# Atmospheric Ekman Flows with Variable Eddy Viscosity

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## Abstract

We revisit the classical problem of the behaviour of the wind in the steady atmospheric Ekman layer. We show, for *general* variable eddy viscosities, that in the Northern Hemisphere the time-averaged ageostrophic wind profile *always* decays in magnitude and turns clockwise with increasing height. This general result is new; all previous work is based on a few explicit, special examples. As part of the development, we present two ways of formulating the problem, one of which is a novel approach (making use of a transformation to polar coordinates) that helps to explain the complex nature of these flows. The two formulations are supported by several examples that show, for instance, how the deflection angle can be other than the familiar \(45^\circ \). These results can be used as the basis for testing, and developing, various models for the height variation of eddy viscosity.

## Keywords

Ageostrophic wind spiral Ekman layer Variable eddy viscosity## 1 Introduction

It is well-established that the atmospheric boundary layer can be divided vertically into essentially three parts (Holton 2004; Marshall and Plumb 2016). The lowest layer—the laminar sublayer—has a thickness of only a few millimetres and is of no relevance for the transfer of wind energy since therein all physical processes are controlled by molecular motion. Above it is the Prandtl (surface) layer, whose vertical extent is about 20–100 m (depending on the thermal stratification of the air) and where turbulence is fully developed, but the influence of the Earth’s rotation may be ignored. The upper layer, which typically covers 90% of the atmospheric boundary layer, having a vertical extent ranging from about 20–100 m to heights in excess of 1000 m, is the Ekman layer, in which the airflow is driven by a three-way balance between frictional effects, pressure gradients and the influence of the Coriolis force. Above the Ekman layer, in the free atmosphere, the small-scale flow is more or less non-turbulent and geostrophic, being controlled by the balance between the pressure gradient and the Coriolis force (since curvature effects are negligible). However, the Earth’s curvature has to be accounted for in studies based on synoptic scales by also including the centrifugal force, thus obtaining a gradient wind balance rather than merely the geostrophic balance (Holton 2004).

*w*, of the velocity vector (

*u*,

*v*,

*w*) is typically much smaller than the zonal and meridional velocity components

*u*and

*v*, the characteristic numerical values for the magnitudes of the horizontal and vertical components of the wind velocity being 10 m s\(^{-1}\) and 0.1 m s\(^{-1}\), respectively (Zdunkowski and Bott 2003). Since in the Prandtl layer the airflow does not change direction with height, with a logarithmic profile commonly used to describe the vertical distribution of the horizontal mean wind speed under neutral conditions (Holton 2004), the main changes of wind direction in the boundary layer occur in the Ekman layer. Our aim here is to present a qualitative study of this motion. In non-equatorial regions of the Northern Hemisphere, the governing equations for mesoscale steady flow in the Ekman layer are conventionally taken to be

*u*,

*v*) is the horizontal

*z*-dependent mean wind velocity, with zonal component

*u*and meridional component

*v*, \(u_g\) and \(v_g\) are the corresponding constant geostrophic wind components, while \(K=K(z)\) is the eddy viscosity; here

*z*is the height above the Prandtl layer (Brown 1974; Pielke 2002; Holton 2004). We have written \(f=2{\Omega }\sin \theta \), the Coriolis parameter at the fixed latitude \(\theta \), in the Northern Hemisphere, where \({\Omega } \approx 7.29 \times 10^{-5}\) s\(^{-1}\) is the angular speed of rotation of the Earth and \(\theta \in (0,\uppi /2]\) is the angle of latitude in right-handed rotating spherical coordinates (with \(\theta =0\) corresponding to the Equator and \(\theta =\uppi /2\) to the North Pole). We invoke the traditional boundary conditions for the system (1)–(2) as

*K*, but field data show that this is an extreme simplification. Although a number of other significant assumptions are required to derive (1)–(2), since, e.g., stratification and spherical geometry effects are ignored and the geostrophic velocity is assumed to be height-independent (i.e. a barotropic atmosphere), the system (1)–(2) is commonly used in meteorology. In this study, we limit ourselves to steady motions, the aim being to emphasize the role played by the vertical variation of the eddy viscosity. There is little doubt that temporal variations need to be included at some stage, but we regard this as a future extension of the current work, although any time dependence on

