2.3. PBL Equations for Mean Flow and Their Applications

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1 .3. PBL Equations for Mean Flow and Their Applications Read Holton Section 5.3!.3.1. The PBL Momentum Equations We have derived the Reynolds averaed equations in the previous section, and they describe the mean flow, takin into account the effect of turbulence motion. Generally, inside the boundary layer, vertical turbulent flux diverence is much bier than horiontal one, because vertical shear/radient is much bier. We often consider the so-called horiontally homoeneous turbulence for which the horiontal turbulent flux diverence is nelected. We therefore have, from (.5), du 1 p wu ' ' = + fv, (.7a) dt ρ x 0 dv 1 p wv ' ' = fu. (.7b) dt ρ y 0 Consider further that the mean variables are for synoptic-scale horiontal flows, the acceleration terms are much smaller than the PGF and Coriolis terms, they can be nelected. 1 p 1 p Further, because fv =, and fu =, we replace the PGF ρ0 x ρ0 y terms usin u and v : wu ' ' f( v v ) = 0, (.8a) wv ' ' f( u u ) = 0. (.8a) 1

2 They represent the balance amon Coriolis force, pressure radient force and the vertical momentum flux diverence (a net stress applied to the air parcel). We will use the above two equations to solve for boundary-layer wind in two different models..3.. The Mixed Layer Model (Section of Holton) As we saw earlier, in the well-mixed boundary layer, the mean wind and mean potential temperature are nearly constant with heiht. If we look at the flux profiles, they decrease linearly with heiht, and to ero at the top of boundary layer (they often o to neative values before becomin ero) where turbulence becomes weak. Accordin to observations, the momentum fluxes at the surface can be accurately represented by the bulk aerodynamic formula (or dra law): ( wu ' ') sfc = Cd V u, (.9) ( wv ' ') = C V v, sfc d where C d = non-dimensional dra coefficient and V = u + v is the wind speed at the surface, or more precisely, at the anemometer heiht (10 m). The above formulae are very important and are used commonly in numerical weather prediction models to model the surface fluxes. Similar formulae are available for heat and moisture. The dra coefficient C d depends on the rouhness of the surface and the stability of the surface layer air. Over ocean, it is on the order of 10-3 and over land on the order of Now that we know the fluxes at the surface (~=0) and at the top of the boundary layer (=h) and know that the fluxes are linear function of heiht. We also assumin that the mean wind is constant with heiht, we can easily interate Equations (.8a,b) vertically, i.e.,

3 h h wu ' ' f ( v v ) d = d 0 0 h h wu ' ' f ( v v ) d = d = w' u ' w' u ' = Cd V u 0 0 h 0 Similarly, u v f ( v v ) h =. (.30a) C V d f ( u u ) h =. (.30b) C V d Without loss of enerality, we can choose our coordinate system so that the x-axis is parallel to u so that v = 0. Therefore, u = u K V v v = K V s where K C /( fh) and has units of s/m. s d s (.31a) (.31b) Since u is the wind speed (positive), we find that inside the mixed layer, the wind speed is subeostrophic (we now see why), i.e., less than the eostrophic wind u. Also we have a component of motion directin towards the low-pressure side (to the left of eostrophic wind in northern hemisphere and the riht in the S.H.) that is proportional to K s which is ero without surface dra. The v component is completely aeostrophic (perpendicular to the isobars), and arises due to the dra/fiction/momentum flux. The stroner is the surface dra, the larer is the anle of the wind to the eostrophic wind / pressure contours. 3

4 v u u Example, u = 10 m/s K s = / ( ) = 0.05 s m -1 u =8.8 m/s, v =3.77 m/s, V = 9.10m/s at all heiht within this well mixed layer. All of this is consistent with what you have learned before about the three-force balance. In a eostrophic flow, the PGF and the Coriolis force are in balance. When friction is present, friction acts in the opposite direction as the velocity vector and is therefore also perpendicular to the Coriolis force. The balance between the three forces are illustrated as follows: P- PGF P 0 φ V F r Cor P + As can be seen above, the wind vector is no loner parallel to the pressure contours, the flow is crossin the isobars to the lower pressure side! In vector form, the balance is expressed as 4

