Ocean DynamicsWrite 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

(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.

(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 M2and the Earth to G2. There are depletion caused by the Coriolis force, greater for higher latitude.

Contribution ofthe Earth's rotation: 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..

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.

Contribution of the Earth's rotation::A projectile fired from the Equator to the North moves to the East, such as theEarth, and to the North with the firing speed. As it moves north, the speed atwhich the Earth moves to the East is smaller, because v=r, is constant andr 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 rationaleapplies in the case of the firing occurs from North to South in the northernhemisphere: 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 effectof apparent force called Coriolis force.

The Coriolis force is an apparent force that acts on moving objects on theEarth's surface, according to a 90-degree angle to the right in the northernhemisphere and to the left in the southern hemisphere. The Coriolis force isnull 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

Coriolis ForceA portion of water at rest on the Equator carries an angular momentum of the Earth's rotation. Whenthis parcel moves towards the poles it carries the angular momentum, but at the same time itsdistance from the axis of rotation is reduced. To conserve its angular momentum it has to increase itsrotation around the axis, in the same way that dancers can increase its speed of rotation by bringingyour arms closer to your body (bringing more mass toward the axis of rotation). The particle begins torotate faster than the rotation of the Earth below it. This means it moves towards the East. Thisresults in a deflection in the linear path to the right in the northern hemisphere and to the left in thesouthern hemisphere. Similarly, a portion of water coming out from the Poles toward the Equator willincrease it distance from the axis of rotation and to conserve angular momentum has to decrease itsspeed of rotation relative to the Earth below; so, begins to move toward the West which again willrepresent a deflection to the right in the northern hemisphere and to the left in the southernhemisphere. Don't forget: v=r!!!

Laboratory experiment that shows how a coordinate systemsthat performs rotation leads to a virtual force:A small ball is moving backwards and forwards under the forceof 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 astraight line back and forth, while the bowl runs under it (aboveleft).When the observer is placed in the bowl, and thereforeperforms rotation with it, the ball seems to move in a circle(right). To explain the circular motion, the observer has to createa force that deflects the ball from it linear motion. This virtualforce is the Coriolis effect that acts on the ocean currents.

sin ‐ angular velocity at latitude 

y

z

cos ‐ tangencialcomponent

Ω +Ω

Coriolis term:

Variation of the Coriolis term with the latitude:

2Ω⋀

Once solved the external product and some approximations applied, the vector of the Coriolis acceleration is given by:

2Ω 2Ω

2Ω Coriolis parameter: 

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)

Classification of the forces 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 ".

THE EQUATION OF MOTION IN OCEANOGRAPHY

The pressure in the ocean is partially affected by the movement of thewater. However, the ocean currents are very slow, and the verticalmovements even slower. Thus, for most of the purposes the pressure indepth is taken as the hydrostatic pressure.

The hydrostatic pressure is the weight of the water column per unit of areaat a depth z.

Considering =constant, the hydrostatic pressure at depth z is given by theHydrostatic Equation,

In the real ocean, changes with depth. We can consider the watercolumn as an infinite number of layers of infinitesimal thickness dz, thatcontributes with an infinitesimal pressure dp to the total hydrostaticpressure at the depth z. Thus, the hydrostatic pressure at the depth z isgiven by:

The pressure at the depth z is the summation (well....the integral....) of allthe individual contributions dp of the several layers.

HYDROSTATIC EQUILIBRIUM

.gzp

.gdzdp

Variation of the hydrostatic pressure with depth: (a) in the realcase, where the density varies with depth; (b) in the case thedensity is assumed as constant.

HYDROSTATIC EQUILIBRIUM

Horizontal pressure gradient and associated force

Lateral borders (coastlines), lateral differences of density and inhomogeneities inthe wind field induce slopes in the sea surface that do vary the hydrostaticpressure along horizontal surfaces at depth in the ocean horizontal pressuregradients.

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: 

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:

Analysis of scales: 

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 CURRENTSHorizontal Pressure Gradient Geostrophic Adjustment

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.

Coastal boundaries and the heterogeneity of thewind field originate slopes in the sea surface, thatinduces variations of the hydrostatic pressure onhorizontal surfaces at depth horizontal pressure gradient.

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).

Initial situation: there is an accelerated motion,

“descending” the pressure gradient.

tan1 gdxdp

Horizontal pressure gradientforce per unit of mass:

tanfgu Geostrophic velocity:

The importance of the Earth rotation: 

The rate of rotation of the Earth (or angular velocity) is:

if εt < 0(1), the effects of the Earth's rotation should be considered. Alternatively, we can use the scale estimated by the speed:

εt is the “local Rossby number" and ε is the “advective Rossby number".

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 of 1 revolution

Time of 1 revolution

Time scale of the motion

Time for a particle to travel L with speed U 

Barotropic Conditions :

• In real conditions, if the ocean is homogeneous, the density increases in depth due to the compression caused by the weight of the overlaying water; the isobaric surfaces are parallel to the sea surface and to the isopycnic surfaces we are in Barotropic Conditions.

• In barotropic conditions, the pressure variation on a horizontal surface, at a given depth, is determined only by the sea surface slope, because the the isobars are parallel to the sea surface.

Baroclinic Conditions:

• Any variation of the density will affect the weight of the overlaying water and, consequently, will affect the pressure that acts on a given horizontal surface. When lateral density variations occur, the isobaric surfaces are not parallel to the sea surface; the isobars intersect the isopycnics, with opposing slopes. The tilt of the isobars relatively to the isopycnics characterize the Baroclinic Conditions.

BAROTROPIC AND BAROCLINIC CONDITIONS

BAROTROPIC AND BAROCLINIC CONDITIONS

In barotropic flow the isopycnic and isobaric surfaces are parallel and their slopes in relation to thehorizontal remain constant with depth. Thus, since the slope of the isobars is constant with depth,the horizontal pressure gradient from B to A, and in consequence the geostrophic current, isconstant with depth.

In a baroclinic flow the isopycnic surfaces intersect the isobaric surfaces. At shallow depths, theisobaric surfaces are parallel to the sea surface, but with the increasing depth their slope becomessmaller, because the average density of a column of water at A is higher than that of a column ofwater at B (in barotropic conditions the average density these two columns is the same). As theisobaric surfaces become increasingly near horizontal, so the horizontal pressure gradientdecreases and so does the geostrophic current, until at some depth the isobaric surfaces arehorizontal and the geostrophic current is zero.

The relationship between isobaric and isopycnic surfaces: (a) barotropicconditions – the desnsity distribution, indicted by the intensity of blue shading,does not influence the shape of isobaric surfaces. (b) baroclinic conditions –lateral variations of density do affect the shape of isobaric surfaces.

BAROTROPIC AND BAROCLINIC CONDITIONS

(a) Barotropic conditions: the slope of the isobars is constant with depth. Geostrophic velocity is the same at all depths: (u: geost. veloc.)tan

fgu

(b) Baroclinic conditions: the slope of the isobars varies with depth. At the reference level, z0, the isobar corresponding to pressure p0 is assumed to be constant. In this case, geostrophic velocity decreases with depth. Anyway, different behaviors may occur.

Profiles of geostrophic current velocity:(a) Baroclinic(b) Combination of baroclinic and barotropic components. In this case the reference level is not a level of no motion.