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Partial and spectral-viscosity models for geophysical flows

Two models based on the hydrostatic primitive equa- tions are proposed. The first model is the primitive equations with partial viscosity only, and is oriented towards large-scale wave structures in the ocean and atmosphere. The second model is the viscous primitive equations with spectral eddy viscosity, and is oriented towards turbulent geophysical flows. For both models, the existence and uniqueness of global strong solutions is estab- lished. For the second model, the convergence of the solutions to the solutions of the classical primitive equations as eddy viscosity parameters tend to zero is also established.


Introduction
We study two models for geophysical flows based on the hydrostatic primitive equations; both are designed to faithfully simulate certain phenomena in the geophysical flows but they are motivated by different physical considerations. A distinctive characteristic of the flows under consideration is that the vertical scale (∼ 10km) is much smaller than the horizontal scale (∼ 6000km). Thanks to this disparity, a hydrostatic approximation is possible, and gives rise of the primitiveequations.
On the mathematical side, the theory for the primitive equations is fairly complete; see, e.g., the pioneering work in [22,23], and the survey article [28]. In particular, in contrast with the Navier-Stokes equations [9,33], the primitive equations have been shown to have unique global strong solutions [7,20,21,18].
The first model we study aims to faithfully simulate large-scale coherent structures including wave phenomena in the ocean and atmosphere. For relevant discussions of this topic, see, e.g., [12,24,27]. For the phenomena that we are interested in, both the ocean and atmosphere are close to being inviscid. Therefore, the inviscid primitive equation is the preferred model. However, it is very costly to simulate the inviscid primitive equations directly due to the small scales embodied within the model. What we propose here is a new model with eddy viscosity added to the small scale (high frequency) part only while keeping the large scale (low frequency) features intact (at least directly). Intuitively, this type of model would reduce the complexity due to the damping on small scales whereas keeping the desirable large scale structures. Such a naive approach may not work all the time due to the cascade of energy induced by the nonlinear advection term. However, we can easily find situations where such cascade is small or negligible. Indeed, it is easy to find exact large-scale solutions to this primitive equation with partial viscosity; see below. Of course, the existence of such examples do not fully justify the model, and extensive numerical experiments are called for which is our future plan. The idea of partial damping on the high frequency components of the system is not alien to the geophysical community (see, e.g., [11,25]) or the mathematical community (see e.g. [26,4,5,10]), although the application to the primitive equations is new here. Our goal in this paper is to demonstrate the global well posedness of this model with partial viscosity.
The second model is oriented at turbulence modeling for geophysical flows. The simulation of three-dimensional turbulent flows is a formidable task due to the need to resolve the small scale fluctuations or eddies that have subtle effects on the large-scale dynamics of the flow. To make this problem computationally tractable, these effects must be modeled. In one approach, the velocity field is averaged over a small radius to derive equations in terms of the averaged velocity. For nonlinear equations, there arises the problem of closure because the product operation is not closed under the averaging process. To obtain a closed system, the average of the nonlinear terms in the equations must be approximated and expressed solely in terms of averaged quantities. The way in which this is done gives rise to a variety of models. The approach we consider, called the eddy-viscosity method, treats the Reynolds stress as a viscous effect caused by the transport and dissipation of energy due to the small-scale eddies. For this reason, this additional viscosity is called the eddy viscosity or turbulent viscosity. The turbulence model of Smagorinsky [31] belongs to this type. For an overall survey on issues related to these models, see [3].
Unfortunately, a straightforward application of the approach described above leads to the over smearing of the large-scale structures in the fluid. To remedy this unwanted effect, it has been proposed that the eddy viscosity be added only to the subgrid scales. In this way, one hopes to prevent the large-scale structure from being smeared away. Here, we examine a particular class of models of this type called spectral-viscosity or spectral-vanishing-viscosity models, in which the scales are defined in terms of Fourier modes. The subgrid viscosity is simply realized as an addition of the artificial viscosity only to the high-frequency modes. The most intuitive way of doing this to insert a high-pass filter to the standard eddy viscosity. This approach was considered in [13] for hyperviscosity on the Navier-Stokes equations and in [15] for nonlinear as well as hyper-viscosity on the Navier Stokes Equations. In both works, the well-posedness of the resulting spectral viscosity is proven, and the consistency of these model with the original Navier-Stokes equations is discussed.
We employ the idea of spectral viscosity to build and analyze a turbulence model for the geophysical flows in the ocean and atmosphere. As mentioned above, the primitive equations, even without any eddy viscosity, have been shown to have unique global solutions, provided the initial and boundary data are sufficiently smooth. For our model, we prove its global well posedness, which should not come as a surprise. In addition, we will show the convergence of the solutions of the model to the solutions of the primitive equations without any eddy viscosity, as the eddy viscosity parameters tend to zero. This is not possible yet for the Navier-Stokes equations because there convergence is only shown to be in a weak sense.
We should point out that this technique, usually under the name of spectral viscosity or spectral-vanishing viscosity, is known in terms of turbulence modelling (see [19,13,2,15,32], and below), and to applications in geophysical fluid dynamics [11]. However, to the best of our knowledge, the well-posedness result for the three-dimensional nonlinear primitive equations with partial viscosity is new.
The paper is organized as follows. In Section 2, we introduce and prove the well posedness of a model with only partial viscosity. In Section 3, we introduce the linear spectral eddy-viscosity model. In Section 3.2, we prove the existence and uniqueness of strong solutions to the model. In Section 3.3 we study the convergence of the solutions of the model to the solutions of the original primitive equations.

