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Chapter 6 The Planetary Boundary Layer
Chapter
6
The Planetary Boundary Layer
In deriving the idealized flow fields discussed in Chapter 3 we assumed in all cases that frictional forces were sufficiently small that they could be neglected. For synoptic scale motions in the free atmosphere (that is, that part of the atmosphere far enough removed from the ground to be free of direct surface friction effects), this approximation is probably quite valid. However, in the lowest kilometer of the atmosphere the vertical viscous force generally is comparable in magnitude to the pressure gradient and Coriolis forces. This region of the atmosphere is referred to as the planetary boundary layer. It contains nearly ten percent of the mass of the atmosphere. Because the planetary boundary layer is a turbulent boundary layer, a rigorous mathematical theory for the structure of the velocity field in the layer is not possible. In order to satisfactorily study the structure of the planetary boundary layer we would require detailed knowledge of the structure and amplitude of the turbulent eddies which are responsible for the vertical momentum transport. Even a cursory examination of this subject is outside the scope of this text. In the present chapter it will be assumed that the frictional force in turbulent flow may be represented in the same manner as done in laminar flow by introducing an eddy uiscosity coefficient. 81
82
6
THE PLANETARY BOUNDARY LAYER
6.1 The Mixing Length Theory The concept of an eddy viscosity coeficient was mentioned briefly in Section 1.2.3. I n this section we will discuss in more detail the heuristic argument of the famous fluid dynamicist L. Prandtl which provides some theoretical basis for estimating the magnitude of the eddy viscosity. Prandtl’s basic idea was that the momentum transport of the small-scale eddy motions may be parameterized in terms of the large-scale mean flow. In order to understand the basis of this parameterization we must first consider the derivation of mean flow equations for a turbulent fluid. (In fact we have been dealing with such equations throughout this book; however, we have not previously explicitly considered the partitioning of the flow between turbulent eddies and mean fields.) In a turbulent fluid the velocity measured at a point generally fluctuates rapidly in time as eddies of various scales pass the point. In order that our velocity measurements be truly representative of the large-scale flow it is thus necessary to average the flow over an interval of time long enough to average out the eddy fluctuations but still short enough to preserve the trends in the large-scale flow field. To do this we use angle brackets to define the mean velocity (V) as the average in time at a given point. The instantaneous velocity is then
v = (V) + V’ where V’ designates the deviation from the mean at any time. V‘ is thus associated with the turbulent eddies. We now apply this averaging scheme to the horizontal equations of motion. We have previously written these equations in the approximate form : 1ap
au au au au -+u-+uy+w--fu=--at
ax
az
ay
P ax
Using the continuity equation, aP + a (pu) at
ax
+a
- (pu)
aY
+a (PW) = 0
(6.3)
aZ
Eqs. (6.1) and (6.2) can be transformed to a more convenient form. If we multiply (6.1) by p and (6.3) by u then add the resulting equations, we obtain theflux form of the x-momentum equation :
a
-(pu) at
+a (pu2) + a (puu) + a (PUW) -fPV ax aY aZ
= -
axaP
(6.4)
6.1
83
THE MIXING LENGTH THEORY
An analogous operation gives the flux form of the y-momentum equation:
a
- (pu)
at
a (puu) + -a (pu’) + -a (puw) + f P U +ax aY aZ
= --
aY
(6.5)
We next substitute in place of u, 2:, and w in (6.4) and (6.5) letting
u = (u)
+ u’,
v
= (2:)
+ v’,
w = (w)
+ w‘
(6.6)
If we neglect the small fluctuations in density associated with the turbulence, the resulting equations can be averaged in time to get a simple partition of the flow between the mean flow and turbulent fields. Thus, for example, the term puw becomes (PUW>
= P(((U>
+ u’>((w> + w’))
= P (u>(w>
+ P(U’W’>
where terms like ( ( u ) w ’ ) and ( u ’ ( w ) ) vanish because ( u ) and (w) are constant over the averaging interval so that, for example, ( ( u > w ‘ ) = ( u > ( w ’ ) = 0 since (w’) = 0. Carrying out this averaging process for all terms in (6.4) and (6.5) we obtain with the aid of the averaged continuity equation:
The terms in square brackets on the right-hand side in (6.7) and (6.8) which depend on the turbulent fluctuations are called the eddy stress terms. In the mixing length theory these eddy stress terms are parameterized in terms of the mean field variables by assuming that the eddy stresses are proportional to the gradient of the mean wind. Since we are here primarily concerned with the planetary boundary layer where vertical gradients are much larger than the horizontal gradients, the discussion will be limited to the vertical eddy stress terms. According to the mixing length hypothesis a parcel of fluid which is displaced vertically will carry the mean horizontal velocity of its original level for a characteristic distance I‘ analogous to the mean free path in molecular viscosity. This displacement will create a turbulent fluctuation whose
84
6
THE PLANETARY BOUNDARY LAYER
magnitude will depend upon 1’ and the shear of the mean velocity. Thus, for example,
where it must be understood that I‘ > 0 for upward parcel displacement and I‘ < 0 for downward parcel displacement. The vertical eddy stress-p(u’w’) can then be written as -p(u‘w‘)
= p(w‘l’) xu>
az
(6.9)
In order to estimate w’ in terms of the mean fields we assume that the vertical stability of the atmosphere is nearly neutral so that buoyancy effects are small (see Section 9.4). The horizontal scale of the eddies should then be comparable to the vertical scale so that I w‘ I I vh’[ and we can set
-
where V,’ and (vh) designate the turbulent and mean parts of the horizontal velocity field, respectively. Here the absolute value of the mean velocity gradient is needed because if I‘ > 0 we must have w‘ > 0 (that is, upward parcel displacement by the eddy fluctuations). Thus the eddy stress can be written (6.10)
