Download HYDROSTATIC EQUILIBRIUM

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 ρ=constant, the hydrostatic pressure at depth z is given by the
Hydrostatic Equation,
p = − ρgz.
In the real ocean, ρ 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 = − ρgdz.
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.
GEOSTROPHIC CURRENTS
Horizontal Pressure Gradient
Geostrophic Adjustment
isobaric
sea surface
⇒
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:
1 dp
= g tan θ
ρ dx
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.
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.
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
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 = g tan θ (u: geost. veloc.)
f
(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.