Chapter 10. Geostrophic Balance. (1) Wind and Height Gradients. Strong winds associated with contours close together; weak winds with contours farther apart. Winds tend to blow (nearly) parallel to height contours. On a contour analysis of a pressure surface:.
The geostrophic wind is parallel to the height gradient; i.e., 90o to the right of the height gradient (in the northern hemisphere).
Right hand Rule: All fingers point in direction of k. Curl fingers in direction of increasing height gradient vector. Thumb points in direction of resultant of cross product.
Thus, we could say that the advection of “A” by the geostrophic wind is inversely proportional to the area of the parallelogram formed by the height contours and the isopleths of “A.”
The geostrophic wind vector is written as:
If “f” is considered constant.
In other words, it is opposite to the pressure gradient (which is directed from low pressure toward high pressure).
A gradient vector always points in the direction of the
greatest rate of increase.
v is the tangential velocity of the ball.
ΔΘ is the angle through which the ball moves (the change in the angle measured from the zero angle position).
r is the radius from the axis of rotation to the ball.(8) Nature of the Coriolis Force
Δv is the change in velocity (acceleration - change in direction) of the ball as it moves in a circular path.
v + Δv is the new velocity of the ball (I.e., because it changed direction).
To compute the acceleration, we consider the change in velocity, Dv, which occurs for a time increment, Dt, during which the ball rotates through an angle DQ.
DQ is also the angle between the vectors v and v + Dv.
Remember: S = rΘ
The spheroid shape of the Earth makes g (the Effective gravity) directed normal to the level surface.
also directed outward, like the Centrifugal Force.
The horizontal force causes a change in the direction of movement of the object (as perceived by someone on a moving coordinate system).
As the object moves equatorward, it will conserve its angular momentum in the absence of torques in the east-west direction.Objects moving north or south.
Since the distance to the axis of rotation, R, increases to R + DR for the object moving equatorward, a relative westward velocity, must develop, (w becomes smaller - or negative), if the object is to conserve its angular momentum.
Arc length =
Dividing by a unit time increment gives the horizontal acceleration in the west-east direction.
Note: “a” is the radius of the Earth.
Acceleration = Pressure Gradient acceleration + Coriolis acceleration + Friction
The change in the wind vector is equal to a vector drawn from the end of the first vector to the end of the second vector (second vector minus the first vector).
The acceleration is obtained by dividing the difference vector (blue) by the time interval.
If only the speed is changing, the direction of the acceleration is parallel to the original direction.
Then the Pressure Gradient force is greater than the Coriolis force (since the Coriolis force - in the northern hemisphere- tries to make the air move to the right.
This is how air flows about a low pressure (low height) region in the absence of friction.(10) Gradient and Cyclostrophic Winds
It is called “balanced” because it flows parallel to the contours, even though the forces are not in balance.
In straight flow, the wind speed must be strong enough for the Coriolis force to balance the Pressure Gradient force.
In curved flow about a low, the Coriolis Force is weaker than the Pressure Gradient force, so the winds are weaker.
The winds are Subgeostrophic.
The contours / isobars about a low will be close together, signifying a strong Pressure Gradient force.
The wind speed is greater than it would be in a straight isobar / contour flow. The winds are Supergeostrophic.
The center of a high will be broad and flat, signifying a weak PGF.