A model with partial viscosity
In this section we study a model with partial high-frequency viscosity only. The lower modes are not damped directly. This feature renders the model suitable for large scale coherent structures in the ocean and atmosphere, because non-physical large scale damping could change the large scale coherent structures over time.
The model reads In the above, u = (u, v) is the horizontal velocity, w is the vertical velocity, µ and ν are the horizontal and vertical kinetic viscosities, respectively. We use ∇ and ∆ to denote the 2D horizontal gradient and Laplacian operators, respectively. The operator (I − P M,N ) represents the high-pass filter and will be defined later on. We consider a rectangular domain Ω = M × (−H, 0), with M = (0, L x ) ×(0, L y ). We consider periodic boundary conditions in both the x and y directions, and free-slip, non-penetration boundary conditions in the vertical directions. More precisely, u(x, y, z, t) = u(x + L x , y, z, t), (2.4) u(x, y, z, t) = u(x, y + L y , z, t), Under the settings just described, we can define the high-pass filter in terms of Fourier frequencies. Specifically, for each function u ∈ (L 2 (Ω)) 2 we let Remark 2.1. The issue of suitable physical boundary conditions for the inviscid primitive equations is an unresolved one. Partial results concerning the linearized primitive equations are available in a series of papers [29], [8], and [30]. There, an infinite set of nonlocal boundary conditions were proposed, which guaranteed the well-posedness of the linearized system. Here we avoid this issue and use the periodic boundary conditions on the lateral boundaries. and u = ∇ ⊥ ψ. With the surface pressure p 0 given by (u, 0) is a set of exact solutions of the inviscid primitive equations, that is, the system (2.1)-(2.8) without the viscosities, or with partial viscosity with M > 4.
2.1. The barotropic and baroclinic modes. As usual, p and w can be expressed in terms of u. Specifically, integrating (2.2) from z to 0, and using the boundary condition (2.8) we obtain p(x, y, z, t) = p 0 (x, y, t) − ρ 0 gz, (2.10) Integrating (2.3) from z to 0, and using the boundary condition (2.7), we obtain w(x, y, z, t) = Setting z = −H in (2.11), and using (2.7) again, we find We substitute (2.10) and (2.11) into (2.1), and obtain a single closed equation for u: 13) It turns out essential to rewrite this equation and work with the following form: (2.14) We note here that P M,N u contains only finite number modes of u. The prognostic variable u satisfies the equation (2.14), the constraint (2.12), and the boundary conditions (2.4)-(2.6). To complete the system, we also require u to satisfy the following initial condition: u(x, y, z, 0) = u 0 (x, y, z). (2.15) The diagnostic variables w and p are given by (2.11) and (2.10) respectively. We now specify the barotropic and baroclinic modes of equation (2.14). We let which denotes the barotropic mode of the primitive variables. We also denote by u ′ (x, y, z, t) = u(x, y, z, t) − u(x, y, t) (2.17) the baroclinic mode of the primitive variables. It is easy to see that u ′ = 0.
(2.18) From (2.4) and (2.5) we derive the boundary conditions for u: u(x, y, t) = u(x + L x , y, t), (2.19) u(x, y, t) = u(x, y + L y , t). (2.20) By (2.12) we see that u satisfies the following constraint: Then u ′ also satisfy the periodic boundary conditions on the lateral boundary, It inherits the boundary conditions for u on the top and bottom, We now derive the equations that u and u ′ satisfy by first taking average of the equation (2.14): By using (2.18), we find that Using (2.12), (2.18) and (2.21), we find Hence the equation for u can be written as The barotropic variable u satisfies the following conditions: Subtracting (2.26) from (2.14) we obtain the equation for the baroclinic mode: Global well-posedness of the model with partial viscosity. In order to have uniqueness for the solution of (2.1)-(2.8) we work with functions that have zero average over Ω. Indeed it can be checked that if the initial data and the forcing have zero average over Ω, then the solutions have zero average over Ω at any time.
We use the conventionL 2 (Ω),Ḣ 1 (Ω), etc. to denote function spaces that have zero average over Ω. We let period L y , periodic and even in z ith period 2H , The inner product and norm of H will be denoted as (·, ·) and | · |, respectively. The space V inherits the inner product and norm of H 1 , which will be denoted as ((·, ·)) and || · ||, respectively.
Since the functions in V has zero spatial averages, we have the following Poincaré inequality for functions in V : 2 is equivalent to the usual H 1 norm, and can be taken as the norm for V .
In what follows we abuse the notation by denoting every generic constant by C. Such constants may depend on the domain Ω and the function spaces in the context, but we omit such dependence in the notation. But if the constant depends on any other parameters, such as M, N etc., we shall use a specific symbol and specify such dependence in the notation.
We shall prove the global existence and uniqueness of strong solutions to the system (2.1)-(2.8).
We shall first obtain some key estimates that will be needed for the proof of Theorem 2.1. The proof of the theorem will be furnished at the end of this subsection.

