea water to become less dense, like in the case of freshwater (for example in lakes). The above might lead us to the belief that the whole ocean water column would have to reach the freezing point for the initiation of ice formation. In reality, since the sea surface layer is separated from the underlying water by the pycnocline (a big density gradient), only the surface layer water needs to reach the freezing temperature to initiate the formation of ice. Once the sea ice forms, it acts as an insulator between the ocean and the atmosphere, reducing the rate of the heat transfer and slowing the ice growth as the thickness of the ice increases. Another factor that can increase the insulating effect of the ice is the accumulation of snow above it, which causes even more delay in the growth of sea ice.
During summer and spring, when the solar radiation reaching the surface of the earth increases, the melting of the ice is initiated. Even if sea ice has a much higher albedo, thus absorbs much lesser and reflects more solar radiation than the open water, the energy provided to the sea ice is enough to initiate melting. The first sign of melting is small puddles of water on the surface of the sea ice. These puddles of water, by having lower albedo than sea ice, will absorb even more solar energy and further increase the melting of the underlying sea ice. In fact, the contact of the water in the puddles with the warm summer air will accelerate the melting of the surrounding sea ice even more due to the heat transfer from the air to the water inside the puddles. Finally the melting rate can also be increased by the drift of sea ice towards warmer waters.
If the sea ice is not attached to the land (coast), the sea ice will not always stay in its place of formation instead it will drift. Two main factors can initiate and affect the motion of sea ice, the wind and the ocean currents. The wind causes the motion of sea ice through frictional drag, acting on the ice surface. The stronger the wind, the faster sea ice moves and typically, if no other forces act on the sea ice, an open pack of ice will move with a speed of approximate 2% the wind speed. Since we are dealing with friction, one can understand that ice packs with rougher surfaces move faster. In the existence of an ocean current we have sea water in motion, as the sea water moves it drifts, along with it, the sea ice. The higher the velocity of a current, the faster sea ice moves. Since the velocity of currents decrease with depth, the greater the depth of a sea ice pack, the slower its motion.
When the top layer of the ocean reaches the freezing point, small needle-like crystals start forming. These crystal forms are called frazil ice and they consist of nearly fresh water since during their formation they dismiss salt into the ocean. In calm ocean conditions, the needle-like ice crystals combine by freezing together and create a thin crust of ice called Nilas. Nilas surface is matte and its thickness does not exceed 10cm. Next Nilas grows thicker at the bottom through a process called congelation growth. On the other hand, in a rough ocean frazil ice is accumulated, with the help of winds and the ocean waves, into circular forms of ice called “pancake ice”. With time, these “pancakes” bond together into a more solid form. In both cases (calm and rough ocean), the sea ice continues to grow and becomes thicker during the winter, with the sea ice product of one winter being called as “first year ice”. During summer, sea ice starts melting; the ice formed during the past winter, that does not melt totally, and instead survive until the next winter, is called as “old year ice”.
During the formation of sea ice the water that freezes, expels salt into the underlying ocean; as a result the surrounding water salinity increases. In contrast, when sea ice melts (temperature of sea ice is raised above freezing point), fresh water is provided to the surrounding ocean, make it less saline. Not all the salt, from the formation of sea ice, is provided immediately to the ocean. As the ice crystals are forming, the salt that is expelled is accumulated into small drops called brine. When the ice crystals combine, brine is trapped into cracks between the crystals. Since brine is more saline, it requires a lower temperature to freeze thus it remains in liquid form in these cracks until it drains out and air replaces it.
1.2.5 Global climate
Sea ice affects as much as it is affected by the global climate. Sea ice and the atmosphere are closely connected and influence each other. Sea ice works as an insulator between the atmosphere and the ocean. If changes in the sea ice occurs, for example if sea ice becomes thin enough, the heat exchange between the ocean and the atmosphere become possible thus the atmosphere can absorb heat from the ocean, in the polar areas. The absorption of heat will result in a rising of the temperature of the atmosphere, thus the global circulation of the atmosphere can be affected. This will result in the further melting of the ice.
