BOUNDARY-LAYER METEOROLOGY

Transcription

1 BOUNDARY-LAYER METEOROLOGY Han van Dop, September 2008 D 08-01

2 Contents 1 INTRODUCTION Atmospheric thermodynamics Statistical aspects of fluid mechanics CONSERVATION LAWS Governing equations THE REYNOLDS EQUATIONS The average flow The mean-flow energy equation The turbulent kinetic energy equation Spectra Closure THE ATMOSPHERIC BOUNDARY LAYER Phenomenology The surface layer; Monin-Obukhov theory The neutral boundary layer The convective boundary layer The stable boundary layer Summary EXCHANGE OF HEAT AND WATER VAPOUR The surface-energy budget The profile method The energy-balance method Estimating the evaporation Air-Sea interaction HETEROGENEOUS BOUNDARY LAYERS Thermal transitions Roughness transitions EXERCISES BOUNDARY LAYERS AND TURBULENCE 77 2

3 Chapter 1 INTRODUCTION This reader acts as a follow-up of the lecture notes of the Bachelor Hydrodynamics and Turbulence course. Some of that material directly relating to boundary-layer meteorology, will be shortly discussed here. 1.1 Atmospheric thermodynamics First Law of Thermodynamics The change of energy dq per unit mass of a closed system equals the sum of the change of the internal energy du, and the amount of work, pdv : dq = du + p dv. (1.1) The specific heat at constant volume, or constant pressure, c v and c p are: (dq) v = (du) v c v dt (dq) p c p dt where (... ) v denotes that volume is held constant, and (... ) p indicates a constant pressure (c v,p in J kg 1 K 1 ). Equation of state Again for a unit mass we have pv = RT (1.2) or, using ρ = 1/V : p = ρrt. (1.3) 3

4 An alternative for the First Law can now be written as: dq = c p dt 1 dp. (1.4) ρ where c v + R = c p, and R is the specific gas constant for dry air. R = R /m a, with R, the universal gas constant (8.314 J K 1 mole 1 ) and m a the molecular weight of (dry) air ( kg). Thus R equals 288 J kg 1 K 1. Adiabatic lapse rate The heat exchange between the atmosphere and its surroundings, for example by molecular conduction or by radiation, is often negligible, implying dq = 0. From the first law of thermodynamics and the hydrostatic equation, it follows that c p dt + g dz = 0 dt dz = g Γ d 10 2 Km 1 c p (1.5) This vertical temperature gradient is commonly referred to as the (dry) adiabatic lapse rate. Entropy The entropy S is defined as ds dq T = c dt p T R dp p = c p d ln T R d ln p. Upon integration, this yields: S = c p ln T R ln p. (1.6) Potential temperature In adiabatic processes, ds = 0. In a process (1 2), this means that : 4

5 c p ln T 1 R ln p 1 = c p ln T 2 R ln p 2. (1.7) If we assign a constant pressure p s (e.g mb) and temperature θ (potential temperature) to the reference level 2, then (1.7) can be written as: c p ln T R ln p = c p ln θ R ln p s, (1.8) where the subscript 1 is omitted. This equation can be rewritten to obtain θ = T ( p s p )κ, (1.9) where κ = R/c p. This expression allows us to determine the potential temperature of any air mass as long as its temperature and pressure are known. In this way, a vertical profile of potential temperature can be constructed. Using (1.6), the entropy now becomes S = c p ln θ R ln p s, (1.10) leading to the conclusion that, in case of an adiabatic process, there is a direct connection between the potential temperature, θ, and the entropy S. Furthermore, Eq. (1.8) implies dθ dz dt dz + Γ d, (1.11) where the dry adiabatic lapse rate is given by Γ d = θ T g c p g c p. (1.12) The above-mentioned equation poses a simple relation between the vertical temperature gradient dt/dz and the vertical gradient of the potential temperature dθ/dz. Standard atmosphere Using (1.5), an atmospheric temperature profile can be described: T T 0 = 1 z H, (1.13) 5

6 T 0 being the surface temperature, and H = c p T 0 /g. We can now invoke the equation of state (p = ρrt ) and the hydrostatic equation to derive equations for vertical atmospheric profiles of pressure and air density: p p 0 = (1 z H ) cp R (1.14) ρ ρ 0 = (1 z H ) cp R 1. (1.15) The atmospheric scale height is denoted by H. This approach implicates that the atmosphere has a finite thickness (H 30 km) which is obviously not correct but nevertheless yields a reasonable estimate of the thickness of the atmosphere. 1.2 Statistical aspects of fluid mechanics The length scales relevant in a turbulent flow cover a wide range. The macroscale and the smallest turbulent scale, known as the Kolmogorov microscale, are related through: l/l K = Re 3 4 (1.16) In the atmospheric boundary layer, typical values of l = 1000 m and l K = m, yield a Reynolds number of O(10 8 ). Solving all length scales up to m on a domain of 10 km x 10 km x 1 km using a numerical program would require grid points, whereas computation on a grid with points is currently feasible. Thus, turbulence equations cannot be solved numerically in the entire domain of length scales. As an alternative, turbulence equations could be solved for averaged quantities only. In this way, the range of length scales, and thus the required amount of grid points, can be reduced dramatically. Stochastic variables, moments, probability density functions In a turbulent flow, kinematic processes occur over a wide range of scales. As a first step in a statistical treatment of turbulence, the average of a variable (the first moment) can be determined. More detailed information of the flow can be obtained by calculating higher moments of variables, or combinations of variables (correlations). When analysing a turbulent flow in a statistical way, correlations between flow variables at different locations or at different times play a very important role. Examples of such correlations are the variance u 2 u(x, t)u(x, t) (1.17) and the covariance 6

7 uw u(x, t)w(x, t), (1.18) being statistical properties of flow velocity at location x and time t. The overbar is used for the so-called ensemble-average of a quantity, i.e. the average obtained by taking the average of a large amount of realisations under the same boundary conditions. If we treat velocity, pressure and other quantities in a turbulent flow as stochastic variables, it is possible to define the flow by its moments. Let u be any variable, and p(u) its probability density function. The moments are then given by u m + p(u)u m du (1.19) A stationary stochastic process is defined as a process where the density function p is invariant under a time translation. If a stochastic process is stationary, p(u 1, u 2 ; t 1, t 2 ) = p(u 1, u 2 ; t 1 t 2 ) applies. The autocorrelation reads u(x, t 1 ) u(x, t 2 ) = ρ(x, t 1 t 2 ). Analogously, the spatial autocorrelation is defined as u(x 1, t)u(x 2, t). In a homogeneous flow this function depends only on the separation: ρ s (x 1 x 2, t) = u(x 1, t)u(x 2, t). where ρ s is the (spatial) velocity autocorrelation function. Reynolds decomposition and averaging processes Reynolds decomposition We can split the motion of a turbulent flow into a large scale and a small scale component (here denoted with a prime). In this way, the average of a variable and its fluctuation are discerned. Let u be the velocity. It can be split as follows u = u + u (1.20) Averaging, denoted by the overbar, means ensemble-averaging in this context. We assume that the averaging process is a Reynolds operator which means that 7

8 u = u (and thus u = 0), and that cu = cu (c is a constant) Averaging Features of ensemble-averaging are: Ensemble-averaging removes all turbulence from the description of the flow: the variables are no longer chaotic. Any information about the structure of the turbulence vanishes. Transport by eddies has to be parameterised, i.e. only the statistical (average) properties of the turbulence are featured in the resulting equations. Such a parameterisation requires a good knowledge of the flow. Averages vary much less in time and space than the turbulence itself. Grid point distances can therefore be chosen relatively large. Time-averaging The average value of a velocity variable over a period T is defined as U T (x, t) = 1 T t+ 1 2 T t 1 2 T u(x, t )dt, (1.21) If T incorporates all turbulence time scales, the time-average would be a good approximation of the ensemble-average (U will then be independent of T ). Volume-averaging Analogous to time-averaging, a variable can also be averaged over a certain volume. The local values are split into a spatial average and a deviation from this average a = {a} + a (1.22) where the volume-average is defined as {a} 1 V V a dv. (1.23) In more general terms, the volume-average can also be expressed as {a}(x) + G(x x)a(x )dx (1.24) 8

9 where the filter function G is given by, for example, G(x x ) = 1 if x x < L = 0 otherwise. Important properties of (1.23), that differ from the properties of ensembleaverages, are {{a}{b}} {a}{b} {{a}} {a} (1.25) In practical applications these inequalities are often ignored, since errors made by assuming that these quantities are equal appear to be small. Characteristics of volume-averaging are: Volume-averaging only removes the scales smaller than V 1 3 ; it conserves the larger scales. A so-called closure is needed for the scales smaller than V 1 3. The advantage of volume-averaging, compared to ensemble-averaging, is that the closure hypothesis is less critical, since the small scales do not carry much energy and have relatively well-known properties. 9

10 Chapter 2 CONSERVATION LAWS We summarize the basic equations as follows: The continuity equation dρ dt + ρ u i x i = 0. (2.1) The momentum equation ρ u i t + ρu u i j = p ρgδ i3 + µ 2 u i. (2.2) x j x i x j x j The temperature equation θ t + u j θ dθ x j dt = κ 2 θ. (2.3) x j x j 2.1 Governing equations The continuity equation As a good approximation, the continuity equation can be expressed as u i x i = 0, also implying that 10

