An Estimation of Normal Mode Oscillation in the Ocean.
The basic property of the Ocean or Atmosphere is that they are highly stratified. Simply speaking, in the case of Oceans, the high density water always stays deeper, because it is heavy. The lighter ones stays one over the other according to its density decreases with height. For instance, if the water in the upper level gets heavier than the lower layer, the system is unstable. The heavier water have to sink down and the lighter water has to come up. This is accomplished by a process called 'vertical convection'. This process is quite common in the surface layer of the ocean, where the water gets heavier (saltier) by evaporation. This stratification in the Ocean (as well as in the atmosphere), and the tendency of it to restore the stability with a stable stratification causes an important property of the system. Any parcel of water displaced vertically from its stably stratified form, the restoring tendency gives a fundamental mod of oscillation within the system. The fundamental quantity, we usually refer to such a system, is so called the 'buoyancy frequency' or Brunt Vaisala frequency. For a layer structure of the Ocean, this can be defined as squar(N) = (g/rho0)(drho/dz), where rho is the density of water in each layer, and rho0 is the mean density and 'g' is the gravity.
Now the real question is, in a stratified ocean, for instance, a small perturbation is produced in any of the interface, what adjustment will be happen in the Ocean. The answer of this question leads to the important concept in the Ocean dynamics, called the 'Normal Modes'.
What are normal modes ?
In order to understand the normal mode theory of the ocean, let us follow a simple experiment in the laboratory. For a while, let us forget about the Oceans, and consider a simple oscillations of a 'spring and mass system' in a laboratory. Follow the figure.1 shown below. It is a combination of three masses and two springs.
The system is composed of three masses M1, m2 and M3 and are connected by two springs K1 and K2, where K1 and K2 are their 'spring constants'. Consider the system is resting on a 'smooth friction less table'. Now, let us squeeze this 'system' and release (or stretch a little and release). It is EASY to imagine that the system will undergo some kind of Oscillation. Although, it is NOT EASY to
imagine, What is the type of Oscillation it executes.
In order to understand the type of Oscillation of the system, we have to consider the Basic Balance Equations of the system. The simple logic is to assign a set of equations which contains the balance, force = mass x acceleration Thus for the three above masses M1, m2 and M3 let us write three equations as given above (Figure.1). Please note that, here the Force means the restoring force of the spring, which is by rule, F = -k.x.
The next thing is to find out the solutions of the above set of three equations. Let us follow the analatical method to find the solution of the above system of equations. i.e. an x = f(t) form for each masses. This will represent the instantaneous displacement of each masses at any time.
How to find these solutions? .
As it is clear that, the system will exicute some kind of oscillation, the necessary solutions will also be in the oscillatory form. With out much explanation, we can attribute the solutions to a set of oscillating sine or cose curves or a combination of it. For analatycal convinience, let us assume each mass is vibrating with a common frequency, say 'w' (omega) Let us find this common frequency. If 'w' is the frequency of oscillation of the system, at any instant, the displacement of each massess can be written in the most general form xi = x0.exp(iwt), i=1,2,3. represents mass 1, 2 and 3 respectively. So this is the general solution of the above system of mass vibrations. Thus the solution contains both the real part and imaginary part. Let us concentrate on the real part of the solution.
Analatycal method to find the solutions .
Since the above system is composed only of limited numbr of mass (three in our mase), it is convinient to find the analatycal solution of the system. If xi = x0. exp(iwt) , is the solution of the system (instantanious displacement of each masses), we can find the second order derivative of this, and substitute substitute back to the equations 1,2 and 3, we will get a set of three equations as follows. A common factor exp(iwt) is divided out from each term in the following equation.
With a simple rearranging we shall get a system of matrix as given below
A close examination of this analatical form of the vibrating system, we can see that this is in the form A.m=(lamda).m, a matrix eigen value problem, that yields eigen values (characteristic roots) 'lamda' and eigen function m . Each Eigen value will represent a 'type of (kind of or a manner of) ' oscillation of the system. Also each 'eigen value' corresponds to an eigen function', which is the solution of the system representing that 'particular manner' of osccilation. and they are called as the Normal Mod oscillation of the system. Here in this experiment, the eigen value 'lamda' is w^2, or the square of frequency of the oscillation. The above set of equation gives 'three' eigen values and each eigen value represents eigenfunction or a mod of oscillation. So, inorder to find the solution of the system of vibration, (i.e. x=f(t) form) we need to find find the Eigen values and the corresponding Eigen functions.
Eigen values and Eigen vector (function)
The eigen values can be found from the coefficient matrix, which is obtained by writing the above system in a secular form .ie. (A-lamda.I) = 0, where I is an unit matrix. The value of lamda is called [/b]'eigen values'[/b]. For any N x N matrix there are 'N' eigen values and for each eigen value there is an
Eigen vector. In the above experiment, the eigen values are (1) w^2 = 0, (2)w^2 = k/M and (3) w^2 = (k/M + 2k/m) . So, there will an associated for Eigen function for these three eigen values and that will give the solution (instantanious displacement of each mass ) of the system.
