The baroclinic response to wind in a small two-basin lake

Abstract

Field data and two linear layered models were used to examine the baroclinic response to wind in a small elongated two-basin lake, Amisk Lake (Alberta, Canada). For the first vertical baroclinic mode, wind-forced horizontal modes were simulated using a dynamic two-layer variable cross-section (TVC) model. The first horizontal mode, H1, was found to dominate the exchange between the two basins of Amisk Lake. The highest velocities associated with H1 occurred in the constricted channel connecting the two basins. This high velocity led to strong damping which brought H1 in near-resonance with the diurnal wind despite a difference in periods. Along with H1, the second horizontal mode, H2, was detected at a thermistor mooring. The TVC showed that H2 resulted from the coupling between along-thalweg wind distribution and the bathymetry of Amisk Lake. The damping for H2 was found to be weaker than for H1, likely because of weaker bottom drag in the connecting channel. The evolution of vertical H1 modes were simulated using a multi-layered box model. In response to wind pulses, the first vertical mode V1H1 initially dominated over higher vertical modes causing two-layer exchange. Following the faster decay of V1H1, higher vertical modes shifted the exchange to three and more layers. Our study shows the importance of the coupling between wind stress distribution and lake bathymetry in exciting horizontal modes, reveals the effect of damping on resonance with wind, and explains the evolution of exchange associated with vertical modes.

Appendix: Model derivation

The objective of this appendix is to give the necessary background for the derivation of the TVC and MLM models. We start with the governing equations for the response to wind in elongated lakes of variable cross-section and N density layers (“Governing equations and assumptions”). The equations are simplified and then limited to two layers for the derivation of the TVC (“Two-layer variable-cross-section model”). The governing equations for N layers are then restricted to uniform width and depth for derivation of the MLM model (“Multi-layer box model”).

Governing equations and assumptions

For a narrow elongated lake with N density layers, the linearized equations of momentum and mass conservation are given by,

$$ \frac{\partial q_{j}}{\partial t}=-gS_{j}\sum_{i=1}^{j}\left(\frac{\rho_{i}- \left(1-\delta_{1i}\right)\rho_{i-1}} {\rho_{j}}\right)\frac{\partial\eta_{i}}{\partial x}+ \frac{1}{\rho_{j}}\left(\delta_{1j}F_{s}-F_{d,j}\right) $$

(15)

$$ \frac{\partial q_{j}}{\partial x}+b_{j}\frac{\partial\eta_{j}}{\partial t}-\left(1-\delta_{Nj}\right)b_{j+1}\frac{\partial\eta_{j+1}}{\partial t}=0 $$

(16)

where δ_1i , δ_1j , and δ _Nj are Kronecker delta, the index $j=1,2\ldots N$ denotes the density layers and interfaces from the surface downwards, q _j (x, t) is the flow in layer j, η _j (x, t) is the deflection of interface j, S _j (x) and ρ _j are the cross-sectional area and density of layer j, b _j (x) is the cross-sectional width at interface j, F _s (x, t) is the wind forcing acting on the surface of the cross-section, and F _d,j (x, t) is the drag force on layer j. It should be noted that, for arriving at Eq. 15, hydrostatic pressure, no-rotation, and negligible cross-thalweg motions were assumed.

Further, by using a Guldberg–Mohn parameterization for the drag τ _d,j  = 2κρ _j q _j as done in many similar studies (e.g., Heaps and Ramsbottom 1966; Bengtsson 1973; Platzman 1978; Bäuerle 1994; Shimizu and Imberger 2008), Eq. 15 simplifies to,

$$ \frac{\partial q_{j}}{\partial t}+2\kappa q_{j}=-gS_{j}\sum_{i=1}^{j}\left(\frac{\rho_{i}- \left(1-\delta_{1i}\right)\rho_{i-1}} {\rho_{j}}\right)\frac{\partial\eta_{i}}{\partial x}+\delta_{1j}\frac{F_{s}}{\rho_{j}} $$

(17)

where κ(x, t) is the drag coefficient.

To eliminate η _i from the above equation, differentiate it with respect to t, 

$$ \frac{\partial^{2}q_{j}}{\partial t^{2}}+2\frac{\partial\kappa q_{j}}{\partial t}=-gS_{j}\sum_{i=1}^{j}\left(\frac{\rho_{i}-\left(1-\delta_{1i}\right) \rho_{i-1}}{\rho_{j}}\right)\frac{\partial^{2}\eta_{i}}{\partial t\partial x}+ \delta_{1j}\frac{1}{\rho_{j}}\frac{\partial F_{s}}{\partial t}, $$

(18)

recast Eq. 16 into the form,

$$ \frac{1}{b_{j}}\sum_{i=j}^{N}\frac{\partial q_{i}}{\partial x}+\frac{\partial\eta_{j}}{\partial t}=0, $$

