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Lực Coriolis là gì? Khái niệm, ví dụ. Phân tích, nhận định ảnh hưởng của lực Coriolis lên biển. Đánh giá, giải pháp...Lực Coriolis là gì? Khái niệm, ví dụ. Phân tích, nhận định ảnh hưởng của lực Coriolis lên biển. Đánh giá, giải pháp...
Ocean Dynamics Write the equation of motion corresponds to write the 2nd Law of Newton (F = ma) in a form that can be applied to Oceanography This Law tells us that as a result of various forces acting on a body of mass m, this body acquires an acceleration, that is a variation on its speed, which is proportional to the resulting force. The acceleration has the direction of the resulting force. If F resulting = 0, then a = 0 and there will be no changes in the motion, that is, the movement remains as it is, but movement still exists. The idea that no forces act is impossible at the surface of the Earth, where at least the gravity force is acting This Law applies to an absolute frame of reference (coordinate system), i.e. the system is at rest or moving at constant speed (relative to what? discuss). The coordinate systems in oceanography are defined with its origin somewhere on the surface. And so, they're not at rest or moving at a constant speed. They follow the rotation of the Earth. If the Newton's second law of motion is applied on these systems, we have to include an apparent or virtual force to take the effect of Earth's rotation into account – the Coriolis force Contribution of the Earth's rotation: (a) A missile launched to the North from the Equator moves to the East such as the Earth, and to the North with the clock speed Effect of the Coriolis force, because the land curve to the poles Result: the movements are deformed – to the right in the N H and to the left in S H (b) trajectory of the missile with respect to the Earth At time T1 the missile moved to M1 and the Earth to G1 In time T2 the missile moved to M2 and the Earth to G2 There are depletion caused by the Coriolis force, greater for higher latitude The bicycle wheel does not rotate at the Equator, but will rotate clockwise with respect to the Earth, each time with higher speed as it approaches the pole Coriolis Force Contribution of the Earth's rotation:: A projectile fired from the Equator to the North moves to the East, such as the Earth, and to the North with the firing speed As it moves north, the speed at which the Earth moves to the East is smaller, because v=r, is constant and r decreases with latitude As a result, the projectile moves not just to the North, but also to the East in relation to the Earth (to its right) The same rationale applies in the case of the firing occurs from North to South in the northern hemisphere: relatively to the Earth moves not only South, but also to its right (West) The same happens to the water masses moving in the Ocean effect of apparent force called Coriolis force The Coriolis force is an apparent force that acts on moving objects on the Earth's surface, according to a 90-degree angle to the right in the northern hemisphere and to the left in the southern hemisphere The Coriolis force is null at the Equator and increases with latitude, being maximum at the poles Horizontal component of the Coriolis force: m2sinVH = mfVH, f – Coriolis parameter Coriolis Force A portion of water at rest on the Equator carries an angular momentum of the Earth's rotation When this parcel moves towards the poles it carries the angular momentum, but at the same time its distance from the axis of rotation is reduced To conserve its angular momentum it has to increase its rotation around the axis, in the same way that dancers can increase its speed of rotation by bringing your arms closer to your body (bringing more mass toward the axis of rotation) The particle begins to rotate faster than the rotation of the Earth below it This means it moves towards the East This results in a deflection in the linear path to the right in the northern hemisphere and to the left in the southern hemisphere Similarly, a portion of water coming out from the Poles toward the Equator will increase it distance from the axis of rotation and to conserve angular momentum has to decrease its speed of rotation relative to the Earth below; so, begins to move toward the West which again will represent a deflection to the right in the northern hemisphere and to the left in the southern hemisphere Don't forget: v=r!!! Laboratory experiment that shows how a coordinate systems that performs rotation leads to a virtual force: A small ball is moving backwards and forwards under the force of gravity along a shallow bowl that is in rotation (below left) When both the bowl and the ball are observed from the outside (a system of absolute coordinates), the ball seems to move in a straight line back and forth, while the bowl runs under it (above left) When the observer is placed in the bowl, and therefore performs rotation with it, the ball seems to move in a circle (right) To explain the circular motion, the observer has to create a force that deflects the ball from it linear motion This virtual force is the Coriolis effect that acts on the ocean currents Variation of the Coriolis term with the latitude: Ω ̂ + Ω ̂ y z cos ̂ ‐ tangencial component sin ‐ angular velocity Coriolis term: at latitude 2 Ω⋀ Once solved the external product and some approximations applied, the vector of the Coriolis acceleration is given by: 2Ω ̂ Coriolis parameter: 2Ω ̂ 2Ω Classification of the forces in Oceanography External forces (applied in the fluid boundaries): (a) tangential forces (tensions)‐e.g. forces exerted by the wind, by the margins, etc (b) Forces