Rotational and divergent winds: The horizontal wind field V vector can be decomp
ID: 106254 • Letter: R
Question
Rotational and divergent winds: The horizontal wind field V vector can be decomposed into the rotational and the divergent parts; V vector V vector_R + V vector_D, according to Helmholtz theorem. Consider a velocity field that can be represented as oy and v JC, where y is called V vector_r = k cap times delta psi or, in Cartesian coordinates u_R = partial psi/partial y and v_R = partial psi/partial x, where psi is called the stream function and VR is called the rotational wind. Further we can define divergent wind as VD VX, where X is called the velocity potential and VD is called the divergent wind. Prove that V. V 0 everywhere Prove that vortices can be given by Vey Prove that divergence can be given by V. V V2X Given the field of vortices, together with appropriate boundary conditions, the inverse of the above equation can be used to obtain the corresponding stream function field y V In a similar manner, we can compute the velocity potential X v-2v V If LP my, where m is a constant, sketch the rotational wind VR using either a simple diagram or 1-2 sentences and compute vortices. If y m(x2 y2), where m is a constant, sketch the rotational wind V using either a simple diagram or 1-2 sentences and compute vortices.Explanation / Answer
(a) A general assumption for micrometeorological measurements is the zero mean vertical wind speed. Generally, zero mean average vertical wind speed is forced as part of the wind vector rotation routine. It is proposed that this may be a good assumption very close to the ground, but it is generally invalid at higher altitudes. The non-zero mean vertical velocity is caused by local thermal circulations, topographically modified flow, divergence in convective cell-like structures or synoptic scale subsidence. The non-zero mean vertical wind speed transports heat, water vapor and carbon dioxide across the plane of the actual measuring height, while this transport is undetectable by the eddy covariance system, which is based on the measurement of the fluctuating signals. Focusing on carbon dioxide, this transport can be severe during nighttime, when carbon dioxide usually accumulating below the inversion layer causing high vertical gradients of CO2 near the ground. As an example, a mean vertical velocity of 5 cm s-1 at the measuring height causes -100 W m2 equivalent energy flux for daytime (or an uncertainty of about 20% of the observed net radiation).
The true mean vertical velocity can be approximated from the following equation:
W = w + a (theta) + b (theta) u,
Where u and w are the measured mean horizontal and vertical velocities in the coordinate system defined by the instrument, respectively. W is the true mean velocity and a and b are the wind direction (theta) dependent coefficients. Since w behaves in a random fashion, the long term data can be used to evaluate a and b with least absolute deviation method.
Here below the defined values - an x, y, z coordinate system which has an origin somewhere on the Earth's surface (say at the equator and the Greenwich meridian), and we measure the three directions in the following way:
x: is the zonal (East-West) direction; positive eastward
y: is the meridional (North-South) direction; positive northward
z: is the vertical (up-down) direction; positive upward.
Thus the frame of reference has curved axes and is located on a rotating surface. This system is convenient to us because this is the frame of reference from which we view the atmospheric or oceanic motion. However, because of its peculiar properties it introduces some difficulties. In particular, we need to consider not only fundamental forces like the pressure force, but also apparent, or inertial, forces that result from the fact that the motion is viewed with respect to an accelerating (rotating) frame of reference. We shall see below how these considerations determine the balance of forces acting on the motion of air. Note that aside from within convection cells or clouds, the horizontal atmospheric motion is much more energetic than the vertical one, thus our section deals mainly with the balance of forces governing horizontal motion. The hydrostatic balance equation is the dominant vertical balance in the types of motion.
Pressure, is the force per unit area exerted by the air molecules on any imaginary surface within the atmosphere. Consider an air parcel suspended in the atmosphere in hydrostatic balance. If pressure on one side of parcel exceeds that on other side, the parcel will experience a net force from high pressure toward low pressure. The force per unit mass acting on the parcel (in Newtons/kg) is given by:
Fpx = - (delta p / delta x) / p
Fpy = - (delta p / delta y) / p
The pressure gradient force is thus given by the ratio of pressure difference to the distance over which it acts, divided by density. The pressure gradient force is the active force in the climate system (friction, see below, is passive because it only exists when motion exists). Thus to the weather forecaster or an observer of motion on Earth, the horizontal distribution of pressure is extremely important. Pressure is routinely measured in land stations, on board ships, and from weather balloons, and the reports are disseminated to all weather centers.
Air is not very viscous ("sticky"), so "real" friction (the one that comes from molecular motion) is only important in a very thin layer of atmosphere next to the surface. However, air is very turbulent. This turbulence generates small-scale up and down motion, which mixes slow air from the friction layer with fast air from above, thereby spreading the effect of molecular friction over a layer a few hundred meters thick (turbulence is the reason for wind gusts). This interaction with the surface slows down atmospheric motion. Thus the frictional force per unit mass is:
Ffx = -au and Ffy = -av
Where uand v are the zonal and meridional components of the wind (in units of m/sec), and is a constant equal to about 2 x10-5 1/sec.
Combining the equations above we find that when an air parcel is subjected to the forces of pressure gradient and friction the equation describing the motion (per unit mass) can be written as:
ax = - (delta p / delat x) / p-au
ay = - (delta p / delat y) / p – av
Here ax and ay are the acceleration of a unit mass in the west-to-east and south-to-north directions. If a balance is achieved between friction and pressure, the left hand terms in these equations are replaced by 0 hence proved that delat VR = 0.
(b)
Vorticity is mathematically defined as the curl of the velocity field and is hence a measure of local rotation of the fluid. This definition makes it a vector quantity. Circulation, on the other hand, is a scalar quantity defined as the line integral of the velocity field along a closed contour. In Fluid Dynamics, Vorticity is a vector quantity and it tells us the tendency of a fluid particle to rotate or circulate at a particular point. It is mathematically defined as the curl of velocity. The vorticity equation of fluid dynamics describes evolution of the vorticity w of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The equation is:
Dw / Dt = aw / at + (u* delta )w
= (w*delta)u-w(delta)+ 1/p2 * delta p* delta p + delat * (delta*r / p) + delat * (B / p)
Where D/Dt is the material derivative operator, also denoted by in capital D notation as D/Dt, u is the flow velocity, p is the local fluid term on the right hand side represents vortex stretching.
The equation is valid in the absence of any concentrated torques and line forces, for a compressible Newtonian fluid.
Using Stoke's theorem, the line integral of the velocity field along the closed path, can be expressed as a surface integral of the curl of the velocity field normal to an arbitrary area bounded by the path. But, as already defined, that curl operation is called vorticity. Hence, circulation can be referred to as flux of vorticity. Conversely, it can also be said that that vorticity at a point is essentially circulation per unit area. The last two statements characterize the two quantities, vorticity and circulation, as microscopic and macroscopic respectively. Both these quantities are essentially a measure of the rotation of the fluid flow. A local rotation of a fluid in a tank as an example element is called the vorticity w. It can be expressed mathematically by looking at how the velocity u changes locally w=delta*u. Now, despite the lack of local rotation, the spinning wheel in the tank was moving the fluid in a circle. This global rotation is the circulation T. It can be expressed mathematically by integrating the velocity around a loop
T = theta u*dsT = thata u*ds.
These two concepts being closely related, the circulation around a loop is the integral of the vorticity inside it. In this example, all that vorticity is found at the spinning wheel in the middle. If you draw a little loop around the cross, you would find no circulation because there is no vorticity inside.
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