(19)

differentiate this form with respect to x, and use the result to eliminate ∂²η _i /∂x∂t from Eq. 18. These steps give the governing equation for layer flow,

$$ \frac{\partial^{2}q_{j}}{\partial t^{2}}+2\frac{\partial\kappa q_{j}}{\partial t}=gS_{j}\sum_{i=1}^{j} \left(\frac{\rho_{i}-\left(1-\delta_{1i}\right)\rho_{i-1}} {\rho_{j}}\right)\frac{\partial}{\partial x} \left(\frac{1}{b_{i}}\sum_{j=i}^{N}\frac{\partial q_{j}}{\partial x}\right)+\delta_{1j}\frac{1}{\rho_{j}}\frac{\partial F_{s}}{\partial t} $$

(20)

Two-layer variable-cross-section model

For two-layer stratification $\left(j=1,2\right),$ Eq. 20 can be recast as,

$$ \frac{\partial^{2}}{\partial t^{2}}\left(q_{2}-\frac{S_{2}}{S_{1}}\left(\frac{\rho_{1}} {\rho_{2}}\right)q_{1}\right)+2\frac{\partial}{\partial t}\kappa\left(q_{2}-\frac{S_{2}} {S_{1}}\left(\frac{\rho_{1}}{\rho_{2}}\right)q_{1}\right)-gS_{2}\left(\frac{\rho_{2}- \rho_{1}}{\rho_{2}}\right)\frac{\partial}{\partial x}\left(\frac{1}{b_{2}} \frac{\partial q_{2}}{\partial x}\right)=-\frac{1}{\rho_{2}}\frac{S_{2}}{S_{1}} \frac{\partial F_{s}}{\partial t}, $$

(21)

which can be simplified to Eq. 2 by using the rigid-lid (q ₁ =  −q ₂) and Boussinesq approximations (Gill 1982).

To solve Eq. 2, we first consider free modal evolution with no damping and assume separable layer flow with sinusoidal temporal dependence, $q_{2}=Q_{n}(x)\sin\left(\omega_{u,n}t\right).$ For this case, Eq. 2 reduces to the eigenvalue problem described by Eq. 3 whose solution yields Q _n and the undamped modal frequency ω _u,n .

Next, consider wind forcing of arbitrary spatial distribution and periodic temporal dependence as given by,

$$ F_{s}=\sum_{n=1}^{\infty}F_{s}^{(n,m)}=-\left(\frac{S_{1}+S_{2}}{S_{2}} \right)\left(\sum_{n=1}^{\infty}A_{n}Q_{n}\right)\left(B_{m}\exp \left(\imath\omega_{f,m} t\right)\right). $$

(22)

Let the layer flow forced by this wind be given by the superposition of modal responses, $q_{2}=\Upsigma_{n=1}^{\infty}M_{n,m}(t)Q_{n}(x),$ use a constant drag coefficient κ _n for each mode such that $\Upsigma_{n=1}^{\infty}\kappa_{n}M_{n,m}Q_{n}=\kappa(x,t)q_{2},$ and substitute for F _s , q ₂, and κq ₂ in Eq. 2 to get,

$$ \sum_{n=1}^{\infty}Q_{n}\left\{ \frac{d^{2}M_{n,m}}{dt^{2}}+2\kappa_{n}\frac{dM_{n,m}}{dt}+ \omega_{n,r}^{2}M_{n,m} \right\} =\frac{1}{\rho_{1}}\sum_{n=1}^{\infty}A_{n}Q_{n}B_{m}\frac{d}{dt}\left\{ \exp\left(\imath\omega_{f,m}t\right)\right\} . $$

(23)

To derive an evolution equation for each mode (i.e., decouple the modes), multiply both sides of the eigenvalue problem (Eq. 3) by (S ⁻¹₁  + S ⁻¹₂ ) and note that the operators on both sides are self-adjoint which leads to the orthogonality property $\int_{0}^{L}(S_{1}^{-1}+S_{2}^{-1})Q_{p}Q_{n}dx=0$ for p≠ n. Multiply both sides of Eq. 23 by (S ⁻¹₁  + S ⁻¹₂ )Q _p , integrate over x, and use the orthogonality property to obtain the ordinary differential equation (ODE) for a single mode,

$$ \frac{d^{2}M_{n,m}}{dt^{2}}+2\zeta_{n}\omega_{u,n} \frac{dM_{n,m}}{dt}+\omega_{u,n}^{2} M_{n,m}=\frac{\imath\omega_{f,m}}{\rho_{1}}A_{n}B_{m} \exp\left(\imath\omega_{f,m}t \right) $$

(24)

which describes the forced response of a damped linear oscillator having a damping ratio $\zeta_{n}=\kappa_{n}\omega_{u,n}^{-1}.$ The solution to this ODE and the derivation of the TVC proceeds as outlined in Eqs. 5–8 and related text.