The following figure will help us to see the (tentative) oscillation of the system. For example, 1) corresponding the to the first eigen value, w^2 = 0.0, the eigenfunction is x1=x2=x3.. This means that there is no relative displacement between the masess or in other way we can say all the masses moves with the same displacement. Thus the motion (oscillation) corresponding to this eigen value is such that all the masses moves in same direction with same displacement. The other way of interpretation is that mod (type or kind) of oscillation of the system curresponding to the eigen value w^2 = 0.0 is, x1=x2=x3.
The second eigen value is w^2 = k/M. The eigenfunction of this mod is x1=-x3 and x2=0. The outer massesmoves oppositely whereas the central mass is at rest. Also the third eigen value is w^2 = (k/M + 2k/m). The eigen function of this mod is x1=x3, x2 = -(2M/m)x1, means the outer masses moves in same direction and the central mass moves opposite to the outer mass. Thus, the system composed of three masses (three equations), gives three eigen value, and represents three normal mod oscillation of the system. The word normal stands for the type of oscillation, it is perpendicular (normal) to the system.
The figure shows the oscillation of three modes separately. The displacement of the each masses is given as a graphical plot (please note that the values of the graphs are arbitrary, not produced from true solutions, but it gives the sketch of the real oscillations). The eigenfunction corresponding to the first mod is a straight line. In this case, all the masses moves in the same direction with same displacement. Thus the entire system behaves unique. The eigenfunction of the second mod shows that the outer massess moves opposite and the central mass is rest. The eigen function for this mod crosses the zero-line in the graph at one point. This is called the nodal point, or the point where amplitude of that mod is zero. Also, the third mod has two nodal points. The important thing should bear in mind that, at any instant the state of the system is a linear combination of this three mod of oscillation. Thus it turns out following coclussive points.
1. At any instant, the state of the system is a linear combination of the above three type of oscillations or mod of oscillations.
2. Thus the state of the system can be represented as the sum of these Normal Modes and it is an important tool to find the evolution of the system.
3. By observing the resultant oscillation of the system, we can not say exactly, which type of osccilation or Mod of osccilation is prominent in the system.
Why do we follow the above experiment ?
The above experiment gives us a simple explanation about the Normal Mod Osccilations. Now let us see, how this experiment help us to understand about the ocean. Let us apply the same principle to the Ocean. We can interpret an analogy in the Ocean very similar to this mass-spring oscillation. Consider, we are tilting the mass-spring system vertically, we can easily understand What is mean by the normal Mode Oscillation in the Ocean. see figure for an analogically view of spring-mass system and the ocean.
In the Ocean, each layer stands for M1, M2 and M3 of the above experiment and restoring forces due to the density difference between the interface of the two layers (or the 'reduced gravity') stands for the springs. Thus, for instance, any small perturbation produced in the ocean can excite oscillations vertically. Certainly for n-layer of ocean, there will be n-vertical (normal) modes.
Why these Oscillations are important ?
In the above experiment, we can see that, the entire oscillation in the mass-spring system can be represented by a 'combination of three mod of oscillation'. This is an effective tool because all the motion governing by the system can be represented by this three modes. Likewise in the case of ocean, with a finite vertical stratification, the solutions can be expressed as a sum of normal modes, each of which has a fixed vertical structure and behaves in the horizontal and time (x,y,t) dimensions in the same way as does a homogeneous fluid with a free surface (Gill, 1982). That means, the vertical oscillaion is accompanied with a fixed horizontal motion. To an extension to the above simple mass-spring system, the Ocean is a fluid body and it state is determined by various parameters such as composition, pressure, velocity etc... More over, the great utility of this Normal Mod construction in the ocean is that, with a sufficient approximation (The propogating waves are very long in horizontal scale compared to the vertical scale or the Hydrostatic approximation (pressure varies only with depth) ), the sysptem possess a seperable solutions into Horizontal and Vertical. And in that case, the each Mod of oscillation behaves indepndently and this also a great utility. Simply saying, at any instant, the state of the ocean, say u = u(x,y,z,t) can be found from the Normal Mod structure. In addition to that, if we apply the above approximations, the state of the system can be further simplified into u=u(x,y,t).u(z) .i.e., we seperated the Normal Mod and Horizontal motions into two.
How to find Normal Mod oscillation of the Ocean ?
In the above spring-mass experiment, we sought a solution of the form, x=x0.exp(iwt). In the case of stratified Ocean, the similar solutions of the perturbation can be expressed as w = w0.exp[i(kx+ly-wt)], where k and l are horizontal wavenumber associated with the vertical perturbationw.. Also, in the case of Oceans, there exists boundaries at the bottom and surface, which confines the energy to a region of finite vertical extent, though the horizontal propagations are permitted. Thus Ocean can be considered as a waveguide that causes energy to propagate horizontally (Gill, 1982). This wave guide property of the Ocean provides only a possible number of eigen values and in the long-wave limit (i.e. with very small values of 'k' and 'l' ) the corresponding eigenfunction are independent of the horizontal wave numbers. Here, more theoretical aspects are not discussed, because this post is mainly discussing 'How to calculate normal mod oscillation in Ocean'. The readers are refereed to (Gill, 1982) for the theoretical discussion of the Normal modes.
( Continues to the next post..... )