induced by thermohaline differences (cooling of the surface, evaporation, etc.) ‐ factors that lead to changes in density that result in changes of the pressure field, thus inducing forces Internal forces: (applied in all the water parcels) (c) field of internal pressure (pressure gradient) (d) tidal forces Forces that slow down the currents: (a) Friction (moment diffusion) ‐ friction of the layers on top of each other (b) Forces induced by diffusion of density (have the effect of changing the pressure gradient) "Apparent" or "virtual“ forces: (a) Coriolis force (b) Centrifugal force (e.g. in oceanic vortices) THE EQUATION OF MOTION IN OCEANOGRAPHY Making the addition of all the forces in Newton's second law for the oceans, it takes the following form: particle acceleration = (‐pressure gradient force + Coriolis force + tidal forces + friction + gravity) / mass The tidal force needs to be considered only in more specific problems; it can be ignored in the discussion of the oceanic general circulation, of large scale, as it is an oscillatory motion The force of gravity does not act as a horizontal force and so cannot produce a horizontal acceleration; It is important in movements that involve vertical displacements (convection, waves) Why is there a negative sign in the pressure gradient? Because the acceleration produced by a pressure gradient is directed in a manner opposite to gradient, so the associated movement of the water “flow down the gradient " HYDROSTATIC EQUILIBRIUM The pressure in the ocean is partially affected by the movement of the water However, the ocean currents are very slow, and the vertical movements even slower Thus, for most of the purposes the pressure in depth is taken as the hydrostatic pressure The hydrostatic pressure is the weight of the water column per unit of area at a depth z Considering =constant, the hydrostatic pressure at depth z is given by the Hydrostatic Equation, p gz In the real ocean, changes with depth We can consider the water column as an infinite number of layers of infinitesimal thickness dz, that contributes with an infinitesimal pressure dp to the total hydrostatic pressure at the depth z Thus, the hydrostatic pressure at the depth z is given by: dp gdz The pressure at the depth z is the summation (well the integral ) of all the individual contributions dp of the several layers HYDROSTATIC EQUILIBRIUM Variation of the hydrostatic pressure with depth: (a) in the real case, where the density varies with depth; (b) in the case the density is assumed as constant Horizontal pressure gradient and associated force Lateral borders (coastlines), lateral differences of density and inhomogeneities in the wind field induce slopes in the sea surface that vary the hydrostatic pressure along horizontal surfaces at depth in the ocean horizontal pressure gradients The equation of motion: horizontal and vertical components Considering the Coriolis force, the pressure gradient force and the friction with the eddy viscosities , the equation of motion in the x direction is: Similarly, in the y direction we have: In the vertical component we have to consider the gravity and we have: Analysis of scales: This hydrodynamic equations systems have great complexity, in addition to difficulties in establishing the initial conditions and boundary. Solutions based on numerical techniques have been obtained in several spatial and temporal scales The analysis of scale allows the estimation of the order of magnitude of each term of the basic hydrodynamic equations, depending on the patterns of the observed motions It is possible to simplify the equations of motion using the following scale analysis: For the open ocean, typical values of the distance L, horizontal velocity U, depth H, Coriolis parameter f, gravity g and density ρ are: From these values one can calculate the typical values of vertical velocity W, pressure P and time T, using the equations of continuity and hydrostatics: Thus, for the equation of the vertical motion we have: so that the vertical equilibrium may be expressed by the hydrostatic equation: The analysis of scale for the equation of motion in the x‐direction indicates that: and so the pressure gradient acceleration balances the Coriolis acceleration, leading to the geostrophic balance equations: GEOSTROPHIC CURRENTS Horizontal Pressure Gradient Geostrophic Adjustment geostrophic equilibrium situation: The pressure gradient force and the Coriolis force are in balance, thus the movement has constant velocity – geostrophic velocity (Law of Inertia) Coastal boundaries and the heterogeneity of the wind field originate slopes in the sea surface, that induces variations of the hydrostatic pressure on horizontal surfaces at depth horizontal pressure gradient Horizontal pressure gradient force per unit of mass: dp g tan dx g tan Geostrophic velocity: u f Initial situation: there is an accelerated motion, “descending” the pressure gradient The water tends to move in order to eliminate the horizontal differences in the pressure field The force that originates this motion is known as the horizontal pressure gradient force If the Coriolis force, that acts on the moving water, is balanced by the horizontal pressure gradient force, the current is in geostrophic equilibrium and is called geostrophic current The importance of the Earth rotation: The rate of rotation of the Earth (or angular velocity) is: Time of 1 revolution If the fluid movement is evolving in a comparable time scale (or greater) than the rotation period, the fluid feels the effect of rotation. So, when calculating the ratio: Time of 1 revolution Time scale of the motion if εt