*K*could be added as a parametric dependence if we use (1) and (2) as the relevant model equations. Note also that the time variation in the atmosphere is usually slow enough (the primary changes over land being driven by the diurnal cycle) for the main characteristics of the steady-state setting to prevail (Mason and Thomson 2015). A discussion of time-dependent Ekman layers can be found in Lewis and Belcher (2004) and Momen and Bou-Zeid (2017).

*K*typically varies rapidly with height near the base of the Ekman layer (Holton 2004). We observe that there appears to be no generally accepted view on the behaviour of this variation, other than the fact that

*K*has to be determined by a combination of dimensional considerations and empirical measurements, a process beset by the technical challenges inherent in parameter estimations in meteorology (Mason and Thomson 2015). The available explicit solutions for a non-constant eddy viscosity are very scarce, being apparently restricted to linear and exponentially decaying profiles only (Madsen 1977; Miles 1994) or quadratic and cubic polynomials (Nieuwstadt 1983; Giometto et. al. 2017), so that for other types of non-constant eddy viscosity we have to rely on case-by-case approximations and numerical simulations. Several types of eddy-viscosity profiles have been determined as being relevant in specific physical contexts and these present quite different monotonicity properties: some increase with height, others decrease, and a few are not even monotonic (see Table 1 and Fig. 1 for examples of eddy viscosities suggested by field data, discussed in Berger and Grisogno (1998), Grisogno (1995), Tan (2001) and Parmhed et al. (2005)). We also note that in some models the eddy viscosity

*K*(

*z*) is taken to be constant above a certain height, with a non-constant polynomial profile below: linear in Estoque (1963), quadratic in Yang (1957), and cubic in O’Brien (1970). In addition, numerical simulations for profiles obtained by multiplying together a polynomial and a height-dependent function are discussed in Grisogno (2011), Large et al. (1994), and Marlatt et al. (2012), while Chai et al. (2004) provide field evidence for a function

*K*(

*z*) with multiple local extrema and inflexion points in the lower part of the Ekman layer.

Some physically relevant height-dependent eddy viscosities over the first 1500 m in the Ekman layer (above this they remain practically constant)

Height-dependent formulae |
| \(k_0\) (m\(^2\) s\(^{-1}\)) | \(\gamma \) (m\(^{-1}\)) | \(\gamma _1\) (m\(^{-1}\)) |
---|---|---|---|---|

\(K_2(z)=k + (k_0-k)\,\mathrm{e}^{-\gamma z}\) | 5.5 | 0.7 | 0.00313 | |

\(K_3(z)=k + (k_0-k)\,\mathrm{e}^{-\gamma z}\) | 0.7 | 1.4 | 0.00389 | |

\(K_4(z)=(1 + \gamma _1 z)k_0\,\mathrm{e}^{-\gamma z}\) | 0.405 | 0.00223 | 0.06 |