The ocean is also affected by sea ice through fresh water or brine release, associated with melting and freezing processes. Sea ice drifts from the Arctic into the North Atlantic, by melting it releases fresh water in the surface layer. The excessive melting of ice in high latitudes will result in the release of excessive amounts of fresh water, which as it is less dense than saline water does not sink. This leads to the suppression of convection and thus the suppression of deep water formation. Hence the large scale ocean circulation, driven by global density gradients created by surface heat and freshwater fluxes (thermohaline circulation), which contribute to a proper temperature balance around the earth, may be impacted (Yang et al 1993).
1.3 Canadian East Coast (below 60oN)
1.3.1 Labrador and Newfoundland shelves
The Labrador and Newfoundland shelves are located on the west border of the Atlantic Ocean, from approximate 60o North to 45o North latitude (fig. 1.3.1). These shelves are generally very shallow in contrast to the very deep interior of the Labrador Sea. Sea Ice is a major feature along the Labrador and Newfoundland shelves, greatly affects the climatic conditions and the marine life of the region. Another aspect that makes sea ice so important in this region is its possible influence on deep water formation, in the interior of Labrador Sea, as the melting of sea ice can provide additional fresh water to the surface layer of the central Labrador Sea.
Fig. 1.3.1: Labrador and Newfoundland shelves. The Labrador Current is represented by light blue arrows.
1.3.2 Sea Ice in the region
Along the Labrador and the Newfoundland shelves, the Labrador Current (fig. 1.3.1) flows with a maximum surface speed of around 0.5 m/s and has a width of 50km. The Labrador Current water originates, mainly, from the Arctic, so it is really fresh and cold. This cold and fresh water is separated by a strong shelf break front from the warmer and more saline waters offshore; thus ice only exists along the coast and it is rarely found in the interior of the Labrador Sea. Some sea ice drifts, from the Arctic, through Davis Straight, in the Labrador shelf with the help of the Labrador Current, but the most amount of the sea ice is primarily formed in the Labrador shelf due to the cold and dry winds that blow above the region. The ice drifts further south with the help of the Labrador Current and the strong northwesterly winds blowing along the coast. In an average year, sea ice appears in December, drifts southward towards the Newfoundland shelf and reaches its maximum quantity by March. Melting usually begins in April and the sea ice disappears completely by the end of June (Canadian Ice Service).
1.3.4 Why study this region
As we already stated, sea ice is a significant component of the climatic system. Knowledge about sea ice in the region, is vital for navigations since ice can be hazardous to ship activity and so it is essential to know how much and how thick the sea ice is, in order to determine the most suitable rout for a ship to follow. For an economy partially based on fishing, as the one of Labrador and Newfoundland regions, we can understand how important the knowledge about the conditions of sea ice is, since they affect the phytoplankton and zooplankton distribution, thus the fish distribution. Another reason that makes this region so interesting is the fact that sea ice disappears completely in the summer; so different issues and procedures must be taken into account when we study it in comparison to places where ice never melts (like Arctic) and where most of the studies are focusing. Finally sea ice can influence the atmosphere and ocean conditions, through its melting and formation, and control the rate of exchange fluxes between the ocean and the atmosphere, thus influence the climatic conditions of the area.
1.4 data assimilation
Forecasting is usually used in the geophysical sciences to predict the state of a system (e.g. atmosphere and ocean) at some time in the future. Generally two methods are used for forecasting, the empirical approach and the dynamical approach. The empirical approach is based on the analysis of historical data and the occurrence of similar situations. The dynamical approach is based upon equations that predict the evolution of a system. In practice, the dynamical approach is based on computer modeling and called numerical prediction. The forecast is computed with the help of mathematical equations describing the physics and the dynamics of a system. Ocean and sea ice models have been used generally to simulate and predict the properties of ocean and sea ice, respectively. Since sea ice and ocean properties are closely linked and can influence each other, ocean and ice models can be combined together (coupled ocean/sea ice models) in order to achieve better forecasting of both the sea ice and the ocean. Thus ocean/sea ice coupled models have been developed and used in areas where the presence of the ice is noticeable, like the east coast of Canada (Ikeda, 1989/ Yao, 2000).
