Frictional boundary layers

Transcription

1 Chapter 12 Frictional boundary layers Turbulent convective boundary layers were an example of boundary layers generated by buoyancy forcing at a boundary. Another example of anisotropic, inhomogeneous turbulence is the turbulence induced by flow over a frictional boundary. Both the atmophere and the ocean are influenced by solid boundaries, near which turbulence is modified, because boundary friction reduces the flow speed near the boundary, introduces a surface stress and generates vorticity. We consider a parallel irrotational flow over a flat boundary. Turbulence is generated because the no-slip condition U = 0 at the boundary means that a shear layer results, and vorticity is introduced into the flow. We can use similar ideas to those of the isotropic homogeneous turbulence spectrum to look at different regions of the boundary layer. 1. The viscous sublayer For distances close to the wall, i.e. z < z f where z f is the distance at which Re = 1, friction is important. This can be compared to lengthscales l < 1/k d in homogeneous turbulence, where viscosity is important. 2. The integral scale The depth of the turbulent boundary layer ν is the maximum size of the eddies, the integral scale. 3. The inertial sublayer At distances further away from the wall, we can neglect viscosity. Similarly, if we are not close to the edge of the boundary layer we can assume that the flow will not depend directly on the size of the boundary layer. Therefore we have an inertial sublayer for z f << z << ν. This region is similar to the inertial range in homogeneous turbulence, where the flow is not affected by or by k 0, the wavenumber of the energy input. 1

2 4. The ambient flow Finally at some distance z > ν, the flow is no longer turbulent and we are in the irrotational ambient flow Equations of motion We will assume a constant background flow U 0, which is independent of distance along the plate x and distance normal to the plate z. We assume 2-dimensional flow (σ/σy = 0), and also assume that downstream evolution is slow. If L is a streamwise lengthscale, we are assuming ν/l << 1, so that we can neglect variations in the streamwise direction compared to those in the vertical (i.e. σ/σx = 0). Given these assumptions, the Reynolds averaged equations become du d du W = w u (12.1) dz dz dz dw = 0 (12.2) dz Now with no normal flow through the boundary, we have W = w = 0 at z = 0, the bottom boundary. Then from eqn 12.2 W = 0 for all z. Then eqn 12.1 becomes: d du w u = 0 (12.3) dz dz Hence if we have a stress φ given by du du φ = = (12.4) dz w u dz z=0 this stress is constant throughout the boundary layer. Near the boundary the stress is dominated by the viscous term. Away from the boundary we will have We can define a velocity scale from this surface stress w u = φ (12.5) 2 u = φ (12.6) where u is the friction velocity. Away from the boundary eqn (12.5) implies that u is the turbulent velocity fluctuation magnitude. Then we see that a frictional length scale can be defined as z f = (12.7) u the lengthscale at which Re = 1, where the transition between the inertial and viscous sublayers occurs.

3 12.2 Law of the wall In the viscous sublayer z < z f, the velocity must depend on z, the distance from the wall, u, the friction velocity and, the viscosity. We can write this relationship as U zu = f (12.8) u Note that U has been nondimensionalized by u, and the distance z has been nondimensionalized by the frictional lengthscale /u. We can rewrite the relation in nondimensional form U + = f (z + ) (12.9) where U + = U /u and z + = zu /. Near a rough wall, instead of being controlled by a frictional length scale, the roughness length z 0 may instead be the relevant parameter, if z 0 > z f The velocity-defect law Outside the viscous layer, we neglect viscosity. Near the edge of the turbulent boundary layer, the flow must depend on the turbulent velocity scale u, the total depth of the boundary layer ν and the height z away from the wall. We can express this dependence as du u z = g (12.10) dz ν ν We know that at z, U U 0. So we integrate from z = in towards the boundary to obtain U : u du z dz = g dz (12.11) z dz ν z ν and hence z U (z) U 0 = u F (12.12) ν or U + + U 0 = F (κ) (12.13) where κ = z/ν. This is a similarity solution for U +, which assumes that as the boundary layer changes size, or for different boundary layers U + has the same form. This similarity solution is only valid outside of the viscous boundary layer, and cannot satisfy the boundary condition U = 0 at the wall.