1 is called the eddy exchange coeficient. Similarly where A , = p1I2 I d(Vh)/dz we may show that the vertical eddy stress due to motion in they direction may be written
In the planetary boundary layer it is usually assumed that A , depends only on the distance from the surface. 6.2 The Ekman Layer Equations
If the horizontal eddy stress terms in (6.7) and (6.8) are neglected and the vertical stresses are parameterized in terms of the mean flow by introducing the eddy exchange coefficient the momentum equations become
6.2
85
THE EKMAN LAYER EQUATIONS
au au au au -+ u - + u - - + w - - f t J = - - - + - at ay ax az
pax
au av av au -+u-+v-+w-+fu=---+-at ax ay aZ
1 a"p
a a, 1 a
p aY
P
1 ap
1
p
aZ
(A
au
-
az) au
(''4 4 -
(6.11) (6.12)
where we have omitted the angle brackets since all dependent variables are here mean flow variables. For midlatitude synoptic scale motions, we showed in Section 2.4 that the acceleration terms dujdr and dv/dt in (6.11) and (6.12) were small compared to the Coriolis force and pressure gradient force terms. Outside the boundary layer the first approximation was then simply geostrophic balance. In the boundary layer the acceleration terms are still small compared to the Coriolis and pressure gradient terms. Thus to a first approximation the planetary boundary layer equations express a three-way balance between the Coriolis force, the pressure gradient force, and the viscosity force :
-'' ; + - - ( = -
1 ap
a p aZ 1
au
A,-
a,)
(6.13) (6.14)
To simplify further development we now assume that the eddy exchange coefficient is independent of height and introduce an eddy viscosity coefficient K = A,/p which is analogous to the molecular kinematic viscosity coefficient in laminar flow. Actually A, cannot be constant all the way to the ground since the value of A, depends on the mixing length which must decrease as one approaches the ground. However, the qualitative nature of the solutions is not changed by including a vertical dependence of A , . The constant coefficient solution is also of interest because it can be modeled easily with laminar flow in the laboratory. Thus, we obtain finally as our approximate equations for the planetary boundary layer: KT
+ f(v
- v,) = O
(6.15)
K ,
- f ( u - u,) = 0
(6.16)
az a% aZ
where we have used the definitions
86
6
THE PLANETARY BOUNDARY LAYER
The Ekman layer equations (6.15) and (6.16) can be solved to determine the departure of the wind field from geostrophic balance in the boundary layer. To solve this set of simultaneous second-order differential equations we must specify boundary conditions on u and L'. At the surface of the earth the velocity must go to zero, whereas in the free atmosphere sufficiently far from the surface the wind must approach its geostrophic value. Therefore, as boundary conditions on (6.15) and (6.16) we choose
u=O, u + u,.
at
v=O t 1 - i
rg
z=O
(6.17)
as z - i co
4-
T o solve this set it is convenient to multiply (6.16) by i = 1 and add the result to (6.15) to obtain a second-order equation in the complex relocity u + ir : K
dZ (u + ir) - if(u + ir) = -if(u, + iL,,) dZ2
(6.18)
For simplicity, we assume that the flow is oriented so that the geostrophic wind is entirely in the zonal direction (rg = 0). Then the general solution of (6.18) may be written
+ ir = A e~p[(if'K)''~z]+ B e ~ p[ -(if iK )'/~ z] It can be shown that ,/i = ( 1 + i) ,/2. Using this relationship and applying zi
the boundary conditions (6.17). we find that A I,
+ j r , = -I,p
- ~ ( +1 i):
=0
and B
+ up
= - u, . Thus,
where y = (fI2K)''Z
Applying the Euler formula e-'' = cos 0 - i sin 0 and separating the real from the imaginary part we obtain u = u,(l - e-" cos yz) 1) =
u g e - y zsin yz
(6.19)
This solution is the famous Ekniari spiral named for the Swedish oceanographer V . W. Ekman who first derived an analogous solution for the surface wind drift current in the ocean. The structure of the solution (6.19) is best illustrated by a hodograph as shown in Fig. 6.1. In this figure the components of the wind velocity are plotted as a function of height. Thus the points on the curve in Fig. 6.1 correspond to u and in (6.19) for values of yz increasing as one moves away from the origin along the spiral. It can be seen from Fig. 6.1 that when z = n:y, the wind is parallel to the geostrophic wind although 13
6.2
87
THE EKMAN LAYER EQUATIONS
0.4
' ,"0.2
\
y;T
I
02
04
06
00
10
Fig. 6.1 Hodograph of the Ekman spiral solution. Points marked on the curve are values of yz which is a nondiniensional measure of height.
slightly greater than geostrophic in magnitude. It is conventional to designate this level as the top of the planetary boundary layer. Thus the depth of the Ekman layer is
De
= n/y
(6.20)
Observations indicate that the wind approaches its geostrophic value at about one kilometer above the ground. Substituting De = 1 km and f = into (6.20), we can solve for the eddy viscosity K . The result is that K z 5 x lo4 cm2 sec-'. Referringbackto(6.10)wesee that K =(Z'2) ld(V,,)/i?zI .Thusif the mean wind shear is of the order of 5 m sec-' km-', the mixing length I' must be about 30 m in order that K = 5 x lo4 cm2 sec-'. This mixing length is small compared to the depth of the boundary layer, as it should be if the mixing length concept is to be useful. Qualitatively the most striking feature of the Ekman layer solution is the fact that the wind in the boundary layer has a component directed toward lower pressure. This is a direct result of the three-way balance between the pressure gradient force, the Coriolis force, and the viscous force. This balance is illustrated in Fig. 6.2 for a level well within the boundary layer. Since the Coriolis force is always normal to the velocity and the frictional force is mainly a retarding force, their sum can only exactly balance the pressure
P
Fig. 6.2 Balance of forces in the Ekman layer. P designates the pressure gradient, Co designates the Coriolis force, Fr designates the frictional force.
