About Ekman currents in shallow waters.

Muntean, Angela ; Moroianu, Corneliu ; Bejan, Mihai 等

1. INTRODUCTION

An accurate understanding of wind-driven circulation of Black Sea water on the Romanian seashore is required. Ekman curent is an important elementof the wind-driven circulation. Romanian Black Sea shore is characterized by a large region with shallow depth. We focus on the study of Ekman currents for shallow basins, using Navier-Stokes equations with friction. Boundary conditions and assumptions are made for Black Sea waters. One aim are to compute Ekman currents characteristics as velocity and direction. Flux and direction of the Ekman transport calculus is another aim of this study.

2. THE EQUATIONS OF MOTION FOR THE WIND-DRIVEN MARINE WATER CIRCULATION

The equation of motion used in oceanography is the Navier-Stokes equation. Among many forces which can be take in consideration in this equation, the most important are: gravity, Coriolis force and, for wind-driven circulation, the friction force. It can be used a coordinate system with the two horizontal axes Ox and Oy on plane tangent to the Earth with Ox to the east and Oy to the north. The third axis Oz is vertical up. In generaly, the friction stress is given by

[tau] = [mu] [partial derivative]u/[partial derivative]z = [rho] [??] [partial derivative]u/[partial derivative]z.

This unit force acts on th surface between two neighboor layers which are moving at different speeds. The friction tends to slow down the faster and to speed up the slower (Pickard & Emery 1990).

The friction force component in Ox direction is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where eddy viscosity for different directions [A.sub.x], [A.sub.y] and [A.sub.z] are used. In the last scalar equation in Navier-Stokes system all the terms are very smaller than the pressure one and g (Stewart 2002).

That's why all these terms can be ignored except the pressure term and g. The vertical equation is, in fact, the hydrostatic equation. When friction is included, the first two equations in the Navier-Stokes system are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

where [F.sub.x], [F.sub.y] are the components of friction force per unuit mass in the fluid.

The forces in equilibrum are: the Coriolis force, pressure force and the friction one (Knauss 2005).

Taking in consideration that it exists few informations on the manner in which [rho][A.sub.z] does vary with depth [rho][A.sub.z] is assumed to be constant and

the friction force per unit mass = [A.sub.z] [[partial derivative].sup.2]u/[partial derivative][z.sup.2].

Boussinesq approximation is used in oceanography and then [rho] can be take as a constant.

The horizontal equations in Navier-Stokes system can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

The vertical equation reduced to the hydrostatic equation. In this situation, the motion is due of two forces:

--the wind friction

--the pressure force given by the density variation. The velocity can be written as a sum of two terms:

u = [u.sub.E] + [u.sub.g], v = [v.sub.E] + [v.sub.g] (3)

where [u.sub.E], [v.sub.E] are the components of the velocity associated with the vertical shear friction (Ekman velocity) and [u.sub.g], [v.sub.g] the components of the geostrophic velocity (Pond & Pickard 2003). Ekman was the first who studied the effect of the frictional stress on the sea water circulation due to the wind blowing over the sea. The equations of motion can be written

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

But f [v.sub.g] = [alpha] [partial derivative]p/[partial derivative]x. Then, the first equation in (4) system becomes

f [v.sub.E] = -[A.sub.z] [[partial derivative].sup.2][u.sub.E]/[[partial derivative].sup.2],

because the term [A.sub.z] [[partial derivative].sup.2][u.sub.g]/[partial derivative][z.sup.2] is very small.

For the system of differential equation obtained up, Ekman used some assumptions (Lacombe 1965):

--no boundaries;

--infinitely deep water;

--[A.sub.z] constant;

--a steady wind blowing for a long time;

--homogeneous water at the sea surface level since density depends only on pressure;

--f constant.

We don't use the second assumption. We considered the basin with a finite depth, equal with H. The equations of motion are

f[v.sub.E] + [A.sub.z] [[partial derivative].sup.2][u.sub.E]/[partial derivative][z.sup.2] = 0, - f[u.sub.E] + [A.sub.z] [[partial derivative].sup.2][v.sub.E]/[partial derivative][z.sup.2] = 0 (5)

In order to achieve the system solutions we need the boundary conditions. We assume the wind blowing in direction given by an angle [[alpha].sub.1] with the north direction (the direction of Oy axis) and the wind stress magnitude is [tau]. The boundary conditions can be written

[[rho].sub.w][A.sub.z] [([du.sub.E]/dZ).sub.z=0] = [tau] sin[[alpha].sub.1] (6)

[[rho].sub.w][A.sub.z] [([dv.sub.E]/dZ).sub.z=0] = [tau] cos[[alpha].sub.1]

where [[rho].sub.w] is the density of marine water.