5 ˆ 1 C d fk V = p V sfc V. (.3) ρ h 0 The last term is sometimes called friction but keep in mind that this is due to turbulent mixin, not molecular viscosity, the latter is much smaller K-Theory: First-order Turbulence Closure (Section 5.3. of Holton) When the boundary layer is neutral or stable, the constant wind speed assumption used in the previous slab model is no loner ood (because of the absence of stron vertical mixin). To solve the equations for the mean variables, we have to express the turbulent fluxes in terms of the mean variables, to close the equation set (so that we have the same number of unknowns as the number of equations). This is called the closure problem. One of the commonly used methods is to express the fluxes in terms of the radient of the mean variables, in a way analoous to the molecular diffusion. The flux is assumed to be proportional to the local radient of the mean: u uw ' ' = K m, v vw ' ' = K m, (.33a) (.33b) and similarly to the potential temperature: θ θ ' w' = Kh. (.33c) 5

6 We call K m the eddy viscosity (has units of m /s) and K h the eddy diffusivity of heat. Both are positive. This closure scheme is often called the K-theory. This closure says that flux is always down-radient. With the closure iven in (.33), we can write equations in (.7) as du 1 p fv K u = + + m dt ρ0 x, (.34a) dv 1 p fu K v = + m dt ρ y. (.34b) 0 We see that the perturbation variables no loner appear in the equations. Given p we can solve for uandv. For the potential temperature, the equation is d θ θ = Kh + Qθ dt. (.34c) We will use this set of equations to obtain the Ekman layer solution in the next section. Note also that the K-theory actually has a number of limitations. Firstly K should be flow-dependent and it is often not easy to determine its value. Secondly, often the eddies that cause most of the mixin have sies that are comparable to the PBL depth and are therefore not so much related to local radient therefore the closure assumption does not work well. For this reason, certain non-local closure schemes have been developed. 6

7 .3.4. The Ekman Layer (Section of Holton) The Ekman layer is the part of the atmosphere that extends from the anemometer level up to the heiht where the radient wind balance holds (PGF + COR + CENTRIF). In this layer, we make the followin assumptions within the context of K-theory: a) The force balance is amon PGF, Coriolis and turbulence. Centrifual force is nelected. b) The flow is steady-state, i.e., / t = 0. c) The atmosphere is horiontally homoeneous, i.e., ()/ x+ ()/ y = 0. this implies that V is independent of heiht, via the thermal wind law. d) Lapse rate is neutral, i.e., θ / = 0. So, the PBL is well mixed. e) K m is constant f) Earth is flat with w =0 at round. ) Flow is incompressible. With these assumptions, our mean flow momentum equations in (.34) become u + f( v v ) 0 = v f( u u ) 0 = (.35a) (.35b) The above two equations are coupled. One way to simplify the solution procedure is to combine the two equations into one, by usin complex variables. Define complex velocities w= u + iv and w = u + iv. (.35a) + i (.35b) ( u + iv) m K + f( v ui v + u i) = 0 7

8 ( u + iv) m K + fi( iv u + iv + u ) = 0 w fi( w w ) 0 = (.36) Since we assumed that w is constant with heiht, (.36) can be rewritten as ( w w ) if ( w w ) 0 = K m (.37) which is a linear, nd-order ODE for ( w w ) with constant coefficients a basic ODE problem. The characteristic equation for (.37) is if λ = 0 if therefore λ =± where λ's are the eienvalues of the PDE. K m The solutions to (.37) are therefore (verify for yourself): if if w w = Ae + Be. (.38) Here A and B are arbitrary constants to be determined by the boundary conditions. The boundary conditions are w =0 at = 0 0 w = A+ B w= w as 0 = Ae A = 0. if Therefore B = w. Solution (.38) becomes 8