6
88
THE PLANETARY BOUNDARY LAYER
gradient force if the wind is directed toward lower pressure (that is, to the left of the geostrophic wind in the northern hemisphere). Furthermore, it is easy to see that as the frictional force becomes increasingly important the cross isobar angle must increase. The ideal Ekman layer discussed here is rarely, if ever, observed in the atmospheric boundary layer partly because, as mentioned above, the eddy mixing coefficient must vary rapidly with height near the ground. An additional reason, however, is that the Ekman layer wind profile is generally unstable for a neutrally buoyant atmosphere. The circulations which develop as a result of this instability have horizontal and vertical scales comparable to the depth of the boundary layer. Thus, it is not possible to parameterize then1 by a simple mixing length theory. However, these circulations do in general transport considerable momentum vertically. The net result is usually to decrease the angle between the boundary layer wind and the geostrophic wind. A typical observed wind hodograph is shown in Fig. 6.3. Although the detailed structure is rather different from the Ekman spiral, the vertically integrated horizontal mass transport in the boundary layer is still directed toward lower pressure. And as we shall see in the next section it is this fact which is of primary importance for synoptic scale systems.
u (msec-')
Fig. 6.3 Mean wind hodograph for Jacksonville, 4 April 1968 (solid line) compared with the Ekman spiral (dashed line). (Adapted from Brown, 1970.)
6.3 Secondary Circulations and Spin-Down In the idealized Ekman spiral solution of (6.19) the component of the wind multiplied by p gives the cross isobaric mass transport per unit area at any level in the boundary layer. Thus the net mass transport toward lower pressure in the Ekman layer for a column of unit width extending vertically through the entire layer is 13
M =
jopu d z = jop u p e - " " l D esin dz De De
De
7cZ
(6.21)
6.3
89
SECONDARY CIRCULATIONS A N D SPIN-DOWN
Neglecting local variations of density in the boundary layer the continuity equation may be written as (6.22) Integrating (6.22) through the depth of the boundary layer we find that
Here we have assumed that the ground is level so that M’ = 0 at z = 0. Substituting from (6.19) we can rewrite this expression for the vertical mass flux at the top of the Ekman layer as (6.23) Comparing (6.23) with (6.21) we see that the vertical flux at the top of the boundary layer is equal to the horizontal convergence of mass in the boundary layer which is simply -2M 2y in the above example. Noting that -?ug/ay = 5, is just the geostrophic vorticity in this case, we have after integrating (6.23) M’( De) =
ig(K/2f)”2
(6.24)
where we have neglected the variation of density with height in the boundary layer. Hence, we obtain the important result that the vertical velocity at the top of the boundary layer is proportional to the geostrophic vorticity. I n this way the effect of friction in the boundary layer is communicated directly to the interior fluid through a forced secondary circulation rather than indirectly by the slow process of viscous diffusion. For a typical synoptic scale system sec-I, and De 1 km, the vertical velocity sec-I, f with 5, given by (6.24) is of the order of a few tenths of a centimeter per second. An analogous secondary circulation is responsible for the decay of the circulationcreated when a c u p o f tea is stirred. Away from the boundary of the cup there is a n approximate balance between the radial pressure gradient and the centrifugal force of the spinning fluid. However, near the bottom of the cup viscosity slows the motion and the centrifugal force is not sufficient to balance the radial pressure gradient. (Note that the radial pressure gradient is independent of depth since tea is an incompressible fluid.) Therefore, radial inflow takes place near the bottom of the cup. Because of this inflow the tea leaves always are observed to cluster near the center at the bottom of the cup if the tea has been stirred. By continuity the radial inflow in the bottom boundary layer requires upward motion and a slow compensating outward radial flow throughout the remaining portion of the cup. This slow outward
-
-
-
90
6
THE PLANETARY BOUNDARY LAYER
flow approximately conserves angular momentum, and by replacing high angular momentum fluid by low angular momentum fluid serves to spin-down the vorticity in the cup far more rapidly than could mere molecular diffusion. This spin-down effect is also important in the atmosphere. It is most easily illustrated in the case of a barotropic atmosphere. We showed previously in Section 5.5 that for synoptic scale motions the vorticity equation could be written approximately as au
ac.
dW
(6.25)
where we have neglected icompared to f i n the divergence term. Neglecting the latitudinal variation offwe next evaluate the integral of (6.25) from the top of the boundary layer, z = De, to the tropopause, z = H .
(6.26) Assuming that w = 0 at z = H a n d that the vorticity may be approximated by its geostrophic value (which is independent of height) we obtain from (6.26)
-’
% - ( H - De) w( De) dt Substituting from (6.24) and noting that H % De we obtain a differential equation for the time dependence of [,:
(6.27) Equation (6.27) may be immediately integrated to give
5,
= i,(O)
exp{ - (fK,’2H2)’/’r}
(6.28)
where c,(O) is the value of the geostrophic vorticity at time t = 0. From (6.28) we see that T~ E H ( 2 f K ) ’ I 2 is the time which it takes a barotropic vortex of height H to spin-down to e-’ of its original value (this “e-folding” time scale is what is meant by the spin-down rime). Taking typical values of the parameters as follows: H = 10 km, f = lop4 sec-’, and K = lo5 cm2 sec-’, we find that T = z 4 days. Thus, for midlatitude synoptic scale disturbances in a barotropic atmosphere the characteristic spin-down time is a few days. This decay time scale should be compared to the time scale for ordinary viscous diffusion. I t can be shown that the time for eddy diffusion to penetrate a depth H is of the order Td H 2 ,K. For the above values of H and K , the diffusion time scale is thus T~ 100 days. Hence, the spin-down process is a far more effective mechanism for destroying vorticity in a rotating atmosphere than is ordinary diffusion.