It only remains to note that, in Eq. 24, A _n is obtained by decomposing the spatial wind-forcing distribution b ₁(x) in terms of Q _n , 

$$ b_{1}(x)=-\sum_{n=1}^{\infty}\left(\frac{S_{1}+S_{2}}{S_{2}}\right) (A_{n}Q_{n}). $$

(25)

This decomposition is done by multiplying both sides of Eq. 25 by Q _p /S ₁, integrating over the thalweg, and using the orthogonality property $\int_{0}^{L}(S_{1}^{-1}+S_{2}^{-1})Q_{p}Q_{n}dx=0$ for p≠ n to get,

$$ A_{n}=-\frac{\int_{0}^{L}S_{1}^{-1}Q_{n}b_{1}(x)dx}{\int_{0}^{L} (S_{1}^{-1}+S_{2}^{-1})Q_{n}^{2}dx}. $$

(26)

Multi-layer box model

For a basin having uniform width and depth, the multi-layer box model (MLM) simulates the response of individual VrHn modes to wind forcing. To simulate the individual responses, the MLM separates vertical modes (Vr) following the model of Monismith (1985) and separates horizontal modes (Hn) following the model of Heaps and Ramsbottom (1966). To derive the MLM, we start from Eq. 20 which governs the flow q _j for N layers. By setting the width of the basin to a uniform width B and S _j  = Bh _j , Eq. 20 reduces to,

$$ \frac{\partial^{2}q_{j}}{\partial t^{2}}+2\frac{\partial\kappa q_{j}}{\partial t}-\sum_{k=1}^{N}\alpha_{jk}\frac{\partial^{2}q_{k}}{\partial x^{2}}=\delta_{1j}\frac{1}{\rho_{j}}\frac{\partial F_{s}}{\partial t} $$

(27)

where h _j is the static-equilibrium thickness of layer j and the matrix α is a function of h _j as defined earlier. Following the procedure outlined above for the TVC, Eq. 27 is reduced to the eigenvalue problem described by Eq. 10 by considering the case of undamped unforced motion and using $q_{j}=R_{j}\sin(n\pi xL^{-1})\sin(\pi\beta tL^{-1}).$ The solution to this eigenvalue problem, β² R = α R, yields the wave celerity β _r and eigenvector R ^(r) for a vertical mode r. Now, consider the wind forcing given by $F_{s}=A_{n}(t)\sin(n\pi xL^{-1})$ and assume a solution of the form $q_{j}=\Upsigma_{r=0}^{\infty}R_{j}^{(r)}M_{n,r}(t)\sin(n\pi xL^{-1}).$ Substituting these expressions for F _s and q _j in Eq. 27 reduces it to,

$$ \sum_{r=0}^{N-1}\left\{ R_{j}^{(r)} \left(\frac{d^{2}M_{n,r}}{dt^{2}}+2\kappa_{n}\frac{dM_{n,r}}{dt}+ \omega_{n,r}^{2}M_{n,r}\right)\right\} =\frac{\delta_{1j}}{\rho_{j}}\frac{dA_{n}}{dt}. $$

(28)

By multiplying both sides by R ^(p) _j ρ _j h ⁻¹ _j (where p denotes a different vertical mode) and summing over j, we get

$$ \frac{d^{2}M_{n,r}}{dt^{2}}+2\kappa_{n} \frac{dM_{n,r}}{dt}+\omega_{n,r}^{2}M_{n,r}= \frac{R_{1}^{(r)}h_{1}^{-1}}{\sum_{j=1}^{N} (R_{j}^{(r)})^{2}\rho_{j}h_{j}^{-1}}\frac{dA_{n}}{dt} $$

(29)

where the orthogonality property ∑ ^N _j=1 R ^(p) _j ρ _j h _j ⁻¹ R ^(r) _j  = δ _pr ∑ ^N _j=1 (R ^(r) _j ) ²ρ _j h ⁻¹ _j was used. This property can be proved by multiplying Eq. 10 by a diagonal matrix ϕ whose diagonal elements are ϕ _jj  = ρ _j h ⁻¹ _j and noting that the matrices ϕ and ϕ⁻¹α are real and symmetric which necessitates that, for the distinct eigenvalues β _r , the corresponding eigenmodes are orthogonal.

For steady wind initiated at t = 0, the solution to Eq. 29 is as given earlier (Eq. 13) and can be used to obtain the solution for general wind conditions A _n (t) using the convolution (Heaps and Ramsbottom 1966),

$$ M_{n,r}=\int\limits_{0}^{t}M_{n,r}^{{\rm st}}(t-t^{\prime})\frac{dA_{n}} {dt^{\prime}}dt^{\prime} $$

(30)

where M ^st _n,r is the solution to steady wind (Eq. 13) and A _n is given by (Heaps and Ramsbottom 1966),

$$ A_{n}(t)=\frac{\int_{0}^{L}\sin(n\pi xL^{-1})F_{s} (x,t)dx}{\int_{0}^{L}\sin^{2}(n\pi xL^{-1})dx}. $$

(31)