a. Formulation

The linear quasigeostrophic vorticity equation is

where p is pressure, f is the Coriolis parameter, β is the northward gradient of f, N is the buoyancy frequency, and Q represents an unspecified potential vorticity source. The (x, y, z) coordinates are eastward, northward, and vertical, while t is time. Subscripted independent variables represent partial differentiation. The equation is to be solved subject to a rigid lid and (assuming infinitesimally thin turbulent boundary layers) an Ekman compatibility surface condition:

where (τ₀^x, τ₀^y) is the wind stress vector. At the sloping bottom, there is also an Ekman compatibility condition:

where α^x and α^y are the bottom slopes in the x and y directions, respectively (i.e., h = h₀ + α^xx + α^yy, with |α^xx|, |α^yy| ≪ h₀), and r is the effective bottom resistance coefficient. In the following, the bottom will often be taken to be flat so that α^x = α^y = 0.

b. Vertical modal solutions

The solution to (1) for free mode n is taken to have the form

so that [from (1)]

where

The unforced boundary conditions are

where

When the stratification is constant, the solution takes the simple form of

where application of (5) leads to

with

Analytical approximate solutions to this problem are detailed in sections 2d and 2e below. The modal functions have the familiar property of no vertical gradient (i.e., no vertical velocity) at the surface, while at depth, higher modes have increasingly sinuous structure, but always with the vertical velocity (hence F_nz) nonzero at the top of the infinitesimal bottom boundary layer (z = −h₀) as long as Γ ≠ 0, for example, in order to accommodate Ekman pumping when r ≠ 0. Thus, the symmetry of the upper and bottom boundary conditions is broken. Further, when bottom friction is nonzero, Γ, ζ_n, ω_n, and F_n are complex and wave phase is no longer uniform in the vertical.

c. A nondimensional parameter

A scaling of the bottom boundary condition [(5b)] helps reveal when frictional effects cease to be perturbations and begin to revamp the wave modal structures. Specifically, the time derivative in (2b) is estimated with the inviscid barotropic Rossby wave frequency ω_0R (the largest frequency, or shortest time scale, available for any wavenumber pairing). Thus, a nondimensional parameter quantifying the frictional effect is

where N_B is the buoyancy frequency at the bottom (z = −h₀). This expression can be thought of as consisting of two ratios. The first, r(h₀ω_0R)⁻¹, is the ratio of the wave time scale to the barotropic spindown time, a measure of the overall importance of bottom friction. The remaining factors in (7) constitute a Burger number for the wave, that is, the ratio of the natural vertical scale in a stratified system (the deformation scale) to the water depth. If the deformation scale is large relative to the water depth, frictional effects are distributed throughout the water column. If the deformation scale is less than the water depth, frictional effects will be trapped near the bottom. This interpretation emphasizes that frictionally adjusted modal structures depend on both the relative strength of damping and the relative importance of the stratification at a given length scale. Thus, for R ≪ 1, modal structures are unperturbed by bottom friction, but for R ≥ O(1), frictional effects lead to substantial modal adjustments.

An expression similar to (7) can be derived by multiplying (4a) by F_n and integrating over depth. The result is

Comparing the two terms on the right-hand side, it is found that frictional modification of modal structure over a flat bottom becomes important when

becomes O(1), where is a depth average. While this expression may not be as intuitive as that leading to (7), it has the advantage of applying to arbitrary distributions of the stratification. This is a concern because the usefulness of (7) is questionable when N_B is extremely small, as is the case with an exponential stratification.

At first, it may be unintuitive that the frictional effect increases with stratification. After all, stratification inhibits the vertical velocity that is the messenger of spindown in the water column. This is indeed the case, but stronger stratification also leads to an increasing tendency to trap any spindown processes closer to the bottom. Further, stronger stratification, by trapping spindown closer to the bottom, can inhibit near-bottom horizontal flow and hence decrease bottom stresses. Thus, since the bottom boundary condition [(5b)] deals only with conditions very near the bottom, there is no contradiction. Moreover, the frictional modifications to the vertical modal structure often serve to decrease wave damping, as will be seen below.

d. Asymptotics: Barotropic and bottom-trapped modes

With constant N² = , a solution to (4a) subject to (5a) can take the form

where

allows solutions for F that are either barotropic or bottom intensified. [Choosing this hyperbolic cosine form in (9a)—as opposed to a simple cosine form—is somewhat arbitrary, since γ is complex, but treating the forms separately is thought to improve clarity. This is equivalent to choosing Real(μ_n²) < 0 in (4b).] Substitution of (9a) into the bottom boundary condition in (5b) yields

If stratification is weak, or if ω falls near the inviscid barotropic Rossby wave resonance, |γh₀| ≪ 1, tanh(γh₀) ≈ γh₀, and (10) simply reduces to

with no further approximations. This same form would result from assuming at the outset that there is no stratification.