The previous considerations motivate the study of the governing equations (1)–(4) for an arbitrary height-dependent eddy viscosity *K*. We aim to present some general observations about this problem, with examples, which will help to clarify the effects of various choices for the model taken for *K*(*z*). This will also provide a framework for testing the relevance and usefulness of different types of model for *K*(*z*). Despite the significant variations in the different models used for the eddy viscosity, all the existing Ekman-type solutions retain two of the fundamental properties of the classical Ekman flow: the horizontal ageostrophic velocity spirals to the right (in the Northern Hemisphere) and the ageostropic speed decreases with height. This raises two important questions: (i) are these two features universally valid? (ii) what effect does the choice of *K*(*z*) have on the details of the Ekman flow? We answer the first question in the affirmative by pursuing an in-depth qualitative study of the governing equations (1)–(4) and provide information that helps to clarify the answer to the second question. Our aim is, primarily, to show that the features covered by question (i) hold whenever \(K: [0,\infty ) \rightarrow [k_-,k_+]\) is a continuous and bounded function with \(K(z) \rightarrow k^*\) as \(z \rightarrow \infty \), at a rate faster than quadratic (see Eq. 17 below), for some \(k^*\in [k_-,k_+]\); here \(k_\pm ,\,k^*>0\) are positive constants. This prescription on *K*(*z*) admits profiles that are non-monotonic (see Sect. 4). The system (1)–(2) is linear, yet we have made the remarkable discovery that the details for the Ekman flow are best developed by transforming it, using polar coordinates, into a nonlinear system. This aspect of the problem underlines the technically intricate nature of the dynamics of Ekman-type solutions with height-dependent eddy viscosities. We also present some explicit examples of particular choices for the eddy viscosity, *K*(*z*), which demonstrate the effect of a viscosity that initially increases, or decreases, above the bottom of the Ekman layer.

## 2 Existence of Ekman-Type Solutions

*s*variable (here and throughout the paper). The boundary conditions (3)–(4) are transformed into the equivalent form

*every*

*K*(

*z*) (as specified). To establish their existence one can proceed as follows. Setting

*K*(

*z*), which is otherwise continuous and bounded.

## 3 Qualitative Features of the Ekman-Type Solutions

The main issue of interest is to elucidate the behaviour of the physically relevant solution \({\Psi }_+\), whose existence was established in Sect. 2. We prove two general results, namely that the ageostrophic horizontal vector (*U*, *V*) rotates clockwise (see Sect. 3.2) and its speed \(\sqrt{U^2+V^2}\) is strictly decreasing with increasing height (see Sect. 3.3) for every \(K(z)>0\) satisfying (17).

### 3.1 Polar Coordinates

*M*is differentiable whenever \(M \ne 0\) and

### 3.2 Monotonicity of the Ageostrophic Speed

*a*,

*b*) with \(M_+(a)=M_+(b)=0\) and \(M_+(s)>0\) for all \(s \in (a,b)\). This is impossible since \(M_+\) is differentiable on (

*a*,

*b*) and convex, given that (21) yields

### 3.3 Rotation of the Horizontal Ageostrophic Flow

We now show that if \({\Psi }_+(s)=M_+(s)\,\mathrm{e}^{i\tau _+(s)}\) with \(\tau _+ \in (-\uppi ,\uppi ]\), then \(\tau _+'(s)<0\) for \(s \ge 0\), that is, the horizontal ageostrophic vector (*U*, *V*) rotates clockwise with increasing height, for *any* choice of *K*(*z*) satisfying (17).

Let us first prove that \(\tau _+'\) has at most one zero in \([0,\infty )\). Indeed, if \(b>a \ge 0\) were points with \(\tau _+'(a)=\tau _+'(b)=0\), then integration of (21b) yields \(0=\int _a^b \alpha (s)d_+^2(s)\,\mathrm{d}s\), which is impossible since \(\alpha >0\) and \(M_+>0\) on \([0,\infty )\), as shown in Sect. 3.2.

### 3.4 The Top of the Ekman Layer

### 3.5 The Deflection Angle

*s*. We note in passing that the direction of the flow near the bottom of the layer is obtained from

*M*(

*s*) and \(\tau (s)\) are both strictly decreasing, with \(M(0)=M_g\); thus we see that

## 4 Examples

We have seen that the equivalent formulations, (7) and (21), of the governing equations (1)–(2), for flows in the atmospheric boundary layer, enable us to produce some general results. But, quite significantly, they also provide a basis for testing various choices of variable eddy viscosity and, furthermore, they lay the foundations for the construction of explicit solutions. Indeed, either of the approaches that we have presented provides a systematic method for extending the choice of the eddy-viscosity function, opening the way to new theoretical observations on the nature of Ekman flows. We now illustrate these ideas by presenting a few examples.

### 4.1 Bessel-Type Solutions

*c*is an arbitrary complex constant. This example is simply an adaptation, to the atmosphere, of the solution found by Madsen (1977) in the oceanographic context.