L 2 estimates
We multiply (2.14) by u, integrate by parts over Ω, using the boundary conditions (2.4)-(2.8), we obtain where w(u) is defined as in (2.11). We can verify that the trilinear operator b(u, u, u # ) is skew symmetric with respect to the last two arguments, that is, Using (2.44) and the Cauchy-Schwarz inequality, we derive from (2.43) that d dt Applying the Gronwall inequality to (2.45) yields We take inner product of (2.30) with |u ′ | 4 u ′ , and integrate by parts over Ω to obtain In the above, the trilinear operator b(·, ·, ·) is defined as in (2.40). We can verify by calculations that Noticing the divergence free condition (2.29) for u and the horizontal periodic boundary conditions (2.27)-(2.28) and (2.31)-(2.32) for u and u ′ respectively, we find For the inner products involving the diffusion terms, we find (2.54) For integrals on the right-hand side of (2.48), we use (2.44) (with k = 2) to find that In the above we have used the interpolation inequality which can be obtained by setting p = 10/3, p 1 = 2, p 2 = 6 in (A.1), and then using (A.5). Similarly, Hence we derive from (2.48) that For the integrals on the right-hand side of (2.56) that involve nonlinear terms, we proceed by integration by parts, using the periodic boundary conditions on u ′ and u when appropriate, and we find that By the Cauchy-Schwarz inequality and the Holder's inequality, we find Using (2.59), we infer from (2.58) that By the Ladyzhenskaya inequality (A.2) for functions in R 2 , we find that .
For the other integral that involves nonlinear terms, By similar use of the Minkowski inequality (A.6) and various interpolation inequalities, we find that By Young's inequality, we have Putting (2.56)-(2.65) together, we find Ignoring the other positive terms on the left hand side of (2.66), and dividing both sides by |u ′ | 4 L 6 , we have Applying the Gronwall inequality to (2.67), and using the L 2 estimate result (2.46), we obtain with J 6 (t) = e C(K 2 1 (t)+K 1 (t)t+K 1 (t)) |u 0 | 2 L 6 (Ω) Integrating (2.66) over [0, t], and using the estimate (2.68), we obtain Estimate |∇u| L 2 (M ) We multiply (2.26) by −∆u and integrate by parts over M to obtain (2.70) We note that, by integration by parts, And thanks to (2.21), M ∇p 0 · ∆u dM = 0.
Following similar steps in the handling of the 2D Navier Stokes equations, we obtain M (u · ∇)u · ∆u dM ≤ C|u|

PARTIAL AND SPECTRAL-VISCOSITY MODELS FOR GEOPHYSICAL FLOWS 17
Applying the Cauchy-Schwarz and Holder inequalities one have For this last integral in (2.70), we proceed by the Cauchy-Schwarz inequality and (2.44) with k = 2, Then we derive from (2.70) that Applying the Gronwall inequality to (2.71), and using the previous estimates (2.46) and (2.69), we obtain To estimate |∂ z u| L 2 (Ω) We multiply (2.14) by ∂ 2 u/∂z 2 and integrate by parts over Ω, and we have 1 2 . (2.73) By Holder's inequality, With regard to the integral in (2.73) that involves the nonlinear convection terms, we find Hence ≤ C|u| L 6 (Ω) |∂ z u| The last two integrals on the right-hand side of (2.73) can be handled by (2.44) with k = 2. Thus we derive from (2.73) that (2.74) An application of the Gronwall inequality to (2.74) readily yields To estimate |∇u| L 2 We multiply (2.14) by −∆u and integrate by parts over Ω, It is easy to see that

