Abstract
In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with classical boundary conditions and the eddy viscosity is an arbitrary height-dependent function with a finite limit value. We present existence and uniqueness and smooth results to justify computing first order approximation of solutions. Using a different argument that in previous works, we construct the Green’s function to derive the solution by a perturbation approach.
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1 Introduction
The atmospheric boundary layer has three parts [1, 2], i.e., the lamina sublayer, surface (Prandtl) layer and the Ekman layer. The Ekman layer covers 90% of the atmospheric boundary layer and it is driven by a three-way balance among frictional effects, pressure gradient and the influence of the coriolis force [1, 3, 4]. In general, textbooks on geophysical fluid dynamics and dynamic meteorology contain the derivation for the Ekman layer with a constant eddy viscosity k [5, 6]. However k usually varies with the height. As a result it is necessary to find the explicit solution of Ekman flows with a non-constant eddy viscosity, but unfortunately explicit solutions are scare in the literature and are restricted to linear [7, 8] or quadratic and cubic poly-nominal [9]. For arbitrary k(z) or k(z, t), we often have to rely on approximation and numerical simulation; the authors in [10,11,12,13] apply the Wentzel, Kramers and Brillouin’s method to get the approximation solution.
Constantin and Johnson [14] studied Ekman flows with variable eddy viscosity k(z) and they derived the explicit solution through an unclosed form and verified the existence of the solution by transformation and the iterative technique. The authors in [15] studied the horizontal wind drift currents which spiral and decay with depth and they obtained the solution by a perturbation approach. In this paper, we adopt the linearization approach to establish existence and uniqueness results and we consider the smoothness of solutions, which justify computing first order approximation of solutions. Also, based on [15], we regard the eddy viscosity k(z) as perturbations of the asymptotic reference value and we perform the variable change and get a linear, non-homogeneous second order differential equation and then we show the existence of a solution using the Green’s function.
2 Model description
The Ekman layer is governed by the following equations, see [1, 2]
where u, v and w are the components of the wind in the x, y and z directions respectively, P is the atmospheric pressure, \(\rho \) is the reference density, \(f=2\Omega sin\theta \) is the Coriolis parameter at the fixed latitude \(\theta \), \(\Omega \approx 7.29\times 10^{-5}\) is the angular speed of the roattion of the earth in the northern Hemisphere, and \(\theta \in (0, \pi /2]\) is the angle of latitude in right-handed rotating spherical cooridates, t is time and k is the eddy diffusivity for momentum.
Assuming a steady state we get \(\frac{Du}{Dt}=0, \frac{Dv}{Dt}=0.\) From the geostrophic balance, we have
From the Flux-Gradient theory, we get
where k is the eddy viscosity coefficient. Then we obtain
where \(u_{g}\) and \(v_{g}\) are the corresponding constant geostrophic wind components. We use the traditional boundary conditions for (1) as
Let \(\Phi =(u-u_{g})+i(v-v_{g})\), and from (1), we will get
If \(k=\) constant, we have
where \(\gamma =\sqrt{\frac{f}{2k}}.\) However, the eddy viscosity k always varies with height [10], so (4) will become
Here, we regard the physically relevant eddy viscosity k(z) as perturbations of the asymptotic reference value \(k_{*}=\lim \limits _{z\rightarrow \infty }k(z)>0,\) so
where \(\varepsilon \ll 1\), and the asymptotic rate of convergence is faster than quadratic [14], that is, there exist constants \(a, b, c>0\) such that
3 Main results
3.1 Existence and uniqueness
Let
so (1) is replaced with
for \(\hat{k}(z)=\frac{1}{k(z)}\). The affine system (8) has a unique equilibrium
with the linearization
When \(\varepsilon =0\), \(k(z)=k_*\) is a constant function, and then the matrix
has eigenvalues
with corresponding eigenvectors
Thus the linear system (10) has a stable space
Introduce a linear subspace
and condition (2) means \((u(0),v(0),x(0),y(0))\in W\). Note
First we study a general affine system
for \(A\in L(\mathbb {R}^n)\), \(B:(-\delta ,\delta )\times \mathbb {R}_+\rightarrow L(\mathbb {R}^n)\), \(\delta >0\) and \(p\in \mathbb {R}^n\), where \(\mathbb {R}_+=[0,\infty )\) and \(L(\mathbb {R}^n)\) is the space of linear maps on \(\mathbb {R}^n\). We suppose
-
(A1)
A is hyperbolic, i.e., the real parts of eigenvalues of A are nonzero.
-
(A2)
\(B(\varepsilon ,z)\) has continuous partial derivatives \(\partial _{\varepsilon }^iB(\varepsilon ,z)\) with
$$\begin{aligned} \sup _{(\varepsilon ,z)\in (-\delta ,\delta )\times \mathbb {R}_+}\Vert \partial _{\varepsilon }^iB(\varepsilon ,z)\Vert <\infty \end{aligned}$$for \(i=0,1,\cdots ,r\).