4 12.4 Logarithmic layer Thus far we have two different laws for U + ; one applies close to the wall in the viscous sublayer, the other further away from the wall. Of course the velocity doesn t suddenly jump from one form to another - there is a transition region. In this transition region we expect both the law of the wall and the velocity defect law to apply. Then du + df = (12.14) dz + dz + from eqn 12.9 and du + dκ df κ df = = (12.15) dz + dz + dκ z + dκ from eqn In this overlap region these two must be equal so df κ df = (12.16) dz + z + dκ so that + df df z = κ (12.17) dz + dκ The right hand side can depend only on κ and the left hand side can depend only on + z. This can only be true for both if they equal a constant: z + df dz = κ df + dκ = 1 (12.18) γ where γ is the Von Karman constant. This implies that du dz = u (12.19) γz so that in this region the only important quantities are u and z. Then in this transition region, the inertial sublayer, the flow is unaware both of viscosity and of the size of the boundary layer ν - just as in the inertial range turbulence is unaware of viscosity or of the integral scale of the forcing. Integrating eqn we have U 1 u z = ln + A (12.20) u γ and U U 0 1 z = ln + B (12.21) u γ ν The region where this applies (κ << 1, z + >> 1) is known as the logarithmic layer. Near a rough boundary, the equivalent of would be U 1 z = ln + A (12.22) u γ z 0 with z 0, the roughness length, taking the place of z f = /u, the frictional lengthscale.

5 12.5 Frictional Drag coefficient The stress at the boundary φ is often written in terms of a frictional drag coefficient c d : 1 2 φ = c d U 2 0 (12.23) where u 2 c d = 2 U 0 (12.24) For a smooth boundary, from equation and we obtain [ ] 2 c d 1 c d 1 U 0 ν = ln + ln + A B (12.25) 2 2γ 2 γ while for a rough boundary [ ] 2 c d 1 ν = ln + A B (12.26) 2 γ z Boundary layer turbulence with buoyancy effects At a boundary, in addition to surface stresses acting as a source of vorticity, we may also have buoyancy forcing at the surface (for example atmospheric heating and cooling at the ocean surface, or radiant heating at the land surface). As we have done before, we might assume that the dynamics of the turbulent boundary layer are unaffected by buoyancy forcing if the buoyancy effects are small compared to the shear effects. In this scenario, the relative measure is the flux Richardson number, w b R f = (12.27) u w σu/σz the ratio between buoyancy production and shear production of TKE. Recall that positive values of R f imply stable stratification, when TKE is lost to PE, and negative values imply unstable stratification. Hence if R f > 1, we expect the surface buoyancy fluxes to suppress the boundary layer turbulence, while if R f < 1 we expect the boundary turbulence to be dominated by convective mixing, rather than shear generated turbulence. For 1 < R f < 1, the shear production of turbulence dominates.

6 Using boundary layer scaling we have σu σz = u γz (12.28) Then 2 uw = u u 3 u 3 (12.29) w b γz R f = (12.30) 3 When R f = 1, then the buoyancy production/loss of TKE is of equal magnitude to shear production. This occurs at a lengthscale u L b = (12.31) γ w b If the buoyancy flux is supplied through a surface flux, then the minimum value of L b is L b = (12.32) γ w b 0 This is the Monin-Obukhov lengthscale. If w b 0 > 0 the flux is destabilising. Then for distances from the boundary z < L b, the shear production dominates, while for distances z > L b b, buoyant convection dominates. If w 0 < 0 for distances z > L b the turbulence is damped by the stable stratification. Further reading: Tennekes and Lumley Ch 2.5, 3.4 and 5; Lesieur Ch IV, section 1.2.6; Hinze Ch 7; Phillips, Ch 6.6.

7 Chapter 13 Parameterizing turbulence in ocean models The model filtered equations for the large-scale In a numerical model we can directly solve for the field variables at the scale of the model grid. Recall that the relevant equations for the velocity components and conserved tracers are: σu i σu i 1 σ σu i + u j = P ν i.j + ρ 0 ρ 0 u j u σt σx j ρ 0 σx i (13.1) j σxj σu j = 0 (13.2) σx j σt σt σ σt σt + u j = γ T σx j σx u j j σxj (13.3) (and similarly for salt). ζ represents the space and time filtered value of ζ, on the model grid scale and timestep, while ζ is the perturbation quantity ζ = ζ + ζ. By definition ζ = 0. The problem of representing turbulence in numerical models of finite grid size and time step amounts to finding a way to express the subgridscale quantities u j u i (the Reynolds stresses) and T u (the turbulent fluxes) in terms of resolved quantities. j In Chapter 8 we learned how the subgridscale tracer fluxes can be written in terms of a symmetric component which is like a effective diffusion, and an antisymmetric part which is like an advective transport. In Chapter 9 we learned more about this antisymmetric component as it relates to mesoscale eddies. Here we will focus 7