By the Young's inequality and the Sobolev interpolation inequality (A.4) for L 3 functions, we have
We appeal to the following inequality for functions in R 3 (the scaling is 2D due to the vertical integration. For a proof, see [6]), The last two integrals in the equation above are handled by (2.44) with k = 2. After these intermediate steps we derive from (2.76) that (2.77) We appeal to the Gronwall inequality, and, with use of (2.46), (2.68), and (2.75), we obtain We now prove Theorem 2.1.
Proof. The short time existence and uniqueness of the strong solutions of (2.1)-(2.8) can be established as for the viscous primitive equations (see [14,17]). Let u be such a strong solution corresponding to the initial data u 0 with the maximal interval of existence [0, T * ). If T * ≥ T , then there is nothing to prove here. Let us suppose that T * < T , and in particular, T * < ∞. Then it is clear that lim sup Otherwise the solution can be extended beyond T * . However the estimates (2.75) and (2.78) indicate that ||u(·, t)|| H 1 < ∞ for all t < T , which contradicts (2.79). Hence the solution must exist for the whole period of [0, T ). It remains to show the continuous dependence of the solution on the data, of which the uniqueness of the solution is a consequence. Let us assume that u 1 and u 2 be two solutions corresponding to the two sets of initial data u 1 0 and u 2 0 , respectively. We let Then u satisfies the following equation (2.80) We multiply (2.80) by u and integrate by parts over Ω to obtain We verify that Again, the last two integrals on the right-hand side of (2.81) are handled by (2.44) with k = 1. Thus we derive from (2.81) that (2.85) Thanks to the a priori estimates (2.75) and (2.78) and the Gronwall inequality, we have This shows that the solution depends on the initial data continuously. When u(·, 0) = u 1 (·, 0) − u 2 (·, 0) = 0, u(·, t) = 0, for all t > 0. (2.87) This shows the uniqueness of the solution.

A spectral-viscosity model for geophysical turbulence
In this section we study a model that has applications in the simulation of geophysical turbulent flows.
3.1. The model. The 3D primitive equations with linear spectral eddy viscosity read with Lu = µ δ ∆(I − P M,N )u + ν δ ∂ 2 ∂z 2 (I − P M,N )u. The other notations being the same as those in Section 2, the newly introduced ones µ δ and ν δ are the eddy closure parameters in the horizontal and vertical directions, respectively. The subscript δ indicates that the parameters depend on the grid resolution.
As for the model with partial viscosity in Section 2, we consider a rectangular domain Ω = M × (−H, 0), with M = (0, L x ) × (0, L y ), and we also consider periodic boundary conditions in both the x and y directions, and free-slip, non-penetration boundary conditions in the vertical directions. More precisely, u(x, y, z, t) = u(x + L x , y, z, t), (3.4) u(x, y, z, t) = u(x, y + L y , z, t), As a consequence of the boundary conditions taken here, the spectral low-pass filter P M,N can be, and is defined as in (2.9). For the model (3.1)-(3.8) we obtain two results. The first is the global well-posedness of the model, and the second is concerned with the convergence of the solutions of (3.1)-(3.8) to those of the viscous primitive equations as µ δ and ν δ tend to zero. The first result shall come as no surprise, and therefore its proof will only be briefly sketched. The second result will be discussed in more details.
As usual, p and w can be expressed in terms of u. Specifically, integrating (3.2) from z to 0, and using the boundary condition (3.8) we obtain p(x, y, z, t) = p 0 (x, y, t) − ρ 0 gz, (3.9) Integrating (3.3) from z to 0, and using the boundary condition (3.7), we obtain (3.10) Setting z = −H in (3.10), and using (3.7) again, we find We substitute (3.9) and (3.10) into (3.1), and obtain a single closed equation for u: 12) The prognostic variable u satisfies the equation (3.1), the constraint (3.11), and the boundary conditions (3.4)-(3.6). To complete the system, we also require u to satisfy the following initial condition: u(x, y, z, 0) = u 0 (x, y, z). (3.13) The diagnostic variables w and p are given by (3.10) and (3.9) respectively. We now specify the barotropic and baroclinic modes of (3.12). We let which denotes the barotropic mode of the primitive variables. We also denote by u ′ (x, y, z, t) = u(x, y, z, t) − u(x, y, t) (3.15) the baroclinic mode of the primitive variables. As in Section 2, we derive the equations and conditions that the barotropic mode u and the baroclinic mode u ′ must satisfy. They are summarized as follows. The equation and boundary conditions for u are u(x, y, t) = u(x, y + L y , t). (3.18) The equation, boundary conditions, and constraint for u ′ are 3.2. Existence and uniqueness of strong solutions. The functional settings are the same as specified in Section 2. A priori estimates are the essential ingredients in the proof of global well-posedness of (3.1)-(3.8). We shall first derive the a priori estimates for u, u and u ′ .
We take inner product of (3.12) with u, integrate by parts over Ω, using the boundary conditions (3.4)-(3.8), we obtain Following similar steps in Section 2, we obtain that We note that K 1 (t) is a non-decreasing positive function in t. This notion will be useful later in the paper.
L 6 estimate on u ′ We rewrite (3.20) as Now (3.28) is in the same form as equation (2.30), and therefore the same techniques from Section (2) can be applied to yield the L 6 estimates on u ′ . We omit the details and state the results as follows.
We also have Estimate |∇u| L 2 (M ) We multiply (3.16) by −∆u and integrate by parts over M to obtain We note that M f k × u · ∆u dM = 0, and M ∇p 0 · ∆u dM = 0.
Following similar steps in the handling of the 2D Navier Stokes equations, we obtain M (u · ∇)u · ∆u dM ≤ C|u| Applying the Cauchy-Schwarz and Holder inequalities one have Utilizing these estimates, and using Youngs inequality again, we derive from (3.31) that Applying the Gronwall inequality to (3.32), and using the previous estimates (3.26) and (3.30), we obtain To estimate |∂ z u| L 2 (Ω) We multiply (3.12) by ∂ 2 u/∂z 2 and integrate by parts over Ω, and utilizing the boundary conditions (3.4)-(3.6), we have By Holder's inequality,