11 1 dρ ρ dt 0. The density along the fluid-particle trajectories is approximately constant. The flow can therefore be considered incompressible. In an adiabatic process, dρ dt = 1 dp c 2 dt applies, where c (the velocity of sound) stands for ( cp ρ ) 1 2. When combined with the Bernoulli equations, this yields ( u i = 1 d 1 2 u ) i 2 x i c 2 + gu 3. dt c v p Choosing U and L as scale sizes of u i and x i respectively, then L U u i x i = O( u2 c 2 ) + O( g L c 2 ). Using the Boussinesq approximations, the right-hand side can be neglected. Boussinesq approximations The above approximations, u/c << 1 and L << c 2 /g, which imply that atmospheric flow can be considered incompressible, form part of the so-called Boussinesq approximations. This however does not mean that density differences are dynamically unimportant. They certainly are in combination with gravity. The resulting equations are referred to as the Boussinesq equations. They originate from a number of subtle arguments which will be given below. We will suppose a reference state of the atmosphere (denoted by the subscript 0). The atmosphere is at rest (u i = 0), and horizontally homogeneous regarding temperature, density and pressure. We further assume that thermodynamical deviations from the reference state in a realistic atmosphere are small. We further suppose that the actual atmospheric state does not deviate much from the reference state, viz. p p 0 + p ρ ρ 0 + ρ θ θ 0 + θ u i 0 + ũ i, where p, ρ, θ and ũ i denote small deviations from the reference state. When we substitute this in the equation of state, p = ρrθ, (we assume that in the ABL 11

12 θ T, see Eq. (1.9)). We obtain in zero order: p 0 = ρ 0 Rθ 0, (2.4) and to first order p = R(ρ 0 θ + ρθ0 ). We assume, confirmed by observations and given the approximations already made, that so that p << θ, ρ, ρ θ. (2.5) ρ 0 θ 0 Now we make the same substitutions in (2.2): (ρ 0 + ρ) dũ i dt = p 0 p (ρ 0 + ρ)gδ i3 + µ 2 ũ i. x i x i In zero order this yields p 0 x i = ρ 0 g δ i3, (2.6) the hydrostatic equation, and to first order: dũ i ρ 0 dt = p ρg δ i3 + µ 2 ũ i, x i x 2 j which can be rewritten, using (2.5), as dũ i dt = 1 p + θ g δ i3 + ν 2 ũ i, ρ 0 x i θ 0 x 2 j where ν = µ/ρ 0, the kinematic viscosity. variable θ yields dũ i dt = 1 ρ 0 p x i + ( θ θ 0 θ 0 x 2 j Back substitution of the original )gδ i3 + ν 2 ũ i x 2. (2.7) j This is a familiar expression of the Boussineq equations for a shallow boundary layer. Essential is that density (temperature) deviations are only important in combination with gravity. Otherwise density can be considered constant. So the Boussinesq-approximations for the (shallow) ABL include: 12

13 u c << 1 L << c2 g θ T p p << 1 and we summarize the equations in Boussinesq form: The continuity equation ũ i x i = 0. (2.8) The momentum equation ũ i t + u ũ i j = 1 p + (θ θ 0) gδ i3 2ɛ ijk Ω j u k + ν 2 ũ i. (2.9) x j ρ 0 x i θ 0 x j x j For the sake of completeness, the Coriolis term has been added, making the equations apply for a rotating coordinate system, with an angular velocity Ω (in rad/s). The temperature equation The temperature equation, neglecting molecular conduction, is changed to d θ dt = 0. (2.10) Together, eqs.(2.8, 2.9 and 2.10) constitute a set of 5 equations (with unknown variables ũ i, θ and p) that serves as a starting point for the study of the atmospheric boundary layer. 13

14 Chapter 3 THE REYNOLDS EQUATIONS We will suppose that a flow consists of a laminar, average flow, and, superimposed, turbulent fluctuations, the so-called Reynolds decomposition (see section 1.2). 3.1 The average flow We will adopt the Navier-Stokes equation, Eq. 2.9, ũ i t + ũ ũ i j = 1 p + (θ θ 0) gδ i3 2ɛ ijk Ω j u k + ν 2 ũ i. (3.1) x j ρ 0 x i θ 0 x j x j and the continuity equation ũ i x i = 0. (3.2) We decompose the velocity ũ i, the pressure p and the temperature θ in a mean and fluctuating (turbulent) component. After substitution of ũ i = U i + u i, p = Π + π and θ = Θ + θ (with this substitution we redefine from hereon θ as a temperature fluctuation) and averaging, we obtain U i t +U U i j = 1 Π + (Θ θ 0) gδ i3 2ɛ ijk Ω j U k +ν 2 U i u iu j, (3.3) x j ρ 0 x i x j x j x j and θ 0 U i x i = 0. (3.4) 14

15 Figure 3.1: Frictional flow along a flat surface The extra terms, u i u j, which represent turbulent momentum fluxes, are called Reynolds stresses and are expressed in known variables. This procedure is called closure (or K-theory ) and will be extensively treated in section 3.5: u i u j = K m ( U i x j + U j x i ) (3.5) The proportionality constant K m takes the dimension of viscosity (m 2 s 1 ), but is entirely determined by the turbulence itself. Let l and u be the length and velocity scales of the turbulence, then it seems reasonable to suppose that K m lu. This can be nicely illustrated by examining a flow along a flat surface. Neutral boundary-layer flow The average flow is stationary and homogeneous in the x-direction ( U The equations of motion (meglecting Coriolis forces) are (see 3.3) ( U ν z z uw) = 1 Π ρ 0 x t, U x = 0). (3.6) 1 Π ρ 0 z w2 + g = 0. (3.7) z By differentiating (3.7) to x, we find that 1 Π ρ 0 x cannot be a function of z. Now, integrating the first equation yields: ( ν U z uw) z. (3.8) 15

16 Suppose that h is the thickness of the boundary layer (the height at which U/ z 0)). Consider two cases: a. Laminar flow close to the wall (uw = 0). Eq. (3.8) gives ν U z = az + b, (3.9) where the constants a and b follow from the boundary conditions at z = 0 and z = h. At z = h we assume U/ z = 0. b represents the friction force at the surface (with dimension velocity squared). This defines the friction velocity u as lim ν U z 0 z u2, (3.10) so that we finally get ν U z u2 (1 z ), (3.11) h implying that the velocity profile is linear for z << h. In this (thin) layer molecular friction dominates and thus Re = O(1). If the thickness of that layer is δ we have thus u δ ν 1, which yields δ ν/u for the thickness of the laminar sub-layer. b. Turbulent flow: the logarithmic wind profile Above the laminar sub-layer (ν/u z h) turbulent friction dominates: uw >> ν U z. Eq. 3.8 now reads uw z which, applying the boundary conditions uw = 0 at z = h and uw = u 2 at z = ν/u, yields uw u 2 (1 z ). (3.12) h (note that ν/u << h). If z/h << 1, the vertical transport of momentum ( uw u 2 ) is approximately constant. This defines another sub-layer, the constant flux layer, ν/u z << z/h. 16

17 We can now use the closure relation (3.5) to express the Reynolds stress in terms of the average gradient in this layer: uw = K m U z = u2. (3.13) According to (3.13), the velocity profile would be linear (using a constant value for K m ). This has not been confirmed experimentally: instead, logarithmic profiles are found. This only follows from (3.13) if the eddy diffusion coefficient K is proportional to z. If we define K m = κzu, where κ is the Von Karman constant), the solution of (3.13) will be U u = 1 κ ln z z 0, (3.14) This is a logarithmic profile 1, where z 0 is an integration constant. The proportionality constant κ has a value of 0.4. The quantity z 0 depends on the roughness of the wall, and is called the roughness length. For an aerodynamically smooth wall, z 0 is defined as the height at which the corresponding Reynolds number has the value of 1: (Re) z0 u z 0 ν, (3.15) leading to z 0 = ν/u. If an aerodynamically rough wall is characterised by irregularities of height h, the Reynolds number is given by (Re) h u h ν. (3.16) A surface is called smooth when Re h < 1 and rough when Re h > 1. A couple of typical values for z 0 are listed in table 3.1. The roughness of a rough water surface is treated later in these notes. Under neutral conditions and over a large area of varying surface types, the wind profile is approximately logarithmic from a height z 0 to a couple of hundreds of metres. If the area is covered with higher objects (trees), it is possible to define a new surface where the wind speed is 0. The wind profile is then given by U = 1 u κ ln z d, (3.17) z 0 1 There is as yet no exact derivation for the logarithmic wind profile. In section 4.3, it is made plausible that logarithmic profiles occur in boundary layers 17

18 Table 3.1: Typical values for the roughness length z 0 Surface type Roughness length (m) Smooth water/ice 10 4 Short grass 10 2 Low vegetation 0.05 Countryside 0.20 Low built-up area 0.6 Forests/cities 1 5 where d is the so-called displacement height. The displacement height equals roughly 80% of the object height (see figure 3.2). Equation (3.17) is very suitable to calculate the wind speed at a given height, if the wind speed is known on any other height: U 2 U 1 = ln(z 2 d)/z 0 ln(z 1 d)/z 0. Drag coefficient If we square (3.14) and subsequently multiply it by the density, we will find the surface shear stress: τ ρ u 2 = ρ κ 2 ln 2 (z/z 0 ) U 2. The drag coefficient is thus defined as C d = implying that κ 2 ln 2 (z/z 0 ), (3.18) τ = ρ C d U 2, (3.19) This is a simple relationship to estimate the surface shear stress from the average wind speed under neutral circumstances. Power law In practice, an algebraic formula for the wind profile is often applied: 18