9 w= w w e if K m Without loss of enerality, we choose our x coordinate axis to be parallel to the eostrophic wind, so that w = u, v =0. w u u e if K m = (.39) Let's see what i is equal to. i= 1, 1+ i= 1+ 1, (1 + i) = (1 + 1) 1+ i (1 + i) = = i = i Our solution is therefore f K (1 i) m w= u (1 e + ) γ γ u + iv = u [1 e cos( γ ) + ie sin( γ )] γ u = u [1 e cos( γ )] (.40a) γ v = u e sin( γ ) (.40b) When plotted the solution on a hodoraph, which is a plot of the wind vector on a polar coordinate diaram, we obtain what is called the Ekman Spiral solution as shown below: 9

10 We see that with this solution, there is an aeostrophic wind component pointin to the left of eostrophic wind, i.e., towards the low-pressure side, this arees with the results of the earlier mixed-layer model, and the solution represents a three-way balance, i.e., a balance between the PGF, Coriolis and frictional forces. Compared to the mixed layer model, we don t assume that the boundary wind is constant with heiht, and we actually obtained the solution for the vertical wind profile. The idealied Ekman spiral wind profile is rarely observed in the atmosphere, however, because of other effects that are not considered in the idealied model. In the surface layer, the equations used to obtain this solution are not valid. One way to take it into account is to match the Ekman layer solution with that in the surface layer, doin so ives us the so called modified Ekman layer solution is that often a better match of the observations. The followin fiure ives such an example. 10

11 .3.5. Secondary Circulation and Spin-Down (Section 5.4 of Holton) From both the Ekman layer solution and the mix-layer solution obtained in section.3. (Equation.31), we know that the friction reduces the boundary layer wind speed below eostrophic wind, and causes it to cross the isobars from hih towards low pressure. In a synoptic situation where the isobars are curved, such as in low and hih pressure systems, the cross-isobaric component of flow near the surface causes risin air in low-pressure systems and descendin air in hihs. The processin of inducin vertical motions by boundary layer friction is called Ekman pumpin. 11

12 Usin Ekman spiral solution, we can quantitatively estimate the vertical motion at the top of boundary layer. Aain assume v = 0. From mass continuity equation and the Ekman spiral solution, the vertical velocity at the top of the boundary layer is D w d = 0 0 D v d y D v D u γ γ 0 (.41) 0 w( D) = d = e sin( ) d y y where D is the boundary layer depth. The mass flux across the isobars in a column of unit width spannin the depth of the boundary layer is D D (.4) γ M = ρ vd = ρ u e sin( γ) d where the depth D can be defined as D = π / γ, the level at which the wind (as iven by the spiral solution) becomes parallel to the eostrophic wind. Comparin (.4) to (.41), we see that M ρ0 wd ( ) = (.43) y therefore the vertical mass flux out of the PBL is equal to the converence of the cross-isobaric mass transport in the layer. u Further, = ξ, the eostrophic vorticity, and substitutin the Ekman y spiral solution into (.41) (notin the eostrophic wind does not chane with heiht), we obtain 1

13 1/ f wd ( ) = ξ f f (.44) hence the vertical mass flux is proportional to the eostrophic vorticity and the square root of the boundary layer eddy viscocity. In the northern hemisphere (f>0), w at Z=D is positive inside a cyclone ( ξ > 0) and neative inside an anticyclone. This w induces a secondary (vertical) circulation as shown below. Above the PBL inside a cyclone, there exists diverence, accordin to anular momentum conservation, the horiontal rotation rate (vorticity) will decrease with time, hence the spindown of the cyclone. Therefore, the spin down is not by simply surface friction and vertical momentum exchane, but throuh a most effective way that involves the vertical (secondary) circulation! A quantitative discussion of this usin a vorticity equation is iven in Holton section 5.4. Read it yourself! 13

14 Tea Cup Example. After the tea in a cup is stirred, the bottom friction slows down the spinnin tea in the cup near the bottom surface, breakin the balance between the PGF and centrifual force. The PGF cause flow towards the center of the cup near the bottom, creatin vertical motion alon the central axis and induces compensatin flow above this layer away from the central axis. To conserve anular momentum, the spin has to slow down. This is analoous to the spin-down of a cyclone. 14

Chapter 2. Turbulence and the Planetary Boundary Layer

Chapter 2. Turbulence and the Planetary Boundary Layer In the chapter we will first have a qualitative overview of the PBL then learn the concept of Reynolds averain and derive the Reynolds averaed equations.