QINGSHAN CHEN, MAX GUNZBURGER AND XIAOMING WANG
With regard to the last integral in (3.34) we find Hence After these intermediate steps, we derive from (3.34) that We note that Therefore, by (3.29) and (3.33), we have |u| 4 L 6 ≤ C K 2 2 (t) + K 2 6 (t) .
An application of the Gronwall inequality to (3.37) readily gives To estimate |∇u| L 2 We multiply (3.12) by −∆u and integrate by parts over Ω, It is easy to see that Therefore we derive from (3.39) that By the Young's inequality and the Sobolev interpolation inequality (A.4) for L 3 functions, we have We appeal to the following inequality, to obtain (by setting f = ∆u, g = ∂ z u) After these steps, we infer from (3.40) that

PARTIAL AND SPECTRAL-VISCOSITY MODELS FOR GEOPHYSICAL FLOWS 31
We are ready to apply the Gronwall inequality, with use of (3.36), (3.38), to obtain Now that the key estimates are in place, with an argument similar to that for Theorem 2.1 in Section 2, we can show Theorem 3.1. For a given T > 0, let F ∈ L 2 (0, T ; L 2 (Ω)), u 0 ∈ V . Then there exists a unique strong solution of the system (3.1)-(3.8) which depends continuously on the initial data.
The proof is omitted.

3.3.
Convergence of the solutions. In this section we take µ δ −→ 0, ν δ −→ 0 in (3.12), and study the convergence of the solution of the system. We first rewrite (3.12) as follows: The superscript δ emphasizes the fact that the solution u δ depends on the spectral viscosity parameters µ δ and ν δ , which themselves are determined by the grid resolution. The proper relation between the parameters µ δ and ν δ and the grid resolution δ is the subject of a separate endeavor, and will be presented elsewhere. In this work we focus on the behavior of the solution u δ of system (3.43) as µ δ and ν δ tend to 0 (corresponding to the scenario when the grid becomes finer and finer).
By Holder's inequality and the interpolation inequalities (A.4) and (A.5) in R 3 , we find that .
We first note that  We notice that the a priori estimates obtained in the previous section are independent of µ δ and ν δ . We apply the Gronwall inequality to We list here some functional inequalities that are frequently used in this paper. L p interpolation inequality Let Ω ⊂ R 3 , and 1 ≤ p 1 ≤ p ≤ p 2 , p 1 = p 2 . Let u ∈ L p 1 (Ω) ∩ L p 2 (Ω). Then u ∈ L p (Ω), and ||u|| L p ≤ ||u|| s 1 L p 1 ||u|| s 2 L p 2 , (A.1) with s 1 = p 1 p p 2 − p p 2 − p 1 , s 2 = p 2 p p − p 1 p 2 − p 1 .