8 entirely on the symmetric, diffusion-like component. In addition, despite the fact that momentum is not a conserved quantity, and we have learned in Chapter 8, that momentum fluxes cannot therefore be written rigorously in terms of an effective viscosity, we will examine parameterizations which do just that Eddy viscosity models The simplest parameterization of these turbulent quantities employs an eddy viscosity assumption: T σu i σu j u j u i = + (13.4) 2 σx j σx i σt (13.5) u j T = γ T σx j This is a first order closure. Models vary in the complexity of the system used to specify T and γ T, the eddy viscosity and eddy diffusivity. µ T, γ T can be specified directly in terms of the large-scale quantities of the flow and/or model grid, or they can be specified in terms of subgridscale quantities for which extra prognostic equations are required. Most eddy viscosity models draw on Prandtl s Mixing length model. Mixing length model Consider a parcel in a 2-D shear flow (U(y), 0, 0), initially at a position y = 0. If the parcel moves due to turbulent motion, up to a position y, and it conserves momentum, then it has a momentum deficit compared to the parcels around it of σu M = [U(y) U(0)] + [u (y) u (0)] [U(y) U(0)] y (13.6) σy The average momentum flux is then Using the approximation σu Mv = u v = yv (13.7) σy yv (v 2 )l (13.8) where l is the mixing length - the distance at which v and y become uncorrelated, we then have σu σu u v = (v 2 T )l (13.9) σy σy

9 where T = c (v 2 )l (13.10) where c is a constant. If the turbulence is isotropic v 2 = u 2 = w 2, we can write this as T = c µ (q)l (13.11) where q/2 is the subgridscale kinetic energy, and c µ is again a constant. Eqn could also be obtained on dimensional grounds, by assuming the turbulent motion is characterised by a single velocity scale q, and a single lengthscale l. This mixing length argument cannot tell us anything about the ratio T /γ T. It is often assumed that the turbulent Prandtl number T /γ T 1, so that turbulent transport of heat and momentum are equally efficient. However, this is one of the adjustable parameters of a turbulence parameterization. If we assume constant eddy viscosity and diffusivity, we are assuming that a single mixing length l and subgridscale kinetic energy v 2 characterise the flow at all points in space and time. Obviously this cannot be true for the ocean, where turbulence is highly inhomogeneous. Now the problem of estimating T and γ T can be reshaped as one of estimating the TKE and the mixing length l. An alternative, equivalent expression for T can be obtained by using Kolmogorov inertial range scaling:e(k) π 2/3 k 5/3. Then where l is a lengthscale. π q 3/2 l (13.12) Then we can write T in terms of q and π: T = c µ q 2 π (13.13) Hence we have two equivalent representations of T in terms of turbulent quantities: eqn (13.11) and eqn (13.13). One or the other of these representations for T form the basis of many eddy viscosity models. Of course we need to know either q and l or q and π to obtain T Smagorinsky model If we assume a local production/dissipation balance exists for the TKE σu i u i u j = π (13.14) σx j