To run and produce a forecast, models must start from an initial state based on observations. To reduce uncertainties in the model it is essential to have an initial model state as close as possible to reality. The discontinuities that can occur in the observations, the lack of data and the errors that exist even in the most modern observation tools, are responsible for the drift of the observation data from reality. This is where data assimilation comes into play, making the first appearance in meteorology about 50 years ago, where all the available observations, for a time period, are merged into the model and, through reanalysis, better initial conditions are prepared. As in meteorology data assimilation is applied in oceanography for the same purpose. Moreover, data assimilation can be used for the improvement of boundary conditions and to some extent compensate for imperfect model physics or inadequate model resolution in order to provide estimates in agreement with nature (Malannotte, 1996).
Data assimilation progresses in analysis cycles. In every cycle the observations, of a current state of a system, are combined with a background information, which usually has the form of a forecast by the model, in order to balance the uncertainties in the forecast of the model and the errors of the observations and produce a more accurate reanalysis. Then the model progresses in time and its results become the forecast for the next cycle and the procedure repeats
The use of data assimilation began first in meteorology due to the availability of measurements for atmospheric properties. In oceanography due to the sparse of observations for ocean properties, data assimilation became vital, later, with the development of satellites and projects that provide high density and more accurate observations.
The first step towards data assimilation was the objective analysis introduced by Panofsky (1949) and modified further by Cressman (1954, evolve of his technique in 1959). In Cressman analysis the corrections of the initial forecasting of every gridpoint (background field), are based on a weighted combination of the differences between the observed values and the predicted values, within a radius of influence. The weight of each difference depends only on the distance between the background gridpoint and the position of the observation. This method was evolutionary in comparison to the previous methods of analysis (subjective analysis) where the forecast corrections were based on individual opinions. The mathematic formulation of a simple Cressman analysis is:
Eq 1.3.1 and eq 1.3.2: xa(j) denotes the analyzed model state at grid point j, xb(j) denotes the background model state at grid point j, y(i) denotes the observation vector at the point i where the observation took place, xb(i) is the background state interpolated to point i, w(i,j) denotes the weight function, R is a constant which represent the influence radius and di,j represents the distance between the grid point j and the observation point i.
The big breakthrough in data assimilation came from Gandin (1963) and the introduction of a statistical approach, referred to as optimal (or statistical) interpolation. The weights used in optimal interpolation are related correctly to the observation errors and on the contrary with objective analysis, the weights are also related with the background errors so now the background fields are used as source of information and not only as an initial state. The analysis equation has the below form:
Eq 1.3.3: xa and xb denote the analyzed and background model state correspondingly, y represents the observations, H is called observation operator and basically it maps from the analysis space to the observation space and W is called the optimal weighted matrix (is explained in eq 1.3.4) .This equation says that the analysis is obtained by adding a correction term to the background field. This equation can be regarded as a list of scalar analysis equations, one per model variable.
Eq 1.3.4: where B denotes the background errors covariances (the background errors denote the difference between the background field, just before the analysis, from the true state), R the observation errors covariances (the observation errors denote errors in the observations, representativeness errors and errors in the design of the operator H), H and HT represent the linearalized observation operator and its transpose. (More information about covariances at Appendix B)
In most recent years (after 1990) variational analysis methods became more popular. The basic idea behind these methods is the use of observation to correct the least accurate points of the model without allowing the modification of the most accurate points. These methods have the advantage of higher accuracy but they have too much computational cost, in comparison with the optimal analysis methods. The variational methods work by looking for the analysis field (xa) that minimizes a cost function. The mathematical form of this cost function is:
Eq 1.3.5: J(x) called cost function of the variational analysis, x and xb denotes the model state and the background model state vector correspondingly, y represents the observation vector, H is the observation operator, B and R denotes the background and the observation error covariances.
2.1 NEMO Sea-Ice/Ocean Coupled Model
As we have already seen sea ice interaction with ocean can influence both ocean and the sea ice properties. Thus a model with two components, one for the sea ice and one for the ocean, that are coupled, can be beneficial for hindcasting and forecasting both the sea ice and the ocean. In our study we used the NEMO (Nucleus for European Modelling of the Ocean) sea ice/ocean coupled model with OPA (Ocean General Circulation model) as the oceanic component and LIM (Louvain-la-Neuve sea Ice Model) as the sea ice component.