--
6.3
SECONDARY CIRCULATIONS AND SPIN-DOWN
91
Physically the spin-down process in the atmospheric case is similar to that described for the teacup, except that in synoptic scale systems it is primarily the Coriolis force which balances the pressure gradient force away from the boundary, not the centrifugal force. Again the role of the secondary circulation driven by viscous forces in the boundary layer is to provide a slow radial flow in the interior which is superposed on the azimuthal circulation of the vortex. This secondary circulation is directed outward in a cyclone so that the horizontal area enclosed by any chain of fluid particles gradually increases. Since the circulation is conserved, the azimuthal velocity at any distance from the vortex center must decrease in time. Or, from another point of view, the Coriolis force for the outward-flowing fluid is directed clockwise, and this force thus exerts a torque opposite to the direction of the circulation of the vortex. In Fig. 6.4 a qualitative sketch of the streamlines of this secondary flow is shown. It should now be obvious exactly what is meant by the term secondary circulation. It is simply a circulation superposed on the primary circulation
Fig. 6.4 Streamlines of the secondary circulation forced by frictional convergence in the planetary boundary layer for a cyclonic vortex in a barotropic atmosphere.
(in this case the azimuthal circulation of the vortex) by the physical constraints of the system. In the case of the boundary layer it is viscosity which is responsible for the presence of the secondary circulation. However, other processes such as temperature advection and diabatic heating may also lead to secondary circulations as we shall see later. The above discussion has concerned only the neutrally stratified barotropic atmosphere. An analysis for the more realistic case of a stably stratified
92
6
THE PLANETARY BOUNDARY LAYER
baroclinic atmosphere would be much more complicated. However, qualitatively the effects of stratification may be easily understood. The buoyancy force (see Section 9.4) will act to suppress vertical motion since air lifted vertically in a stable environment will be denser than the environmental air. As a result the interior secondary circulation will be restricted in vertical extent as shown in Fig. 6.5. Most of the return flow will take place just above the
I
De
Fig. 6.5 Streamlines of the secondary circulation forced by frictional convergence in the planetary boundary layer for a cyclonic vortex in a stably stratified baroclinic atrnosphere.
boundary layer. This secondary flow will rather quickly spin-down the vorticity at the top of the Ekman layer without appreciably affecting the higher levels. When the geostrophic vorticity at the top of the boundary layer is reduced to zero, the “pumping” action of the Ekman layer is eliminated. The result is a baroclinic vortex with a vertical shear of the azimuthal velocity which is just strong enough to bring Cg to zero at the top of the boundary layer. This vertical shear of the geostrophic wind requires a radial temperature gradient to satisfy the thermal wind relationship. This radial temperature gradient is in fact produced during the spin-down phase by the adiabatic cooling of the air forced out of the Ekman layer. Thus, the secondary circulation in the baroclinic atmosphere serves two purposes: (1) it changes the azimuthal velocity field of the vortex through the action of the Coriolis force, and (2) it changes the temperature distribution so that a thermal wind balance is always maintained between the vertical shear of the azimuthal velocity and the radial temperature gradient.
SUGGESTED REFERENCES
93
Problems 1. Verify by direct substitution that the Ekman spiral expression (6.19) is indeed a solution of the boundary layer equations (6.15) and (6.16).
2. Derive the Ekman spiral solution for the more general case where the geostrophic wind has both x and y components, us and u g . 3. Letting the Coriolis parameter and density be constants, show that Equation (6.24) is correct for the more general Ekman spiral solution obtained in Problem 2. 4. For laminar flow in a rotating cylindrical vessel filled with water (molecular kinematic viscosity v = 0.01 cm2 sec-'), compute the depth of the Ekman layer and the spin-down time if the depth of the fluid is 30 cm and the rotation rate of the tank is ten revolutions per minute. 5. For the situation of Problem 4, how small would the radius of the tank have to be in order that the time scale for viscous diffusion from the sidewalls be comparable to the spin-down time? 6. In a homogeneous barotropic fluid show that the secondary radial circulation associated with a geostrophic vortex does not vary in strength with height. 7. Derive an expression for the wind driven surface Ekman layer in the ocean. Assume that the wind stress T~ is constant and directed along the x-axis. Continuity of stress at the air-sea interface ( z = 0) requires that the wind stress equals the water stress so that the boundary condition at the surface becomes
where K is the eddy viscosity in the ocean (assumed constant). As a lower boundary condition assume that u, L' + 0 as z + - co. If K = 10 cm2 sec-' what is the depth of the surface Ekman layer? 8. Show that the vertically integrated mass transport in the wind driven oceanic surface Ekman layer is directed 90" to the right of the surface wind stress in the Northern Hemisphere. Explain this result physically.
Suggested References Batchelor, An Introduction to Fluid Dynamics, has a useful discussion of the molecular basis of viscosity as well as a succinct derivation of the complete stress tensor. This book also has a n excellent treatment of viscous boundary layers including the Ekman layer.
94
6
THE PLANETARY BOUNDARY LAYER
Cole, Perturbation Methods it1 Applied Mathetnatics provides an excellent graduate level introduction to the mathematical foundations of boundary layer theory. Greenspan, The Theory of Rotating Fluids gives a unified account of the role of viscosity in rotating fluids at an advanced level. This is the only book available which focuses solely on phenomena which occur only in rotating fluids. Luniley and Panofsky, The Strrictirre of Atniospheric Turbulence covers both observational and theoretical aspects of the subject including the effects of thermal stratification in the boundary layer.