19 Figure 3.2: Sketch of the wind speed profile over a homogeneous forest. U 2 U 1 = ( z2 z 1 ) p. (3.20) In neutral conditions, the following rule approximately holds: ( ) p ln 1 z1 z 2. (3.21) z 0 This relation can be derived by the requirement that at the geometric mean value of z 1 and z 2, (z 1 z 2 ), the wind velocity and the first derivative are equal in both formulations. A frequently used, but not necessarily correct value of p is 1/7 (based on z 1 z 2 10m and a roughness length of 8 cm). 3.2 The mean-flow energy equation The starting point is (3.3). We shall neglect the Coriolis force since it does not play a role in energy budget considerations: U i t + U U i j = 1 Π + (Θ θ 0) gδ i3 + ν 2 U i u iu j, x j ρ 0 x i x j x j x j Multiplying by U i yields: θ 0 19

20 E t + U E j x j = 1 Π U i + U i(θ θ 0 ) gδ i3 + U i {νu i U i u i u j } ρ 0 x i θ 0 x j x j + u i u j U i x j ν( U i x j ) 2, (3.22) where E 1 2 U i 2. If the Reynolds number is high, the viscous terms can be omitted. When integrated over the volume of the flow, the result is de tot dt = V U i u i u j dv + x j V U i (Θ θ 0 ) θ 0 gδ i3 dv. (3.23) The integrand of the first integral is negative in a shear flow, and is called the deformation work. This term provides the translation of the average flow energy to its fluctuations (i.e. the coupling between average flow and turbulence). The second integral represents the conversion of potential into kinetic energy by the mean motion in a stratified flow. 3.3 The turbulent kinetic energy equation The energy transfer from the average flow to turbulence is provided by the deformation work. We will now derive an expression for the average kinetic energy of the fluctuations, q 1 2 u i 2. We shall start from (2.9): ũ i t + ũ ũ i j = 1 p + (θ θ 0) gδ i3 + ν 2 ũ i, x j ρ 0 x i θ 0 x j x j and for the temperature θ t + ũ θ j = κ 2 θ, x j x j x j As in the derivation of the equation for the mean energy (3.3), we make again the substitutions ũ i = U i + u i, etc. In order to arrive at an equation for the fluctuations, we substract from this equation the equation for the mean flow (3.3). We multiply the resulting equation with u i. Taking the average of this equation, we finally obtain the equation for the energy of the fluctuation q 1 2 u i 2 : q t +U q U i j = u i u j + g wθ { x j x j θ 0 x 1 2u i u i u j + 1 πu j ν 1 2 u2 i } ɛ. (3.24) j ρ 0 x j 20

21 The first term on the right-hand side of (3.24) is the mechanical production term, supplied by the average flow (normally positive). The second term represents the thermal production of turbulent fluctuations due to differences in density of the flow. In a gravity field, this term can transform potential energy into kinetic and vice versa. The ratio between mechanical and thermal production of turbulence is defined as the flux-richardson number: Ri f = We discern three cases: g θ 0 wθ u i u j U i x j (3.25) wθ > 0 unstable flow Ri f < 0 wθ = 0 neutral flow Ri f = 0 wθ < 0 stable flow Ri f > 0 The use of the flux-richardson number can be illustrated by the atmospheric boundary layer that is heated by the Earth surface during the day, so that wθ > 0. At daytime, the boundary layer is thus generally unstable. During the night, the Earth surface will cool down by radiating out its energy. Thus, wθ < 0, meaning that the boundary layer will be stable with little turbulence. We also distinguish the gradient Richardson number, Ri, by applying K-theory to the fluxes in Eq Similar to Eq. 3.5 we may write for wθ: wθ = K h Θ z, (3.26) so that Eq becomes Ri f = K h K m Ri, (3.27) where Ri is given by Ri = g θ 0 Θ/ z ( U i / x j + U j / x i ) Ui x j. (3.28) Apart from the flux-richardson number, we also use the bulk-richardson number, which follows from (3.28), by approximating gradients by finite differences and assuming a uniform windfield, U(z), Ri b = g θ 0 Θ z. (3.29) ( U) 2 21

22 Returning to Eq. 3.24, we observe that the redistribution term between braces contains 3 energy fluxes: by turbulent fluctuations, by pressure fluctuations, and by molecular fluctuations (the latter has already been omitted in the equation). The last term represents the energy dissipation. It is the only negative term, and therefore necessarily of the same order of magnitude as the other terms. If not, then q could grow unlimitedly. In a neutral, semi-stationary situation, (3.24) implies P u i u j U i x j ν( u i x j ) 2 ɛ, so P ɛ. The production and dissipation of turbulent energy are approximately balanced. Also note that P = O( u3 l ), which means that ɛ = O( u3 ). (3.30) l 3.4 Spectra Turbulence can be viewed upon as a set of eddies of different sizes between l K and L. In a turbulent flow, energy of a particular scale is transferred towards larger and smaller scales. The average flow provides the energy for the turbulence at the macroscale. The larger eddies are unstable and disintegrate into smaller eddies of various sizes. This process is repeated several times. The smallest eddies will ultimately lose its energy by molecular viscosity. The net effect is that eddies tranfer energy from larger to smaller scales, known as the energy cascade. A turbulent velocity field consists of a superposition of fluctuations with a wide range of temporal and spatial scales. To analyse the contribution of each scale, a Fourier analysis can be exploited. Let the autocorrelation of a continuous velocity function u(t) be defined as 1 + T 2 ρ(τ) = u(t) u(t + τ) = lim T T T 2 u(t) u(t + τ) dt. (3.31) We have assumed stationary conditions so that ρ depends on the time difference τ only. The energy spectrum S(ω) is, by definition, the Fourier transform of the correlation function: S(ω) 1 2π + with the associated inverse operation ρ(τ) + e iωτ ρ(τ) dτ, (3.32) e iωτ S(ω) dω, (3.33) 22

23 From (3.33), it immediately follows that for τ = 0 u 2 + S(ω) dω. The lefthand side represents the turbulent kinetic energy and this equation shows why S is actually called the energy spectrum, since S(ω) equals the contribution to u 2 of S between the frequencies ω and ω + dω. 23

24 In practice the energy or power spectrum of the function u(t) is determined in a more direct way: we rewrite Eq as Z 1 + ρ(τ) = lim v(t) v(t + τ) dt, (3.34) T T where v(t) is defined as v(t) = u(t) for t < T/2 v(t) = 0 otherwise. The Fourier tranform of v(t) is ṽ(ω) = 1 Z + e iωt v(t) dt, (3.35) 2π with the inverse transformation Z + v(t) = e iωt ṽ(ω) dω. We use the last expression to rewrite Eq as Z 1 + Z + ρ(τ) = lim v(t) ṽ(ω) e iω(t+τ) dω dt. T T Rearranging terms we get Z 1 + Z + ρ(τ) = lim ṽ(ω) e iωτ v(t)e iωt dt dω, T T or Z 2π + ρ(τ) = lim ṽ(ω) ṽ( ω) e iωτ dω. T T Since ṽ( ω) = ṽ (ω) we have Z 2π + ρ(τ) = lim ṽ(ω) 2 e iωτ dω. T T Comparing this result with Eq yields S(ω) = lim T which can be rewritten as 2π S(ω) = lim T T 2π T ṽ(ω) 2, Z + T 2 T 2 2 u(t) e iωt dt. (3.36) This equation, known as the Wiener-Khintchin theorem, provides a direct relation between the velocity signal u(t) and its power spectrum. The Fast Fourier Transform (FFT) is an efficient numerical way to determine the spectrum of an arbitrary stationary process based on a discrete representation of Eq Some of the properties of S are S(ω) = S( ω) 24

25 S(ω) is real and 0. It also follows from the definition that S(0) = 1 π ρ(τ) dτ = 1 0 π (u2 T u ), where T u denotes the time scale of the signal u(t). Spectral gap During the 1950s, the ideas about boundary-layer meteorology were dominated by the assumption that (small-scale) turbulent kinetic energy mainly occurred at scales comparable to the boundary layer height (1-2 km). At larger scales, hardly any energy would be available, whereas the energy would increase at mesoscale and sub-synoptic scales. This implied that the turbulence energy spectrum would contain a minimum, separating boundary-layer turbulence from turbulence at larger scales. This would also justify the process of Reynolds decomposition. Now that much more experimental data are available, this view has become challenged. An example of a spectral analysis of the energy, recorded during an airplane flight, is given in figure 3.4. In this figure, the spectral energy of the u and v velocity components continue to increase at larger scales. Other turbulent variables, like temperature, water vapour content, and liquid water concentration show comparable behaviour. Only the spectrum of the vertical velocity w seems to level off towards larger scales. That the spectral energy at larger scales does not vanish, has some inconvenient implications for the integral properties of the spectrum. For example, the total variance of u, being defined as u 2 0 E u (k)dk, (3.37) cannot be determined if E u (k) continues to grow for k 0. In these cases, the ogive is defined as Og u (k 0 ) k 0 E u (k)dk, (3.38) so that integral properties of the spectrum can be determined anyway. choice of k 0 is an arbitrary one. The 3.5 Closure Expressing the stress in terms of the average flow is called closure. More generally, closure means that higher moments are expressed in terms of lower ones. It is called closure because the set of equations of motion can be solved after closure. In the previous sections, some simple examples of 1 st -order closure have already been demonstrated. More elaborate ways of closure exist, which will be shortly discussed below. 25