It is useful to note that this representation, (31), of a solution for variable eddy viscosity contains valuable information. So, for example, for \(2\sqrt{af}/|b| \approx 1.69\), the deflection of the flow near the bottom of the layer, relative to the direction of the geostrophic flow, can be as large as about \(55^\circ \); the classical \(45^\circ \) is approached for larger values of this parameter. Correspondingly (in the Northern Hemisphere) the wind rotates to the right and its speed relative to the ambient state tends to zero, for increasing height through the layer. This property of solution (31) is shown in Fig. 3, demonstrating that the familiar Ekman structure is recovered; specifically, we plot the ratio of the wind speed at any height to that at the bottom of the Ekman layer, and the deflection angle relative to the wind direction at the bottom, both against a normalized (non-dimensional) depth in the form *az* / *b*. More pronounced differences between the classical results and these Bessel-based ones will become evident when both are plotted against *z* (which is non-dimensional) for various *a*, *b* and *f* (which do not need to be specified for the plots shown here). The foregoing description of the flow properties is obtained simply by computing the real and imaginary parts of \({ \Phi }\) and its derivative.

### 4.2 Extended Bessel-Type Solution

*a*and \(\lambda \), and \(b \in {{\mathbb {R}}}\). We see that

*K*(

*z*) as

*K*, we may have a deflection of the ageostropic wind direction in excess of \(45^\circ \) to the right (Northern Hemisphere), the maximum possible angle varying with \(\nu \). On the other hand, for

*K*increasing with height (but bounded), the deflection angle can be less than \(45^\circ \): as \(|\nu |\) increases, so the angle decreases e.g., using the Maple mathematical package, at \(|\nu |=10\) the deflection angle is about \(30^\circ \). In all cases, as \(|b|/\lambda \) increases, so the deflection angle approaches the classical result: \(45^\circ \), this limit corresponding to \(\lambda \rightarrow 0\), which recovers constant

*K*.

### 4.3 Asymptotic Solution Near the Bottom of the Layer

*K*(

*z*) is otherwise arbitrary; we construct the associated asymptotic solution valid as \(z \rightarrow 0\) for \(a>0\) and \(b \in {\mathbb R}\), with \(a+bz_0>0\). This choice of eddy viscosity enables us to identify the behaviour at the bottom of the layer and so determine its effect on the flow properties. Thus we obtain

*c*an arbitrary complex constant) because, although we want the solution only as \(z \rightarrow 0\), this specification enables us to impose the condition that the solution of interest must be such that the velocity field can approach geostrophic balance away from \(z = 0\) (i.e. \(|{\Psi }(z)|\) decreases as

*z*increases) . We find that the required solution corresponds to the choice

*K*decreasing away from the bottom (\(b<0\)), and from zero to 45\(^\circ \) for

*K*increasing. (Note that \(\gamma \rightarrow \infty \) corresponds to \(|b|\rightarrow 0\), the constant-viscosity case.)

This asymptotic property extends the results obtained in the two preceding examples (and also that given in the next example) by emphasising the behaviour of the solution at the bottom of the layer. The apparent inconsistency with Example 4.1 arises because there we used *K* linear in *z* throughout a bounded, and non-zero, domain; here we used only the value of *K*, and its derivative, on \(z = 0\). These calculations, in isolation, suggest that, for suitable choices of *K*(*z*), it is possible to obtain solutions for which the angle of deflection of the wind relative to that of the applied stress (required to maintain the no-slip condition) can take any value (to the right in the Northern Hemisphere) between zero (i.e. parallel) and \(\uppi /2\). However, we should be more circumspect. Combining these asymptotic results (appropriate on \(z = 0\)) with those based on more complete prescriptions for *K*(*z*) (in Example 4.1, Example 4.2 and also in Example 4.4 below) indicate that the particular choice of *K*(*z*) throughout the Ekman layer does affect the deflection angle. Indeed, we conjecture that the form of *K*(*z*) above \(z = 0\) restricts the deflection angle from the extremes predicted from the behaviour on \(z = 0\) alone.