2.1.1 OPA model
As a fluid the ocean can be described through equations used in fluid dynamic. In order to be able to solve these equations the following assumptions have to be made (OPA 8.1 Reference manual, NEMO ocean engine):
Spherical Earth approximation: The gravity is assumed to be parallel to the earth radius.
Thin shell approximation: The ocean depth can be assumed small enough to be neglected in compare with the earth’s radius, thus is neglected. This means that in every depth of the ocean (from the bottom to the surface) g is constant.
Turbulence closure hypothesis: The unresolved processes (small scale processes) are represented in resolved parameterization.
Boussinesq hypothesis: The density differences are small enough to be neglected except in the buoyancy term.
Incompressibility hypothesis: The divergence of velocity of the fluid flow is zero (divU=0).
Hydrostatic Hypothesis: The vertical momentum equation is reduced to an equation where the buoyancy force is balanced by the vertical pressure gradient. This leads to the removal of the convective term from the Navier-Stoke equation (this term must be parameterized).
Using the above assumptions we obtain the following six equations correspondingly, the momentum balance (eq. 2.1.1), the hydrostatic equilibrium (eq. 2.1.2), the incompressibility equation (eq. 2.1.3), the heat conservation (eq.2.1.4), the salt conservation (eq. 2.1.5) and an equation of state (2.1.6). The equations are represented in a coordinate system with orthogonal set of unit vectors (i,j,k), such as k is the local upward vector and i,j are horizontal with i directed to the North and j directed to the East (OPA 8.1 Reference manual, NEMO ocean engine):
Where U is the velocity with Uh horizontal vector component and wk vertical vector component, T is the potential temperature of the ocean, S the salinity of the ocean, ρ the density and ρo a reference density of the ocean, g the gravitation acceleration, p the pressure and f=2Ωk the Coriolis acceleration (Ω is the Earth angular velocity vector).Where DU, DT and DS are the parameterizations of small scale physic physics for momentum temperature and salinity correspondingly and FU, FT, FS are surface forcing terms.
The small scale motions (turbulence) cannot be solved by the usual large model resolution; thus the turbulence motions are parameterized (subgrid scale physics DU, DT and DS). Due to the strong anisotropy between the vertical and the lateral motion the subgrid scale physics is divided into lateral and vertical part. The vertical part parameterization is analogous to that of molecular and diffusion and dissipation. For the lateral subgrid scale physics a spatially varying GM parameterization is used for salinity and temperature (Gent and McWilliam, 1990)…. (Momentum and vertical).
The ocean exchanges fluxes with atmosphere, solid earth and sea ice. These fluxes are exchanged at the interface (boundary) of the ocean with the rest earth system components and presented briefly as (OPA 8.1 Reference manual, NEMO ocean engine):
Coast/land-ocean interface: The major exchange of flux that can be taken into account is the freshwater provision, to the ocean, though the discharges. This freshwater supply can alter locally the surface salinity of the ocean. For short range integrations this flux can be neglected, like in our case.
Bottom of the Ocean-Ocean interface: The model neglect heat and salt exchange between the ocean and its bottom. Since there is no flow in or out solid boundaries the ocean’s velocity at the bottom is parallel to the solid boundaries. On the other hand the ocean exchange momentum with the solid boundaries through friction and since this is a small scale process it needs to be parameterized.
Atmosphere-Ocean interface: Heat, momentum (wind stress) and freshwater (Precipitation-Evaporation) is exchanged between the Atmosphere and the Ocean.
Sea ice- Ocean interface: Ocean and Sea ice exchange freshwater, momentum, heat and salt. We discuss further these exchanges in section 2.1.3.
For better representation a curvilinear geopotential system is used (three dimensional orthogonal grids on a sphere). All the equations used by the model are transformed in the above coordinate system and then numerically solved based on the traditional centered second order finite difference approximation (further details on appendix C). The ocean model uses a C grid arrangement (further details on appendix C).
APPENDIX A (basic linear matrix algebra)
A matrix is an array of numbers. An A n x p matrix contain n rows and p columns and its element can be denoted as ai,j for (i=1:n and for j=1:p), where i is the row index and j the column index. Specifically this matrix can be expressed as:
-Square matrix: A matrix is called squared when the number of its rows is equal to the number of its columns.
