## 1. Introduction

Ekman (1905) presented the classic geophysical model of the planetary boundary layer. This model consists of a force balance between the Coriolis, frictional, and pressure gradient forces in which a constant eddy diffusivity is employed. The model predicts a hodograph that veers with height with a cross-isobaric flow toward lower pressure near the surface that asymptotes to a geostrophic flow aloft. The ageostrophic component of the flow can be convergent and lead to rising motion at the top of the boundary layer. This so-called Ekman pumping is the primary mechanism whereby the interior flow feels the influence of the boundary layer (Charney and Eliassen 1949).

The Ekman model neglects the effect of the acceleration of the flow. Benton et al. (1964) presented solutions for a class of steady-state models that included the convective accelerations. Their findings show that Ekman theory overestimates the boundary layer pumping for cyclonic flow. Young (1973) found that Ekman theory also overestimates the pumping for cyclonic flow if the local acceleration terms are included.

A variety of theories have been advanced to approximate the equations of motion by eliminating gravity waves in order to describe a balanced motion. The purpose of this paper is to compare the effect of the various approximations on the boundary layer pumping. This comparison is accomplished for two test cases that are solved both with and without approximation. Comparison of these solutions helps us understand the role of various processes in the boundary layer response to flow acceleration.

Section 2 describes the theories. Section 3 presents the first of two cases. In this linear case the flow transience is retained by allowing the geostrophic wind to increase exponentially with time. This choice is a crude model of the growth of a front or a baroclinically unstable flow. Section 4 presents the results for the second test case for which the flow is assumed to be steady but the nonlinear convective accelerations are retained. Section 5 summarizes the findings.

## 2. The models

*f*plane with a Cartesian coordinate system. The effects of friction are incorporated using a constant eddy viscosity in the vertical. The flow is driven by a zonal pressure gradient force. The associated geostrophic wind in the meridional (

*y*) direction is assumed to bewhere

*V*

_{0}, L, and

*Ïƒ*are constants. The subscript

*g*denotes a geostrophic quantity. Here a positive value of

*Ïƒ*models a growing geostrophic flow; a negative value models a decaying geostrophic flow. The atmosphere is horizontally unbounded. The linear dependence in the zonal (

*x*) direction in (2.1) enables separable solutions. We assume

*V*

_{0}is positive. Then a positive value of

*L*models a cyclonic shear; a negative value models an anticyclonic shear.

*x*) and vertical (

*z*) spatial coordinates and of time (

*t*). Then the horizontal momentum equations and the continuity equation becomewhere

*f*is the constant Coriolis parameter (assumed positive),

*Îº*is the constant vertical eddy diffusivity, and the material derivative isHenceforth we refer to the set (2.2)â€“(2.4) as the nongeostrophic (NG) equations. The energetics are summarized bywhereis the frictional force. Equation (2.6) states that changes in the kinetic energy of the wind arise from an imbalance between the work done by the frictional and by the pressure gradient forces.

*uF*

_{x}

*Ï…F*

_{y}

*fV*

_{g}

*u,*

*V*

_{g}. The energy equation takes the formHere the kinetic energy is approximated by that of the geostrophic wind and the work done by the frictional forces is only that associated with the geostrophic flow.

*U*

_{E},

*V*

_{E}) is the solution to the Ekman equations (2.8) and (2.9). The energetics of this set are given byThis equation is similar to (2.6) except that the kinetic energy and frictional working are evaluated using the Ekman wind. However, the last two terms in (2.18) are unphysical Coriolis contributions; they represent additional energy sources/sinks with no counterpart in the nongeostrophic energy equation (2.6).

## 3. Linear transient case

*Îº*= 0 in (2.2) and (2.3) with (3.1) yields the inviscid (denoted with the subscript

*I*) equations, which have the solutionwhereis a nondimensional measure of the ageostrophy introduced by the time dependence of the geostrophic wind (2.1). In writing (3.2) we have neglected any transient solution (e.g., an inertial oscillation) associated with unbalanced initial conditions. We do not consider here the general initial value problem but assume the time dependence (2.1).

*Î·*in the boundary layer. The velocity fields with a circumflex in (3.4) are nondimensional.

*Î·*= âˆž has the formwhereare the two roots of the equationwith negative real parts. HereThe constants

*A*

_{+}and

*A*

_{âˆ’}in (3.7) are determined from the no-slip boundary conditions at

*Î·*= 0. The result is

*r*

^{1/2}is the decay rate of the boundary layer flow while

*Î¸*/2 is the crossing angle (or angle that the surface flow makes to the left of the interior flow). From (3.10a)the decay rate increases for both accelerating or decelerating flow, implying a shallower boundary layer. From (3.10b) the crossing angle for the individual spiral is greater for decelerating flow (

*Ï„*< 0) than accelerating flow (

*Ï„*> 0). However, the crossing angle relative to the isobars for the total solution is typically larger for the accelerating case whose isallobaric component is down the pressure gradient.

*Ï„*

^{2})

^{âˆ’1}while the reduction in the decay rate of the spiral contributes a factor (1 +

*Ï„*

^{2})

^{âˆ’1/4}.

