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
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.
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
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
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:
‐ angular velocity
Once solved the external product and some approximations applied, the vector of
the Coriolis acceleration is given by:
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
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
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
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,
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 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
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
The pressure at the depth z is the summation (well....the integral....) of all
the individual contributions dp of the several layers.
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 do vary the hydrostatic
pressure along horizontal surfaces at depth in the ocean horizontal pressure
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:
Horizontal Pressure Gradient
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
Horizontal pressure gradient
force per unit of mass:
Geostrophic velocity: u
Initial situation: there is
an accelerated motion,
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
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 < 0(1), the effects of the Earth's rotation should be considered. Alternatively,
we can use the scale estimated by the speed:
Time of 1 revolution
Time for a particle to travel L with speed U
εt is the “local Rossby number" and ε is the “advective Rossby number".
BAROTROPIC AND BAROCLINIC CONDITIONS
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.
• 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
In barotropic flow the isopycnic and isobaric surfaces are parallel and their slopes in relation to the
horizontal 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, is
constant with depth.
In a baroclinic flow the isopycnic surfaces intersect the isobaric surfaces. At shallow depths, the
isobaric surfaces are parallel to the sea surface, but with the increasing depth their slope becomes
smaller, because the average density of a column of water at A is higher than that of a column of
water at B (in barotropic conditions the average density these two columns is the same). As the
isobaric surfaces become increasingly near horizontal, so the horizontal pressure gradient
decreases and so does the geostrophic current, until at some depth the isobaric surfaces are
horizontal and the geostrophic current is zero.
BAROTROPIC AND BAROCLINIC CONDITIONS
The relationship between isobaric and isopycnic surfaces: (a) barotropic
conditions – 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.
(a) Barotropic conditions: the slope of the isobars is
constant with depth. Geostrophic velocity is the same
at all depths: u tan (u: geost. veloc.)
(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:
(b) Combination of baroclinic and barotropic components.
In this case the reference level is not a level of no motion.