Recall that (A1) is equivalent to the existence of constants \(K>0\), \(\alpha >0\) and a splitting \(\mathbb {R}^n=X_s\oplus X_u\) such that \(A(X_s)=X_s\), \(A(X_u)=X_u\) along with \(\Vert e^{A_sz}\Vert \le Ke^{-\alpha z}\) and \(\Vert e^{-A_uz}\Vert \le Ke^{-\alpha z}\) for \(z\in \mathbb {R}_+\), where \(A_s=A/X_s\) and \(A_u=A/X_u\). We consider a projection \(P:\mathbb {R}^n\rightarrow \mathbb {R}^n\) with \(\ker P=X_u\) and \({\text {im}\,}P=X_s\).
Let \(W\subset \mathbb {R}^n\) be a linear subspace such that \(\mathbb {R}^n=X_s\oplus W\), i.e., W is transversal to \(X_s\). We look for solutions of (12) satisfying
Let \(Q:\mathbb {R}^n\rightarrow \mathbb {R}^n\) be a projection with \(\ker Q=W\) and \({\text {im}\,}Q=X_s\).
Proposition 3.1
Assume (A1) and (A2). Set
For any \(\varepsilon \in (-\delta ,\delta )\) satisfying,
there is a unique solution \(q(\varepsilon ,z)\) of (12) satisfying (13). Moreover, \(q(\varepsilon ,z)\) is \(C^r\)-smooth in \(\varepsilon \).
Proof
Following [16], any bounded solution of (12) is given by
for \(q_s\in X_s\). We consider a Banach space \(C_b(\mathbb {R}_+,\mathbb {R}^n)\) of all bounded and continuous functions \(q:\mathbb {R}_+\rightarrow \mathbb {R}^n\) endowed with a norm \(\Vert q\Vert =\sup _{z\in \mathbb {R}_+}|q(z)|\). A solution of (15) is a fixed point of the map
Now H is linear in q and \(C^r\)-smooth in \(\varepsilon \). For any \(q_1,q_2\in C_b(\mathbb {R}_+,\mathbb {R}^n)\), we have
Condition (14) guarantees the contraction of \(H(\varepsilon ,q_s,q)\) in q, so the Banach fixed point theorem ensures a unique solution \(\hat{q}(\varepsilon ,q_s,z)\) of (15). Next, for any \(q_{s1},q_{s2}\in X_s\), we have
Consequently, \(\hat{q}(\varepsilon ,q_s,z)\) is globally Lipschitz in \(q_s\) with a constant
and \(C^r\)-smooth in \(\varepsilon \). To finish the proof, we need to solve
We set
From (15), for any \(q_{s1},q_{s2}\in X_s\), we have
Clearly (16) is equivalent to
Assumption (14) ensures that the map
is a contraction. Thus (17) has a unique solutions \(q_s(\varepsilon )\). Next, by using [16, Proposition 1, p. 34], we see that (12) is dichotomous for any \(\varepsilon \) satisfying (14). Consequently, \(q(\varepsilon ,z)=\hat{q}(\varepsilon ,q_s(\varepsilon ,z)\) is the desired unique solution of (12) satisfying (13). The proof is finished. \(\square \)
Now we apply Proposition 3.1 to (8). We already verified (A1). To show (A2), we note that
for
Thus supposing that \(k_1(z)\ge 0\) is continuous with \(\sup _{z\in \mathbb {R}_+}|k_1(z)|<\infty \), condition (A2) holds. Consequently, we have the following result.
Theorem 3.2
Assume \(k_1(z)\ge 0\) is continuous with \(\sup _{z\in \mathbb {R}_+}|k_1(z)|<\infty \). Then for any \(\varepsilon \) small, there is a unique solution of (8) satisfying (13) with p and W given by (9) and (11), respectively. This solution is \(C^\infty \) smooth in \(\varepsilon \).
Since k(z) is continuous, here we have a solution u(z), v(z) of (12) such that u(z), v(z), \(\frac{\partial u(z)}{\partial z}\), \(\frac{\partial v(z)}{\partial z}\), \(\frac{\partial }{\partial z}(k(z)\frac{\partial u(z)}{\partial z})\) and \(\frac{\partial }{\partial z}(k(z)\frac{\partial u(z)}{\partial z})\) exist and are continuous on \(\mathbb {R}_+\).