### 4.4 Solutions in Terms of Hypergeometric Functions

*a*,

*b*and

*c*are positive constants, with \(k^*=a(\mathrm{e}^{-bz_0}-c)>0\); this exponential form corresponds to that used by Miles (1994). We introduce the change of variable defined by

*c*is an arbitrary (complex) constant and \({\mathfrak {F}}\) is Gauss’ hypergeometric function (Abramowitz and Stegun 1964).

The details of the solution follow similar lines to those noted in the Examples 4.1 and 4.2, based on the Bessel function. So here, for example, we find that for \(b^{-1}\sqrt{f/(2ac)} \approx 0.774\), the deflection angle is about 65\(^\circ \) (and approaches the classical 45\(^\circ \) as this parameter is increased), and the rotation and approach to ambient conditions at higher altitude are also repeated.

### 4.5 Solutions Related to Fuchsian Equations

### 4.6 Another Method to Generate Explicit Solutions

*s*. This result can be used to find solutions by first choosing

*M*(

*s*), so that \(\alpha (s)\) is determined directly, and then integrating to obtain \(\tau (s)\) from (21) i.e. from \(\tau '=-\sqrt{M''/M}\). We are concerned only with those solutions that steadily approach the ambient state (the geostrophic balance) at higher altitudes. The most straightforward way to accomplish this is to restrict ourselves to bounded intervals, and then it suffices to select

*M*(

*s*) as any convex, strictly decreasing function, chosen to ensure that (37) produces a positive expression for \(\alpha \). We present two solutions that show how this method can be employed.

#### 4.6.1 Exponential Decay of the Ageostrophic Velocity with Height

*z*, is obtained from the definitions (6) and (10),

*z*(

*s*) by quadrature. This slight complication here is avoided in the next example, but at the expense of a very simple variation of eddy viscosity with height and an associated simple flow structure.

#### 4.6.2 Slow Decay of the Ageostrophic Velocity with Height

*M*(

*s*), which then leads to the determination of \(\alpha (s)\) and \(\tau (s)\), and those presented earlier that led directly to the construction of \({\Psi }=U+\hbox {i}V\), show the potential of these ideas. There are clearly many opportunities for future investigation as other, possibly more sophisticated, choices are made.

## 5 Discussion

This general approach to the classical problem of the atmospheric Ekman flow has significantly extended and enhanced what is currently known. Most importantly, we have shown, for any eddy viscosity that is bounded and tends to a constant finite value at high altitude, that the decay and spiralling of the flow upwards is an enduring property. The spiral is clockwise in the Northern Hemisphere; the corresponding result is anticlockwise in the Southern Hemisphere, although we have not discussed this case explicitly here: it simply requires a change of sign in places. All previous work on this problem, starting from the seminal model proposed by Ekman, has been driven by a few explicit examples; these have provided the basis for conjecture, but we have proved that the familiar flow structure is *always* present. The overall picture of the classical Ekman spiral is unaltered by the details of the varying eddy viscosity, which is slightly surprising. We might have expected that viscosity profiles with a number of local maxima and minima, for example, would produce a significant distortion of the familiar structure of the flow; this is not the case.

The ideas and techniques that have been described here can be used to investigate these flows further. So, for example, the procedures that we have developed enable models for the variation of eddy viscosity with height to be checked, tested or invented. This is essentially because explicit solutions can be constructed in one of two ways: either directly from the classical formulation (based on the Ekman model in the *f*-plane approximation), written in complex form, or from the reformulation of this method using a polar-coordinate transformation. This latter approach helps to explain why the underlying structure of this problem, with variable viscosity, is so involved. We have provided a number of examples in each case, a small number of which are new. One particular result that we have obtained, both from explicit, exact solutions, and from an asymptotic analysis valid near the bottom of the Ekman layer, is that angles of deflection other than the usual 45\(^\circ \) are possible, and we are able to give some guidance as to whether the deflection is initially above or below this value. And all that we have presented is only the start of a detailed investigation. A number of different avenues can be explored because it is clear that we have barely scratched the surface; so, for example, we might investigate how stratification and baroclinicity modulate the solutions described here.