26 Figure 3.3: Kaimal s schematic representation of the energy spectrum. (A) production subrange; (B) the inertial subrange, where production and dissipation are not important; (C) dissipation subrange. The second Kolmogorov hypothesis conjectures that in the inertial subrange, the dissipation ɛ and the wavenumber k are the dominating parameters. The spectrum E(k) can be determined using dimensional analysis based on ɛ and k only. The result is E(k) = C K ɛ 2 3 k 5 3. First-order closure (K-theory) First-order closure is often applied since it is a fast method. It implies that the K-theory is used to determine the Reynolds-stress term (see, e.g., Eq. 3.5 ). The diffusion coefficient K is typical for the flow and may thus change in time and space. At the surface layer, we acceptably showed that K m = κzu (see chapter 3.3, Eq. 3.13). Above the surface layer, there are many possible ways to follow. At the top of the boundary layer, the vertical exchange is small, leaving K m approximatly zero. Somewhere within the boundary layer, K m should have a maximum. A possible formulation for K m is: [ ] 1 ( K m = l 2 U ) 2 ( V ) (3.39) z z In this equation, l is the mixing length, that will be proportional to the height in the surface layer, l = κz, and converge to a constant value, say λ, for greater heights. This choice for K m will give the desired behaviour in the surface layer, where V 0 and uw are constant (u 2 ). It now follows from uw = K m U z (3.40) 26

27 Figure 3.4: 1-D spectra of the horizontal (u and v) and vertical (w) wind-speed components as a function of the wave number k. Measurements from an airplane at approx. 150 m above the sea. 27

28 after substitution in (3.39), that U z = u κz, (see Eq. 3.13) which provides the desired logarithmic wind profile. The heightdependent growth of the length scale can be delimited by choosing the following parameterisation: l = κz 1 + κz λ (3.41) where λ is an empirical heigth scale (typically 100 m). The eddie diffusion parameter K m (3.39) can easily be corrected for the stability by adding an empirical function F : [ ] 1 ( K = l 2 U ) 2 ( V ) F (Ri), (3.42) z z F being a function of the stability through the Richardson number. This parameterisation from 1979 is still used in present-day global circulation models like the ECMWF-model, that calculate weather forecasts. 1.5-order closure This type of closure makes use of the energy equation (3.24), which is a secondmoment equation. Suppose that K m obeys K m lq 1 2, (3.43) where q is given by the turbulent kinetic-energy equation (3.24). The terms present in that equation can be approximated in the following way, if the flow is assumed to be horizontally homogeneous: U i ( U u i u j = K m x j z g wθ = g ( Θ) Kh θ 0 θ 0 z ( 1 x 2u i u i u j + πu j ) = j ρ 0 z K q m z ɛ = q 3 2 l 1. Substituting the above equations into the stationary-energy equation yields: ) 2 28

29 ( U ) 2 g Θ 0 = K m + K h z θ 0 z + z K q m z q 3 2 l 1. (3.44) An equation for the turbulent kinetic energy q emerges. The set of equations (3.3, 3.5, 3.41, 3.43 and 3.44) can now be solved. The name 1.5-order closure is used, since only one second-order equation is used for the closure algorithm. Second-order closure In this case, all second moments, like u i u j, u i θ, are incorporated in the closure algorithm. These equations contain terms of the third order, which are either parameterised or neglected. As a result, the number of equations that are to be solved increases rapidly. Sometimes, this method yields better results. On the other hand, the mathematical complexity, the large number of empirical constants and the lack of a solid theory make second-order closure techniques less favourable. Large Eddy Simulation (LES) LES makes use of a fundamentally different approach. The variables in the Navier-Stokes equation are averaged over a small volume. Contrary to the variables in the Reynolds-averaged equations, the LES-variables describe the turbulent behaviour of the flow at scales larger than the scales of the averaging process. The starting point is the momentum equation (see 3.1), where we take for simplicity a neutral flow and neglect the Coriolis-term and molecular viscosity. The volume-average is: {ũ i } t + {ũ j } {ũ i} x j = 1 ρ 0 { p} x i + R ij x j, (3.45) where R ij is a tensor that describes the impact of the flow behaviour within the volume on that outside the volume. This is called the subgrid scale stress, defined as R ij = {ũ i }{ũ j } {ũ i ũ j }. Comparable to Reynolds-averaging, this term is approximated by relating it to volume-averaged variables: ( {ũi } R ij = ν t + {ũ ) j} x j x i (3.46) In a mathematical sense, LES is almost identical to the method of Reynoldsaveraging. The difference between the two processes can be found in the way the eddy viscosity is formulated. Reynolds-averaging appoints an order of magnitude u.l to K, so that the Reynolds number, based on K, is of the order of 1. The choice for the eddy viscosity in LES, ν t, depends on the volume over which the averaging process takes place. This volume is usually of the same order of 29

30 magnitude as the grid point distance of the numerical scheme,. The Reynolds number, ul/ν t l/ is now much larger than 1 (approximatly 10 2 to 10 3 using current computer capacity). A well-known choice for the eddy viscosity is formulated by Smagorinsky (1963), which is based on a mixing hypothesis (like the K-theory): ν t = (C s ) 2 S (3.47) where C s is the Smagorinsky constant (C s = 0.2), S = (2S ij S ij ) 1 2 and S ij = 1 2 ( {ũi} x j + {ũj} x i ), the deformation tensor. The most important applications of LES can be found in the simulation of convective turbulence, where transports are dominated by large-scale motions in the atmosphere. Direct Numerical Simulation (DNS) This technique aims to find solutions for the Navier-Stokes equation itself, without introducing closure or averaging. At large Reynolds numbers, this approach faces difficulties, since the range of turbulence scales increases rapidly. In order to find numerical solutions, the grid points distances have to be approximately equal to the smallest scale. For calculating practical situations, a large number of grid points are required. Flows with Reynolds numbers up to can be simulated with currently available computer power. 30

31 Chapter 4 THE ATMOSPHERIC BOUNDARY LAYER 4.1 Phenomenology Boundary-layer meteorology comprises the dynamics and physics of the atmospheric layer that is closest to the Earth surface. Exchange processes between the Earth surface and the free atmosphere occur in the boundary layer. The state of the boundary layer is influenced by flow in the free atmosphere on the one hand, and by boundary conditions imposed by the Earth surface on the other hand. Dominating boundary layer processes are the vertical exchange of momentum τ = ρ 0 uw, heat H = ρ 0 c p wθ and water vapour E = ρ 0 wq. The free atmosphere (see figure 4.3) is the layer above the uppermost boundary of the (turbulent) boundary layer. The behaviour of the boundary layer has an enormous impact on values of the maximum and minimum temperature, wind speed gradient (wind shear), fog and cloudiness. In aviation and architecture (wind engineering), both wind shear, fog and the occurrence of wind gusts are important factors to keep in mind. Turbulence, radiation, surface properties and thermodynamics are the key ingredients of boundary-layer meteorology. We will first give a qualitative overview. Slightly above the Earth surface, a thin laminar layer exists: the viscous sublayer. This layer adjusts the balance between the Earth surface and the surface layer. The surface layer (or inner layer) is a couple of tens of metres thick. The surface layer is topped by the (planetary) boundary layer, sometimes also referred to as the Ekman layer or outer layer. The transition between the surface layer and the Ekman layer is smooth. The scales in the surface layer are very different from those in the Ekman layer. The boundary-layer dynamics are not only influenced by the average horizontal 31

32 Figure 4.1: Schematic overview of the troposphere. 32

33 flow, but also by turbulent processes. The most important of these processes are: a. Mechanical turbulence b. Buoyant or convective turbulence. The stability of the atmosphere influences the boundary layer, which has a typical height of m. The flux-richardson number is a measure for the atmospheric stability. It is the ratio between buoyancy production and mechanical production Ri f = g θ 0 wθ u i u j U i x j (see section 3.3). The surface layer is heated when the surface heat flux wθ 0 is positive. This happens during daytime as a consequence of solar radiation. Convection may also take place above a warm sea surface, over which cooler air is flowing. If the heat flux is negative, the boundary layer will cool down. This happens during the night. These two situations differ substantially, having quite some implications for the dynamics (e.g. the growth of the boundary layer height: if wθ > 0, then Ri f < 0, so the atmosphere is unstable. This implies that gravity is generating turbulence, and the boundary layer height increases as a consequence. On the other hand, if wθ < 0, so Ri f > 0, the boundary layer is stable. Gravity will suppress turbulence. The boundary layer height will not change notably. We conclude that there is a daily cycle of boundary layer height over land surfaces (see figure 4.2). Unstable boundary layer Incoming shortwave radiationwill heat the Earth surface which will heat and lift the surface-layer air: convection (thermals) will start to emerge. In this manner, the boundary layer becomes unstable. Normally, there is a stable layer on top of the boundary layer, which limits the growth of convection (see figure 4.3). When convective cells penetrate into a stable layer, they will lose their kinetic energy and won t ascend any further. These convective cells mix the air of the boundary layer with the air above. In this way, the thickness of the boundary layer increases. This process is called entrainment. The strong turbulence makes the mixing effective, resulting in a rather uniform distribution of momentum, heat and water vapour. A (moist) convective cell that reaches its LCL (Lifting Condensation Level) will condensate and form a cumulus cloud. As soon as the cell reaches its LFC (Level of Free Convection) the clouds can grow upward until the next stable atmospheric layer. The limit of this entrainment process is at the level of the tropopause, an extremely stable layer at a height of about kms. 33

34 Figure 4.2: Typical diurnal progress of the boundary layer over a land surface. (From: Stull, An Introduction to Boundary-Layer Meteorology, 1988). Figure 4.3: Characteristic profiles of average potential temperature, wind speed, vapour and a random trace gas in an unstable boundary layer (From: Stull, An Introduction to Boundary-Layer Meteorology, 1988). 34