On the other hand, for stronger stratification, Re(γ)h₀ ≫ 1, bottom trapping occurs and tanh(γh₀) → 1. Thus, (10) becomes

so that

When R becomes large, the physically realizable (decaying) solution to (13) is then

A physical interpretation of this scale is that it represents the inverse spindown time of a region having vertical extent f[N₀(k² + l²)^1/2]⁻¹.

e. Asymptotics: Baroclinic modes

Solutions to (4a) with constant stratification subject to (5a) can be written in the form

where

is a complex quantity. The form (15a) is equivalent to choosing Real(μ_n²) > 0 in (4b). The parameter η takes on discrete values in accord with the bottom boundary condition, and so

For small R, an approximate solution is found by seeking a perturbation around the r = 0 values:

Assuming small, O(R) corrections, the expansions

and

(where ω⁽⁰⁾ is the frictionless frequency) are used. Thus, (5b) yields

and, after Taylor expansion and using tan (η⁽⁰⁾h₀) = 0,

that is, the correction to the vertical trapping scale increases with the square of scalar wavenumber, as might be expected from (7). Using this form in (16) and expanding η² ≈ η⁽⁰⁾² + 2η⁽⁰⁾η⁽¹⁾ leads to the solution

for small R (i.e., long waves or weak friction).

In the limit of large R, a similar expansion is carried out for small R⁻¹. In this case, we anticipate that interior velocity near the bottom becomes weak (see section 2f below) so that υ → 0 at z = −h₀. Thus,

Then, the correction (imaginary) frequency is

and

for large R.

f. Computed results

When there is no bottom friction, Γ is real and it is straightforward to obtain the infinity of solutions for F_n by solving (6c) for constant stratification. From a practical standpoint, however, solutions of the form (6a) with r ≠ 0 raise computational difficulties because solving (6c) involves searches in the complex plane, such that there is sometimes a question as to whether all of the desired roots have been found. For example, it is usually easy to find the solution having the simplest vertical structure, but then some higher-mode solutions can be harder to isolate. A messier, but more algorithmically certain, approach is to expand F_n in terms of the complete set of inviscid, flat-bottom baroclinic modes and thus replace (6) with an algebraic eigenvalue problem that is straightforward to solve with readily available software.

For the case of constant N² = N₀², the inviscid, flat-bottom (where Γ = 0) modes are

It is also well known that analytical expressions for Γ = 0 modal structures can also be found for exponential stratification:

where J₀, J₁, Y₀, and Y₁ are Bessel functions, a_m is a constant,

q_mB is q_m evaluated at z = −h₀, and the multiple values of λ_m are found by solving

In either case, the modes are normalized so that

The unforced problem for Γ ≠ 0 is then solved by expressing the solution in terms of the inviscid modes; namely,

where Λ is a suitably large integer (Λ = 50 or more in the following calculations), and the notation F_n anticipates the multiple solutions to the problem. Multiplication of (4a) by G_m, followed by vertical integration and application of (5a) and (5b) then yields an algebraic eigenvalue problem for ω_n:

(for n = 1, 2, 3, …, Λ), where

and c_m is the long gravity wave speed associated with inviscid mode m. For example, for constant stratification,

and for exponential stratification [(25a)],

The problem in (28a) is solved using a MATLAB function, and the eigenvalues and eigenvectors having the largest absolute values of ω_n are saved. The first three solutions are identified as ω₀, ω₁, and ω₂, and each is associated with a set b_0m, b_1m, and b_2m. Using these, the frictional modal structures F₀, F₁, and F₂ are readily constructed.