One obvious route to follow is the hunt for other exact solutions, most easily accomplished by choosing suitable ageostrophic-wind-speed functions *M*(*s*); this leads directly to the corresponding variable eddy viscosity and the fully detailed flow structure. A case is made in some of the literature, and we have alluded to this earlier, that time dependence needs further investigation, although we note that we can always accommodate significant time variation simply by superimposing a parametric time dependence on the solutions described here. A more fundamental issue is the use of the classical Ekman equations; these do not capture the effects of the Earth’s curvature, so a more comprehensive study, along the lines developed here, but based on the full Navier–Stokes equation, in rotating, spherical coordinates, is worth pursuing; a preliminary discussion of Ekman-type flows in a rotating, spherical coordinate system, can be found in Constantin and Johnson (2018). Our work, we submit, has enhanced our understanding of the classical Ekman flow, in the presence of variable eddy viscosity, to the extent that a number of new results and methods have been introduced. Finally, we note that all that we have described here is equally applicable to the corresponding oceanic flow model in the *f*-plane approximation, for wind-driven currents in the upper layer of the oceans; see Vallis (2006) and Marshall and Plumb (2016) for a theoretical discussion, and Röhrs and Christensen (2015) for recent field data.

## Notes

### Acknowledgements

Open access funding provided by University of Vienna. This research was supported by the WWTF Research Grant MA16-009. The authors are grateful for the suggestions and comments made by the referees.