35 Figure 4.4: Typical profiles of potential temperature and wind speed in a stable boundary layer (From: Stull, An Introduction to Boundary-Layer Meteorology, 1988). Stable boundary layer A stable boundary layer emerges when the net longwave radiation of the Earth surface is negative, which cools down the air above it. Vertical motions are often hampered in a stable boundary layer, since the (negative) buoyancy slows them down. The turbulent kinetic energy and the length scales are much smaller than in an unstable boundary layer. The turbulence is not well-structured. The height of a stable boundary layer is determined by the balance between turbulence production and dissipation. Since turbulence is suppressed, little mixing takes place in the boundary layer. Gradients of heat, momentum, and water vapour are therefore much larger (see figure 4.4). 4.2 The surface layer; Monin-Obukhov theory The Monin-Obukhov length In a non-neutral boundary layer, the surface heat flux (wθ) 0 is not zero. Together with the variables z and u, the surface heat flux, or rather the buoyancy g θ 0 (wθ) 0 plays a significant role. Since u and g θ 0 (wθ) 0 can be more or less constant over an hour or longer, they are considered to have a big influence on the dynamical structure of the surface layer. Based on these variables, we can derive a stability parameter, the Monin-Obukhov (MO) length scale L: u 3 L =. κ(g/θ 0 )(wθ) 0 The turbulent kinetic energy equation (3.24) can be simplified to 35

36 0 = uw U z + g θ 0 wθ ɛ, and it can be made dimensionless through multiplication by κ z/u 3 : 0 = uw κ z U u 2 u z z L φ ɛ. We used the definition of L to arrive at the latter equation, and denoted the dimensionless dissipation of energy by φ ɛ. Since in the surface layer uw u 2, we find: 0 = φ m z L φ ɛ, in which we wrote the dimensionless wind speed profile as φ m = κ z u U z. (4.1) In this simplified form, the energy equation can be interpreted as a balance between mechanical and buoyancy production on the one hand, and the dissipation of energy on the other. The equation is expressed in only a few scales (z, L, and u ). From dimensional analysis, we can also define a dimensionless temperature gradient φ h : φ h (z/l) = κz θ dθ dz. (4.2) The temperature scale θ is defined as θ wθ 0 /u. The functions φ m and φ h have been determined experimentally. In case of instability (z/l < 0), the following relations have been found: φ m = (1 16z/L) 1/4 φ h = (1 16z/L) 1/2 and for the stable case (z/l > 0): φ m = φ h = z L. An analogous function for the water vapour profile is often set equal to the expression for φ h. 36

37 Figure 4.5: A summary of dimensionless gradients of wind speed (a) and temperature (b) as they have been experimentally determined for the surface layer (From: Yaglom, blm 1977). Upon integration of (4.1) and (4.2), we obtain U(z) = u κ {ln(z/z 0) Ψ m (z/l)} Θ(z) Θ 0 = θ κ {ln(z/z 0) Ψ h (z/l)} If z/l < 0, the expressions for Ψ m and Ψ h become ( ) ( ) 1 + x 1 + x 2 Ψ m = 2 ln + ln 2 tan 1 x + π/2 2 2 ( ) 1 + y Ψ h = 2 ln, 2 and otherwise (if z/l > 0) Ψ m = Ψ h = 5z/L, where x = (1 15z/L) 1/4 and y = (1 9z/L) 1/2 (see figure 4.5). If, for whatever practical reason, a power law for the wind speed profile is preferred, the exponent p is derived from the previous equations using (3.21): p = φ m (z 12 /L) ln z 12 /z 0 Ψ(z 12 /L), where z 12 = z 1 z 2. Turbulence in the surface layer 37

38 The following (empirical) relations hold in a stable, flat surface layer: σ u /u = 2.39 ± 0.03 σ v /u = 1.92 ± 0.03 σ w /u = 1.25 ± 0.03 In unstable conditions, this relation becomes ( σ w = z ) 1/3. u L In the unstable situation, the horizontal-velocity fluctuations are too strongly influenced by variations on a larger scale. The Monin-Obukhov theory starts to lose its validity. For the temperature variance we have σ θ / θ = 0.95( z L ) 1 3. Expressions for the unstable situation, that also apply to the surface layer, will be given in section 4.4. (see also Garratt section 3.5) Spectra There exists a plethora of spectra and spectral data in the surface layer, for example in?. For our purposes, it suffices to know that fs u (f) u 2 fs v (f) u 2 fs w (f) u 2 = = = 102 n (1 + 33n) 5/3 17 n ( n) 5/3 2.1 n ( n) 5/3, (4.3) in which n is the dimensionless frequency (n = fz/u). 4.3 The neutral boundary layer The Ekman layer In this section, we will consider a neutral, homogeneous and stationary boundary layer. Unlike previously, the Coriolis force will also be taken into account. We will assume that the flow above the boundary layer is geostrophic, i.e. the 38

39 Figure 4.6: Force balance in the Ekman layer. P is the pressure gradient, Co the Coriolis force, and F r is friction. pressure gradient and the Coriolis force balance each other. From the momentum equation, Eq. 3.3, neglecting the buoyancy term, it follows that 0 = f(v V g ) uw z 0 = f(u U g ) vw z (4.4) (see figure 4.6). U g and V g are the components of the geostrophic wind, by definition being equal to the wind speed while neglecting the turbulent stress in (4.4). These components are expressed as U g = 1 Π ρ 0 f y 1 Π V g = ρ 0 f x. The Coriolis parameter, f, equals 2Ω sin φ, where φ represents the latitude, Ω the angular velocity (rad/s) of the Earth axis. In order to solve eqs. (4.4), K-theory is applied. Suppose that uw = K m U z vw = K m V z. After substitution into (4.4), we obtain 0 = f(v V g ) + K m 2 U z 2 0 = f(u U g ) + K m 2 V z 2, 39

40 Figure 4.7: The Ekman spiral, or the Leipzig wind profile (Mildner, 1932). with the solution U = U g (1 e γz cos γz) V = U g e γz sin γz. (4.5) (γ = f/2k m ). These equations are solved by multiplying one of them by i (= 1) and add it to the other. Using the substitution w = U U g + i(v V g), the result is 2 w z 2 = i f K m w, In a coordinate system where G is chosen along the x-axis (so that V g = 0; see figure 4.7), the boundary conditions for w are given by w = U g at z = 0, and w = 0 at z =. The solution which satisfies the boundary conditions is w = U g exp (i 1)γz, with γ equal to p f/2k m. Equating the real and imaginary parts yields the solution (4.5). The Ekman spiral is only sporadically observed, since the conditions of the theoretical derivation are seldomly met in practice. The assumption that K m is constant in the entire layer is incorrect, as are the assumptions of first-order closure. (For example, K m would rather be proportional to z close to the surface). Using the parameter γ defined in the box above, we can define the boundary layer height, for example as the height at which the wind speed is within 5 % of the geostrophic wind. In this way, we can choose a boundary layer height like h (K m /f) 1/2. If, on the other hand, the eddy diffusivity parameter is supposed to be the product of a length scale (h) and a velocity scale (u ), K m = hu, then the boundary layer height can be expressed as: 40

41 h = c 1 u /f. (4.6) If we substitute the values c 1 = 0.3, f 10 4 s 1, and u = 0.3m/s, then the neutral boundary layer height amounts to approximately 1000 m. The equations for the Ekman profile can now be rewritten by using (4.6): U U g = U g e c 1/2 z/h cos( c 1 /2 z/h) V = U g e c 1/2 z/h sin( c 1 /2 z/h). As a result, the wind profile can be written as the product of the geostrophic wind speed and dimensionless functions of z/h. This solution doe not hold for the surface layer, where a logarithmic wind speed profile is found, both in theory and in practice. Apparently, other parameters apply close to the surface. We will come back to this is the next section. Resistance laws In this paragraph, we will discuss the relation betwen the surface layer parameter u, and the external pressure gradient G. A stationary, homogeneous and neutral boundary layer is considered. Resistance laws for non-neutral boundary layers will be shortly touched later on. The starting point is again a stationary, homogeneous and neutral boundary layer. The negative x-axis is directed towards the shear stress. The equations of motion are 0 = f(v V g ) uw z 0 = f(u U g ) vw z, (4.7) The boundary conditions that apply are given in table 4.1. Note that the lower boundary of the layer is denoted by z = z 0, the roughness length. We are looking for a relation between the friction at the Earth surface (u ), and the geostrophic wind speed G U 2 g + V 2 g. Dimensional analysis shows that, apart from G/u, only one other dimensionless parameter can be found, namely Ro = u /fz 0, the so-called friction Rossby number, implying that U g u = k x (Ro) (4.8) V g u = k y (Ro) where k x,y, are unknown function of ρ 0. 41

42 Table 4.1: Boundary conditions z = z 0 z = h U = 0 U = U g V = 0 V = V g 2 uw = u uw = 0 vw = 0 vw = 0 These relations are called resistance laws. We can construct these by adapting the solution for z h to the solution for z = z 0. This procedure is called matching. Let us define two dimensionless height coordinates as η = z/h, and ξ = z/z 0 where h = u /f, the boundary layer height. We implicitly assume that z 0 << h = u /f, being equivalent to taking Ro. When z h, the dynamical equations (see 4.4) in this coordinate system are: 2 U U g = dvw/u g x (η) (4.9) u dzf/u V V g u = duw/u 2 dzf/u g y (η). (4.10) In case z 0 z << h and τ( uw 0, 0) (note that the shear stress at the surface is chosen along the negative x-axis): U u = f x (ξ) (4.11) V u 0. (4.12) If we subtract (4.9) from (4.11), we are left with U g u = f x (ξ) g x (η), (4.13) which can be proven to be equal to U g u = 1 κ (ln Ro A 0). (4.14) 42