*Ï„*tends to 0. Then the inviscid QG solutions [cf. (3.2)] consist of only the geostrophic flow

*Ï…*=

*V*

_{g}. The QG approximation overestimates the meridional wind and fails to include the isallobaric zonal component (3.2a). The QG vertical motion field has no inviscid isallobaric component like (3.2c), and the pumping term (3.14) reduces to the well-known value, given in the present notation aswhich is independent of the ageostrophic parameter

*Ï„.*

*O*(

*Ï„*

^{2}) terms in (3.2) are dropped. In contrast with the QG approximation, the GM solution retains an isallobaric flow that is subject to the no-slip boundary condition. This feature is its major improvement over the QG result. The vertical motion field is the sum of (3.2c)[with the

*O*(

*Ï„*

^{2}) term neglected], and the approximation to (3.14) isThe transience decreases the boundary layer pumping in a growing (

*Ïƒ*> 0) disturbance but increases the pumping in a decaying (

*Ïƒ*< 0) disturbance. This behavior shows that friction reduces the isallobaric component of the flow in the boundary layer, thereby reducing the isallobaric convergence into a deepening low pressure system (Young 1973).

*Ï„.*The result (3.17) uses (2.16) and (2.17). If, following Tan and Wu (1993), we make the additional assumption that the frictional terms may be evaluated using the Ekman velocity field, then the boundary layer pumping is identical to the GM result (3.16).

Figure 1 compares the boundary layer pumping terms for the various approximations. All the solutions yield the Ekman result *W* = 1/2 for the steady-state case *Ïƒ*= 0. The slopes of the curves at *Ï„* = 0 are âˆ’0.750 for the nongeostrophic and Ekman momentum solutions, âˆ’0.625 for the semigeotriptic solution, and âˆ’0.500 for the geostrophic momentum solution. In general, the approximations tend to overestimate the boundary layer pumping. A major reason for this overestimation is that the approximate solutions overestimate the interior flow. Mathematically they neglect the factor (1 + *Ï„*^{2})^{âˆ’1} in (3.2). The NG solution (3.14) without this factor is plotted as the dashed-dotted curve labeled NGâ€² in Fig. 1. Only the SG approximation contains an estimate of the effect of the transience on the boundary layer decay rate and crossing angle. For a growing pressure gradient, *Ï„*> 0, the EM and SG approximations are the better of the group compared. For a decaying pressure gradient, the SG approximation is better than most, but the error increases for large negative *Ï„.*

## 4. Nonlinear steady-state case

*Ïƒ*= 0 and the material derivative may be replaced with the convective derivative

*x*and

*z*dependence and the nondimensional vertical structure functions of the ageostrophic flow components are denoted with a subscript,

*a.*Solving for the ageostrophic zonal wind yieldswhere the Rossby number, Ro =

*V*

_{0}/

*fL,*is a nondimensional measure of the ageostrophy of the flow. Note that section 3 considers the limit of vanishing Rossby number. Since

*w*

_{a}is related to

*u*

_{a}by(4.3) is a single equation for the vertical structure function of the ageostrophic zonal wind

*u*

_{a}.

*O*(Ro

^{2}) terms. The EM result (4.8) is found from the NG equation by dropping the

*O*(Ro

^{2}) terms and evaluating some of the functions with their Ekman values.

Figure 2 compares the boundary layer pumping predictions for the various approximations. The NG results are consistent with the analytic series solution of Benton et al. (1964). This agreement provides verification of the numerical method. All the solutions yield the Ekman result of 1/2 for the case of no shear (*L* = âˆž). Qualitatively each theory predicts an increase in the pumping for anticyclonic (Ro < 0) and a decrease for cyclonic shear (Ro > 0). [Note that the sign of the pumping, denoting either ascent or descent, is given by the coefficient in (4.2c).] The SG and EM theories track the behavior of the exact NG result slightly better than does the GM theory. The approximations are better for cyclonic than for anticyclonic shear.

*W*

_{QG}= 1/2, and shows no effects of the flow acceleration. The GM approximation only allows for the zonal advection of the geostrophic meridional wind. Wu and Blumen (1982) show that this meridional acceleration decreases (increases) the cross-isobaric flow and the depth of the boundary layer for cyclonic (anticyclonic) flow. The analytic solution of (4.7) implies a pumping of the formFigure 2 shows that this approximation overestimates the effect of the acceleration on the pumping. In contrast, the GM case includes the full meridional acceleration (i.e., both the zonal and vertical advection of the meridional wind) and produces closer agreement with the NG case. The difference between the SG and NG cases arise from the neglect of the zonal acceleration by the former. This neglect is significant only for anticyclonic flow. The EM approximation, which includes the acceleration of the Ekman wind, predicts a pumping comparable to the SG approximation.

## 5. Conclusions

We have tested three different approximate sets of planetary boundary equations that include the flow acceleration. Two test cases isolate the effect of the approximations in terms of the local and convective accelerations. Despite the special nature of these test cases, the results are believed to be representative. All the theories capture qualitatively the reduction of the boundary layer pumping for a growing geostrophic flow and for a cyclonic geostrophic flow.

All the results presented here have used the no-slip lower boundary condition. Use of the Taylor (1915)condition that the surface stress is parallel to the wind leads to similar qualitative behavior of the approximate theories as presented here. Quantitatively the boundary layer pumping is reduced with the Taylor condition.

The semigeotriptic theory of Cullen (1989) is the most accurate of the approximations over the range of ageostrophy tested (see Figs. 1 and 2). The Ekman momentum theory of Tan and Wu (1993) has similar accuracy except for the decelerating flow case (see Fig. 1for *Ï„* < 0) but its energy equation contains spurious Coriolis contributions. It should be noted that both the two- and three-dimensional versions of Cullenâ€™s theory produce the same results for the two-dimensional test cases investigated here. Further work should consider three-dimensional tests.

## Acknowledgments

Partial financial support was provided by the National Science Foundation (NSF) under NSF Grant ATM-9424462.

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