## References

- Abramowitz M, Stegun IA (1964) Handbook of mathematical functions with formulas, graphs, and mathematical tables. National Bureau of Standards, applied mathematics series, vol 55. U.S. Government Printing Office, WashingtonGoogle Scholar
- Berger BW, Grisogno B (1998) The baroclinic, variable eddy viscosity layer. Boundary-Layer Meteorol 87:363–380CrossRefGoogle Scholar
- Brown RA (1974) Analytical methods in planetary boundary-layer modelling. Wiley, New YorkGoogle Scholar
- Chai T, Lin CL, Newsom RK (2004) Retrieval of microscale flow structures from high-resolution Doppler lidar data using an adjoint model. J Atmos Sci 61:1500–1520CrossRefGoogle Scholar
- Constantin A (2011) Nonlinear water waves with applications to wave-current interactions and tsunami. SIAM, PhiladephiaCrossRefGoogle Scholar
- Constantin A, Johnson RS (2018) Steady large-scale ocean flows in spherical coordinates. Oceanography 31:42–50CrossRefGoogle Scholar
- Deift P, Trubowitz E (1979) Inverse scattering on the line. Commun Pure Appl Math 32:121–251CrossRefGoogle Scholar
- Drazin PG, Johnson RS (1989) Solitons: an introduction. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Etling D (2011) Theoretische Meteorologie. Springer, BerlinGoogle Scholar
- Estoque MA (1963) A numerical model of the atmospheric boundary layer. J Geophys Res 68:1103–1113CrossRefGoogle Scholar
- Giometto MG, Grandi R, Fang J, Monkewitz PA, Parlange MB (2005) Katabatic flow: a closed-form solution with spatially-varying eddy diffusivities. Boundary-Layer Meteorol 162:307–317CrossRefGoogle Scholar
- Grachev AA, Fairall CW, Persson POG, Andreas EL, Guest PS (2005) Stable boundary-layer scaling regimes: the SHEBA data. Boundary-Layer Meteorol 116:201–235CrossRefGoogle Scholar
- Grisogno B (1995) A generalized Ekman layer profile with gradually varying eddy diffusivities. Q J R Meteorol Soc 121:445–453CrossRefGoogle Scholar
- Grisogno B (2011) The angle of the near-surface wind-turning in weakly stable boundary layers. Q J R Meteorol Soc 137:700–708CrossRefGoogle Scholar
- Holton JR (2004) An introduction to dynamic meteorology. Academic Press, New YorkGoogle Scholar
- Large WG, McWilliams JC, Doney SC (1994) An improved Ekman layer approximation for smooth eddy diffusivity profiles. Rev Geophys 32:363–403CrossRefGoogle Scholar
- Lewis DM, Belcher SE (2004) Time-dependent, coupled, Ekman boundary layer solutions incorporating Stokes drift. Dyn Atmos Oceans 37:313–351CrossRefGoogle Scholar
- Madsen OS (1977) A realistic model of the wind-induced Ekman boundary layer. J Phys Oceanogr 7:248–255CrossRefGoogle Scholar
- Maier RS (2007) The 192 solutions of the Heun equation. Math Comput 76:811–843CrossRefGoogle Scholar
- Mak M (2011) Atmospheric dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Marlatt S, Waggy S, Biringen S (2012) Direct numerical simulation of the turbulent Ekman layer: evaluation of closure models. J Atmos Sci 69:1106–1117CrossRefGoogle Scholar
- Marshall J, Plumb RA (2016) Atmosphere, ocean and climate dynamics: an introductory text. Academic Press, New YorkGoogle Scholar
- Mason PJ, Thomson DJ (2015) Boundary layer (atmospheric) and pollution. In: North GR, Pyle J, Zhang F (eds) Encyclopedia of atmospheric sciences. Elsevier, New York, pp 220–226CrossRefGoogle Scholar
- Miles J (1994) Analytical solutions for the Ekman layer. Boundary-Layer Meteorol 67:1–10CrossRefGoogle Scholar
- Momen M, Bou-Zeid E (2017) Analytical reduced models for the non-stationary diabatic atmospheric boundary layer. Boundary-Layer Meteorol 164:383–399CrossRefGoogle Scholar
- Nieuwstadt FTM (1983) On the solution of the stationary, baroclinic Ekman-layer equations with a finite boundary-layer height. Boundary-Layer Meteorol 26:377–390CrossRefGoogle Scholar
- O’Brien JJ (1977) A realistic model of the wind-induced Ekman boundary layer. J Atmos Sci 27:1213–1215CrossRefGoogle Scholar
- Parmhed O, Kos I, Grisogono B (2005) An improved Ekman layer approximation for smooth eddy diffusivity profiles. Boundary-Layer Meteorol 115:399–407CrossRefGoogle Scholar
- Peña A, Floors R, Sathe A, Gryning SE, Wagner R, Courtney MS, Larsén Hahmann AN, Hasager CB (2016) Ten years of boundary-layer and wind-power meteorology at Høvsøre, Denmark. Boundary-Layer Meteorol 158:1–26CrossRefGoogle Scholar
- Pielke RA (2002) Mesoscale meteorological modeling. Academic Press, New YorkGoogle Scholar
- Röhrs R, Christensen KH (2015) Drift in the uppermost part of the ocean. Geophys Res Lett 42:10349–10356CrossRefGoogle Scholar
- Rysman JP, Lahellec A, Vignon E, Genthon C, Verrier S (2016) Characterization of atmospheric Ekman spirals at Dome C, Antarctica. Boundary-Layer Meteorol 160:363–373CrossRefGoogle Scholar
- Tan ZM (2001) An approximate analytical solution for the baroclinic and variable eddy diffusivity semi-geostrophic Ekman viscosity layer. Boundary-Layer Meteorol 98:361–385CrossRefGoogle Scholar
- Yang T (1957) Velocity distribution of lower wind in Beijing. Acta Meteorol Sin 28:185–197Google Scholar
- Vallis GK (2006) Atmosphere and ocean fluid dynamics. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Zdunkowski W, Bott A (2003) Dynamics of the atmosphere. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Zhang G, Xu X, Wang J (2003) A dynamic study of Ekman characteristics by using 1998 SCSMEX and TIPEX boundary layer data. Adv Atmos Sci 20:349–356CrossRefGoogle Scholar

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