43 Proof: using Eq. 4.8, Eq can be written as k x(ro) = f x(ro η) g x(η). (4.15) Note that ξ = Ro η. Now, f, g and k must be continuous and logarithmic functions. This can be demonstrated for f by first differentiating (4.15) to Ro and subsequently to η. We then obtain the differential equation 0 = f x + Ro η f x, that has a logarithmic solution. A similar procedure shows that also g is a logarithmic function. Substituting the logarithmic solutions into (4.9) yields: 1 Ug ln ξ = 1 ln η + a constant. κ u κ which can be rewritten to U g u = 1 κ (ln Ro A 0), (4.16) where A 0 is a constant. For V g we have V g u = k y (Ro) which for Ro, must converge to a constant. If we assign to this constant the value B 0 /κ, the equation can be written as V g = B 0 u κ Combining the above-mentioned equations and substituting G = gives: (4.17) U g 2 + V g 2 u G = κ. (4.18) (ln Ro A0 ) 2 + B0 2 This is the resistance law under neutral circumstances. A and B The relation between frictional velocity and the geostrophic wind forcing is only roughly approximated by resistance laws. If the layer is not neutral, the constants A 0 and B 0 in (4.18) are simply assumed to relate to h/l, implying the following empirical relation (see figure 4.8): u G = κ (ln Ro A(h/L))2 + B 2 (h/l), in which h stands for u /f, Ro = u /(z 0 f) and τ = uw 2 + vw 2. 43

44 The angle between the geostrophic wind and the shear stress at the surface is given by α = arctan(v g /U g ). Substitution leads to α = arctan B ln Ro A. Thus, the angle α is a measure for the wind turning in the boundary layer. The constants A and B have been determined experimentally (see figure 4.8). 4.4 The convective boundary layer The dynamics in the convective boundary layer are determined by wind shear and convection (wθ > 0). As soon as the convection becomes so strong that the contribution of the wind shear to the turbulence becomes negligible, the convection is usually called free convection. The velocity scale u is then of minor importance. Relevant parameters in this layer are the boundary layer height (h) and the surface heat flux. Using these, we can define a convective velocity and temperature scale as follows: w = ( ) 1/3 g (wθ) 0 h θ 0 T = (wθ) 0 /w. Besides these, z and h are the length scales. The time scale h/w amounts to approximately 1000 s, which is one order of magnitude smaller than 1/f. The influence of the Coriolis force on the dynamics in the convective boundary layer is thus only important in processes that play at a scale of hours or more. Concerning the wind profile in the convective layer, it can only be remarked that the wind profile gradient will be small due to the strong mixing that will occur. Usually, a uniform (constant) wind is assumed. Turbulence characteristics In the convective boundary layer, the following expressions are used. They are derived using dimensional analysis, the constants follow from observations: σ w /u = ( z/l) 1/3, and in the upper part of the boundary layer: σ w /u = ( h/l) 1/3. 44

45 Figure 4.8: The functions A and B are dependent on the stability parameter µ h/l, given that L 0. The curves numbered 1-5 are based on empirical data. The curves labeled 6 are the relations A(µ) = ln(1 + 3 µ 3.4 ) 2.55 µ and B(µ) = µ 1.7. The hatched areas represent the spread in the observations. (from:?). 45

46 Figure 4.9: Schematic representation of the flow, temperature and eddy structure in the convective boundary layer. The horizontal variances are expressed empirically as follows: σ u σ ( v = h ) 1/3 u u L These equations also apply to the surface layer. For the temperature variance we have σ θ ( z ) 1/3 = 1.34, θ h which applies in the lower half of the CBL. Dynamics In a convective boundary layer, the air is lifted locally (thermals). When these pockets of air reach the uppermost boundary layer, they penetrate into a more stable part of the atmosphere. As a consequence, they will start to descend again. The net vertical transport is approximately zero. This process also causes warmer air to be pulled into the turbulent layer. This is called entrainment. In this way, the boundary between strong and weak turbulence, the inversion layer, can move upward. The mechanical turbulence may also contribute to the decay of the stable layer above, and thus giving way to the growth of the mixing layer. Because the convection is controlled by the heat flux at the surface, which has a strong diurnal cycle, this process is highly unstationary. In the course of the morning, a convective layer will build up if sufficient solar radiation is available. 46

47 Figure 4.10: Structure of the unstable boundary layer (From: Holtslag and Nieuwstadt, 1986). Its vertical size will reach a maximum at the end of the afternoon. For this growth and decay, a simple mathematical models exists, that is based on three simplified dynamical equations: one for the boundary layer temperature (θ), the inversion strength ( ), and the boundary layer height (h). In figure 4.11a two subsequent temperature profiles, separated by a time difference t, in the CBL are drawn. The surface δh represents the amount of heat that is captured by the boundary layer by entrainment, so that the total amount of heat absorbed by the boundary layer in a period t equals hδθ = (wθ) 0 δt + δh, (4.19) where the surface heat flux, (wθ) 0, is supposed to be constant. In a differential form: h θ t = (wθ) 0 + h t. (4.20) From geometric considerations, the two subsequent profiles imply: t = γ h t θ t, (4.21) 47

48 where γ is the temperature gradient above the inversion ( θ z ). Concludingly, we will assume that the ratio between entrainment and surface heat flux, ε, remains constant, leaving us with h t (wθ) i = ε(wθ) 0. (4.22) These equations, containing the unknown variables h, and θ provide an expression for h: ( h h 0 ) 2 [ 1 2ε ( h ] ) 1/ε 1 h 0 = 2(1 2ε)(wθ) 0 γh 2 t. (4.23) 0 In this equation, h(t) represents the inversion height, h 0 its initial value, γ the initial temperature gradient (= dθ/dz), and ε is the ratio between the heat flux at the top and the bottom of the boundary layer (see figure 4.11), (wθ) i /(wθ) 0. In the derivation of (4.23), it is assumed that (t = 0) = 0. If the mechanical turbulence cannot be neglected as a fraction of the thermally produced turbulence, the entrainment process is parameterised using: u 3 (wθ) i = ε (wθ) 0 + ε m. hg/θ 0 The constants ε and ε m have the values 0.2 and 2.5, respectively. 4.5 The stable boundary layer In a stable boundary layer, the turbulence is suppressed by Archimede or buoyancy forces. Especially during nighttime above a land surface, the cooling of the Earth surface induces a negative heat flux (directed to the surface). The turbulent energy will decrease dramatically, and vertical motion in the boundary layer will be suppressed. This process will ultimately result in a cooled boundary layer and a stable potential-temperature profile (dθ/dz > 0). Nocturnal boundary-layer height The vertical exchange being small, the growth of the stable layer is much weaker than the growth of an unstable layer. Observations indicate that a stable layer rapidly builds up in the beginning of the evening, which remains approximately the same during the rest of the night. The thickness of the layer can be estimated using dimensional analysis: h = c 3 u 0 L/f. 48

49 Figure 4.11: Geometry of an expanding boundary layer. In the uppermost panel, the idealized potential temperature profile is drawn at two subsequent moments. In the rightmost picture, the heat flux as a function of height is depicted. The entrainment process is symbolically depicted by the hatched area. The middle and lower panes give a more realistic picture of the entrainment process (From:?). 49

50 Figure 4.12: Structure of the stable boundary layer (From: Holtslag and Nieuwstadt, 1986). The constant c 3 approximately equals 0.4. A more general estimate would be the solution of ( f h C n u ) 2 + h C s L + N h C i u = 1, where C n,s,i are 0.5, 10 and 20, respectively, and N frequency. The local length scale (Λ) g θ 0 Θ z, the Brunt-Vaisala In a stable boundary layer, the vertical exchange is small. As a consequence, the dynamics are governed by local parameters. We will therefore introduce the local Monin-Obukhov length: Λ τ 3/2 κ(g/θ 0 )wθ, where τ and wθ are now dependent on z. As a result of the high stability, the vertical exchange over large distances is very limited. Small eddies having a more isotropic structure dominate the turbulent processes. The y-component of the shear stress will be significant as well. 50

51 For that reason, the total shear stress τ (= uw 2 + vw 2 ) is preferred to the frictional velocity u. Vertical exchange within the stable boundary layer can be so small that the distance to the surface becomes a meaningless parameter. In theoretical treatments, the Richardson number is assigned its maximal constant value of Another occurence is called intermittant turbulence. In that case, flow can become laminar at times. Wavelike processes are sometimes observed as well. Wind maximum in the stable boundary layer It regularly happens that, especially during nighttime above a land surface, the wind speed relatively close to the surface can be m/s, while at the surface, it is very calm. This is when the so-called low-level jet occurs. It is a consequence of the transition from a convective turbulent boundary layer to a stable boundary layer at nightfall. The boundary layer rapidly becomes stable, reducing the vertical transport of momentum strongly. As starting equations, we will take the time-dependent boundary layer equations in a horizontally homogeneous situation: U t V t = fv = f(u G). We chose a coordinate system with the positive x-axis directed towards the geostrophic wind, implying G = (G, 0)). Multiply the latter equation by i, add both, and introduce the new variable w U + iv. This will result in: U G = V 0 sin ft + (U 0 G) cos ft (4.24) V = V 0 cos ft (U 0 G) sin ft. As starting condition, it is assumed that at t = 0 : (U, V ) = (U 0, V 0 ). Squaring and adding these equations will lead to (U G) 2 + V 2 = (U 0 G) 2 + V 2 0. This equation can be represented by a circle in a (U,V)-coordinate system, as depicted in figure The wind vector will move in a clockwise direction over the circle in a period 2π/f (approx. 15 hours at middle latitudes). It can thus become supergeostrophic ( U > G). The radius of the circle ( V (U 0 G) 2 ) increases when the difference between U 0 and the geostrophic wind is larger (close to the surface). The boundary condition U = 0 at z = 0, will cause a maximum for the wind speed at a certain height (see figure 4.14). 51