6
The Planetary Boundary Layer
In deriving the idealized flow fields discussed in Chapter 3 we assumed in all cases that frictional forces were sufficiently small that they could be neglected. For synoptic scale motions in the free atmosphere (that is, that part of the atmosphere far enough removed from the ground to be free of direct surface friction effects), this approximation is probably quite valid. However, in the lowest kilometer of the atmosphere the vertical viscous force generally is comparable in magnitude to the pressure gradient and Coriolis forces. This region of the atmosphere is referred to as the planetary boundary layer. It contains nearly ten percent of the mass of the atmosphere. Because the planetary boundary layer is a turbulent boundary layer, a rigorous mathematical theory for the structure of the velocity field in the layer is not possible. In order to satisfactorily study the structure of the planetary boundary layer we would require detailed knowledge of the structure and amplitude of the turbulent eddies which are responsible for the vertical momentum transport. Even a cursory examination of this subject is outside the scope of this text. In the present chapter it will be assumed that the frictional force in turbulent flow may be represented in the same manner as done in laminar flow by introducing an eddy uiscosity coefficient. 81
82
6
THE PLANETARY BOUNDARY LAYER
6.1 The Mixing Length Theory The concept of an eddy viscosity coeficient was mentioned briefly in Section 1.2.3. I n this section we will discuss in more detail the heuristic argument of the famous fluid dynamicist L. Prandtl which provides some theoretical basis for estimating the magnitude of the eddy viscosity. Prandtl’s basic idea was that the momentum transport of the small-scale eddy motions may be parameterized in terms of the large-scale mean flow. In order to understand the basis of this parameterization we must first consider the derivation of mean flow equations for a turbulent fluid. (In fact we have been dealing with such equations throughout this book; however, we have not previously explicitly considered the partitioning of the flow between turbulent eddies and mean fields.) In a turbulent fluid the velocity measured at a point generally fluctuates rapidly in time as eddies of various scales pass the point. In order that our velocity measurements be truly representative of the large-scale flow it is thus necessary to average the flow over an interval of time long enough to average out the eddy fluctuations but still short enough to preserve the trends in the large-scale flow field. To do this we use angle brackets to define the mean velocity (V) as the average in time at a given point. The instantaneous velocity is then
v = (V) + V’ where V’ designates the deviation from the mean at any time. V‘ is thus associated with the turbulent eddies. We now apply this averaging scheme to the horizontal equations of motion. We have previously written these equations in the approximate form : 1ap
au au au au -+u-+uy+w--fu=--at
ax
az
ay
P ax
Using the continuity equation, aP + a (pu) at
ax
+a
- (pu)
aY
+a (PW) = 0
(6.3)
aZ
Eqs. (6.1) and (6.2) can be transformed to a more convenient form. If we multiply (6.1) by p and (6.3) by u then add the resulting equations, we obtain theflux form of the x-momentum equation :
a
-(pu) at
+a (pu2) + a (puu) + a (PUW) -fPV ax aY aZ
= -
axaP
(6.4)
6.1
83
THE MIXING LENGTH THEORY
An analogous operation gives the flux form of the y-momentum equation:
a
- (pu)
at
a (puu) + -a (pu’) + -a (puw) + f P U +ax aY aZ
= --
aY
(6.5)
We next substitute in place of u, 2:, and w in (6.4) and (6.5) letting
u = (u)
+ u’,
v
= (2:)
+ v’,
w = (w)
+ w‘
(6.6)
If we neglect the small fluctuations in density associated with the turbulence, the resulting equations can be averaged in time to get a simple partition of the flow between the mean flow and turbulent fields. Thus, for example, the term puw becomes (PUW>
= P(((U>
+ u’>((w> + w’))
= P (u>(w>
+ P(U’W’>
where terms like ( ( u ) w ’ ) and ( u ’ ( w ) ) vanish because ( u ) and (w) are constant over the averaging interval so that, for example, ( ( u > w ‘ ) = ( u > ( w ’ ) = 0 since (w’) = 0. Carrying out this averaging process for all terms in (6.4) and (6.5) we obtain with the aid of the averaged continuity equation:
The terms in square brackets on the right-hand side in (6.7) and (6.8) which depend on the turbulent fluctuations are called the eddy stress terms. In the mixing length theory these eddy stress terms are parameterized in terms of the mean field variables by assuming that the eddy stresses are proportional to the gradient of the mean wind. Since we are here primarily concerned with the planetary boundary layer where vertical gradients are much larger than the horizontal gradients, the discussion will be limited to the vertical eddy stress terms. According to the mixing length hypothesis a parcel of fluid which is displaced vertically will carry the mean horizontal velocity of its original level for a characteristic distance I‘ analogous to the mean free path in molecular viscosity. This displacement will create a turbulent fluctuation whose
84
6
THE PLANETARY BOUNDARY LAYER
magnitude will depend upon 1’ and the shear of the mean velocity. Thus, for example,
where it must be understood that I‘ > 0 for upward parcel displacement and I‘ < 0 for downward parcel displacement. The vertical eddy stress-p(u’w’) can then be written as -p(u‘w‘)
= p(w‘l’) xu>
az
(6.9)
In order to estimate w’ in terms of the mean fields we assume that the vertical stability of the atmosphere is nearly neutral so that buoyancy effects are small (see Section 9.4). The horizontal scale of the eddies should then be comparable to the vertical scale so that I w‘ I I vh’[ and we can set
-
where V,’ and (vh) designate the turbulent and mean parts of the horizontal velocity field, respectively. Here the absolute value of the mean velocity gradient is needed because if I‘ > 0 we must have w‘ > 0 (that is, upward parcel displacement by the eddy fluctuations). Thus the eddy stress can be written (6.10)