A sample calculation with constant N², a flat bottom, and representative parameters illustrates the frictional effects. Specifically, = 2.58 × 10⁻⁶ s⁻², h₀ = 4500 m, l = 2 × 10⁻⁶ m⁻¹, r = 1 × 10⁻⁴ m s⁻¹, f = 0.73 × 10⁻⁴ s⁻¹, β = 2 × 10⁻¹¹ (s m)⁻¹, and k is varied over a wide range. The is chosen so that c₁ = 2.3 m s⁻¹ in (29a), while f and β are representative of 30° latitude. Because r depends on the strength of total bottom currents (including tides; e.g., Wright and Thompson 1983), it is not obvious what an appropriate deep-sea value might be, but r = 5 × 10⁻⁴ m s⁻¹ is commonly used over the continental shelf (e.g., Chapman 1987), and one might expect abyssal currents in most of the ocean to be weaker than representative shelf conditions. Hence the choice of r = 1 × 10⁻⁴ m s⁻¹ for these calculations.

Results of the calculation are summarized in the dispersion curves of Fig. 1. For small k, the right-hand side of (2b) is small (i.e., the curl of the bottom stress is small), so that frictional effects are weak, and the modal structures (Fig. 2) are essentially G_n, that is, the forms found in the complete absence of bottom friction. The smaller imaginary part of the modal structure gives rise to phase shifts as a function of the vertical. The real part of frequency ω_R is essentially the inviscid result, while the imaginary part of frequency ω_I (Fig. 1b) behaves as one would expect from a perturbation expansion for small r (see sections 2d and 2e). However, as k increases, ω_I increases for two of the modes, and then ω_I for two modes decreases for larger k. At the same time, ω_I for the other mode abruptly begins to increase dramatically for k > 0.7 × 10⁻⁴ m⁻¹. What is happening? Examination of the modal structures for large k (Fig. 3) clarifies the situation. Two of the modes adjust so that there is a node in horizontal velocity near the bottom; that is, they can be thought of as becoming modes n = 1/2 and n = 3/2. The boundary condition [(2b) or (5b)] is being met by having the bottom stress become small even though r(k² + l²) is growing. This sort of behavior, where a linear wave mode structure adjusts so that the effect of bottom friction is minimized, is not unusual in oceanographic problems (e.g., Allen 1984; Power et al. 1989; Brink 2006). On the other hand, the wave mode that has damping increasing with k has reached a state, for large k, that is strongly bottom trapped (Fig. 3). With a large bottom velocity, there is nothing to mitigate the growing r(k² + l²), and so the wave damping grows strongly as k increases and bottom trapping becomes more pronounced. In this case, stratification increases the wave damping because it leads to intensified (rather than weakened) near-bottom currents. It is worth pointing out that this bottom-trapping happens with a flat bottom.

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Frequency vs east–west wavenumber k for Rossby waves with constant N². (a) Real part of frequency ω_R and (b) imaginary part of frequency ω_I (in red, with asymptotic expressions as blue or green broken lines). The blue curves are for modes that are barotropic or bottom intensified (section 2d), and the green curves are for baroclinic modes (section 2e). Expressions for small k are dotted lines and for large k are dashed lines. Computed with N₀² = 2.58 × 10⁻⁶ s ⁻², h = 4500 m, l = 2 × 10⁻⁶ m⁻¹, r = 1 × 10⁻⁴ m s⁻¹, f = 0.73 × 10⁻⁴ s⁻¹, and β = 2 × 10⁻¹¹ (s m)⁻¹. Note that the vertical axes have different scales in the two panels.

Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-18-0070.1

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Long-wave vertical modal structures for conditions as in Fig. 1 and k = 2.1 × 10⁻⁷ m⁻¹. The wave frequencies are as follows: ω₀ (solid curves) = 1.04 × 10 ⁻⁶ +i2.22 × 10⁻⁸ s⁻¹, ω₁ (dashed) = 4.15 × 10⁻⁹ + i1.78 × 10⁻¹⁰ s⁻¹, and ω₂ (dash–dotted) = 1.04 × 10⁻⁹ + i4.49 × 10⁻¹¹ s⁻¹. Blue lines indicate the real part of the modal structure, and red lines indicate the accompanying imaginary parts.

Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-18-0070.1

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Short-wave vertical modal structures for conditions as in Fig. 1 and k = 1.5 × 10⁻⁴ m⁻¹. The wave frequencies are: ω₀ (solid curves) = 7.01 × 10⁻⁸ + i3.033 × 10⁻⁷ s⁻¹, ω₁ (dashed) = 1.32 × 10⁻⁷ + i7.93 × 10⁻¹¹ s⁻¹, and ω₂ (dash-dot) = 1.21 × 10⁻⁷ + i5.54 × 10⁻¹⁰ s⁻¹. Blue lines indicate the real part of the modal structure, and red lines indicate the accompanying imaginary parts.

Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-18-0070.1

The asymptotic expressions for the imaginary part of frequency for barotropic or bottom-trapped modes [(11) and (14)] are overplotted as blue broken lines in Fig. 1b. The weak-friction limit in (11) is seen to replicate the calculations quite well for k < 0.4 × 10⁻⁴ m⁻¹, while the strong-friction limit (the blue dashed line for ω_I > 1.5 × 10⁻⁷ s⁻¹) captures only the right magnitude and trend in this example.

The asymptotic results for baroclinic modes [(21) and (23)] are overplotted as green broken lines in Fig. 1b. They both provide excellent agreement with the direct calculations in the appropriate R range. Note that three of the expressions for wave damping rates—(11), (14), and (21)—do not depend on wave orientation, although the large R, higher-mode expression in (23) does depend on orientation through ω⁽⁰⁾.

Up to this point, all results have been for the case of constant N². One might ask whether the results change substantially when one uses a more realistic, surface-intensified exponential stratification. The inviscid baroclinic modes in this case obey (25), and we choose z_S = 350 m (following SLF16) and = 7.75 × 10⁻⁵ s⁻² in order to obtain a first internal mode gravity wave speed of 2.3 m s⁻¹ as in the case with constant N². The resulting modal structures (such as the dashed line in the left panel of Fig. 5) have larger amplitudes near the surface and relatively constant values in the lower portions of the water column where N² is small. After solving (28), Fig. 4 is obtained for the imaginary part of the frequency. We emphasize that the only difference between this and the calculations leading to the lower panel of Fig. 1 is the use of exponential stratification. Two points stand out. First, the long-wave damping is much weaker (note the changed vertical scale) with exponential stratification than in the case for constant stratification. This is not surprising in light of the relatively weak (compared to the upper water column) bottom expression of the baroclinic modes with weak deep stratification. Second, the transition between inviscid modal structures (for longer waves) and the bottom- or surface-intensified large-friction (short wave) modes occurs at roughly the same wavelengths. For example, the long-wave barotropic wave’s flat ω_I curve begins to bend at about k = 3 × 10⁻⁵ m⁻¹ with constant N² (Fig. 1) and around k = 4 × 10⁻⁵ m⁻¹ with the exponential stratification (Fig. 4). Similarly, the first baroclinic mode reaches its maximum ω_I at k = 6.4 × 10⁻⁵ m⁻¹ for constant N² (Fig. 1) and at 6.2 × 10⁻⁵ m⁻¹ for exponential stratification (Fig. 4). We conclude from this insensitivity that the vertical structure of the density stratification does not substantially affect either 1) our conclusions about the importance of bottom friction for changing modal structures or 2) the utility of the R_I parameter [(8)]. [It is evidently more appropriate to use a depth-averaged N² than the actual near-bottom value (7).]

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Imaginary part of frequency for conditions as in Fig. 1, but the stratification is exponential ( = 7.75 × 10⁻⁵ s⁻²; z_S = 350 m). Note the change in vertical scale relative to Fig. 1b.

Citation: Journal of Physical Oceanography 48, 9; 10.1175/JPO-D-18-0070.1

The results presented here are fairly representative of many calculations (not detailed here), executed with both constant and exponential N², in that, in all cases, most wave modes evolve toward a state with weak near-bottom velocities and with decreasing wave damping as r(k² + l²) increases. However, in every calculation, there is always one wave mode that is increasingly bottom intensified and where the damping becomes large. In some calculations, the bottom-trapped wave mode is continuous with the inviscid barotropic mode as k varies (e.g., Fig. 4), but in some cases (such as Fig. 1), the bottom-trapped mode is continuous with one of the small-r baroclinic modes. Also, for a given set of parameters, the transition from the nearly inviscid modal structure toward the strongly frictional structures tends to occur at roughly the same wavenumber range (e.g., where k ≈ 0.7 × 10⁻⁴ m ⁻¹ and R_I = 3.7 for Fig. 1b) for each of the three gravest modes. This transition occurs where the deformation scale [≈2πf/(N₀k) = 4100 m here] is comparable to the ocean’s 4500-m depth. Finally, even with a sloping bottom (α^x and/or α^y ≠ 0), the results do not change qualitatively: for large r(k² + l²), there is a single, strongly damped, bottom-trapped wave mode while all other modes adjust so that near-bottom velocity is small.