52 V (U0,V0) G U Figure 4.13: Graphical representation of the behaviour of the wind vector during a low-level jet. The initial situation is depicted (subscript 0). The wind vector will start to move clockwise over the circle, and will in the course of the process become larger than the geostrophic wind G. Figure 4.14: Simple representation of the low-level jet. 52

53 4.6 Summary An overview of the most important parameters in the atmospheric boundary layer is given in the following table. The wind profile It is not easy to formulate general expressions describing the wind profile in the boundary layer. It does turn out, however, that the wind in the surface layer can be extrapolated into the layers above satisfactorily. An empirical relation that performs quite well is that from?: ( ) ln z2 /z 0 Ψ m (z 2 /L) U 2 = U 1. ln(z 1 /z 0 ) Ψ m (z 1 /L) This expression can be used up to 200 m, both in neutral and in diabatic situations. The deviations from this equation are usually less than 1 m/s. Table 4.2: Anatomy of the atmospheric boundary layer. STABILITY PARAMETERS HEIGHT INTERVAL viscous sublayer all u, ν 0 ν/u turbulent sublayer all u, z 0 ν/u z 0 surface layer all z, u, θ z 0 << z < L quasi-neutral boundary layer neutral h, z, u, f 0.1h - h convective boundary layer unstable h, w 0.1h - h free convection layer unstable z, w 0.01h - 0.1h local scaling layer stable z, τ, wθ 0.1h-0.5h z-less scaling layer very stable τ, wθ 0.1h-h intermittency layer very stable N 0-h entrainment layer stable U, θ, u, w 0.8h - 1.2h 53

54 Chapter 5 EXCHANGE OF HEAT AND WATER VAPOUR 5.1 The surface-energy budget Radiation is the driving force of heating and evaporation at the Earth surface (see figure 5.1). It is composed of longwave radiation originating from the Earth and the atmosphere itself, and of shortwave solar radiation. Furthermore, a distinction is made between incoming (downward) and outgoing (upward) radiation. The incoming shortwave radiation (S ) equals the sum of the directly incoming solar radiation and the diffuse radiation due to scattering processes (by clouds and atmospheric molecules). The outgoing shortwave radiation (S ) is the upward reflected shortwave solar radiation. The longwave incoming radiation at the Earth surface (L ) has its origin in the atmospheric water vapour. It is mainly determined by the amount of water vapour and the temperature of the atmosphere. The outgoing longwave radiation at the Earth surface is directly linked to the surface temperature of the Earth. The net radiation (R N ) at the surface is defined positive when the radiation is directed towards the surface, and given by the sum of all radiation components: R N = S S +L L. The ratio of outgoing and incoming shortwave radiation is called the albedo, a, in formula: a = S S. The albedo (reflectivity) of a grass surface is approximately 0.2, it is for a water surface and for fresh snow. The incoming and outgoing radiation exhibit a daily cycle. For a clear day, the typical behaviour of R N is shown in figure

55 Figure 5.1: Longwave and shortwave radiation in the atmosphere. R n H λe G Figure 5.2: The energy balance. 55

56 At climatological time scales, we assume that the energy budget at the Earth surface is approximately in a state of thermal equilibrium. The components of the energy balance are the net radiation (R N ) on the one hand, and on the other hand the energy that is removed by flow or diffusion: the sensible heat flux (H), evaporation (E), and the surface heat flux (G). The total energy balance at the Earth surface can then be expressed as: R N = H + λe + G, (5.1) (see figure 5.3), where λ represents the latent heat of evaporation, and E is the water vapour flux. The terms on the right-hand side of 5.1 are given by H = ρ 0 c p (wθ) 0 E = ρ 0 wq ( 0 ) Ts G = k z, 0 where k is the molecular diffusion coefficient of the soil. R N has a daily cycle above a land surface, see fig As a result of this, λe, H and G also exhibit a daily cycle. During the day, R N is positive. A typical value at a clear summer day would be 400 Wm 2. H is also positive during daytime, as well as λe. G is positive, but very small. It amounts to around 10 % of the net radiation. During nighttime, R N is negative, typically 50 Wm 2 in a clear night. Because there is only little evaporation during night (rather would there be dew formation), λe is practically zero. At night, H is negative, while G can be relatively large (50 % of the net radiation). Several possibilities exist to determine λe and H from measurements of other quantities. For example, temperature and vapour profile measurements can be used, or measuring all other components of the energy balance. Other estimates have been made by Penman and Monteith. A summary will be given below. 5.2 The profile method When using the profile method, temperature and humidity are measured at two levels above the surface. Wind speed is measured at one level, and the surface roughness is supposed to be known. Using Monin-Obukhov similarity theory, we find that: U(z 2 ) = u κ {ln(z 2/z 0 ) Ψ m (z 2 /L)} Θ(z 2 ) Θ(z 1 ) = θ κ {ln(z 2/z 1 ) Ψ h (z 2 /L) + Ψ h (z 1 /L)} (5.2) q(z 2 ) q(z 1 ) = q κ {ln(z 2/z 1 ) Ψ q (z 2 /L) + Ψ q (z 1 /L)}. 56

57 Figure 5.3: Daily cycle of the components of the energy balance on a clear day over a maize crop. Source:?. u 2 T 2 q 2 T 1 q 1 z 0 Figure 5.4: The profile method experimental setup. At level 1, temperature and humidity are measured. At level 2, also the wind speed is recorded. 57

58 Figure 5.5: The setup for the Bowen-ratio method. Net radiation, surface heat flux, temperature, and humidity (at two levels) are measured. By measuring wind, temperature and humidity at two levels, the three unknown parameters (u, θ and q ) can be determined. From these equation, the heat and water-vapour fluxes immediately follow: H = ρ 0 c p u θ E = ρ 0 u q 5.3 The energy-balance method This method will also exploit the energy balance equation(eq. 5.1). The experimental setup is as follows: at two levels, temperature and humidity are measured. At one level, net radiation is measured. At the surface, the surface heat flux G is recorded. We introduce the Bowen ratio, β H/(λE). From the energy balance, it follows that: λ E = H = β (R N G) β 1 + β (R N G). The Bowen ratio is calculated from the temperature and humidity-difference measurements. From the definition of the Bowen ration, it can be concluded that: β = γ θ q, 58

59 with γ = cp λ and where (5.2) and (5.3), as well as the assumption that Ψ m Ψ q. 5.4 Estimating the evaporation In this section, we will present some methods to calculate the evaporation. This can applied usefully in, for example, agriculture. In all different formulations, the type and state of the surface play an important role: the Penman formulae can be applied for a wet surface without vegetation, and the Penman-Monteith formulae can be used for vegetation. Other practical evaporation formulae have been developed by Priestly and Taylor. All of these formulations will be discussed below. The Penman method It is assumed that R N G is known, and that the evaporation pressure of a wet surface equals the saturation value, q 0 = q s (T 0 ). The temperature and humidity of the atmosphere and the surface, as well as the wind speed, are supposed to be known. From (5.2), the following applies in a stratified surface layer: θ = θ u U q = q u U. Note that q = q 1 q 0, which combines with the definitions of H and λe to give: H = ρ 0 c p θ u2 U (5.3) λe = ρ 0 λ q u2 U. (5.4) If we define a drag coefficient as c d (u /U) 2, we get: H = ρ 0 c p θ c d U (5.5) λe = ρ 0 λ q c d U. (5.6) Using the following equations, we can obtain a simple estimate of the evaporation rate. We will use Taylor expansion (subscripts 0 and 1 refer to the two measurements levels): q s (T 1 ) = q s (T 0 ) + (T 1 T 0 )( q s T ) , (5.7) which we ll rewrite using the notation s qs T 0, and the substitution T 1 T 0 θ 1 θ 0 θ = H/(ρ 0 c p c d U): q s (T 1 ) q s (T 0 ) = θ q s, T 0 59

60 yielding q s (T 0 ) = q s (T 1 ) + sh ρ 0 c p c d U. (5.8) The difference in relative humidity, q, can now be written as: q = q(t 1 ) q s (T 0 ) = q(t 1 ) q s (T 1 ) sh ρ 0 c p c d U. (5.9) Substituting this equation into(5.4), replacing H by R N G λe and reordering yields: λe = γ s + γ ρ 0λc d U[q s (T 1 ) q(t 1 )] + s s + γ (R N G). (5.10) The term c p /λ is denoted as γ. The first term on the right-hand side in (5.10) is called the aerodynamical evaporation, whereas the second term represents the so-called thermodynamical evaporation. In order to calculate the evaporation, in addition to R N G, both temperature, humidity and wind speed have to be known at the reference level (1). These are standard meteorological data that are collected every 1 to 3 hours worldwide. The Penman-Monteith method This approach is best suited for a surface with complete coverage of vegetation. Heat exchange and evaporation within the vegetation layer is not taken into account. On the other hand, evaporation is assumed to depend on the vegetation type. The crops have their own resistance against evaporation through stomata (pores) in the leaves. The vapour level is saturated inside the stomata of the crop leaves. Depending on the amount of available water, the stomata will be more or less closed. In this way, there will be more or less resistance against evaporation. This does not apply to the exchange of heat. If, analogous to Ohm s law, the heat and vapour difference is written as the product of the heat (or vapour) flux and a resistance, we can rewrite (5.6) as H = ρ 0 c p θ r h (5.11) λe = ρ 0 λ q r h, where r h, the resistance, equals 1/(c d U). As a next step, we have to realize that transport of moisture within the crops is an extra resistance. This can be easily accounted for by replacing r h with r h + r s in the second equation, where r s is the so-called crop resistance. Using the procedure of the previous section, we get an expression for the evaporation: 60