-Diagonal matrix: A matrix is diagonal when all its non diagonal elements are equal to zero (ai,j=0, i≠j) and at least one of the diagonal elements is non zero. A diagonal matrix with all his diagonal elements equals to 1 is called an identity matrix I.
-Size and equality: Two matrices are the same size if they have the same number of columns and rows (same dimensions). Two matrices A and B are equal when they are same size and all their elements are equal (ai,j=bi,j for every i,j).
-Transpose: A transpose of a n x p A matrix is a p x n matrix denoted as AT, whose every ai,j element is equal to the aj,i element of A. A matrix is characterized as symmetric if A= AT. From this one can understand that diagonal matrices are symmetric matrixes.
-Summation: The summation of two same size matrices A and B, gives a same size matrix C=A+B, with elements ci,j=ai,j+bi,j. The addition is commutative i.e. A+B=B+A. As a rule: (A+B)T=AT+BT.
-Multiplication: A n x p matrix A times a scalar c, is defined as a same size matrix cA with elements cai,j. Generally (cA)T=cAT. Given two matrices A and B, the matrix multiplication is defined only when the number of the columns of A is equal to the number of the rows of B. The product between a n x p matrix A and a p x q matrix B is defined as a n x q matrix C=AB with elements ci,j given as:
As a rule: AB≠BA, A(BC)=(AB)C, A(B+C)=AB+AC and (AB)T=BTAT
A special type of matrix multiplication that is called Kronecker product, can defined for matrices without size restrictions. For two matrices A (n x p) and B (m x q) Kronecker product is defined as the nm x pq matrix:
The Kronecker product is not commulative (A⊗B≠B⊗A).
-Invertible matrix: A square n x n matrix A is invertible if a matrix n x n A-1 , called inverse, exists by satisfying A-1A=A A-1=I. As a rule: (AB)-1=B-1A-1, (AT)-1=(A-1)T.
-Determinant The determinant of an A square matrix, with Aij denoting the sub-matrix obtained from A by deleting the ith row and the jth column; is defined as:
The determinant of a 2 x 2 matrix A is simply a11a22-a12a21. A square matrix with a non-zero determinant is called nonsingular and a square matrix with a zero determinant is called singular.
- Trace: The trace of a square matrix is the sum of its diagonal elements and is defined as the scalar:
As a rule: Tr(AB)=Tr(BA), Tr(AT)=Tr(A) and Tr(B-1AB)=Tr(A)
-Eigenvalues, eigenvectors: For a square matrix A, if we have Ax=λx for a scalar λ and a non-zero vector x, then x is an eigenvector of A corresponding to the eigenvalue λ. The system (A-λI)x=0 has a non trivial solution if, and only if, det(A-λI)=0 (A-λI is singular).
APPENDIX B (STATISTIC)
Let us assume that we have an X set, of n data. Then the mean of this set of data is:
And the standard deviation of the data set is:
The standard deviation of a data set is a measure of how spread out the data is. Another measurement of the spread of data and almost identical as the standard deviation is the Variance of the data set:
Standard deviation and variance operate only in 1-dimension. If we have a data set of 2-dimensions and more, we can only calculate independently the variance and the standard deviation of each dimension. In order to be able to see how much two dimensions (or two variables) change together we have to introduce another term, the covariance, which is measured between two dimensions (two variables). The variance is actually a special case of covariance where we find the covariance of a variable with itself. The formula for the covariance is:
We can understand from the above formula that cov(X,Y)=cov(Y,X). In a multidimensional system, a practical way to gather the covariances between each dimension is to put them all in a square matrix, the so called covariance matrix. If a vector has n dimensions (variables) then the covariance matrix will be an n x n matrix, where its diagonal contains variances of each variable of the vector and its off-diagonal contains cross-covariances between each pair of variables of the vector. To make it more clear let us assume a 3-dimensional data set X of dimensions x, y and z (X(x,y,z)). The covariance matrix is illustrated as:
The matrix is symmetrical about the main diagonal. The above lead us to the general definition of the covariance matrix of an X data set (where we have already subtracted the mean) as:
The correlation coefficient between two random variables is very similar to the covariance. Its formula is:
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