Those are the conditions for the surface basin. At the bottom

[u.sub.E] = [v.sub.E] = 0, (7)

for z = -H [2]. We use the Gongenheim and Saint-Guilly method (Lacombe 1965) with w = [u.sub.E] + i [v.sub.E].

We use the assumption that [u.sub.E], [v.sub.E] depend only on the level (z). It was multiplied the second equation of (4) by i and the two equations are added. It obtained a differential equation with a single unknown, w:

[d.sup.2] w/[dz.sup.2] - I f/[A.sub.z] w = 0.

With [a.sup.2] = f/2[A.sub.z], the equation (7) becomes

[d.sup.2]w/[dz.sup.2] - 2i [a.sup.2] w = 0. (8)

We solved the differentiaa equation (8), with the boundary conditions (6) and (7) [4]. It was obtained

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

We did a study regarding the variation of wind-driven current with the sea level and, for the same level, with the depth.

The magnitude of velocity is given by the formulas

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

The magnitude of Ekman velocity for the same stress wind, at every level z depends on the basin depth. The angle [theta] between the velocity of surface Ekman currents direction and north direction is given by the formulas

tan[theta] = sh2aH - sin2aH tan([pi]/4 - [[alpha].sub.1]) / sh2aH tan([pi]/4 [[alpha].sub.1]) + sin2aH (11)

We study the variation of the velocity and the angle with the basin depth [3]. For the same z, we conclude that the velocity magnitude increases for values less then about H = 1180 m. After this, the magnitude decreases a little. The angle, too, increases to 47[degrees], for about H = 1200 m. After ths, the angle decreases to 45[degrees], value which corresponds to the infinite ocean basin depth [4]. It is known that, for deep basin, the surface current flows at 45[degrees] to the right of the wind direction in northen hemisphere.

With the wind stress magnitude given by [tau] = [[rho].sub.a] [C.sub.D] [W.sup.2] [1], where [[rho].sub.a] is the of air, [C.sub.D] the drag coefficient and W is the wind speed, the Ekman current speed is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Using the values [[rho].sub.a] [congruent to] = 1.3 kg/[m.sup.3], CD = 1.4 x [10.sup.-3], [[rho].sub.w] = 1000kg/[m.sup.3], [A.sub.z] [congruent to] [10.sup.-1] [m.sup.2]/s [1], and it is considered the region of Romanian Black Sea shore, one can takes [phi] = 45[degrees], and / = 2 [ohm] sin[phi] = [10.sup.-4] [s.sup.-1], the (11) formulas becomes

V = 5,75 x [10.sup.-4] [W.sup.2] [square root of ch2a(z + H) - cos2a(z + H)/ch2aH + cos2aH] (13)

If [V.sub.0] is the magnitude of surface Ekman current , the surface urrent and the wind speed are related as

[V.sub.0]/W = 5,75 x [10.sup.-4] [W.sup.2] [square root of ch2aH - cos2aH/ch2aH + cos2aH] (14)

The net transport of Ekman current is given by the formulas

[PHI] = [tau]/2[mu][a.sup.2] chaH - cosaH/[square root of [cos.sup.2] aH + [sh.sup.2] aH] (15)

With the same values used in (12) formulas, the net flux becomes

[PHI] = 0.0182 [W.sup.2] chaH - cosaH/[square root of [cos.sup.2] aH + [sh.sup.2] aH] (16)

In the shallow basin, the angle between the wind direction and the Ekman transport i

tan[[theta].sub.1] = ch2aH + cos2aH - 2 cosaH chaH/2 sinaH shah (17)

3. CONCLUSIONS

The depth of marine basin is very important for all Ekman current characteristics as theoretical results reveal. Marine activities require Ekman current characteristics in Black Sea shallow waters. The core of this study was the determination of relations for this particular zone. Further studies can be orientated to validate the theoretical model through physical determinations.

4. REFERENCES

Knauss, J. (2005). Physical Oceanography. Waveland Press, inc., ISBN 1-57766-429-9, Long Grove, Illinois

Lacombe, H. (1965). Physical Oceanography, Gautier-Villars, Paris

Pickard, G. & Emery, W. (1990). Physical Oceanography. Buttenworth-Heinemann, ISBN 075062759X, Oxford

Pond, S. & Pickard, G. (2003). Introductory Dynamical Oceanography, Butterworth Heinemann, ISBN 0 750 62496 5. Butterworth Heinemann, Vancover

Stewart, R. (2002). Introduction to Physical Oceanography, Available from: http://www.ig.utexas.edu/library/ Accesed: 2008-01-15