3.2 First order approximation
We use Theorem 3.2 to consider first order approximations of solutions. Consider the following second order boundary-value problem
Lemma 3.3
If \(w(\cdot ):=G(\cdot ,t),t\in (0,\infty )\) (called the Green’s function) solves (18), then
where
and
Proof
Note \(x_{1}\) and \(x_{2}\) are two linearly independent solutions of \(w''(s)-\frac{i\cdot f}{k_{*}}w(s)=0\). Suppose
is a solution of (18). Then using the boundary conditions one obtains
and
Using the continuity of w at \(s=t\), we find
From the above we get the value of \(a_{1},~a_{2},~b_{1}\) and \(b_{2}\), and therefore we obtain
The proof is complete. \(\square \)
Let
Then
Let
where
Set
where \(\Psi _{0}(s)\) is the classical Ekman solution
Theorem 3.4
The function defined by (22) is the solution of (6) with (2) and (3) if
Proof
Suppose for simplicity that k(z) is \(C^2\)-smooth. From the definition of \(\Psi (s)\), we have
and
and the corresponding boundary conditions are
Inserting (22) and (7) into (26) one obtains
From the definition of \(\Psi _{0}(s)\), we have
Dividing by \(\epsilon \) and letting \(\epsilon \rightarrow 0\), this yields a second order differential equation for \(\varphi \), namely
The boundary conditions are transformed to
From Lemma 3.3, we know that \(G(\cdot ,\cdot )\) is the Green’s function of (27) and (28), and the solution can be expressed as the convolution
Therefore the solution to (27) and (28) is given by (23). \(\square \)
4 An example
Constantin and Johnson [14] considered the case of an eddy viscosity which is constant above a certain height, with a non-constant below, that is
where \(a>0,~z_{0}>0\), \(b\in R\). Motivated by [14], we change (29) to
here \(a>0,~z_{0}>0\), and we assume \(a+bz_{0}=k_{*}+e^{-z_{0}}\) for continuity. Here, \(\varepsilon =1\). From (19), we get
with \(s_{0}=\frac{k_{*}}{b}\ln \frac{k_{*}}{a}>0\). Note that
with \( z_{0}=\frac{a}{b}(e^{\frac{bs_{0}}{k^{*}}}-1). \)
For the solution in the lower part of the layer \(s\in [0, s_{0}]\), we set
Then (26) will be transformed into the Bessel equation
Let \(\beta = \sqrt{-\frac{4iaf}{b^{2}}}=\frac{2\sqrt{af}}{|b|}e^{-\frac{\pi }{4}i}\), and the general solution of above Bessel equation can be expressed as
where \(J_{0}(x)\) and \(Y_{0}(x)\) are the first and second kind of the Bessel functions with order 0 respectively. In the upper part of the layer, \(s\in (s_{0}, \infty )\), we obtain the solution \(\varphi (s)\) by (23).
Also motivated by [14] and (7) with
we could change (29) to
with \(k_{\star }>0, b>0\) and \(z_0>0\). It follows from (19) that
with \(s_{0}=\frac{k_{*}}{b\varepsilon }\ln \frac{k_{*}+e^{-z_{0}}\varepsilon }{k_{*}+e^{-z_{0}}\varepsilon -b\varepsilon z_{0}}>0\). From (20), we see that
then we obtain the solution \(\varphi (s)\) by (23) and (31).
Next, we change (29) to
where \(z_{0}\) satisfies \(k_{*}=5\cdot e^{-z_{0}}-2>0.\) From (19), we have
with \(s_{0}=-\frac{k_{*}}{2}\ln \frac{5-2e^{z_{0}}}{3}>0\). Then we have
Therefore
For the solution in the upper part of the layer, \(s>s_{0}\), we have
so the general solution can be expressed by
For the solution in the lower part of the layer \(s\in [0,s_{0}]\), we set
From (32), \(k_{3}(s)\) can be expressed by
Now (26) can be written as
and then we get the hypergeometric equation
If we let
then the solution of (33) is
where c is an arbitrary complex constant and \(\digamma \) is the Gauss hypergeometric function.
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Communicated by Adrian Constantin.
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This work is partially supported by the National Natural Science Foundation of China (11661016), Training Object of High Level and Innovative Talents of Guizhou Province ((2016)4006), Major Research Project of Innovative Group in Guizhou Education Department ([2018]012), Guizhou Data Driven Modeling Learning and Optimization Innovation Team ([2020]5016), the Slovak Research and Development Agency under the Contract No. APVV-18-0308, and the Slovak Grant Agency VEGA No. 1/0358/20 and No. 2/0127/20.
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Fečkan, M., Guan, Y., O’Regan, D. et al. Existence and uniqueness and first order approximation of solutions to atmospheric Ekman flows. Monatsh Math 193, 623–636 (2020). https://doi.org/10.1007/s00605-020-01414-7
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DOI: https://doi.org/10.1007/s00605-020-01414-7