61 λe = γδ sδ + γ ρ 0λc d U[q(T 1 ) q s (T 1 )] + sδ sδ + γ (R N G), (5.12) where δ = r h /(r h + r s ). If sufficient water is available in the upper soil layer (where the plant roots are), the plant will optimize its evaporation by minimizing the crop resistance. Each crop is characterized by its own crop resistance. The maximum evaporation is the potential evaporation rate of the crop. At this rate, the crops grow fastest. This is an important quantity in agriculture: if there is drought, the crops have to be irrigated until the potential evaporation rate is reached. Any irrigation surplus is useless since the plant has already reached the potential evaporation rate. The Priestley-Taylor method The Priestley-Taylor evaporation method makes use of the Penman-Monteith equation (5.12). The crop resistance is supposed to be zero, and the relative humidity is chosen at 100 %: q(t 1 ) = q s (T 1 ). From (5.12), it can be deduced that the equilibrium evaporation is given by: λe = s s + γ (R N G). (5.13) 5.5 Air-Sea interaction The surface layer theory also applies over a water surface. There are some aspects that have to be kept in mind, however. When estimating the stability of the surface layer, evaporation plays an important role, since it influences the atmospheric density (at a constant temperature, humid air has a lower density than dry air). To simulate this density effect, a somewhat higher temperature is assigned to humid air, related to the water vapour content in the humid air. This temperature is called virtual temperature, T v. Without deriving it here, the following expression is generally valid for the virtual temperature: T v = T ( q) T θ 0 q, (5.14) where q is the water vapour mixing ration (in kg/kg), and θ 0 is a reference temperature (e.g. 287 K). In the surface layer, the same formulation holds for the virtual potential temperature. θ v : θ v = θ( q) θ θ 0 q, (5.15) If we replace θ by θ v in all Navier-Stokes equations, we also take dynamical effects of humid air into account. This excludes the mechanism of condensation! Should one want to formulate a stability criterion, it is again possible 61

62 to use (wθ) v > 0 for an unstable atmosphere, etcetera (cf. section 4.1). The Richardson number is now given by: Ri f = g θ 0 wθ v u i u j U i x j. (5.16) (see Eq. (3.25)). The virtual-temperature flux follows from (5.15): wθ v = wθ θ 0 w q. In order to determine the potential temperature, the water-vapour flux wq has to be known. It can be estimated from the water-surface temperature and the humidity (see section 5.4). The atmosphere will usually be neutral above a sea surface, since the air flow above it will gradually adapt to the water temperature, which only varies slightly in time and place. The exchange above a sea surface can also be crudely estimated by using drag coefficient relations, where transport is proportional to temperature and to humidity differences. If we take the wind shear stress ρ 0 u 2 and the drag coefficient c d (u /U 10 ) 2 (using wind speed at 10 m), the momentum exchange can be described by: τ ρ 0 u 2 = ρ 0 c d U Measurements show that the drag coefficient is for neutral conditions. In a stable surface layer, we find lower values, and higher values can be found under unstable circumstances. A simple stability correction is c d = c dn (1 Ri), where the Richardson number can be estimated from: Ri = 10 g θ 0 (T 10 T s ) U 10 2 again assuming the reference height at 10 m. Similar to a land surface, the heat and water-vapour transport can be estimated using: and H = ρ 0 c p c h U 10 (T 10 T s ) (5.17) λe = ρ 0 λ c q U 10 (q 10 q s ) (5.18) The coefficients c h and c q both adapt a constant value of

63 Unlike at open sea, stability effects do play a role near the coast. We will come back to that later. A complicating difference between land and sea surfaces is the waves at the sea surface. The observed wave pattern usually is a result of both wind-generated waves (wind waves) and waves that have been generated elsewhere (swell). On top of the wave pattern lies a turbulent layer, again characterized by a vertical momentum flux ρ 0 uw, analogous to the situation above a land surface. Also at sea, a logarithmic wind profile in the surface layer is a suitable assumption: U(z) = 1 u κ ln z + C. (5.19) r (see 3.14). We chose on purpose another parameters, since the length scale r and the constant C will depend on the state of the sea surface and atmospheric turbulence. Because the exact theory is very complicated, we will use scale analysis. A spectral analysis of wind waves can give us a wave number k p that corresponds to the characteristic wave length of the wind waves, 2π/k p (e.g. the value where the spectrum shows a maximum). In this way, the phase velocity C p is determined: C p = g/k p. Its dimensionless counterpart C p /u is sometimes called wave age. The average wave height of a stationary windy sea, H av, can be derived using scale analysis: H av u 2 /g. Another important wind wave parameters is the so-called fetch, the length of the trajectory that a wave has travelled from its birth to its full-grown size. There exists a dimensionless relation between the fetch and the phase velocity. The presence of waves enhances the momentum exchange from the atmosphere to the ocean. Short, steep waves play a relatively important role, since pressure forces can easily clamp to the waves (unlike the land surface, where only the shear stress accounts for the momentum exchange). Momentum exchange above sea The functions C and r in (5.19) depend on u, g and C p, according to the previous section. An alternative formulation of 5.19 would be: U(z) u = 1 κ gz ln u 2 + C, (5.20) where r u 2 /g from dimensional considerations. The proportionality constant has been included in the new constant C, and may still depend on the wave age (C p /u ). A constant value of 11.3 turns out to apply reasonably well, as will be shown later. If C is assumed constant, it is also possible to include C into the logarithm, and rewrite (3.17) analogous to the land surface case: 63

64 Figure 5.6: An ensemble of measurements of the wind-shear stress relation. (a) lower wind speeds, and (b) moderate to strong winds. The dashed line represents Eq. (5.20). (From?) U(z) u = 1 κ ln z z 0, where z 0 = u 2 /g represents the roughness of a windy sea. This formulation (Charnock s Law) combines the parameters r and C without taking into account their functional dependence on C p /u. Under non-neutral circumstances, these formulations can be adjusted in very much the same way as over a land surface (see section 4.2). Charnock s formulation is pretty succesful, despite its simplicity (see figure 5.6). We can conclude that Charnock s Law (including stability correction!) applies for a relatively large range of wind speeds (6-25 m/s), giving a satisfying relation between U and u. The influence of wave age has been studied in more detail. We then assume that z 0 = A c u 2 /g, where A c depends on C p /u. A large number of experiments support the applicability of the following empirical relation (see figure 5.7): A c = 1.89( C p u ) 1.59 [ ( C p ) ( C ] p ) (5.21) u u A simple model for ocean-atmosphere exchange We will assume a homogeneous, thin, and well-mixed layer (for example, the 64

65 Figure 5.7: Several measurements of C p and z 0. The solid line represents the relation in (From?) 65

66 H LE T a q a U Shallow Lake T s ρ s d Figure 5.8: Schematic representation of ocean-atmosphere exchange. oceanic mixing layer, or a shallow lake). Heat and water vapour transport are the most important processes. The water temperature T s can be described by: ρ s c ps d T s t = (H + λe), (5.22) where the subscript s indicates sea-related variables (see fig. 5.8). Using the definition from the previous section (5.17, 5.18), we know that: H + λe = ρ 0 c p c h U 10 {T s T a + λ } (q s (T s ) q a ). (5.23) c p We will introduce the wet-bulb temperature, T w T a λ c p (q s (T w ) q a ). This expression is directly obtained by integrating the First Law of thermodynamics for an adiabatic process, c p dt + λdq = 0, from the initial condition T, q a until the state T w, q s(t w). This describes the physical process in an air parcel in which liquid water evaporates until it saturates. Developing q s into a Taylor series q s (T s ) = q s (T w ) + (T s T w ) ( q s T ) s +..., (5.24) and substituting into (5.23) yields ρ s c ps d T s t = ρ 0 c h U 10 (T s T w ) (c p + sλ). (5.25) This is a differential equation exhibiting an exponential behaviour if all coefficients are assumed constant. From (5.25), we will find the time constant τ: ρ s c ps d τ = ρ 0 c h U 10 (c p + sλ), (5.26) 66

67 which is proportional to d/u 10. The cooling rate is proportional to the layer thickness, and inversely proportional to the wind speed. We can also include the heat loss through radiation. The energy budget then becomes: T s t = Q T s T w τ (5.27) where Q represents the net longwave radiation term due to radiation loss of the water layer ( T 4 s ) and the atmospheric back-radiation ( T 4 a ), implying T 4 s T s t = αta 4 βts 4 T s T w. (5.28) τ can be approximated as yielding T 4 s = (T w + T s T w ) 4 T 4 w + 4 T 3 w (T s T w ), T s t = αt 4 a βt 4 w T s T w τ, (5.29) with a time constant τ = τ/(1 + 4βT 3 w τ). The response time turns out to decrease due to the longwave radiation. The value of τ on mid-latitudes turns out to be 1 day for a depth of 1 m, and 10 days for a depth of 10 m. At these depths, water temperatures can apparently fluctuate quite rapidly. Finally, we will give an example of the effect of a periodic forcing of the water temperature by the atmosphere, q cos ωt, where q is the temperature forcing (in Km 1 ): T s t = q cos ωt T s T eq, (5.30) τ where ω may represent the annual cycle of temperature, and T eq the annual average temperature. The general solution of this equation is: T s T eq = Ce t/τ qτ qτ 1 + (ωτ) 2 e t/τ + cos(ωt + φ), (5.31) 1 + (ωτ) 2 where C is an integration constant, and φ = arctan ωτ. If t >> τ, this relation can be rewritten as 67

68 Figure 5.9: Phase difference between the annual atmosperic temperature cycle and the water temperature at two depths (5 and 15 m). 68