10 and rewrite eqn as 2 1 σu q 3/2 i σu j T + = π = (13.15) 2 σx j σx i l and use eqn to eliminte q: 2 T = l 2 1 σu i σu j m ( + ) (13.16) 2 σxj σx i where 3/4 l m = c µ l (13.17) Now we just need to decide on l, the mixing length. In the Smagorinsky model, l is assumed to be proportional to the largest length scale of the unresolved motion, i.e. the grid scale x. Then, from eqn σu j T = (C S x) 2 ( 1 σu i + ) (13.18) 2 σx j σx i The value of the constant C S is chosen by assuming that the cutoff wavenumber k C = α/ x lies within a k 5/3 inertial range of a Kolmogorov-type energy spectrum. We must have the subgridscale kinetic energy dissipation equal to π. Then where C K is the Kolmogorov constant. 1 3/4 3CK C S (13.19) α 2 It is useful here, since the Smagorinsky model is widely used in oceanography, to remind ourselves of the assumptions we have used to get here. We have assumed: 1. A local balance between shear production and dissipation of subgridscale kinetic energy, i.e. ignoring buoyant production and transport of TKE. 2. A single characteristic lengthscale x 3. x is within an inertial range appropriate to isotropic homogeneous turbulence. These assumptions will be most clearly violated when the model resolution is so coarse that the inertial range is not even partially resolved, when there are strong inhomogeneities in the turbulence, so that transport of TKE is important, when the subgridscale turbulence cannot be characterised as isotropic homogeneous turbulence (i.e. stratification and rotation are important) and when buoyancy is important in generating or removing TKE. Because this model is not appropriate when buoyancy is important on the subgrid scales, it is typically only used in coarser resolution ocean models for the horizontal components of viscosity and diffusivity. If the model resolution is finer, so that buoyancy production of kinetic energy is resolved (i.e. an LES model), it might be justified to assume a balance between shear production and dissipation for the smaller subgrid scales, and hence use the Smagorinsky model for vertical diffusivities/viscosities too.

11 Pacanowski and Philander model A contrasting first order closure for the eddy viscosity and diffusivity is that of Pacanowski and Philander (1981). They prescribe T and γ T of the form T = γ T = where Ri, the Richardson number is given by 0 (1 + τri) n + b (13.20) T (1 + τri) + γ b (13.21) N 2 Ri = 2 U z + V 2 z (13.22) and b, γ b, 0, n and τ are all adjustable parameters. Obviously, provided τ > 0, this model gives greater values of T and γ T when Ri is smaller, attempting to parameterise Kelvin-Helmholtz instability. The functional dependence on Ri is a fit to laboratory measurements of mixing in a shear layer. Note that T /γ T = 1, but instead γ T decreases more rapidly as Ri increases. The reasoning behind this is that mixing is not necessary for momentum exchange, but it is necessary for exchange of tracer properties. This parameterization gives good results in regions where the vertical shear of the large-scale currents is the principal cause of vertical mixing (e.g. the Equatorial undercurrent), but it is wholely empirical, with no physical basis for the choice of the arbitrary parameters. Good results are obtained by careful tuning of these parameters Boundary layer models A large group of models has been developed to parameterize turbulent mixing in boundary layers - e.g. the surface mixed layer of the ocean, bottom boundary layers above topography, and the bottom boundary layer of the atmosphere. A distinguishing feature of these models is that the boundary layer region is treated differently from the rest of the ocean/atmosphere, whereas the models above are applied anywhere, without any knowledge of the location. In boundary layers, the mixing processes depend on the distance from the boundary - the inhomogeneity of turbulence must be included in any parameterization. The processes in a boundary layer can essentially be split into two: (i) Convectively driven mixing - as discussed in chapter 7, where surface buoyancy forcing leads to developement of a well-mixed layer bounded by an interface with the stratified fluid. (ii) Stress driven boundary layer - this might include mixing induced by flow over a frictional boundary, roughness effects, and mixing induced by the wind stress at the

12 surface of the ocean. Eddy viscosity/diffusivity based boundary layer schemes: (a) convective adjustment Convective adjustment acts to eliminate static instability - e.g. when σb/σz < 0, mixing is introduced to eliminate or reduce the negative buoyancy gradient. The eddy diffusivity application of convective adjustment imposes a greatly enhanced vertical diffusivity whenever σb/σz < 0. (The actual value of this enhanced diffusivity does not appear to matter, so long as it is large enough that vertical mixing is effectively instantaneous). (b) frictional bottom bbl parameterizations The commonest, simplest way to account for frictional drag near a boundary is by a bottom stress: φ i,k = C d (U 2 )U i (13.23) where i = 1, 2 and k = 3. This stress is applied at the bottom, and the assumption is therefore that none of the frictional boundary layer is resolved. C d is a constant, which implies a fixed ratio of roughness length to depth of the bottom boundary layer. (c) K-Profile Parameterization (KPP) This boundary layer scheme due to Large, McWilliams and Doney (1994), prescribes a profile of eddy diffusivities as a function of depth relative to the total oceanic boundary layer depth. It accounts for both wind-stirring and convection. The main controlling parameter is σ = d/h where d is the distance from the surface, and h is the depth of the boundary layer. Other controlling parameters are the surface fluxes of momentum and buoyancy, w u and w b, and the Monin-Obukhov length-scale L b, and the nondimensional ratio κ = d/l b. Non-local closure All of the methods we have discussed thus far for parameterizing the turbulent fluxes of heat and momentum involve local closures - i.e. a relationship between the local values of the large scale gradient and the turbulent flux. An alternative is a nonlocal closure, where the flux might be independent of the local gradients. KPP incorporates such a nonlocal closure. The vertical fluxes are parameterized as σx w x = K x ρ x (13.24) σz where ρ x is the nonlocal flux term. Nonlocal fluxes of scalars are important in convective boundary layers, i.e. when buoyant production is the dominant source of TKE. Then scalars are largely homogenised over the convective layer, but fluxes are still finite. ρ x is set to zero when there is no buoyant convection (w b 0 0), but is nonzero for scalars when w b 0 > 0: w T 0 ρ T = C s (13.25) w(σ)h