1 is called the eddy exchange coeficient. Similarly where A , = p1I2 I d(Vh)/dz we may show that the vertical eddy stress due to motion in they direction may be written
In the planetary boundary layer it is usually assumed that A , depends only on the distance from the surface. 6.2 The Ekman Layer Equations
If the horizontal eddy stress terms in (6.7) and (6.8) are neglected and the vertical stresses are parameterized in terms of the mean flow by introducing the eddy exchange coefficient the momentum equations become
6.2
85
THE EKMAN LAYER EQUATIONS
au au au au -+ u - + u - - + w - - f t J = - - - + - at ay ax az
pax
au av av au -+u-+v-+w-+fu=---+-at ax ay aZ
1 a"p
a a, 1 a
p aY
P
1 ap
1
p
aZ
(A
au
-
az) au
(''4 4 -
(6.11) (6.12)
where we have omitted the angle brackets since all dependent variables are here mean flow variables. For midlatitude synoptic scale motions, we showed in Section 2.4 that the acceleration terms dujdr and dv/dt in (6.11) and (6.12) were small compared to the Coriolis force and pressure gradient force terms. Outside the boundary layer the first approximation was then simply geostrophic balance. In the boundary layer the acceleration terms are still small compared to the Coriolis and pressure gradient terms. Thus to a first approximation the planetary boundary layer equations express a three-way balance between the Coriolis force, the pressure gradient force, and the viscosity force :
-'' ; + - - ( = -
1 ap
a p aZ 1
au
A,-
a,)
(6.13) (6.14)
To simplify further development we now assume that the eddy exchange coefficient is independent of height and introduce an eddy viscosity coefficient K = A,/p which is analogous to the molecular kinematic viscosity coefficient in laminar flow. Actually A, cannot be constant all the way to the ground since the value of A, depends on the mixing length which must decrease as one approaches the ground. However, the qualitative nature of the solutions is not changed by including a vertical dependence of A , . The constant coefficient solution is also of interest because it can be modeled easily with laminar flow in the laboratory. Thus, we obtain finally as our approximate equations for the planetary boundary layer: KT
+ f(v
- v,) = O
(6.15)
K ,
- f ( u - u,) = 0
(6.16)
az a% aZ
where we have used the definitions
86
6
THE PLANETARY BOUNDARY LAYER
The Ekman layer equations (6.15) and (6.16) can be solved to determine the departure of the wind field from geostrophic balance in the boundary layer. To solve this set of simultaneous second-order differential equations we must specify boundary conditions on u and L'. At the surface of the earth the velocity must go to zero, whereas in the free atmosphere sufficiently far from the surface the wind must approach its geostrophic value. Therefore, as boundary conditions on (6.15) and (6.16) we choose
u=O, u + u,.
at
v=O t 1 - i
rg
z=O
(6.17)
as z - i co
4-
T o solve this set it is convenient to multiply (6.16) by i = 1 and add the result to (6.15) to obtain a second-order equation in the complex relocity u + ir : K
dZ (u + ir) - if(u + ir) = -if(u, + iL,,) dZ2
(6.18)
For simplicity, we assume that the flow is oriented so that the geostrophic wind is entirely in the zonal direction (rg = 0). Then the general solution of (6.18) may be written
+ ir = A e~p[(if'K)''~z]+ B e ~ p[ -(if iK )'/~ z] It can be shown that ,/i = ( 1 + i) ,/2. Using this relationship and applying zi
the boundary conditions (6.17). we find that A I,
+ j r , = -I,p
- ~ ( +1 i):
=0
and B
+ up
= - u, . Thus,
where y = (fI2K)''Z
Applying the Euler formula e-'' = cos 0 - i sin 0 and separating the real from the imaginary part we obtain u = u,(l - e-" cos yz) 1) =
u g e - y zsin yz
(6.19)
This solution is the famous Ekniari spiral named for the Swedish oceanographer V . W. Ekman who first derived an analogous solution for the surface wind drift current in the ocean. The structure of the solution (6.19) is best illustrated by a hodograph as shown in Fig. 6.1. In this figure the components of the wind velocity are plotted as a function of height. Thus the points on the curve in Fig. 6.1 correspond to u and in (6.19) for values of yz increasing as one moves away from the origin along the spiral. It can be seen from Fig. 6.1 that when z = n:y, the wind is parallel to the geostrophic wind although 13
6.2
87
THE EKMAN LAYER EQUATIONS
0.4
' ,"0.2
\
y;T
I
02
04
06
00
10
Fig. 6.1 Hodograph of the Ekman spiral solution. Points marked on the curve are values of yz which is a nondiniensional measure of height.
slightly greater than geostrophic in magnitude. It is conventional to designate this level as the top of the planetary boundary layer. Thus the depth of the Ekman layer is
De
= n/y
(6.20)
Observations indicate that the wind approaches its geostrophic value at about one kilometer above the ground. Substituting De = 1 km and f = into (6.20), we can solve for the eddy viscosity K . The result is that K z 5 x lo4 cm2 sec-'. Referringbackto(6.10)wesee that K =(Z'2) ld(V,,)/i?zI .Thusif the mean wind shear is of the order of 5 m sec-' km-', the mixing length I' must be about 30 m in order that K = 5 x lo4 cm2 sec-'. This mixing length is small compared to the depth of the boundary layer, as it should be if the mixing length concept is to be useful. Qualitatively the most striking feature of the Ekman layer solution is the fact that the wind in the boundary layer has a component directed toward lower pressure. This is a direct result of the three-way balance between the pressure gradient force, the Coriolis force, and the viscous force. This balance is illustrated in Fig. 6.2 for a level well within the boundary layer. Since the Coriolis force is always normal to the velocity and the frictional force is mainly a retarding force, their sum can only exactly balance the pressure
P
Fig. 6.2 Balance of forces in the Ekman layer. P designates the pressure gradient, Co designates the Coriolis force, Fr designates the frictional force.
