Vertically integrated horizontal velocity

Here it is the vector of the horizontal velocity, U the vertically integrated horizontal velocity, w is the vertical velocity, r is the sea-level elevation, V/, the horizontal nabla operator, q a source term of water flux, T the temperature, S the salinity, p the pressure, and p is the density. Moreover,/is the inertial frequency,/= 2 1 sin 0, where 1 Zn (1 I 1/365.2425)724 h is the earth s angular velocity and 0 is the latitude. Turbulent viscosity is indicated by the term D, . Wind forcing enters the scheme as a vertical boundary condition. The equations are solved usually in spherical coordinates, but are written here for simplicity in Cartesian form. 

The horizontal velocity vector is found by vertical integration of the flow law (Eq. 2) ... 

The above reduced boundary layer problem [(5.6a,b,c), (5.7a,b,c,d)l remains a complicated boundary value problem. This problem is, in fact, very similar to problem (4.14a,b,c)), but it is formulated for a two-dimensional free-falling vertical film. When M = 0 (in this case the thermal field is decoupled to dynamical one), Shkadov (1967), using the integral method have reduced the problem to a system of two averaged equations for H(t, x) and q(T , x), to use the self-similarity assumption of the horizontal velocity component u. 

In the next step, we calculate the total flow through a horizontal cross section at a vertical position z. It can be obtained by integrating the velocity... 

The vertical motion of the plume to the height where it becomes horizontal is known as the plume rise, (refer back to Figure 1). The plume rise is assumed to be a function primarily of the emission conditions of release, (i.e. velocity and temperature characteristics). A velocity in the vertical plane gives the gases an upward momentum causing the plume to rise until atmospheric turbulence disrupts the integrity of the plume. At this point the plume ceases to rise. This... 

In these equations fi is the coluirm mass of dry air, V is the velocity (u, v, w), and (jf) is a scalar mixing ratio. These equations are discretized in a finite volume formulation, and as a result the model exactly (to machine roundoff) conserves mass and scalar mass. The discrete model transport is also consistent (the discrete scalar conservation equation collapses to the mass conservation equation when = 1) and preserves tracer correlations (c.f. Lin and Rood (1996)). The ARW model uses a spatially 5th order evaluation of the horizontal flux divergence (advection) in the scalar conservation equation and a 3rd order evaluation of the vertical flux divergence coupled with the 3rd order Runge-Kutta time integration scheme. The time integration scheme and the advection scheme is described in Wicker and Skamarock (2002). Skamarock et al. (2005) also modified the advection to allow for positive definite transport. 

Since horizontal transport across the boundaries of the column is neglected and since it moves with the average ground-level horizontal wind velocity, the column may be mathematically represented as a horizontally uniform but vertically non-uniform column with a time-varying source of pollutants at the base. Thus, the only independent variables are time t and vertical distance z. The concentration of species i at time t and height z in the column Ci z,t) (From this point on we use Ci to denote , the mean concentration of species i.) is determined by integration of the abridged form of (7),... 

To evaluate the integrals in (5 49), we would need to determine the components of the rate-of-strain tensor in a Cartesian coordinate system with axes in the horizontal and vertical directions from the velocity components u -) and u(q, given by (5 40a) and (5-40b). This is not a difficult task. However, in the present case, we will be content to show that F-v) is 0(e ]) and is thus asymptotically small compared with l< (yp). To see this, we note that... 

Currents transport, or advect, the oil from a spill site. Oil slicks travel downwind at 2.5. 0% of the wind velocity (Kennish, 1999). Light oils spread faster than heavy oils. The spreading rate is more rapid at higher than lower temperatures and depends on the volume and density of the oil. As wind and wind waves develop, the slick breaks up into distinct patches of oil that drift slowly apart by horizontal eddy diffusion. Subsurface advection mixes the oil in subsurface waters to depths of about 10 m vertical diffusion plays a less integral role in subsurface mixing of the oil. 

A and B are empirical positive coefficients larger than unity (of the order of ten), y and z denote the horizontal and vertical coordinates of the two points in the plane normal to the mean speed, and U denotes the mean velocity [12, 13]. According to this expression, for distances r and frequencies n larger than zero the coherence is smaller than unity. This reflects the physical fact that, at any instant, the velocity fluctuations are not quite the same at two distinct points. The smaller the integral scales of turbulence in the y and z directions, the larger are the constants A and B, and the faster is the decay of the coherence function with increasing n and r. 

The troublesome baseUne error, which for some decades has made direct integration difficult if not impossible, is recovered and foimd to be embedded in the low-frequency fling. It manifests as an acceleration transient ( spike ) in aU three translational components, which on integration shifts the DC level of the velocity and causes linear trends in displacement It has been shown in Graizer (2005) and in Chanerley et al. (2013) that the vertical component is insensitive to tilts therefore, any spikes in the vertical direction in the acceleration time history are usually attributed to instrument noise and indeed are usually small when compared with the spikes of the horizontal components, which are attributed to ground rotation. This use of the undecimated wavelet transform recovers the... 

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