13 Here w T 0 is the surface heat loss. The vertical velocity scale w(σ) depends on position within the boundary layer (i.e. whether or not σh is greater than or smaller than the Monin-Obukhov lengthscale). In the convective limit (h >> L b ) then w T 0 ρ T = C w h (13.26) where w is the convective velocity scale and B 0 is the surface flux. w (B 0 h) 1/3 (13.27) K x, the turbulent diffusivity, is given as a function of the depth of the turbulent boundary layer h (which is diagnosed from the mean density field), the turbulent velocity scale w, and a non-dimensional shape function G(σ), where σ = z/h. K z (σ) = hw(σ)g(σ) (13.28) We can see that this is of a similar form to eqn 13.11, but now w(σ) is determined diagnostically, and the turbulent lengthscale is a function of position within the boundary layer, and scales with h, the depth of the boundary layer. Substituting in eqn we see that in the convective limit (dt/dz = 0), the heat flux is given by C w T = w T 0 (13.29) G(σ) so that the flux is proportional to the surface flux, scaled by a function of the position within the mixed layer. Wind-stirred limit The velocity scales w(σ) are given in terms of nondimensional functions φ(κ) where φ(κ) are nondimensionalized flux profiles, defined by γu w(σ) = (13.30) φ(κ) γd φ m (κ) σ z (U 2 + V 2 ) 1/2 (13.31) u for momentum. These φ are given as empirically determined functions of κ. For κ = 0 (no buoyancy forcing) then φ = 1 to give the log layer profile U (d) U 0 1 = ln(d/z 0 ) (13.32) u γ in the near-surface layer. Then w(σ) = γu. and the diffusivity is then K x (σ) = hγu G(σ) (13.33)

14 Boundary layer depth One of the most important aspects of KPP is the dependence of parameters on the total boundary layer depth. This adds another non-local aspect to the parameterization. The boundary layer depth is determined as the location where the bulk Richardson number exceeds some critical value, with the Richardson number Ri(d) defined as (b r b(d))d Ri(d) = (13.34) V r V (d) 2 + V t 2 (d) where V r and b r are the near surface resolved velocity and buoyancy (averaged over the surface layer, the top 10% of the boundary layer). V t /d is the turbulent shear, which has to be parameterized. This V t term is very important, reducing the Ri and hence extending the depth of the boundary layer over which enhanced diffusivities are applied into the stably stratified region. If V t is set to zero, then KPP does not reproduce penetrative convection. The value of V t depends on the ratio of the reverse (penetrative) buoyancy flux at the base of the convective layer to the forcing w b 0, which is known empirically. For stable forcing w b 0 < 0, so that L b > 0, upper limits are imposed on the boundary layer depth: h < L b ; h < h E = 0.7u /f (13.35) where h E is the Ekman layer depth. Momentum versus tracers Momentum and tracer fluxes are parameterized in a similar fashion in KPP except for the following: φ m > φ t when L b < 0 (i.e. convective forcing), so that tracers are mixed more efficiently than momentum. ρ m = 0 - no counter-gradient fluxes of momentum in convective scenarios equation models From eqn we could calculate the eddy viscosity by prescribing the length scale l and calculating q using a prognostic equation. Such a model, which includes one extra prognostic equation are called 1-equation models. Examples in use in oceanographic modeling include the LES models of Skyllingstand and Denbo, and Garwood, both of which are based on a model developed for the atmosphere by Deardorff (1980), and used extensively by Moeng and coworkers. The exact prognostic equation for the subgridscale (i.e. turbulent) kinetic energy q/2 is σ σ q σ 1 σ q + U j = u i p ν i,j + σxj 2 u j u i u σt σx i j 2 σx j ρ0