6
88
THE PLANETARY BOUNDARY LAYER
gradient force if the wind is directed toward lower pressure (that is, to the left of the geostrophic wind in the northern hemisphere). Furthermore, it is easy to see that as the frictional force becomes increasingly important the cross isobar angle must increase. The ideal Ekman layer discussed here is rarely, if ever, observed in the atmospheric boundary layer partly because, as mentioned above, the eddy mixing coefficient must vary rapidly with height near the ground. An additional reason, however, is that the Ekman layer wind profile is generally unstable for a neutrally buoyant atmosphere. The circulations which develop as a result of this instability have horizontal and vertical scales comparable to the depth of the boundary layer. Thus, it is not possible to parameterize then1 by a simple mixing length theory. However, these circulations do in general transport considerable momentum vertically. The net result is usually to decrease the angle between the boundary layer wind and the geostrophic wind. A typical observed wind hodograph is shown in Fig. 6.3. Although the detailed structure is rather different from the Ekman spiral, the vertically integrated horizontal mass transport in the boundary layer is still directed toward lower pressure. And as we shall see in the next section it is this fact which is of primary importance for synoptic scale systems.
u (msec-')
Fig. 6.3 Mean wind hodograph for Jacksonville, 4 April 1968 (solid line) compared with the Ekman spiral (dashed line). (Adapted from Brown, 1970.)
6.3 Secondary Circulations and Spin-Down In the idealized Ekman spiral solution of (6.19) the component of the wind multiplied by p gives the cross isobaric mass transport per unit area at any level in the boundary layer. Thus the net mass transport toward lower pressure in the Ekman layer for a column of unit width extending vertically through the entire layer is 13
M =
jopu d z = jop u p e - " " l D esin dz De De
De
7cZ
(6.21)
6.3
89
SECONDARY CIRCULATIONS A N D SPIN-DOWN
Neglecting local variations of density in the boundary layer the continuity equation may be written as (6.22) Integrating (6.22) through the depth of the boundary layer we find that
Here we have assumed that the ground is level so that M’ = 0 at z = 0. Substituting from (6.19) we can rewrite this expression for the vertical mass flux at the top of the Ekman layer as (6.23) Comparing (6.23) with (6.21) we see that the vertical flux at the top of the boundary layer is equal to the horizontal convergence of mass in the boundary layer which is simply -2M 2y in the above example. Noting that -?ug/ay = 5, is just the geostrophic vorticity in this case, we have after integrating (6.23) M’( De) =
ig(K/2f)”2
(6.24)
where we have neglected the variation of density with height in the boundary layer. Hence, we obtain the important result that the vertical velocity at the top of the boundary layer is proportional to the geostrophic vorticity. I n this way the effect of friction in the boundary layer is communicated directly to the interior fluid through a forced secondary circulation rather than indirectly by the slow process of viscous diffusion. For a typical synoptic scale system sec-I, and De 1 km, the vertical velocity sec-I, f with 5, given by (6.24) is of the order of a few tenths of a centimeter per second. An analogous secondary circulation is responsible for the decay of the circulationcreated when a c u p o f tea is stirred. Away from the boundary of the cup there is a n approximate balance between the radial pressure gradient and the centrifugal force of the spinning fluid. However, near the bottom of the cup viscosity slows the motion and the centrifugal force is not sufficient to balance the radial pressure gradient. (Note that the radial pressure gradient is independent of depth since tea is an incompressible fluid.) Therefore, radial inflow takes place near the bottom of the cup. Because of this inflow the tea leaves always are observed to cluster near the center at the bottom of the cup if the tea has been stirred. By continuity the radial inflow in the bottom boundary layer requires upward motion and a slow compensating outward radial flow throughout the remaining portion of the cup. This slow outward
-
-
-
90
6
THE PLANETARY BOUNDARY LAYER
flow approximately conserves angular momentum, and by replacing high angular momentum fluid by low angular momentum fluid serves to spin-down the vorticity in the cup far more rapidly than could mere molecular diffusion. This spin-down effect is also important in the atmosphere. It is most easily illustrated in the case of a barotropic atmosphere. We showed previously in Section 5.5 that for synoptic scale motions the vorticity equation could be written approximately as au
ac.
dW
(6.25)
where we have neglected icompared to f i n the divergence term. Neglecting the latitudinal variation offwe next evaluate the integral of (6.25) from the top of the boundary layer, z = De, to the tropopause, z = H .
(6.26) Assuming that w = 0 at z = H a n d that the vorticity may be approximated by its geostrophic value (which is independent of height) we obtain from (6.26)
-’
% - ( H - De) w( De) dt Substituting from (6.24) and noting that H % De we obtain a differential equation for the time dependence of [,:
(6.27) Equation (6.27) may be immediately integrated to give
5,
= i,(O)
exp{ - (fK,’2H2)’/’r}
(6.28)
where c,(O) is the value of the geostrophic vorticity at time t = 0. From (6.28) we see that T~ E H ( 2 f K ) ’ I 2 is the time which it takes a barotropic vortex of height H to spin-down to e-’ of its original value (this “e-folding” time scale is what is meant by the spin-down rime). Taking typical values of the parameters as follows: H = 10 km, f = lop4 sec-’, and K = lo5 cm2 sec-’, we find that T = z 4 days. Thus, for midlatitude synoptic scale disturbances in a barotropic atmosphere the characteristic spin-down time is a few days. This decay time scale should be compared to the time scale for ordinary viscous diffusion. I t can be shown that the time for eddy diffusion to penetrate a depth H is of the order Td H 2 ,K. For the above values of H and K , the diffusion time scale is thus T~ 100 days. Hence, the spin-down process is a far more effective mechanism for destroying vorticity in a rotating atmosphere than is ordinary diffusion.