15 2 σui σ u u σx j i U i + b w (13.36) j σx j (with summation over all i and j). Recall that the first three terms on the right hand side are transport terms, expressed as the divergence of a flux,.f, the next term is equal to the dissipation π and the final two terms are turbulent kinetic energy production terms P + B (which also appear in the mean field equations for U j and T.) Then we can write eqn more succinctly as D q =.F π + P + B (13.37) Dt To close this equation we obviously have to parameterize the production terms in the identical fashion to the way in which they are parameterized in the mean field equations: and B = γ T σb σz = T P r T σb σz = c µ (q)l P r T σb σz (13.38) P = u u 1 σu j σu i 1 σu j σu i j i = T + = cµ (q)l + (13.39) σu i /σx j 2 σx i σx j 2 σx i σx j Furthermore the TKE transport terms can be parameterized in a similar way to other subgridscale fluxes, i.e. by a downgradient eddy diffusion: T q F = (13.40) σ q 2 where σ q = T /γ q is the Schmidt number, the ratio of turbulent eddy viscosity to turbulent diffusivity of TKE. Finally we need to parameterize π, the dissipation. One-equation models use eqn 13.12, with l prescribed. (Remember that this implies Kolmogorov scaling). Then the prognostic equation for q is σ σ q σ q 3/2 T σ q + U j = + σt σx j 2 σx j σ q σx j 2 l 1 σu j σu i σ T σb T + U i 2 σx i σxj σx j P rt σz (13.41) The 1-equation model contains three empirical constants: c µ and σ q, P r T. In order to implement this scheme, the turbulent length scale l has to be prescribed. Usually, if the grid scale lies within the turbulent regime (i.e. a Large-Eddy simulation), l = C δ x is used (where c δ is a prescribed constant). This is consistent with the integral over wavenumbers down to the cutoff wavenumber implied by eqn If

16 stratification is strong enough that the turbulent length scale is reduced below the grid stratification, l is reduced to the Osmidov lengthscale: l = c stab q/n, where c stab is another constant. These prescriptions of lengthscales do not work well near the boundary - they do not reproduce the law of the wall. Corrections designed to give the correct logarithmic velocity field have to be included. From eqn (12.19), in the log layer of wall-bounded turbulence, an appropriate form for the lengthscale is l = γz, where z is distance from wall and γ is the Von Karman constant. We also have from eqn (12.8) that in the viscous boundary layer the turbulent mixing must tend to zero, i.e. l m 0 as z/z f 0, where z f = /u. An appropriate form for l m is therefore l = γz(1 exp( z + /A + )) where A + is a constant. In a turbulent plume or jet l = ν, the width of the turbulent region (which increases downstream through entrainment). 1-equation models therefore require some prior knowledge regarding typical turbulent velocity scales (unless we are in the inertial range, when l = x can be used) Two-equation models Two-equation models also use eqn to calculate the TKE. However, the turbulent length scale l is not specified a priori. Instead, l is found by use of an additional prognostic equation. In the class of models known as k-epsilon models (where k is used for the turbulent kinetic energy - we have used q to avoid confusion with the wavenumber k) the additional equation is for π, from which l is determined using Alternatively, eqn is used to specify π in the q prognostic equation, and another prognostic equation is added for ql n. An example of the latter approach is the Mellor-Yamada level 2.5 model. Because both model classes make use of equation they are in fact equivalent. (a) k-epsilon models The equation for π is of the same form as the approximated form of the q equation eqn σ σ σ T σ π 2 P B + U j π = π C ɛ2 + C ɛ1 π + C ɛ3 (13.42) σt σx j σxj σ ɛ σx j q q q where the first term on the right hand side is the turbulent transport term (parameterised by an eddy diffusivity), the second term on the right hand side is the

17 destruction of dissipation, and the third and fourth terms are the production of dissipation. C ɛ1, C ɛ2 and C ɛ3 are constants (empirically determined), and P + B is the subgridscale kinetic energy production (from eqn 13.41). σ ɛ is the ratio T /γ ɛ. In the k-epsilon model, we therefore have 7 empirical constants: σ q, σ ɛ, P r T, c µ, C ɛ1, C ɛ2, and C ɛ3. These parameters are tuned to give best results when compared to flows for which data is available - but this does not mean they should have the same value for some other completely different flow! As for 1-equation models, modifications have to be introduced to the k-π model in stable stratification. Imposing an upper limit on the length-scale proportional to the Ozmidov scale in a stably stratified flow corresponds to a lower limit on π: π 2 C OZ q 2 N 2 N 2 > 0 (13.43) In addition a minimum background level of q might be needed in stably stratified flows. (b) Mellor-Yamada level 2.5 model The Mellor-Yamada model typically employs a boundary layer approximation, such that σ/σz >> σ/σx, σ/σy. Then the subgridscale kinetic energy equation is D(q/2) σ T σ q = π + P + B (13.44) Dt σz σ q σz 2 where π = q 3/2 /l as before, and P + B is the production term, now including only vertical derivatives and fluxes: σu σv P + B = w u v + b w (13.45) σz w σz Additionally, a second prognostic equation has to be included to determine the turbulent lengthscale: D(ql) σ T σ q 3/2 2 l = (ql) + le 1 P q 1 + E 2 (13.46) Dt σz σ l σz B 1 γl where P q is the subgridscale kinetic energy production. B 1, E 1, E 2 and σ l are constants to be empirically determined (by comparison with data). The third term on the right is again, required to reproduce the boundary layer scaling close to a solid wall: γ is the Von Karman constant, and L is a measure of the distance from the boundary.

18 Stability functions Both k π and q l models calculate the turbulent viscosities and diffusivities using c µ T = c µ (q)l ; γ T = (q)l = c µ (q)l (13.47) P r T The parameters c µ and c µ are known as stability functions since they are often prescribed to depend on stratification. Many alternative derivations of the dependence of c constant c and c. µ and c µ have been made. In addition many implementations of k π model use µ µ nd order closure schemes So far we have discussed only eddy viscosity models, where the subgridscale flux is parameterised in the mean field equations by eqns 13.4 and An alternative is to write out the exact prognostic equations for the second order correlations u i u j and i T u. These of course involve third order quantities of the form u i u j u i. Then some closure assumption, relating these third order quantities to the 2nd order correlations must be made. An example of a 2nd order closure scheme is the Mellor-Yamada level 4 model. The closure assumptions are: ( ) σu i u 3 j σu i u k σuju u u lq 1/2 S k k u i j = q + + (13.48) 5 σx k σx j σx i σu k T σu j T u k u j T = lq 1/2 S ut + σx j σx k where S q and S ut are empirical constants. (13.49) Using these closure assumptions, the evolution of the second order moments can be calculated using the prognostic equations. Note the high cost of such a scheme: for each of the 6 u iu j and 3 u i T terms, a new equation is added - 9 additional prognostic equations! The closure assumptions are similar to those of the first order scheme - essentially a downgradient diffusion, so the problems in dealing with nonlocal fluxes are not necessarily cured by these schemes. However, if a countergradient flux of a first order quantity (e.g. T, U) is a consequence of a downgradient flux of a higher order quantity, then the 2nd order scheme will work better than the first order scheme.

19 Summary Development of parameterizations is a far from exact science. No matter how sophisticated the model, closure assumptions have to be invoked ultimately. The models are very sensitive to the closures used. Particular problems occur in (a) stable stratification (e.g. the base of the mixed layer, the ocean interior) (b) the near-boundary regions. Further reading Metais, Ch XII Introduction to the modeling of turbulence, lectures from the Von Karman Institute for Fluid Dynamics, 1993 Mellor and Yamada, 1982: Development of a turbulence closure model for geophysical fluid problems. Reviews of Geophysics and Space Physics, 20, Large, McWilliams and Doney, 1994: Oceanic vertical mixing: a review and a model with a nonlocal boundary layer parameterization. Reviews of Geophysics and Space Physics, 32, Skyllingstad, Eric D., W. D. Smyth, J. N. Moum, H. Wijesekera, 1999: Upper-Ocean Turbulence during a Westerly Wind Burst: A Comparison of Large-Eddy Simulation Results and Microstructure Measurements. Journal of Physical Oceanography: Vol. 29, No. 1, pp Burchard, Peterson and Rippeth, 1998: Comparing the performance of the Mellor- Yamada and k π two-equation turbulence models. J. Geophys. Res., ,543-10,554.