--
6.3
SECONDARY CIRCULATIONS AND SPIN-DOWN
91
Physically the spin-down process in the atmospheric case is similar to that described for the teacup, except that in synoptic scale systems it is primarily the Coriolis force which balances the pressure gradient force away from the boundary, not the centrifugal force. Again the role of the secondary circulation driven by viscous forces in the boundary layer is to provide a slow radial flow in the interior which is superposed on the azimuthal circulation of the vortex. This secondary circulation is directed outward in a cyclone so that the horizontal area enclosed by any chain of fluid particles gradually increases. Since the circulation is conserved, the azimuthal velocity at any distance from the vortex center must decrease in time. Or, from another point of view, the Coriolis force for the outward-flowing fluid is directed clockwise, and this force thus exerts a torque opposite to the direction of the circulation of the vortex. In Fig. 6.4 a qualitative sketch of the streamlines of this secondary flow is shown. It should now be obvious exactly what is meant by the term secondary circulation. It is simply a circulation superposed on the primary circulation
Fig. 6.4 Streamlines of the secondary circulation forced by frictional convergence in the planetary boundary layer for a cyclonic vortex in a barotropic atmosphere.
(in this case the azimuthal circulation of the vortex) by the physical constraints of the system. In the case of the boundary layer it is viscosity which is responsible for the presence of the secondary circulation. However, other processes such as temperature advection and diabatic heating may also lead to secondary circulations as we shall see later. The above discussion has concerned only the neutrally stratified barotropic atmosphere. An analysis for the more realistic case of a stably stratified
92
6
THE PLANETARY BOUNDARY LAYER
baroclinic atmosphere would be much more complicated. However, qualitatively the effects of stratification may be easily understood. The buoyancy force (see Section 9.4) will act to suppress vertical motion since air lifted vertically in a stable environment will be denser than the environmental air. As a result the interior secondary circulation will be restricted in vertical extent as shown in Fig. 6.5. Most of the return flow will take place just above the
I
De
Fig. 6.5 Streamlines of the secondary circulation forced by frictional convergence in the planetary boundary layer for a cyclonic vortex in a stably stratified baroclinic atrnosphere.
boundary layer. This secondary flow will rather quickly spin-down the vorticity at the top of the Ekman layer without appreciably affecting the higher levels. When the geostrophic vorticity at the top of the boundary layer is reduced to zero, the “pumping” action of the Ekman layer is eliminated. The result is a baroclinic vortex with a vertical shear of the azimuthal velocity which is just strong enough to bring Cg to zero at the top of the boundary layer. This vertical shear of the geostrophic wind requires a radial temperature gradient to satisfy the thermal wind relationship. This radial temperature gradient is in fact produced during the spin-down phase by the adiabatic cooling of the air forced out of the Ekman layer. Thus, the secondary circulation in the baroclinic atmosphere serves two purposes: (1) it changes the azimuthal velocity field of the vortex through the action of the Coriolis force, and (2) it changes the temperature distribution so that a thermal wind balance is always maintained between the vertical shear of the azimuthal velocity and the radial temperature gradient.
SUGGESTED REFERENCES
93
Problems 1. Verify by direct substitution that the Ekman spiral expression (6.19) is indeed a solution of the boundary layer equations (6.15) and (6.16).
2. Derive the Ekman spiral solution for the more general case where the geostrophic wind has both x and y components, us and u g . 3. Letting the Coriolis parameter and density be constants, show that Equation (6.24) is correct for the more general Ekman spiral solution obtained in Problem 2. 4. For laminar flow in a rotating cylindrical vessel filled with water (molecular kinematic viscosity v = 0.01 cm2 sec-'), compute the depth of the Ekman layer and the spin-down time if the depth of the fluid is 30 cm and the rotation rate of the tank is ten revolutions per minute. 5. For the situation of Problem 4, how small would the radius of the tank have to be in order that the time scale for viscous diffusion from the sidewalls be comparable to the spin-down time? 6. In a homogeneous barotropic fluid show that the secondary radial circulation associated with a geostrophic vortex does not vary in strength with height. 7. Derive an expression for the wind driven surface Ekman layer in the ocean. Assume that the wind stress T~ is constant and directed along the x-axis. Continuity of stress at the air-sea interface ( z = 0) requires that the wind stress equals the water stress so that the boundary condition at the surface becomes
where K is the eddy viscosity in the ocean (assumed constant). As a lower boundary condition assume that u, L' + 0 as z + - co. If K = 10 cm2 sec-' what is the depth of the surface Ekman layer? 8. Show that the vertically integrated mass transport in the wind driven oceanic surface Ekman layer is directed 90" to the right of the surface wind stress in the Northern Hemisphere. Explain this result physically.
Suggested References Batchelor, An Introduction to Fluid Dynamics, has a useful discussion of the molecular basis of viscosity as well as a succinct derivation of the complete stress tensor. This book also has a n excellent treatment of viscous boundary layers including the Ekman layer.
94
6
THE PLANETARY BOUNDARY LAYER
Cole, Perturbation Methods it1 Applied Mathetnatics provides an excellent graduate level introduction to the mathematical foundations of boundary layer theory. Greenspan, The Theory of Rotating Fluids gives a unified account of the role of viscosity in rotating fluids at an advanced level. This is the only book available which focuses solely on phenomena which occur only in rotating fluids. Luniley and Panofsky, The Strrictirre of Atniospheric Turbulence covers both observational and theoretical aspects of the subject including the effects of thermal stratification in the boundary layer.